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(i)a^6+b^6+c^6+d^6−6abcd >= −2 (ii) a^k +b^k +c^k +d^k −kabcd >= M_k

Source: Austrian Federal Competition For Advanced Students 2004, Part 2, p1

August 30, 2019
algebrainequalities4-variable inequality

Problem Statement

Prove without using advanced (differential) calculus: (a) For any real numbers a,b,c,d it holds that a6+b6+c6+d6−6abcd≥−2a^6+b^6+c^6+d^6-6abcd \ge -2. When does equality hold? (b) For which natural numbers kk does some inequality of the form ak+bk+ck+dk−kabcd≥Mka^k +b^k +c^k +d^k -kabcd \ge M_k hold for all real a,b,c,da,b,c,d? For each such kk,
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