RO-NMO 2021 Problem 94

Let $A$ be a finite set of non-negative integers. Determine all functions $f:\mathbb{Z}_{\ge 0} \to A$ such that $$f(|x-y|)=|f(x)-f(y)|$$ for each $x,y\in\mathbb Z_{\ge 0}$ .

Andrei Bâra