Question

В компании 10 человек. Каждому из них нравится не меньше 5 человек в этой компании. Докажите, что найдутся 2 человека, которые нравятся друг другу.

Answer 1

Problem Statement

In a company with 10 people, each person likes at least 5 other people in the company. We need to prove that there are 2 people who like each other.

Proof by Contradiction

To prove that there are 2 people who like each other, we will use a proof by contradiction. We will assume that there are no 2 people who like each other and show that this assumption leads to a contradiction.

Assumption

Let's assume that there are no 2 people in the company who like each other.

Analysis

Since each person in the company likes at least 5 other people, we can conclude that there are at least 10 * 5 = 50 total likes within the company.

Counting Likes

Let's count the number of likes in the company. Since each person likes at least 5 other people, the total number of likes should be at least 10 * 5 = 50.

Contradiction

However, if there are no 2 people who like each other, then each person can only like a maximum of 9 other people in the company (since there are only 9 other people to like). Therefore, the total number of likes would be at most 10 * 9 = 90.

Conclusion

We have reached a contradiction. On one hand, the total number of likes should be at least 50, but on the other hand, it can be at most 90. This contradiction arises from our initial assumption that there are no 2 people who like each other.

Proof by Contradiction

Therefore, we can conclude that our initial assumption is false. There must be at least 2 people in the company who like each other.

Note: The proof by contradiction method is commonly used in mathematics to prove statements. By assuming the opposite of what we want to prove and showing that it leads to a contradiction, we can conclude that the opposite is false and the original statement is true.