## 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 .

## SEEMOUS 2016 – Problems

**Problem 1.** Let be a continuous and decreasing real valued function defined on . Prove that

When do we have equality?

**Problem 2.** a) Prove that for every matrix there exists a matrix such that .

b) Prove that there exists a matrix such that for all .

**Problem 3.** Let be idempotent matrices () in . Prove that

where and is the set of matrices with real entries.

**Problem 4.** Let be an integer and set

Prove that

a)

b) .

Some hints follow.

## Vojtech Jarnik Competition 2015 – Problems Category 2

**Problem 1.** Let and be two matrices with real entries. Prove that

provided all the inverses appearing on the left-hand side of the equality exist.

**Problem 2.** Determine all pairs of positive integers satisfying the equation

**Problem 3.** Determine the set of real values for which the following series converges, and find its sum:

**Problem 4.** Find all continuously differentiable functions , such that for every the following relation holds:

where

## Existence of Sylow subgroups

Let be a finite group such that where is a prime number, and . Then there exists a subgroup such that . (such a subgroup is called a Sylow subgroup).

## Agregation 2014 – Mathematiques Generales – Parts 4-6

This is the second part of the Mathematiques Generales French Agregation written exam 2014. For the complete notation list and the first three parts look at this post.

**Part 4 – Reduced form of permutations**

For we denote the set of pairs such that . We call the set of inversions of a permutation the set

and we denote the cardinal of .

**1.** For which permutations is the number maximum?

For we denote the transposition which changes and .

**4.2** (a) Let . Prove that

and that is obtained from by adding or removing an element of .

(b) Find explicitly in function of the element of which makes it differ from .

Let . We call word a finite sequence of elements of . We say that is the length of and that the elements are the letters of . The case of a void word is authorized.

A writing of a permutation is a word such that . We make the convention that the permutation which corresponds to the void word is the identity.

## Agregation 2014 – Mathematiques Generales – Parts 1-3

This post contains the first three parts of the the Mathematiques Generales part French Agregation contest 2014.

**Introduction and notations**

For we denote . For an integer we denote the group of permutations of .

We say that a square matrix is inferior (superior) *unitriangular* if it is inferior (superior) triangular and all its diagonal elements are equal to .

For two integers and we denote the family of -element subsets of .

Let be two positive integers and a matrix with elements in a field . (all fields are assumed commutative in the sequel) A minor of is the determinant of a square matrix extracted from . We can define for and the minor

where (respectively ) are the elements of (respectively ) arranged in increasing order. We denote this minor .

## Sierpinski’s Theorem for Additive Functions

We say that is an additive function if

1. Prove that there exist additive functions which are discontinuous with or without the *Darboux Property*.

2. Prove that for every additive function there exist two functions which are additive, have the *Darboux Property*, and .

The second part is similar to Sierpinski’s Theorem which states that every real function can be written as the sum of two real functions with Darboux property.

(A function has the *Darboux property* if for every , is an interval.)