Question 143
Two sequences of integers, $a_1, a_2, a_3, \dots$ and $b_1, b_2, b_3, \dots$, 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 $a_k = a_{k+2008}$.
$$
(a_n - a_{n-1})(a_n - a_{n-2}) + (b_n - b_{n-1})(b_n - b_{n-2}) = 0
$$
for each integer $n$ greater than $2$. Prove that there is a positive integer $k$ such that $a_k = a_{k+2008}$.