10.1. The math teacher agreed with the eleven students who came to the elective that he would leave the classroom, and the students would agree among themselves who would be a liar (always lie) and who would be a knight (always tell the truth). When the teacher returned to the class, he asked each student to say about each of the others whether they were a knight or a liar. Out of 110 answers, 56 were "He is a liar." What is the maximum number of knights that could have been among the students?
Answer: 7.

Solution: The phrase "He is a knight" would be said by a knight about a knight and by a liar about a liar, while the phrase "He is a liar" would be said by a knight about a liar and by a liar about a knight. Therefore, in each pair of "knight-liar," the phrase "He is a liar" will be said twice. Since this phrase was said 56 times in total, there are 28 pairs of "knight-liar," from which we find that there are 7 knights and 4 liars, or vice versa.

Comment: A correct answer without justification - 0 points.