Baltic Way 1998 · Problem 7

Answer: $f(x) \equiv 0$ is the only such function.

Choose an arbitrary real number $x_{0}$ and denote $f\left(x_{0}\right)=c$. Setting $x=y=x_{0}$ in the equation we obtain $f\left(c^{2}\right)=2 c$. For $x=y=c^{2}$ the equation now gives $f\left(4 c^{2}\right)=4 c$. On the other hand, substituting $x=x_{0}$ and $y=4 c^{2}$ we obtain $f\left(4 c^{2}\right)=5 c$. Hence $4 c=5 c$, implying $c=0$. As $x_{0}$ was chosen arbitrarily, we have $f(x)=0$ for all real numbers $x$.

Obviously, the function $f(x) \equiv 0$ satisfies the equation. So it is the only solution.