For convenience, denote the schools by $A = S_0, B = S_1, C = S_2$. Lemma: There cannot exist students $s_0, s_1, \ldots, s_{3n-1}$ such that i) student $s_i$ comes from school $S_j$ where $i \equiv j \bmod{3}$ ii) for each $0 \leq i \leq 3n-1$, the students $s_i$ and $s_{i+1}$ don't know each other, where we set $s_{3n} = s_0$. We'll use induction on $n$. If $n = 1$ then such a sequence would just be three students, one from each school, such that no two know each other, contradicting the problem statement. If $n > 1$ then, supposing such a sequence exists: - since $s_0$ doesn't know $s_1$ and $s_1$ doesn't know $s_2$, we find that $s_0$ knows $s_2$ - since $s_2$ doesn't know $s_3$ and $s_3$ doesn't know $s_4$, we find that $s_2$ knows $s_4$ - since $s_0$ knows $s_2$ and $s_2$ knows $s_4$, we find that $s_0$ doesn't know $s_4$ and $s_0, s_4, s_5, \ldots, s_{3n-1}$ is a sequence of $3(n-1) - 1$ students which can't exist by the induction hypothesis. -- end lemma -- Now if none of the three given statements hold, for any student in $S_i$, we can find a student in $S_{i+1}$ that that student doesn't know (where of course $S_3 = S_0$). Repeating this process starting from an arbitrary student, since there are only finitely many students in each school, eventually we'll select a student we've selected before, and this forms a loop which can't exist by the lemma. Thus at least one of the three given statements hold.