Let $S=x_1x_2+x_3x_4+\cdots+x_{2015}x_{2016},$ where $x_1,x_2,\ldots,x_{2016}\in\{\sqrt{3}-\sqrt{2},\sqrt{3}+\sqrt{2}\}.$ Can $S$ be equal to $2016?$ Cristian Lazăr

Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. Lucian Petrescu

Let $ABC$ be a triangle with $AB < AC,$ $I$ its incenter, and $M$ the midpoint of the side $BC$. If $IA=IM,$ determine the smallest possible value of the angle $AIM$.

A unit square is removed from the corner of an $n\times n$ grid where $n \geq 2$. Prove that the remainder can be covered by copies of the "L-shapes" consisting of $3$ or $5$ unit square, as depicted in the figure below. Every square must be covered once and the L-shapes must not go over the bounds of the grid. [asy][asy] import geometry; draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0)); draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy][/asy]Estonian Olympiad, 2009

1.Let $ABC$ be a triangle, $D$ the foot of the altitude from $A$ and $M$ the midpoint of the side $BC$. Let $S$ be a point on the closed segment $DM$ and let $P, Q$ the projections of $S$ on the lines $AB$ and $AC$ respectively. Prove that the length of the segment $PQ$ does not exceed one quarter the perimeter of the triangle $ABC$.

A bank has a set S of codes formed only with 0 and 1,each one with length n.Two codes are 'friends' if they are different on only one position.We know that each code has exactly k 'friends'.Prove that: 1)S has an even number of elements 2)S contains at least $2^k$ codes

3. Let n > 1 be an integer and $a_1, a_2, . . . , a_n$ be positive reals with sum 1. a) Show that there exists a constant c ≥ 1/2 so that $\sum \frac{a_k}{1+(a_0+a_1+...+a_{k-1})^2}\geq c$, where $a_0 = 0$. b) Show that ’the best’ value of c is at least $\frac{\pi}{4}$.

4.Prove that there exist only finitely many positive integers n such that $(\frac{n}{1}+1)(\frac{n}{2}+2)...(\frac{n}{n}+n)$ is an integer.