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Inequality for n integers

Source: Greek National Mathematical Olympiad 2008 - P4

November 17, 2011
inequalitiesinequalities unsolved

Problem Statement

If a1,a2,…,ana_1, a_2, \ldots , a_n are positive integers and k=max⁡{a1,…,an}k = \max\{a_1, \ldots, a_n\}, t=min⁡{a1,…,an}t = \min\{a_1,\ldots, a_n\}, prove the inequality (a12+a22+⋯+an2a1+a2+⋯+an)knt≥a1a2⋯an.\left(\frac{a_1^2+a_2^2+\cdots+a_n^2}{a_1+a_2+\cdots+a_n}\right)^{\frac{kn}{t}} \geq a_1a_2\cdots a_n. When does equality hold?
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