More problems

Problem 1. Ten students solved a total of 35 problems in a Math Olympiad. Each problem was solved by exactly one student. There is at least one student who solved exactly one problem, at least one student who solved exactly two problems, and at least one student who solved exactly three problems. Prove that there must be at least one student who has solved at least five problems.

Problem 2. Prove that there exists a power of 3 which ends with the digits 001 (in decimal notation).