2020 OIM Problems/Problem 3

Problem

Let $n \ge 2$ be an integer. A sequence $\alpha = (a_1, a_2, \cdots , a_n)$ of $n$ integers is "Limeña" (from Lima, Perú) if

$gcd \left\{ a_i - a_j | a_i > a_j, 1 \le i, j \le n\right\} = 1$

An "operation" consists of choosing two elements $a_k$ and $a_l$ of a sequence, with $k \ne l$ , and replace $a_l$ with $a'_l$ Show that, given a collection of $2^n-1$ Limeñan sequences, each formed by $n$ integers numbers, there are two of them, say $\beta$ and $\gamma$ , such that it is possible to transform $\beta$ into $\gamma$ by a finite number of operations. Clarification: If all the elements of a sequence are equal, then that sequence is not Limeña.