9.1. At a round table, 2015 people are sitting, each of them is either a knight or a liar. Knights always tell the truth, and liars always lie. Each person was given a card with a number on it; all the numbers on the cards are different. After looking at their neighbors' cards, each person said: "My number is greater than the number of each of my two neighbors." After this, $k$ of the people said: "My number is less than the number of each of my two neighbors." For what largest $k$ could this have happened?

(O. Podlipsky)
Answer: 2013.

Solution. Let $A$ and $B$ be the people who received the cards with the largest and smallest numbers, respectively. Since both of them said the first phrase, $A$ is a knight, and $B$ is a liar. However, if they had said the second phrase, $A$ would have lied, and $B$ would have told the truth; this is impossible. Therefore, $A$ and $B$ cannot say the second phrase, and $k \leqslant 2013$.

We will show that a situation where the remaining 2013 people can say the second phrase is possible. Suppose the people sitting at the table received (in a clockwise direction) cards with the numbers $1, 2, 3, \ldots$, 2015; with the card numbered 2015 going to a knight, and the rest to liars. Then the first phrase can be said by everyone, and the second phrase by everyone except those with cards numbered 1 and 2015.

Remark. There are other examples of card distributions where 2013 people can say the second phrase.

Comment. Only the answer - 0 points.

An example of seating arrangement is provided, where $k$ can be equal to $2013-3$ points.

Only proved that $k \leqslant 2013-4$ points.

Only proved that $k \leqslant 2014-1$ point.

It is claimed that the people with the largest and smallest numbers cannot say the second phrase, but this claim is not justified - 1 point is deducted.