US-TST 2008 Problem 5

Two sequences of integers, $a_1, a_2, a_3, \ldots$ and $b_1, b_2, b_3, \ldots$ , satisfy the equation $$(a_n - a_{n - 1})(a_n - a_{n - 2}) + (b_n - b_{n - 1})(b_n - b_{n - 2}) = 0$$ for each integer $n$ greater than $2$ . Prove that there is a positive integer $k$ such that $a_k = a_{k + 2008}$ .