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Do there exist such 2n positive integers?

Source: Chinese Mathematical Olympiad 1998 Problem 2

August 22, 2013
inequalities unsolvedinequalities

Problem Statement

Given a positive integer n>1n>1, determine with proof if there exist 2n2n pairwise different positive integers a1,…,an,b1,…bna_1,\ldots ,a_n,b_1,\ldots b_n such that a1+…+an=b1+…+bna_1+\ldots +a_n=b_1+\ldots +b_n and n−1>∑i=1nai−biai+bi>n−1−11998.n-1>\sum_{i=1}^{n}\frac{a_i-b_i}{a_i+b_i}>n-1-\frac{1}{1998}.
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