MathNet 2022-part-02 — 197 problems (EN–VI) Question 178

Let $P(x)$ be a polynomial with integer coefficients. Define a sequence $\{a_n\}$ by $a_0 = 0$ and $a_n = P(a_{n-1})$ for all $n \ge 1$. Prove that if there exists a positive integer $m$ for which $a_m = 0$, then $a_1 = 0$ or $a_2 = 0$.