MathDB

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.

Problem Statement