###### Editorial Board Members

**EDUCATIONAL MATTERS**

### About the Vector Space of Third Order
Magic Squares

(in Recreational Mathematics)

**Absract.** The present paper is about magic squares and in particular about the vector space of third order magic squares. The aim is to answer some questions as: which sequence of nine numbers could be “formatted” into a magic square and how this could be done; whether the fact that the number sequence is a nine numbers arithmetic sequence is a necessary and sufficient condition for constructing a magic square; could any number be a magic square characteristic number; is there a “formula” for finding the proper numbers for constructing a magic square and so on. To answer these questions we use linear algebra elementary means for solving systems of linear equations.

*Keywords:* magic squares; vector space; recreational mathematics

**1)Prof. Dr. Zdravko Lalchev, 2)Prof. Dr. Zdravko Lalchev**

**1)Assist. Prof. Irina Voutova**

*1)Department of Classics
Faculty of Classical and Modern Philology
University of Sofia*

*2) University of VelikoTarnovo, VelikoTarnovo, Bulgaria*

### Some New Questions about Sequences of Semi-Prime and Prime Numbers

**Abstract.** Everywhere in Nature and life, prime numbers are used very often: cicadas build their life cycles on them, watchmakers use them to calculate ticks, and in aircraft engines with their help the frequency of air pulses is balanced. However, all these areas of application fade against the backdrop of a fact familiar to every cryptographer: prime numbers are in the very heart of modern computer security, that is, they are directly responsible for protecting everything. If you see the lock in the address bar of the browser, it means that a two-key “handshake” is used, based on primes. And our credit cards are also protected using cryptography based on primes.

During many years prime numbers pull the attention of many mathematicians in the world. There is a number of well-known open questions concerning prime and semi-prime numbers. In this paper, some statements are proposed and new open questions are presented concerning sequences of semi-prime and prime numbers.

*Keywords:* prime numbers; semi-prime numbers; applications

**Mark Applebaum**

*Kaya Academic College of Education – Beer Sheva (Israel)*

### A Paper-Matrix as a Realization Form of the Functions of the Net Research Project “Encyclopedia of Flat Curves: We Write by Ourselves” (on the Work of the Moderator)

**Abstract.** The present paper is devoted to the moderator’s intellectual work on the international research project “Encyclopedia of flat curves”. A process of creating the so called paper-matrix is described. One unexpected circumstance is brought to light: mainly, the content of a paper-matrix is determined by the project general functions, in spite of its reflection on the moderator’s personal views.

*Keywords:* network research project; crowd-sourcing; functions of a project; paper-matrix

**1)Prof. Alexander Yastrebov, 2)Assoc. Prof. Gennady Klekovkin**

*Yaroslavl State Pedagogical University, *

*Yaroslavl, Russian Federation*

*Samara Affiliate of Moscow City Pedagogical University *

*Samara, Russian Federation*

### Concurrent Perpendiculars, Determined by Regular Polygons

**Absract.** The paper is a secondary student elaboration under the guidance of Assoc. Prof. Veselin Nenkov. It contains new results on the topic, which brought the first award to the authors during the national round of the International competition MITE (Methodology and information technologies in education) in

February, 2020. The elaboration is dedicated to concurrent lines generated by points on Euler lines of in-polygons.

*Keywords:* polygon; circle; Euler line; Euler circle; perpendicular; isogonal points

**Ms. Yoana Hristova,
Ms. Genoveva Marinova,
Mr. Nikola Kushev, **

**Mr. Svetoslav Apostolov,
Mr. Tsvetomir Ivanov,**

*Mathematical High School – Lovech*

**EDUCATIONAL TECHNOLOGIES**

### A New Proof of the Feuerbach Theorem

**Abstract.** By using the Paskalev-Tchobanov distance formula, we give a new proof of the famous Feuerbach theorem.

*Keywords:* Paskalev-Tchobanov distance formula; Feuerbach theorem

**1)Sava Grozdev, Hiroshi Okumura, Deko Dekov**

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

### Problem 3 in the IMO’2019 Paper

**Abstract.** The aim of the present note is to discuss Problem 3 on the IMO’2019 paper. The 60th edition of the International Mathematical Olympiad (IMO) took place in the city of Bath, United Kingdom, 11–22 July 2019, with the participation of 621 students from 112 countries. The event is the most prestigious scientific Olympiad for high school students. Problem 3 is one of the difficult ones on the paper. It was solved fully (7 points) by 28 participants, 4 students were marked with 6 points, 9 with 5 points, 5 with 4 points, 6 with 3 points, 3 with 2 points, 46 with 1 point and 520 with 0 points. The mean result of all the 621 participants in the Olympiad is 0, 572, which shows the high level of the problem difficulty.

*Keywords:* Olympiad; problem solving; graph; vertex; edge; connectivity; complete graph

**1)Sava Grozdev, 2)Veselin Nenkov**

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

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

### The Challenge – National Program “Training in IT Career”

**Absract.** The article comments the education of secondary school students during their participation in a computer programming course for obtaining the professional qualification “Application programmer”. European and international recommendations for the elaboration of learning programs for students in the field of Computer Science are analyzed and their comparison with the curriculum and learning programs for the occupation of “Application programmer” is realized. The aims of the presented teaching, the methods and the instruments of the process during the course and the preparation of teachers in the field of Computer sciences are described.

*Keywords:* education; IT career; application programmer; computer science

**Mr. Muharem Mollov, Dr. Gencho Stoitsov**

*University of Plovdiv “Paisii Hilendarski, Plovdiv, Bulgaria*

### Using Educational Mobile Technologies and Applications – A Survey in Trakia University

**Absract.** The knowledge, readiness and perceptions of the trainees for their possible participation in the process of mobile learning have not been studied so far for the students of Trakia University, Stara Zagora. This paper presents the results of a study on the use of educational mobile applications among students from different faculties of the university. Conclusions on the knowledge, experience and necessary mobile applications for the various specialties are drawn, as well as recommendations for expanding the application of mobile applications in higher education

*Keywords:* mobile learning; mobile applications; education; survey

**Dr. Nadezhda Angelova**

*Trakia University -Stara Zagora*

### Impact on Research Visibility Using Structured Data and Social Media Integration

**Abstract.** The paper presents a study on applicability of structured data (metadata) in increasing of scientific articles’ visibility, published on the Web. The usage of structured data in the description of the scientific content is discussed. The brief analyses of the used approaches for scientific publishing and some academic publishing systems are presented. The special attention is given to the metadata schemas (Metadata Object Description Schema, Dublin Core Metadata Element Set and Schema.org) used successfully for the publishing purposes. The appropriate integration of the information on scientific articles within social media (with Open Graph and Twitter Card tags) is also examined.

*Keywords:* structured data; scientific publishing; metadata schemas; academic publishing systems; social media integration

**Nikolay Kasakliev, Elena Somova, Margarita Gocheva**

*University of Plovdiv “Paisii Hilendarski” (Bulgaria)*

### On Some Randomized Algorithms and their Evaluation

**Abstract.** The paper considers implementations of some randomized algorithms in connection with a random Sudoku matrix with the programming language C++. For this purpose we describe the set of all matrices, consisting of elements of the set , such that every row is a permutation. We emphasize the relationship between-matrices and the Sudoku matrices. An algorithm to obtain random matrices is presented. Several auxiliary algorithms that are related to the underlying problem have been described. We evaluated all algorithms according to two criteria – probability evaluation, and time for the generation of random objects and checking a belonging to a specific set. These evaluations are interesting from both theoretical and practical points of view because they are particularly useful in the analysis of computer programs.

*Keywords:* randomized algorithm; random object; permutation; binary matrix; algorithm evaluation; Sudoku matrix

**Krasimir Yordzhev**

*South-West University – Blagoevgrad (Bulgaria)*

**CONTEST PROBLEMS**

### Contest Problems of this Issue

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