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Inequality with arithmetic mean

Source: Czech and Slovak Olympiad 1973, National Round, Problem 3

July 16, 2024
inequalitiesalgebramean

Problem Statement

Let (ak)k=1∞\left(a_k\right)_{k=1}^\infty be a sequence of real numbers such that ak−1+ak+1≥2aka_{k-1}+a_{k+1}\ge2a_k for all k>1.k>1. For n≥1n\ge1 denote An=1n(a1+⋯+an).A_n=\frac1n\left(a_1+\cdots+a_n\right). Show that also the inequality An−1+An+1≥2AnA_{n-1}+A_{n+1}\ge2A_n holds for every n>1.n>1.
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