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Omni-MATH / Amy and Bob play the game. At the beginning, Amy writes down a positive integer…

Problem

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problem

Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number nn on the blackboard with a number of the form n−a2n-a^2, where aa is a positive integer. On any move of hers, Amy replaces the number nn on the blackboard with a number of the form nkn^k, where kk is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob’s win?
Plain-text mathematical notation (without MathML)
Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number n on the blackboard with a number of the form n−a², where a is a positive integer. On any move of hers, Amy replaces the number n on the blackboard with a number of the form n^(k), where k is a positive integer. Bob wins if the number on the board becomes zero.
Can Amy prevent Bob’s win?
Original LaTeX notation
Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero.
Can Amy prevent Bob’s win?

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Source and history

task_id
c8398745-b092-572b-af05-7afe9534067c
task_key
test--c8398745-b092-572b-af05-7afe9534067c
task_revision_id
1
config / split
/ test

Official source

initial import