AM-GM-HM inequality problem

Question

Prove that $a^5 + b^5 +c^5 >abc (ab+bc+ac) $ for all positive distinct values of a, b & c.

Answer 1

$a^5+b^5+c^5+a^5+b^5>5abcab$

$a^5+b^5+c^5+b^5+c^5>5abcbc$

$a^5+b^5+c^5+a^5+c^5>5abcac$

Strict inequalities hold since $a$, $b$ and $c$ are distinct.

Add them up and the result follows.