TITLE: Every question solved by one of two students
QUESTION [6 upvotes]: $200$ students take a test with $6$ questions. For each question, at least $120$ students got it correct. Show that for some two students, every question is done correctly by at least one of them.
From the given, the average number of questions that a student got correct is at least $\dfrac{120\cdot 6}{200}=3.6$. But how does this imply the desired conclusion?

REPLY [9 votes]: Since each question got at least 120 correct answers there were at least 720 correct answers, this means the average number of correct answers per student was 3.6. It follows there are students who got more than 3 correct answers.
If any student got 6 correct answers, they can be paired with any other student.
If any student got 5 correct answers, then we know there at least 120 others who got the question they missed, and that they can be paired with.
If any student got 4 correct answers, then of the 200 students no more than 80 got each of the questions that that student missed wrong. Hence at least 200 - 80 - 80 = 40 must've got both right. So there are at least 40 students that student can be paired with.