CMO 2013

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. i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$ ). ii) Prove that the lengths of $CA$ , $AP$ and $PE$ are the side lengths of a right triangle.

Attachments:

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

Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality $$\max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td$$ holds for all (not necessarily distinct) $x,y,z\in X$ , all real numbers $a$ and all positive real numbers $d$ .

Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition $$\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \forall i,j \in \left\{ {1,2,...,n} \right\}.$$ Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|}$ .

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

Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$ , then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$ .