Diding M.

In a country of knights (who always tell the truth) and liars (who always lie), 10 people are sitting at the vertices of a regular decagon around a round table, and among them, there are liars. A traveler can stand somewhere and ask the seated people: "What is the distance from me to the nearest liar among you?" After this, each of them answers him. What is the minimum number of questions the traveler must ask to guarantee finding out who the liars are at the table? (There are no outsiders nearby, and the traveler cannot stand on the table. Consider people as points. Everyone, including the traveler, can accurately measure any distance.)
Example. The first question is asked from an arbitrary point. If all answers to it are the same, then everyone sitting at the table is a liar, since a knight and a liar give different answers.

Otherwise, there will be neighbors who answered differently. We will stand in the middle of the arc between them. Since at least one of the two is a liar, the distance to the nearest liar is known. Then those who name it are knights, and the rest are liars.

Evaluation. Suppose only one question was asked. We divide the people into groups based on their distance from the traveler. There will be more than one group, since one cannot stand at the center of the table. Suppose the nearest group pointed to the next one, and the rest pointed to the first. It is easy to see that the first group could have been knights, and the rest liars, but it could have been the other way around.

## Answer

Two questions.