2019 USAMO Problems/Problem 1: Difference between revisions

Revision as of 15:13, 5 May 2019

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

$f^m(k)=k$

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

$m=f(k)=k$

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.

@@ Line 15: / Line 15: @@
 <math>\implies p=q</math> since <math>p,q>0</math>.
-Therefore, <math>f</math> is injective.
+Therefore, <math>f</math> is injective. It follows that <math>f^r</math> is also injective.
@@ Line 23: / Line 23: @@
 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.
+<math>f^r(b)=a=f^r(a)</math> which implies <math>b=a</math> by injectivity of <math>f^r</math>.

2019 USAMO (Problems • Resources)
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2019 USAMO Problems/Problem 1: Difference between revisions

Revision as of 15:13, 5 May 2019

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