Difference between revisions of "2021 USAJMO Problems"
Line 22: | Line 22: | ||
==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== | ||
− | + | Carina has three pins, labeled <math>A, B</math>, and <math>C</math>, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance <math>1</math> away. What is the least number of moves that Carina can make in order for triangle <math>ABC</math> to have area 2021? | |
− | (A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.) | + | (A lattice point is a point <math>(x, y)</math> in the coordinate plane where <math>x</math> and <math>y</math> are both integers, not necessarily positive.) |
[[2021 USAJMO Problems/Problem 4|Solution]] | [[2021 USAJMO Problems/Problem 4|Solution]] |
Revision as of 14:24, 16 June 2021
Contents
[hide]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.)
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:
2021 USAJMO (Problems • Resources) | ||
Preceded by 2020 USOJMO |
Followed by 2022 USAJMO | |
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.