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IMC 2016 – Day 1 – Problem 2

July 27, 2016 Leave a comment

Problem 2. Let {k} and {n} be positive integers. A sequence {(A_1,...,A_k)} of {n\times n} matrices is preferred by Ivan the Confessor if {A_i^2 \neq 0} for {1\leq i \leq k}, but {A_iA_j = 0} for {1\leq i,j \leq k} with {i \neq j}. Show that if {k \leq n} in al preferred sequences and give an example of a preferred sequence with {k=n} for each {n}.

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Vojtech Jarnik Competition 2015 – Problems Category 2

March 27, 2015 Leave a comment

Problem 1. Let {A} and {B} be two {3 \times 3} matrices with real entries. Prove that

\displaystyle A - (A^{-1}+(B^{-1}-A)^{-1})^{-1} = ABA,

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

Problem 2. Determine all pairs {(n,m)} of positive integers satisfying the equation

\displaystyle 5^n = 6m^2+1.

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

\displaystyle \sum_{n=1}^\infty \left( \sum_{k_i \geq 0, k_1+2k_2+...+nk_n = n} \frac{(k_1+...+k_n)!}{k_1!...k_n!} x^{k_1+...+k_n}\right).

Problem 4. Find all continuously differentiable functions {f : \Bbb{R} \rightarrow \Bbb{R}}, such that for every {a \geq 0} the following relation holds:

\displaystyle \int_{D(a)} xf\left( \frac{ay}{\sqrt{x^2+y^2}}\right) dxdydz = \frac{\pi a^3}{8}(f(a)+\sin a -1),

where {D(a) = \left\{ (x,y,z) : x^2+y^2+z^2 \leq a^2,\ |y| \leq \frac{x}{\sqrt{3}}\right\}}

Seemous 2015 Problems

March 11, 2015 Leave a comment

Problem 1. Prove that for every {x \in (0,1)} the following inequality holds:

\displaystyle \int_0^1 \sqrt{1+\cos^2 y}dy > \sqrt{x^2+\sin^2 x}.

Problem 2. For any positive integer {n}, let the functions {f_n : \Bbb{R} \rightarrow \Bbb{R}} be defined by {f_{n+1}(x)=f_1(f_n(x))}, where {f_1(x)=3x-4x^3}. Solve the equation {f_n(x)=0}.

Problem 3. For an integer {n>2}, let {A,B,C,D \in \mathcal{M}_n(\Bbb{R})} be matrices satisfying:

\displaystyle AC-BD = I_n,

\displaystyle AD+BC = O_n,

where {I_n} is the identity matrix and {O_n} is the zero matrix in {\mathcal{M}_n(\Bbb{R})}.

Prove that:

  • (a) {CA-DB = I_n} and {DA+CB = O_n}.
  • (b) {\det(AC) \geq 0} and {(-1)^n \det(BD) \geq 0}.

Problem 4. Let {I\subset \Bbb{R}} be an open interval which contains {0} and {f: I \rightarrow \Bbb{R}} be a function of class {C^{2016}(I)} such that {f(0)=0, f'(0)=1, f''(0) = ... = f^{(2015)}(0)=0,\ f^{(2016)}(0)<0}.

  • (i) Prove that there exists {\delta>0} such that {0<f(x)<x} for {x \in (0,\delta)}.
  • (ii) With {\delta} determined at (i) define the sequence {(a_n)} by

    \displaystyle a_1 = \frac{\delta}{2},\ a_{n+2}=f(a_n),\ n \geq 1.

    Study the convergence of the series {\displaystyle \sum_{n=1}^\infty a_n^r} for {r \in \Bbb{R}}.

Hints: 

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Categories: Olympiad Tags: , , ,

IMC 2014 Day 2 Problem 2

August 1, 2014 5 comments

Let {A=(a_{ij})_{i,j=1}^n} be a symmetric {n \times n} matrix with real entries, and let {\lambda_1,...,\lambda_n} denote its eigenvalues. Show that

\displaystyle \sum_{1 \leq i<j\leq n} a_{ii}a_{jj} \geq \sum_{1 \leq i<j\leq n}\lambda_i \lambda_j

and determine all matrices for which equality holds.

IMC 2014 Day 2 Problem 2

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Two more mind games solved using Matlab

March 31, 2014 1 comment

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.

1. Three in a row

You have a {2n \times 2n} table {(n \geq 3)} and on each row and column you need to have {n} squares coloured with color {A} and {n} squares coloured with color {B} 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.

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Agregation 2014 – Mathematiques Generales – Parts 4-6

March 21, 2014 Leave a comment

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 {n \geq 2} we denote {\Gamma} the set of pairs {(i,j)} such that {1\leq i<j \leq n}. We call the set of inversions of a permutation {\sigma \in S_n} the set

\displaystyle I(\sigma) = \{(i,j) \in \Gamma : \sigma(i)>\sigma(j)\}

and we denote {N(\sigma)} the cardinal of {I(\sigma)}.

1. For which permutations {\sigma \in S_n} is the number {N(\sigma)} maximum?

For {k \in [1..n-1]} we denote {\tau_k \in S_k} the transposition which changes {k} and {k+1}.

4.2 (a) Let {(k,\sigma) \in [1..n-1]\times S_n}. Prove that

\displaystyle N(\tau_k \circ \sigma) = \begin{cases} N(\sigma)+1 & \text{ if }\sigma^{-1}(k) < \sigma^{-1}(k+1)\\ N(\sigma)-1 & \text{ if }\sigma^{-1}(k) > \sigma^{-1}(k+1), \end{cases}

and that {I(\tau_k \circ \sigma)} is obtained from {I(\sigma)} by adding or removing an element of {\Gamma}.

(b) Find explicitly {\sigma^{-1} \circ \tau_k \circ \sigma} in function of the element of {I(\tau_k \circ \sigma)} which makes it differ from {I(\sigma)}.

Let {T= \{ \tau_1,...,\tau_{n-1} \}}. We call word a finite sequence {m=(t_1,...,t_l)} of elements of {T}. We say that {l} is the length of {m} and that the elements {t_1,..,t_l} are the letters of {m}. The case of a void word {(l=0)} is authorized.

A writing of a permutation {\sigma \in S_n} is a word {m=(t_1,...,t_l)} such that {\sigma =(t_1,...,t_l)}. We make the convention that the permutation which corresponds to the void word is the identity.

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Agregation 2014 – Mathematiques Generales – Parts 1-3

March 20, 2014 1 comment

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

Introduction and notations

For {m \leq n} we denote { [m..n] =\{m,m+1,..,n\} }. For an integer {n \geq 1} we denote {S_n} the group of permutations of {[1..n]}.

We say that a square matrix is inferior (superior) unitriangular if it is inferior (superior) triangular and all its diagonal elements are equal to {1}.

For two integers { n \geq 1} and {k \geq 0} we denote {\mathcal{P}_k(n)} the family of {k}-element subsets of {[1..n]}.

Let {m,n} be two positive integers and {A} a {m\times n} matrix with elements in a field {\Bbb{K}}. (all fields are assumed commutative in the sequel) A minor of {A} is the determinant of a square matrix extracted from {A}. We can define for {k \in [1..\min(m,n)]} and {(I,J) \in \mathcal{P}_k(m) \times \mathcal{P}_k(n)} the minor

\displaystyle \left| \begin{matrix}a_{i_1,j_1} & ... & a_{i_1,j_k} \\ \vdots & \ddots & \vdots \\ a_{i_k,j_1} & ... & a_{i_k,j_k} \end{matrix} \right|

where {i_1,..,i_k} (respectively {j_1,..,j_k}) are the elements of {I} (respectively {J}) arranged in increasing order. We denote this minor {\Delta_{I,J}(A)}.

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