IMO Shortlist 1992 problem 2
Dodao/la:
arhiva2. travnja 2012. Let
![\mathbb{R}^+](/media/m/4/d/d/4dd6182efc1bb170a565248a692ee278.png)
be the set of all non-negative real numbers. Given two positive real numbers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
and
![b,](/media/m/2/4/9/249ba90824bebecff6103a8730af063f.png)
suppose that a mapping
![f: \mathbb{R}^+ \mapsto \mathbb{R}^+](/media/m/4/2/2/42289b446eb14b2354a4fb62d763ee9b.png)
satisfies the functional equation:
Prove that there exists a unique solution of this equation.
%V0
Let $\mathbb{R}^+$ be the set of all non-negative real numbers. Given two positive real numbers $a$ and $b,$ suppose that a mapping $f: \mathbb{R}^+ \mapsto \mathbb{R}^+$ satisfies the functional equation:
$$f(f(x)) + af(x) = b(a + b)x.$$
Prove that there exists a unique solution of this equation.
Izvor: Međunarodna matematička olimpijada, shortlist 1992