BG-MO 1968 Problem 2

Find all functions $f:\mathbb R \to \mathbb R$ such that $xf(y)+yf(x)=(x+y)f(x)f(y)$ for all reals $x$ and $y$ .

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.