IMO Shortlist 1994 problem A5
Dodao/la:
arhiva2. travnja 2012. Let
![f(x) = \frac{x^2+1}{2x}](/media/m/5/c/e/5cec25f6ef81eeac6d2f3082718ce507.png)
for
![x \neq 0.](/media/m/8/4/3/84338a23525e72f1548cb6f4afe0800c.png)
Define
![f^{(0)}(x) = x](/media/m/a/a/5/aa57cf4ffe0c6dd5a34e20781f19f043.png)
and
![f^{(n)}(x) = f(f^{(n-1)}(x))](/media/m/8/4/f/84f26fbd7cc3f04fb2c76a6f1fb268b7.png)
for all positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and
![x \neq 0.](/media/m/8/4/3/84338a23525e72f1548cb6f4afe0800c.png)
Prove that for all non-negative integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
and
%V0
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