IR-MO 2001

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 $m$ : $$b_{m}+b_{2m}+b_{3m}+\dots=\frac1m$$