# 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>