Find all functions $f:(0,\infty)\to (0,\infty)$ 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 $x,y,w,z$ satisfying $wx = yz$.
When I saw this question I was just blown away. Can anybody please explain the answer in detail?
This is from the 2008 shortlist: