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Best constant!

Source: Romania TST Day 3 Problem 3

January 21, 2015
inequalitiesinequalities unsolved

Problem Statement

Determine the smallest real constant cc such that ∑k=1n(1k∑j=1kxj)2≤c∑k=1nxk2\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2 for all positive integers nn and all positive real numbers x1,⋯ ,xnx_1,\cdots ,x_n.
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