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Showing properties about subsets of positive integers.

Source: IMO 2018 Shortlist A3

July 17, 2019
algebraIMO Shortlistrational numberSets

Problem Statement

Given any set SS of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets FF and GG of SS such that ∑x∈F1/x=∑x∈G1/x\sum_{x\in F}1/x=\sum_{x\in G}1/x;
(2) There exists a positive rational number r<1r<1 such that ∑x∈F1/x≠r\sum_{x\in F}1/x\neq r for all finite subsets FF of SS.
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