## IMC 2017 – Day 1

**Problem 1.** Determine all complex numbers for which there exists a positive integer and a real matrix such that and is an eigenvalue of .

**Problem 2.** Let be a differentiable function and suppose that there exists a constant such that

for all . Prove that

holds for all .

**Problem 3.** For any positive integer denote by the product of positive divisors of . For every positive integer define the sequence

Determine whether for every set there exists a positive integer such that the following condition is satisfied:

*For every with the number is a perfect square if and only if .*

**Problem 4.** There are people in a city and each of them has exactly friends (friendship is mutual). Prove that it is possible to select a group of people such that at least persons in have exactly two friends in .

**Problem 5.** Let and be positive integers with and let

be a polynomial with complex coefficients such that

Prove that and have at most common roots.