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

Let a.b.c be positive reals such that their sum is 1. Prove that $ \frac{a^{2}b^{2}}{c^{3}(a^{2}-ab+b^{2})}+\frac{b^{2}c^{2}}{a^{3}(b^{2}-bc+c^{2})}+\frac{a^{2}c^{2}}{b^{3}(a^{2}-ac+c^{2})}\geq \frac{3}{ab+bc+ac}$

$ 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 $ \displaystyle\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\}$

There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then on every $ 2k+1th$ move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every $ 2k+2th$ move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every $ k>0$. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.