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MEMO 2010, Problem T-2: Stronger than Cauchy

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September 12, 2010
inequalitiesCauchy Inequalityinequalities proposed

Problem Statement

For each integer n⩾2n\geqslant2, determine the largest real constant CnC_n such that for all positive real numbers a1,…,ana_1, \ldots, a_n we have \frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}
(4th Middle European Mathematical Olympiad, Team Competition, Problem 2)
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