DE-MO 2012

Define a sequence $(a_n)$ by $a_0 =-4 , a_1 =-7$ and $a_{n+2}= 5a_{n+1} -6a_n$ for $n\geq 0.$ Prove that there are infinitely many positive integers $n$ such that $a_n$ is composite.

Find the maximal number of edges a connected graph $G$ with $n$ vertices may have, so that after deleting an arbitrary cycle, $G$ is not connected anymore.

Let $ABC$ a triangle and $k$ a circle such that: (1) The circle $k$ passes through $A$ and $B$ and touches the line $AC.$ (2) The tangent to $k$ at $B$ intersects the line $AC$ in a point $X\ne C.$ (3) The circumcircle $\omega$ of $BXC$ intersects $k$ in a point $Q\ne B.$ (4) The tangent to $\omega$ at $X$ intersects the line $AB$ in a point $Y.$ Prove that the line $XY$ is tangent to the circumcircle of $BQY.$

Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$ , then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$

Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$ . Prove that $$r<\frac{ab}{2(a+b)}.$$

Let $a_1$ and $a_2$ be postive real numbers. Let $a_{n+2}=1+\frac{a_{n+1}}{a_{n}}$ Prove that $|a_{2012}-2|<10^{-200}$