MathDB

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

\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

\mathbf b

-harmonic and

\mathbf b

\mathbf a

-harmonic. Prove that at least

N+1

of the numbers

a_1, \ldots, a_N,b_1, \ldots, b_N

are zero.

Problem Statement