IMO Shortlist 1985 problem 17
The sequence
of functions is defined for
recursively by
Prove that there exists one and only one positive number
such that
for all integers
%V0
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 $n \geq 1.$
Source: Međunarodna matematička olimpijada, shortlist 1985
Školjka 
