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Inequality on pairs of variables in [0,1]

Source: ISL 2022 A4

July 9, 2023
inequalitiesalgebra

Problem Statement

Let n⩾3n \geqslant 3 be an integer, and let x1,x2,…,xnx_1,x_2,\ldots,x_n be real numbers in the interval [0,1][0,1]. Let s=x1+x2+…+xns=x_1+x_2+\ldots+x_n, and assume that s⩾3s \geqslant 3. Prove that there exist integers ii and jj with 1⩽i<j⩽n1 \leqslant i<j \leqslant n such that 2j−ixixj>2s−3.2^{j-i}x_ix_j>2^{s-3}.
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