9.1. At a round table, 10 people are sitting, some of them are knights, and the rest are liars (knights always tell the truth, while liars always lie). It is known that among them, there is at least one knight and at least one liar. What is the maximum number of people sitting at the table who can say: "Both of my neighbors are knights"? (A false statement is one that is at least partially not true.)

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# Answer. 9.

Solution. Note that all 10 could not have said such a phrase. Since at the table there is both a knight and a liar, there will be a liar and a knight sitting next to each other. But then this knight does not have both neighbors as knights. If, however, at the table there are 9 liars and 1 knight, then each of these 9 liars could say the phrase "Both of my neighbors are knights," since each liar has a liar among their neighbors.

Comment. It has been proven that all 10 people could not have said the required phrase - 4 points.

It has been shown that with a certain seating arrangement, 9 people could have said the required phrase - 3 points.