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Omni-MATH / Let n be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses…

Problem

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problem

Let nn be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, 2n+12^{n+1} competitors, numbered 1 through 2n+12^{n+1} clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a 50%50 \% probability of victory. The tournament unfolds in n+1n+1 rounds. In each round, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players 1 and 2n2^{n} face each other in the last round? Express your answer in terms of nn.
Plain-text mathematical notation (without MathML)
Let n be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, 2^(n+1) competitors, numbered 1 through 2^(n+1) clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a 50% probability of victory. The tournament unfolds in n+1 rounds. In each round, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players 1 and 2^(n) face each other in the last round? Express your answer in terms of n.
Original LaTeX notation
Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle. They are equals in Pushover: whenever two of them face off, each has a $50 \%$ probability of victory. The tournament unfolds in $n+1$ rounds. In each round, the referee randomly chooses one of the surviving players, and the players pair off going clockwise, starting from the chosen one. Each pair faces off in Pushover, and the losers leave the circle. What is the probability that players 1 and $2^{n}$ face each other in the last round? Express your answer in terms of $n$.

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Source and history

task_id
c080c1e9-1af4-5be9-9c4d-cd80b25e27d5
task_key
test--c080c1e9-1af4-5be9-9c4d-cd80b25e27d5
task_revision_id
1
config / split
/ test

Official source

initial import