Question 4
In a contest each participant drew a $99 \times 100$ grid board and wrote a $1$ or a $-1$ in each cell, as they chose. Next, each participant wrote beside each row the result of multiplying the $100$ numbers of that row, and below each column the result of multiplying the $99$ numbers of that column. Finally, they added the $99$ row results and the $100$ column results and obtained their final number.
If in this contest all participants obtained different final numbers, determine the maximum possible number of participants and, for the maximum number found, give the final numbers of all the participants.
If in this contest all participants obtained different final numbers, determine the maximum possible number of participants and, for the maximum number found, give the final numbers of all the participants.