Poddlisky 0. .  Around 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 of them 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 those sitting said: "My number is less than the number of each of my two neighbors." For what maximum $k$ could this have happened?
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. Therefore, neither of them could have said the second phrase. Thus, $k \leq 2013$.

The 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, ..., 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.