Balkan Mathematical Olympiad – 2016 Problems

Problem 1. Find all injective functions {f: \mathbb R \rightarrow \mathbb R} such that for every real number {x} and every positive integer {n},

\displaystyle \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016

Problem 2. Let {ABCD} be a cyclic quadrilateral with {AB<CD}. The diagonals intersect at the point {F} and lines {AD} and {BC} intersect at the point {E}. Let {K} and {L} be the orthogonal projections of {F} onto lines {AD} and {BC} respectively, and let {M}, {S} and {T} be the midpoints of {EF}, {CF} and {DF} respectively. Prove that the second intersection point of the circumcircles of triangles {MKT} and {MLS} lies on the segment {CD}.

Problem 3. Find all monic polynomials {f} with integer coefficients satisfying the following condition: there exists a positive integer {N} such that {p} divides {2(f(p)!)+1} for every prime {p>N} for which {f(p)} is a positive integer.

Problem 4. The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of {1201} colours so that no rectangle with perimeter {100} contains two squares of the same colour. Show that no rectangle of size {1\times1201} or {1201\times1} contains two squares of the same colour.

SEEMOUS 2016 Problem 4 – Solution

March 6, 2016 3 comments

Problem 4. Let {n \geq 1} be an integer and set

\displaystyle I_n = \int_0^\infty \frac{\arctan x}{(1+x^2)^n}dx.

Prove that

a) {\displaystyle \sum_{i=1}^\infty \frac{I_n}{n} =\frac{\pi^2}{6}.}

b) {\displaystyle \int_0^\infty \arctan x \cdot \ln \left( 1+\frac{1}{x^2}\right) dx = \frac{\pi^2}{6}}.

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SEEMOUS 2016 – Problems

March 5, 2016 3 comments

Problem 1. Let {f} be a continuous and decreasing real valued function defined on {[0,\pi/2]}. Prove that

\displaystyle \int_{\pi/2-1}^{\pi/2} f(x)dx \leq \int_0^{\pi/2} f(x)\cos x dx \leq \int_0^1 f(x) dx.

When do we have equality?

Problem 2. a) Prove that for every matrix {X \in \mathcal{M}_2(\Bbb{C})} there exists a matrix {Y \in \mathcal{M}_2(\Bbb{C})} such that {Y^3 = X^2}.

b) Prove that there exists a matrix {A \in \mathcal{M}_3(\Bbb{C})} such that {Z^3 \neq A^2} for all {Z \in \mathcal{M}_3(\Bbb{C})}.

Problem 3. Let {A_1,A_2,...,A_k} be idempotent matrices ({A_i^2 = A_i}) in {\mathcal{M}_n(\Bbb{R})}. Prove that

\displaystyle \sum_{i=1}^k N(A_i) \geq \text{rank} \left(I-\prod_{i=1}^k A_i\right),

where {N(A_i) = n-\text{rank}(A_i)} and {\mathcal{M}_n(\Bbb{R})} is the set of {n \times n} matrices with real entries.

Problem 4. Let {n \geq 1} be an integer and set

\displaystyle I_n = \int_0^\infty \frac{\arctan x}{(1+x^2)^n}dx.

Prove that

a) {\displaystyle \sum_{i=1}^\infty \frac{I_n}{n} =\frac{\pi^2}{6}.}

b) {\displaystyle \int_0^\infty \arctan x \cdot \ln \left( 1+\frac{1}{x^2}\right) dx = \frac{\pi^2}{6}}.

Some hints follow.

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FreeFem++ Tutorial – Part 1

February 24, 2016 Leave a comment

First of all, FreeFem is a numerical computing software which allows a fast and automatized treatment of a variety of problems related to partial differential equations. Its name, FreeFem, speaks for itself: it is free and it uses the finite element method. Here are a few reasons for which you may choose to use FreeFem for a certain task:

  1. It allows the user to easily define 2D (and 3D) geometries and it does all the work regarding the construction of meshes on these domains.
  2. The problems you want to solve can be easily written in the program once we know their weak forms.
  3. Once we have variables defined on meshes or solutions to some PDE, we can easily compute all sorts of quantities like integral energies, etc.

Before showing a first example, you need to install FreeFem. If you are not familiar with command line work or you just want to get to work, like me, you can install the visual version of FreeFem which is available here. Of course, you can find example programs in the FreeFem manual or by making a brief search on the internet.

I’ll present some basic stuff, which will allow us in the end to solve the Laplace equation in a circular domain. Once we have the structure of the program, it is possible to change the shape of the domain in no time.

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Print your Matlab models in 3D

December 13, 2015 Leave a comment

The emergence of 3D printers opens a whole new level of creation possibilities. Any computer generated model could be materialized as soon as it can be transformed in a language that the 3D printer can use. This is also the case with objects and structures which emerge from various mathematical research topics. Since I’m working on shape optimization problems I have lots of structures that would look nice printed in 3D. Below you can see an example of a 3D model and its physical realization by a 3D printer.

I want to show below how can  you can turn a Matlab coloured patch into a file which can be used by a 3D printer. The first step is to export the Matlab information regarding the position of the points, the face structure and the colours into an obj file format. This is not at all complicated. Vertex information is stored on a line of the form

\displaystyle v\ x\ y\ z\ R\ G\ B

where {v} is exactly the character {v}, {x,y,z} give the coordinates of the points and {R,G,B} give the colour associated to the point in the RGB format. The face information can be entered in a similar fashion:

\displaystyle f\ t_1\ t_2\ t_3

where {t_1, t_2, t_3} are the indices of the points in the corresponding face. Once such an obj file is created, it can be imported in MeshLab (a free mesh editing software). Once you’re in MeshLab you should be able to export the structure into any file format you want, which can be understood by a 3D printer (like STL). Once you have the stl file, you can go on a 3D printing website like Sculpteo and just order your 3D object.

Miklos Schweitzer 2015 Problems

October 31, 2015 Leave a comment

Problem 1. Let {K} be a subset of the closed unit sphere in { \Bbb{R}^3} such that a dense system of chords of the unit sphere is disjoint from {K}. Prove that there exists a closed subset {H} of the unit sphere such that a chord connecting any two points of {H} is disjoint from {K}.

Problem 2. Let {\{x_n\}} be a van der Jorput series, that is, if the binary representation of the positive integer is {n = \sum_i a_i2^i}, then {x_n = \sum_i a_i2^{-i-1}}. Let {V} be the set of points on the plane of the form {(n,x_n)}. Let {G} be the graph with vertex set {V} that is connecting any two points {(p,q)} if and only if there is a rectangle {R} which lies in a parallel position to the axes and {R\cap V = \{p,q\}}. Prove tha tthe chromatic number of {G} is finite.

Problem 3. Let {A} be a finite set and {\rightarrow} be a binary relation on it such that for any {a,b,c \in A}, if {a\neq b}, {a \rightarrow c} and {b \rightarrow c} then either {a \rightarrow b} or {b \rightarrow a} (or possibly both). Let {B \subset A} be minimal such that for any {a \in A \setminus B} there exists {b \in B} such that either {a \rightarrow b} or {b \rightarrow a} (or possibly both). Supposing that {A} has at most {k} elements that are pairwise not in relation {\rightarrow}, prove that {B} has at most {k} elements.

Problem 4. Let {a_n} be a series of positive integers with {a_1 = 1} and for any arbitrary prime number {p} the set {\{a_1,a_2,...,a_p\}} is a complete remainder system modulo {p}. Prove that {\lim_{n\rightarrow \infty} a_n/n = 1}.

Problem 5. We denote {f(x) = x^n+x^{n-1}+...+x+1} for an integer {n \geq 1}. For which {n} are there polynomials {g,h} with real coefficients and degrees smaller than {n} such that {f(x) = g(h(x))}.

Problem 6. Let {G} be a permutation group on the finite set {\Omega}. Consider {S \subset G} such that {1 \in S} and for any {x,y \in \Omega} there is a unique {\sigma \in S} such that {\sigma(x)=y}. Prove that if the elements of {S \setminus \{1\}} are conjugate in {G} then {G} is {2}-transitive on {\Omega}.

Problem 7. We call a bar of width {w} on the surface of the unit sphere {\Bbb{S}^2} centered at the origin a spherical segment which has width {w} and is symmetric with respect to the origin. Prove that there exists a constant {c>0} such that for any positive integer {n} the surface {\Bbb{S}^2} can be covered with {n} bars of the same width so that any point is contained in no more than {c\sqrt{n}} bars.

Problem 8. Prove that all continuous solutions of the functional equation

\displaystyle (f(x)-f(y))\left( f\left(\frac{x+y}{2}\right) -f(\sqrt{xy})\right) \equiv 0,\ x,y \in (0,\infty)

are constant functions.

Problem 9. For a function {u} defined on {G \subset \Bbb{C}} let us denote {Z(u)} the neignborhood of unit raduis of the set of roots of {u}. Prove that for any compact set {K \subset G} there exists a constant {C} such that if {u} is an arbitrary real harmonic function on {G} which vanishes in a point of {K} then

\displaystyle \sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.

Problem 10. Let {f: \Bbb{R} \rightarrow \Bbb{R}} be a continuously differentiable, strictly convex function. Let {H} be a complex Hilbert space and {A,B} be bounded, self adjoint linear operators on {H}. Prove that if {f(A)-f(B) = f'(B)(A-B)} then {A=B}. (not sure if this is right, since I can’t imagine what {f(A)} means)

Problem 11. For {[0,1] \subset E \subset [0,\infty)}, where {E} is composed of a finite number of closed intervals, we start a two dimensional Brownian motion from the point {x<0} terminating when we first hit {E}. Let {p(x)} be the probability of this finish point being in {[0,1]}. Prove that {p(x)} is increasing on the interval {[-1,0)}.

Many thanks to Eles Andras for the translation. Solutions should not be discussed until the end of the competition: November the 2nd 2015.

Source: http://www.bolyai.hu/schweitzer_feladatsor_2015.pdf

Categories: Uncategorized

Even small errors can be fatal

October 14, 2015 Leave a comment

Machines and computers represent numbers as sequences of zeros and ones called bits. The reason for doing this is the simplicity of constructing circuits dealing with two states. This fact coupled with limited memory capacities means that from the start we cannot represent all numbers in machine code. It is true that we can get as close as we want to any number with great memory cost when precision is important, but in fact we always use a fixed precision. In applications this precision is fixed to a number of bits (16, 32, 64) which correspond to significant digits in computations. Doing math operations to numbers represented on bits may lead to loss of information. Consider the following addition using {15} significant digits:

\displaystyle 5.00000000000002+6.00000000000003 = 11.0000000000000,

notice that using only the first {15} significant digits we have made an error of {5\times 10^{-14}}. This may seem small, but if we do not pay attention and we let errors of this kind accumulate we may have a final error which is unacceptable. This is what happened in the case of the Patriot missile case which I’ll discuss briefly below.

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