RS-SMO 2015

Consider circle inscribed quadriateral $ABCD$ . Let $M,N,P,Q$ be midpoints of sides $DA,AB,BC,CD$ .Let $E$ be the point of intersection of diagonals. Let $k1,k2$ be circles around $EMN$ and $EPQ$ . Let $F$ be point of intersection of $k1$ and $k2$ different from $E$ . Prove that $EF$ is perpendicular to $AC$ .

Let $k$ be fixed positive integer . Let $Fk(n)$ be smallest positive integer bigger than $kn$ such that $Fk(n)*n$ is a perfect square . Prove that if $Fk(n)=Fk(m)$ than $m=n$ .

For integer $a$ , $a \neq 0$ , $v_2(a)$ is greatest nonnegative integer $k$ such that $2^k | a$ . For given $n \in \mathbb{N}$ determine highest possible cardinality of subset $A$ of set $\{1,2,3,...,2^n \}$ with following property: For all $x, y \in A$ , $x \neq y$ , number $v_2(x-y)$ is even.

In nonnegative set of integers solve the equation: $$(2^{2015}+1)^x + 2^{2015}=2^y+1$$