Question 26
(USA 2) Let $P$ be a cubic polynomial with rational coefficients, and let $q_1,q_2,q_3,\ldots$ be a sequence of rational numbers such that $q_n=P(q_{n+1})$ for all $n\ge1$. Prove that there exists $k\ge1$ such that for all $n\ge1$, $q_{n+k}=q_n$.