International Mathematical Olympiad 2008 Problem 4

Find all functions $f: (0, \infty) \to (0, \infty)$ (so, $f$ is a function from the positive real numbers to the positive real numbers) such that $$\frac{(f(w))^2 + (f(x))^2}{f(y^2) + f(z^2)} = \frac{w^2 + x^2}{y^2 + z^2}$$ for all positive real numbers $w, x, y, z$, satisfying $wx = yz$.