Difference between revisions of "2016 USAMO Problems"
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Prove that for any positive integer <math>k,</math> | Prove that for any positive integer <math>k,</math> | ||
| − | <cmath>\left(k^2\right)!\cdot\ | + | <cmath>\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}</cmath> |
is an integer. | is an integer. | ||
Revision as of 00:48, 27 April 2016
Contents
[hide]Day 1
Problem 1
Let 
be a sequence of mutually distinct nonempty subsets of a set 
. Any two sets 
and 
are disjoint and their union is not the whole set , that is, 
and 
, for all 
. Find the smallest possible number of elements in .
Problem 2
This problem has not been edited in. Help us out by adding it.
Prove that for any positive integer 
![\[\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}\]](http://latex.artofproblemsolving.com/4/e/d/4ed23f598fe1573fe79161056cfcad1ffcb0f1fd.png)
is an integer.
Problem 3
This problem has not been edited in. Help us out by adding it.
Day 2
Problem 4
Find all functions 
such that for all real numbers 
and 
, ![\[(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.\]](http://latex.artofproblemsolving.com/2/d/d/2ddcde1699356cbca5812e17244428064eb74546.png)
Problem 5
This problem has not been edited in. Help us out by adding it.
Problem 6
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These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
| 2016 USAMO (Problems • Resources) | ||
| Preceded by 2015 USAMO |
Followed by 2017 USAMO | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
