Question 156
Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(0) = 0$ and
$$
2f\left(-\frac{1}{2}xy + f(x + y)\right) = x f(y) + y f(x)
$$
for all $x, y \in \mathbb{R}$. Prove that $f(2) = f(-2) = 0$.
$$
2f\left(-\frac{1}{2}xy + f(x + y)\right) = x f(y) + y f(x)
$$
for all $x, y \in \mathbb{R}$. Prove that $f(2) = f(-2) = 0$.