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<h1>一个常规的递推数列不动点法问题及其引申 `zhangzhiyuan|20171125`</h1>
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<p><mark>TODO: 本文暴露出了我曾经的数学基础的弱点, 需要修改</mark></p>
<p>数列 \(\{a_n\}\) 满足 \(a_1 = 2\), \(a_{n+1} = \dfrac{a_n}2 + \dfrac1{a_n}\), 求其通项公式.</p>
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<h1>lixinyi 20171119</h1>
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<p>已知函数 \(f(x) = \sqrt{x^2+1}-ax\) (\(a \in \mathbb R\)).</p>
<p>(1) 当 \(a=1\) 时, 判断 \(f(x)\) 在 \(\mathbb R\) 上的点调性;<br/>
(2) 求实数 \(a\) 的取值范围, 使得函数 \(f(x)\) 在 \(\mathbb R^+\) 上是单调函数.</p>
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<h1>libinglun 20171112</h1>
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<p>设 \(p\) 和 \(q\) 是两个质数, 求不能表示为 \(mp + nq\) (其中, \(m,n\) 是自然数) 的最大正整数 (用 \(p\) 和 \(q\) 表示).</p>
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<h1>组合极值/组合构造问题 `kongyuqing|20171024`</h1>
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<p>100 个数围成一圈, \(a_{3} > a_{2} + a_{1}\), \(a_{4} > a_{3} + a_{2}\), \(\dots\) , \(a_{100} > a_{99} + a_{98}\), \(a_{1} > a_{100} + a_{99}\), \(a_{2} > a_{1} + a_{100}\), 问其中最多有多少个正数?</p>
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<span class="date">2017/10/25</span>
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<h1>递推数列推出组合数的问题 `duyan@bj#4|20171012`</h1>
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<p>已知 \(S_n\) 为数列 \(\{a_n\}\) 的前 \(n\) 项和, 规定 \(S_0 = 0\), 若对任意 \(n \in \mathbb N^*\), 均有 \(\dfrac{a_n}{2017} = - \dfrac{2017+S_{n-1}}{n}\), 则 \(\sum\limits_{n=1}^{2017} 2^n a_n = \) <code>______</code>.</p>
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<h1>鲁列斯三角形问题</h1>
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<p><mark>TODO: 微博图床已挂, 本篇缺图, 需要找回</mark></p>
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<h1>三圆轮换对称不等式问题 `mawentao|20170914`</h1>
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<p><code>问题</code> \(a,b,c \ge 0\), 求证:<br/>
\[<br/>
\begin{aligned}<br/>
&\sqrt{a^2 - ab + b^2} \cdot \sqrt{b^2 - bc + c^2} \\<br/>
&{}+ \sqrt{b^2 - bc + c^2} \cdot \sqrt{c^2 - ca + a^2} \\<br/>
&{}+ \sqrt{c^2 - ca + a^2} \cdot \sqrt{a^2 - ab + b^2}<br/>
\ge a^2 + b^2 + c^2<br/>
\end{aligned}<br/>
\]</p>
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<h1>201710月考复习 高一</h1>
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