1. In a toy store, 125 plush bears are sold in $k$ different colors and six different sizes. For what largest $k$ can we assert that there will be at least three identical bears? (i.e., matching both in color and size) (20 points)
Solution: If $k=10$, then there are 60 varieties of teddy bears in the store. If there are no more than two of each type, then there are no more than 120 teddy bears in the store - a contradiction. Therefore, there will be at least three identical teddy bears among them.

If $k=11$, then there may not be three identical teddy bears. Indeed, there will be at least 66 varieties of teddy bears, among which 59 types can be sold with 2 copies each, and another 7 types - with one copy each. For $k \geq 11$, there will be more varieties of teddy bears, so as a counterexample, one can remove the second copies of the existing types and add one new copy of each added type.

Answer: 10