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Mediterranean M.C. 2010 Problem 2

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July 11, 2010
inequalitiesinequalities proposedMediterraneann-variable inequality

Problem Statement

Given the positive real numbers a1,a2,…,an,a_{1},a_{2},\dots,a_{n}, such that n>2n>2 and a1+a2+⋯+an=1,a_{1}+a_{2}+\dots+a_{n}=1, prove that the inequality a2⋅a3⋅⋯⋅ana1+n−2+a1⋅a3⋅⋯⋅ana2+n−2+⋯+a1⋅a2⋅⋯⋅an−1an+n−2≤1(n−1)2 \frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}
does holds.
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