Chinese
- 给定平面上两个圆$\omega_1$，$\omega_2$，设$S$是满足下面条件的 $\Delta ABC$ 的集合：$\Delta ABC$ 内接于圆
$\omega_1$ ，且圆
$\omega_2$ 是
$\Delta ABC$ 的与边 $BC$ 相切的旁切圆. 设圆与直线 $BC$、$CA$、$AB$ 分别相切于点 $D$、$E$、$F$.
证明：若$S$非空，则
$\Delta DEF$ 的重心是平面上的定点.
- 设 $d(n)$ 表示正整数 $n$ 的正约数个数. 正整数$n$称为超级数，如果对任何小于 $n$ 的正整数 $m$，均有 $d(m)<d(n)$.
证明：对任意给定的正整数 $k$，除有限的超级数外，每个超级数都能被 $k$ 整除
- 给定整数 $n$，求所有的函数 $f:\mathbb{Z}\mapsto\mathbb{Z}$，满足对任意整数 $x,y$，均有$f(x+y+f(y))=f(x)+ny$，其中$\mathbb{Z}$是整数集.
English
- For two given circles $\omega_1,\omega_2$ in the plane, let $S$ be the set of all triangles $\Delta ABC$ satisfying the following condition: $\Delta ABC$ is inscribed in circle $\omega_1$, and $\omega_2$ is the excircle of $\Delta ABC$ opposite $A$, and $\omega_2$ meets lines $BC, CA, AB$ at $D,E,F$, respectively. Prove that if $S$ is non-empty, the centroid of $\Delta DEF$ is a fixed point in the plane.
- Let $d(n)$ denote the number of positive integer divisors of $n$. A positive integer $n$ is called super if for every positive integer $m<n$, we have $d(m)<d(n)$. Prove that for each positive integer $k$, only finitely many good numbers are not divisible by $k$.
- Given a positive integer $n$, find all functions $f:\mathbb{Z}\mapsto\mathbb{Z}$ such that for every pair of integers $x,y$, we have $f(x+y+f(y))=f(x)+ny$. (Here $\mathbb{Z}$ denotes the set of integers.)
Chinese
- 设$x_1,x_2,\ldots,x_n;y_1,y_2,\ldots,y_n$均为模等于1的复数，$z_i=xy_i+yx_i-x_iy_i$ $(i=1,2,\ldots,n)$，其中 $x=\frac1n\displaystyle\sum_{i=1}^{n}x_i, y=\frac1n\displaystyle\sum_{i=1}^{n}y_i$. 证明: $\displaystyle\sum_{i=1}^n \vert z_i\vert\le n$.
- 如图，三角形$ABC$三边互不相同，其内切圆与三边的切点别为$D,E,F$. 设$L$为$D$关于直线$EF$的对称点，$M$为$E$关于直线$FD$的对称点，$N$为$E$关于直线$DE$的对称点. 直线$AL$与直线$BC$交与$P$，直线$BM$与直线$CA$交与$Q$，直线$CN$与直线$AB$交与$R$. 证明：$P,Q,R$ 三 点共线.
- 设$x_n=C^n_{2n}(n=1,2,\ldots)$，证明： 存在无穷多对有限集合 $A,B\subset N^*$，使得 $A\cap B=\varnothing$，且$\displaystyle\frac{\displaystyle\prod_{i\in A}x_i}{\displaystyle\prod_{i\in B}x_i}=2012$.
English
- Let $x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ be complex numbers with magnitude 1. For $1\le i\le n$, let $z_i=xy_i+yx_i-x_iy_i$, where $x=\frac1n\displaystyle\sum_{i=1}^{n}x_i$ and $y=\frac1n\displaystyle\sum_{i=1}^{n}y_i$. Prove that $\displaystyle\sum_{i=1}^n \vert z_i\vert\le n$.
- The incircle of scalene triangle $ABC$ meets sides $BC,CA,AB$ at $D,E,F$, respectively. Let $L$ be the reflection of $D$ across $EF$, $M$ be the reflection of $E$ across $FD$, and $N$ be the reflection of $F$ across $DE$. Suppose lines $AL$ and $BC$ meet at $P$, $BM$ and $CA$ meet at $Q$, and $CN$ and $AB$ meet at $R$. Prove that $P,Q,$ and $R$ are collinear.
- For each positive integer $n$, let $x_n=\binom{2n}{n}$ . Prove that there exist infinitely many pairs $(A,B)$ of finite sets of positive integers such that $A\cap B=\varnothing$ and $\displaystyle\frac{\displaystyle\prod_{i\in A}x_i}{\displaystyle\prod_{i\in B}x_i}=2012$.
