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Functional equation on the real numbers between 0 and 1

Source: South African MO 2009 Q6

May 26, 2012

functionalgebralinear equationabsolute valuealgebra unsolved

Problem Statement

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}