2017 USAMO Problems/Problem 2

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Problem

Let $m_1,\dotsc,m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1,\dotsc,a_n)$ and any permutation $w_1,\dotsc,w_n$ of $m_1,\dotsc,m_n$, define an $A-inversion$ of $w$ to be a pair of entries $w_i,w_j$ with $i<j$ for which one of the following condition holds: