(Following Benoît's course 2021)

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy:

$$
\forall x, y \in \mathbb{R}, \quad f(2 f(x)+f(y))=2 x+f(y)
$$
We take $y=0$, we have $f(2 f(x)+f(0))=2 x+f(0)$. Let $a$ and $b$ be two real numbers such that $f(a)=f(b)$, then $f(2 f(a)+f(0))=f(2 f(b)+f(0))$ hence $2 a+f(0)=2 b+f(0)$ and $a=b$, so $f$ is injective. We consider $x=0$, we then have $f(2 f(0)+f(y))=f(y)$, by injectivity, $f(y)=y-2 f(0)$, in particular $f(0)=-f(0)$ so $f(0)=0$ and $f$ is the identity. It works, so it is the unique solution.