4

Part of 2016 Middle European Mathematical Olympiad

Problems(2)

students taking exams

Source: MEMO 2016 T4

An exam was taken by some students. Each problem was worth

1

point for the correct answer, and

0

points for an incorrect one.
For each question, at least one student answered it correctly. Also, there are two students with different scores on the exam.
Prove that there exists a question for which the following holds: The average score of the students who answered the question correctly is greater than the average score of the students who didn't.

combinatoricscombinatorics proposed

functional equation on integers with divisibility

Source: MEMO 2016 I4

f : \mathbb{N} \to \mathbb{N}

f(a) + f(b)

2(a + b - 1)

a, b \in \mathbb{N}

\mathbb{N} = \{ 1, 2, 3, \ldots \}

denotes the set of the positive integers.

number theorynumber theory proposedfunctional equationfunctionDivisibilityalgebra