Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

a) Find all prime numbers $p, q, r$ satisfying $3 \nmid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares. b) Do there exist prime numbers $p, q, r$ such that $3 \mid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares?

Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves the ball to an adjacent box. If $B$ finds the ball, then $B$ wins. Find the least number of steps for which $B$ can guarantee to win.

For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that \[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]

Let $a, b, c ,d$ be real numbers greater than $1$ and $x, y$ be real numbers such that \[ a^x+b^y = (a^2+b^2)^x \quad \text{and} \quad c^x+d^y = 2^y(cd)^{y/2} \] Prove that $x<y$.

Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.

In a convex quadrilateral $ABCD$ diagonals intersect at $E$ and $BE = \sqrt{2}\cdot ED, \: \angle BEC = 45^{\circ}.$ Let $F$ be the foot of the perpendicular from $A$ to $BC$ and $P$ be the second intersection point of the circumcircle of triangle $BFD$ and line segment $DC$. Find $\angle APD$.

In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.