CMO 2009

Given an acute triangle $PBC$ with $PB\neq PC.$ Points $A,D$ lie on $PB,PC,$ respectively. $AC$ intersects $BD$ at point $O.$ Let $E,F$ be the feet of perpendiculars from $O$ to $AB,CD,$ respectively. Denote by $M,N$ the midpoints of $BC,AD.$ $(1)$ : If four points $A,B,C,D$ lie on one circle, then $EM\cdot FN = EN\cdot FM.$ $(2)$ : Determine whether the converse of $(1)$ is true or not, justify your answer.

Find all the pairs of prime numbers $(p,q)$ such that $pq|5^p+5^q.$

Given two integers $m,n$ satisfying $4 < m < n.$ Let $A_{1}A_{2}\cdots A_{2n + 1}$ be a regular $2n+1$ polygon. Denote by $P$ the set of its vertices. Find the number of convex $m$ polygon whose vertices belongs to $P$ and exactly has two acute angles.

Given an integer $n > 3.$ Let $a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $min |a_{i} - a_{j}| = 1, 1\le i\le j\le n.$ Find the minimum value of $\sum_{k = 1}^n|a_{k}|^3.$

Let $P$ be a convex $n$ polygon each of which sides and diagnoals is colored with one of $n$ distinct colors. For which $n$ does: there exists a coloring method such that for any three of $n$ colors, we can always find one triangle whose vertices is of $P$ ' and whose sides is colored by the three colors respectively.

Given an integer $n > 3.$ Prove that there exists a set $S$ consisting of $n$ pairwisely distinct positive integers such that for any two different non-empty subset of $S$ : $A,B, \frac {\sum_{x\in A}x}{|A|}$ and $\frac {\sum_{x\in B}x}{|B|}$ are two composites which share no common divisors.