2012 China TST 1, Day 2

Chinese

1. 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.
2. 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
3. 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

1. 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.
2. 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 $k$.
3. 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.)