Question 9
(IRN 1) Let $f(x)$ be a polynomial with rational coefficients and $\alpha$ be a real number such that $\alpha^3-\alpha=f(\alpha)^3-f(\alpha)=33^{1992}$. Prove that for each $n\ge1$,
$$(f^{(n)}(\alpha))^3-f^{(n)}(\alpha)=33^{1992},$$
where $f^{(n)}(x)=f(f(\ldots f(x)))$, and $n$ is a positive integer.
$$(f^{(n)}(\alpha))^3-f^{(n)}(\alpha)=33^{1992},$$
where $f^{(n)}(x)=f(f(\ldots f(x)))$, and $n$ is a positive integer.