IrMO 2009

Let $ABCD$ be a square. The line segment $AB$ is divided internally at $H$ so that $|AB|\cdot |BH|=|AH|^2$ . Let $E$ be the midpoints of $AD$ and $X$ be the midpoint of $AH$ . Let $Y$ be a point on $EB$ such that $XY$ is perpendicular to $BE$ . Prove that $|XY|=|XH|$ .

Find all positive integers $n$ for which $n^8+n+1$ is a prime number.

For any positive integer $n$ define $$E(n)=n(n+1)(2n+1)(3n+1)\cdots (10n+1).$$ Find the greatest common divisor of $E(1),E(2),E(3),\dots ,E(2009).$

In the triangle $ABC$ we have $|AB|<|AC|$ . The bisectors of the angles at $B$ and $C$ meet $AC$ and $AB$ at $D$ and $E$ respectively. $BD$ and $CE$ intersect at the incenter $I$ of $\triangle ABC$ .

Prove that $\angle BAC=60^\circ$ if and only if $|IE|=|ID|$