UA-UMO 2005 Problem 105

Find all functions $f: \mathbb{R}^{+}\rightarrow \mathbb{R}$ that satisfy:

$f(x)f(y)=f(xy)+2005 \left( \frac{1}{x}+\frac{1}{y}+2004 \right)$ for all $x,y>0.$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.