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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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<span class="date">2020/04/08</span>
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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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<span class="date">2018/08/29</span>
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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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<span class="date">2018/01/13</span>
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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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<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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