Difference between revisions of "2016 USAMO Problems"
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===Problem 2=== | ===Problem 2=== | ||
{{problem}} | {{problem}} | ||
| + | Prove that for any positive integer <math>k,</math> | ||
| + | <cmath>\left(k^2\right)!\cdot\product_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}</cmath> | ||
| + | is an integer. | ||
| + | |||
===Problem 3=== | ===Problem 3=== | ||
{{problem}} | {{problem}} | ||
Revision as of 23:45, 26 April 2016
Contents
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. If you know this problem, please help us out by adding it.
Prove that for any positive integer 
\[\left(k^2\right)!\cdot\product_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}\] (Error compiling LaTeX. Unknown error_msg)
is an integer.
Problem 3
This problem has not been edited in. If you know this problem, please 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. If you know this problem, please help us out by adding it.
Problem 6
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The problems on this page are copyrighted by the Mathematical Association of America's 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 | ||
