MathDB

17

Part of 1985 IMO Shortlist

Problems(1)

Prove that there exist one number a

Source:

8/28/2010
The sequence f1,f2,⋯ ,fn,⋯f_1, f_2, \cdots, f_n, \cdots of functions is defined for x>0x > 0 recursively by f_1(x)=x ,   f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right) Prove that there exists one and only one positive number aa such that 0<fn(a)<fn+1(a)<10 < f_n(a) < f_{n+1}(a) < 1 for all integers n≥1.n \geq 1.
functionrecurrence relationIMO 1985IMO ShortlistFixed pointcalculuslimit