Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$, perpendicular to $CD$ at $N$, with $AN\gt NB$. A circle $\Gamma_2$ centred at $C$ with radius $CN$ intersects $\Gamma_1$ at $P,Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$.
Let $a_n,b_n$, $n=1,2,\ldots$, be two sequences of integers defined by $a_1=1,b_1=0$ and, for $n\ge1$,
$$a_{n+1}=7a_n+12b_n+6,\qquad b_{n+1}=4a_n+7b_n+3.$$
Prove that $a_n^2$ is the difference of two consecutive cubes.
Suppose $a_1,\ldots,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d\gt 0$. If $a_1\gt 15$, prove that $d\gt 30000$.
Let $n$ be a positive integer. Find the smallest positive integer $k$ such that, for every colouring of the squares of a $2n\times k$ chessboard with $n$ colours, there are two columns and two rows whose four intersection squares have the same colour.
Let $p$ be a prime number and let $x,y,z$ be positive integers with $0\lt x\lt y\lt z\lt p$. Suppose $x^3,y^3,z^3$ have the same remainder when divided by $p$. Show that $x^2+y^2+z^2$ is divisible by $x+y+z$.