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IMC 2016, Problem 6

Source: IMC 2016

July 28, 2016
IMCIMC 2016college contestsSequencesSummation

Problem Statement

Let (x1,x2,…)(x_1,x_2,\ldots) be a sequence of positive real numbers satisfying ∑n=1∞xn2n−1=1{\displaystyle \sum_{n=1}^{\infty}\frac{x_n}{2n-1}=1}. Prove that ∑k=1∞∑n=1kxnk2≤2. \displaystyle \sum_{k=1}^{\infty} \sum_{n=1}^{k} \frac{x_n}{k^2} \le2.
(Proposed by Gerhard J. Woeginger, The Netherlands)
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