2021 IMO Shortlist C1

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a, b, c, d \in S$ with $\operatorname{gcd}(a, b) \neq \operatorname{gcd}(c, d)$. Prove that there exist three pairwise distinct $x, y, z \in S$ such that $\operatorname{gcd}(x, y)=\operatorname{gcd}(y, z) \neq \operatorname{gcd}(z, x)$.

令 $S$ 为正整数的无限集合，使得 S$中存在四个成对不同的$a, b, c, d 和 $\operatorname{gcd}(a, b) \neq \operatorname{gcd}(c, d)$。证明 S$中存在三个成对不同的$x, y, z 使得 $\operatorname{gcd}(x, y)=\operatorname{gcd}(y, z) \neq \operatorname{gcd}(z, x)$。