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Prove the given inequality and how that K is non-negative

Source: Balkan MO ShortList 2009 A8

April 6, 2020

Problem Statement

For every positive integer mm and for all non-negative real numbers x,y,zx,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*}
[*] Prove that Km≥0K_m \geq 0 for every odd positive integer mm [*] Let MM =∏cyc(x−y)2= \prod_{cyc} (x-y)^2. Prove, K7+M2K1≥MK4K_7+M^2 K_1 \geq M K_4
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