IMO Shortlist 1994 problem A5


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2. travnja 2012.
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Let f(x) = \frac{x^2+1}{2x} for x \neq 0. Define f^{(0)}(x) = x and f^{(n)}(x) = f(f^{(n-1)}(x)) for all positive integers n and x \neq 0. Prove that for all non-negative integers n and x \neq \{-1,0,1\}

\frac{f^{(n)}(x)}{f^{(n+1)}(x)} = 1 + \frac{1}{f \left( \left( \frac{x+1}{x-1} \right)^{2n} \right)}.
Izvor: Međunarodna matematička olimpijada, shortlist 1994