2012 China TST 1, Day 1

Chinese

1. 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: $\displaystyle\sum_{i=1}^n \vert z_i\vert\le n$.
2. 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.
3. 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

1. 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 $\displaystyle\sum_{i=1}^n \vert z_i\vert\le n$. 
2. 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.
3. 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 $\displaystyle\frac{\displaystyle\prod_{i\in A}x_i}{\displaystyle\prod_{i\in B}x_i}=2012$.