# 2001 IMO Shortlist Problems/A4

## Problem

Find all functions , satisfying

for all .

## Solution

Assume . Take . We get , so . This is a solution, so we can take it out of the way: assume .

. We either have or , so for every , . In particular, .

Assume . We get . This means that ( is defined because ). Assume now that and . We get , and after replacing everything we get , so . Assume now . From we get , and after applying again to we get . We can now see that combine to .

Let . and simply say that is a subgroup of .

Conversely, let be a subgroup of the multiplicative group . Take $f(x) = \left\{\begin{array}{c}f(1)x,\ x\in G \\ 0,\ x\not \in G\end{array}$ (Error compiling LaTeX. ! Missing \right. inserted.). It's easy to check the condition .