IMO 2008 Problem 4

Determine all functions $f : (0, \infty) \to (0, \infty)$ such that $$\frac{(f(p))^2 + (f(q))^2}{f(r^2) + f(s^2)} = \frac{p^2 + q^2}{r^2 + s^2}$$ for all positive real numbers $p$, $q$, $r$, $s$ satisfying $pq = rs$.