Problem

Source: 2019 ELMO Shortlist N2

Tags: number theory, function



Let $f:\mathbb N\to \mathbb N$. Show that $f(m)+n\mid f(n)+m$ for all positive integers $m\le n$ if and only if $f(m)+n\mid f(n)+m$ for all positive integers $m\ge n$. Proposed by Carl Schildkraut