Find the smallest integer $n$ that there must be either three mutual friends or three mutual strangers among $n$ people, suppose each pair of people are either friends or strangers? ...

In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects. ...

Various ways of distributing balls into cells. ...

It is extremely difficult, and often impossible, to prevent errors when data are stored, retrieved, operated on, or transmitted. ...

Any mathematical system containing all the theorems of arithmetic is an incomplete system. ...

问题： 证明 $1+\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+\cdots=\dfrac{\pi^2}{6}$ ...

问题： 在圆上任取 $n$ 个点，将每对点用直线连接，并规定三条线不能交于一点，这些直线会将圆分割成多少份？ ...

想象你的麦克风在录制一段由四个纯音同时播放的音频，由于其只能捕捉气压——时间图像，因此最后的结果看起来相当复杂： Pressure-Time 如果给定这样一段音频，该如何将其分解为不同纯音的叠加？ ...

问题： 在球面上随机选择四点组成四面体，问球心落在该四面体内部的概率？ ...

问题： 一条项链上有 $n$ 种类型的珠宝，每种珠宝的数量均为偶数。问至少需要切几刀，可以将所有珠宝均分？ ...