olymmath

ISL 2018 A3

    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. (Luxembourg)

    Dataset text NuminaMath olympiads. Quoted with source.

    Official text: IMO 2018 Shortlist booklet, page 5.

    Solutions

    Official solution Solution 1 by the IMO Problem Selection Committee

    The terms of the source let olymmath link to this solution but not copy it. Read the solution (IMO official site).

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