NMC 2021 Problem 2

Find all functions $f:R->R$ satisfying that for every $x$ (real number): $f(x)(1+|f(x)|)\geq x \geq f(x(1+|x|))$

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.