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href="/atom.xml" class="dropdown-menu button">RSS</a></div></div></header><cover></cover><main><div class="post-content"><div class="post-title"><h1 class="post-title__text">NOIP 2021 游记&退役记</h1><div class="post-title__meta"><a href="/archives/2021/11/" class="post-meta__date button">2021-11-21</a> <span class="separate-dot"></span> <a href="/categories/%E6%B0%B4/" class="button"><span class="post-meta__cats">水</span></a><style>.post-meta__pv{color:var(--t-l);visibility:hidden;opacity:0;transition:.2s}</style><span class="separate-dot"></span> <span class="post-meta__pv"></span></div></div><aside class="post-side"><div class="post-side__toc"><div class="toc-title">文章目录</div><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#day-0"><span class="toc-text">Day 0</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#day-1"><span class="toc-text">Day 1</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#t1-%E6%8A%A5%E6%95%B0-number"><span class="toc-text">T1 报数 number</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#t2-%E6%95%B0%E5%88%97-sequence"><span class="toc-text">T2 数列 sequence</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#t3-%E6%96%B9%E5%B7%AE-variance"><span class="toc-text">T3 方差 variance</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#t4-%E6%A3%8B%E5%B1%80-chess"><span class="toc-text">T4 棋局 chess</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%BB%93%E6%9D%9F"><span class="toc-text">结束</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%BB%B6%E7%BB%AD"><span class="toc-text">延续</span></a></li></ol></div></aside><a class="btn-toc button" id="btn-toc" tabindex="0"><svg viewBox="0 0 1024 1024" width="32" height="32" xmlns="http://www.w3.org/2000/svg"><path d="M128 256h64V192H128zM320 256h576V192H320zM128 544h64v-64H128zM320 544h576v-64H320zM128 832h64v-64H128zM320 832h576v-64H320z" fill="currentColor"></path></svg></a><div class="toc-menus" id="toc-menus"><div class="toc-title">文章目录</div><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#day-0"><span class="toc-text">Day 0</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#day-1"><span class="toc-text">Day 1</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#t1-%E6%8A%A5%E6%95%B0-number"><span class="toc-text">T1 报数 number</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#t2-%E6%95%B0%E5%88%97-sequence"><span class="toc-text">T2 数列 sequence</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#t3-%E6%96%B9%E5%B7%AE-variance"><span class="toc-text">T3 方差 variance</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#t4-%E6%A3%8B%E5%B1%80-chess"><span class="toc-text">T4 棋局 chess</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%BB%93%E6%9D%9F"><span class="toc-text">结束</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%BB%B6%E7%BB%AD"><span class="toc-text">延续</span></a></li></ol></div><article class="post post__with-toc card"><div class="post__header"><img alt="Cover Image" class="lazy" src="https://hans362-img.oss.0vv0.top/2019/11/03/noi.png?width=1920" srcset="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABAQMAAAAl21bKAAAABlBMVEXMzMyWlpYU2uzLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAACklEQVQImWNgAAAAAgAB9HFkpgAAAABJRU5ErkJggg==" data-srcset="https://hans362-img.oss.0vv0.top/2019/11/03/noi.png?width=1920"><div class="post__expire" id="post-expired-notify"><p><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 16 16" style="fill:#f5a623;stroke:#f5a623"><path fill-rule="evenodd" d="M8.893 1.5c-.183-.31-.52-.5-.887-.5s-.703.19-.886.5L.138 13.499a.98.98 0 0 0 0 1.001c.193.31.53.501.886.501h13.964c.367 0 .704-.19.877-.5a1.03 1.03 0 0 0 .01-1.002L8.893 1.5zm.133 11.497H6.987v-2.003h2.039v2.003zm0-3.004H6.987V5.987h2.039v4.006z"></path></svg> 本文最后更新于 <span id="expire-date"></span> 天前，文中部分描述可能已经过时。</p></div><script>(()=>{var e=Date.parse("2021-11-21"),t=(new Date).getTime(),t=Math.floor((t-e)/864e5);120<=t&&(document.querySelectorAll("#expire-date")[0].innerHTML=t,document.querySelectorAll("#post-expired-notify")[0].style.display="block")})()</script></div><div class="post__content"><html><head><script>var meting_api="https://api-v2.hans362.cn/vip/?server=:server&type=:type&id=:id&r=:r"</script><script class="meting-secondary-script-marker" src="/js/Meting.min.js"></script></head><body><p>11月20日下午一点，考场内清脆的结束铃响起，我按下了屏幕上的提交按钮，短暂的三年 OI 生涯终于随 NOIP 2021 的落幕画上了句号，我的心里感慨万千。这应该是这个博客的最后一篇 NOI 系列活动游记了，因为，无论这次考得怎样，我都真的彻底退役了，高三了，老了。</p><span id="more"></span><h2 id="day-0"><a class="markdownIt-Anchor" href="#day-0"></a> Day 0</h2><p>没想到 NOIP 竟然不在自己学校考，大概是因为比起 CSP-S2 人要少很多，我被分配到了比较远的华师大二附中某分校，路上要花不少时间。</p><p>考试前一天周五晚上六点放学（毕竟高三了），回到家都已经七点了，写了会儿作业就开始看 OI Wiki，复习了一些模版和 STL，十点多就睡觉了。</p><h2 id="day-1"><a class="markdownIt-Anchor" href="#day-1"></a> Day 1</h2><p>一大早六点起床，七点四十五到达考点。门口工作人员正在悬挂 NOIP（上海赛区）的横幅，排了会儿队测完体温就进去了。</p><p>进了机房找到座位开始试机。不愧是四大名校之一，华二的机房硬件设施比我校的高到不知道哪里去了。电脑装的是 Windows 10 精简版本，内存配了8个G，VirtualBox 装了 NOI Linux 2.0，总算不会出现 CSP-S2 时在我校虚拟机都开不起来的情况了。唯一一点不好的就是这精简系统把 Windows 自带的计算器给阉割了，要算个东西还得写代码。</p><p>到时间了下发密码，建文件夹，解压试题。看了眼编译参数，是开 O2 优化的，说明以后 O2 优化将成为常态。扫了一眼四道题，果然又没看到什么签到题，T1相对而言感觉是最好下手的，决定从 T1 开始。</p><h3 id="t1-报数-number"><a class="markdownIt-Anchor" href="#t1-报数-number"></a> T1 报数 number</h3><p>读完题想到了三种思路。第一种就是喜闻乐见的打表。幸亏我在考前仔细阅读了相关的技术条例，在 CCF 一份2011年的文件中明确规定了代码的长度不得超过 100KB，而这道题 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">10^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></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:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span> 的数据量显然会超出大小限制而爆零，果断否定。考完后发现傻乎乎打表的人竟然还真有，上海有个人打出了 23MB 的表。</p><p>第二种想法是预处理 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">10^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></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:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span> 以内所有含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">7</span></span></span></span> 的数字，打上不可报标记，同时对于每个含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">7</span></span></span></span> 的数字其整数倍对应的数字也打上不可报标记，这些都在预处理过程中完成，然后再针对每个询问进行查询。这种方法实际上就是所谓的「埃拉托斯特尼筛法」，貌似是本题的正解。我竟然在没有学过线性筛法的情况下自己想出了这种做法。悲催的是我错误地估计了时间复杂度，觉得预处理时间复杂度很高，查询又不需要使用到预处理的所有数据，会进行很多没有必要的预处理，从而超时。于是我在<strong>未加尝试的情况下</strong>竟然毅然决然地否定掉了正解。我根本没有意识到这样的查询实际上是离线查询，查询时间复杂度是 <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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span>，预处理时间复杂度也不过 <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>l</mi><mi>o</mi><mi>g</mi><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nloglogn)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mord mathnormal" style="margin-right:.01968em">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:.03588em">g</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，是完全可行的。这也为我在这场比赛的失败埋下了伏笔。</p><p>然后我就瞎搞出了第三种做法，预处理只处理 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>7</mn></msup></mrow><annotation encoding="application/x-tex">10^7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8141079999999999em;vertical-align:0"></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:.8141079999999999em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span> 以内所有含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">7</span></span></span></span> 的数字，打上不可报标记。然后针对每个查询数字 <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:.65952em;vertical-align:0"></span><span class="mord mathnormal">i</span></span></span></span> 再去校验 <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:.74285em;vertical-align:-.08333em"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span> 及以后的数是否可报。我洋洋得意，自以为这样的做法似乎避免了不必要的预处理，但赛后发现实际上查询时间复杂度高得离谱。一测发现前三个样例 AC，样例4 TLE。于是我又加了个记忆化数组，保存校验结果减少对同一个数的重复校验，然而样例4依旧 TLE。一看时间已经过去一个多小时了，想着算了先往后做吧。</p><h3 id="t2-数列-sequence"><a class="markdownIt-Anchor" href="#t2-数列-sequence"></a> T2 数列 sequence</h3><p>一看真就没啥思路，感觉和组合数学有关，估计是什么数学题。思索了很久实在没想法就 DFS 暴力开搞，逐个枚举数列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{a_n\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></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:.151392em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 中每一项的值，再求相应的 <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:.68333em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.05764em">S</span></span></span></span>，用 <code>__builtin_popcount()</code> 统计二进制中 <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:.64444em;vertical-align:0"></span><span class="mord">1</span></span></span></span> 的个数（今年起开放了下划线开头的函数），如果满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\leq k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7719400000000001em;vertical-align:-.13597em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.69444em;vertical-align:0"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> 就计算权值。同时要时刻注意取模避免爆 <code>long long</code>。测了一下样例1 AC，样例2 WA，而且我的答案竟然是负数，怀疑哪里溢出了检查了好几遍都没发现问题，该取模的地方也都取了，我只好决定放在一边接着往下做。</p><h3 id="t3-方差-variance"><a class="markdownIt-Anchor" href="#t3-方差-variance"></a> T3 方差 variance</h3><p>一看这题，我靠，怎么又是数学题，搁这上机数学竞赛呢？莫不是全国高联 CCF 限定版？</p><p>众所周知方差是反映一组数据波动程度的统计量，方差最小那不就是要数据波动尽可能小嘛，那这道题就是要通过若干次这种操作减少数据的波动程度。我打算先研究一下题干中给出的这种操作的一些性质，花了点时间发现了下面两条：</p><ol><li>对 <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:.58056em;vertical-align:-.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:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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:.15em"><span></span></span></span></span></span></span></span></span></span> 进行连续的两次操作将使其复原。（非常显然）</li><li>将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.638891em;vertical-align:-.208331em"></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:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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:.208331em"><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>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:.58056em;vertical-align:-.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:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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:.15em"><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>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.638891em;vertical-align:-.208331em"></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:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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:.208331em"><span></span></span></span></span></span></span></span></span></span> 放在一条平滑的曲线上，发现操作的直观意义在于使 <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:.58056em;vertical-align:-.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:.31166399999999994em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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:.15em"><span></span></span></span></span></span></span></span></span></span> 变为其关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.638891em;vertical-align:-.208331em"></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:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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:.208331em"><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>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a_{i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.638891em;vertical-align:-.208331em"></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:.311664em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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:.208331em"><span></span></span></span></span></span></span></span></span></span> 的均值的对称点。（也比较显然）</li></ol><p>然后就没了，<s>好不容易</s>证出来的结论似乎对解题依然没什么帮助。考虑了一下贪心、差分，感觉都不太对，也没敢贸然下手。最后只好忍痛放弃，拿了 <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 \leq 4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7719400000000001em;vertical-align:-.13597em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">4</span></span></span></span> 的12分跑路。</p><h3 id="t4-棋局-chess"><a class="markdownIt-Anchor" href="#t4-棋局-chess"></a> T4 棋局 chess</h3><p>看完题目就没有什么做的欲望了，再加上给了 1024MB 内存，4s 时限，估计不是什么好做的题。写 DFS 当成大模拟做我觉得划不来，吭哧吭哧写完调半天可能也没几分，果断放弃。</p><p>回过头又在 T1、T2、T3 间反复横跳，一道都没能突破，直到最后都没意识到自己把 T1 正解给否决了。走出考场的那一刻我知道自己已经凉了，突然很庆幸去年拿了二等奖，今年恐怕连奖都没了，在高一高二同学面前我这个高三的半退役选手真的是身败名裂。同时也深刻认识到了数学是自己的最大软肋，今年三道数学题一搞就拿不到什么分了。</p><p>回到家测了下民间数据，洛谷 62pts，小图灵 82pts，InfOJ 82pts，市排名 200+/371，凉透了。写这篇游记来记录我在考场上的真实反应，也是希望后人能吸取我的教训，引以为戒。</p><h2 id="结束"><a class="markdownIt-Anchor" href="#结束"></a> 结束</h2><p>我的高中三年 OI 生涯就在这一场失败的比赛中结束了，感觉自己又回到了2019年第一次打 CSP-S（那年 NOIP 取消）的时候，也是拿了几十分滚粗了，还真是首尾呼应啊（笑）。不管怎么样，至少高二的时候拿过 CSP-S 一等，拿过 NOIP 二等奖，也算是满意了。</p><p>更何况，自始至终我从来不指望 OI 能给我带来升学上的任何实质性帮助，也并不只是为了那么几个奖打 OI，更是因为热爱它，愿意为它付出时间和精力。三年下来，我收获了独属于我的一段人生经历与体验，也深刻认识到了自己是多么的菜，比自己厉害的人数不胜数，因而时刻心怀谦卑，这就足矣。</p><p>借此也特别感谢我校的两位信竞教练，把我们领进了 OI 的世界，高一高二时坚持每周两个晚上的集训无偿给我们上课，真的付出了很多。还要感谢我的班主任，在刚进高中家访的时候得知我有这方面的兴趣，二话不说当场打电话给了我现在的教练，把我拉进了校队。</p><h2 id="延续"><a class="markdownIt-Anchor" href="#延续"></a> 延续</h2><p>12月底还有一场上海市高三信息科技竞赛，仅限上海市高三学生参加，教练推荐了我去，到那时就要真正和 OI 说再见了。</p><p>不过，在9个月之后，当我考完高考，当我步入大学的校园时，说不定还能再次重逢，也许我会去打蓝桥杯，也许会成为 ACMer，又有谁说得准呢。</p><p>OI，后会有期。</p></body></html></div><div class="license"><div class="license-title">NOIP 2021 游记&退役记</div><div class="license-link"><a href="https://blog.hans362.cn/post/noip-2021/">https://blog.hans362.cn/post/noip-2021/</a></div><div class="license-meta"><div class="license-meta-item"><div class="license-meta-title">本文作者</div><div class="license-meta-text">Hans362</div></div><div class="license-meta-item"><div 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