Question 164
Define the sequences $\{a_n\}_{n \ge 0}$, $\{b_n\}_{n \ge 0}$ as follows: $a_0, b_0$ are natural numbers and
$$a_{n+1} = a_n + \lfloor \sqrt{b_n} \rfloor$$
$$b_{n+1} = b_n + \lfloor \sqrt{a_n} \rfloor$$
for $n = 0, 1, 2, \dots$. Prove that there exists a nonnegative $n$ such that $a_n = b_n$.
$$a_{n+1} = a_n + \lfloor \sqrt{b_n} \rfloor$$
$$b_{n+1} = b_n + \lfloor \sqrt{a_n} \rfloor$$
for $n = 0, 1, 2, \dots$. Prove that there exists a nonnegative $n$ such that $a_n = b_n$.