Difference between revisions of "2019 USAMO Problems/Problem 1"
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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(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}</cmath>for all positive integers <math>n</math>. Given this information, determine all possible values of <math>f(1000)</math>. | 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(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}</cmath>for all positive integers <math>n</math>. Given this information, determine all possible values of <math>f(1000)</math>. | ||
Revision as of 23:06, 19 April 2019
Problem
Let 
be the set of positive integers. A function 
satisfies the equation ![\[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]](http://latex.artofproblemsolving.com/2/e/7/2e73ff2f75b103e9d6a4004a42e4ed7f4ee64a67.png)
for all positive integers 
. Given this information, determine all possible values of 
.
Solution
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 | ||
