Problem 1118

Answer: No, they cannot.

Solution. For each cell of one of the main diagonals, we will consider a set of twenty-five cells: the cell itself and all cells that are in the same row or column with it. We will call such a set a "cross."

Consider all "crosses" formed by the cells of the selected main diagonal. Note that any cell of this diagonal belongs to only one "cross" (its own), while any other cell of the table belongs to exactly two such "crosses."

We can reason in different ways:

First method. Consider a natural number from 1 to 25 that is missing from the selected main diagonal (such a number will definitely exist, as there are only 13 cells on the diagonal). Let all cells of the main diagonal be "good," then this number appears in each of the thirteen "crosses" exactly once. But any number outside the main diagonal must appear in two "crosses," so the "crosses" should be paired, which is impossible for thirteen "crosses." Contradiction.

Second method. For the number 1 to appear in each of the 13 "crosses," it must be written in the table at least seven times. This can be said about each of the twenty-five given numbers. Therefore, to make all cells of the considered main diagonal "good," it would require filling at least $7 \cdot 25 = 175$ cells. But the table has only $13 \cdot 13 = 169$ cells. Contradiction.

Thus, all cells of the main diagonal cannot be "good."

## Grading Criteria:

+ A complete and well-reasoned solution is provided

± A generally correct reasoning is provided, with minor inaccuracies or gaps

- A correct answer is provided with an incorrect justification (including by considering any number of specific cases or by an incomprehensible reference to the oddness of the number 13)
- Only the answer is provided
- The problem is not solved or is solved incorrectly