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Concave Sequence [Iran TST 2010]

Source:

May 10, 2010
inductioninequalitiesconcavity

Problem Statement

Sequence of real numbers a0,a1,…,a1389a_0,a_1,\dots,a_{1389} are called concave if for each 0<i<13890<i<1389, ai≥ai−1+ai+12a_i\geq\frac{a_{i-1}+a_{i+1}}2. Find the largest cc such that for every concave sequence of non-negative real numbers: ∑i=01389iai2≥c∑i=01389ai2\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2
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