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Difficult Inequality

Source: Saudi Arabia IMO TST Day IV Problem 1

July 22, 2014
inequalitiesinductioninequalities unsolved

Problem Statement

Let a1,…,ana_1,\dots,a_n be a non increasing sequence of positive real numbers. Prove that a12+a22+⋯+an2≤a1+a22+1+⋯+ann+n−1.\sqrt{a_1^2+a_2^2+\cdots+a_n^2}\le a_1+\frac{a_2}{\sqrt{2}+1}+\cdots+\frac{a_n}{\sqrt{n}+\sqrt{n-1}}. When does equality hold?
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