Difference between revisions of "2023 USAMO Problems"
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===Problem 3=== | ===Problem 3=== | ||
− | Consider an <math>n</math>-by-<math>n</math> board of unit squares for some odd positive integer <math>n</math>. We say that a collection <math>C</math> of identical dominoes is a maximal grid-aligned configuration on the board if <math>C</math> consists of <math>(n^2-1)/2</math> dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: <math>C</math> 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 <math>k(C)</math> be the number of distinct maximal grid-aligned configurations obtainable from <math>C</math> by repeatedly sliding dominoes. Find | + | Consider an <math>n</math>-by-<math>n</math> board of unit squares for some odd positive integer <math>n</math>. We say that a collection <math>C</math> of identical dominoes is a maximal grid-aligned configuration on the board if <math>C</math> consists of <math>(n^2-1)/2</math> dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: <math>C</math> 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 <math>k(C)</math> be the number of distinct maximal grid-aligned configurations obtainable from <math>C</math> by repeatedly sliding dominoes. Find all possible values of <math>k(C)</math> as a function of <math>n</math>. |
[[2023 USAMO Problems/Problem 3|Solution]] | [[2023 USAMO Problems/Problem 3|Solution]] | ||
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===Problem 6=== | ===Problem 6=== | ||
− | + | Let ABC be a triangle with incenter <math>I</math> and excenters <math>I_a</math>, <math>I_b</math>, <math>I_c</math> opposite <math>A</math>, <math>B</math>, and <math>C</math>, respectively. Given an arbitrary point <math>D</math> on the circumcircle of <math>\triangle ABC</math> that does not lie on any of the lines <math>II_a</math>, <math>I_bI_c</math>, or <math>BC</math>, suppose the circumcircles of <math>\triangle DII_a</math> and <math>\triangle DI_bI_c</math> intersect at two distinct points <math>D</math> and <math>F</math>. If <math>E</math> is the intersection of lines <math>DF</math> and <math>BC</math>, prove that <math>\angle BAD = \angle EAC</math>. | |
[[2023 USAMO Problems/Problem 6|Solution]] | [[2023 USAMO Problems/Problem 6|Solution]] | ||
− | {{USAMO newbox|year= 2023 |before=[[2022 USAMO]]|after=[[2024 USAMO]]}} | + | ==See Also== |
+ | {{USAMO newbox|year=2023|before=[[2022 USAMO Problems]]|after=[[2024 USAMO Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 22:57, 9 March 2024
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.