2023 USAMO Problems
Contents
Day 1
Problem 1
In an acute triangle , let be the midpoint of . Let be the foot of the perpendicular from to . Suppose the circumcircle of triangle intersects line at two distinct points and . Let be the midpoint of . Prove that .
Problem 2
Let be the set of positive real numbers. Find all functions such that, for all ,
Problem 3
Consider an -by- board of unit squares for some odd positive integer . We say that a collection of identical dominoes is a maximal grid-aligned configuration on the board if consists of dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let be the number of distinct maximal grid-aligned configurations obtainable from by repeatedly sliding dominoes. Find the maximum value of as a function of .
Day 2
Problem 4
Problem 5
Let be an integer. We say that an arrangement of the numbers , , , in a table is row-valid if the numbers in each row can be permuted to form an arithmetic progression, and column-valid if the numbers in each column can be permuted to form an arithmetic progression. For what values of is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?
Problem 6
2023 USAMO (Problems • Resources) | ||
Preceded by 2022 USAMO |
Followed by 2024 USAMO | |
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.