Minimum number of participants who must have the same total score

In a math competition, there are two types of questions:

Multiple-choice questions: There are 8 questions, each worth 4 points for a correct answer and 0 points for an incorrect or unanswered question.

Fill-in-the-blank questions: There are 6 questions, each worth 7 points for a correct answer and 0 points for an incorrect or unanswered question.

There are 400 participants in the competition. What is the minimum number of participants who must have the same total score?

Solution:

\(\boxed{8}\)

The lowest score is \(0\) and the full score is \(74\). By the Chicken McNugget Theorem, when \(m\) and \(n\) are co-prime, there are exactly \(\frac{(m - 1)(n - 1)}{2}\) positive integers which cannot be expressed as in the form of \(am + bn\). In this case, \(m=4, n=7\), hence on lower end, there are \(9\) scores which can not be achieved, e.g. \(1,2,3\). By symmetry, there are also \(9\) scores which cannot be achieved, e.g. \(73, 72, 71\). Hence, the total number of possible scores is \(75-18=57\).

Finally by the Pigeonhole Principle, the minimum number of participants who must have the same total score is \[ \lceil \dfrac{400}{57} \rceil = 8. \]