KöMaL 2007 B3997

Prove that if the product of the real numbers $x$, $y$, $z$ is 1, then $x^4$+$y^4$+$z^4$+$x^2 y^2$+$y^2 z^2$+$z^2 x^2 \ge$2($x$+$y$+$z$).