2019 USAMO Problems/Problem 1
Let be the set of positive integers. A function satisfies the equation for all positive integers . Given this information, determine all possible values of .
Let denote the result when is applied to times. If , then and
Therefore, is injective.
Lemma 1: If and , then .
Otherwise, set , , and to a counterexample of the lemma, such that is minimized. By injectivity, , so . If , then and , a counterexample that contradicts our assumption that is minimized, proving Lemma 1.
Lemma 2: If , and is odd, then .
Let . Since , . So, . .
This proves Lemma 2.
I claim that for all odd .
Otherwise, let be the least counterexample.
Since , either
, contradicted by Lemma 1 since .
, also contradicted by Lemma 1.
and , which implies that by Lemma 2. This proves the claim.
By injectivity, is not odd. I will prove that can be any even number, . Let , and for all other . If is equal to neither nor , then . This satisfies the given property.
If is equal to or , then since is even and . This satisfies the given property.
|2019 USAMO (Problems • Resources)|
|First Problem||Followed by|
|1 • 2 • 3 • 4 • 5 • 6|
|All USAMO Problems and Solutions|