IrMO 1991 Problem 9

Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$ . Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$ .