Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2019 USAMO Problems/Problem 1

Revision as of 22:45, 19 April 2019 by Superram (talk | contribs) (Created page with "==Problem 1== Let <math>\mathbb{N}</math> be the set of positive integers. A function <math>f:\mathbb{N}\to\mathbb{N}</math> satisfies the equation <cmath>\underbrace{f(f(\ldo...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem 1

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_USAMO_Problems/Problem_1&oldid=105401"