BALKAN 2009

We start by observing that $z$ must be even, so $z^{2}=3^{x}-5^{y} \equiv(-1)^{x}-1(\bmod 4)$ is divisible by 4 , which implies that $x$ is even, say $x=2 t$. Then our equation can be rewritten as $\left(3^{t}-z\right)\left(3^{t}+z\right)=5^{y}$, which means that both $3^{t}-z=5^{k}$ and $3^{t}+z=5^{y-k}$ for some nonnegative integer $k$. Since $5^{k}+5^{y-k}=2 \cdot 3^{t}$ is not divisible by 5 , it follows that $k=0$ and

$$2 \cdot 3^{t}=5^{y}+1$$

Suppose that $t \geq 2$. Then $5^{y}+1$ is divisible by 9 , which is only possible if $y \equiv 3$ $(\bmod 6)$. However, in this case $5^{y}+1 \equiv 5^{3}+1 \equiv 0(\bmod 7)$, so $5^{y}+1$ is also divisible by 7 , which is impossible. Therefore we must have $t \leq 1$, which yields a (unique) solution $(x, y, z)=(2,1,2)$.

In a triangle $A B C$, points $M$ and $N$ on the sides $A B$ and $A C$ respectively are such that $M N \| B C$. Let $B N$ and $C M$ intersect at point $P$. The circumcircles of triangles $B M P$ and $C N P$ intersect at two distinct points $P$ and $Q$. Prove that $\angle B A Q=\angle C A P$. (Moldova)

A $9 \times 12$ rectangle is divided into unit squares. The centers of all the unit squares, except the four corner squares and the eight squares adjacent (by side) to them, are colored red. Is it possible to numerate the red centers by $C_{1}, C_{2}, \ldots, C_{96}$ so that the following two conditions are fulfilled: $1^{\circ}$ All segments $C_{1} C_{2}, C_{2} C_{3}, \ldots C_{95} C_{96}, C_{96} C_{1}$ have the length $\sqrt{13}$; $2^{\circ}$ The poligonal line $C_{1} C_{2} \ldots C_{96} C_{1}$ is centrally symmetric? (Bulgaria)

Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfying

$$f\left(f(m)^{2}+2 f(n)^{2}\right)=m^{2}+2 n^{2} \text { for all } m, n \in \mathbb{N} . \text { (Bulgaria) }$$

Time allowed: 270 minutes. Each problem is worth 10 points.