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 that
Prove 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.
2020 USOJMO (Problems • Resources) | ||
Preceded by 2019 USAJMO |
Followed by 2021 USAJMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |