Question 54
Let $(a_0, a_1, a_2, ...)$ and $(b_0, b_1, b_2, ...)$ be two infinite sequences of integers such that
$$
(a_n - a_{n-1})(a_n - a_{n-2}) + (b_n - b_{n-1})(b_n - b_{n-2}) = 0,
$$
for all integers $n \ge 2$. Prove that there exists a positive integer $K$ such that
$$
a_{K+2011} = a_{K+(2011)^{2011}}.
$$
$$
(a_n - a_{n-1})(a_n - a_{n-2}) + (b_n - b_{n-1})(b_n - b_{n-2}) = 0,
$$
for all integers $n \ge 2$. Prove that there exists a positive integer $K$ such that
$$
a_{K+2011} = a_{K+(2011)^{2011}}.
$$