## IMC 2016 – Day 2 – Problem 8

**Problem 8.** Let be a positive integer and denote by the ring of integers modulo . Suppose that there exists a function satisfying the following three properties:

- (i) ,
- (ii) ,
- (iii) for all .

Prove that modulo .

## IMC 2016 – Day 2 – Problem 7

**Problem 7.** Today, Ivan the Confessor prefers continuous functions satisfying for all . Fin the minimum of over all preferred functions.

## IMC 2016 – Day 2 – Problem 6

**Problem 6.** Let be a sequence of positive real numbers satisfying . Prove that

## IMC 2016 Problems – Day 2

**Problem 6.** Let be a sequence of positive real numbers satisfying . Prove that

**Problem 7.** Today, Ivan the Confessor prefers continuous functions satisfying for all . Fin the minimum of over all preferred functions.

**Problem 8.** Let be a positive integer and denote by the ring of integers modulo . Suppose that there exists a function satisfying the following three properties:

- (i) ,
- (ii) ,
- (iii) for all .

Prove that modulo .

**Problem 9.** Let be a positive integer. For each nonnegative integer let be the number of solutions of the inequality . Prove that for every we have .

**Problem 10.** Let be a complex matrix whose eigenvalues have absolute value at most . Prove that

(Here for every matrix and for every complex vector .)

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## IMC 2016 – Day 1 – Problem 2

**Problem 2.** Let and be positive integers. A sequence of matrices is *preferred* by Ivan the Confessor if for , but for with . Show that if in al preferred sequences and give an example of a preferred sequence with for each .

## IMC 2016 – Day 1 – Problem 3

**Problem 3.** Let be a positive integer. Also let and be reap numbers such that for . Prove that

## IMC 2016 – Day 1 – Problem 1

**Problem 1.** Let be continuous on and differentiable on . Suppose that has infinitely many zeros, but there is no with .

- (a) Prove that .
- (b) Give an example of such a function.