Problem 7. On an island, there live knights who always tell the truth, and liars who always lie. One day, 15 natives, among whom were both knights and liars, stood in a circle, and each said: "Of the two people standing opposite me, one is a knight, and the other is a liar." How many of them are knights?
Answer: 10 knights.

Solution. Consider any knight. He tells the truth, which means that opposite him stand a knight and a liar. One of the people opposite the found liar is the initial knight, so next to him stands another knight. Opposite this new knight stands the previously found liar and another person, who must be a knight. Repeating these considerations, we see that in the circle, there are two knights, then one liar, then again two knights, then another liar, and so on. This means that the liars make up a third of the total number of participants. Therefore, the number of liars is $15 \div 3 = 5$, and the number of knights is $15 - 5 = 10$.