7. On an island of knights, who always tell the truth, and liars, who always lie, a school was opened. All $2 N$ students of different heights lined up in pairs (i.e., in two columns). The first two people said: "I am taller than two people: my partner and the person behind me." The last two said: "I am also taller than two people: my partner and the person in front of me." Finally, everyone else said: "I am taller than three people: my partner, the person in front of me, and the person behind me."

a) What is the maximum number of knights that can study in the school?

b) Can the school have only liars?
Solution. A) In each pair, there is no more than one knight, so there are no more than $N$ knights (the example is achieved by placing $N$ taller students in a checkerboard pattern).

B) Since all students are of different heights, the tallest of them is definitely taller than their neighbors, so they are a knight, that is, they cannot all be liars.

Criteria. Correctly done part a) - 5 points. Only the estimate is given - 2 points, only the example is given - 2 points. A specific example is considered as justification - 0 points. Part b) is completed - 2 points.