2024 USAMO Problems
Contents
Day 1
Problem 1
Find all integers such that the following property holds: if we list the divisors of in increasing order as , then we have
Problem 2
Let be finite sets of integers whose intersection is not empty. For each non-empty , the size of the intersection of the sets in is a multiple of the number of sets in . What is the least possible number of elements that are in at least 50 sets?
Problem 3
Let be a positive integer. A triangulation of a polygon is -balanced if its triangles can be colored with colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the colors. Find all positive integers for which there exists an -balanced triangulation of a regular -gon.
Note: A triangulation of a convex polygon with sides is any partitioning of into triangles by diagonals of that do not intersect in the polygon's interior.
Day 2
Problem 4
Let and be positive integers. A circular necklace contains beads, each either red or blue. It turned out that no matter how the necklace was cut into blocks of consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair .
Problem 5
Point is selected inside acute triangle so that and . Point is chosen on ray so that . Let be the midpoint of . Show that line is tangent to the circumcircle of triangle .
Problem 6
Let be an integer and let . A collection of (not necessarily distinct) subsets of is called -large if for all . Find, in terms of and , the largest real number such that the inequality holds for all positive integers , all nonnegative real numbers , and all -large collections of subsets of .
Note: For a finite set denotes the number of elements in .
See Also
2024 USAMO (Problems • Resources) | ||
Preceded by 2023 USAMO Problems |
Followed by 2025 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.