ÖMO 2000 Problem 4

We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$ . (a) Determine the terms of the sequence for $u_1 = 1$ . (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.