Question 148
Find all functions $f: (0, +\infty) \to (0, +\infty)$ ($f$ is a function mapping positive real numbers to 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
$$wx = yz.$$
$$
\frac{(f(w))^2 + (f(x))^2}{f(y^2) + f(z^2)} = \frac{w^2 + x^2}{y^2 + z^2}
$$
for all positive real numbers $w, x, y, z$ satisfying
$$wx = yz.$$