1. On an island, there live knights who always tell the truth, and liars who always lie. In a room, 15 islanders gathered. Each of those in the room said two phrases: "Among my acquaintances in this room, there are exactly six liars" and "among my acquaintances in this room, there are no more than seven knights." How many knights can there be in this room?
Answer: 9 knights

Solution. If there is a liar in the room, then his statement "among my acquaintances in this room, there are no more than seven knights" is a lie, and there are no fewer than 8 knights in the room. If there is a knight in the room, then his statement "among my acquaintances in this room, there are exactly six liars" is true, and there are no fewer than 6 liars in the room. Therefore, there are both knights and liars in the room, with no fewer than 8 knights and no fewer than 6 liars. This means there are no more than $15-6=9$ knights. Suppose there are exactly 8 knights. Then there are 7 liars in the room, and each of them has more than 7 acquaintances who are knights. Therefore, each of them is acquainted with all the knights in the room. This means that each knight is also acquainted with all the liars in the room, i.e., each knight has 7 acquaintances who are liars. But then he would be lying. Therefore, there cannot be 8 knights. Thus, there are 9 knights in the room. Such a situation is possible if each knight knows each liar, and no one else is acquainted with anyone else.