US-TST 2015 Problem 4

Let $f: \mathbb{Q} \rightarrow \mathbb{Q}$ be a function such that for any $x, y \in \mathbb{Q}$, the number $f(x+y)-$ $f(x)-f(y)$ is an integer. Decide whether there must exist a constant $c$ such that $f(x)-c x$ is an integer for every rational number $x$.