RS-SMO 2014

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$ , $y \in \mathbb{R}$ hold: $$f(xf(y)-yf(x))=f(xy)-xy$$

Proposed by Dusan Djukic

On sides $BC$ and $AC$ of $\triangle ABC$ given are $D$ and $E$ , respectively. Let $F$ ( $F \neq C$ ) be a point of intersection of circumcircle of $\triangle CED$ and line that is parallel to $AB$ and passing through C. Let $G$ be a point of intersection of line $FD$ and side $AB$ , and let $H$ be on line $AB$ such that $\angle HDA = \angle GEB$ and $H-A-B$ . If $DG=EH$ , prove that point of intersection of $AD$ and $BE$ lie on angle bisector of $\angle ACB$ .

Proposed by Milos Milosavljevic

We call natural number $n$ $crazy$ iff there exist natural numbers $a$ , $b >1$ such that $n=a^b+b$ . Whether there exist $2014$ consecutive natural numbers among which are $2012$ $crazy$ numbers?

Proposed by Milos Milosavljevic

Regular $n$ -gon is divided to triangles using $n-3$ diagonals of which none of them have common points with another inside polygon. How much among this triangles can there be the most not congruent?

Proposed by Dusan Djukic