Question 74
Let $S$ be a finite set of positive rational numbers. Let $x_1, x_2, x_3, \dots$ be a sequence such that $x_1 = 0$ and for every positive integer $n$ there exists some $q_n \in S$ such that $x_{n+1} = \sqrt{x_n + q_n}$. Suppose that all the numbers $x_1, x_2, x_3, \dots$ are rational. Show that there are only finitely many distinct members in the sequence $x_1, x_2, x_3, \dots$.