Question 58
The rows of a $50 \times 50$ table are labelled with numbers $a_1, \dots, a_{50}$, while the columns are labelled with numbers $b_1, \dots, b_{50}$. These 100 numbers are mutually distinct, and exactly 50 of them are rational. The table is filled so that the number $a_i + b_j$ is written in the $(i, j)$ cell, for $i, j = 1, 2, \dots, 50$. Determine the largest possible number of rational numbers written in the table cells.