Problem 1: (Putnam and Beyond, 296) Show that if a (noncommutative) ring with identity has three elements such that
- for any , implies
then the ring cannot be finite.
Solution: Suppose the ring is finite. Let denote left multiplication by . By condition (2), is an injective function from to . Since is finite, this must be a bijective function. Thus there exists such that , or , i.e. the right inverse exists; then exists.
Next note by (3), . Since is bijective, and are functions , must also be bijective. Thus by the same argument as before exist. Then conjugating by gives using (1); this is a contradiction. Thus is infinite.
Problem 2: (Miklos Schweitzer, 1987/1) The numbers 1 to N are colored with 3 colors such that each color appears more than times. Prove that the solution has a solution with distinct colors.