US-TST 2014 Problem 2

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of integers, with the property that every consecutive group of $a_{i}$ 's averages to a perfect square. More precisely, for all positive integers $n$ and $k$, the quantity $$\frac{a_{n}+a_{n+1}+\cdots+a_{n+k-1}}{k}$$ is always the square of an integer. Prove that the sequence must be constant (all $a_{i}$ are equal to the same perfect square).