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Omni-MATH / Let k be a positive real. A and B play the following game: at the start, there are 80 zeroes arrange around a circle. Each turn, A increases some of these 80 numbers, such that the…

Problem

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problem

Let

k

be a positive real.

A

and

B

play the following game: at the start, there are

80

zeroes arrange around a circle. Each turn,

A

increases some of these

80

numbers, such that the total sum added is

1

. Next,

B

selects ten consecutive numbers with the largest sum, and reduces them all to

0

A

then wins the game if he/she can ensure that at least one of the number is

\geq k

at some finite point of time. Determine all

k

such that

A

can always win the game.

Plain-text mathematical notation (without MathML)

Let k be a positive real. A and B play the following game: at the start, there are 80 zeroes arrange around a circle. Each turn, A increases some of these 80 numbers, such that the total sum added is 1. Next, B selects ten consecutive numbers with the largest sum, and reduces them all to 0. A then wins the game if he/she can ensure that at least one of the number is ≥k at some finite point of time. 

Determine all k such that A can always win the game.

Original LaTeX notation

Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest sum, and reduces them all to $0$. $A$ then wins the game if he/she can ensure that at least one of the number is $\geq k$ at some finite point of time. 

Determine all $k$ such that $A$ can always win the game.

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task_id: b62063a6-51bc-5db7-a6d6-3086660ec1e8
task_key: test--b62063a6-51bc-5db7-a6d6-3086660ec1e8
task_revision_id: 4
config / split: / test

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