Question 31
Initially, numbers $2, 3, \ldots, 99$ are written on the board. In each step, one of the following operations would be performed;
i. We choose an integer $i$, $2 \le i \le 89$ and if numbers $i$ and $i+10$ are both on the board, we remove both of them from the board;
ii. We choose an integer $i$, $2 \le i \le 98$, $10 \nmid i$ and if the numbers $i$ and $i+1$ are both on the board, we remove both of them from the board;
Having performed these operations, determine the maximum number of numbers that can be removed from the board?
i. We choose an integer $i$, $2 \le i \le 89$ and if numbers $i$ and $i+10$ are both on the board, we remove both of them from the board;
ii. We choose an integer $i$, $2 \le i \le 98$, $10 \nmid i$ and if the numbers $i$ and $i+1$ are both on the board, we remove both of them from the board;
Having performed these operations, determine the maximum number of numbers that can be removed from the board?