10.5. In each cell of a 5 by 5 table, a letter is written such that in any row and in any column there are no more than three different letters. What is the maximum number of different letters that can be in such a table
Answer: 11.

Solution: If each row contains no more than two different letters, then the total number of letters does not exceed $10=5 * 2$. Further, we can assume that the first row contains exactly three different letters. If each of the remaining rows has at least one letter in common with the first, then the total number of letters does not exceed $3+4 * 2=11$. Suppose there is a row, which we can consider the second, with three different letters, distinct from the letters in the first row. Then, in each column, apart from the letters in the first and second rows, there can be no more than one new letter, totaling no more than $3+3+5 * 1=11$. An example of arranging 11 different letters: along the main diagonal of the table from the bottom left corner to the top right, the first five different letters are written, along the next lower diagonal - the next four, in the top left corner - the tenth, and in the remaining cells - the eleventh letter.

Grading criteria. The maximum of 11 is proven: 5 points. Example for 11: 2 points. Any incorrect answer and an attempt to prove it: 0 points.