**Belarus
**Viachaslau Murashka

Periodical Fractions and Sequences

In this work I considered periodicity of rational fractions and sequences (ai), constructed by the way p, q, r ÎN and p isn’t divisor of r and qаi ≡ r (mod рi), in p-ary number system. I don’t consider as periods (0) and (p–1). In this work I made criterion of periodicity of rational fractions in p-ary number system. Also I found rational fraction’s period (if it is periodical) and preperiod lengths and period. I studied properties of (ai) and showed that most of them are equal to properties of. I showed that if (ai) is periodical so is. I made criterion of equality of (ai) and periods. As corollary from the work I made algorithm of solving congruence qаi ≡ r (mod рi) when i is rather big.

*Stand 17*

**Bulgaria
**Rumen Dangovski

Convex Sets Among Points in an Euclidean Space

We consider the different methods of proving one interesting conjecture. A configuration is being constructed which leads to some new results. The following problem is a new generalization of an open problem in the discrete and computational geometry of *Erdős*-Szekeres type. Let r(n,k) denotes the minimum number of convex k-gons among n points in general position. With the help of geometrical apparatus we set and analyze bounds for r(n,k). A generalization of the problem in higher dimensional space is shown. The author's main contributions are proving that the number of convex k-gons r(n,k) ∈O(n^{k}), considering ideas for computer proof of *Erdős* -Szekeres problem and introducing a higher dimensional equivalent. The ongoing research is focused on improving the bounds for r(n,k) and analyzing the feasibility of the conjecture of *Erdős* -Szekeres.

*Stand 18*

**Bulgaria
**Anton Belev

Kaloyan Bukovski

New Constructions Connected with Isogonal Conjugation. 'Isogonal' Triangles

Let ΔABC be a triangle and k(O) is the circumcircle. Point P is an arbitrary point on the circle k, different of the ΔABC’s vertices and point X is also an arbitrary point, which doesn’t belong to circle k - we construct the isogonal conjugate points A_{XP}, B_{XP}and C_{XP}of point P with respect to ΔBCX, ΔCAX and ΔABX. We call the triangles A_{XP}B_{XP}C_{XP}’isogonal’ triangles of point X with respect to ΔABC. The loci of these triangles’ vertices are shown in the project. It is proven that all isogonal triangles from the set are similar to the pedal triangle of a given point with respect to the given triangle. It is proven also that the isogonal triangles of two inverse points with respect to the circumcircle are congruent. For each point from the plane of the triangle the respective generalized Napoleon’s triangles are constructed. They are similar to the pedal triangles of these points. Some characteristics of the isogonal triangles of remarkable triangle centers are given.

*Stand 19*

**Czech Republic
**David Majer

Martin Prchal

Tereza Konecná

Geometric Reconstructions of Gothic Monuments

The objective of this work is to acquaint with basic gothic elements of geometric view and show right processes during reconstructions of these buildings of Gothic period on one of the Czech gothic monuments – basilica “Assumption Virgin Mary” in Brno. Specifically on window above the main portal of this basilica, which is complicated because of its traceries and we can see then master’s knowledge. Unfortunately, the most of technical plans of this basilica and other monuments, which need to be reconstructed, didn’t survived. We show some processes so gothic monuments can

have their origin meaning for next hundreds years.

*Stand 20*

**Georgia
**Irakli Saralidze

Exact Solutions of Interval Equations of the First Power

In practice frequently it’s impossible to build an exact mathematical model of several physical processes. This motivates us to apply several approximation methods, in practice we have to find out numerals in non exact measuring conditions, for which it is effectively used the methods of interval arithmetic. In 1966 R.Moore initiated uncertainties study by the mathematical methods of interval analyses [1]. Initially he tried to introduce arithmetic operations for uncertainties of interval types. That leads to solutions of numerous problems in computer science, financial analyses, biology etc. In the progress of internal mathematics great role played by E.Hansen [2] and K. Nickel [3]. Main problem of this subject was that there were no formulas which could lead us to exact solutions of equations such as: After some work I have solved this problem and I will show it my topic.

*Stand 21*

**Germany
**Danial Sanusi

Xianghui Zhong

Fabian Henneke

(Almost) Unit-Distance Points in the Polychromatic Plane - Colourings of the n-Dimensional Space

In 1950, Edward Nelson sought to discover the minimum number of colours needed to colour all points of the plane, such that no two points with distance 1 have the same colour. A proof that this number lies between 4 and 7 can be delivered using only elementary methods, but better bounds are not known up to today. However, in this project, Fabian Henneke, Xianghui Zhong and Danial Sanusi show that even a colouring of the plane with 5 colours contains equally coloured pairs of points with distance 'arbitrarily close' to 1. For this, concepts originating from the mathematical field of topology are of unexpected help. They also treat generalisations of this setup to higher-dimensional spaces. But not only colourings of infinitely small points are interesting objects of research: they address colourings of cubic grids, applying these topological tools to prove a theorem related to the fact that the board game 'Hex' always ends with a winner.

*Stand 22*

**Poland
**Michal Miskiewicz

The Charm of the 'mi' Set

My work discusses the following property of a point X in the plane: for a given finite set A the sum of squared distances of points in A from a line through X does not depend on the choice of the line. Among other results, I prove that for any finite set A there are exactly one or two such points. In the first case it is the center of mass of a chosen set, in the latter these two points are symmetric with respect to the center of mass. I provide a few examples of applications of this theorem and I consider a number of special cases, in particular a set of vertices of a triangle.

*Stand 23*

**Russia
**Igor Penkov

The Secret of One Unproved Inequality

Actuality: in the 21st century we often encounter with notions as arithmetical mean, geometric mean and quadratic mean, these notions are applied in physics, mathematics and computer science but unfortunately we don’t know much about relation between them and besides how they connect with each other.

Methods and techniques: Rearrangements of summands; Method of mathematical induction; Dividing inequality for some parts; Method of comparison.

*Stand 24*

**Ukraine
**Anton Pugach

On Inequalities for the Lengths of Perpendiculars Drawn from a Point to the Sides of a Triangle

The given project deals with a research of maximal value of geometric mean and harmonic mean of the power of the distances from the point to the sides of the many-dimensional simplex and polygon on the plane. These values have been assessed. The construction of the points, for which this value is obtained, has also been described.

*Stand 25*

**Ukraine
**Yuriy Zakharyan

Research Varignon’s Theorem, Generalization Wittenbauer’s and Varignon’s Theorems, Development of them and use Discoveries in Practice

Mathematics is science. It helps understand us many things in nature, engineering etc. Also it helps us in deciding problems. Most of them we cannot decide without mathematics. Also it develop logic of the human, it is a gymnastic for our brains. So the development of this science is very important for us. Attractiveness of the mathematic is simplicity and versatility. For development of it, we can research not only difficult things. We can also develop classical mathematics. One of the classical geometrical problem – Varignon’s theorem is our object for research. Much literature was studied, many Olympiad tasks were decided, some of the famous facts were proved. Also some authorial tasks and theorems were created. This is development of the geometry. But also we must search practical utility. This is our main problem in near future.

*Stand 26*