IMO 2003

Determine all pairs of positive integers $(a,b)$ such that $$\dfrac{a^2}{2ab^2-b^3+1}$$ is a positive integer.

Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

Let $ABCD$ be a cyclic quadrilateral. Let $P$ , $Q$ , $R$ be the feet of the perpendiculars from $D$ to the lines $BC$ , $CA$ , $AB$ , respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$ .

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Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$ , the number $n^p-p$ is not divisible by $q$ .

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