BG-MO 2006

Consider the set $A=\{1,2,3\ldots ,2^n\}, n\ge 2$ . Find the number of subsets $B$ of $A$ such that for any two elements of $A$ whose sum is a power of $2$ exactly one of them is in $B$ .

Aleksandar Ivanov

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be a function that satisfies for all $x>y>0$ $$f(x+y)-f(x-y)=4\sqrt{f(x)f(y)}$$ a) Prove that $f(2x)=4f(x)$ for all $x>0$ ; b) Find all such functions.

Nikolai Nikolov, Oleg Mushkarov

Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$ . Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors.

Aleksandar Ivanov

The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$ . Prove that if $X$ lies on the ray $AC$ , $Y$ lies on the ray $BC$ and $OX=BY$ , where $O$ is the circumcentre of triangle $ABC$ , then $S_{XY}$ passes through a fixed point.

Emil Kolev