Question 11
There are 2025 white balls and 2025 black balls arranged in a row. A balanced segment is a contiguous nonempty segment of balls that contains the same number of white balls as black balls. A balanced removal is the operation of selecting a balanced segment S, removing all the balls in S, and shifting left the balls that were to the right of S so that the remaining balls form a contiguous row.
Determine the smallest positive integer $N$ satisfying the following property: for every configuration of balls and for every integer $k$ satisfying $0 \leq k \leq 2025$, there exists a sequence of at most $N$ balanced removals after which there are exactly $2k$ remaining balls.
Determine the smallest positive integer $N$ satisfying the following property: for every configuration of balls and for every integer $k$ satisfying $0 \leq k \leq 2025$, there exists a sequence of at most $N$ balanced removals after which there are exactly $2k$ remaining balls.