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<h1>平面几何的全等和相似符号到底该怎么写</h1>
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<blockquote>
<p>这篇文章的原名本来计划是「\(\rm\LaTeX\) 中怎么输入平面几何的全等和相似符号」, 听上去满满的技术范, 但其实是一篇<mark>吐槽</mark>文章, 只为收集一点<mark>吐槽能量</mark>.</p>
</blockquote>
<table>
<thead>
<tr>
<th style="text-align: center">问题: 下图中有几种全等和相似符号的写法, 哪种才是对的呢?</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: center"><img src="media/15864160277643/15864171158667.jpg" alt="" style="width:399px;"/></td>
</tr>
</tbody>
</table>
<p>我不知道你们看到这个问题的时候是什么感觉, 反正我发现这个居然会成为一个问题的时候, 心里感受到了一整个族群的神兽奔腾而过, 然后默念了一句「<mark>mdzz</mark>」。</p>
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<h1>数列不等式 `lhc@jinan|20200407`</h1>
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<p>已知数列 \(\{a_n\}\) 满足 \(a_1 = 1\), \(a_n a_{n+1} = n\), \(n = 1,2,3,\dots\). 求证: <br/>
\[<br/>
\frac1{a_1} + \frac1{a_2} + \frac1{a_3} + \dots + \frac1{a_n} \ge 2 \sqrt n - 1.<br/>
\]</p>
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<h1>如何判断复合根式是否可以进一步化简</h1>
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<p><img src="media/15863166036092/15863168475561.jpg" alt="" style="width:324px;"/></p>
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<h1>Simon Tatham's Portable Puzzle Collection</h1>
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<p><img src="https://is2-ssl.mzstatic.com/image/thumb/Purple3/v4/c4/4b/49/c44b49fd-a2ff-e8e1-c762-a03cbb855aec/mzl.aehgjxxn.png/460x0w.png" alt=""/></p>
<p><a href="https://www.chiark.greenend.org.uk/%7Esgtatham/puzzles/"><strong>Simon Tatham's Portable Puzzle Collection</strong></a> (以下简称 SGT Puzzles), 顾名思义是一个谜题的合集 (Puzzle Collection), Simon Tatham 是它的作者, 而 Portalble 在这里我认为有两重意思:</p>
<ol>
<li>这些谜题游戏都很小巧, 它们都是一个人玩的, 并且一局的时间通常只有几分钟;</li>
<li>这个谜题合集作为一个软件包, 可以轻易的运行在多种不同的操作系统上, 即计算机专业说的 portable (可移植).</li>
</ol>
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<h1>Books</h1>
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<p>这个页面中是我推荐的书籍.</p>
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<h1>Links</h1>
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<p>这个页面中是一些我比较喜欢的网站.</p>
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<h1>二元有理函数最值</h1>
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<p>已知 \(x,y > 0\), 求 <br/>
\[<br/>
\frac{(2x+1) (y+1)}{2x^2 + 5y^2 + 7}<br/>
\]<br/>
的最大值.</p>
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<h1>两个数列不等式问题</h1>
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<ol>
<li>已知实数 \(x_1, x_2, \dots, x_{10}\), \(\sum\limits_{k=1}^{10} k x_k = 1\). 求 \(\left( \sum\limits_{k=1}^{10} x_k \right)^2 + \sum\limits_{k=1}^{10} x_k^2\) 的最小值.</li>
<li>设 \(a_n = 1 + \dfrac12 + \dfrac13 + \dots + \dfrac1n\). 求证: 对 \(n\ge 2\) 有 \(a_n^2 < 2 \left( \dfrac{a_2}2 + \dfrac{a_3}3 + \dots \dfrac{a_n}n \right) + \dfrac{n+3}{2n+2}\).</li>
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<h1>CMO 2018 题 3</h1>
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<p><mark>TODO: 微博图床已挂, 本篇缺图, 需要找回</mark></p>
<p>\(\triangle ABC\) 中, \(AB < AC\), \(O\) 为外心, \(D\) 是 \(\angle BAC\) 平分线上一点, \(E\) 在 \(BC\) 上, 满足 \(OE \parallel AD\), \(DE \perp BC\). 在射线 \(EB\) 上取点 \(K\) 满足 \(EK = EA\), \(\triangle ADK\) 外接圆与 \(BC\) 交于另一点 \(P \ne K\), \(\triangle ADK\) 外接圆与 \(\triangle ABC\) 外接圆交于另一点 \(Q \ne A\). 求证: \(PQ\) 与 \(\triangle ABC\) 外接圆相切.</p>
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<h1>两个二次函数问题</h1>
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<ol>
<li>已知二次函数 \(f(x) = a x^2 + b x + c\) 的图象过点 \((2,8)\), 且对一切实数 \(x\) 恒有 \(2x + 3 \le f(x) \le 2x^2 - 2x + 5\), 求 \(f(x)\).</li>
<li>在平面直角坐标系中, 不论 \(m\) 取何值时, 抛物线 \(y = mx^2 + (2m+1) x - (3m+2)\) 都不通过的直线 \(y = -x+1\) 上的点有哪些? (写出全部符合条件点的坐标)</li>
</ol>
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<h1>构造高次多项式解决多元方程问题</h1>
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<p>已知 \(x_1, x_2, \dots, x_9\) 满足方程组<br/>
\[<br/>
\begin{cases}<br/>
\dfrac{x_1}{11} + \dfrac{x_2}{12} + \dots + \dfrac{x_9}{19} = 1 \\<br/>
\dfrac{x_1}{21} + \dfrac{x_2}{22} + \dots + \dfrac{x_9}{29} = 1 \\<br/>
\dfrac{x_1}{31} + \dfrac{x_2}{32} + \dots + \dfrac{x_9}{39} = 1 \\<br/>
\dfrac{x_1}{41} + \dfrac{x_2}{42} + \dots + \dfrac{x_9}{49} = 1 \\<br/>
\dfrac{x_1}{51} + \dfrac{x_2}{52} + \dots + \dfrac{x_9}{59} = 1 \\<br/>
\dfrac{x_1}{61} + \dfrac{x_2}{62} + \dots + \dfrac{x_9}{69} = 1 \\<br/>
\dfrac{x_1}{71} + \dfrac{x_2}{72} + \dots + \dfrac{x_9}{79} = 1 \\<br/>
\dfrac{x_1}{81} + \dfrac{x_2}{82} + \dots + \dfrac{x_9}{89} = 1 \\<br/>
\dfrac{x_1}{91} + \dfrac{x_2}{92} + \dots + \dfrac{x_9}{99} = 1 \\<br/>
\end{cases}<br/>
\]<br/>
试求 \(x_1 + x_2 + \dots + x_9\).</p>
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<h1>伪装成函数方程的数论问题</h1>
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<div class="a-content-text">
<p>已知函数 \(f(x,y)\) 定义在正整数集上, 满足 \(\forall x,y \in \mathbb N^*\),</p>
<ol>
<li>\(f(x,x) = x\),</li>
<li>\(f(x,y) = f(y,x)\),</li>
<li>\((x+y) \cdot f(x,y) = y \cdot f(x, x+y)\).</li>
</ol>
<p>求证: \(f(x,y) = [x,y]\) (这里, \([x,y]\) 表示 \(x\) 与 \(y\) 的最小公倍数).</p>
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<h1>二次根式化简</h1>
<div class="a-content">
<div class="a-content-text">
<p>计算:<br/>
\[<br/>
\dfrac{<br/>
1 + \sqrt{2-\sqrt2} + \sqrt{2-\sqrt3}<br/>
}{<br/>
\sqrt3 + \sqrt{2+\sqrt2} + \sqrt{2+\sqrt3}<br/>
}<br/>
\]</p>
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<h1>liujihang 20180801</h1>
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<div class="a-content-text">
<p><mark>TODO: 微博图床已挂, 本篇缺图, 需要找回</mark></p>
<p>无穷数列 \(P \colon a_1, a_2, \dots, a_n, \dots\) 满足 \(a_i \in \mathbb N^*\), 且 \(a_i \le a_{i+1}\) (\(i\in \mathbb N^*\)). 对于数列 \(P\), 记 \(T_k (P) = \min \{ n | a_n \ge k \}\) (\(k\in \mathbb N^*\)), 其中 \(\min \{ n | a_n \ge k \}\) 表示集合 \(\{ n | a_n \ge k \}\) 中最小的数.</p>
<ol>
<li>若数列 \(P\colon 1,3,4,7,\dots\), 写出 \(T_1(P), T_2(P), \dots, T_5(P)\);</li>
<li>若 \(T_k (P) = 2k-1\), 求数列 \(P\) 的前 \(n\) 项之和;</li>
<li>已知 \(a_{20} = 46\), 求 \(s = a_1 + a_2 + \dots + a_{20} + T_1 (P) + T_2 (P) + \dots + T_{46} (P)\) 的值.</li>
</ol>
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<h1>不动点为复数值的递推数列问题</h1>
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<p>正数数列 \(\{a_n\}\) 满足 \(S_n + 1 = \dfrac{S_n+4}{2a_n - S_n}\), 其中 \(S_n\) 是 \(\{a_n\}\) 的前 \(n\) 项和, 求 \(\{a_n\}\) 的通项公式.</p>
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<h1>三角不等式/极值问题</h1>
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<p><mark>TODO: 本题确认伪证了</mark></p>
<p>已知 \(A,B,C\) 为三角形三内角. 求 \(\dfrac{\cos^2 A}{1+\cos A} + \dfrac{\cos^2 B}{1+\cos B} + \dfrac{\cos^2 C}{1+\cos C}\) 的最小值.</p>
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<h1>一组高中联赛二试模拟/选拔题 `zhangboxin|20180516`</h1>
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<div class="a-content-text">
<p><mark>TODO: 微博图床已挂, 本篇缺图, 需要找回</mark></p>
<p><mark>TODO: 本篇内容也还没完成</mark></p>
<p>这是二试难度的问题 (rdfz 20180516 高联选拔题)</p>
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<h1>Poncelet 定理: 圆和抛物线 `zhangboxin|20180409`</h1>
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<p>已知圆 \(C_1 \colon x^2+y^2 = 1\) 和抛物线 \(C_2 \colon y = x^2 - 2\). \(P,Q,R\) 是抛物线 \(C_2\) 上的三个不同的点, 且直线 \(PQ\) 和 \(PR\) 都是圆 \(C_1\) 的切线. 求证: \(QR\) 也是圆 \(C_1\) 的切线.</p>
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<span class="date">2018/04/16</span>
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<h1>平面几何 `duyan|20180123|2`</h1>
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<p><mark>TODO: 微博图床已挂, 本篇缺图, 需要找回</mark></p>
<p>已知 \(H\) 为 \(\triangle ABC\) 的垂心, 过 \(H\) 的直线交 \(BC, AB\) 于 \(D,Z\), 过 \(H\) 且垂直于 \(ZH\) 的另一条直线交 \(BC, AC\) 于 \(E,X\), 点 \(Y\) 使得 \(DY \parallel AC, EY \parallel AB\). 求证: \(X,Y,Z\) 三点共线.</p>
<p><img src="https://ws2.sinaimg.cn/large/006tKfTcly1fnqjnu4ou9j30iu0h7js7.jpg" alt=""/></p>
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<span class="date">2018/01/23</span>
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<h1>USAMO19xx #5, 高斯函数, 不等式, 第二数学归纳法</h1>
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<p>证明不等式: <br/>
\[<br/>
[nx] \ge \dfrac{[x]}1 + \dfrac{[2x]}2 + \dfrac{[3x]}3 + \dots +\dfrac{[nx]}n<br/>
\]<br/>
其中, \(n \in \mathbb N^*\), \([x]\) 表示不大于 \(x\) 的最大整数.</p>
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<h1>一道难度其实不大的平面几何长度计算类证明题 `zhangboxin|20171218`</h1>
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<p><mark>TODO: 微博图床已挂, 本篇缺图, 需要找回</mark></p>
<p>在 \(\triangle ABC\) 中, \(\angle A = 4 \angle C\), \(\angle B = 2 \angle C\), 试证: \((BC+CA) \cdot AB = BC \cdot CA\).</p>
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<h1>一个`多项式`版本的线性空间问题</h1>
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<p><mark>TODO: 本篇内容似乎也还没完成</mark></p>
<p>试求所有的实系数多项式 \(p(x)\), 使得对满足 \(ab + bc + ca = 0\) 的所有实数 \(a、b、c\), 均有: <br/>
\[<br/>
p(a - b) + p(b - c) + p(c - a) = 2p(a + b + c).<br/>
\]</p>
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<h1>一个很有意思的数列变换问题</h1>
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<p>已知数列 \(\{a_n\}\) 和 \(\{b_n\}\), 其中 \(n \ge 0\) (即两个数列的首项分别为 \(a_0\) 和 \(b_0\)). 若满足<br/>
\[<br/>
b_n = \sum_{k=0}^n (-1)^k C_n^k a_k<br/>
\,, \quad \forall n \in \mathbb N.<br/>
\]<br/>
求证:<br/>
\[<br/>
a_n = \sum_{k=0}^n (-1)^k C_n^k b_k<br/>
\,, \quad \forall n \in \mathbb N.<br/>
\]</p>
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<p>本页面用于记载这个 blog 的代办事项.</p>
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