Twenty math students are comparing grades on their first two

Twenty math students are comparing grades on their first two quizzes of the year. They discover the whenever any pair of students consult with one another, these two students recieved the same grade on their first quiz of they received the same grade on their second quiz(or both). Prove that the entire class received the same grade on at least one of the two quizzes. *Hint* If all students received the same grade on the first quiz then we are done. Otherwise, two students got different grades on the first quiz; call them A and B. So we know that A and B got the same grade on the second quiz. Now show that any other student C must have gotten the same grade as A and B on the second quiz.

Solution

If two students consulted then they get the same grade either on first quiz or second quiz or both.

If possible let all students did not get the same grade on the first quiz.

Then A and B (say) got different grades on their first quiz.

Since they did not get the same score on the first quiz, according to the given conditionn that for any pair of students received the same grade in any one quiz.

Hence A and B must have got same grade in the second quiz.

Consider a student C who got same marks either with A or with B in the first quiz.

If he gets the same grade in the second quiz, then A, B and C get the same grade in the second quiz.

Now try for another person D and on same argument it follows that A,B, C and D got the same grade on second quiz and so on.

Thus if all students either got same grade in the first quiz, or if atleast two students did not get the same grade, it follows that they got the same grade in the second quiz.

Thus proved.