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(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) >= 2(a_1^3 + ...+ a_n^3)^2

Source: Romania TST 1995 3.1

February 17, 2020
SumSum of powersinequalitiesalgebra

Problem Statement

Let a1,a2,....,ana_1, a_2,...., a_n be distinct positive integers. Prove that (a15+...+an5)+(a17+...+an7)≥2(a13+...+an3)2(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2 and find the cases of equality.
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