Difference between revisions of "2019 USAMO Problems/Problem 1"
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Proof: | Proof: | ||
− | + | <math>f^r(b)=a=f^r(a)</math> which implies <math>b=a</math> by applying injectivity of <math>f</math> <math>r</math> times. | |
Line 50: | Line 50: | ||
Since <math>f^2(m)\cdot f^{f(m)}(m)</math>, either | Since <math>f^2(m)\cdot f^{f(m)}(m)</math>, either | ||
− | <math>(1) f^2(m)=k<m</math>, contradicted by Lemma 1 since <math>f(k)=k</math>. | + | <math>(1) f^2(m)=k<m</math>, contradicted by Lemma 1 since <math>k</math> is odd and <math>f(k)=k</math>. |
− | <math>(2) f^{f(m)}(m)=k<m</math>, also contradicted by Lemma 1. | + | <math>(2) f^{f(m)}(m)=k<m</math>, also contradicted by Lemma 1 by similar logic. |
<math>(3) f^2(m)=m</math> and <math>f^{f(m)}(m)=m</math>, which implies that <math>f(m)=m</math> by Lemma 2. | <math>(3) f^2(m)=m</math> and <math>f^{f(m)}(m)=m</math>, which implies that <math>f(m)=m</math> by Lemma 2. |
Revision as of 01:18, 25 April 2019
Problem
Let be the set of positive integers. A function satisfies the equation for all positive integers . Given this information, determine all possible values of .
Solution
Let denote the result when is applied to times. If , then and
since .
Therefore, is injective.
Lemma 1: If and , then .
Proof:
which implies by applying injectivity of times.
Lemma 2: If , and is odd, then .
Proof:
Let . Since , . So, . .
Since ,
This proves Lemma 2.
I claim that for all odd .
Otherwise, let be the least counterexample.
Since , either
, contradicted by Lemma 1 since is odd and .
, also contradicted by Lemma 1 by similar logic.
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.
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
See also
2019 USAMO (Problems • Resources) | ||
First Problem | Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |