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

Does there exist a sequence $ \{b_{i}\}_{i=1}^\infty$ of positive real numbers such that for each natural $ m$: \[ b_{m}+b_{2m}+b_{3m}+\dots=\frac1m\]