CZ-SK-MO 2004 Problem 6

Find all functions $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that for all positive real numbers $x,y$ , $x^2[f(x)+f(y)]=(x+y)f(yf(x)).$

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.