South-Afrika, 2011

Algebra Difficulty 4.7 AIME Prove it South Africa

Find all functions $f : \mathbb{N} \to \mathbb{N}$ satisfying $f(n+1)f(n+2) = f(n)^2$ for all $n \in \mathbb{N}$ .

Solution

Suppose that the function is non-constant. Then it will have a least element $f(k)$ . Then $f(k+1) \neq f(k)$ , or the function would be constant, and by definition $f(k+1) > f(k)$ . But $f(k+2) = \frac{f(k)^2}{f(k+1)} = f(k)\frac{f(k)}{f(k+1)} < f(k)$ , which is a contradiction. So $f(n)$ constant is the only allowed function.

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