ISL 2008 A1

Find all functions $f:(0, \infty) \rightarrow(0, \infty)$ such that $$\frac{f(p)^{2}+f(q)^{2}}{f\left(r^{2}\right)+f\left(s^{2}\right)}=\frac{p^{2}+q^{2}}{r^{2}+s^{2}}$$ for all $p, q, r, s>0$ with $p q=r s$.