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

a) What is the maximum number of knights that can study at 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) If it turns out that all students are of the same height, then everyone is lying.

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