Ten students attempted to solve a total of 35 problems. Each problem was solved by one student only. There is at least one student who solved only one problem, at least one who solved only two problems and at least one who solved exactly three problems. Prove that there is also at least one student who has solved at least 5 problems.
If the pigeons are the exercises and the nests are the students 35>10. Is this argument valid for the exercise?
Best Answer
At least $1+2+3=6$ problemas were solved by the students mentioned in the problem statement. Therefore, there $29$ problems left to be solved, and $7$ students to account for them. If each students had solved only $4$ problems, then there would have been only $28$ problems solved. Therefore, one student must have solved at least $5$ problems.