CMOQR 2020 Problem 3

Let $N$ be a positive integer and $A = a_1, a_2, ... , a_N$ be a sequence of real numbers. Define the sequence $f(A)$ to be $$f(A) = \left( \frac{a_1 + a_2}{2},\frac{a_2 + a_3}{2}, ...,\frac{a_{N-1} + a_N}{2},\frac{a_N + a_1}{2}\right)$$ and for $k$ a positive integer define $f^k (A)$ to be $f$ applied to $A$ consecutively $k$ times (i.e. $f(f(... f(A)))$ ) Find all sequences $A = (a_1, a_2,..., a_N)$ of integers such that $f^k (A)$ contains only integers for all $k$ .