8.5. At a round table, 410 deputies sat down, each of whom was either a knight, who always tells the truth, or a liar, who always lies. Each of the deputies said: "Among my twenty neighbors to the left and twenty neighbors to the right, there are exactly 20 liars in total." It is known that at least half of the people at the table are liars. How many knights are there at the table?
Answer: None.

Solution: Let's divide all the people sitting at the table into ten groups of 41 people each. Then, at least one group will have at least 21 liars. Otherwise, in each group, there would be a maximum of 20, i.e., no more than \(20 \times 10 = 200\) in total, which is less than half of the total number. Consider this group.

If a knight is sitting in the center, he tells the truth, and there are exactly 20 liars in this group. But we have already established that there are at least 21. Contradiction. Therefore, a liar is sitting in the center, and there are at least 22 liars in the group (there cannot be exactly 21, as otherwise, the central person would be telling the truth). Now consider the left neighbor of the central liar. Among his 40 neighbors, there are at least 21 liars, since we lost one extreme neighbor (who might be a liar) by shifting to the left, but gained a liar neighbor who was previously central. Repeating the reasoning for him, we get that he is also a liar. Shift left again and continue this process. Eventually, we will get that everyone sitting at the table is a liar.

## Criteria:

Only the answer, answer with verification - 0 points.

Proved that there will be a group with 21 liars - 2 points.

Proved that in this group, a liar is sitting in the center - another 1 point.