2016 USAJMO Problems

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.

Solution

Problem 2

Prove that there exists a positive integer such that has six consecutive zeros in its decimal representation.

Solution

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 .

Solution

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 .

Solution

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.

Solution

Problem 6

Find all functions such that for all real numbers and ,

Solution

2016 USAJMO (Problems • Resources)
Preceded by 2015 USAJMO Problems	Followed by 2017 USAJMO Problems
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.