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Bisecting a set into 2-adic squares and non-squares

Source: 2017 Korea Winter Program Practice Test 1 Day 2 #4

January 21, 2017
combinatoricsnumber theory

Problem Statement

For a nonzero integer kk, denote by ν2(k)\nu_2(k) the maximal nonnegative integer tt such that 2t∣k2^t \mid k. Given are n(≥2)n (\ge 2) pairwise distinct integers a1,a2,…,ana_1, a_2, \ldots, a_n. Show that there exists an integer xx, distinct from a1,…,ana_1, \ldots, a_n, such that among ν2(x−a1),…,ν2(x−an)\nu_2(x - a_1), \ldots, \nu_2(x - a_n) there are at least n/4n/4 odd numbers and at least n/4n/4 even numbers.
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