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AIME 2024 / AIME 2024 train 8881eca3-7b10-561a-9739-70ea81085358

Problem

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Problem

Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is

\tfrac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

Plain-text mathematical notation (without MathML)

Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is (m)/(n), where m and n are relatively prime positive integers. What is m+n?

Original LaTeX notation

Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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Source and history

task_id: 8881eca3-7b10-561a-9739-70ea81085358
task_key: default--train--8881eca3-7b10-561a-9739-70ea81085358
task_revision_id: 2
config / split: default / train

Official source

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