Difference between revisions of "2021 USAJMO Problems"
Line 9: | Line 9: | ||
[[2021 USAJMO Problems/Problem 2|Solution]] | [[2021 USAJMO Problems/Problem 2|Solution]] | ||
===Problem 3=== | ===Problem 3=== | ||
− | An equilateral triangle <math>\Delta</math> of side length <math>L>0</math> is given. Suppose that <math>n</math> equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside <math>\Delta</math>, such that each unit equilateral triangle has sides parallel to <math>\Delta</math>, but with opposite orientation. (An example with <math>n=2</math> is drawn below.) | + | An equilateral triangle <math>\Delta</math> of side length <math>L>0</math> is given. Suppose that <math>n</math> equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside <math>\Delta</math>, such that each unit equilateral triangle has sides parallel to <math>\Delta</math>, but with opposite orientation. (An example with <math>n=2</math> is drawn below.) |
+ | <asy> | ||
+ | draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); | ||
+ | filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); | ||
+ | filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); | ||
+ | </asy> | ||
+ | Prove that<cmath>n \leq \frac{2}{3} L^{2}.</cmath> | ||
+ | |||
[[2021 USAJMO Problems/Problem 3|Solution]] | [[2021 USAJMO Problems/Problem 3|Solution]] | ||
==Day 2== | ==Day 2== |
Revision as of 23:43, 15 April 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 Solution
Problem 2
Rectangles and are erected outside an acute triangle Suppose thatProve that lines and are concurrent. Solution
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.) Solution
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 . Solution
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 |
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 |