DE-BUNDESWETTBEWERB 2011 Problem 4

Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$ . Find all pairs $(a, b)$ for which $q^2 + r = 2011$ .