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GCD and COMB

Source: 2021 ISL C1

July 12, 2022
combinatorics

Problem Statement

Let SS be an infinite set of positive integers, such that there exist four pairwise distinct a,b,c,d∈Sa,b,c,d \in S with gcd⁡(a,b)≠gcd⁡(c,d)\gcd(a,b) \neq \gcd(c,d). Prove that there exist three pairwise distinct x,y,z∈Sx,y,z \in S such that gcd⁡(x,y)=gcd⁡(y,z)≠gcd⁡(z,x)\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x).
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