RMM 2025 Problem 4

Let $\mathbb{Z}$ denote the set of integers, and let $S \subset \mathbb{Z}$ be the set of integers that are at least $10^{100}$ . Fix a positive integer $c$ . Determine all functions $f: S \to \mathbb{Z}$ satisfying $f(xy + c) = f(x) + f(y)$ for all $x, y \in S$ .