## 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 .

## 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

## Seemous 2015 Problems

**Problem 1.** Prove that for every the following inequality holds:

**Problem 2.** For any positive integer , let the functions be defined by , where . Solve the equation .

**Problem 3.** For an integer , let be matrices satisfying:

where is the identity matrix and is the zero matrix in .

Prove that:

- (a) and .
- (b) and .

**Problem 4.** Let be an open interval which contains and be a function of class such that .

- (i) Prove that there exists such that for .
- (ii) With determined at (i) define the sequence by
Study the convergence of the series for .

**Hints: **

## IMC 2014 Day 2 Problem 2

Let be a symmetric matrix with real entries, and let denote its eigenvalues. Show that

and determine all matrices for which equality holds.

**IMC 2014 Day 2 Problem 2**

## Two more mind games solved using Matlab

After doing this, solving the Sudoku puzzle with Matlab, I wondered what else can we do using integer programming? Many of the mind games you can find on the Internet can be modelled as matrix problems with linear constraints. I will present two of them, as well as their solutions, below. The game ideas are taken from www.brainbashers.com. Note that in order to use the pieces of code I wrote you need to install the YALMIP toolbox.

You have a table and on each row and column you need to have squares coloured with color and squares coloured with color such that no three squares taken vertically or horizontally have the same color. In the begining you are given the colors of some of the square such that it leads to a unique solution satisfying the given properties. The goal is to find the colors corresponding to each small square starting from the ones given such that the end configuration satisfies the given properties.

## 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 .