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Ineq of the selection in sets

Source: Pre-VMO 2012 - Problem 7

November 27, 2011
inductioncombinatorics proposedcombinatorics

Problem Statement

There are nn students. Denoted the number of the selections to select two students (with their weights are aa and bb) such that ∣a−b∣≤1\left| {a - b} \right| \le 1 (kg) and ∣a−b∣≤2\left| {a - b} \right| \le 2 (kg) by A1A_1 and A2A_2, respectively. Prove that A2<3A1+nA_2<3A_1+n.
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