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
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
,
Problem 5
This problem has not been edited in. Help us out by adding it.
Problem 6
This problem has not been edited in. Help us out by adding it.
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 |