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Inequality involving 3 non-negative real numbers among 9

Source: Baltic Way 1994

December 22, 2011
inequalitiestrigonometryalgebrapolynomialinequalities proposed

Problem Statement

Let a1,a2,…,a9a_1,a_2,\ldots ,a_9 be any non-negative numbers such that a1=a9=0a_1=a_9=0 and at least one of the numbers is non-zero. Prove that for some ii, 2≤i≤82\le i\le 8, the inequality ai−1+ai+1<2aia_{i-1}+a_{i+1}<2a_i holds. Will the statement remain true if we change the number 22 in the last inequality to 1.91.9?
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