2024 USAJMO Problems
Contents
Day 1
Problem 1
Let be a cyclic quadrilateral with and . Points and are selected on segment such that . Points and are selected on segment such that . Prove that is a cyclic quadrilateral.
Problem 2
Let and be positive integers. Let be the set of integer points with and . A configuration of rectangles is called happy if each point in is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Problem 3
Let be the sequence defined by for each integer . Suppose that is prime and is a positive integer. Prove that some term of the sequence is divisible by .
Day 2
Problem 4
Let be an integer. Rowan and Colin play a game on an grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is if:
- no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and
- no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring;
In terms of , how many orderly colorings are there?
Problem 5
Find all functions that satisfy for all .
Problem 6
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 .
See also
2024 USAJMO (Problems • Resources) | ||
Preceded by 2023 USAJMO Problems |
Followed by 2025 USAJMO Problems | |
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.