1. 200 people are standing in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist who stands next to two conformists always tells the truth. A conformist who stands next to at least one liar can either tell the truth or lie. 100 of those standing said: "I am a liar," and the other 100 said: "I am a conformist." Find the maximum possible number of conformists among these 200 people. (R. Zhendarov, S. Berlov)
Answer: 150. Solution: A liar cannot say, "I am a liar." Therefore, 100 people who said, "I am a liar," are conformists. All of them lied, so next to each of them stands a liar. Since next to a liar there can be a maximum of two conformists, there are no fewer than 50 liars. Thus, there are no more than 150 conformists. Example. Place 50 liars in a circle. In each of the 50 gaps between the liars, place three conformists. The middle one of these three tells the truth, the two at the ends lie that they are liars, and all liars lie that they are conformists.