In a scalene triangle $ABC$, $\alpha$ and $\beta$ respectively denote the interior angles at vertixes $A$ and $B$. The bisectors of these two angles meet the opposite sides of the triangle at points $D$ and $E$, respectively. Prove that the acute angle between the lines $DE$ and $AB$ does not exceed $ \frac{ | \alpha - \beta |}{3}$ . Proposed by Dusan Djukic

Find the smallest natural number which is a multiple of $2009$ and whose sum of (decimal) digits equals $2009$ Proposed by Milos Milosavljevic

Determine the largest positive integer $n$ for which there exist pairwise different sets $\mathbb{S}_1 , ..., \mathbb{S}_n$ with the following properties: $1$) $|\mathbb{S}_i \cup \mathbb{S}_j | \leq 2004$ for any two indices $1 \leq i, j\leq n$, and $2$) $\mathbb{S}_i \cup \mathbb{S}_j \cup \mathbb{S}_k = \{ 1,2,...,2008 \}$ for any $1 \leq i < j < k \leq n$ Proposed by Ivan Matic

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. Proposed by Vidan Govedarica, Serbia

Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$. Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$. When does equality occur? Proposed by Marko Radovanovic

Triangle ABC has incircle w centered as S that touches the sides BC,CA and AB at P,Q and R respectively. AB isn't equal AC, the lines QR and BC intersects at point M, the circle that passes through points B and C touches the circle w at point N, circumcircle of triangle MNP intersects with line AP at L (L isn't equal to P). Then prove that S,L and M lie on the same line