1. On an island, only knights, who always tell the truth, and liars, who always lie, live. One fine day, 450 islanders sat around a round table. Each person at the table said: "Of the person sitting to my right and the person immediately after them, exactly one is a liar." How many liars can be sitting at the table?
Answer: 150 or 450.

Solution. All the liars can sit at the table, then there are 450 of them. Suppose there is at least one knight sitting at the table. Then there is a knight who has a liar sitting to their right (otherwise, they are all knights, which is impossible). The person to the right of this liar must be a knight. But to the right of the second knight, there must also be a knight, since the liar is lying. The second knight tells the truth, so to the right of the third knight, there must be a liar. Reasoning this way, we find that all those sitting at the table are divided into triples of the form knight, liar, and knight. Therefore, the liars are exactly one third.