MathDB

IMO ShortList 2008, Number Theory problem 5

Source: IMO ShortList 2008, Number Theory problem 5, German TST 6, P2, 2009

July 9, 2009

functionnumber theorymodular arithmeticdivisorIMO Shortlistfunctional equation

Problem Statement

For every

n\in\mathbb{N}

let

d(n)

denote the number of (positive) divisors of

n

. Find all functions

f: \mathbb{N}\to\mathbb{N}

with the following properties: [*] d\left(f(x)\right) \equal{} x for all

x\in\mathbb{N}

. [*]

f(xy)

divides (x \minus{} 1)y^{xy \minus{} 1}f(x) for all

x

y\in\mathbb{N}

. Proposed by Bruno Le Floch, France