2007 IMO Shortlist Problems/A4
Revision as of 08:25, 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 , .