2001 IMO Shortlist Problems/A4
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Problem
Find all functions , satisfying
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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
. It's easy to check the condition
.