Task 10.4. In a class, there are $N$ students: several excellent students and 8 troublemakers. Excellent students always tell the truth, while troublemakers always lie.

One day, all the students in this class sat around a round table, and each of them declared to everyone else: “At least a third of you are troublemakers!”

What can $N$ be? List all possible options.
Answer: 23, 24, or 25.

Solution.

The condition that the troublemaker made this statement is equivalent to the inequality $\frac{7}{N-1} < \frac{1}{3}$, which is equivalent to $22 < N$.

The condition that the excellent student made this statement is equivalent to the inequality $\frac{8}{N-1} \geqslant \frac{1}{3}$, which is equivalent to $25 \geqslant N$.

Since there are troublemakers in the class, $N > 22$. Then there are also excellent students in the class, so $25 \geqslant N$. Therefore, all possible natural values of $N$ are $23, 24, 25$.