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combo in interval, sums of given form

Source: Yugoslav TST 1982 P3

May 29, 2021
combinatorics

Problem Statement

Let there be given real numbers xi>1 (i=1,2,…,2n)x_i>1~(i=1,2,\ldots,2n). Prove that the interval [0,2][0,2] contains at most (2nn)\binom{2n}n sums of the form α1x1+…+α2nx2n\alpha_1x_1+\ldots+\alpha_{2n}x_{2n}, where αi∈{−1,1}\alpha_i\in\{-1,1\} for all ii.
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