## Task A-2.7.

The rows of a $50 \times 50$ table are labeled with numbers $a_{1}, \ldots, a_{50}$, and the columns with numbers $b_{1}, \ldots, b_{50}$. These 100 numbers are all distinct, and exactly 50 of them are rational numbers. The table is filled such that for $i, j=1,2, \ldots, 50$, the number $a_{i}+b_{j}$ is written in the cell $(i, j)$. Determine the maximum possible number of rational numbers written in the cells of the table.
Scoring: A correct answer without explanation earns 1 point. A solution that contains only an example achieving 1250 rational numbers earns a maximum of 4 points. A solution that shows $N \leqslant 1250$ but does not include an example where this number is actually achieved earns a maximum of 6 points.

## SCHOOL/CITY COMPETITION IN MATHEMATICS  Additional notes for graders

## 3rd grade - high school - A variant

## January 25, 2018.