5. In a health camp, 225 schoolchildren are resting. Some of the children know each other, while others do not. It is known that among any six schoolchildren, there are three non-intersecting pairs who know each other. What is the minimum number of pairs of schoolchildren who could be resting in the camp?
Answer: 24750.

Solution. Suppose that some student (let's call him Vasya) is not acquainted with at least five other students. Then Vasya and the five students he is not acquainted with form a group of six that does not satisfy the condition of the problem. Therefore, each student can have no more than four unknowns. Thus, each student is acquainted with at least 220 students, and the number of pairs of acquaintances is no less than $\frac{225 \cdot 220}{2}=24750$.

Now, let's show that the number of pairs of acquaintances can be exactly 24750. Seat all the students around a large round table and introduce everyone except those sitting next to each other and those sitting one apart. Let's check that this construction satisfies the condition of the problem. Since each student is not acquainted with exactly four other students, each student is acquainted with exactly 220 students, and the total number of pairs of acquaintances is $\frac{225 \cdot 220}{2}=24750$.

![](https://cdn.mathpix.com/cropped/2024_05_06_66368caef28b3de8e403g-24.jpg?height=294&width=275&top_left_y=1412&top_left_x=959)

Now consider any group of six students. Leave only them at the large round table. Notice that each of them is acquainted with the one sitting "opposite" him (i.e., the one sitting two seats away), since at the large table, he could not be either his neighbor or the one sitting one seat away. Thus, we have selected three non-overlapping pairs of acquaintances.

![](https://cdn.mathpix.com/cropped/2024_05_06_66368caef28b3de8e403g-24.jpg?height=483&width=425&top_left_y=1917&top_left_x=884)