Problem

Source: Romanian NMO 2021 grade 10 P4

Tags: functional equation, algebra



Determine all nonzero integers $a$ for which there exists two functions $f,g:\mathbb Q\to\mathbb Q$ such that \[f(x+g(y))=g(x)+f(y)+ay\text{ for all } x,y\in\mathbb Q.\]Also, determine all pairs of functions with this property. Vasile Pop