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Sum of all products of pairs

Source: Canada 1972, Problem 2

June 24, 2006

Problem Statement

Let a1,a2,…,ana_1,a_2,\ldots,a_n be non-negative real numbers. Define MM to be the sum of all products of pairs aiaja_ia_j (i<j)(i<j), <spanclass=′latex−italic′>i.e.</span><span class='latex-italic'>i.e.</span>, M=a1(a2+a3+⋯+an)+a2(a3+a4+⋯+an)+⋯+an−1an. M = a_1(a_2+a_3+\cdots+a_n)+a_2(a_3+a_4+\cdots+a_n)+\cdots+a_{n-1}a_n. Prove that the square of at least one of the numbers a1,a2,…,ana_1,a_2,\ldots,a_n does not exceed 2M/n(n−1)2M/n(n-1).
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