IMC 2007 · Problem 5

Let $n$ be a positive integer and $a_1, \dots, a_n$ be arbitrary integers. Suppose that a function $f : \mathbb{Z} \to \mathbb{R}$ satisfies $\sum\limits_{i=1}^{n} f(k + a_i \ell) = 0$ whenever $k$ and $\ell$ are integers and $\ell \ne 0$. Prove that $f = 0$.