parmenides51 wrote: Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$. Let $P(x,y,z)$ be the assertion $f(f(x)f(y)+f(y)f(z)+f(z)f(x))=f(x)+f(y)+f(z)$ Let $a=f(0)$ $P(0,0,0)$ $\implies$ $f(3a^2)=3a$ $P(3a^2,0,0)$ $\implies$ $f(7a^2)=5a$ $P(7a^2,0,0)$ $\implies$ $f(11a^2)=7a$ $P(11a^2,0,0)$ $\implies$ $f(15a^2)=9a$ $P(3a^2,3a^2,0)$ $\implies$ $f(15a^2)=7a$ And so (subtracting these two last lines) $a=0$ $P(x,0,0)$ $\implies$ $\boxed{f(x)=0\quad\forall x}$ which indeed is a solution.