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 \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 - f^{n}(\alpha) = 33^{1992},\] where $ f^{n}(x) = f(f(\cdots f(x))),$ and $ n$ is a positive integer.
orl wrote:
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 \geq 1,$
\[ \left [ f^{n}(\alpha) \right]^3 - f^{n}(\alpha) = 33^{1992},
\]
where $ f^{n}(x) = f(f(\cdots f(x))),$ and $ n$ is a positive integer.
Since the equation $ x^3 -x - c = 0$ has only one real root for every $ c >
\frac{2}{3\sqrt{3}}$ , $ \alpha$ is the unique real root of $ x^3-x-33^{1992} = 0$. Hence $ f^n(\alpha) =
f(\alpha) = \alpha$ ,as desired