Difference between revisions of "2023 USAJMO Problems"
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Two players, <math>B</math> and <math>R</math>, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with <math>B</math>. On <math>B</math>'s turn, <math>B</math> selects one white unit square and colors it blue. On <math>R</math>'s turn, <math>R</math> selects two white unit squares and colors them red. The players alternate until <math>B</math> decides to end the game. At this point, <math>B</math> gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score <math>B</math> can guarantee? | Two players, <math>B</math> and <math>R</math>, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with <math>B</math>. On <math>B</math>'s turn, <math>B</math> selects one white unit square and colors it blue. On <math>R</math>'s turn, <math>R</math> selects two white unit squares and colors them red. The players alternate until <math>B</math> decides to end the game. At this point, <math>B</math> gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score <math>B</math> can guarantee? | ||
− | (A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes. | + | (A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.) |
[[2023 USAJMO Problems/Problem 4|Solution]] | [[2023 USAJMO Problems/Problem 4|Solution]] | ||
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[[2023 USAJMO Problems/Problem 6|Solution]] | [[2023 USAJMO Problems/Problem 6|Solution]] | ||
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+ | == See also == | ||
{{USAJMO box|year=2023|before=[[2022 USAJMO Problems]]|after=[[2024 USAJMO Problems]]}} | {{USAJMO box|year=2023|before=[[2022 USAJMO Problems]]|after=[[2024 USAJMO Problems]]}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:38, 19 March 2024
Contents
Day 1
Problem 1
Find all triples of positive integers that satisfy the equation
Problem 2
In an acute triangle , let
be the midpoint of
. Let
be the foot of the perpendicular from
to
. Suppose that the circumcircle of triangle
intersects line
at two distinct points
and
. Let
be the midpoint of
. Prove that
.
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
Two players, and
, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with
. On
's turn,
selects one white unit square and colors it blue. On
's turn,
selects two white unit squares and colors them red. The players alternate until
decides to end the game. At this point,
gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score
can guarantee?
(A simple polygon is a polygon (not necessarily convex) that does not intersect itself and has no holes.)
Problem 5
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 6
Isosceles triangle , with
, is inscribed in circle
. Let
be an arbitrary point inside
such that
. Ray
intersects
again at
(other than
). Point
(other than
) is chosen on
such that
. Line
intersects rays
and
at points
and
, respectively. Prove that
.
See also
2023 USAJMO (Problems • Resources) | ||
Preceded by 2022 USAJMO Problems |
Followed by 2024 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.