Problem 5.1. At a physical education class, 27 seventh-graders arrived, some of them brought one ball each. Sometimes during the class, one of the seventh-graders would give their ball to another seventh-grader who didn't have one.

At the end of the class, $N$ seventh-graders said: "I received balls less frequently than I gave them away!". Find the maximum possible value of $N$, given that none of the students lied.
Answer: 13.

Solution. If a seventh-grader received the ball less frequently than he gave it away, then he originally had the ball, but no longer had it at the end. Thus, at the beginning of the lesson, the students collectively had at least $N$ balls, which ultimately ended up with some of the remaining $27-N$ students. Therefore, $N \leqslant 27-N$, from which we get that $N \leqslant 13$ (since $N$ is an integer). It remains to show that the value $N=13$ is possible. For convenience, let's number the seventh-graders from 1 to 27. Suppose students with numbers $1,2, \ldots, 13$ brought a ball to the lesson, and during the lesson, the seventh-grader with number $i$ passed the ball to the seventh-grader with number $13+i$ (for all $i=1,2, \ldots, 13$). It is clear that the condition of the problem is satisfied.