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IMO ShortList 2003, algebra problem 5

Source: IMO ShortList 2003, algebra problem 5

October 4, 2004
functionalgebrafunctional equationIMO Shortlist

Problem Statement

Let R+\mathbb{R}^+ be the set of all positive real numbers. Find all functions f:R+→R+f: \mathbb{R}^+ \to \mathbb{R}^+ that satisfy the following conditions:
- f(xyz)+f(x)+f(y)+f(z)=f(xy)f(yz)f(zx)f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx}) for all x,y,z∈R+x,y,z\in\mathbb{R}^+;
- f(x)<f(y)f(x)<f(y) for all 1≤x<y1\le x<y.
Proposed by Hojoo Lee, Korea
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