SMO 2010

Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$ . A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$ . The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$ ; and the extension of $NK$ meets $\Gamma_2$ at $L$ . Prove that $PQ$ is perpendicular to $AL$

Let $(a_n), (b_n)$ , $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$ $a_{n+1} = 7a_n + 12b_n + 6$ $b_{n+1} = 4a_n + 7b_n + 3$ Prove that $a_n^2$ is the difference of two consecutive cubes.

Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$

Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.