BALKAN-SL 2009 A8

For every positive integer $m$ and for all non-negative real numbers $x,y,z$ denote \begin{align*} K_m =x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m \end{align*} [list=a] [] Prove that $K_m \geq 0$ for every odd positive integer $m$ [] Let $M$ $= \prod_{cyc} (x-y)^2$ . Prove, $K_7+M^2 K_1 \geq M K_4$