2007 IMO Shortlist Problems/A4
(Thailand) Find all functions such that for all . (Symbol denotes the set of all positive real numbers [sic].)
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.
- 2007 IMO Shortlist Problems
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