Question 122
Oleg has labelled all the columns and all the rows of a $50 \times 50$ table with $100$ distinct numbers $a_1, \dots, a_{50}$ and $b_1, \dots, b_{50}$, respectively; exactly $50$ of these numbers are rational. Then he has placed into each cell $(i, j)$ the number $a_i + b_j$. Find the greatest possible number of rational numbers placed into the cells.