Question 236, Oleg wrote a real number above each column square (outside the table) and to the left of each row square (outside the table) on a blank $50 \times 50$ grid. It is known that all 100 real numbers he wrote are distinct, and there are exactly 50 rational and 50 irrational numbers among them. Then, Oleg wrote the sum of the number to the left of the row and the number above the column in each cell (creating an addition table). How many rational numbers can appear in the grid at most?
Question 236, Solution: Suppose Oleg wrote $\mathrm{x}$ rational numbers and $50-\mathrm{x}$ irrational numbers on the left side of each row, then he wrote $\mathrm{x}$ irrational numbers and $50-\mathrm{x}$ rational numbers above each column. Note that the sum of a rational number and an irrational number is always irrational, so the number of irrational numbers in the grid is at least $x^{2}+(50-x)^{2} \geq 2 \times 25^{2}=1250$. Therefore, the number of rational numbers in the grid does not exceed $50^{2}-1250=1250$.

On the other hand, if we sequentially write $1, 2, \ldots, 25, 1+\sqrt{2}, 2+\sqrt{2}, \ldots, 25+\sqrt{2}$ on the left side of each row; and $-1, -2, \ldots, -25, 1-\sqrt{2}, 2-\sqrt{2}, \ldots, 25-\sqrt{2}$ above each column, then the grid will contain exactly 1250 rational numbers.