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[f(x)]^3 - f(x) = 33^1992

Source: IMO Shortlist 1992, Problem 9

August 13, 2008
algebrapolynomialfunctional equationIterationIMO ShortlistIMO Longlist

Problem Statement

Let f(x) f(x) be a polynomial with rational coefficients and α \alpha be a real number such that \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}. Prove that for each n≥1, n \geq 1, \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992}, where f^{n}(x) \equal{} f(f(\cdots f(x))), and n n is a positive integer.
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