# Problem 1. (2 points)

In a class, each student has either 5 or 7 friends (friendship is mutual), and any two friends have a different number of friends in the class. What is the smallest number of students, greater than 0, that can be in the class?

Answer: 12
Solution:

Let's look at some person. Suppose he has five friends. Then each of these five people has seven friends. Similarly, there are at least seven more people with five friends each. In total, there are 12 people.

It is quite easy to construct an example: two groups of 5 and 7 people, people from different groups are friends with each other, but not within the same group.

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