Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2023 USAJMO"

(→‎Problem 4)
 
(11 intermediate revisions by the same user not shown)
Line 1: Line 1:
−
Do not post these problems until after March 21 and March 22, 2023.
+
The 14th USAJMO was held on March 22 and March 23, 2023. The first link will contain the full set of test problems. The rest will contain each individual problem and its solution.
−
==Day 1==
 
  
−
==Problem 1==
+
[[2023 USAJMO Problems]]
−
==Problem 2==
+
* [[2023 USAJMO Problems/Problem 1]]
−
==Problem 3==
+
* [[2023 USAJMO Problems/Problem 2]]
 +
* [[2023 USAJMO Problems/Problem 3]]
 +
* [[2023 USAJMO Problems/Problem 4]]
 +
* [[2023 USAJMO Problems/Problem 5]]
 +
* [[2023 USAJMO Problems/Problem 6]]
  
−
==Day 2==
+
== See Also ==
 +
* [[Mathematics competitions]]
 +
* [[Mathematics competition resources]]
 +
* [[Math books]]
 +
* [[USAJMO]]
  
−
==Problem 4==
+
{{USAJMO newbox|year= 2023 |before=[[2022 USAJMO]]|after=[[2024 USAJMO]]}}
−
(David Torres) 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.
 
−
 
 
−
==Problem 5==
 
−
A positive integer <math>a</math> 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 <math>n</math> on the board with <math>n+a</math>, and on Bob's turn he must replace some even integer <math>n</math> on the board with <math>n/2</math>. 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 <math>a</math> and these integers on the board, the game is guaranteed to end regardless of Alice's or Bob's moves.
 
−
 
 
−
==Problem 6==
 

Latest revision as of 12:33, 24 March 2023

The 14th USAJMO was held on March 22 and March 23, 2023. The first link will contain the full set of test problems. The rest will contain each individual problem and its solution.

2023 USAJMO Problems

See Also

2023 USAJMO (Problems • Resources)
Preceded by
2022 USAJMO 		Followed by
2024 USAJMO
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2023_USAJMO&oldid=191256"