Baltic Way Problem Shortlist, 2011

Assume the contrary and let $a$ be a student from $A$ knowing as many students from $B$ as possible. As $a$ does not know all students from $B$, there is a student $b$ from $B$ not known to $a$. Similarly, we may pick a student $c$ from $C$ not known to $b$ and then a student $a'$ from $A$ not known to $c$. Applying the assumption to the sets of students $\{a, b, c\}$ and $\{a', b, c\}$, we learn that $a$ and $c$ know each other, and so do $a'$ and $b$. As $b$ knows $a'$ but not $a$, we have $a \neq a'$. Moreover, the maximality condition imposed on $a$ tells us that some student $b'$ from $B$ is known to $a$ but not to $a'$. Now if $b'$ and $c$ knew each other, then any two students from $\{a, b', c\}$ would know one another, which is not possible. Thus $b'$ and $c$ do not know each other, but this means that no two students from $\{a', b', c\}$ know one another, which is likewise impossible. Thereby the problem is solved.