CMO 2013

1. Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$, let $C$ and $D$ be two points on $K_1$ and $K_2$, respectively, such that $A$ is the midpoint of the segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$, respectively.
$(1)$ Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$).
$(2)$ Prove that the lengths of $CA$, $AP$ and $PE$ are the side lengths of a right triangle.

2. Find all nonempty sets $S$ of integers such that $3m-2n\in S$ for all (not necessarily distinct) $m,n\in S$.

3. Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality

$\displaystyle \max\{\mid x-(a-d)\mid, \mid y-a\mid, \mid z-(a+d)\mid\}>td$

holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$.

4. $n \geqslant 2$, there are $n$ finite sets ${A_1},{A_2},...,{A_n}$ satisfy that for any $i,j \in \left\{ {1,2,...,n} \right\}$, $\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right|$. Try to find the  the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|}$.

5. For any positive integer $n$ and $0 \leqslant i \leqslant n$, denote $C_n^i \equiv c(n,i)(\bmod 2)$, where $c(n,i) \in \left\{ {0,1} \right\}$, define $f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}}$$m,n,q$ are positive integers,  and $q + 1 \ne {2^\alpha }$ for any $\alpha\in N$.  Prove that if $f(m,q)\left| {f(n,q)} \right.$, then $f(m,r)\left| {f(n,r)} \right.$ for any positive integer $r$.

6. $m,n$ are positive integers, find the minimum positive integer $N$  satisties that if $S$ is a set of integers contains a complete residue system of $m$ and $\left| S \right| = N$, then there exit a nonempty set $A \subseteq S$, that $n\left| {\sum\limits_{x \in A} x } \right.$.