FI-MO 2006

Determine all pairs $(x, y)$ of positive integers for which the equation $$x + y + xy = 2006$$ holds.

The numbers $p, 4p^2 + 1,$ and $6p^2 + 1$ are primes. Determine $p.$

Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.