## 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 – Day 1 – Problem 3

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

## IMO 2015 Day 2

**Problem 4.** Triangle has circumcircle and circumcenter . A circle with center intersects the segment at points and , such that , , , and are all different and lie on line in this order. Let and be the points of intersection of and , such that , , , , and lie on in this order. Let be the second point of intersection of the circumcircle of triangle and the segment . Let be the second point of intersection of the circumcircle of triangle and the segment .

Suppose that the lines and are different and intersect at the point . Prove that lies on the line .

**Problem 5.** Let be the set of real numbers. Determine all functions that satisfy the equation

for all real numbers and .

**Problem 6.** The sequence of integers satisfies the conditions:

(i) for all , (ii) for all . Prove that there exist two positive integers and for which

for all integers and such that .

## Nice characterization side-lengths of a triangle

Find the greatest such that and implies that are the side-lengths of a triangle.

## Best approximation of a certain square root

Let be a real number such that the inequality

holds for an infinity of pairs of natural numbers. Prove that .

## Agreg 2012 Analysis Part 1

**Part 1. Finite dimension**

The goal is to prove the following theorem:

**Theorem 1.** Let be a square matrix with non-negative coefficients. Suppose that for every with non-negative coordinates, the vector has strictly positive components. Then

- (i) the spectral radius is a simple eigenvalue for ;
- (ii) there exists an eigenvector of associated to with strictly positive coordinates.
- (iii) any other eigenvalue of verifies ;
- (iv) there exists an eigenvector of associated to with strictly positive components.

1. Consider such that . Prove that for distinct we have . Deduce that there exists such that .

2. Prove that the coefficients of are strictly positive.

3. For we denote . Prove that if and only if there exists such that .

4. Denote . Consider and denote . Prove that

5. Denote . Prove that is an interval which does not reduces to , it is bounded and closed.

6. Denote . Prove that if verifies then we have . Deduce that is an eigenvalue of and that for this eigenvalue there exists an eigenvector with coordinates strictly positive.

7. Consider . Prove that and implies . Deduce that and every other eigenvalue of verifies .

8. Prove that every eigenvector of which has positive coordinates is proportional to .