Problem 694

Students who perform better than most of their friends are called good. Let $n$ be the number of good students, and $k$ be the number of friends each student has. We will prove that $n \leq 25$. For this, we will consider two cases.

1) $k \geq 8$. Then the five worst students in the class are not good.
2) $k \leq 7$. The best student in the class is the best in $k$ pairs of friends, and any other good student is the best in at least ${ }^{k+1 / 2}$ pairs. Therefore, good students are the best in at least $\left.k+(n-1)\right)^{k+1} / 2$ pairs. This number cannot exceed the number of all pairs of friends in the class, which is $15 k$. Hence, $2 k+(n-1)(k+1) \leq 30 k$, which means $n-1 \leq 28 \cdot k / k+1=28-28 / k+1 \leq 28-28 / 8=24.5$. Therefore, in this case, $n \leq 25$.

We will show that $n$ can equal 25. Number the students from 1 to 30 in the order of decreasing performance and arrange the numbers in a $6 \times 5$ table as shown in the figure.

A pair of students is a pair of friends if their numbers are arranged in one of three ways: a) in adjacent rows and in different columns; b) in the same column and one of the numbers is in the bottom row; c) in the top row. In this case, as can be easily verified, all the required conditions are met.

## Answer

25 students.