CWMO 2003 Problem 5

The sequence $\{a_n\}$ satisfies $a_0 = 0, a_{n + 1} = ka_n + \sqrt {(k^2 - 1)a_n^2 + 1}, n = 0, 1, 2, \ldots$ , where $k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $2k$ divides $a_{2n}, n = 0, 1, 2, \ldots$ .