Question 21
Determine whether there exists a $2011 \times 2011$ matrix with the following properties:
- Every cell is filled with an integer from $1$ to $4021$.
- For every integer $i$ ($1 \leq i \leq 2011$), the $i$th row and the $i$th column together contain every integer from $1$ to $4021$.
- Every cell is filled with an integer from $1$ to $4021$.
- For every integer $i$ ($1 \leq i \leq 2011$), the $i$th row and the $i$th column together contain every integer from $1$ to $4021$.