Let $a_1, a_2,...., a_n$ be distinct positive integers. Prove that $(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2$ and find the cases of equality.
Source: Romania TST 1995 3.1
Tags: Sum, Sum of powers, inequalities, algebra
Let $a_1, a_2,...., a_n$ be distinct positive integers. Prove that $(a_1^5 + ...+ a_n^5) + (a_1^7 + ...+ a_n^7) \ge 2(a_1^3 + ...+ a_n^3)^2$ and find the cases of equality.