ÖMO 2004 Problem 1

Prove without using advanced (differential) calculus: (a) For any real numbers a,b,c,d it holds that $a^6+b^6+c^6+d^6-6abcd \ge -2$ . When does equality hold? (b) For which natural numbers $k$ does some inequality of the form $a^k +b^k +c^k +d^k -kabcd \ge M_k$ hold for all real $a,b,c,d$ ? For each such $k$ ,