BG-MO 2013

Find all $f : \mathbb{R}\to \mathbb{R}$ , bounded in $(0,1)$ and satisfying: $x^2 f(x) - y^2 f(y) = (x^2-y^2) f(x+y) -xy f(x-y)$ for all $x,y \in \mathbb{R}$ Proposed by Nikolay Nikolov

The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$ .

Proposed by Nikolay Beluhov

EDIT: It was the problem 3 (not 2), corrected the source title.

Suppose $\alpha,\beta,\gamma \in [0.\pi/2)$ and $\tan \alpha + \tan\beta + \tan \gamma \leq 3$ . Prove that: $$\cos 2\alpha + \cos 2\beta + \cos 2\gamma \ge 0$$

Proposed by Nikolay Nikolov

Consider acute $\triangle ABC$ with altitudes $AA_1, BB_1$ and $CC_1$ ( $A_1 \in BC,B_1 \in AC,C_1 \in AB$ ). A point $C'$ on the extension of $B_1A_1$ beyond $A_1$ is such that $A_1C' = B_1C_1$ . Analogously, a point $B'$ on the extension of A $_1C_1$ beyond $C_1$ is such that $C_1B' = A_1B_1$ and a point $A'$ on the extension of $C_1B_1$ beyond $B_1$ is such that $B_1A' = C_1A_1$ . Denote by $A'', B'', C''$ the symmetric points of $A' , B' , C'$ with respect to $BC, CA$ and $AB$ respectively. Prove that if $R, R'$ and R'' are circumradiii of $\triangle ABC, \triangle A'B'C'$ and $\triangle A''B''C''$ , then $R, R'$ and $R''$ are sidelengths of a triangle with area equals one half of the area of $\triangle ABC$ .

Given $m\in\mathbb{N}$ and a prime number $p$ , $p>m$ , let $$M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\, \text{is a perfect square} \}$$ Prove that $|M|$ does not depend on $p$ .

Proposed by Aleksandar Ivanov