10. 8 Class

30 students in total, each
student in the class
has the same number of friends.
How many students at most can there be who are
better than the majority of their friends' performance? (Assuming for any two students in the class, their performance can be compared to determine who is better or worse.)


保留源文本的换行和格式，直接输出翻译结果。
10.8 䇾案, 25 个.  
We refer to a student who performs better than the majority of their friends as a "good student". Let there be $x$ "good students", and each student has $k$ friends.

The best student is better in all $k$ "friend pairs", while each of the other "good students" is better in at least $\left[\frac{k}{2}\right]+1$ (the integer part). Therefore, the "good students" are better in at least $k+$ non-overlapping "friend pairs" in total, which is $\frac{30 k}{2}=15 k$. From this, we know that
$$
x \leqslant 28 \cdot \frac{k}{k+1}+1 .
$$
The number of students is no more than $30-x$. Therefore, we also have
$$
\frac{k+1}{2} \leqslant 30-x \text {. }
$$

The right side of inequality (1) is a decreasing function of $k$, and inequality (2) is equivalent to the condition
$$
k \leqslant 59-2 x \text {. }
$$

Combining (1) and (3), we get
$$
x \leqslant 28 \cdot \frac{59-2 x}{60-2 x}+1 \text {, }
$$

which simplifies to $x^{2}-59 x+856 \geqslant 0$.
The largest integer satisfying (4) and the condition $x \leqslant 30$ is $x=25$, so the number of "good students" does not exceed 25.
Next, we prove that 25 can be achieved.
\begin{tabular}{|c|c|c|c|c|}
\hline 1 & 2 & 3 & 4 & 5 \\
\hline 6 & 7 & 8 & 9 & 10 \\
\hline 11 & 12 & 13 & 14 & 15 \\
\hline 16 & 17 & 18 & 19 & 20 \\
\hline 21 & 22 & 23 & 24 & 25 \\
\hline 26 & 27 & 28 & 29 & 30 \\
\hline
\end{tabular}
35. We number the students in the class from 1 to 30 in the order of their performance, and list these numbers in a $6 \times 5$ table (as shown).

Assume that two students are friends if their numbers in the table fall into one of the following three cases: 1) they are in adjacent rows and different columns; 2) they are in the same column, with one in the bottom row; 3) they are in the top row (i.e., any two students in these three cases are friends). It is not difficult to verify that all the given conditions are satisfied.