2007 IMO Shortlist Problems/A4
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
. The given condition then translates to
or
Lemma 1. The function is injective.
Proof. Suppose . Then
as desired.
Lemma 2. If , then
.
Proof. Set ,
.
Lemma 3. For all ,
.
Proof. Pick an arbitrary positive real . Then by Lemma 2,
Since
is injective, it follows that
. The lemma then follows.
Now, let be any positive real; pick some
. Then by Lemmata 3 and 2,
Hence
and
. Therefore the function
is the only possible solution to the problem. Since this function evidently satisfies the problem's condition, it is the unique solution, as desired.
Resources
- 2007 IMO Shortlist Problems
- <url>viewtopic.php?p=1165901#1165901 Discussion on AoPS/MathLinks</url>