Difference between revisions of "2023 USAJMO"
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==Problem 4== | ==Problem 4== | ||
==Problem 5== | ==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. | ||
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| + | 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. | ||
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==Problem 6== | ==Problem 6== | ||
Revision as of 12:21, 24 March 2023
Do not post these problems until after March 21 and March 22, 2023.
Day 1
Problem 1
Problem 2
Problem 3
Day 2
Problem 4
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.
