Let
![f(x)](/media/m/3/f/4/3f40d68090aa4fb60a440be4675c7aca.png)
be a polynomial with rational coefficients and
![\alpha](/media/m/f/c/3/fc35d340e96ae7906bf381cae06e4d59.png)
be a real number such that
![\alpha^3 - \alpha = [f(\alpha)]^3 - f(\alpha) = 33^{1992}.](/media/m/2/1/6/216d3c56b635fb22da2f4ad1fda4d7c5.png)
Prove that for each
![\left [ f^{n}(\alpha) \right]^3 - f^{n}(\alpha) = 33^{1992},](/media/m/4/1/b/41bc0f447a9c314abc3b67cbc0dd14e5.png)
where
![f^{n}(x) = f(f(\cdots f(x))),](/media/m/8/c/5/8c5dbcab8d03f4d6551833b37eb05bc6.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
is a positive integer.
%V0
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.