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2019 USAMO Problems/Problem 1

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 $f^{r-1}(x)$, where $f^1(x)=f(x)$. $\hfill \break \hfill \break$ 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. It follows that $f^r$ is also 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 injectivity of $f^r$.


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

$\implies f^m(k)=k$

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

$\implies f(k)=k$

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 $f^2(k)=k$.

$(2) f^{f(m)}(m)=k<m$, 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 (Problems • Resources)
First Problem Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
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