IMO Shortlist 1985 problem 17


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2. travnja 2012.
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The sequence f_1, f_2, \cdots, f_n, \cdots of functions is defined for x > 0 recursively by
f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)
Prove that there exists one and only one positive number a such that 0 < f_n(a) < f_{n+1}(a) < 1 for all integers n \geq 1.
Izvor: Međunarodna matematička olimpijada, shortlist 1985