Question

В классе каждый ученик — либо болтун, либо молчун, причем каждый болтун дружит хотя бы с одним молчуном. Болтун молчит, если в кабинете находится нечётное число его друзей — молчунов. Докажите, что учитель может пригласить на факультатив не менее половины класса так, чтобы все присутствующие на факультативе болтуны молчали.

Answer 1

8 болтунов и 7 молчунов

Answer 2

Problem Analysis

The problem states that in a class, each student is either a talkative person or a quiet person, and every talkative person is friends with at least one quiet person. It is also mentioned that a talkative person remains quiet if there is an odd number of their friends (quiet people) in the classroom. The task is to prove that the teacher can invite at least half of the class to an optional class in such a way that all the students present in the optional class remain quiet.

Proof

To prove that the teacher can invite at least half of the class to the optional class in a way that all the students present remain quiet, we can use the following approach:

1. Assume that there are a total of n students in the class. 2. Let's consider the worst-case scenario where there are m talkative students and n-m quiet students in the class. 3. Since each talkative student is friends with at least one quiet student, we can assume that each talkative student has exactly one quiet friend. 4. In order for a talkative student to remain quiet, there must be an odd number of their quiet friends in the classroom. 5. Let's consider the case where the teacher invites k talkative students to the optional class. Since each talkative student has exactly one quiet friend, there will be k quiet students present in the optional class as well. 6. In order for all the students present in the optional class to remain quiet, the number of quiet students (k) must be odd. 7. Therefore, the teacher can invite any odd number of talkative students to the optional class, and all the students present will remain quiet.

This proves that the teacher can invite at least half of the class (since half of the class is an odd number) to the optional class in such a way that all the students present remain quiet.

Note: The proof assumes that there are at least as many talkative students as there are quiet students in the class. If the number of talkative students is less than the number of quiet students, the teacher can simply invite all the talkative students to the optional class, and they will remain quiet.

I hope this explanation helps! Let me know if you have any further questions.