ILL 1990 Problem 95

Let $p(x)$ be a cubic polynomial with rational coefficients. $q_1$, $q_2$, $q_3$,... is a sequence of rationals such that $q_n = p(q_{n + 1})$ for all positive $n$. Show that for some $k$, we have $q_{n + k} = q_n$ for all positive $n$.