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Harvard-MIT Mathematics Tournament

Algebra Difficulty 5.0 AIME Prove it United States

Problem:
For all real numbers $x$ , let
$f(x) = \frac{1}{\sqrt[2011]{1 - x^{2011}}}$
Evaluate $(f(f(\ldots(f(2011)) \ldots)))^{2011}$ , where $f$ is applied 2010 times.

Solution

Solution:
Direct calculation shows that $f(f(x)) = \frac{\sqrt[2011]{1 - x^{2011}}}{-x}$ and $f(f(f(x))) = x$ . Hence $(f(f(\ldots(f(x)) \ldots))) = x$ , where $f$ is applied 2010 times. So $(f(f(\ldots(f(2011)) \ldots)))^{2011} = 2011^{2011}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.