(D) Find the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$
f(f(y)+2 x)=x+f(x)+f(y)
$$
Let $x=0$ and $f(f(y))=f(y)+f(0)=f(y)+a$.

Let $x=2 w-a, y=0$ and $f(4 w-a)=2 w-a+f(2 w-a)+a$ and thus every $w$ can be written as $w=\frac{f(u)-f(v)}{2}$ for well-chosen $u, v$.

Now let $x=\frac{f(u)-f(v)}{2}$ and $y=v$ and we have

$$
f(f(u))=f(u)+a=\frac{f(u)-f(v)}{2}+f\left(\frac{f(u)-f(v)}{2}\right)+a
$$

and thus finally

$$
f(x)=x+a
$$

which is indeed a solution for any real $a$.