2016 USAJMO Problems
Contents
Day 1
Problem 1
The isosceles triangle , with , is inscribed in the circle . Let be a variable point on the arc that does not contain , and let and denote the incenters of triangles and , respectively.
Prove that as varies, the circumcircle of triangle passes through a fixed point.
Problem 2
Prove that there exists a positive integer such that has six consecutive zeros in its decimal representation.
Problem 3
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 .
Day 2
Problem 4
Find, with proof, the least integer such that if any elements are removed from the set , one can still find distinct numbers among the remaining elements with sum .
Problem 5
Let be an acute triangle, with as its circumcenter. Point is the foot of the perpendicular from to line , and points and are the feet of the perpendiculars from to the lines and , respectively.
Given that prove that the points and are collinear.
Problem 6
Find all functions such that for all real numbers and ,
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2016 USAJMO (Problems • Resources) | ||
Preceded by 2015 USAJMO |
Followed by 2017 USAJMO | |
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