Problem

Source: 2019 USAJMO 2, by Ankan

Tags: AMC, USA(J)MO, USAJMO, function, algebra, functional equation, Hi



Let $\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f \colon \mathbb{Z}\rightarrow \mathbb{Z}$ and $g \colon \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying \[ f(g(x))=x+a \quad\text{and}\quad g(f(x))=x+b \]for all integers $x$. Proposed by Ankan Bhattacharya