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 00:45, 27 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
,
Problem 5
This problem has not been edited in. If you know this problem, please help us out by adding it.
Problem 6
This problem has not been edited in. If you know this problem, please help us out by adding it.
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 |