Estonian Mathematical Olympiad

Taking $y=f(x)$ in the given equation, we get $$f(x)+f(f(f(x)))+f(f(f(x)))=x+f(x),$$ from which, by expressing $f(f(f(x)))$, we find $$f(f(f(x)))=\frac{x}{2}.(7)$$ Taking $y=0$ in the given equation, we get $f(x)+f(f(0))+f(f(f(x)))=x+0$; using the equality (7) here, we get after simplification $$f(x)=\frac{x}{2}-f(f(0)).(8)$$ Substituting $x=f(f(0))$ into equation (8) and applying equation (7) on the left hand side, we get $$\frac{0}{2}=\frac{f(f(0))}{2}-f(f(0)),(9)$$ which after simplification gives $f(f(0))=0$. Consequently, equation (8) takes the form $f(x)=\frac{x}{2}$. But in this case, $f(f(f(x)))=\frac{x}{8}$, whereas from equation (7) $f(f(f(x)))=\frac{x}{2}$. Since in general $\frac{x}{2}\ne \frac{x}{8}$, there are no functions that satisfy the given functional equation.