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Differences of the sequences ordered/unordered

Source: Baltic Way 1993

June 15, 2012
algebra proposedalgebra

Problem Statement

Let a1,a2,…,ana_1,a_2,\ldots,a_n and b1,b2,…,bnb_1,b_2,\ldots,b_n be two finite sequences consisting of 2n2n real different numbers. Rearranging each of the sequences in increasing order we obtain a1′,a2′,…,an′a_1',a_2',\ldots,a_n' and b1′,b2′,…,bn′b_1',b_2',\ldots,b_n'. Prove that max⁡1≤i≤n∣ai−bi∣≥max⁡1≤i≤n∣ai′−bi′∣.\max_{1\le i\le n}|a_i-b_i|\ge\max_{1\le i\le n}|a_i'-b_i'|.
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