4. In a certain school, every tenth-grader either always tells the truth or always lies. The principal called several tenth-graders to his office and asked each of them about each of the others, whether they are a truth-teller or a liar. In total, 44 answers of "truth-teller" and 28 answers of "liar" were received. How many truthful answers could the principal have received?
Answer: 16 or 56.

Solution. If $n$ tenth-graders are called, then $n(n-1)=44+28=72$ answers are given, from which $n=9$. Let $t$ be the number of truth-tellers and $(9-t)$ be the number of liars among these 9 students. The answer "liar" can only be given by a liar about a truth-teller and a truth-teller about a liar, and there were $2t(9-t)=28$ such phrases, from which $t=2$ or $t=7$. If there are two truth-tellers, they gave $2 \cdot 8=16$ truthful answers. If there are seven truth-tellers, they gave $7 \cdot 8=56$ truthful answers.

Comment. Note that from the condition, it follows that half of the answers "liar" are truthful. However, it is not immediately clear what the proportion of truthful answers "truth-teller" is.

## Grading Criteria.

- $\quad$ Full solution - 7 points.
- Correctly found both cases (how many truth-tellers and liars), but the number of truthful answers is calculated incorrectly - 4 points.
- There are 2 situations described in the problem. If only one is correctly analyzed, then 3 points.
- $\quad$ Both answers are provided without explanation - 1 point.
- Only one of the answers is provided - 0 points.