MathDB

Function Equation on Positive Rationals

Source: 1991 IrMO Paper 2 Problem 4

October 1, 2017

algebrafunctional equationfunctional equation in Q

Problem Statement

Let

\mathbb{P}

be the set of positive rational numbers and let

f:\mathbb{P}\to\mathbb{P}

be such that

f(x)+f\left(\frac{1}{x}\right)=1

and

f(2x)=2f(f(x))

for all

x\in\mathbb{P}

. Find, with proof, an explicit expression for

f(x)

for all

x\in \mathbb{P}