Question

В классе 36 учеников. Для поднятия настроения учитель начал хвалить от- дельных учеников. Выяснилось, что в любой четвёрке учеников одинаковое количество похваленных учеников. Более того, в любой тройке учеников коли- чество похваленных учеников также одинаковое. Докажите, что добрый учи- тель похвалил всех своих учеников.

Answer 1

Допустим, он похвалил всех, кроме 1. Тогда можно составить четверку,
в которой похвалены в 4, а можно - в которой только 3.
Тоже самое с тройками. В одной похвалены все 3, а в другой только 2.
Тоже самое будет, если он не похвалит 2, 3, и так далее учеников.
Поэтому он или похвалил всех (в четверках - по 4, в тройках - по 3),
или не похвалил никого (и в четверках, и в тройках - по 0).

Answer 2

Problem Analysis

To solve this problem, we need to show that the teacher praised all 36 students. We know that the teacher praised an equal number of students in every group of 3 and every group of 4. This implies that the number of students praised must be a multiple of both 3 and 4, and the only number that satisfies this condition between 1 and 36 is 12. Therefore, we need to show that the teacher praised 12 students in each group of 4 and 3 students in each group of 3.

Proof

Let's start by proving that the teacher praised 12 students in each group of 4.

Assume the opposite, that the teacher did not praise 12 students in any group of 4. This means the teacher praised 0, 1, 2, or 3 students in each group of 4. However, this is impossible because the number of students praised must be a multiple of 4, and the only number between 1 and 36 that is a multiple of 4 is 12. Therefore, the teacher must have praised 12 students in each group of 4.

Now, let's prove that the teacher praised 3 students in each group of 3.

Assume the opposite, that the teacher did not praise 3 students in any group of 3. This means the teacher praised 0, 1, or 2 students in each group of 3. However, this is impossible because the number of students praised must be a multiple of 3, and the only number between 1 and 36 that is a multiple of 3 is 12. Therefore, the teacher must have praised 3 students in each group of 3.

Since the teacher praised 12 students in each group of 4 and 3 students in each group of 3, it follows that the teacher praised all 36 students.

Therefore, the good teacher praised all of his students.

I hope this explanation helps! If you have further questions, feel free to ask.