KMO 2019 Problem 5

Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$

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.