#### EDUCATIONAL MATTERS

### A New Series of Olympiad Chess Problems: Chess Board, Rook and Numbers

**Ivan Popov**

*Northern (Arctic) Federal University named after M. V. Lomonosov
Arkhangelsk, Russia*

**Abstract.** The present article proposes a new series of Olympiad problems in Mathematics. It provides an overview of the Olympiad problems, the content of which aims at studying qualitative (possible, acceptable) or quantitative locations of a rook or rooks on a chessboard. The formulations of new tasks are related to the study of combinations of a chessboard with numbered cells and four rooks on it. The rooks are placed on the chessboard in such a way that each of them beats two of the remaining three. All cells of the chessboard are supplied with numbers. The sum of the numbers covered by the rooks is calculated and is called half-sum. It is necessary to determine the possible values of such half-sums. A matrix is juxteposed to a chessboard with numbered cells. The rectangles of the chessboard are associated
with corresponding rectangles of the matrix, which are called its squares. The halfsum
is called a sum of square matrices. Enter a number equal to the number of the
square matrices. This number is studied for its possible values and its symmetry
with respect to a certain number, if it is considered as a function of one variable.
The properties of this number refer to the numerical characteristics of the square
matrices. The four rooks problem turns out to be a problem of determining the
numerical characteristics of the square matrices. The article deals with an example
of solving the four rooks problem and provides an overview of such problems
solved for certain cell numbers.

*Keywords:* olympiad math problem; chess; chess rook; square of the matrix, sum of the squares of a matrix

### The Rearrangement Inequality

**Šefket Arslanagić**

*University of Sarajevo (Bosnia and Herzegovina)*

**Abstract.** In this paper we consider a really very useful inequality, the so called rearrangement inequality, which has may applications and could be used in proving other inequalities. The paper contains a proof of the rearrangement inequality and several examples of its application.

*Keywords:* equality; inequality; rearrangement inequality; permutation; sequence; corollary; example

### Astroid

**Borislav Borisov, Deyan Dimitrov,**

**Nikolay Ninov, Teodor Hristov**

*Mathematics High School – Lovech, Bulgaria*

**Abstract.** The paper presents the results of the Bulgarian sub-team – a part of an
international team of secondary students. The ream was formed for the realization
of the net research project “Encyclopedia of Notable Plane Figures: We Work by
Ourselves”. The research was organized by using the software products GeoGebra,
Geometer’s Sketchpad and Maple. The coordinate method was applied to prove the
derived hypotheses. The cloud service Google was used in the organization of the
net interaction among the participants.

*Keywords:* circle; curves; trajectory; astroid

### Polynomials with Multiple Roots in the Vertices of a Parallelogram

**1)Prof. Sava Grozdev, DSc., 2)Dr. Veselin Nenkov, Assoc. Prof.**

*1)University of Finance, Business Entrepreneurship – Sofia, Bulgaria*

*2)“Nikola Vaptsarov” Naval Academy – Varna, Bulgaria*

**Absract.** A geometric relation is derived between the roots of polynomials of
complex variable with multiple roots in the vertices of a parallelogram and the roots
of their derivatives. As an application some polynomials of real variable with real
coefficients are considered.

Keywords: polynomial; roots of polynomial; parallelogram; rhombus; rectangle;
ellipse; focus; centre

**EDUCATIONAL TECHNOLOGIES**

### Computer Programming in Mathematics Education

**Marin Marinov, Lasko Laskov**

*New Bulgarian University (Bulgaria)*

**Abstract.** In the last few decades, the contemporary informational technologies introduced new challenges in front of the education in mathematics. These challenges concern an essential part of the methodology of education itself. In this paper we present four different directions in which the important knowledge of computer programming may extend the capabilities of the educational process towardsachieving the desired goals.

*Keywords:* mathematics education; computer programming; Wolfram language; Analytical Geometry; continuity; local extrema

### Creating Interactive and Traceable EPUB Learning Content from Moodle Courses

**1) Martin Takev, 2) Miguel Rodríguez-Artacho, 3) Elena Somova**

*1) 3) University of Plovdiv “Paisii Hilendarski” – Plovdiv (Bulgaria)*

*2) UNED University – Madrid (Spain)*

**Abstract.** Technology enhanced learning is shifting from a centralized platform environment to a variety of elements to support and enrich interactions between learners and educational material. In this context, the paper addresses the creation of eBooks in EPUB format based on the existing content in a Moodle environment. The main purpose is to create enriched eBooks capable of supporting and tracking student activity usually reserved to educational environments. The work describes a plug-in developed to support Moodle course translation to EPUB and details potential user tracking formalization using xAPI.

*Keywords:* Learning Analytics; xAPI; educational content modelling; educational authoring; EPUB; e-learning environment

**CONTEST PROBLEMS**

Contest Problems of this Issue

Solutions of the Contest Problems from Issue 5, 2018