2007 IMO Shortlist Problems/A4
Revision as of 09:25, 4 September 2008 by Boy Soprano II (talk | contribs) (arrgh, didn't have time to finish writing this solution)
Problem
(Thailand)
Find all functions such that
for all
. (Symbol
denotes the set of all positive real numbers [sic].)
Solution
We will show that is the unique solution to this equation. To this end, let
.
Lemma 1. The function is injective.
Proof. Suppose . Then
as desired.
Lemma 2. If , then
.
Proof. Set ,
.
Lemma 3. For all ,
.