IMC 2016 · Problem 4

Let $k$ be a positive integer. For each nonnegative integer $n$, let $f(n)$ be the number of solutions $(x_1, \ldots, x_k) \in \mathbb{Z}^k$ of the inequality $|x_1| + \ldots + |x_k| \le n$. Prove that for every $n \ge 1$, we have $f(n - 1)f(n + 1) \le f(n)^2$.