IMO Shortlist 2003 problem A5
Dodao/la:
arhiva2. travnja 2012. Let
![\mathbb{R}^+](/media/m/4/d/d/4dd6182efc1bb170a565248a692ee278.png)
be the set of all positive real numbers. Find all functions
![f: \mathbb{R}^+ \longrightarrow \mathbb{R}^+](/media/m/8/6/0/8602671ab9e8f2e5a62e5ee72b77c234.png)
that satisfy the following conditions:
-
![f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})](/media/m/8/1/5/815398d6079438439c0307e65c3f8617.png)
for all
![x,y,z\in\mathbb{R}^+](/media/m/7/f/2/7f2c7dec861501fa71650f58fd2caacf.png)
;
-
![f(x)<f(y)](/media/m/3/9/3/3936e87ef5d871fcfd52af9ba2644698.png)
for all
![1\le x<y](/media/m/0/e/9/0e9ba819a0d83afb030ade73379683de.png)
.
%V0
Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \longrightarrow \mathbb{R}^+$ that satisfy the following conditions:
- $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$;
- $f(x)<f(y)$ for all $1\le x<y$.
Izvor: Međunarodna matematička olimpijada, shortlist 2003