Difference between revisions of "2019 USAMO Problems/Problem 1"
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Let <math>f^r(x)</math> denote the resulr when <math>f</math> is applied to <math>x</math> <math>r</math> times. | Let <math>f^r(x)</math> denote the resulr when <math>f</math> is applied to <math>x</math> <math>r</math> times. | ||
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If <math>f(p)=f(q)</math>, then <math>f^2(p)=f^2(q)</math> and <math>f^{f(p)}(p)=f^{f(q)}(q)\newline\implies p^2=f^2(p)\cdot f^{f(p)}(p)=f^2(q)\cdot f^{f(q)}(q)=q^2\newline\implies p=\pm q\newline\implies p=q</math> since <math>p,q>0</math>. | If <math>f(p)=f(q)</math>, then <math>f^2(p)=f^2(q)</math> and <math>f^{f(p)}(p)=f^{f(q)}(q)\newline\implies p^2=f^2(p)\cdot f^{f(p)}(p)=f^2(q)\cdot f^{f(q)}(q)=q^2\newline\implies p=\pm q\newline\implies p=q</math> since <math>p,q>0</math>. | ||
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Revision as of 21:48, 24 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 resulr when
is applied to
times.
If
, then
and
since
.
\newline
Therefore,
is injective.
Lemma 1: If
and
, then b=a.
Proof:
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
.
Proof:
Let
. Since
,
. So,
.
.
Since
, $\newlinef^{f(k)}(k)=k$ (Error compiling LaTeX. Unknown error_msg)
This proves Lemma 2.
I claim that
for all odd
.
Otherwise, let
be the least counterexample.
Since
, either
(1)
, contradicted by Lemma 1 since
.
(2)
, also contradicted by Lemma 1.
(3)
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 |