BALKAN-SL 2018 A5

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a concave function and let $g: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Given that

$$f(x+y)+f(x-y)-2 f(x)=g(x) y^{2}$$

for all $x, y \in \mathbb{R}$, prove that $f$ is a quadratic function. (Bulgaria)