IMO Shortlist 1985 problem 17
Dodao/la:
arhiva2. travnja 2012. The sequence
![f_1, f_2, \cdots, f_n, \cdots](/media/m/d/d/b/ddb90b9dbc3b4d6c7e33ac068e22399d.png)
of functions is defined for
![x > 0](/media/m/1/9/4/1945adeeed2d2765b5d8b1595074b738.png)
recursively by
![f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)](/media/m/0/2/5/02579c6b05ae77c8416ff76006a9c768.png)
Prove that there exists one and only one positive number
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
such that
![0 < f_n(a) < f_{n+1}(a) < 1](/media/m/6/8/f/68f3c21f38455d110e08c50340b0c738.png)
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 $n \geq 1.$
Izvor: Međunarodna matematička olimpijada, shortlist 1985