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Prove that the least of the numbers does not exceed 1/10

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September 22, 2010
InequalitypolynomialdistanceminimizationoptimizationIMO Shortlist

Problem Statement

The sum of the squares of five real numbers a1,a2,a3,a4,a5a_1, a_2, a_3, a_4, a_5 equals 11. Prove that the least of the numbers (ai−aj)2(a_i - a_j)^2, where i,j=1,2,3,4,5i, j = 1, 2, 3, 4,5 and i≠ji \neq j, does not exceed 110.\frac{1}{10}.
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