2016 USAMO Problems
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. If you know this problem, please help us out by adding it. Prove that for any positive integer is an integer.
Problem 3
This problem has not been edited in. If you know this problem, please help us out by adding it. Let be an acute triangle, and let and denote its -excenter, -excenter, and circumcenter, respectively. Points and are selected on such that and Similarly, points and are selected on such that and
Lines and meet at Prove that and are perpendicular.
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. An equilateral pentagon is inscribed in triangle such that and Let be the intersection of lines and Denote by the angle bisector of
Prove that is parallel to where is the circumcenter of triangle and is the incenter of triangle
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 |