Question 58
Prove that the inequality
$$
\frac{(x-y)^7 + (y-z)^7 + (z-x)^7 - (x-y)(y-z)(z-x)((x-y)^4 + (y-z)^4 + (z-x)^4)}{(x-y)^5 + (y-z)^5 + (z-x)^5} \ge 3
$$
holds for all pairwise different integers $x$, $y$, $z$. When does equality hold?
$$
\frac{(x-y)^7 + (y-z)^7 + (z-x)^7 - (x-y)(y-z)(z-x)((x-y)^4 + (y-z)^4 + (z-x)^4)}{(x-y)^5 + (y-z)^5 + (z-x)^5} \ge 3
$$
holds for all pairwise different integers $x$, $y$, $z$. When does equality hold?