4. (13 points) Sixteen people are standing in a circle: each of them is either a truth-teller (he always tells the truth) or a liar (he always lies). Everyone said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle?
Answer: 10.

Solution. A truthful person can only be next to liars. Three liars in a row cannot stand, so between any two nearest truthful persons, there is one or two liars. Then, if there are 5 or fewer truthful persons, in the intervals between them, there can be no more than 10 liars in total, so there are no more than 15 people in total. This leads to a contradiction. Therefore, there are no fewer than 6 truthful persons, and no more than 10 liars. An example with 10 liars is easily provided.

Evaluation Criteria. Full solution - 13 points. Example constructed - 3 points. Estimate proven - 9 points. Correct estimate named but not justified by considering examples - 2 points.