EGMO 2014 Problem 6

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition

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

for all real numbers $x$ and $y$. Proposed by Daniël Kroes, NLD The answer is the functions $f(x)=x, f(x)=-x$ and $f(x)=\frac{1}{2}-x$.