VWO 2006 Problem 4

Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that

$f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}.$

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.