Question 1
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 $p, q, r, s\gt 0$ with $p q=r s$.
$$
\frac{f(p)^{2}+f(q)^{2}}{f\left(r^{2}\right)+f\left(s^{2}\right)}=\frac{p^{2}+q^{2}}{r^{2}+s^{2}}
$$
for all $p, q, r, s\gt 0$ with $p q=r s$.