Estonian Math Competitions

Denote $f(1)=a$ and $f(-1)=b$. Choosing $x=y=1$ and $x=y=-1$ yields the system of equations $$\{\begin{array}{l}2a^3+b=a\\ 2b^3+a=b\end{array}$$ Adding the equations and cancelling equal terms, we get $2b^3=-2a^3$. Therefore $b=-a$. Substituting this into the first equation yields $2a^3-2a=0$. So $f(1)$ is one of the numbers $-1$, $0$, $1$. These values can be obtained by the functions $f(x)=-x|x|$, $f(x)=0$ and $f(x)=x|x|$ respectively, which satisfy the equations.