Question 169
Find, with proof, all functions $f: \mathbb{R} \setminus \{0\} \to \mathbb{R}$ such that
$$
f(x)^2 - f(y) f(z) = x(x+y+z)(f(x) + f(y) + f(z))
$$
for all real $x, y, z$ such that $x y z = 1$.
$$
f(x)^2 - f(y) f(z) = x(x+y+z)(f(x) + f(y) + f(z))
$$
for all real $x, y, z$ such that $x y z = 1$.