▪ Greek Mathematical Olympiad TST 2005

Problem 1 

Examine if we can place $9$ convex $6$-angled polygons (the one next to the other) to construct a convex $39$ -angled.
Problem 2 
Prove that: For each $x,y,z$ in the $R$ have power that 

$\frac{x^2 - y^2}{2x^2+1} + \frac{y^2 - z^2}{2y^2+1}+ \frac{z^2 - x^2}{2z^2+1}\leq0$.

Problem 3 

Let the midpoint $M$ of the side $AB$ of an 4sided, inscribed in circle, $ABCD$. Let P the point of section of $MC$ with $BD$. Let the parallel from the point $C$ to the $AP$ which intersects the $BD$ at $S$. If

   $\angle{CAD}=\angle{PAB}=\frac{1}{2}{\angle{BMC}}$.

prove that BP= SD.
Problem 4 
Find all the positive integers $n$, $n ≥ 3$ such that $\frac{n}{(n-2)!}$.