During a secret meeting, 20 interns elect their favorite animator. Each votes for two animators. It is known that if you take any two interns, there is always at least one animator for whom they have both voted. What is the minimum number of votes received by the animator who wins the election?
Suppose there is no animator who has 20 votes (which is not a case that minimizes the winner's score in the election). Let's take an intern $s_{1}$ and denote $A$ and $B$ as the two people he voted for. By hypothesis, he has a colleague $s_{2}$ who did not vote for $A$, but according to the statement, he voted for $B$. Let $C$ be his other vote. Similarly, there exists an intern $s_{3}$ who did not vote for $B$ but has a vote in common with $s_{1}$, so he voted for $A$, and a vote in common with $s_{2}$, so he also voted for $C$. Now take any other intern $s$. Since he has a vote in common with each of $s_{1}, s_{2}, s_{3}$, his two votes are among $A, B$, and $C$. Thus, $A, B$, and $C$ share the 40 votes cast. According to the pigeonhole principle, one of them has at least 14 votes.

Finally, we verify that this is indeed the minimum as follows: interns 1 to 14 voted for $A$, interns 15 to 20 and 1 to 7 voted for $B$, and the remaining 13 votes are for $C$.