MathDB

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})

\frac{1}{2}\le x<1

.
Prove that
(a)

f(x)+f(1-x)\ge \frac{2}{3}

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}

6