2008 IMO Shortlist 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 $p q=r s$.

查找所有函数 $f:(0, \infty) \rightarrow(0, \infty)$ 使得 $$\frac{f(p)^{2}+f(q)^{2}}{f\left(r^{2}\right)+f\left(s^{2}\right)}=\frac{p^{2}+q^{2}}{r^{2}+s^{2}}$$ 对于所有 $p, q, r, s>0$ 且 $p q=r s$。