IR-MO 2001 Problem 1

Find all functions $f: \mathbb Q\longrightarrow\mathbb Q$ such that: $f(x)+f(\frac1x)=1$ $2f(f(x))=f(2x)$

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.