2022 USAMO Problems
Contents
Day 1
Problem 1
Let and be positive integers. The cells of an grid are colored amber and bronze such that there are at least amber cells and at least bronze cells. Prove that it is possible to choose amber cells and bronze cells such that no two of the chosen cells lie in the same row or column.
Problem 2
Let and be fixed integers, and . Given are identical black rods and identical white rods, each of side length 1.
We assemble a regular -gon using these rods so that parallel sides are the same color. Then, a convex -gon is formed by translating the black rods, and a convex -gon is formed by translating the white rods. An example of one way of doing the assembly when and is shown below, as well as the resulting polygons and .
Prove that the difference of the areas of and depends only on the numbers and , and not on how the -gon was assembled.
Problem 3
Let be the set of all positive real numbers. Find all functions such that for all we have
Day 2
Problem 4
Find all pairs of primes for which and are both perfect squares.
Problem 5
A function is essentially increasing if holds whenever are real numbers such that and .
Find the smallest integer such that for any 2022 real numbers there exist essentially increasing functions such that
Problem 6
There are 2022 users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.)
Starting now, Mathbook will only allow a new friendship to be formed between two users if they have at least two friends in common. What is the minimum number of friendships that must already exist so that every user could eventually become friends with every other user?
See Also
2022 USAMO (Problems • Resources) | ||
Preceded by 2021 USAMO Problems |
Followed by 2023 USAMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.