Question 65
Alex and Betty play a game with a row of consecutive $2022$ cells. At the start of the game, the name of Alex is written in the $1$st, $3$rd, ..., $2021$st cells from the left and the name of Betty is written in the $2$nd, $4$th, ..., $2022$nd cells from the left. Starting from Alex, two players do the following operation in turn.
Choose two cells with his or her own name which are not adjacent such that all cells between the two have the opponent's name. Then replace opponent's names with his or her own names for all cells between the chosen two.
The game ends when one player is not able to do the operation. Determine the largest positive integer $m$ satisfying the following condition.
No matter how Betty does the operations, Alex can play such that there are $m$ or more cells with Alex's name at the end of the game.
Choose two cells with his or her own name which are not adjacent such that all cells between the two have the opponent's name. Then replace opponent's names with his or her own names for all cells between the chosen two.
The game ends when one player is not able to do the operation. Determine the largest positive integer $m$ satisfying the following condition.
No matter how Betty does the operations, Alex can play such that there are $m$ or more cells with Alex's name at the end of the game.