CN-TST 2019 Problem 6

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.