ZA-SAMO 2009 Problem 6

Let $A$ denote the set of real numbers $x$ such that $0\le x<1$ . A function $f:A\to \mathbb{R}$ has the properties:

(i) $f(x)=2f(\frac{x}{2})$ for all $x\in A$ ;

(ii) $f(x)=1-f(x-\frac{1}{2})$ if $\frac{1}{2}\le x<1$ .

Prove that

(a) $f(x)+f(1-x)\ge \frac{2}{3}$ if $x$ is rational and $0<x<1$ .

(b) There are infinitely many odd positive integers $q$ such that equality holds in (a) when $x=\frac{1}{q}$ .