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AIME 2024 / AIME 2024 train af5bf059-eeab-5ff3-92fa-75bd78c29ce3

Problem

Answer published by the source. Consult the official source to check your work against its answer.

Problem

Alice and Bob play the following game. A stack of

n

tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either

1

token or

4

tokens from the stack. Whoever removes the last token wins. Find the number of positive integers

n

less than or equal to

2024

for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alice's play.

Plain-text mathematical notation (without MathML)

Alice and Bob play the following game. A stack of n tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either 1 token or 4 tokens from the stack. Whoever removes the last token wins. Find the number of positive integers n less than or equal to 2024 for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alice's play.

Original LaTeX notation

Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for which there exists a strategy for Bob that guarantees that Bob will win the game regardless of Alice's play.

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Source and history

task_id: af5bf059-eeab-5ff3-92fa-75bd78c29ce3
task_key: default--train--af5bf059-eeab-5ff3-92fa-75bd78c29ce3
task_revision_id: 2
config / split: default / train

Official source

initial import