52. In the table, there are 25 columns and 300 rows, and Kostya painted all its cells in three colors. Then Lesha, looking at the table, names one of the three colors for each row and marks all the cells of this color in that row. (If there are no cells of the specified color in the row, he marks nothing in it.) After that, all columns that contain at least one marked cell are erased from the table. Kostya wants as few columns as possible to remain in the table, while Lesha wants as many as possible. What is the maximum number of columns that Lesha can guarantee to leave?
52. Answer: Two columns. To leave at least two columns, Lesha must for each row name a color that does not appear in the first two cells of that row. With this strategy, the first two columns will not be crossed out.

Note now that $300=C_{25^{2}}$. Using this observation, we can associate each row with its own pair of columns. Let Kostya paint in each row the cells in the corresponding two columns in the 1st and 2nd colors, and the rest of the cells in the third color. If Lesha chooses the 3rd color in some row, this will immediately cross out everything except two columns. If, however, Lesha always chooses the 1st or 2nd color, then at most one column will remain! Indeed, suppose some two columns remain. Since there is a row in which the cells of these columns are painted in the 1st and 2nd colors, one of these columns must be destroyed by Lesha's choice for that row!