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Prove that there exists an integer k

Source: MMC 2014 Problem 1

June 8, 2014
inductioninequalitiestriangle inequalityalgebran-variable inequality

Problem Statement

Let a1,…,ana_1,\ldots,a_n and b1…,bnb_1\ldots,b_n be 2n2n real numbers. Prove that there exists an integer kk with 1≤k≤n1\le k\le n such that ∑i=1n∣ai−ak∣  ≤  ∑i=1n∣bi−ak∣. \sum_{i=1}^n|a_i-a_k| ~~\le~~ \sum_{i=1}^n|b_i-a_k|. (Proposed by Gerhard Woeginger, Austria)
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