Podlipsky o.k.

Oleg drew an empty $50 \times 50$ table and wrote a non-zero number above each column and to the left of each row. It turned out that all 100 written numbers are distinct, with 50 of them being rational and the other 50 being irrational. Then, in each cell of the table, he wrote the product of the numbers written next to its row and its column ("multiplication table"). What is the maximum number of products in this table that could be rational numbers?
Evaluation. The fact that there are no more than 1250 rational numbers in the table is proven in the same way as in problem 66016 (since the product of a non-zero rational and an irrational number is irrational).

Example, when there are exactly 1250 rational numbers. Place along the left side the numbers $1,2, \ldots, 24,25$,

$\sqrt{2}, 2 \sqrt{2}, \ldots, 25 \sqrt{2}$, and along the top - the numbers $26,27, \ldots, 49,50,26 \sqrt{2}, 27 \sqrt{2}, \ldots, 50 \sqrt{2}$. Then only $2 \cdot 25^{2}=1250$ products of a rational and an irrational number will be irrational.

## Answer

1250 products.

Submit a comment