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Finding three elements of the product set with certain divisibility

Source: Baltic Way 2020, Problem 20

November 14, 2020
combinatoricscombinatorics proposed

Problem Statement

Let AA and BB be sets of positive integers with ∣A∣≥2|A|\ge 2 and ∣B∣≥2|B|\ge 2. Let SS be a set consisting of ∣A∣+∣B∣−1|A|+|B|-1 numbers of the form abab where a∈Aa\in A and b∈Bb\in B. Prove that there exist pairwise distinct x,y,z∈Sx,y,z\in S such that xx is a divisor of yzyz.
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