<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
  <title>Sanhai&#39;s Blogs</title>
  
  
  <link href="http://example.com/atom.xml" rel="self"/>
  
  <link href="http://example.com/"/>
  <updated>2026-05-16T03:30:19.615Z</updated>
  <id>http://example.com/</id>
  
  <author>
    <name>Sanhai</name>
    
  </author>
  
  <generator uri="https://hexo.io/">Hexo</generator>
  
  <entry>
    <title>Hexo添加最后在线时间</title>
    <link href="http://example.com/2026/05/16/hello-world(1)/"/>
    <id>http://example.com/2026/05/16/hello-world(1)/</id>
    <published>2026-05-16T03:27:32.229Z</published>
    <updated>2026-05-16T03:30:19.615Z</updated>
    
    <content type="html"><![CDATA[<p>Welcome to <a href="https://sanhaikeji.github.io/">Sanhai’s Blogs</a>!</p><h2 id="快速开始">快速开始</h2><p>根据搜索结果，安知鱼（AnZhiYu） 是一款基于 Hexo 框架的博客主题。要在该主题中添加"最后更新时间"功能，我给你提供以下技术方案：</p><h1>实现方案</h1><h2 id="方案一：在文章-Front-matter-中添加更新时间">方案一：在文章 Front-matter 中添加更新时间</h2><p>在每篇 Markdown 文章的头部添加 updated 字段：</p><h2 id="Yaml">Yaml</h2><h2 id="title-文章标题date-2026-01-01-10-00-00updated-2026-05-16-11-30-00-最后更新时间">title: 文章标题<br>date: 2026-01-01 10:00:00<br>updated: 2026-05-16 11:30:00  # 最后更新时间</h2><p>然后在主题配置文件中启用显示：</p><p>Yaml</p><h1>_config.anzhiyu.yml</h1><p>post_meta:<br>updated: true  # 显示更新时间</p><h2 id="方案二：使用-Hexo-自动生成更新时间插件">方案二：使用 Hexo 自动生成更新时间插件</h2><p>安装插件<br>Bash<br>npm install hexo-generator-update --save<br>配置插件 在 Hexo 根目录的 _config.yml 中添加：</p><figure class="highlight yaml"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="attr">update_generator:</span></span><br><span class="line">  <span class="attr">enable:</span> <span class="literal">true</span></span><br><span class="line">  <span class="attr">name:</span> <span class="string">'updated'</span></span><br><span class="line">  <span class="attr">default_value:</span> <span class="string">''</span></span><br></pre></td></tr></tbody></table></figure><p>修改主题模板 在 themes/anzhiyu/layouts/_partial/post-meta.pug 中添加：</p><figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">- if post.updated</span><br><span class="line">  .post-meta-item</span><br><span class="line">    i.fa.fa-clock-o</span><br><span class="line">    |  更新时间：#{date(post.updated, 'YYYY-MM-DD HH:mm')}</span><br></pre></td></tr></tbody></table></figure><h2 id="方案三：手动修改主题文件显示更新时间">方案三：手动修改主题文件显示更新时间</h2><p>找到文章元信息模板文件<br><code>themes/anzhiyu/layouts/_partial/post-meta.pug</code></p><h3 id="添加更新时间代码">添加更新时间代码</h3><figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">// 在已有日期显示代码后添加</span><br><span class="line">if post.updated</span><br><span class="line">  .post-updated</span><br><span class="line">    i.fa.fa-history</span><br><span class="line">    span 最后更新：#{date(post.updated, 'YYYY-MM-DD')}</span><br></pre></td></tr></tbody></table></figure><h3 id="重新生成博客">重新生成博客</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">hexo clean &amp;&amp; hexo g &amp;&amp; hexo s</span><br></pre></td></tr></tbody></table></figure><h2 id="主题配置示例">主题配置示例</h2><p>在 _config.anzhiyu.yml 中配置：</p><figure class="highlight yaml"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 文章元信息</span></span><br><span class="line"><span class="attr">post_meta:</span></span><br><span class="line">  <span class="attr">item_date:</span> <span class="literal">true</span></span><br><span class="line">  <span class="attr">item_updated:</span> <span class="literal">true</span>  <span class="comment"># 启用更新时间</span></span><br><span class="line">  <span class="attr">item_category:</span> <span class="literal">true</span></span><br><span class="line">  <span class="attr">item_tag:</span> <span class="literal">true</span></span><br><span class="line">  <span class="attr">position:</span> <span class="string">left</span></span><br><span class="line"></span><br><span class="line"><span class="comment"># 时间格式</span></span><br><span class="line"><span class="attr">date_format:</span> <span class="string">'YYYY-MM-DD'</span></span><br><span class="line"><span class="attr">updated_format:</span> <span class="string">'YYYY-MM-DD HH:mm'</span></span><br></pre></td></tr></tbody></table></figure>]]></content>
    
    
      
      
    <summary type="html">&lt;p&gt;Welcome to &lt;a href=&quot;https://sanhaikeji.github.io/&quot;&gt;Sanhai’s Blogs&lt;/a&gt;!&lt;/p&gt;
&lt;h2 id=&quot;快速开始&quot;&gt;快速开始&lt;/h2&gt;
&lt;p&gt;根据搜索结果，安知鱼（AnZhiYu） 是一款基于 Hexo 框架的博</summary>
      
    
    
    
    
  </entry>
  
  <entry>
    <title>肚子的爷爷操纵天气使小 $&#92;zeta$ 挂大分</title>
    <link href="http://example.com/2026/05/01/TUMMY/"/>
    <id>http://example.com/2026/05/01/TUMMY/</id>
    <published>2026-05-01T13:34:09.000Z</published>
    <updated>2026-05-18T01:38:38.149Z</updated>
    
    <content type="html"><![CDATA[<h1>肚子的爷爷操纵天气使小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 挂大分</h1><p><strong>本文内容在作者本人经历下参考 <a href="https://haisan.club">这里</a> 文学化改编，如小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 、肚子的 等名字在现实生活中如有雷同，实属巧合，无影射现实的任何成分。</strong></p><p>五月，即使已过谷雨，冰冷的寒气尚未完全褪去，但也一天天热下去，仿佛在对气温求导一般一层层如链式法则地剥去寒气的外壳。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 是一个 <em>生中初</em> ，上学期在 <em>中期试考</em> 中表现平平，获得了<strong>高达</strong>一个 <em>名四十第语英</em> 和一个 <em>名六十第分总</em> 。浑浑噩噩度过一年之后，他再次踏上征途。</p><p>他为了不被 <em>肚子的</em> 甩开太远，觉得顺便参加一下 Day1 下午的英语考试，摸一摸底（被打击），当作 Day2 上午数学的安慰赛。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 对此心里完全没底，觉得自己能及格的概率低于投掷一枚质地均匀的硬币结果立在地上的概率——虽然他知道后者几乎为零。</p><p>天气不佳。肚子的 大手使前一天晚上起就下起了淅淅沥沥的小雨。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 预感着，选手们的分数就将要如同如雨水一般不断流逝，<em>肚子的</em> 大手将如这风一般夺去选手们仅剩的尊严，挂掉更多的分。而他，大概会是挂得最多的那个。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 早早来到考场，却发现早上操场上聚集着一群 <em>生学小</em> ，叽叽喳喳的。他把自己的物品在架子上摆得整齐之后，毕恭毕敬地站着。旁边几个 _生学小_一边大吵大闹，抛出的一枚硬币立在了地上。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 叹了口气，心想这大概就是他今天唯一的运气了。</p><p>在 Day1 的 <em>试考文语</em> 前，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的 <em>文语老师</em> 有三节课。然而这三节课中前两节课小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的老师在讲课，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 努力聚精会神的听着，作为上次期中考试全班第一名，所以压力非常的大；等到最后一节课时，老师让小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 自己复习，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 摸了摸书包——发现自己的幸运符不见了。他深吸一口气，在 <em>书文语</em> 上狠狠写入数个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>F**k&nbsp;**F</mtext></mrow><annotation encoding="application/x-tex">\text{F**k **F}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.75em;"></span><span class="mord text"><span class="mord">F**k&nbsp;**F</span></span></span></span></span> ，然后又连续用笔划掉了——。</p><p>开考了，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 在考试发卷子时拼命想注意前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>6</mtext></mrow><annotation encoding="application/x-tex">\text{6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">6</span></span></span></span></span> 个选择的答案，但脑子一片空白。于是他将卷子翻到了反面。他两手十指交错，支撑着头部，模仿碇源堂一般思考着。他以惊人的注意力确认了他确实什么也不会。难道要在一题 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>4</mtext></mrow><annotation encoding="application/x-tex">\text{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">4</span></span></span></span></span> 分的 <em>读阅小</em> 里丢分吗？小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 绝望的想着。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 在这题里面想了许久许久，直到他在开考前的勇气全然失去时，他才勉勉强强，哆哆嗦嗦的写下了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="red"><mi>A</mi><mi>D</mi><mi>D</mi></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{red}{ADD}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="color:red;">A</span><span class="mord mathnormal" style="margin-right:0.0278em;color:red;">D</span><span class="mord mathnormal" style="margin-right:0.0278em;color:red;">D</span></span></span></span> 的答案——尽管他连题目都没看懂。</p><p>接着，他看向一题 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>4</mtext></mrow><annotation encoding="application/x-tex">\text{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">4</span></span></span></span></span> 分的 <em>读阅文诗古</em> ，或许由于前三题，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 十分慌张，全然忘记了老师之前说在 <em>中初</em> ， 王维的诗被认为是<strong>积极正面的</strong>，反手认为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>B</mtext></mrow><annotation encoding="application/x-tex">\text{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">B</span></span></span></span></span> 选项“第二句体现出王维心中的落寞与孤寂”是正确的，去选择了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>C</mtext></mrow><annotation encoding="application/x-tex">\text{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">C</span></span></span></span></span> ，直到对答案才意识到这一点——当然，这是后话了，当时他连对答案的勇气都没有。</p><p>然后小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 看到 B 卷。第一题的第一小问便十分迷惑：</p><figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line">正确的是（）：</span><br><span class="line"></span><br><span class="line">A. 鲁肃独不然（然：这样）。</span><br><span class="line">B. 请为将军筹之（筹：谋划）。</span><br><span class="line">C. 为操后患（患：祸患）。</span><br><span class="line">D. 竟败操于赤壁（竟：最终）。</span><br></pre></td></tr></tbody></table></figure><p>四个都是对的。</p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>A&nbsp;Fake&nbsp;Observation.</mtext></mrow><annotation encoding="application/x-tex">\text{A Fake Observation.}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">A&nbsp;Fake&nbsp;Observation.</span></span></span></span></span><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 急了：这肯定是 <em>肚子的</em> 使用了大风修改了选项描述，肯定是他，一定是他。望着窗外愈下愈大的雨，他沉默着，灰心的写着后面的题目。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 看到作文，想起来这是曾经自己小学写过的，还登上了文刊。但是 <em>肚子的</em> 却操作小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的记忆，使得自己直接忘记了这篇自己写过的作文。</p><p>肯定是 <em>肚子的</em> 的阴谋。那无形的大手，那无法逃脱的大手。</p><p>直到考试结束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>-75</mtext></mrow><annotation encoding="application/x-tex">\text{-75}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">-75</span></span></span></span></span> 分钟，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 才回过神来，去检查前面的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="red"><mi>A</mi><mi>D</mi><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{red}{ADC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="color:red;">A</span><span class="mord mathnormal" style="margin-right:0.0278em;color:red;">D</span><span class="mord mathnormal" style="margin-right:0.0715em;color:red;">C</span></span></span></span> 和被 <em>肚子的</em> 修改的题目——当然，检查了半天，修改了4道题，其中现代文阅读改为了正确，同时把古诗文的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>B</mtext></mrow><annotation encoding="application/x-tex">\text{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">B</span></span></span></span></span> 改成了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>C</mtext></mrow><annotation encoding="application/x-tex">\text{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">C</span></span></span></span></span> ， 另外两题全部改错，随后小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 想要修改他那作文，_肚子的_操控了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的精神使得小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 没有注意到时间已经不够，导致小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 在收卷子的前0分钟没有写完这个修改，从而让作文语句不顺， 暴扣4分。尽管他知道，这一场决定了未来 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>500</mtext></mrow><annotation encoding="application/x-tex">\text{500}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">500</span></span></span></span></span> 年的历史，但他也明白自己已经无力回天，无力再在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>烈酒&nbsp;(spirit)</mtext></mrow><annotation encoding="application/x-tex">\text{烈酒 (spirit)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">烈酒</span><span class="mord">&nbsp;(spirit)</span></span></span></span></span> 上与 <em>肚子的</em> 对抗，于是倒头就睡……反正也做不出来了。</p><p>走出考场。</p><p>大雨疯狂的下着，仿佛整座楼里考生内心的祈愿般不停坠落——除了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> ，他知道自己没什么可祈愿的。</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>98</mn><mi>p</mi><mi>t</mi><mi>s</mi><mo>=</mo><mn>24</mn><mo>+</mo><mn>0</mn><mo>+</mo><mn>0</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>12</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>30</mn><mo>≤</mo><mi>S</mi><mi>c</mi><mi>o</mi><mi>r</mi><msub><mi>e</mi><mtext>文语</mtext></msub><mo>≤</mo><mn>24</mn><mo>+</mo><mn>12</mn><mo>+</mo><mn>0</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>12</mn><mo>+</mo><mn>6</mn><mo>+</mo><mn>20</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>40</mn><mo>=</mo><mn>133</mn><mi>p</mi><mi>t</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">98pts=24+0+0+10+12+3+10+9+30 \le Score_{\text{文语}} \le 24+12+0+10+12+6+20+9+40=133pts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">98</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">24</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">30</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mord mathnormal" style="margin-right:0.0278em;">cor</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">文语</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">24</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">20</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">40</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">133</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span></span></span></span></span><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 决定效仿汉文帝汉景帝休养生息，但主要是想逃避面对肚子的侵扰。</p><p>下午，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的幸运符依旧不在。</p><p>不仅如此，就连下午 <em>试考语英</em> 前，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的老师也被 <em>肚子的</em> 大手控制了，不断地压力小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span>  ，说他并不比出这套卷子的校学 <em>SLFW</em> 并且扬言小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 考不好的话他的美妙的，梦想的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="blue"><mtext>五一假期</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{blue}{五一假期}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord cjk_fallback" style="color:blue;">五一假期</span></span></span></span> 就要遭殃。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 终于还是在 <em>肚子的</em> 大手下屈服了，虽然他知道自己比 <em>SLFW</em> 的人差得远，但还是开启了他悲惨的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="red"><mtext>陪跑全场比赛</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{red}{陪跑全场比赛}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord cjk_fallback" style="color:red;">陪跑全场比赛</span></span></span></span> 。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 拿到那可恶的WFLS的卷子，发现有足足65道单选， 而且剩下的全是小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的弱项————填词</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>这是肚子的大手，我用尽解数无法逃脱的大手。</mtext></mrow><annotation encoding="application/x-tex">\text{这是肚子的大手，我用尽解数无法逃脱的大手。}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">这是肚子的大手，我用尽解数无法逃脱的大手。</span></span></span></span></span></span><p>下午的小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 状态极差。从语文交卷那一刻起，他就感到头昏脑涨，仿佛 <em>肚子的</em> 大手直接捏住了他的太阳穴。他随之开题，结果 肚子的 通过天气操纵了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的听力音频，使听力音频<strong>响度忽高忽低，频率忽近忽远，读出来的单词仿佛窗外下落的雨点，一个连着一个，一个还没有听懂一个又来了</strong>。再加上小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 头昏目眩，听力部分几乎全军覆没，浪费了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 大量时间——虽然小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 本来也听不懂多少。</p><p>接着，等到小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 看到单选 T(我也忘了) 时， <em>肚子的</em> 通过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>比上方谷之雨还要好的好雨</mtext></mrow><annotation encoding="application/x-tex">\text{比上方谷之雨还要好的好雨}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">比上方谷之雨还要好的好雨</span></span></span></span></span> 临时改变了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的记忆，让他忘记他前一天刚复习 时看到的单词 。并且以此影响小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的心态，让他在看到 <em>读阅形完</em> 时被长难句所误导，错失高额分数。头昏的小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 连题目都看不清楚，眼前的字母仿佛在跳舞。</p><p>可怜的小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> ，在考试的最后也没想到单选和 <em>读阅形完</em> 的是啥意思。直接导致了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的挂分，是小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 与 肚子的 战役的导火索——虽然这更像是一场单方面的屠杀。</p><p>赛后，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 麻木地走出考场，知道的人说他自己觉得有 90pt，不知道的还以为他看懂了 <em>读阅形完</em> ，爆切 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>CD</mtext></mrow><annotation encoding="application/x-tex">\text{CD}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">CD</span></span></span></span></span> 篇，并且还在原文填空中记起了原文是什么呢。其实他只是瞎蒙的。</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>72</mn><mi>p</mi><mi>t</mi><mi>s</mi><mo>=</mo><mn>12</mn><mo>+</mo><mn>6</mn><mo>+</mo><mn>3</mn><mo>+</mo><mn>20</mn><mo>+</mo><mn>11</mn><mo>+</mo><mn>5</mn><mo>+</mo><mn>10</mn><mo>+</mo><mn>5</mn><mo>≤</mo><mi>S</mi><mi>c</mi><mi>o</mi><mi>r</mi><msub><mi>e</mi><mtext>语英</mtext></msub><mo>≤</mo><mn>18</mn><mo>+</mo><mn>18</mn><mo>+</mo><mn>5</mn><mo>+</mo><mn>28</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>8</mn><mo>+</mo><mn>15</mn><mo>+</mo><mn>8</mn><mo>=</mo><mn>102</mn><mi>p</mi><mi>t</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">72pts=12+6+3+20+11+5+10+5 \le Score_{语英} \le 18+18+5+28+2+8+15+8=102pts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">72</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">20</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">11</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mord mathnormal" style="margin-right:0.0278em;">cor</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord cjk_fallback mtight">语英</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">18</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">28</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">15</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">102</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span></span></span></span></span><p>结束了。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 发现自己并没有看懂 <em>读阅形完</em> ，也没有爆切 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>CD</mtext></mrow><annotation encoding="application/x-tex">\text{CD}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">CD</span></span></span></span></span> 篇，并且不记得原文是什么——他甚至不记得自己填了什么。</p><p>肯定是 <em>肚子的</em> 的阴谋。那无形的大手，那无法逃脱的大手。</p><p>Day1 晚上， <em>肚子的</em> 大手继续操控着一切。 <em>肚子的</em> 将小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的 <em>学数老师</em> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>HL</mtext></mrow><annotation encoding="application/x-tex">\text{HL}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">HL</span></span></span></span></span> 传送过来并控制 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>HL</mtext></mrow><annotation encoding="application/x-tex">\text{HL}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">HL</span></span></span></span></span> ，以此消磨小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的时间——虽然小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的时间本来就不值钱。</p><p>更可怕的是，Day1 night ，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 发烧了。不是进入梦乡，而是被高烧直接击倒在床上。他浑身发烫，额头滚烫，脑子里一片混沌。 <em>肚子的</em> 大手连他的免疫系统都不放过。</p><p>他梦见了，他自己勉强及格，没有垫底，成功躲过了 <em>肚子的</em> 的嘲讽，在被称为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>文科第一强科</mtext></mrow><annotation encoding="application/x-tex">\text{文科第一强科}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">文科第一强科</span></span></span></span></span> 的 <em>学数</em> 上拿了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Not&nbsp;Last&nbsp;One</mtext></mrow><annotation encoding="application/x-tex">\text{Not Last One}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Not&nbsp;Last&nbsp;One</span></span></span></span></span> ，以及 <em>语英，文语</em> 上了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>90</mn><mi>p</mi><mi>t</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">90pts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">90</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span></span></span></span> ，同时 <em>理物</em> 灵感迸发，成功 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>AC</mtext></mrow><annotation encoding="application/x-tex">\text{AC}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">AC</span></span></span></span></span> 了一半。</p><p>他梦见了， <em>肚子的</em> 在阴暗的角落里操控着一切，想要夺走小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 所拥有的一切——虽然小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 本来也没多少。</p><p>他梦见了， 他成功 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>AK&nbsp;ION&nbsp;and&nbsp;IOI</mtext></mrow><annotation encoding="application/x-tex">\text{AK ION and IOI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">AK&nbsp;ION&nbsp;and&nbsp;IOI</span></span></span></span></span> ，保送清北，在校园里无忧无虑，成为人生赢家。然后他醒了，因为高烧让他头痛欲裂。</p><p>但梦是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="green"><mtext>True</mtext></mstyle><mstyle mathcolor="gray"><mtext>&nbsp;or&nbsp;</mtext></mstyle><mstyle mathcolor="red"><mtext>False</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{green}{\text{True}}\textcolor{gray}{\text{ or }}\textcolor{red}{\text{False}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text" style="color:green;"><span class="mord" style="color:green;">True</span></span><span class="mord text" style="color:gray;"><span class="mord" style="color:gray;">&nbsp;or&nbsp;</span></span><span class="mord text" style="color:red;"><span class="mord" style="color:red;">False</span></span></span></span></span> ？</p><p>Day2 早晨，阳光明媚，太阳射进了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的家。</p><p><em>肚子的</em> 意念操控着自然光，让小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 比平时更早醒来——大概是想让他多受一会儿折磨。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 看了一眼窗外。他预感到，即使是 Day2 ，选手们的分数依然像阳光一样，被肆意的散发出去，被 <em>肚子的</em> 控制。而他的分数，大概会像阴影一样，被彻底忽略。</p><p>来到考场，操场上依旧聚集着 <em>生学小</em> ，他们也被 <em>肚子的</em> 意念控制，决定要在小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 试考时排练<strong>运动会</strong>。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 对他的 <em>学数</em> 毫无信心，认为他能及格的概率高于一个质地均匀的瓶子从 10m 高空坠落还能立起来的概率。紧接着，旁边的一名 <em>生学小</em> 从地上抛出瓶子，瓶子在 20m 高空做了一个完美的抛体运动后，恰好在地面上立起来了。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 苦笑了一下，心想这运气要是能分他一点就好了。</p><p>早上刚来，HL直接传送的班上，开始给我们划分进度以及时间</p><p>早上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>8:00</mtext></mrow><annotation encoding="application/x-tex">\text{8:00}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">8:00</span></span></span></span></span> ，准时开考。</p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>HL</mtext></mrow><annotation encoding="application/x-tex">\text{HL}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">HL</span></span></span></span></span> 老师给小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 定的进度是：<figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">T16: 12~18min</span><br><span class="line">T21: 35~45min</span><br><span class="line">T25: 50~58min</span><br></pre></td></tr></tbody></table></figure><p>然而，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 在提前发卷子时注意到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>6</mtext></mrow><annotation encoding="application/x-tex">\text{6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">6</span></span></span></span></span> 个选择的答案。这次考试 T1~T25 都是氵题——对小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 来说全是难题。以下是小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 真实进度：</p><figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line">T16: 25min（卡住了）</span><br><span class="line">T21: 40min（空了三题）</span><br><span class="line">T25: 65min（还有两题没看）</span><br></pre></td></tr></tbody></table></figure><p>然而，到了 T26 ，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 发现自己居然在上面卡了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>30</mtext></mrow><annotation encoding="application/x-tex">\text{30}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">30</span></span></span></span></span> 分钟！这一定是 <em>肚子的</em> ，是 <em>d <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> d</em> 篡改了我的草稿纸上的二次方程！小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 愤愤的想着——其实是因为他根本不会解。等他做完这题时，已经过去了 <code>95min</code> 。</p><p>然后是 T27 和 T28 。这次的几何题目 T27 和一次函数大题 T28 十分的氵。然而， <em>肚子的</em> 的大手阴魂不散，在这最后，也是最重要的关头，用尽了全身招数对付小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 本来就 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>水火皆惧</mtext></mrow><annotation encoding="application/x-tex">\text{水火皆惧}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">水火皆惧</span></span></span></span></span> ，面对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>天下无敌</mtext></mrow><annotation encoding="application/x-tex">\text{天下无敌}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">天下无敌</span></span></span></span></span> 并且有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>三百里无人敢惹结界</mtext></mrow><annotation encoding="application/x-tex">\text{三百里无人敢惹结界}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">三百里无人敢惹结界</span></span></span></span></span> 的 <em>肚子的</em> 时，更是被 <em>肚子的</em> 意念操控得神志不清。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 不仅将 T27 中题目问的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>A</mi><mi>B</mi></mrow><mrow><mi>A</mi><mi>G</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\frac{AB}{AG}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal">G</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 看成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>A</mi><mi>B</mi></mrow><mrow><mi>B</mi><mi>G</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle\frac{AB}{BG}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mord mathnormal">G</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> ，让小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 痛失 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>5</mtext></mrow><annotation encoding="application/x-tex">\text{5}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord text"><span class="mord">5</span></span></span></span></span> 分；还忘记了以前学习 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>k</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y=kx+b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 时经常用的面积转换法——其实他就没学会过，浪费了大量时间。最终，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 在考试结束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mi>π</mi><mn>3</mn></msup><mo>−</mo><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mrow><mn>8</mn><msup><mi>x</mi><mn>5</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">(</mo><mn>324889</mn><mo>−</mo><mn>120736</mn><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo><msup><mrow><mi>ln</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>x</mi></mrow><mrow><mn>445625</mn><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>x</mi></mstyle></mrow><annotation encoding="application/x-tex">-\pi^3-\displaystyle\int_0^1 \frac{8x^5(1-x)^2(324889-120736 x^2)\ln^2x}{445625(1+x^2)}dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.5114em;vertical-align:-0.936em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.4445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.564em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5754em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">445625</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">324889</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">120736</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">ln</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8984em;"><span style="top:-3.1473em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span> 时才写完卷子——当然，是蒙完的。之后的时间内小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 虽然认真的检查了所有题目，但只是把错误答案又确认了一遍。最终，在对答案时，发现了自己将 T27 看错的可怕事实……以及 一堆 全错的事实。</p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>How&nbsp;can&nbsp;he&nbsp;get&nbsp;Not&nbsp;Last&nbsp;One?</mtext></mrow><annotation encoding="application/x-tex">\text{How can he get Not Last One?}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">How&nbsp;can&nbsp;he&nbsp;get&nbsp;Not&nbsp;Last&nbsp;One?</span></span></span></span></span><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>111</mn><mi>p</mi><mi>t</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">111pts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">111</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span></span></span></span></span> <p>毁了，一切都毁了——其实也没多少可毁的。</p><p>半小时之后是试考理物。小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span>  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>（勇气）Spirit</mtext></mrow><annotation encoding="application/x-tex">\text{（勇气）Spirit}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord cjk_fallback">（勇气）</span><span class="mord">Spirit</span></span></span></span></span> 直接崩溃，根本不想动弹，更别提出去散步了。他趴在桌子上，试图让发烧的脑袋冷静一点。</p><p>开考了，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 惊恐的发现， <em>肚子的</em> 虽然没能刮走小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 实力的象征 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathcolor="orange"><mtext>橡皮</mtext></mstyle></mrow><annotation encoding="application/x-tex">\textcolor{orange}{橡皮}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord cjk_fallback" style="color:orange;">橡皮</span></span></span></span> ，但小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 带着橡皮也不会用。</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>这是肚子的的阴谋！一定是！</mtext></mrow><annotation encoding="application/x-tex">\text{这是肚子的的阴谋！一定是！}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">这是肚子的的阴谋！一定是！</span></span></span></span></span></span><p>然而，奇迹发生了。或许是被 <em>肚子的</em> 折磨了一整天，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的求生本能被彻底激发；又或许是发烧让他的大脑进入了某种奇特的亢奋状态。接下来的试考，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 如有神助，选择题全对，填空题只错一个，大题思路清晰，仿佛 <em>肚子的</em> 的大手在物理这一科突然松开了。他奋笔疾书，提前交卷，神采奕奕地走出考场——这是他两天来第一次挺直腰板。</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>96</mn><mi>p</mi><mi>t</mi><mi>s</mi><mo>≤</mo><mi>S</mi><mi>c</mi><mi>o</mi><mi>r</mi><msub><mi>e</mi><mtext>理物</mtext></msub><mo>≤</mo><mn>98</mn><mi>p</mi><mi>t</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">96pts \le Score_{理物} \le 98pts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">96</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.0576em;">S</span><span class="mord mathnormal" style="margin-right:0.0278em;">cor</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord cjk_fallback mtight">理物</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">98</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span></span></span></span></span><p>……</p><p>Day2 下午， <em>肚子的</em> 大手依然离不开 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>HL</mtext></mrow><annotation encoding="application/x-tex">\text{HL}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">HL</span></span></span></span></span> ， <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Force</mtext></mrow><annotation encoding="application/x-tex">\text{Force}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Force</span></span></span></span></span> 其在中午改完了试卷，出了成绩。</p><p>全班一共16人。</p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>111pts&nbsp;。</mtext></mrow><annotation encoding="application/x-tex">\text{111pts 。}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">111pts&nbsp;</span><span class="mord cjk_fallback">。</span></span></span></span></span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>倒数第一。</mtext></mrow><annotation encoding="application/x-tex">\text{倒数第一。}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">倒数第一。</span></span></span></span></span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Where&nbsp;is&nbsp;his&nbsp;Not&nbsp;Last&nbsp;One?</mtext></mrow><annotation encoding="application/x-tex">\text{Where is his Not Last One?}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord text"><span class="mord">Where&nbsp;is&nbsp;his&nbsp;Not&nbsp;Last&nbsp;One?</span></span></span></span></span><p><em>肚子的</em> 击碎了小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 的梦，他第一个梦——虽然那梦本身就挺寒酸的。</p><p>然而，物理成绩也出了。</p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>98pts&nbsp;。</mtext></mrow><annotation encoding="application/x-tex">\text{98pts 。}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">98pts&nbsp;</span><span class="mord cjk_fallback">。</span></span></span></span></span><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>全班第一。</mtext></mrow><annotation encoding="application/x-tex">\text{全班第一。}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">全班第一。</span></span></span></span></span><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 愣住了。他第一次感受到，原来 <em>肚子的</em> 的大手也有漏网之鱼。</p><p>Day3 ，放学时。</p><p>各科成绩都出了： <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>122</mn><mi>p</mi><mi>t</mi><mi>s</mi><mo separator="true">,</mo><mn>111</mn><mi>p</mi><mi>t</mi><mi>s</mi><mo separator="true">,</mo><mn>102</mn><mi>p</mi><mi>t</mi><mi>s</mi><mo separator="true">,</mo><mn>98</mn><mi>p</mi><mi>t</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">122pts,111pts,102pts,98pts</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">122</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">111</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">102</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">98</span><span class="mord mathnormal">pt</span><span class="mord mathnormal">s</span></span></span></span> 。</p><p>语文122pts；英语102pts，全班16人中倒数第三；数学111pts，全班16人中倒数第一；物理98pts，全班16人中正数第一。</p><p>Day3 night 。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 趴在床上，烧已经退了，但他还是想哭。他梦见自己没有挂分，参加了高考和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>I</mi><mi>O</mi><mi>N</mi></mrow><annotation encoding="application/x-tex">ION</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0785em;">I</span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span> ，勇夺金牌保送名牌大学，然后在校园里无忧无虑，成为人生赢家。但是梦是假的。梦被 <em>肚子的</em> 大手击碎了。他跌落在成为人生赢家怀抱的第一步上——其实他还差得远，连门槛都没摸到。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 只是一个炮灰、垫脚石而已，是别人成功路上的杂草、泥沙，将要被大雨冲刷殆尽的。</p><p>小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 再次趴着哭了。他梦见了 <em>肚子的</em>，梦见了 <em>肚子的</em> 在背后操纵的一切。</p><p>梦是假了。<em>肚子的</em> 永远不会面见像小  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span>  这样的炮灰、垫脚石。</p><p>但梦是真的，小 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0738em;">ζ</span></span></span></span> 挂掉的分是实实在在的，一去不复返的。而且，他本来也没多少分可挂。</p><p>只是，那全班第一的物理98pts，像一把 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>小</mtext><mi>ω</mi><mstyle mathcolor="orange"><mtext>小尺直</mtext></mstyle></mrow><annotation encoding="application/x-tex">小\omega\textcolor{orange}{小尺直}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord cjk_fallback">小</span><span class="mord mathnormal" style="margin-right:0.0359em;">ω</span><span class="mord cjk_fallback" style="color:orange;">小尺直</span></span></span></span> 一样，笔直地插在他满是泥泞的成绩单上，刺眼得让人想哭。</p>]]></content>
    
    
      
      
    <summary type="html">&lt;h1&gt;肚子的爷爷操纵天气使小 &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mi&gt;ζ&lt;/mi&gt;&lt;</summary>
      
    
    
    
    
    <category term="总结" scheme="http://example.com/tags/%E6%80%BB%E7%BB%93/"/>
    
  </entry>
  
  <entry>
    <title>Vscode配置c环境之利用dev c++或codeblock环境，无缝升级VScode编辑器教程</title>
    <link href="http://example.com/2026/04/15/Vscode%E9%85%8D%E7%BD%AEc%E7%8E%AF%E5%A2%83%E4%B9%8B%E5%88%A9%E7%94%A8dev%20c++%E6%88%96codeblock%E7%8E%AF%E5%A2%83%EF%BC%8C%E6%97%A0%E7%BC%9D%E5%8D%87%E7%BA%A7VScode%E7%BC%96%E8%BE%91%E5%99%A8%E6%95%99%E7%A8%8B-3BPM%E7%9A%84%E6%96%87%E7%AB%A0/"/>
    <id>http://example.com/2026/04/15/Vscode%E9%85%8D%E7%BD%AEc%E7%8E%AF%E5%A2%83%E4%B9%8B%E5%88%A9%E7%94%A8dev%20c++%E6%88%96codeblock%E7%8E%AF%E5%A2%83%EF%BC%8C%E6%97%A0%E7%BC%9D%E5%8D%87%E7%BA%A7VScode%E7%BC%96%E8%BE%91%E5%99%A8%E6%95%99%E7%A8%8B-3BPM%E7%9A%84%E6%96%87%E7%AB%A0/</id>
    <published>2026-04-15T02:49:58.440Z</published>
    <updated>2024-03-03T02:42:00.000Z</updated>
    
    <content type="html"><![CDATA[<p><strong>24年上完计科专业更新 本地windows的c语言环境，建议自行安装msys或采用 wsl2环境</strong>（主推）Vmware VirtualBox<strong>等虚拟机方案，现在的wsl2带x11转发不是一般的好用</strong></p><p><strong>也可采用visual studio的非常完整的环境</strong></p><p><strong>直接使用devcpp的环境只是便捷之举，msys2很强。而且自己写json这种运行方法太不适合大项目了 cmake 、makefile才能自动化 建议直接找个用vscode的cmake教程自己学，，一步到位 免得将来看大项目发憷</strong></p><p>因为许多大学生学习c语言时接触到的第一个IDE都是dev c++或者codeblock，而相对来讲VScode功能更强，使用更便捷，界面更现xuan代ku。</p><p>因为某些原因国内下mingw的速度比较慢，而且devcpp自带的编译器会更加贴合课程的要求。（而且基于下载的mingw的vscodec环境搭建教程已经很详尽了 但是步骤较多）</p><p>本教程仅仅三步无缝从devc升级为vscode…</p><p>#1 找到devc 集成的mingw路径</p><p>通常为：C:\Program Files (x86)\Dev-Cpp\MinGW64\bin（gcc,g++,gdb所在路径）</p><p><img src="https://pic4.zhimg.com/v2-ed5e690bf1b472be4612425e0cdf44a9_r.jpg" alt=""></p><p>并将其添加到环境变量中</p><p><img src="https://pic4.zhimg.com/v2-b0c1a59b5d26f23991be771a91b02c07_r.jpg" alt=""></p><p><img src="https://pic2.zhimg.com/v2-652079b09ff917abb08fff2b0e8f3175_r.jpg" alt=""></p><p>这时打开cmd 键入gcc -v就有了神奇的结果</p><p><img src="https://pic2.zhimg.com/v2-74d1feb11e4251d75b43758297a9a9cf_r.jpg" alt=""></p><p>顺便再通过命令行查看自己的配置路径：<code>gcc -v -E -x c++ -</code> 并记住，后面会用到。</p><p><img src="https://pic2.zhimg.com/v2-b3e61472fbb3d7bd62099728d2e1cfe7_r.jpg" alt="有/.../不是bug！如果后续如果不这么写FILE *fp 会有问题的"></p><p>#2官网下载安装vscode</p><p><a href="https//code.visualstudio.com/">link</a></p><p><img src="https://pic4.zhimg.com/v2-45c70a6282393c88a1807a5486eba3b7_r.jpg" alt=""></p><p>安装好后打开并安装中文和c/c++插件</p><p><img src="https://pic1.zhimg.com/v2-454e9f296ae3ef3ee2c93846be0de11a_r.jpg" alt=""></p><p><img src="https://pic1.zhimg.com/v2-63d373ab456c69efd1b36e1996f0c1ec_r.jpg" alt=""></p><p>#3 找一个你想要把代码放进去的文件夹（<strong>一定要全英文且没有特殊符号，不然没法运行！</strong>）用vscode打开文件夹（左上角文件-&gt;打开文件夹 或者ctrl+k +o）</p><p><img src="https://pica.zhimg.com/v2-6c2f13977b15eeed933295ca95c087a0_r.jpg" alt="比如此文件夹"></p><p>然后在文件夹下（会自动生成）的.vscode文件夹下新建 三个文件（<strong>这三个文件极其重要本人也摸索了半天</strong>，把我的结果贴在这，需要稍加改动即可正常调试。）</p><p><img src="https://pic4.zhimg.com/v2-c9d99e1ba6798bbb821e4e339d6cddf7_1440w.jpg" alt=""></p><p>launch.json</p><figure class="highlight json"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br></pre></td><td class="code"><pre><span class="line"><span class="punctuation">{</span></span><br><span class="line">    <span class="attr">"version"</span><span class="punctuation">:</span> <span class="string">"0.2.0"</span><span class="punctuation">,</span></span><br><span class="line">    <span class="attr">"configurations"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">        <span class="punctuation">{</span></span><br><span class="line">            <span class="attr">"name"</span><span class="punctuation">:</span> <span class="string">"(gdb) 启动"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"type"</span><span class="punctuation">:</span> <span class="string">"cppdbg"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"request"</span><span class="punctuation">:</span> <span class="string">"launch"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"program"</span><span class="punctuation">:</span> <span class="string">"${fileDirname}\\${fileBasenameNoExtension}.exe"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"args"</span><span class="punctuation">:</span> <span class="punctuation">[</span><span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"stopAtEntry"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">false</span></span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"cwd"</span><span class="punctuation">:</span> <span class="string">"${fileDirname}"</span><span class="punctuation">,</span><span class="comment">//指默认文件操作的目录（如fopen的默认寻找位置），现在是c文件的目录，修改为${workspaceFolder}就是工作文件夹目录</span></span><br><span class="line">            <span class="attr">"environment"</span><span class="punctuation">:</span> <span class="punctuation">[</span><span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"externalConsole"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">true</span></span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"MIMode"</span><span class="punctuation">:</span> <span class="string">"gdb"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"miDebuggerPath"</span><span class="punctuation">:</span> <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/gdb.exe"</span><span class="punctuation">,</span><span class="comment">//路径要修改</span></span><br><span class="line">            <span class="attr">"setupCommands"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">                <span class="punctuation">{</span></span><br><span class="line">                    <span class="attr">"description"</span><span class="punctuation">:</span> <span class="string">"为 gdb 启用整齐打印"</span><span class="punctuation">,</span></span><br><span class="line">                    <span class="attr">"text"</span><span class="punctuation">:</span> <span class="string">"-enable-pretty-printing"</span><span class="punctuation">,</span></span><br><span class="line">                    <span class="attr">"ignoreFailures"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">true</span></span></span><br><span class="line">                <span class="punctuation">}</span></span><br><span class="line">            <span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"preLaunchTask"</span><span class="punctuation">:</span> <span class="string">"C/C++: gcc.exe 生成活动文件"</span></span><br><span class="line">        <span class="punctuation">}</span></span><br><span class="line">    <span class="punctuation">]</span></span><br><span class="line"><span class="punctuation">}</span></span><br></pre></td></tr></tbody></table></figure><p>tasks.json</p><figure class="highlight json"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br></pre></td><td class="code"><pre><span class="line"><span class="punctuation">{</span></span><br><span class="line">    <span class="attr">"version"</span><span class="punctuation">:</span> <span class="string">"2.0.0"</span><span class="punctuation">,</span></span><br><span class="line">    <span class="attr">"command"</span><span class="punctuation">:</span> <span class="string">"gcc"</span><span class="punctuation">,</span></span><br><span class="line">    <span class="attr">"args"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">        <span class="string">"-g"</span><span class="punctuation">,</span></span><br><span class="line">        <span class="string">"${file}"</span><span class="punctuation">,</span></span><br><span class="line">        <span class="string">"-o"</span><span class="punctuation">,</span></span><br><span class="line">        <span class="string">"${file}.exe"</span></span><br><span class="line">    <span class="punctuation">]</span><span class="punctuation">,</span> <span class="comment">// 编译命令参数</span></span><br><span class="line">    <span class="attr">"problemMatcher"</span><span class="punctuation">:</span> <span class="punctuation">{</span></span><br><span class="line">        <span class="attr">"owner"</span><span class="punctuation">:</span> <span class="string">"cpp"</span><span class="punctuation">,</span></span><br><span class="line">        <span class="attr">"fileLocation"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">            <span class="string">"relative"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="string">"${workspaceRoot}"</span></span><br><span class="line">        <span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">        <span class="attr">"pattern"</span><span class="punctuation">:</span> <span class="punctuation">{</span></span><br><span class="line">            <span class="attr">"regexp"</span><span class="punctuation">:</span> <span class="string">"^(.*):(\\d+):(\\d+):\\s+(warning|error):\\s+(.*)$"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"file"</span><span class="punctuation">:</span> <span class="number">1</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"line"</span><span class="punctuation">:</span> <span class="number">2</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"column"</span><span class="punctuation">:</span> <span class="number">3</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"severity"</span><span class="punctuation">:</span> <span class="number">4</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"message"</span><span class="punctuation">:</span> <span class="number">5</span></span><br><span class="line">        <span class="punctuation">}</span></span><br><span class="line">    <span class="punctuation">}</span><span class="punctuation">,</span></span><br><span class="line">    <span class="attr">"presentation"</span><span class="punctuation">:</span> <span class="punctuation">{</span></span><br><span class="line">        <span class="attr">"echo"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">true</span></span><span class="punctuation">,</span></span><br><span class="line">        <span class="attr">"reveal"</span><span class="punctuation">:</span> <span class="string">"always"</span><span class="punctuation">,</span></span><br><span class="line">        <span class="attr">"focus"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">false</span></span><span class="punctuation">,</span></span><br><span class="line">        <span class="attr">"panel"</span><span class="punctuation">:</span> <span class="string">"new"</span><span class="punctuation">,</span></span><br><span class="line">        <span class="attr">"showReuseMessage"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">true</span></span><span class="punctuation">,</span></span><br><span class="line">        <span class="attr">"clear"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">false</span></span></span><br><span class="line">    <span class="punctuation">}</span><span class="punctuation">,</span></span><br><span class="line">    <span class="attr">"tasks"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">        <span class="punctuation">{</span></span><br><span class="line">            <span class="attr">"type"</span><span class="punctuation">:</span> <span class="string">"cppbuild"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"label"</span><span class="punctuation">:</span> <span class="string">"C/C++: gcc.exe 生成活动文件"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"command"</span><span class="punctuation">:</span> <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/gcc.exe"</span><span class="punctuation">,</span><span class="comment">//路径要修改</span></span><br><span class="line">            <span class="attr">"args"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">                <span class="string">"-g"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"${file}"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"-o"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"${fileDirname}\\${fileBasenameNoExtension}.exe"</span></span><br><span class="line">            <span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"options"</span><span class="punctuation">:</span> <span class="punctuation">{</span></span><br><span class="line">                <span class="attr">"cwd"</span><span class="punctuation">:</span> <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin"</span><span class="comment">//路径要修改</span></span><br><span class="line">            <span class="punctuation">}</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"problemMatcher"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">                <span class="string">"$gcc"</span></span><br><span class="line">            <span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"group"</span><span class="punctuation">:</span> <span class="string">"build"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"detail"</span><span class="punctuation">:</span> <span class="string">"编译器: \"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/gcc.exe"</span><span class="comment">//路径要修改</span></span><br><span class="line">        <span class="punctuation">}</span></span><br><span class="line">    <span class="punctuation">]</span></span><br><span class="line"><span class="punctuation">}</span></span><br></pre></td></tr></tbody></table></figure><p>c_cpp_properties.json</p><figure class="highlight json"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br></pre></td><td class="code"><pre><span class="line"><span class="punctuation">{</span></span><br><span class="line">    <span class="attr">"configurations"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">        <span class="punctuation">{</span></span><br><span class="line">            <span class="attr">"name"</span><span class="punctuation">:</span> <span class="string">"Win32"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"includePath"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">                <span class="string">"${workspaceFolder}/**"</span><span class="punctuation">,</span>            </span><br><span class="line">                <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/../lib/gcc/x86_64-w64-mingw32/4.9.2/include/c++"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/../lib/gcc/x86_64-w64-mingw32/4.9.2/include/c++/x86_64-w64-mingw32"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/../lib/gcc/x86_64-w64-mingw32/4.9.2/include/c++/backward"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/../lib/gcc/x86_64-w64-mingw32/4.9.2/include"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/../lib/gcc/x86_64-w64-mingw32/4.9.2/../../../../include"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/../lib/gcc/x86_64-w64-mingw32/4.9.2/include-fixed"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/../lib/gcc/x86_64-w64-mingw32/4.9.2/../../../../x86_64-w64-mingw32/include"</span></span><br><span class="line">    <span class="comment">//路径要修改</span></span><br><span class="line">            <span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"defines"</span><span class="punctuation">:</span> <span class="punctuation">[</span></span><br><span class="line">                <span class="string">"_DEBUG"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"UNICODE"</span><span class="punctuation">,</span></span><br><span class="line">                <span class="string">"_UNICODE"</span></span><br><span class="line">            <span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"compilerPath"</span><span class="punctuation">:</span> <span class="string">"C:/Program Files (x86)/Dev-Cpp/MinGW64/bin/gcc.exe"</span><span class="punctuation">,</span><span class="comment">//路径要修改</span></span><br><span class="line">            <span class="attr">"cStandard"</span><span class="punctuation">:</span> <span class="string">"c99"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"cppStandard"</span><span class="punctuation">:</span> <span class="string">"c++98"</span><span class="punctuation">,</span></span><br><span class="line">            <span class="attr">"intelliSenseMode"</span><span class="punctuation">:</span> <span class="string">"gcc-x64"</span></span><br><span class="line">        <span class="punctuation">}</span></span><br><span class="line">    <span class="punctuation">]</span><span class="punctuation">,</span></span><br><span class="line">    <span class="attr">"version"</span><span class="punctuation">:</span> <span class="number">4</span></span><br><span class="line"><span class="punctuation">}</span></span><br></pre></td></tr></tbody></table></figure><p>愉快的开始新建new.c 编程吧（注意文件名也不能有中文和括号）</p><p>注意有一个问题我一直没有解决就是F5debug运行到最后不会停住，在命令行里看不到结果，就闪退了，只能在程序结尾自行添加system(“PAUSE”);（引入stdlib.h）。</p><p>自动格式化代码需要改setting.json具体网上搜索。</p>]]></content>
    
    
      
      
    <summary type="html">&lt;p&gt;&lt;strong&gt;24年上完计科专业更新 本地windows的c语言环境，建议自行安装msys或采用 wsl2环境&lt;/strong&gt;（主推）Vmware VirtualBox&lt;strong&gt;等虚拟机方案，现在的wsl2带x11转发不是一般的好用&lt;/strong&gt;&lt;/p&gt;
&lt;p</summary>
      
    
    
    
    
  </entry>
  
  <entry>
    <title>实用CCW拓展记录</title>
    <link href="http://example.com/2026/02/24/%E5%AE%9E%E7%94%A8CCW%E6%8B%93%E5%B1%95%E8%AE%B0%E5%BD%95/"/>
    <id>http://example.com/2026/02/24/%E5%AE%9E%E7%94%A8CCW%E6%8B%93%E5%B1%95%E8%AE%B0%E5%BD%95/</id>
    <published>2026-02-24T10:27:00.000Z</published>
    <updated>2026-02-24T10:28:49.817Z</updated>
    
    <content type="html"><![CDATA[<h1>UI类</h1><p><code>OpenUITextEditor</code> – 文本编辑器</p><p><code>mdui</code> -类安卓风格复杂UI弹窗系统</p>]]></content>
    
    
      
      
    <summary type="html">&lt;h1&gt;UI类&lt;/h1&gt;
&lt;p&gt;&lt;code&gt;OpenUITextEditor&lt;/code&gt; – 文本编辑器&lt;/p&gt;
&lt;p&gt;&lt;code&gt;mdui&lt;/code&gt; -类安卓风格复杂UI弹窗系统&lt;/p&gt;
</summary>
      
    
    
    
    
  </entry>
  
  <entry>
    <title>Hydro-ai</title>
    <link href="http://example.com/2026/01/28/Hydro-ai/"/>
    <id>http://example.com/2026/01/28/Hydro-ai/</id>
    <published>2026-01-28T10:51:54.000Z</published>
    <updated>2026-01-28T10:53:05.926Z</updated>
    
    <content type="html"><![CDATA[<p>当一道 timber 函数式编程思维题摆在你面前，读懂题面就花了 20 分钟，debug 时发现理解错了题意，这种痛苦相信每个 OIer 都懂。 HydroOJ 的题面往往承载着完整的背景故事，但对于初赛选手或是跨语言学习者来说，<strong>理解题意本身就是一道门槛</strong>。</p><p>今天开源的 <strong>hydrooj-plugin-ai-reader</strong>（AI 题面理解助手），不是来抢走你 AC 的成就感的，而是帮你<strong>用最短的时间抓住题目的命门</strong>。</p><h2 id="痛点：题面太长，抓不住重点">痛点：题面太长，抓不住重点</h2><p>以一道典型的 OI 题为例：</p><blockquote><p><em>“Alice 和 Bob 在一张 n 个点 m 条边的无向图上玩游戏。每个点有一个权值……（500 字背景）……要求你输出一个最小的 k 使得对于所有满足……（数学定义）……的方案，都能……（条件判断）……”</em></p></blockquote><p>读完这段文字，你的大脑需要：</p><ol><li>过滤掉 Alice 和 Bob 的故事线</li><li>提取形式化定义：图、点权、阈值 k</li><li>识别约束条件：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≤</mo><msup><mn>10</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">n \le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span> 意味着什么？</li></ol><p>这不仅耗时，还容易看错。特别是在多校训练中，时间宝贵，读懂题意就赢了一半。</p><h2 id="解决方案：AI-题面理解助手">解决方案：AI 题面理解助手</h2><p>这个插件在 HydroOJ 的每道题目右侧新增一个"🤖 AI 题面理解"按钮。点击后，你会看到：</p><h3 id="1-极简题意概括">1. 极简题意概括</h3><p>用 1-2 句话说明"给定什么，求什么"，去掉所有背景故事。<br>*示例：*给定无向图，点带点权，求满足连通块最大值限制的最小删边代价。</p><h3 id="2-关键信息提取">2. 关键信息提取</h3><ul><li><strong>数据范围速查</strong>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≤</mo><msup><mn>10</mn><mn>5</mn></msup><mo>⇒</mo></mrow><annotation encoding="application/x-tex">n \le 10^5 \Rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⇒</span></span></span></span> 需要线性或线性对数算法</li><li><strong>隐藏约束</strong>：无重边、权值非负、保证有解等容易忽视的前提</li><li><strong>边界情况</strong>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">m=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>、权值相等时的特殊处理</li></ul><h3 id="3-思考路径（苏格拉底式提问）">3. 思考路径（苏格拉底式提问）</h3><p>不会告诉你"用动态规划"或"建最短路"，而是问：</p><ul><li>“如果固定答案 k，能否快速检验可行性？”（暗示二分 or 贪心检验）</li><li>“观察数据范围，什么样的复杂度会 TLE？”（暗示算法复杂度选择）</li></ul><p>这是一种<strong>启发式学习</strong>，既保护了解题乐趣，又避免了在错误的方向上浪费时间。</p><h2 id="技术架构：与-HydroOJ-深度融合">技术架构：与 HydroOJ 深度融合</h2><h3 id="域级开关控制（安全第一）">域级开关控制（安全第一）</h3><p>与市面上其他插件不同，<strong>默认全站关闭</strong>，需在「控制面板-域设置」中手动开启。这保证了：</p><ul><li>正式比赛期间可完全禁用，维护公平性</li><li>不同训练域可灵活配置（入门域开启，高阶域关闭）</li><li>防止非目标用户误用（如将 OJ 用于就业笔试时）</li></ul><h3 id="智能缓存机制">智能缓存机制</h3><p>同一道题的 AI 分析只需生成一次，存储于 MongoDB，后续所有用户共享。<strong>减少 API 调用成本</strong>的同时，也保证了同一道题提供的思考角度具有共识性。</p><h3 id="零侵入式集成">零侵入式集成</h3><ul><li><strong>前端</strong>：通过 <code>NamedPage</code> 注入侧边栏，符合 Hydro 原生 UI 风格（使用 Hydro 的卡片、按钮、排版系统）</li><li><strong>后端</strong>：独立路由 <code>/p/:pid/ai</code>，权限继承 Hydro 的 <code>PRIV_USER_PROFILE</code></li><li><strong>风格统一</strong>：输出采用 Markdown + KaTeX，与题面渲染一致</li></ul><h3 id="提词工程（Prompt-Engineering）">提词工程（Prompt Engineering）</h3><p>核心提示词设计遵循**“教渔不授鱼”**原则：</p><figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">角色：资深 OI 教练</span><br><span class="line">禁止词：动态规划、最短路、线段树、ST 表...</span><br><span class="line">必做：将数学抽象描述为"是否具有某种性质"而非"使用某种数据结构"</span><br><span class="line">必做：每个思考点以疑问句结尾</span><br></pre></td></tr></tbody></table></figure><p>实际调用的 DeepSeek-V3 模型在此约束下，输出质量足以媲美教练手写 hint。</p><h2 id="快速开始">快速开始</h2><h3 id="安装">安装</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 进入服务器</span></span><br><span class="line"><span class="built_in">cd</span> /path/to/hydrooj/addons</span><br><span class="line">git <span class="built_in">clone</span> https://github.com/your-repo/hydrooj-plugin-ai-reader.git</span><br><span class="line"><span class="built_in">cd</span> hydrooj-plugin-ai-reader</span><br><span class="line"></span><br><span class="line"><span class="comment"># HydroOJ 自动识别插件</span></span><br><span class="line">hydrooj addon add $(<span class="built_in">pwd</span>)</span><br><span class="line">pm2 reload hydrooj</span><br></pre></td></tr></tbody></table></figure><h3 id="配置（管理员）">配置（管理员）</h3><p>进入「控制面板-系统设置」，添加 AI 配置（支持 GitCode/OpenRouter/DeepSeek 等）：</p><figure class="highlight yaml"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line"><span class="attr">ai-summary:</span></span><br><span class="line">  <span class="attr">apiKey:</span> <span class="string">"sk-your-key"</span></span><br><span class="line">  <span class="attr">model:</span> <span class="string">"deepseek-ai/DeepSeek-V3.2-Exp"</span></span><br><span class="line">  <span class="attr">apiUrl:</span> <span class="string">"https://api-ai.gitcode.com/v1/chat/completions"</span></span><br></pre></td></tr></tbody></table></figure><h3 id="启用功能">启用功能</h3><p>进入「域设置-🤖 AI 题面理解设置」，勾选<strong>启用</strong>。此时该域下的所有题目右侧都会出现 AI 按钮。</p><h2 id="实际效果对比">实际效果对比</h2><p><strong>原题面片段（300 字）：</strong><br><em>小 K 是一位著名的嗜睡病患者，他每天都在与无垠的睡意作斗争。在这一天，他收到了 n 个任务，第 i 个任务有一个截止时间 d_i 和需要的时间 t_i……</em></p><p><strong>AI 简化后：</strong></p><blockquote><p><strong>题意</strong>：单资源调度问题，n 个任务有截止时间和耗时，求最大可完成的任务数。<br><strong>关键</strong>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">n \le 2×10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>t</mi><mi>i</mi></msub><mo>≤</mo><msup><mn>10</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">d_i,t_i \le 10^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span>，暗示需 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 或更优。<br><strong>思考</strong>：“如果有个任务耗时很长但截止时间很晚，它是否一定比耗时短且截止时间早的任务优先级低？”（暗示贪心按截止时间排序）</p></blockquote><p><strong>价值</strong>：20 秒抓住题目是"带截止时间的调度"，而非浪费 3 分钟去同情小 K 的嗜睡症。</p><h2 id="使用建议-边界">使用建议 &amp; 边界</h2><h3 id="适合开启的场景">适合开启的场景</h3><ul><li><strong>新手入门域</strong>：帮助初学者学习如何读题、提取形式化模型</li><li><strong>英文题面训练</strong>：快速获取中文题意概括，降低语言门槛</li><li><strong>题单复习</strong>：快速回顾题目核心，验证自己对题意的记忆是否准确</li></ul><h3 id="建议关闭的场景">建议关闭的场景</h3><ul><li><strong>正式比赛期间</strong>：维护公平性（可在比赛配置中单独关闭）</li><li><strong>原创题校考</strong>：防止外校选手利用 AI 快速理解题意优势</li></ul><h3 id="局限性">局限性</h3><ul><li><strong>无法替代独立思考</strong>：不提供算法代码，不提供具体转移方程</li><li><strong>可能有幻觉</strong>：AI 可能对复杂题意理解有偏差，最终以原题面为准</li><li><strong>API 成本</strong>：生成一次约消耗 1-2k tokens，请自行评估成本</li></ul><h2 id="结语">结语</h2><p>hydrooj-plugin-ai-reader 不是来帮你 AC 的，而是帮你<strong>更快地进入状态</strong>。</p><p>OI 训练的本质是思维训练，而不是阅读理解题。当你能在 30 秒内通过 AI 摘要确认"这题是线段树维护区间众数"，然后立即开始思考具体怎么维护，你的训练效率就已经超过了还在读背景故事的同学。</p><p><strong>GitHub 地址</strong>：<a href="https://github.com">github.com/your-repo/hydrooj-plugin-ai-reader</a><br><strong>讨论/反馈</strong>：欢迎提 Issue 或加 HydroOJ 用户群交流</p><p><em>祝你在 AI 的辅助下，把更多时间花在真正的算法思考上。</em></p>]]></content>
    
    
      
      
    <summary type="html">&lt;p&gt;当一道 timber 函数式编程思维题摆在你面前，读懂题面就花了 20 分钟，debug 时发现理解错了题意，这种痛苦相信每个 OIer 都懂。 HydroOJ 的题面往往承载着完整的背景故事，但对于初赛选手或是跨语言学习者来说，&lt;strong&gt;理解题意本身就是一道门槛&lt;/</summary>
      
    
    
    
    
  </entry>
  
  <entry>
    <title>糖果公园 - 题解</title>
    <link href="http://example.com/2025/12/27/P4070%E7%B3%96%E6%9E%9C%E5%85%AC%E5%9B%AD%E9%A2%98%E8%A7%A3/"/>
    <id>http://example.com/2025/12/27/P4070%E7%B3%96%E6%9E%9C%E5%85%AC%E5%9B%AD%E9%A2%98%E8%A7%A3/</id>
    <published>2025-12-27T13:47:20.000Z</published>
    <updated>2025-12-31T03:46:38.866Z</updated>
    
    <content type="html"><![CDATA[<h1>糖果公园 题解</h1><h2 id="题目大意">题目大意</h2><p>给定一棵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个节点的树，每个节点有一种糖果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (共 m 种)。每种糖果有美味值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，品尝第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span></span></span></span> 次同种糖果的新奇感为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">W_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。游客从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> 走到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span></span></span></span> 的愉悦度为路径上所有节点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><msub><mi>C</mi><mi>i</mi></msub></msub></mrow><annotation encoding="application/x-tex">V_{C_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9334em;vertical-align:-0.2501em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.2222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span></span></span> × <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mrow><mi>c</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">W_{count(C_i)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0385em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">co</span><span class="mord mathnormal mtight">u</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> ​之和，其中</p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>o</mi><mi>u</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><msub><mi>C</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">count(C_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">co</span><span class="mord mathnormal">u</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0715em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是此种糖果在当前路径上出现的次数。<p>需要支持两种操作：</p><p>修改某个节点的糖果种类。<br>查询某条路径的愉悦度。</p><h2 id="算法分析">算法分析</h2><p>一看就指了解这题是 <strong>树上带修改莫队</strong> 问题。与序列莫队不同，本解法采用 DFN序结合路径移动 的方式，直接处理树上路径。<br>核心思路：DFN序结合带修莫队<br>DFN序预处理：通过DFS得到每个节点的DFN序（深度优先搜索序），用于莫队的块划分。<br>路径移动：使用 work 函数通过不断移动到父节点来改变当前路径。对于路径 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u, v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">v</span><span class="mclose">)</span></span></span></span>，通过移动端点到LCA来添加或移除节点。<br>处理修改：增加时间维度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span>，记录修改操作。在莫队移动左右指针时，同时移动时间指针来处理修改。<br>LCA处理：关键点在于，路径移动后，LCA节点可能未被包含，因此需要在计算答案前临时添加LCA点，计算后移除。<br>块大小设置：通常取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>=</mo><msup><mi>n</mi><mrow><mn>2</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex">B = n^{2/3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.0502em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2/3</span></span></span></span></span></span></span></span></span></span></span></span>，总时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mrow><mn>5</mn><mi mathvariant="normal">/</mi><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^{5/3})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5/3</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。</p><h2 id="CODE">CODE</h2><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br><span class="line">124</span><br><span class="line">125</span><br><span class="line">126</span><br><span class="line">127</span><br><span class="line">128</span><br><span class="line">129</span><br><span class="line">130</span><br><span class="line">131</span><br><span class="line">132</span><br><span class="line">133</span><br><span class="line">134</span><br><span class="line">135</span><br><span class="line">136</span><br><span class="line">137</span><br><span class="line">138</span><br><span class="line">139</span><br><span class="line">140</span><br><span class="line">141</span><br><span class="line">142</span><br><span class="line">143</span><br><span class="line">144</span><br><span class="line">145</span><br><span class="line">146</span><br><span class="line">147</span><br><span class="line">148</span><br><span class="line">149</span><br><span class="line">150</span><br><span class="line">151</span><br><span class="line">152</span><br><span class="line">153</span><br><span class="line">154</span><br><span class="line">155</span><br><span class="line">156</span><br><span class="line">157</span><br><span class="line">158</span><br><span class="line">159</span><br><span class="line">160</span><br><span class="line">161</span><br><span class="line">162</span><br><span class="line">163</span><br><span class="line">164</span><br><span class="line">165</span><br><span class="line">166</span><br><span class="line">167</span><br><span class="line">168</span><br><span class="line">169</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">define</span> int long long</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"><span class="type">const</span> <span class="type">int</span> N = <span class="number">200005</span>;</span><br><span class="line"><span class="type">int</span> n, M, m, dfn[N], idx, vv[N], ww[N], b[N], a[N], lg[N], f[N][<span class="number">20</span>], d[N], res, ans[N];</span><br><span class="line"><span class="type">int</span> B = <span class="number">200</span>;</span><br><span class="line">bitset&lt;N&gt;ex;</span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">node</span>{</span><br><span class="line">    <span class="type">int</span> l, r, id, t;</span><br><span class="line">}q[N], qr[N];</span><br><span class="line">vector&lt;<span class="type">int</span>&gt;v[N];</span><br><span class="line"><span class="type">bool</span> <span class="keyword">operator</span> &lt; (node x, node y) {</span><br><span class="line">    <span class="type">int</span> bx = dfn[x.l] / B, by = dfn[y.l] / B;</span><br><span class="line">    <span class="keyword">if</span> (bx != by) <span class="keyword">return</span> bx &lt; by;</span><br><span class="line">    <span class="type">int</span> cx = dfn[x.r] / B, cy = dfn[y.r] / B;</span><br><span class="line">    <span class="keyword">if</span> (cx != cy) <span class="keyword">return</span> cx &lt; cy;</span><br><span class="line">    <span class="keyword">return</span> x.t &lt; y.t;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">dfs</span><span class="params">(<span class="type">int</span> x, <span class="type">int</span> fa)</span> </span>{</span><br><span class="line">    dfn[x] = ++idx;</span><br><span class="line">    d[x] = d[fa] + <span class="number">1</span>;</span><br><span class="line">    f[x][<span class="number">0</span>] = fa;</span><br><span class="line">    <span class="keyword">for</span> (<span class="keyword">auto</span> to : v[x]) {</span><br><span class="line">        <span class="keyword">if</span> (to != fa) {</span><br><span class="line">            <span class="built_in">dfs</span>(to, x);</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">lca</span><span class="params">(<span class="type">int</span> x, <span class="type">int</span> y)</span> </span>{</span><br><span class="line">    <span class="keyword">if</span> (d[x] &lt; d[y]) <span class="built_in">swap</span>(x, y);</span><br><span class="line">    <span class="type">int</span> diff = d[x] - d[y];</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; diff; ++i) {</span><br><span class="line">        <span class="keyword">if</span> (diff &amp; <span class="number">1</span>) x = f[x][i];</span><br><span class="line">        diff &gt;&gt;= <span class="number">1</span>;</span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">if</span> (x == y) <span class="keyword">return</span> x;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = lg[n]; i &gt;= <span class="number">0</span>; i--) {</span><br><span class="line">        <span class="keyword">if</span> (f[x][i] != f[y][i]) {</span><br><span class="line">            x = f[x][i];</span><br><span class="line">            y = f[y][i];</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">return</span> f[x][<span class="number">0</span>];</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">add</span><span class="params">(<span class="type">int</span> x)</span> </span>{</span><br><span class="line">    b[a[x]]++;</span><br><span class="line">    res += vv[a[x]] * ww[b[a[x]]];</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">del</span><span class="params">(<span class="type">int</span> x)</span> </span>{</span><br><span class="line">    res -= vv[a[x]] * ww[b[a[x]]];</span><br><span class="line">    b[a[x]]--;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">work</span><span class="params">(<span class="type">int</span> x, <span class="type">int</span> t)</span> </span>{</span><br><span class="line">    <span class="type">int</span> l = <span class="built_in">lca</span>(x, t);</span><br><span class="line">    <span class="keyword">while</span> (x != l) {</span><br><span class="line">        <span class="keyword">if</span> (ex[x]) {</span><br><span class="line">            ex[x] = <span class="number">0</span>;</span><br><span class="line">            <span class="built_in">del</span>(x);</span><br><span class="line">        } <span class="keyword">else</span> {</span><br><span class="line">            ex[x] = <span class="number">1</span>;</span><br><span class="line">            <span class="built_in">add</span>(x);</span><br><span class="line">        }</span><br><span class="line">        x = f[x][<span class="number">0</span>];</span><br><span class="line">    }</span><br><span class="line">    x = t;</span><br><span class="line">    <span class="keyword">while</span> (x != l) {</span><br><span class="line">        <span class="keyword">if</span> (ex[x]) {</span><br><span class="line">            ex[x] = <span class="number">0</span>;</span><br><span class="line">            <span class="built_in">del</span>(x);</span><br><span class="line">        } <span class="keyword">else</span> {</span><br><span class="line">            ex[x] = <span class="number">1</span>;</span><br><span class="line">            <span class="built_in">add</span>(x);</span><br><span class="line">        }</span><br><span class="line">        x = f[x][<span class="number">0</span>];</span><br><span class="line">    }</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">change</span><span class="params">(<span class="type">int</span> x, <span class="type">int</span> y)</span> </span>{</span><br><span class="line">    <span class="keyword">if</span> (ex[x]) {</span><br><span class="line">        <span class="built_in">del</span>(x);</span><br><span class="line">        a[x] = y;</span><br><span class="line">        <span class="built_in">add</span>(x);</span><br><span class="line">    } <span class="keyword">else</span> {</span><br><span class="line">        a[x] = y;</span><br><span class="line">    }</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="type">signed</span> <span class="title">main</span><span class="params">()</span> </span>{</span><br><span class="line">    cin &gt;&gt; n &gt;&gt; M &gt;&gt; m;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">2</span>; i &lt;= n; i++) {</span><br><span class="line">        lg[i] = lg[i &gt;&gt; <span class="number">1</span>] + <span class="number">1</span>;</span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= M; i++) cin &gt;&gt; vv[i];</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= n; i++) cin &gt;&gt; ww[i];</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt; n; i++) {</span><br><span class="line">        <span class="type">int</span> x, y;</span><br><span class="line">        cin &gt;&gt; x &gt;&gt; y;</span><br><span class="line">        v[x].<span class="built_in">push_back</span>(y);</span><br><span class="line">        v[y].<span class="built_in">push_back</span>(x);</span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];</span><br><span class="line"></span><br><span class="line">    idx = <span class="number">0</span>;</span><br><span class="line">    <span class="built_in">dfs</span>(<span class="number">1</span>, <span class="number">0</span>);</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= lg[n]; i++) {</span><br><span class="line">        <span class="keyword">for</span> (<span class="type">int</span> j = <span class="number">1</span>; j &lt;= n; j++) {</span><br><span class="line">            f[j][i] = f[f[j][i - <span class="number">1</span>]][i - <span class="number">1</span>];</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="type">int</span> qcnt = <span class="number">0</span>, tcnt = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">if</span>(<span class="built_in">pow</span>((<span class="type">double</span>)n, <span class="number">2.0</span>/<span class="number">3.0</span>) &gt; <span class="number">1</span>){</span><br><span class="line">        B = <span class="built_in">pow</span>((<span class="type">double</span>)n, <span class="number">2.0</span>/<span class="number">3.0</span>);</span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">else</span>{</span><br><span class="line">        B = <span class="number">1</span>;</span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= m; i++) {</span><br><span class="line">        <span class="type">int</span> type, x, y;</span><br><span class="line">        cin &gt;&gt; type &gt;&gt; x &gt;&gt; y;</span><br><span class="line">        <span class="keyword">if</span> (type == <span class="number">0</span>) {</span><br><span class="line">            tcnt++;</span><br><span class="line">            qr[tcnt].l = x;</span><br><span class="line">            qr[tcnt].r = a[x];</span><br><span class="line">            qr[tcnt].id = y;</span><br><span class="line">            qr[tcnt].t = tcnt;</span><br><span class="line">            a[x] = y;</span><br><span class="line">        } <span class="keyword">else</span> {</span><br><span class="line">            qcnt++;</span><br><span class="line">            q[qcnt].l = x;</span><br><span class="line">            q[qcnt].r = y;</span><br><span class="line">            q[qcnt].id = qcnt;</span><br><span class="line">            q[qcnt].t = tcnt;</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = tcnt; i &gt;= <span class="number">1</span>; --i) {</span><br><span class="line">        a[ qr[i].l ] = qr[i].r;</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="built_in">sort</span>(q + <span class="number">1</span>, q + qcnt + <span class="number">1</span>);</span><br><span class="line"></span><br><span class="line">    <span class="type">int</span> l = <span class="number">1</span>, r = <span class="number">1</span>, t = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= n; i++) ex[i] = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= M; i++) b[i] = <span class="number">0</span>;</span><br><span class="line">    res = <span class="number">0</span>;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= qcnt; i++) {</span><br><span class="line">        <span class="built_in">work</span>(l, q[i].l);</span><br><span class="line">        l = q[i].l;</span><br><span class="line">        <span class="built_in">work</span>(r, q[i].r);</span><br><span class="line">        r = q[i].r;</span><br><span class="line">        <span class="keyword">while</span> (t &lt; q[i].t) {</span><br><span class="line">            t++;</span><br><span class="line">            <span class="built_in">change</span>(qr[t].l, qr[t].id);</span><br><span class="line">        }</span><br><span class="line">        <span class="keyword">while</span> (t &gt; q[i].t) {</span><br><span class="line">            <span class="built_in">change</span>(qr[t].l, qr[t].r);</span><br><span class="line">            t--;</span><br><span class="line">        }</span><br><span class="line">        <span class="type">int</span> lc = <span class="built_in">lca</span>(l, r);</span><br><span class="line">        <span class="built_in">add</span>(lc); <span class="comment">// 临时添加LCA点</span></span><br><span class="line">        ans[q[i].id] = res;</span><br><span class="line">        <span class="built_in">del</span>(lc); <span class="comment">// 移除LCA点以恢复状态</span></span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= qcnt; i++) {</span><br><span class="line">        cout &lt;&lt; ans[i] &lt;&lt; endl;</span><br><span class="line">    }</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure>]]></content>
    
    
      
      
    <summary type="html">&lt;h1&gt;糖果公园 题解&lt;/h1&gt;
&lt;h2 id=&quot;题目大意&quot;&gt;题目大意&lt;/h2&gt;
&lt;p&gt;给定一棵 &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Math/Mat</summary>
      
    
    
    
    <category term="题解" scheme="http://example.com/categories/%E9%A2%98%E8%A7%A3/"/>
    
    
    <category term="题解" scheme="http://example.com/tags/%E9%A2%98%E8%A7%A3/"/>
    
  </entry>
  
  <entry>
    <title>方格染色 - 题解</title>
    <link href="http://example.com/2025/12/25/P9478-NOI2023-%E6%96%B9%E6%A0%BC%E6%9F%93%E8%89%B2A%E9%A2%98%E8%A7%A3/"/>
    <id>http://example.com/2025/12/25/P9478-NOI2023-%E6%96%B9%E6%A0%BC%E6%9F%93%E8%89%B2A%E9%A2%98%E8%A7%A3/</id>
    <published>2025-12-25T15:07:36.000Z</published>
    <updated>2025-12-31T03:49:20.399Z</updated>
    
    <content type="html"><![CDATA[<h1>子任务 A 题解（纯水平染色）</h1><blockquote><p>说句实话，我写的题解真的是不如AI。那么作为一个<s>OI选手</s>人类，我们能做的是什么呢？只有如何想出来的思路罢了。所以本篇题解主要注意的是思想过程，以及遇到一道题目时我的反应。</p></blockquote><h2 id="一、首先">一、首先</h2><h3 id="1-1-从长题面中提取">1.1 从长题面中提取</h3><p>看着很长，最关键的是什么呢？那肯定是提炼（说了句废话）。</p><p>通过看题，我们可以找到以下关键点：</p><blockquote><p><strong>注意</strong>：因为我们只做A，所以我们只需要考虑横的情况</p></blockquote><p>所以最后就变成了：一开始全白，给操作，每次把一行中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[x_1, x_2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> 这个范围变黑。</p><p>好的那我们就成功的完成了缩小（是不是很简单）</p><h3 id="1-2-看到问题首先想到什么">1.2 看到问题首先想到什么</h3><p><strong>第一反应</strong>：这是一个二维平面染色问题。但操作仅限于水平方向，显然不同行之间没有关系。</p><p>可是SAM说禁用显然，这可怎么办，那只能证明了v-v</p><ul><li>操作1：染黑第3行的[1,5]</li><li>操作2：染黑第5行的[2,4]</li><li>第3行的操作会影响第5行吗？<strong>显然不会</strong></li></ul><blockquote><p><strong>核心观察</strong>：不同行的操作完全独立！每行的染色结果只取决于该行的操作集合。</p></blockquote><p>于是问题转化为： <strong>对每一行，已知若干区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[x_1, x_2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>，求被覆盖的格子总数</strong>。</p><h2 id="二、那么我们可以想到什么方法呢">二、那么我们可以想到什么方法呢</h2><blockquote><p><strong>提示</strong>：首先我们要考虑的是：因为可能出现相同的地方重复染色，所以自然，我们可以想到以下算法</p></blockquote><h3 id="2-1-方法一：暴力法">2.1 方法一：暴力法</h3><p><strong>思路</strong>：对每行，创建一个大小为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的布尔数组，遍历每个操作，将对应位置标记为 <code>true</code>，最后统计 <code>true</code> 的数量。</p><p><strong>缺点</strong>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>≤</mo><msup><mn>10</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">m \le 10^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span>，数组太大，空间爆炸。</p><p><strong>暴力</strong></p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">// 此方法仅用于理解问题</span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>{</span><br><span class="line">    <span class="type">int</span> c; <span class="type">long</span> <span class="type">long</span> n, m; <span class="type">int</span> q;</span><br><span class="line">    cin &gt;&gt; c &gt;&gt; n &gt;&gt; m &gt;&gt; q;</span><br><span class="line">    </span><br><span class="line">    <span class="comment">//空间占用超标</span></span><br><span class="line">    vector&lt;vector&lt;<span class="type">bool</span>&gt;&gt; <span class="built_in">board</span>(n + <span class="number">1</span>, <span class="built_in">vector</span>&lt;<span class="type">bool</span>&gt;(m + <span class="number">1</span>, <span class="literal">false</span>));</span><br><span class="line">    </span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; q; i++) {</span><br><span class="line">        <span class="type">int</span> t; <span class="type">long</span> <span class="type">long</span> x1, y1, x2, y2;</span><br><span class="line">        cin &gt;&gt; t &gt;&gt; x1 &gt;&gt; y1 &gt;&gt; x2 &gt;&gt; y2;</span><br><span class="line">        <span class="keyword">if</span> (t == <span class="number">1</span>) {</span><br><span class="line">            <span class="keyword">for</span> (<span class="type">long</span> <span class="type">long</span> x = x1; x &lt;= x2; x++) {</span><br><span class="line">                board[y1][x] = <span class="literal">true</span>;</span><br><span class="line">            }</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> ans = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> y = <span class="number">1</span>; y &lt;= n; y++) {</span><br><span class="line">        <span class="keyword">for</span> (<span class="type">long</span> <span class="type">long</span> x = <span class="number">1</span>; x &lt;= m; x++) {</span><br><span class="line">            <span class="keyword">if</span> (board[y][x]) ans++;</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line">    cout &lt;&lt; ans &lt;&lt; endl;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><h3 id="2-2-方法二：线段树-树状数组">2.2 方法二：线段树/树状数组</h3><p><strong>思路</strong>：对每行建立线段树或树状数组，支持区间染色和查询。每次操作调用区间更新，最后查询整行的覆盖长度。</p><p><strong>缺点</strong>：需要为每行维护一棵线段树，空间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>⋅</mo><mi>log</mi><mo>⁡</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \cdot \log m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 或需要动态开点，实现复杂且效率低下。</p><p><strong>线段</strong></p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"></span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">SegTree</span> {</span><br><span class="line">    <span class="type">int</span> n;</span><br><span class="line">    vector&lt;<span class="type">int</span>&gt; tree, lazy;</span><br><span class="line">    </span><br><span class="line">    <span class="built_in">SegTree</span>(<span class="type">int</span> _n) {</span><br><span class="line">        n = _n;</span><br><span class="line">        tree.<span class="built_in">assign</span>(<span class="number">4</span> * n + <span class="number">5</span>, <span class="number">0</span>);</span><br><span class="line">        lazy.<span class="built_in">assign</span>(<span class="number">4</span> * n + <span class="number">5</span>, <span class="number">0</span>);</span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    <span class="function"><span class="type">void</span> <span class="title">push</span><span class="params">(<span class="type">int</span> node, <span class="type">int</span> l, <span class="type">int</span> r)</span> </span>{</span><br><span class="line">        <span class="keyword">if</span> (lazy[node]) {</span><br><span class="line">            tree[node] = (r - l + <span class="number">1</span>);</span><br><span class="line">            <span class="keyword">if</span> (l != r) {</span><br><span class="line">                lazy[node * <span class="number">2</span>] = lazy[node * <span class="number">2</span> + <span class="number">1</span>] = <span class="number">1</span>;</span><br><span class="line">            }</span><br><span class="line">            lazy[node] = <span class="number">0</span>;</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    <span class="function"><span class="type">void</span> <span class="title">update</span><span class="params">(<span class="type">int</span> node, <span class="type">int</span> l, <span class="type">int</span> r, <span class="type">int</span> ul, <span class="type">int</span> ur)</span> </span>{</span><br><span class="line">        <span class="built_in">push</span>(node, l, r);</span><br><span class="line">        <span class="keyword">if</span> (ul &gt; r || ur &lt; l) <span class="keyword">return</span>;</span><br><span class="line">        <span class="keyword">if</span> (ul &lt;= l &amp;&amp; r &lt;= ur) {</span><br><span class="line">            lazy[node] = <span class="number">1</span>;</span><br><span class="line">            <span class="built_in">push</span>(node, l, r);</span><br><span class="line">            <span class="keyword">return</span>;</span><br><span class="line">        }</span><br><span class="line">        <span class="type">int</span> mid = (l + r) &gt;&gt; <span class="number">1</span>;</span><br><span class="line">        <span class="built_in">update</span>(node * <span class="number">2</span>, l, mid, ul, ur);</span><br><span class="line">        <span class="built_in">update</span>(node * <span class="number">2</span> + <span class="number">1</span>, mid + <span class="number">1</span>, r, ul, ur);</span><br><span class="line">        tree[node] = tree[node * <span class="number">2</span>] + tree[node * <span class="number">2</span> + <span class="number">1</span>];</span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    <span class="function"><span class="type">int</span> <span class="title">query</span><span class="params">()</span> </span>{</span><br><span class="line">        <span class="keyword">return</span> tree[<span class="number">1</span>];</span><br><span class="line">    }</span><br><span class="line">};</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>{</span><br><span class="line">    <span class="type">int</span> c; <span class="type">long</span> <span class="type">long</span> n, m; <span class="type">int</span> q;</span><br><span class="line">    cin &gt;&gt; c &gt;&gt; n &gt;&gt; m &gt;&gt; q;</span><br><span class="line">    </span><br><span class="line">    <span class="comment">// 需要为每行建线段树，空间太大</span></span><br><span class="line">    unordered_map&lt;<span class="type">long</span> <span class="type">long</span>, SegTree*&gt; rowTrees;</span><br><span class="line">    </span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; q; i++) {</span><br><span class="line">        <span class="type">int</span> t; <span class="type">long</span> <span class="type">long</span> x1, y1, x2, y2;</span><br><span class="line">        cin &gt;&gt; t &gt;&gt; x1 &gt;&gt; y1 &gt;&gt; x2 &gt;&gt; y2;</span><br><span class="line">        <span class="keyword">if</span> (t == <span class="number">1</span>) {</span><br><span class="line">            <span class="keyword">if</span> (rowTrees.<span class="built_in">find</span>(y1) == rowTrees.<span class="built_in">end</span>()) {</span><br><span class="line">                rowTrees[y1] = <span class="keyword">new</span> <span class="built_in">SegTree</span>(m);</span><br><span class="line">            }</span><br><span class="line">            rowTrees[y1]-&gt;<span class="built_in">update</span>(<span class="number">1</span>, <span class="number">1</span>, m, x1, x2);</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> ans = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span> (<span class="keyword">auto</span>&amp; p : rowTrees) {</span><br><span class="line">        ans += p.second-&gt;<span class="built_in">query</span>();</span><br><span class="line">    }</span><br><span class="line">    cout &lt;&lt; ans &lt;&lt; endl;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><p>&lt;-本代码由AI辅助生成-&gt;</p><h3 id="2-3-方法三：区间合并法">2.3 方法三：区间合并法</h3><p><strong>思路</strong>：将每行的所有区间收集起来，排序后合并重叠或相邻的区间，直接计算并集长度。<br>那这个就是最后的代码了，直接看下面把</p><h2 id="三、区间合并法">三、区间合并法</h2><h3 id="3-1-整体流程">3.1 整体流程</h3><p><strong>1.读入</strong><br>将水平操作（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">t=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>），存入数组 <code>seg[]</code>。</p><p><strong>2.排序</strong><br>按行号<code>h</code>对 <code>seg[]</code> 排序，使同一行的区间连续。</p><p><strong>3.逐行处理</strong><br>对每一行执行：</p><ol><li>将该行所有区间复制到临时数组 <code>tmp[]</code></li><li>按左端点<code>l</code>对 <code>tmp[]</code> 排序</li><li><strong>合并扫描</strong>：<ul><li>维护当前区间 <code>[curL, curR]</code></li><li>遍历排序后的区间 <code>[L, R]</code>：<ul><li>如果 <code>L &lt;= curR + 1</code>：可合并，<code>curR = max(curR, R)</code></li><li>否则：结算当前区间，<code>ans += curR - curL + 1</code>，重置 <code>[curL, curR] = [L, R]</code></li></ul></li><li>遍历结束后再结算最后一个区间</li></ul></li></ol><p>是不是简单高效的方法呢～</p><h2 id="复杂度分析">复杂度分析</h2><ul><li><strong>时间</strong>：主要开销是排序。按行号排序 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>c</mi><mi>n</mi><mi>t</mi><mi>log</mi><mo>⁡</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(cnt \log cnt)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>，每行内部排序的总长度不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">cnt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span></span></span></span>，故整体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(q \log q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mclose">)</span></span></span></span>。</li><li><strong>空间</strong>：两个大小为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span></span></span></span> 的静态数组 <code>seg[]</code> 和 <code>tmp[]</code>，空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">q</span><span class="mclose">)</span></span></span></span>。</li></ul><h2 id="CODE">CODE</h2><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"></span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">QD</span> {</span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> h;<span class="comment">//行号</span></span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> l;<span class="comment">//左端点</span></span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> r;<span class="comment">//右端点</span></span><br><span class="line">};</span><br><span class="line"></span><br><span class="line">QD seg[<span class="number">100005</span>];</span><br><span class="line">QD tmp[<span class="number">100005</span>];</span><br><span class="line"></span><br><span class="line"><span class="comment">// 按行号排序</span></span><br><span class="line"><span class="function"><span class="type">bool</span> <span class="title">cmp1</span><span class="params">(<span class="type">const</span> QD&amp; x, <span class="type">const</span> QD&amp; y)</span> </span>{</span><br><span class="line">    <span class="keyword">return</span> x.h &lt; y.h;</span><br><span class="line">}</span><br><span class="line"><span class="comment">// 按左端点排序</span></span><br><span class="line"><span class="function"><span class="type">bool</span> <span class="title">cmp2</span><span class="params">(<span class="type">const</span> QD&amp; x, <span class="type">const</span> QD&amp; y)</span> </span>{</span><br><span class="line">    <span class="keyword">return</span> x.l &lt; y.l;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>{</span><br><span class="line">    <span class="type">int</span> c; </span><br><span class="line">    cin &gt;&gt; c;</span><br><span class="line">    </span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> n, m;</span><br><span class="line">    <span class="type">int</span> q;</span><br><span class="line">    cin &gt;&gt; n &gt;&gt; m &gt;&gt; q;</span><br><span class="line">    </span><br><span class="line">    <span class="type">int</span> cnt = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; q; i++) {</span><br><span class="line">        <span class="type">int</span> t; <span class="type">long</span> <span class="type">long</span> x1, y1, x2, y2;</span><br><span class="line">        cin &gt;&gt; t &gt;&gt; x1 &gt;&gt; y1 &gt;&gt; x2 &gt;&gt; y2;</span><br><span class="line">        <span class="keyword">if</span> (t == <span class="number">1</span>) {</span><br><span class="line">            seg[cnt++] = {y1, x1, x2};</span><br><span class="line">        }</span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    <span class="comment">//按行号排序，让同一行的区间连续</span></span><br><span class="line">    <span class="built_in">sort</span>(seg, seg + cnt, cmp1);</span><br><span class="line">    </span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> ans = <span class="number">0</span>;</span><br><span class="line">    </span><br><span class="line">    <span class="comment">//遍历每一行</span></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; cnt; ) {</span><br><span class="line">        <span class="type">int</span> j = i;</span><br><span class="line">        <span class="comment">// 找到同一行的区间，j指向下一行的开头</span></span><br><span class="line">        <span class="keyword">while</span> (j &lt; cnt &amp;&amp; seg[j].h == seg[i].h) {</span><br><span class="line">            j++;</span><br><span class="line">        }</span><br><span class="line">        </span><br><span class="line">        <span class="comment">//按左端点排序</span></span><br><span class="line">        <span class="type">int</span> len = j - i;</span><br><span class="line">        <span class="keyword">for</span> (<span class="type">int</span> k = <span class="number">0</span>; k &lt; len; k++) {</span><br><span class="line">            tmp[k] = seg[i + k];</span><br><span class="line">        }</span><br><span class="line">        <span class="built_in">sort</span>(tmp, tmp + len, cmp2);</span><br><span class="line">        </span><br><span class="line">        <span class="comment">//合并区间并累加长度</span></span><br><span class="line">        <span class="type">long</span> <span class="type">long</span> curL = tmp[<span class="number">0</span>].l;</span><br><span class="line">        <span class="type">long</span> <span class="type">long</span> curR = tmp[<span class="number">0</span>].r;</span><br><span class="line">        <span class="keyword">for</span> (<span class="type">int</span> k = <span class="number">1</span>; k &lt; len; k++) {</span><br><span class="line">            <span class="type">long</span> <span class="type">long</span> L = tmp[k].l;</span><br><span class="line">            <span class="type">long</span> <span class="type">long</span> R = tmp[k].r;</span><br><span class="line">            <span class="keyword">if</span> (L &lt;= curR + <span class="number">1</span>) {<span class="comment">//可合并</span></span><br><span class="line">                <span class="keyword">if</span> (R &gt; curR) curR = R;</span><br><span class="line">            } <span class="keyword">else</span> {<span class="comment">//不能合并，结算</span></span><br><span class="line">                ans += curR - curL + <span class="number">1</span>;</span><br><span class="line">                curL = L;</span><br><span class="line">                curR = R;</span><br><span class="line">            }</span><br><span class="line">        }</span><br><span class="line">        ans += curR - curL + <span class="number">1</span>;<span class="comment">//别忘了最后一个区间</span></span><br><span class="line">        </span><br><span class="line">        i = j;<span class="comment">// 跳到下一行</span></span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    cout &lt;&lt; ans &lt;&lt; endl;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><hr><p><strong>🎌 结语</strong></p><p>OK啊，分拿到了。本题解结束</p>]]></content>
    
    
      
      
    <summary type="html">&lt;h1&gt;子任务 A 题解（纯水平染色）&lt;/h1&gt;
&lt;blockquote&gt;
&lt;p&gt;说句实话，我写的题解真的是不如AI。那么作为一个&lt;s&gt;OI选手&lt;/s&gt;人类，我们能做的是什么呢？只有如何想出来的思路罢了。所以本篇题解主要注意的是思想过程，以及遇到一道题目时我的反应。&lt;/p&gt;
&lt;/</summary>
      
    
    
    
    <category term="题解" scheme="http://example.com/categories/%E9%A2%98%E8%A7%A3/"/>
    
    
    <category term="题解" scheme="http://example.com/tags/%E9%A2%98%E8%A7%A3/"/>
    
  </entry>
  
  <entry>
    <title>Hello World</title>
    <link href="http://example.com/2025/12/25/hello-world/"/>
    <id>http://example.com/2025/12/25/hello-world/</id>
    <published>2025-12-25T12:57:17.626Z</published>
    <updated>2025-11-25T09:22:14.953Z</updated>
    
    <content type="html"><![CDATA[<p>Welcome to <a href="https://app-36819-sanhai.cn24.ugdocker.link/">Sanhai’s Blogs</a>! This is your very first post. Check <a href="https://hexo.io/docs/">documentation</a> for more info. If you get any problems when using Hexo, you can find the answer in <a href="https://hexo.io/docs/troubleshooting.html">troubleshooting</a> or you can ask me on <a href="https://github.com/hexojs/hexo/issues">GitHub</a>.</p><h2 id="快速开始">快速开始</h2><h3 id="新建一篇文章">新建一篇文章</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">$ hexo new <span class="string">"My New Post"</span></span><br></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/writing.html">Writing</a></p><h3 id="开始服务">开始服务</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">$ hexo server</span><br></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/server.html">Server</a></p><h3 id="Generate-static-files">Generate static files</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">$ hexo generate</span><br></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/generating.html">Generating</a></p><h3 id="Deploy-to-remote-sites">Deploy to remote sites</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">$ hexo deploy</span><br></pre></td></tr></tbody></table></figure><p>More info: <a href="https://hexo.io/docs/one-command-deployment.html">Deployment</a></p>]]></content>
    
    
      
      
    <summary type="html">&lt;p&gt;Welcome to &lt;a href=&quot;https://app-36819-sanhai.cn24.ugdocker.link/&quot;&gt;Sanhai’s Blogs&lt;/a&gt;! This is your very first post. Check &lt;a href=&quot;https:</summary>
      
    
    
    
    
  </entry>
  
  <entry>
    <title>学习小组 - 题解</title>
    <link href="http://example.com/2025/12/11/T706110%E5%AD%A6%E4%B9%A0%E5%B0%8F%E7%BB%84%E9%A2%98%E8%A7%A3/"/>
    <id>http://example.com/2025/12/11/T706110%E5%AD%A6%E4%B9%A0%E5%B0%8F%E7%BB%84%E9%A2%98%E8%A7%A3/</id>
    <published>2025-12-11T15:07:36.000Z</published>
    <updated>2025-12-11T07:17:30.202Z</updated>
    
    <content type="html"><![CDATA[<h1>T706110 学习小组</h1><h2 id="题意简述">题意简述</h2><p>给定长度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex"> n </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex"> n \leq 2 \times 10^5 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span>）的序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex"> s </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span>，统计满足下标严格递增且</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>s</mi><mi>i</mi></msub><mo>&lt;</mo><msub><mi>s</mi><mi>j</mi></msub><mo>&lt;</mo><msub><mi>s</mi><mi>k</mi></msub><mspace width="1em"></mspace><mtext>或</mtext><mspace width="1em"></mspace><msub><mi>s</mi><mi>i</mi></msub><mo>&gt;</mo><msub><mi>s</mi><mi>j</mi></msub><mo>&gt;</mo><msub><mi>s</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">s_i &lt; s_j &lt; s_k \quad \text{或} \quad s_i &gt; s_j &gt; s_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8252em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mord text"><span class="mord cjk_fallback">或</span></span><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8252em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0315em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><p>的三元组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (i, j, k) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0315em;">k</span><span class="mclose">)</span></span></span></span> 的数量。</p><h2 id="算法思路">算法思路</h2><p>将三元组问题转化为“中间人”问题：</p><p>对于每个位置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex"> j </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span>，统计：</p><ul><li>前面比 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> s_j </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 小的数量：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">s</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathrm{preLess}[j] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">preLess</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span></span></span></span></li><li>前面比 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> s_j </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 大的数量：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathrm{preGreater}[j] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">preGreater</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span></span></span></span></li><li>后面比 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> s_j </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 小的数量：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">s</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathrm{postLess}[j] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">postLess</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span></span></span></span></li><li>后面比 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> s_j </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0572em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 大的数量：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathrm{postGreater}[j] </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">postGreater</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span></span></span></span></li></ul><p>则每个位置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex"> j </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span></span></span></span> 的贡献为：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">s</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>×</mo><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>+</mo><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">G</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo>×</mo><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">L</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">s</mi></mrow><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{preLess}[j] \times \mathrm{postGreater}[j] + \mathrm{preGreater}[j] \times \mathrm{postLess}[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">preLess</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">postGreater</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">preGreater</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathrm">postLess</span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.0572em;">j</span><span class="mclose">]</span></span></span></span></span><p>总答案为所有位置贡献之和。</p><h2 id="实现方法">实现方法</h2><ol><li><strong>离散化</strong>：将值域映射到连续区间，方便树状数组处理。</li><li><strong>树状数组</strong>：两次扫描（正序 + 倒序）统计前后关系。</li><li><strong>累加贡献</strong>：线性统计每个位置的贡献。</li></ol><h2 id="复杂度分析">复杂度分析</h2><ul><li><strong>时间复杂度</strong>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> O(n \log n) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.0139em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li><strong>空间复杂度</strong>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> O(n) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul><h2 id="参考代码">参考代码</h2><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"><span class="keyword">using</span> ll = <span class="type">long</span> <span class="type">long</span>;</span><br><span class="line"></span><br><span class="line"><span class="type">const</span> <span class="type">int</span> N = <span class="number">2e5</span> + <span class="number">10</span>;</span><br><span class="line"></span><br><span class="line"><span class="type">int</span> n, s[N], tmp[N], m;</span><br><span class="line">ll preL[N], preG[N], postL[N], postG[N], ans;</span><br><span class="line"><span class="type">int</span> c[N];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="type">void</span> <span class="title">add</span><span class="params">(<span class="type">int</span> x, <span class="type">int</span> v = <span class="number">1</span>)</span> </span>{</span><br><span class="line">    <span class="keyword">for</span> (; x &lt;= m; x += x &amp; -x) c[x] += v;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="type">int</span> <span class="title">ask</span><span class="params">(<span class="type">int</span> x)</span> </span>{</span><br><span class="line">    <span class="type">int</span> s = <span class="number">0</span>;</span><br><span class="line">    <span class="keyword">for</span> (; x; x -= x &amp; -x) s += c[x];</span><br><span class="line">    <span class="keyword">return</span> s;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>{</span><br><span class="line">    ios::<span class="built_in">sync_with_stdio</span>(<span class="literal">false</span>);</span><br><span class="line">    cin.<span class="built_in">tie</span>(<span class="literal">nullptr</span>);</span><br><span class="line">    cin &gt;&gt; n;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= n; ++i) {</span><br><span class="line">        cin &gt;&gt; s[i];</span><br><span class="line">        tmp[i] = s[i];</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 离散化</span></span><br><span class="line">    <span class="built_in">sort</span>(tmp + <span class="number">1</span>, tmp + n + <span class="number">1</span>);</span><br><span class="line">    m = <span class="built_in">unique</span>(tmp + <span class="number">1</span>, tmp + n + <span class="number">1</span>) - tmp - <span class="number">1</span>;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 正序</span></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= n; ++i) {</span><br><span class="line">        <span class="type">int</span> r = <span class="built_in">lower_bound</span>(tmp + <span class="number">1</span>, tmp + m + <span class="number">1</span>, s[i]) - tmp;</span><br><span class="line">        preL[i] = <span class="built_in">ask</span>(r - <span class="number">1</span>);</span><br><span class="line">        preG[i] = <span class="built_in">ask</span>(m) - <span class="built_in">ask</span>(r);</span><br><span class="line">        <span class="built_in">add</span>(r);</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 倒序</span></span><br><span class="line">    <span class="built_in">memset</span>(c, <span class="number">0</span>, <span class="built_in">sizeof</span>(c));</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = n; i &gt;= <span class="number">1</span>; --i) {</span><br><span class="line">        <span class="type">int</span> r = <span class="built_in">lower_bound</span>(tmp + <span class="number">1</span>, tmp + m + <span class="number">1</span>, s[i]) - tmp;</span><br><span class="line">        postL[i] = <span class="built_in">ask</span>(r - <span class="number">1</span>);</span><br><span class="line">        postG[i] = <span class="built_in">ask</span>(m) - <span class="built_in">ask</span>(r);</span><br><span class="line">        <span class="built_in">add</span>(r);</span><br><span class="line">    }</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 累加答案</span></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= n; ++i)</span><br><span class="line">        ans += preL[i] * postG[i] + preG[i] * postL[i];</span><br><span class="line">    cout &lt;&lt; ans &lt;&lt; <span class="string">'\n'</span>;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><h2 id="提示">提示</h2><ul><li>树状数组常数较小，可直接通过。</li><li>注意离散化时处理重复值。</li></ul>]]></content>
    
    
      
      
    <summary type="html">&lt;h1&gt;T706110 学习小组&lt;/h1&gt;
&lt;h2 id=&quot;题意简述&quot;&gt;题意简述&lt;/h2&gt;
&lt;p&gt;给定长度为 &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1998/Ma</summary>
      
    
    
    
    <category term="题解" scheme="http://example.com/categories/%E9%A2%98%E8%A7%A3/"/>
    
    
    <category term="题解" scheme="http://example.com/tags/%E9%A2%98%E8%A7%A3/"/>
    
  </entry>
  
  <entry>
    <title>降水 - 题解</title>
    <link href="http://example.com/2025/12/04/T706068%E9%99%8D%E6%B0%B4%E9%A2%98%E8%A7%A3/"/>
    <id>http://example.com/2025/12/04/T706068%E9%99%8D%E6%B0%B4%E9%A2%98%E8%A7%A3/</id>
    <published>2025-12-04T13:40:36.000Z</published>
    <updated>2025-12-10T00:50:56.730Z</updated>
    
    <content type="html"><![CDATA[<p><strong>题意理解</strong></p><p>有 N 座山围成一圈（N 是奇数），每座山 i 会下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 升雨。这些雨水会平均分给左右两个大坝：左边的大坝 i-1 和右边的大坝 i 各得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">p_i/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span></span> 升。大坝 i 收集到的总水量就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。</p><p>所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的关系就是：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>p</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">a_i = \frac{p_i + p_{i+1}}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.9463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span><p>两边乘 2：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>p</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mi>a</mi><mi>i</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(1)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">p_i + p_{i+1} = 2a_i \tag{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">1</span></span><span class="mord">)</span></span></span></span></span></span><p><strong>核心思路</strong></p><p>这个式子很关键！它告诉我们：只要知道 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，就能算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">p_{i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span>：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><msub><mi>p</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mi>a</mi><mi>i</mi></msub><mo>−</mo><msub><mi>p</mi><mi>i</mi></msub></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>(2)</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">p_{i+1} = 2a_i - p_i \tag{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(</span><span class="mord"><span class="mord">2</span></span><span class="mord">)</span></span></span></span></span></span><p>问题就变成了：怎么求出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>？</p><p>因为 N 是奇数，我们可以绕一圈用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 把自己表示出来。从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 开始，反复用公式 (2) 推下去：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>p</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>&nbsp;</mtext><msub><mi>p</mi><mn>3</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><msub><mi>a</mi><mn>2</mn></msub><mo>−</mo><msub><mi>p</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><msub><mi>a</mi><mn>2</mn></msub><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><msub><mi>p</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>&nbsp;</mtext><msub><mi>p</mi><mn>4</mn></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>2</mn><msub><mi>a</mi><mn>3</mn></msub><mo>−</mo><msub><mi>p</mi><mn>3</mn></msub><mo>=</mo><mn>2</mn><msub><mi>a</mi><mn>3</mn></msub><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><msub><mi>p</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><mn>2</mn><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msub><mi>a</mi><mn>3</mn></msub><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p_2 &amp;= 2a_1 - p_1 \\\p_3 &amp;= 2a_2 - p_2 = 2a_2 - (2a_1 - p_1) = -2a_1 + 2a_2 + p_1 \\\p_4 &amp;= 2a_3 - p_3 = 2a_3 - (-2a_1 + 2a_2 + p_1) = 2a_1 - 2a_2 + 2a_3 - p_1\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.2em;vertical-align:-1.85em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.51em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.51em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.51em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.51em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span></span></span></span></span></span><p>我们注意到一个规律：奇数项是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 加上一堆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><mn>2</mn><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\pm 2a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">±</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，偶数项是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">-p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 加上一堆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><mn>2</mn><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\pm 2a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">±</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。</p><p>因为 N 是奇数，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">p_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 也是奇数项，可以写成：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mi>N</mi></msub><mo>=</mo><msub><mi>p</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><msub><mi>a</mi><mn>3</mn></msub><mo>−</mo><msub><mi>a</mi><mn>4</mn></msub><mo>+</mo><mo>⋯</mo><mo>−</mo><msub><mi>a</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_N = p_1 + 2(a_1 - a_2 + a_3 - a_4 + \dots - a_{N-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span><p>注意最后一个方程 (1) 当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">i=N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span></span></span></span> 时是：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mi>N</mi></msub><mo>+</mo><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><msub><mi>a</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">p_N + p_1 = 2a_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><p>把 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">p_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的表达式代入：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>−</mo><msub><mi>a</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><msub><mi>a</mi><mi>N</mi></msub></mrow><annotation encoding="application/x-tex">(p_1 + 2(a_1 - a_2 + \dots - a_{N-1})) + p_1 = 2a_N</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><p>两边除以 2，得到：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><msub><mi>a</mi><mi>N</mi></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><mo>−</mo><msub><mi>a</mi><mn>3</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">p_1 = a_N - a_1 + a_2 - a_3 + \dots + a_{N-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span><p>也就是：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo>=</mo><msub><mi>a</mi><mi>N</mi></msub><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msup><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_1 = a_N + \sum_{i=1}^{N-1} (-1)^{i+1} a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:3.106em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.109em;">N</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span><p><strong>算法流程</strong></p><ol><li>用公式算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li><li>从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，用递推式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mi>a</mi><mi>i</mi></msub><mo>−</mo><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_{i+1} = 2a_i - p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 算出所有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li></ol><p>复杂度是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mclose">)</span></span></span></span> 时间，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(N)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.109em;">N</span><span class="mclose">)</span></span></span></span> 空间（存 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 数组）。</p><p><strong>代码实现</strong></p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"></span><br><span class="line"><span class="type">long</span> <span class="type">long</span> a[<span class="number">100005</span>];</span><br><span class="line"><span class="type">long</span> <span class="type">long</span> p[<span class="number">100005</span>];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>{</span><br><span class="line">    ios::<span class="built_in">sync_with_stdio</span>(<span class="literal">false</span>);</span><br><span class="line">    cin.<span class="built_in">tie</span>(<span class="literal">nullptr</span>);</span><br><span class="line">    </span><br><span class="line">    <span class="type">int</span> N;</span><br><span class="line">    cin &gt;&gt; N;</span><br><span class="line"></span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; N; ++i) cin &gt;&gt; a[i];</span><br><span class="line">    </span><br><span class="line">    <span class="comment">// 计算 p1：交错和</span></span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> p1 = a[N];  <span class="comment">// 先放 a_N</span></span><br><span class="line">    <span class="type">int</span> sign = <span class="number">+1</span>;            <span class="comment">// a_1 的系数是 +1</span></span><br><span class="line">    </span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; N - <span class="number">1</span>; ++i) {</span><br><span class="line">        p1 += sign * a[i];</span><br><span class="line">        sign = -sign;         <span class="comment">// 正负交替</span></span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    p[<span class="number">0</span>] = p1;</span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt; N; ++i) {</span><br><span class="line">        p[i] = <span class="number">2LL</span> * a[i - <span class="number">1</span>] - p[i - <span class="number">1</span>];</span><br><span class="line">    }</span><br><span class="line">    </span><br><span class="line">    <span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">0</span>; i &lt; N; ++i) {</span><br><span class="line">        <span class="keyword">if</span> (i) cout &lt;&lt; <span class="string">' '</span>;</span><br><span class="line">        cout &lt;&lt; p[i];</span><br><span class="line">    }</span><br><span class="line">    cout &lt;&lt; <span class="string">'\n'</span>;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></tbody></table></figure><p><strong>一些小细节</strong></p><ul><li>数据范围最大到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>10</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">10^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span>，N 最大到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>10</mn><mn>5</mn></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10^5-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 可能很大，要用 long long</li><li>N 是奇数这个条件非常关键，如果 N 是偶数，方程组可能无解或有无穷多解</li></ul>]]></content>
    
    
      
      
    <summary type="html">&lt;p&gt;&lt;strong&gt;题意理解&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;有 N 座山围成一圈（N 是奇数），每座山 i 会下 &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/199</summary>
      
    
    
    
    <category term="题解" scheme="http://example.com/categories/%E9%A2%98%E8%A7%A3/"/>
    
    
    <category term="题解" scheme="http://example.com/tags/%E9%A2%98%E8%A7%A3/"/>
    
  </entry>
  
  <entry>
    <title>最小的x - 题解</title>
    <link href="http://example.com/2025/12/04/T706109%E6%9C%80%E5%B0%8F%E7%9A%84x%E9%A2%98%E8%A7%A3/"/>
    <id>http://example.com/2025/12/04/T706109%E6%9C%80%E5%B0%8F%E7%9A%84x%E9%A2%98%E8%A7%A3/</id>
    <published>2025-12-04T13:40:36.000Z</published>
    <updated>2025-12-10T00:50:26.962Z</updated>
    
    <content type="html"><![CDATA[<h2 id="题解">题解</h2><p><strong>思路</strong></p><p>题目要求最小的正整数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">n+x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是一个完全平方数。</p><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>n</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">y^2=n+x^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，移项后平方差公式直接带走：</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>n</mi><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">y^2-x^2=n \implies (y-x)(y+x)=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.549em;vertical-align:-0.024em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span></span><p>令 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>=</mo><mi>y</mi><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">a=y-x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>=</mo><mi>y</mi><mo>+</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">b=y+x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>⋅</mo><mi>b</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">a\cdot b=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4445em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>。显然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&lt;</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a&lt;b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>，而且因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,b,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span> 都是整数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 必须同奇偶（同为奇数或同为偶数）。</p><p>一旦找到满足条件的因子对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span>，就有</p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><mn>2</mn></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mi>y</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">x=\frac{b-a}{2},\quad y=\frac{a+b}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span><p>为了让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 最小，需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">b-a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 最小。注意到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mo>=</mo><mi>n</mi><mi mathvariant="normal">/</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">b=n/a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/</span><span class="mord mathnormal">a</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 越大，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 越小，差就越小。所以我们从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌊</mo><msqrt><mi>n</mi></msqrt><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">\lfloor\sqrt n\rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0503em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="padding-left:0.833em;">n</span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span><span class="mclose">⌋</span></span></span></span> 开始<strong>倒序</strong>枚举 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>，第一个合法的因子对一定对应最小的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>。</p><p><strong>无解</strong></p><p>两个同奇偶性的数相乘，结果要么是奇数，要么是 4 的倍数。所以如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n\equiv 2\pmod 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4637em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.4444em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.3333em;"></span><span class="mord">4</span><span class="mclose">)</span></span></span></span>，一定无解。</p><p>另外，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n=4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 时，唯一的因子对是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>，算出来 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，不符合正整数要求。这两种情况直接输出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>。</p><p>其余情况必然有解。</p><p><strong>代码</strong></p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line"><span class="type">long</span> <span class="type">long</span> n; cin&gt;&gt;n;</span><br><span class="line"><span class="keyword">if</span> (n%<span class="number">4</span>==<span class="number">2</span> || n==<span class="number">1</span> || n==<span class="number">4</span>) { cout&lt;&lt;<span class="number">-1</span>; <span class="keyword">return</span> <span class="number">0</span>; }</span><br><span class="line"><span class="type">long</span> <span class="type">long</span> ans=<span class="number">-1</span>;</span><br><span class="line"><span class="keyword">for</span> (<span class="type">long</span> <span class="type">long</span> a=<span class="built_in">sqrt</span>(n); a&gt;=<span class="number">1</span>; --a) {</span><br><span class="line">    <span class="keyword">if</span> (n%a!=<span class="number">0</span>) <span class="keyword">continue</span>;</span><br><span class="line">    <span class="type">long</span> <span class="type">long</span> b=n/a;</span><br><span class="line">    <span class="keyword">if</span> ((a%<span class="number">2</span>)==(b%<span class="number">2</span>) &amp;&amp; a&lt;b) {</span><br><span class="line">        ans=(b-a)/<span class="number">2</span>;</span><br><span class="line">        <span class="keyword">break</span>;</span><br><span class="line">    }</span><br><span class="line">}</span><br><span class="line">cout&lt;&lt;ans;</span><br></pre></td></tr></tbody></table></figure>]]></content>
    
    
      
      
    <summary type="html">&lt;h2 id=&quot;题解&quot;&gt;题解&lt;/h2&gt;
&lt;p&gt;&lt;strong&gt;思路&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;题目要求最小的正整数 &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w3.org/1</summary>
      
    
    
    
    <category term="题解" scheme="http://example.com/categories/%E9%A2%98%E8%A7%A3/"/>
    
    
    <category term="题解" scheme="http://example.com/tags/%E9%A2%98%E8%A7%A3/"/>
    
  </entry>
  
  <entry>
    <title>Sanhai-loading</title>
    <link href="http://example.com/2025/11/29/Sanhai-loading/"/>
    <id>http://example.com/2025/11/29/Sanhai-loading/</id>
    <published>2025-11-29T13:40:50.000Z</published>
    <updated>2025-11-29T13:52:06.942Z</updated>
    
    <content type="html"><![CDATA[<h3 id="✅-一、插件目录（官方结构）">✅ 一、插件目录（官方结构）</h3><figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><span class="line">/root/addon/sanhai-loading/</span><br><span class="line">├── package.json</span><br><span class="line">├── template.yaml</span><br><span class="line">├── index.js</span><br><span class="line">├── templates/</span><br><span class="line">│   └── partials/</span><br><span class="line">│       └── loading.html</span><br><span class="line">└── static/</span><br><span class="line">    └── sanhai-loading.css</span><br></pre></td></tr></tbody></table></figure><h3 id="✅-二、核心文件">✅ 二、核心文件</h3><p><strong>package.json</strong></p><figure class="highlight json"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line"><span class="punctuation">{</span></span><br><span class="line">  <span class="attr">"name"</span><span class="punctuation">:</span> <span class="string">"sanhai-loading"</span><span class="punctuation">,</span></span><br><span class="line">  <span class="attr">"version"</span><span class="punctuation">:</span> <span class="string">"1.0.0"</span><span class="punctuation">,</span></span><br><span class="line">  <span class="attr">"description"</span><span class="punctuation">:</span> <span class="string">"SanhaiOJ 7 款弹性加载动画 · 后台可切换"</span><span class="punctuation">,</span></span><br><span class="line">  <span class="attr">"main"</span><span class="punctuation">:</span> <span class="string">"index.js"</span><span class="punctuation">,</span></span><br><span class="line">  <span class="attr">"template"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">true</span></span><span class="punctuation">,</span></span><br><span class="line">  <span class="attr">"style"</span><span class="punctuation">:</span> <span class="literal"><span class="keyword">true</span></span></span><br><span class="line"><span class="punctuation">}</span></span><br></pre></td></tr></tbody></table></figure><p><strong>template.yaml</strong></p><figure class="highlight yaml"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line"><span class="attr">partials/loading.html:</span> <span class="string">./templates/partials/loading.html</span></span><br></pre></td></tr></tbody></table></figure><p><strong>index.js</strong></p><figure class="highlight javascript"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line"><span class="variable language_">module</span>.<span class="property">exports</span> = <span class="keyword">async</span> <span class="keyword">function</span> (<span class="params">ctx</span>) {</span><br><span class="line">  <span class="comment">// 后台配置项（动画类型 + 速度）</span></span><br><span class="line">  ctx.<span class="property">config</span>.<span class="title function_">set</span>(<span class="string">'sanhai_loading_type'</span>, <span class="string">'dots'</span>, <span class="literal">true</span>);</span><br><span class="line">  ctx.<span class="property">config</span>.<span class="title function_">set</span>(<span class="string">'sanhai_loading_speed'</span>, <span class="number">1</span>, <span class="literal">true</span>);</span><br><span class="line">};</span><br></pre></td></tr></tbody></table></figure><h3 id="✅-三、模板文件（官方路径）">✅ 三、模板文件（官方路径）</h3><p><strong>templates/partials/loading.html</strong></p><figure class="highlight html"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">&lt;!-- SanhaiOJ 弹性加载动画 · 7 选 1 --&gt;</span></span><br><span class="line"><span class="tag">&lt;<span class="name">div</span> <span class="attr">id</span>=<span class="string">"sanhai-loading"</span> <span class="attr">class</span>=<span class="string">"sanhai-loader {{ config.get('sanhai_loading_type', 'dots') }}"</span> <span class="attr">style</span>=<span class="string">"--speed:{{ config.get('sanhai_loading_speed', 1) }}s"</span>&gt;</span><span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line"><span class="tag">&lt;<span class="name">link</span> <span class="attr">rel</span>=<span class="string">"stylesheet"</span> <span class="attr">href</span>=<span class="string">"{{ url('static', 'sanhai-loading.css') }}"</span>&gt;</span></span><br></pre></td></tr></tbody></table></figure><h3 id="✅-四、集中样式（7-款动画）">✅ 四、集中样式（7 款动画）</h3><p><strong>static/sanhai-loading.css</strong></p><figure class="highlight css"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">/* ---- 基础覆盖 ---- */</span></span><br><span class="line"><span class="selector-id">#sanhai-loading</span>{</span><br><span class="line">  <span class="attribute">position</span>:fixed;<span class="attribute">inset</span>:<span class="number">0</span>;<span class="attribute">z-index</span>:<span class="number">9999</span>;</span><br><span class="line">  <span class="attribute">display</span>:flex;<span class="attribute">align-items</span>:center;<span class="attribute">justify-content</span>:center;</span><br><span class="line">  <span class="attribute">background</span>:<span class="number">#fff</span>;     <span class="comment">/* 官方白底 */</span></span><br><span class="line">  <span class="attr">--speed</span>:<span class="number">1s</span>;          <span class="comment">/* 后台速度变量 */</span></span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="comment">/* ---- 7 款动画 ---- */</span></span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.dots</span>{</span><br><span class="line">  <span class="attribute">display</span>:flex;<span class="attribute">gap</span>:<span class="number">8px</span>;</span><br><span class="line">}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.dots</span><span class="selector-pseudo">::before</span>,</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.dots</span><span class="selector-pseudo">::after</span>,</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.dots</span>{</span><br><span class="line">  <span class="attribute">content</span>:<span class="string">''</span>;<span class="attribute">width</span>:<span class="number">10px</span>;<span class="attribute">height</span>:<span class="number">10px</span>;<span class="attribute">background</span>:<span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">50%</span>;</span><br><span class="line">  <span class="attribute">animation</span>:dotsJump <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">45</span>,.<span class="number">03</span>,.<span class="number">51</span>,<span class="number">1.4</span>) infinite;</span><br><span class="line">}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.dots</span><span class="selector-pseudo">::before</span>{<span class="attribute">animation-delay</span>:.<span class="number">1s</span>;}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.dots</span><span class="selector-pseudo">::after</span>{<span class="attribute">animation-delay</span>:.<span class="number">2s</span>;}</span><br><span class="line"><span class="keyword">@keyframes</span> dotsJump{<span class="number">0%</span>,<span class="number">80%</span>,<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">translateY</span>(<span class="number">0</span>) <span class="built_in">scale</span>(<span class="number">1</span>);}<span class="number">40%</span>{<span class="attribute">transform</span>:<span class="built_in">translateY</span>(-<span class="number">12px</span>) <span class="built_in">scale</span>(<span class="number">1.2</span>);}}</span><br><span class="line"></span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.ring</span>{</span><br><span class="line">  <span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">height</span>:<span class="number">40px</span>;<span class="attribute">border</span>:<span class="number">4px</span> solid <span class="number">#165DFF</span>;<span class="attribute">border-top-color</span>:transparent;<span class="attribute">border-radius</span>:<span class="number">50%</span>;</span><br><span class="line">  <span class="attribute">animation</span>:ringElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">@keyframes</span> ringElastic{<span class="number">0%</span>{<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">0deg</span>) <span class="built_in">scale</span>(<span class="number">1</span>);}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">180deg</span>) <span class="built_in">scale</span>(<span class="number">1.2</span>);}<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">360deg</span>) <span class="built_in">scale</span>(<span class="number">1</span>);}}</span><br><span class="line"></span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.bar</span>{</span><br><span class="line">  <span class="attribute">display</span>:flex;<span class="attribute">gap</span>:<span class="number">4px</span>;<span class="attribute">height</span>:<span class="number">32px</span>;<span class="attribute">align-items</span>:flex-end;</span><br><span class="line">}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.bar</span><span class="selector-pseudo">::before</span>,</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.bar</span><span class="selector-pseudo">::after</span>,</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.bar</span>{</span><br><span class="line">  <span class="attribute">content</span>:<span class="string">''</span>;<span class="attribute">width</span>:<span class="number">6px</span>;<span class="attribute">height</span>:<span class="number">100%</span>;<span class="attribute">background</span>:<span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">3px</span>;</span><br><span class="line">  <span class="attribute">animation</span>:barElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">45</span>,.<span class="number">03</span>,.<span class="number">51</span>,<span class="number">1.4</span>) infinite;</span><br><span class="line">}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.bar</span><span class="selector-pseudo">::before</span>{<span class="attribute">animation-delay</span>:.<span class="number">1s</span>;}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.bar</span><span class="selector-pseudo">::after</span>{<span class="attribute">animation-delay</span>:.<span class="number">2s</span>;}</span><br><span class="line"><span class="keyword">@keyframes</span> barElastic{<span class="number">0%</span>,<span class="number">80%</span>,<span class="number">100%</span>{<span class="attribute">height</span>:<span class="number">20%</span>;}<span class="number">40%</span>{<span class="attribute">height</span>:<span class="number">100%</span>;}}</span><br><span class="line"></span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.pulse</span>{</span><br><span class="line">  <span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">height</span>:<span class="number">40px</span>;<span class="attribute">background</span>:<span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">50%</span>;</span><br><span class="line">  <span class="attribute">animation</span>:pulseElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">@keyframes</span> pulseElastic{<span class="number">0%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">0.95</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1.3</span>);<span class="attribute">opacity</span>:<span class="number">0.7</span>;}<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">0.95</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}}</span><br><span class="line"></span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.orbit</span>{</span><br><span class="line">  <span class="attribute">position</span>:relative;<span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">height</span>:<span class="number">40px</span>;</span><br><span class="line">}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.orbit</span><span class="selector-pseudo">::before</span>,</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.orbit</span><span class="selector-pseudo">::after</span>{</span><br><span class="line">  <span class="attribute">content</span>:<span class="string">''</span>;<span class="attribute">position</span>:absolute;<span class="attribute">top</span>:<span class="number">0</span>;<span class="attribute">left</span>:<span class="number">0</span>;<span class="attribute">width</span>:<span class="number">100%</span>;<span class="attribute">height</span>:<span class="number">100%</span>;<span class="attribute">border</span>:<span class="number">3px</span> solid <span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">50%</span>;</span><br><span class="line">  <span class="attribute">animation</span>:orbitElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">45</span>,.<span class="number">03</span>,.<span class="number">51</span>,<span class="number">1.4</span>) infinite;</span><br><span class="line">}</span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.orbit</span><span class="selector-pseudo">::after</span>{<span class="attribute">animation-delay</span>:.<span class="number">2s</span>;<span class="attribute">border-color</span>:<span class="number">#14C4D9</span>;}</span><br><span class="line"><span class="keyword">@keyframes</span> orbitElastic{<span class="number">0%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">0.3</span>);<span class="attribute">opacity</span>:<span class="number">0</span>;}<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}}</span><br><span class="line"></span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.text</span>{</span><br><span class="line">  <span class="attribute">font-size</span>:<span class="number">20px</span>;<span class="attribute">font-weight</span>:<span class="number">600</span>;<span class="attribute">color</span>:<span class="number">#165DFF</span>;<span class="attribute">letter-spacing</span>:<span class="number">2px</span>;</span><br><span class="line">  <span class="attribute">animation</span>:textElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">@keyframes</span> textElastic{<span class="number">0%</span>,<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1</span>);}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1.15</span>);}}</span><br><span class="line"></span><br><span class="line"><span class="selector-class">.sanhai-loader</span><span class="selector-class">.arc</span>{</span><br><span class="line">  <span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">height</span>:<span class="number">40px</span>;<span class="attribute">background</span>:transparent;</span><br><span class="line">  <span class="attribute">background-image</span>:<span class="built_in">url</span>(<span class="string">"data:image/svg+xml,%3Csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 48 48'%3E%3Cpath d='M24 4 A20 20 0 1 1 24 44' stroke='%23165DFF' stroke-width='4' fill='none' stroke-linecap='round' stroke-dasharray='94.2' stroke-dashoffset='94.2'/%3E%3C/svg%3E"</span>);</span><br><span class="line">  <span class="attribute">animation</span>:arcElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">@keyframes</span> arcElastic{<span class="number">0%</span>{<span class="attribute">stroke-dashoffset</span>:<span class="number">94.2</span>;<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">0deg</span>);}<span class="number">50%</span>{<span class="attribute">stroke-dashoffset</span>:<span class="number">0</span>;<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">540deg</span>);}<span class="number">100%</span>{<span class="attribute">stroke-dashoffset</span>:<span class="number">94.2</span>;<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">1080deg</span>);}}</span><br></pre></td></tr></tbody></table></figure><h3 id="✅-五、启用-后台配置">✅ 五、启用 &amp; 后台配置</h3><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">hydrooj addon add /root/addon/sanhai-loading</span><br><span class="line">pm2 restart hydrooja</span><br></pre></td></tr></tbody></table></figure><p>进入 <strong>管理后台 → 系统 → 配置</strong>，新增两项：</p><table><thead><tr><th>键</th><th>值</th><th>说明</th></tr></thead><tbody><tr><td><code>sanhai_loading_type</code></td><td><code>dots</code> / <code>ring</code> / <code>bar</code> / <code>pulse</code> / <code>orbit</code> / <code>text</code> / <code>arc</code></td><td>7 款动画可选</td></tr><tr><td><code>sanhai_loading_speed</code></td><td><code>0.5</code> ~ <code>3</code>（秒）</td><td>实时调速</td></tr></tbody></table><p>刷新任意页面 → 官方 <code>partials/loading.html</code> 已被插件替换，<strong>升级无损</strong>。</p><h1>适用选择页面</h1><figure class="highlight html"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br><span class="line">124</span><br><span class="line">125</span><br><span class="line">126</span><br><span class="line">127</span><br><span class="line">128</span><br><span class="line">129</span><br><span class="line">130</span><br><span class="line">131</span><br><span class="line">132</span><br><span class="line">133</span><br><span class="line">134</span><br><span class="line">135</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">&lt;!DOCTYPE <span class="keyword">html</span>&gt;</span></span><br><span class="line"><span class="tag">&lt;<span class="name">html</span> <span class="attr">lang</span>=<span class="string">"zh-CN"</span>&gt;</span></span><br><span class="line"><span class="tag">&lt;<span class="name">head</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;<span class="name">meta</span> <span class="attr">charset</span>=<span class="string">"UTF-8"</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;<span class="name">title</span>&gt;</span>SanhaiOJ 加载动画实验室（白底完整）<span class="tag">&lt;/<span class="name">title</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;<span class="name">meta</span> <span class="attr">name</span>=<span class="string">"viewport"</span> <span class="attr">content</span>=<span class="string">"width=device-width, initial-scale=1.0"</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;<span class="name">style</span>&gt;</span><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="selector-pseudo">:root</span>{<span class="attr">--speed</span>:<span class="number">1s</span>;<span class="attr">--accent</span>:<span class="number">#165DFF</span>;}</span></span><br><span class="line"><span class="language-css">    *{<span class="attribute">box-sizing</span>:border-box;<span class="attribute">margin</span>:<span class="number">0</span>;<span class="attribute">padding</span>:<span class="number">0</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-tag">body</span>{</span></span><br><span class="line"><span class="language-css">      <span class="attribute">min-height</span>:<span class="number">100vh</span>;<span class="attribute">display</span>:flex;<span class="attribute">flex-direction</span>:column;<span class="attribute">align-items</span>:center;<span class="attribute">justify-content</span>:center;</span></span><br><span class="line"><span class="language-css">      <span class="attribute">background</span>:<span class="number">#ffffff</span>;<span class="attribute">font-family</span>:-apple-system,BlinkMacSystemFont,<span class="string">"Segoe UI"</span>,Roboto,<span class="string">"Helvetica Neue"</span>,Arial,sans-serif;</span></span><br><span class="line"><span class="language-css">      <span class="attribute">padding</span>:<span class="number">40px</span> <span class="number">24px</span>;</span></span><br><span class="line"><span class="language-css">    }</span></span><br><span class="line"><span class="language-css">    <span class="selector-tag">header</span>{<span class="attribute">font-size</span>:<span class="number">24px</span>;<span class="attribute">font-weight</span>:<span class="number">600</span>;<span class="attribute">margin-bottom</span>:<span class="number">24px</span>;<span class="attribute">color</span>:<span class="number">#165DFF</span>;}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="comment">/* 速度滑块 */</span></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.control-bar</span>{<span class="attribute">margin-bottom</span>:<span class="number">24px</span>;<span class="attribute">font-size</span>:<span class="number">14px</span>;<span class="attribute">color</span>:<span class="number">#4e5969</span>;<span class="attribute">display</span>:flex;<span class="attribute">align-items</span>:center;<span class="attribute">gap</span>:<span class="number">8px</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-tag">input</span><span class="selector-attr">[type=<span class="string">"range"</span>]</span>{<span class="attribute">width</span>:<span class="number">160px</span>;<span class="attribute">cursor</span>:pointer;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-id">#speedValue</span>{<span class="attribute">display</span>:inline-block;<span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">text-align</span>:center;<span class="attribute">font-weight</span>:<span class="number">600</span>;<span class="attribute">color</span>:<span class="number">#165DFF</span>;}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="comment">/* 容器布局（原排列方式） */</span></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.container</span>{<span class="attribute">display</span>:flex;<span class="attribute">gap</span>:<span class="number">24px</span>;<span class="attribute">flex-wrap</span>:wrap;<span class="attribute">justify-content</span>:center;}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="comment">/* 卡片样式（白底） */</span></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.card</span>{</span></span><br><span class="line"><span class="language-css">      <span class="attribute">width</span>:<span class="number">160px</span>;<span class="attribute">height</span>:<span class="number">160px</span>;<span class="attribute">border-radius</span>:<span class="number">8px</span>;<span class="attribute">background</span>:<span class="number">#f7f8fa</span>;<span class="attribute">border</span>:<span class="number">1px</span> solid <span class="number">#e5e6eb</span>;</span></span><br><span class="line"><span class="language-css">      <span class="attribute">display</span>:flex;<span class="attribute">flex-direction</span>:column;<span class="attribute">align-items</span>:center;<span class="attribute">justify-content</span>:center;<span class="attribute">cursor</span>:pointer;</span></span><br><span class="line"><span class="language-css">      <span class="attribute">transition</span>:transform .<span class="number">25s</span> ease,box-shadow .<span class="number">25s</span> ease;</span></span><br><span class="line"><span class="language-css">    }</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.card</span><span class="selector-pseudo">:hover</span>{<span class="attribute">transform</span>:<span class="built_in">translateY</span>(-<span class="number">4px</span>);<span class="attribute">box-shadow</span>:<span class="number">0</span> <span class="number">4px</span> <span class="number">12px</span> <span class="built_in">rgba</span>(<span class="number">22</span>,<span class="number">93</span>,<span class="number">255</span>,.<span class="number">15</span>);}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.label</span>{<span class="attribute">margin-top</span>:<span class="number">8px</span>;<span class="attribute">font-size</span>:<span class="number">13px</span>;<span class="attribute">color</span>:<span class="number">#4e5969</span>;}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="comment">/* 动画集合（确保元素可见） */</span></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.dots</span>{<span class="attribute">display</span>:flex;<span class="attribute">gap</span>:<span class="number">8px</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.dots</span> <span class="selector-tag">span</span>{<span class="attribute">width</span>:<span class="number">10px</span>;<span class="attribute">height</span>:<span class="number">10px</span>;<span class="attribute">background</span>:<span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">50%</span>;<span class="attribute">animation</span>:dotsJump <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">45</span>,.<span class="number">03</span>,.<span class="number">51</span>,<span class="number">1.4</span>) infinite;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.dots</span> <span class="selector-tag">span</span><span class="selector-pseudo">:nth-child</span>(<span class="number">2</span>){<span class="attribute">animation-delay</span>:.<span class="number">1s</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.dots</span> <span class="selector-tag">span</span><span class="selector-pseudo">:nth-child</span>(<span class="number">3</span>){<span class="attribute">animation-delay</span>:.<span class="number">2s</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="keyword">@keyframes</span> dotsJump{<span class="number">0%</span>,<span class="number">80%</span>,<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">translateY</span>(<span class="number">0</span>) <span class="built_in">scale</span>(<span class="number">1</span>);}<span class="number">40%</span>{<span class="attribute">transform</span>:<span class="built_in">translateY</span>(-<span class="number">12px</span>) <span class="built_in">scale</span>(<span class="number">1.2</span>);}}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.ring</span>{<span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">height</span>:<span class="number">40px</span>;<span class="attribute">border</span>:<span class="number">4px</span> solid <span class="number">#165DFF</span>;<span class="attribute">border-top-color</span>:transparent;<span class="attribute">border-radius</span>:<span class="number">50%</span>;<span class="attribute">animation</span>:ringElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;}</span></span><br><span class="line"><span class="language-css">    <span class="keyword">@keyframes</span> ringElastic{<span class="number">0%</span>{<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">0deg</span>) <span class="built_in">scale</span>(<span class="number">1</span>);}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">180deg</span>) <span class="built_in">scale</span>(<span class="number">1.2</span>);}<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">360deg</span>) <span class="built_in">scale</span>(<span class="number">1</span>);}}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.bar</span>{<span class="attribute">display</span>:flex;<span class="attribute">gap</span>:<span class="number">4px</span>;<span class="attribute">height</span>:<span class="number">32px</span>;<span class="attribute">align-items</span>:flex-end;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.bar</span> <span class="selector-tag">span</span>{<span class="attribute">width</span>:<span class="number">6px</span>;<span class="attribute">height</span>:<span class="number">100%</span>;<span class="attribute">background</span>:<span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">3px</span>;<span class="attribute">animation</span>:barElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">45</span>,.<span class="number">03</span>,.<span class="number">51</span>,<span class="number">1.4</span>) infinite;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.bar</span> <span class="selector-tag">span</span><span class="selector-pseudo">:nth-child</span>(<span class="number">2</span>){<span class="attribute">animation-delay</span>:.<span class="number">1s</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.bar</span> <span class="selector-tag">span</span><span class="selector-pseudo">:nth-child</span>(<span class="number">3</span>){<span class="attribute">animation-delay</span>:.<span class="number">2s</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.bar</span> <span class="selector-tag">span</span><span class="selector-pseudo">:nth-child</span>(<span class="number">4</span>){<span class="attribute">animation-delay</span>:.<span class="number">3s</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.bar</span> <span class="selector-tag">span</span><span class="selector-pseudo">:nth-child</span>(<span class="number">5</span>){<span class="attribute">animation-delay</span>:.<span class="number">4s</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="keyword">@keyframes</span> barElastic{<span class="number">0%</span>,<span class="number">80%</span>,<span class="number">100%</span>{<span class="attribute">height</span>:<span class="number">20%</span>;}<span class="number">40%</span>{<span class="attribute">height</span>:<span class="number">100%</span>;}}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.pulse</span>{<span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">height</span>:<span class="number">40px</span>;<span class="attribute">background</span>:<span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">50%</span>;<span class="attribute">animation</span>:pulseElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;}</span></span><br><span class="line"><span class="language-css">    <span class="keyword">@keyframes</span> pulseElastic{<span class="number">0%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">0.95</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1.3</span>);<span class="attribute">opacity</span>:<span class="number">0.7</span>;}<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">0.95</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.orbit</span>{<span class="attribute">position</span>:relative;<span class="attribute">width</span>:<span class="number">40px</span>;<span class="attribute">height</span>:<span class="number">40px</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.orbit</span><span class="selector-pseudo">::before</span>,<span class="selector-class">.orbit</span><span class="selector-pseudo">::after</span>{<span class="attribute">content</span>:<span class="string">''</span>;<span class="attribute">position</span>:absolute;<span class="attribute">top</span>:<span class="number">0</span>;<span class="attribute">left</span>:<span class="number">0</span>;<span class="attribute">width</span>:<span class="number">100%</span>;<span class="attribute">height</span>:<span class="number">100%</span>;<span class="attribute">border</span>:<span class="number">3px</span> solid <span class="number">#165DFF</span>;<span class="attribute">border-radius</span>:<span class="number">50%</span>;<span class="attribute">animation</span>:orbitElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">45</span>,.<span class="number">03</span>,.<span class="number">51</span>,<span class="number">1.4</span>) infinite;}</span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.orbit</span><span class="selector-pseudo">::after</span>{<span class="attribute">animation-delay</span>:.<span class="number">2s</span>;<span class="attribute">border-color</span>:<span class="number">#14C4D9</span>;}</span></span><br><span class="line"><span class="language-css">    <span class="keyword">@keyframes</span> orbitElastic{<span class="number">0%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">0.3</span>);<span class="attribute">opacity</span>:<span class="number">0</span>;}<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1</span>);<span class="attribute">opacity</span>:<span class="number">1</span>;}}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.text</span>{<span class="attribute">font-size</span>:<span class="number">20px</span>;<span class="attribute">font-weight</span>:<span class="number">600</span>;<span class="attribute">color</span>:<span class="number">#165DFF</span>;<span class="attribute">letter-spacing</span>:<span class="number">2px</span>;<span class="attribute">animation</span>:textElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;}</span></span><br><span class="line"><span class="language-css">    <span class="keyword">@keyframes</span> textElastic{<span class="number">0%</span>,<span class="number">100%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1</span>);}<span class="number">50%</span>{<span class="attribute">transform</span>:<span class="built_in">scale</span>(<span class="number">1.15</span>);}}</span></span><br><span class="line"><span class="language-css"></span></span><br><span class="line"><span class="language-css">    <span class="selector-class">.arc</span> <span class="selector-tag">svg</span>{<span class="attribute">animation</span>:arcElastic <span class="built_in">var</span>(--speed) <span class="built_in">cubic-bezier</span>(.<span class="number">68</span>,-.<span class="number">55</span>,.<span class="number">27</span>,<span class="number">1.55</span>) infinite;}</span></span><br><span class="line"><span class="language-css">    <span class="keyword">@keyframes</span> arcElastic{<span class="number">0%</span>{<span class="attribute">stroke-dashoffset</span>:<span class="number">94.2</span>;<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">0deg</span>);}<span class="number">50%</span>{<span class="attribute">stroke-dashoffset</span>:<span class="number">0</span>;<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">540deg</span>);}<span class="number">100%</span>{<span class="attribute">stroke-dashoffset</span>:<span class="number">94.2</span>;<span class="attribute">transform</span>:<span class="built_in">rotate</span>(<span class="number">1080deg</span>);}}</span></span><br><span class="line"><span class="language-css">  </span><span class="tag">&lt;/<span class="name">style</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;<span class="name">script</span>&gt;</span><span class="language-javascript"></span></span><br><span class="line"><span class="language-javascript">    <span class="keyword">const</span> root=<span class="variable language_">document</span>.<span class="property">documentElement</span>;</span></span><br><span class="line"><span class="language-javascript">    <span class="keyword">const</span> range=<span class="variable language_">document</span>.<span class="title function_">getElementById</span>(<span class="string">'speedRange'</span>);</span></span><br><span class="line"><span class="language-javascript">    <span class="keyword">const</span> value=<span class="variable language_">document</span>.<span class="title function_">getElementById</span>(<span class="string">'speedValue'</span>);</span></span><br><span class="line"><span class="language-javascript">    range.<span class="property">oninput</span>=<span class="function">()=&gt;</span>{</span></span><br><span class="line"><span class="language-javascript">      <span class="keyword">const</span> v=<span class="built_in">parseFloat</span>(range.<span class="property">value</span>).<span class="title function_">toFixed</span>(<span class="number">1</span>);</span></span><br><span class="line"><span class="language-javascript">      root.<span class="property">style</span>.<span class="title function_">setProperty</span>(<span class="string">'--speed'</span>,v+<span class="string">'s'</span>);</span></span><br><span class="line"><span class="language-javascript">      value.<span class="property">textContent</span>=v;</span></span><br><span class="line"><span class="language-javascript">    };</span></span><br><span class="line"><span class="language-javascript">  </span><span class="tag">&lt;/<span class="name">script</span>&gt;</span></span><br><span class="line"><span class="tag">&lt;/<span class="name">head</span>&gt;</span></span><br><span class="line"><span class="tag">&lt;<span class="name">body</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;<span class="name">header</span>&gt;</span>SanhaiOJ 加载动画实验室（白底完整）<span class="tag">&lt;/<span class="name">header</span>&gt;</span></span><br><span class="line"></span><br><span class="line">  <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"control-bar"</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">label</span> <span class="attr">for</span>=<span class="string">"speedRange"</span>&gt;</span>速度：<span class="tag">&lt;/<span class="name">label</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">input</span> <span class="attr">type</span>=<span class="string">"range"</span> <span class="attr">id</span>=<span class="string">"speedRange"</span> <span class="attr">min</span>=<span class="string">"0.5"</span> <span class="attr">max</span>=<span class="string">"3"</span> <span class="attr">step</span>=<span class="string">"0.1"</span> <span class="attr">value</span>=<span class="string">"1"</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">span</span> <span class="attr">id</span>=<span class="string">"speedValue"</span>&gt;</span>1.0<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line"></span><br><span class="line">  <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"container"</span>&gt;</span></span><br><span class="line">    <span class="comment">&lt;!-- 1. 点跳 --&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"card"</span> <span class="attr">data-anim</span>=<span class="string">"dots"</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"dots"</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">span</span> <span class="attr">class</span>=<span class="string">"label"</span>&gt;</span>Dots<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">    <span class="comment">&lt;!-- 2. 圆环 --&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"card"</span> <span class="attr">data-anim</span>=<span class="string">"ring"</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"ring"</span>&gt;</span><span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">span</span> <span class="attr">class</span>=<span class="string">"label"</span>&gt;</span>Ring<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">    <span class="comment">&lt;!-- 3. 柱状 --&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"card"</span> <span class="attr">data-anim</span>=<span class="string">"bar"</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"bar"</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">span</span>&gt;</span><span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">span</span> <span class="attr">class</span>=<span class="string">"label"</span>&gt;</span>Bar<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">    <span class="comment">&lt;!-- 4. 脉冲 --&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"card"</span> <span class="attr">data-anim</span>=<span class="string">"pulse"</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"pulse"</span>&gt;</span><span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">span</span> <span class="attr">class</span>=<span class="string">"label"</span>&gt;</span>Pulse<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">    <span class="comment">&lt;!-- 5. 轨道 --&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"card"</span> <span class="attr">data-anim</span>=<span class="string">"orbit"</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"orbit"</span>&gt;</span><span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">span</span> <span class="attr">class</span>=<span class="string">"label"</span>&gt;</span>Orbit<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">    <span class="comment">&lt;!-- 6. 文字 --&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"card"</span> <span class="attr">data-anim</span>=<span class="string">"text"</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"text"</span>&gt;</span>SanhaiOJ<span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">span</span> <span class="attr">class</span>=<span class="string">"label"</span>&gt;</span>Text<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">    <span class="comment">&lt;!-- 7. 弧形回旋 --&gt;</span></span><br><span class="line">    <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"card"</span> <span class="attr">data-anim</span>=<span class="string">"arc"</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">div</span> <span class="attr">class</span>=<span class="string">"arc"</span>&gt;</span><span class="tag">&lt;<span class="name">svg</span> <span class="attr">width</span>=<span class="string">"48"</span> <span class="attr">height</span>=<span class="string">"48"</span> <span class="attr">viewBox</span>=<span class="string">"0 0 48 48"</span>&gt;</span><span class="tag">&lt;<span class="name">path</span> <span class="attr">d</span>=<span class="string">"M24 4 A20 20 0 1 1 24 44"</span> <span class="attr">stroke</span>=<span class="string">"#165DFF"</span> <span class="attr">stroke-width</span>=<span class="string">"4"</span> <span class="attr">fill</span>=<span class="string">"none"</span> <span class="attr">stroke-linecap</span>=<span class="string">"round"</span> <span class="attr">stroke-dasharray</span>=<span class="string">"94.2"</span> <span class="attr">stroke-dashoffset</span>=<span class="string">"94.2"</span>/&gt;</span><span class="tag">&lt;/<span class="name">svg</span>&gt;</span><span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">      <span class="tag">&lt;<span class="name">span</span> <span class="attr">class</span>=<span class="string">"label"</span>&gt;</span>Arc<span class="tag">&lt;/<span class="name">span</span>&gt;</span></span><br><span class="line">    <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line">  <span class="tag">&lt;/<span class="name">div</span>&gt;</span></span><br><span class="line"></span><br><span class="line">  <span class="tag">&lt;<span class="name">script</span>&gt;</span><span class="language-javascript"></span></span><br><span class="line"><span class="language-javascript">    <span class="keyword">const</span> root=<span class="variable language_">document</span>.<span class="property">documentElement</span>;</span></span><br><span class="line"><span class="language-javascript">    <span class="keyword">const</span> range=<span class="variable language_">document</span>.<span class="title function_">getElementById</span>(<span class="string">'speedRange'</span>);</span></span><br><span class="line"><span class="language-javascript">    <span class="keyword">const</span> value=<span class="variable language_">document</span>.<span class="title function_">getElementById</span>(<span class="string">'speedValue'</span>);</span></span><br><span class="line"><span class="language-javascript">    range.<span class="property">oninput</span>=<span class="function">()=&gt;</span>{</span></span><br><span class="line"><span class="language-javascript">      <span class="keyword">const</span> v=<span class="built_in">parseFloat</span>(range.<span class="property">value</span>).<span class="title function_">toFixed</span>(<span class="number">1</span>);</span></span><br><span class="line"><span class="language-javascript">      root.<span class="property">style</span>.<span class="title function_">setProperty</span>(<span class="string">'--speed'</span>,v+<span class="string">'s'</span>);</span></span><br><span class="line"><span class="language-javascript">      value.<span class="property">textContent</span>=v;</span></span><br><span class="line"><span class="language-javascript">    };</span></span><br><span class="line"><span class="language-javascript">  </span><span class="tag">&lt;/<span class="name">script</span>&gt;</span></span><br><span class="line"><span class="tag">&lt;/<span class="name">body</span>&gt;</span></span><br><span class="line"><span class="tag">&lt;/<span class="name">html</span>&gt;</span></span><br></pre></td></tr></tbody></table></figure>]]></content>
    
    
      
      
    <summary type="html">&lt;h3 id=&quot;✅-一、插件目录（官方结构）&quot;&gt;✅ 一、插件目录（官方结构）&lt;/h3&gt;
&lt;figure class=&quot;highlight plaintext&quot;&gt;&lt;table&gt;&lt;tbody&gt;&lt;tr&gt;&lt;td class=&quot;gutter&quot;&gt;&lt;pre&gt;&lt;span class=&quot;line&quot;</summary>
      
    
    
    
    <category term="插件" scheme="http://example.com/categories/%E6%8F%92%E4%BB%B6/"/>
    
    <category term="OJ" scheme="http://example.com/categories/%E6%8F%92%E4%BB%B6/OJ/"/>
    
    
    <category term="SanhaiOJ加载动画" scheme="http://example.com/tags/SanhaiOJ%E5%8A%A0%E8%BD%BD%E5%8A%A8%E7%94%BB/"/>
    
  </entry>
  
  <entry>
    <title>Sam的打字计划 - 题解</title>
    <link href="http://example.com/2025/11/27/Sam%E7%9A%84%E6%89%93%E5%AD%97%E8%AE%A1%E5%88%92/"/>
    <id>http://example.com/2025/11/27/Sam%E7%9A%84%E6%89%93%E5%AD%97%E8%AE%A1%E5%88%92/</id>
    <published>2025-11-27T08:12:26.000Z</published>
    <updated>2025-11-30T06:23:56.911Z</updated>
    
    <content type="html"><![CDATA[<h2 id="T4-Sam的打字计划-题解">T4 Sam的打字计划 - 题解</h2><p>一看数据就知道可以搜，但是暴力搜索有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mn>9</mn><mn>9</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(9^9)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 直接 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mi>L</mi><mi>E</mi></mrow><annotation encoding="application/x-tex">TLE</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.1389em;">T</span><span class="mord mathnormal">L</span><span class="mord mathnormal" style="margin-right:0.0576em;">E</span></span></span></span> 了。</p><p>但是这道题是<strong>没有办法</strong>进行剪枝或者优化的，那么其实虽然是搜索，但本质是枚举。</p><p>枚举所有方案有个优化，叫<strong>折半搜索</strong>，标准的空间换时间的思路。</p><p>先枚举前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span> 个数字会导致的所有局面，再枚举后 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span> 个数字会导致的所有局面。</p><p>上下匹配就可以将复杂度降低到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mo>×</mo><msup><mn>9</mn><mn>5</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2\times9^5)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0278em;">O</span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。</p><p>当然这道题还有第二种做法：</p><p>将这个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mo>×</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">5\times4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 的方格分成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>13</mn></mrow><annotation encoding="application/x-tex">13</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">13</span></span></span></span> 个区块：</p><figure class="highlight plaintext"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><span class="line">1  2  2  3</span><br><span class="line">4  .  .  5</span><br><span class="line">6  7  7  8</span><br><span class="line">9  .  .  10</span><br><span class="line">11 12 12 13</span><br></pre></td></tr></tbody></table></figure><p>每个区块的覆盖次数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span>。</p><p>不难发现 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[13]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span></span></span></span> 所有数字都会用，而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 不会用，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[1]=a[13]-a[1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span></span></span></span>。</p><p>同理：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">1,7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span></span></span></span> 不用，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[6]=a[13]-cnt[1]-cnt[7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">7</span><span class="mclose">]</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>7</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[7]=a[13]-a[6]-cnt[1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">7</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span></span></span></span>。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">1,4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span></span></span></span> 不用，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[2]=a[13]-cnt[1]-cnt[4]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">4</span><span class="mclose">]</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[4]=a[13]-a[2]-cnt[1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">4</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span></span></span></span>。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>10</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[10]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">10</span><span class="mclose">]</span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 不用，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>10</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[10]=a[13]-cnt[2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">10</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mn>10</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[2]=a[13]-a[10]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">10</span><span class="mclose">]</span></span></span></span>。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[4]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">4</span><span class="mclose">]</span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">1,2,3,7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span></span></span></span> 不用，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[4]=a[13]-cnt[1]-cnt[2]-cnt[3]-cnt[7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">4</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">3</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">7</span><span class="mclose">]</span></span></span></span>，解得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[3]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">3</span><span class="mclose">]</span></span></span></span>。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">7</span><span class="mclose">]</span></span></span></span> 只有  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo separator="true">,</mo><mn>7</mn><mo separator="true">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1,7,0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span></span></span></span> 不用，解得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[0]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span></span></span></span>。</li></ul><p>综上，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>7</mn><mo stretchy="false">]</mo><mo separator="true">,</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[1],cnt[2],cnt[3],cnt[4],cnt[7],cnt[0]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">3</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">4</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">7</span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span></span></span></span> 可以直接算出。接下来只需要计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mo separator="true">,</mo><mn>6</mn><mo separator="true">,</mo><mn>8</mn><mo separator="true">,</mo><mn>9</mn></mrow><annotation encoding="application/x-tex">5,6,8,9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">8</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">9</span></span></span></span>。</p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">5</span><span class="mclose">]</span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn><mo separator="true">,</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">5,6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">5</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span></span></span></span> 不用，解得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>5</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[5]+cnt[6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">5</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span></span></span></span> 的值，设为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>9</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[9]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">9</span><span class="mclose">]</span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo separator="true">,</mo><mn>6</mn><mo separator="true">,</mo><mn>8</mn><mo separator="true">,</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">2,6,8,0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">8</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span></span></span></span> 会用，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><mn>9</mn><mo stretchy="false">]</mo><mo>=</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>8</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">a[9]=cnt[2]+cnt[6]+cnt[8]+cnt[0]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">9</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">8</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span></span></span></span>，解得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>8</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[6]+cnt[8]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">8</span><span class="mclose">]</span></span></span></span>，设为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span></span></span></span>。<p>又因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>5</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>8</mn><mo stretchy="false">]</mo><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>9</mn><mo stretchy="false">]</mo><mo>=</mo><mi>a</mi><mo stretchy="false">[</mo><mn>13</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>3</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>7</mn><mo stretchy="false">]</mo><mo>−</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[5]+cnt[6]+cnt[8]+cnt[9]=a[13]-cnt[1]-cnt[2]-cnt[3]-cnt[4]-cnt[7]-cnt[0]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">5</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">8</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">9</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord">13</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">1</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">2</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">3</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">4</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">7</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">0</span><span class="mclose">]</span></span></span></span>，设为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span>，所以我们现在有三个方程四个未知数。</p><p>要让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>5</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[5]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">5</span><span class="mclose">]</span></span></span></span> 最小，则最大化 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[6]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span></span></span></span>，而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mi>z</mi><mo>+</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>9</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">cnt[6]=x+y-z+cnt[9]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">9</span><span class="mclose">]</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo><mo stretchy="false">{</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo stretchy="false">[</mo><mn>6</mn><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>9</mn><mo separator="true">,</mo><mi>min</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mi>z</mi><mo>+</mo><mn>9</mn><mo separator="true">,</mo><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\max\{cnt[6]\}=\min(9,\min(x+y-z+9,min(x,y)))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">{</span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mopen">[</span><span class="mord">6</span><span class="mclose">]}</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">min</span><span class="mopen">(</span><span class="mord">9</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">9</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">min</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mclose">)))</span></span></span></span>。</p><p>再利用之前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x,y,z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.044em;">z</span></span></span></span> 可解得所有数字的个数。</p>]]></content>
    
    
      
      
    <summary type="html">&lt;h2 id=&quot;T4-Sam的打字计划-题解&quot;&gt;T4 Sam的打字计划 - 题解&lt;/h2&gt;
&lt;p&gt;一看数据就知道可以搜，但是暴力搜索有 &lt;span class=&quot;katex&quot;&gt;&lt;span class=&quot;katex-mathml&quot;&gt;&lt;math xmlns=&quot;http://www.w</summary>
      
    
    
    
    <category term="题解" scheme="http://example.com/categories/%E9%A2%98%E8%A7%A3/"/>
    
    
    <category term="题解" scheme="http://example.com/tags/%E9%A2%98%E8%A7%A3/"/>
    
  </entry>
  
</feed>
