2021 USAJMO Problems
Contents
Day 1
For any geometry problem whose statement begins with an asterisk , the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 1
Let denote the set of positive integers. Find all functions such that for positive integers and
Problem 2
Rectangles and are erected outside an acute triangle Suppose thatProve that lines and are concurrent.
Problem 3
An equilateral triangle of side length is given. Suppose that equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside , such that each unit equilateral triangle has sides parallel to , but with opposite orientation. (An example with is drawn below.) Prove that
Day 2
Problem 4
Carina has three pins, labeled , and , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance away. What is the least number of moves that Carina can make in order for triangle to have area 2021? (A lattice point is a point in the coordinate plane where and are both integers, not necessarily positive.) Carina has three pins, labeled , and , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance away. What is the least number of moves that Carina can make in order for triangle to have area 2021?
(A lattice point is a point in the coordinate plane where and are both integers, not necessarily positive.)
Problem 5
A finite set of positive integers has the property that, for each and each positive integer divisor of , there exists a unique element satisfying . (The elements and could be equal.) Given this information, find all possible values for the number of elements of .
Problem 6
Let be an integer. Find all positive real solutions to the following system of equations:
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2021 USAJMO (Problems • Resources) | ||
Preceded by 2020 USOJMO |
Followed by 2022 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |