RMM 2020 Problem 2

Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$ , let $a_i = a_k$ and $b_i = b_k$ , where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$ . We say $\mathbf a$ is $\mathbf b$ -harmonic if each $a_i$ equals the following arithmetic mean: $$a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.$$ Suppose that neither $\mathbf a$ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$ -harmonic and $\mathbf b$ is $\mathbf a$ -harmonic.

Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.