IZhO 2023 Problem 3

Let $a_1, a_2, \cdots, a_k$ be natural numbers. Let $S(n)$ be the number of solutions in nonnegative integers to $a_1x_1 + a_2x_2 + \cdots + a_kx_k = n$ . Suppose $S(n) \neq 0$ for all big enough $n$ . Show that for all sufficiently large $n$ , we have $S(n+1) < 2S(n)$ .