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 all possible values of as a function of .
Day 2
Problem 4
A positive integer is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer on the board with , and on Bob's turn he must replace some even integer on the board with . Alice goes first and they alternate turns. If on his turn Bob has no valid moves, the game ends.
After analyzing the integers on the board, Bob realizes that, regardless of what moves Alice makes, he will be able to force the game to end eventually. Show that, in fact, for this value of and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
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
Let ABC be a triangle with incenter and excenters , , opposite , , and , respectively. Given an arbitrary point on the circumcircle of that does not lie on any of the lines , , or , suppose the circumcircles of and intersect at two distinct points and . If is the intersection of lines and , prove that .
See Also
2023 USAMO (Problems • Resources) | ||
Preceded by 2022 USAMO Problems |
Followed by 2024 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.