Romanian Selection 2011

Determine all real functions such that for all real $x, y$:

$$
2 f(x)=f(x+y)+f(x+2 y)
$$
We perform 4 substitutions: $(2 x, x),(x, x),(0, x),(0,2 x)$. By combining them, we find $f(x)=f(0)$, and conversely the constant functions are solutions.