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.