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2019 USAMO Problems/Problem 1

Revision as of 00:24, 25 April 2019 by Blizzardwizard (talk | contribs) (Solution)

Problem

Let $\mathbb{N}$ be the set of positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the equation \[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]for all positive integers $n$. Given this information, determine all possible values of $f(1000)$.

Solution

Let $f^r(x)$ denote the result when $f$ is applied to $x$ $r$ times. $\hfill \break \hfill \break$ If $f(p)=f(q)$, then $f^2(p)=f^2(q)$ and $f^{f(p)}(p)=f^{f(q)}(q)$

$\implies p^2=f^2(p)\cdot f^{f(p)}(p)=f^2(q)\cdot f^{f(q)}(q)=q^2$

$\implies p=\pm q$

$\implies p=q$ since $p,q>0$.

Therefore, $f$ is injective.


Lemma 1: If $f^r(b)=a$ and $f(a)=a$, then $b=a$.


Proof:

$f^r(b)=a=f^r(a)$ which implies $b=a$ by applying injectivity of $f$ $r$ times.


Lemma 2: If $f^2(m)=f^{f(m)}(m)=m$, and $m$ is odd, then $f(m)=m$.


Proof:

Let $f(m)=k$. Since $f^2(m)=m$, $f(k)=m$. So, $f^2(k)=k$. $\newline f^2(k)\cdot f^{f(k)}(k)=k^2$.

Since $k\neq0$, $f^{f(k)}(k)=k$

$f^m(k)=k$

$f^{gcd(m, 2)}=k$

$m=f(k)=k$

This proves Lemma 2.


I claim that $f(m)=m$ for all odd $m$.

Otherwise, let $m$ be the least counterexample.

Since $f^2(m)\cdot f^{f(m)}(m)=m^2$, either

$(1) f^2(m)=k<m$, contradicted by Lemma 1 since $k$ is odd and $f(k)=k$.

$(2) f^{f(m)}(m)=k<m$, also contradicted by Lemma 1 by similar logic.

$(3) f^2(m)=m$ and $f^{f(m)}(m)=m$, which implies that $f(m)=m$ by Lemma 2. This proves the claim.


By injectivity, $f(1000)$ is not odd. I will prove that $f(1000)$ can be any even number, $x$. Let $f(1000)=x, f(x)=1000$, and $f(k)=k$ for all other $k$. If $n$ is equal to neither $1000$ nor $x$, then $f^2(n)\cdot f^{f(n)}(n)=n\cdot n=n^2$. This satisfies the given property.

If $n$ is equal to $1000$ or $x$, then $f^2(n)\cdot f^{f(n)}(n)=n\cdot n=n^2$ since $f(n)$ is even and $f^2(n)=n$. This satisfies the given property.



The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

See also

2019 USAMO (ProblemsResources)
First Problem Followed by
Problem 2
1 2 3 4 5 6
All USAMO Problems and Solutions
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