TR-MO 2008

Given an acute angled triangle $ABC$ , $O$ is the circumcenter and $H$ is the orthocenter.Let $A_1$ , $B_1$ , $C_1$ be the midpoints of the sides $BC$ , $AC$ and $AB$ respectively. Rays $[HA_1$ , $[HB_1$ , $[HC_1$ cut the circumcircle of $ABC$ at $A_0$ , $B_0$ and $C_0$ respectively.Prove that $O$ , $H$ and $H_0$ are collinear if $H_0$ is the orthocenter of $A_0B_0C_0$

$a - )$ Find all prime $p$ such that $\dfrac{7^{p - 1} - 1}{p}$ is a perfect square

$b - )$ Find all prime $p$ such that $\dfrac{11^{p - 1} - 1}{p}$ is a perfect square

$f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions

$a)$ $f(0,0)=1$ , $f(0,1)=1$ ,

$b)$ $\forall k \notin \left\{0,1\right\}$ $f(0,k)=0$ and

$c)$ $\forall n \geq 1$ and $k$ , $f(n,k)=f(n-1,k)+f(n-1,k-2n)$

find the sum $\sum_{k=0}^{\binom{2009}{2}}f(2008,k)$

A circle $\Gamma$ and a line $\ell$ is given in a plane such that $\ell$ doesn't cut $\Gamma$ .Determine the intersection set of the circles has $[AB]$ as diameter for all pairs of $\left\{A,B\right\}$ (lie on $\ell$ ) and satisfy $P,Q,R,S \in \Gamma$ such that $PQ \cap RS=\left\{A\right\}$ and $PS \cap QR=\left\{B\right\}$