Question

Пожалуйста помогите решить задачу : 14 учеников на олимпиаде решили 58 задач. Однако известно, что среди них есть ученики, решившие 2 задачи, 3 задачи, 4 задачи. Докажите, что есть ученики, решившие не менее 5 задач.

Answer 1

Возьмем наибольшее количество задач, которое могли решить, если решали по 4:
14×4=56
56 задач из 58, значит, не менее двух учеников решило 5 задач

Answer 2

Problem Analysis

To solve this problem, we can use the principle of the pigeonhole principle. The pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. In this context, the "pigeons" are the solved problems, and the "pigeonholes" are the students.

Application of Pigeonhole Principle

Given that 14 students solved a total of 58 problems, we can assume that each student solved an equal number of problems. If we divide the total number of problems by the number of students, we get an average of approximately 4.14 problems per student. Since a student can't solve a fraction of a problem, this means that some students must have solved more than 4 problems.

Conclusion

Therefore, based on the pigeonhole principle, there must be at least one student who solved 5 or more problems.

This conclusion is based on the application of the pigeonhole principle, which guarantees that at least one student solved 5 or more problems given the total number of problems and students.