Question 64
Let $f(x)$ be a polynomial with rational coefficients, and let $\alpha$ be a real number. If
$$
\alpha^3 - 2019\alpha = (f(\alpha))^3 - 2019f(\alpha) = 2021,
$$
prove that $(f^n(\alpha))^3 - 2019f^n(\alpha) = 2021$ for any positive integer $n$.
(Here, we define $f^n(x) = \underbrace{f(f(\dots f(x)\dots))}_{n \text{ times}}$.)
$$
\alpha^3 - 2019\alpha = (f(\alpha))^3 - 2019f(\alpha) = 2021,
$$
prove that $(f^n(\alpha))^3 - 2019f^n(\alpha) = 2021$ for any positive integer $n$.
(Here, we define $f^n(x) = \underbrace{f(f(\dots f(x)\dots))}_{n \text{ times}}$.)