BALKAN-SL 2021 A1

Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ and $g: \mathbb{R}^{+} \rightarrow \mathbb{R}$ such that

$$f\left(x^{2}+y^{2}\right)=g(x y)$$

holds for all $x, y \in \mathbb{R}^{+}$.