MathDB

1

Part of 2014 Mediterranean Mathematics Olympiad

Problems(1)

Prove that there exists an integer k

Source: MMC 2014 Problem 1

6/8/2014
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)
inductioninequalitiestriangle inequalityalgebran-variable inequality