{"id":40184,"date":"2015-08-04T20:59:00","date_gmt":"2015-08-04T17:59:00","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/"},"modified":"2019-06-24T17:33:14","modified_gmt":"2019-06-24T14:33:14","slug":"mathinfocontent042015","status":"publish","type":"post","link":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles\/mathinfocontent042015\/","title":{"rendered":"Mathematics and Informatics, Number 4\/2015, Volume 58"},"content":{"rendered":"


 <\/a>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 <\/strong><\/p>\n

 <\/p>\n

EUCATIONAL MATTERS\/<\/strong>
EDUCATIONAL ISSUES<\/strong><\/p>\n

\u0421\u044a\u0441\u0442\u0430\u0432\u043d\u0438 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u0421\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043d\u043e-\u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0447\u0435\u043d \u043c\u043e\u0434\u0435\u043b \u0438 MZ-\u043a\u0430\u0440\u0442\u0430 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430. \u0422\u0435\u043a\u0441\u0442\u043e\u0432\u0438 \u0437\u0430\u0434\u0430\u0447\u0438
[Composite Arithmetic Problems. Structure-Technological Model and MZ-map of a Problem. Word Problems]<\/a> \/
\u0417\u0434\u0440\u0430\u0432\u043a\u043e \u041b\u0430\u043b\u0447\u0435\u0432, \u041c\u0430\u0440\u0433\u0430\u0440\u0438\u0442\u0430 \u0412\u044a\u0440\u0431\u0430\u043d\u043e\u0432\u0430 \/ Zdravko Lalchev, Margarita Varbanova<\/em> – 343<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

\u0415\u0434\u0438\u043d \u0443\u0440\u043e\u043a \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0438 \u043a\u043b\u0430\u0441
[A Mathematical Class in Fourth Grade] <\/a>\/
\u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430 \/ Diana Stefanova<\/em> – 375<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

\u0417\u043b\u0430\u0442\u043d\u043e\u0442\u043e \u0441\u0435\u0447\u0435\u043d\u0438\u0435 \u0438 \u0447\u0438\u0441\u043b\u0430\u0442\u0430 \u043d\u0430 \u0424\u0438\u0431\u043e\u043d\u0430\u0447\u0438 \u2013 \u0441\u0438\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u043d\u0438 \u0432\u0440\u044a\u0437\u043a\u0438 \u043c\u0435\u0436\u0434\u0443 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430, \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u043c\u0443\u0437\u0438\u043a\u0430
[The Golden Section and the Fibonacci Numbers \u2013 Synergetic Relationship Between Mathematics, Informatics and Music] <\/a>\/
\u041a\u0430\u043c\u0435\u043b\u0438\u044f \u041a\u043e\u043b\u0435\u0432\u0430, \u0412\u0430\u043b\u0435\u043d\u0442\u0438\u043d \u0411\u0430\u043a\u043e\u0435\u0432 \/ Kamelia Koleva, Valentin Bakoev<\/em> – 382<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

\u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u043e\u0431\u043b\u0430\u0441\u0442\u043d\u0438\u044f \u043a\u0440\u044a\u0433 \u043d\u0430 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0442\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u043e \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u0435 \u201cE\u0432\u0440\u043e\u043f\u0435\u0439\u0441\u043a\u043e \u043a\u0435\u043d\u0433\u0443\u0440\u0443\u201d \u0437\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438
[Analysis of the Paper for the Regional Round of the International Mathematical Competition for University Students \u201cEuropean Kangaroo\u201d] <\/a>\/
\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432, \u0419\u043e\u0440\u0434\u0430\u043d \u041f\u0435\u0442\u043a\u043e\u0432 \/ Veselin Nenkov, Yordan Petkov<\/em> – 403<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

EDUCATIONAL TECHNOLOGY \/<\/strong>
EDUCATIONAL TECHNOLOGIES<\/strong><\/p>\n

\u0414\u0432\u0435 \u0441\u0438\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u043f\u043e\u043b\u044f\u0440\u0438 \u0438 \u0434\u0432\u0430 \u0445\u0430\u0440\u043c\u043e\u043d\u0438\u0447\u043d\u043e \u0441\u043f\u0440\u0435\u0433\u043d\u0430\u0442\u0438 \u043f\u043e\u043b\u044e\u0441\u0430
[Two Symmetric Polar Lines and Two Harmonically Conjugated Poles] <\/a>\/
\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \/ Sava Grozdev, Veselin Nenkov<\/em> – 415<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

Computer-Discovered Mathematics: Cevian Corner Products <\/a>\/
Sava Grozdev, Deko Dekov<\/em> – 426<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

CONTEST PROBLEMS \/<\/strong>
CONTEST PROBLEMS<\/strong><\/p>\n

Contest Problems of this Issue
[Contest Problems of the Issue] – 437<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 \u0431\u0440\u043e\u0439 3, 2015
[Solutions of the Contest Problems from issue 3, 2015] – 438<\/p>\nLog in to read the full text<\/a>\n

 <\/p>\n

READ IN THE LATEST ISSUES OF \u201eAZ BUKI\u201c JOURNALS \/<\/strong>
\u0412 \u041d\u041e\u0412\u0418\u0422\u0415 \u0411\u0420\u041e\u0415\u0412\u0415 \u041d\u0410 \u0421\u041f\u0418\u0421\u0410\u041d\u0418\u042f\u0422\u0410 \u041d\u0410 \u0418\u0417\u0414\u0410\u0422\u0415\u041b\u0421\u0422\u0412\u041e \u201e\u0410\u0417 \u0411\u0423\u041a\u0418\u201c \u0427\u0415\u0422\u0415\u0422\u0415<\/strong> – 441<\/p>\n

GUIDE FOR AUTHORS \/ \u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415<\/strong> – 443<\/p>\n

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 <\/a>\u0421\u042a\u0421\u0422\u0410\u0412\u041d\u0418 \u0410\u0420\u0418\u0422\u041c\u0415\u0422\u0418\u0427\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418. \u0421\u0422\u0420\u0423\u041a\u0422\u0423\u0420\u041d\u041e-\u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0427\u0415\u041d \u041c\u041e\u0414\u0415\u041b \u0418 MZ-\u041a\u0410\u0420\u0422\u0410 \u041d\u0410 \u0417\u0410\u0414\u0410\u0427\u0410\u0422\u0410. \u0422\u0415\u041a\u0421\u0422\u041e\u0412\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong><\/p>\n

1)<\/sup>\u0417\u0434\u0440\u0430\u0432\u043a\u043e \u041b\u0430\u043b\u0447\u0435\u0432, 2)<\/sup>\u041c\u0430\u0440\u0433\u0430\u0440\u0438\u0442\u0430 \u0412\u044a\u0440\u0431\u0430\u043d\u043e\u0432\u0430<\/p>\n

1)<\/sup>Sofia University<\/p>\n

2)<\/sup>St. Cyril and St. Methodius University of Veliko Turnovo<\/p>\n

Abstract.<\/strong> \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430\u0442\u0430 \u0435 \u0441\u044a\u0441\u0440\u0435\u0434\u043e\u0442\u043e\u0447\u0435\u043d\u0430 \u0432\u044a\u0440\u0445\u0443 \u0441\u044a\u0441\u0442\u0430\u0432\u043d\u0438\u0442\u0435 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0432 \u043d\u0430\u0447\u0430\u043b\u043d\u0430\u0442\u0430 \u0443\u0447\u0438\u043b\u0438\u0449\u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430. \u0410\u0432\u0442\u043e\u0440\u0438\u0442\u0435 \u0441\u0435 \u0432\u043e\u0434\u044f\u0442 \u043e\u0442 \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u043d\u0435\u0442\u043e, \u0447\u0435 \u0441\u044a\u0441\u0442\u0430\u0432\u043d\u0430\u0442\u0430 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u0435 \u043a\u043e\u043c\u043f\u043e\u0437\u0438\u0446\u0438\u044f \u043e\u0442 \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0430\u0440\u043d\u0438 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u0412 \u0442\u0430\u0437\u0438 \u0432\u0440\u044a\u0437\u043a\u0430 \u0441\u0430 \u0432\u044a\u0432\u0435\u0434\u0435\u043d\u0438 \u043f\u043e\u043d\u044f\u0442\u0438\u044f\u0442\u0430 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u0430 \u0438 \u0437\u0430\u0434\u0430\u0447\u0430-\u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u0430 \u043d\u0430 \u0441\u044a\u0441\u0442\u0430\u0432\u043d\u0430 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430. \u0414\u0438\u0430\u043b\u0435\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0442\u043e \u0435\u0434\u0438\u043d\u0441\u0442\u0432\u043e \u043d\u0430 \u043f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0430 \u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u0435 \u043d\u0430 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u0435 \u043e\u0442\u0440\u0430\u0437\u0435\u043d\u043e \u0432 \u043f\u043e\u043d\u044f\u0442\u0438\u0435\u0442\u043e \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043d\u043e-\u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0447\u0435\u043d \u043c\u043e\u0434\u0435\u043b \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430 \u0432 \u043d\u0430\u0447\u0430\u043b\u043d\u0430\u0442\u0430 \u0443\u0447\u0438\u043b\u0438\u0449\u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430. \u0421 \u0446\u0435\u043b \u0433\u0440\u0430\u0444\u0438\u0447\u043d\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f\u043d\u0435 \u043d\u0430 \u043c\u043e\u0434\u0435\u043b\u0430 \u043f\u0440\u0438 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430 \u0430\u0432\u0442\u043e\u0440\u0438\u0442\u0435 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430\u0442 \u0434\u0438\u0434\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043f\u043e\u0434\u0445\u043e\u0434, \u043a\u043e\u0439\u0442\u043e \u0435 \u0441\u0432\u044a\u0440\u0437\u0430\u043d \u0441 \u043f\u043e\u0441\u0442\u0440\u043e\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 MZ-\u043a\u0430\u0440\u0442\u0430 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430. \u0421 \u0440\u0435\u0434\u0438\u0446\u0430 \u043f\u0440\u0438\u043c\u0435\u0440\u0438 \u043d\u0430 \u0441\u044a\u0441\u0442\u0430\u0432\u043d\u0438 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0435 \u043f\u043e\u043a\u0430\u0437\u0430\u043d\u0430 \u043e\u0431\u044f\u0441\u043d\u0438\u0442\u0435\u043b\u043d\u0430\u0442\u0430 \u0440\u043e\u043b\u044f \u043d\u0430 MZ-\u043a\u0430\u0440\u0442\u0430\u0442\u0430 \u043f\u0440\u0438 \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u043d\u0435 \u0438 \u043d\u0435\u0439\u043d\u0430\u0442\u0430 \u0435\u0432\u0440\u0438\u0441\u0442\u0438\u0447\u043d\u0430 \u0440\u043e\u043b\u044f \u043f\u0440\u0438 \u0442\u044a\u0440\u0441\u0435\u043d\u0435 \u043d\u0430 \u0438\u0434\u0435\u044f \u0437\u0430 \u0440\u0435\u0448\u0430\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430. \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430\u0442\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0432\u0430 \u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u0442\u043e \u0437\u0430 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438.<\/p>\n

Keywords: primary school mathematics, elementary and composite arithmetic problem, constructive and technological model, MZ-map, arithmetic transformation, inversion<\/p>\n

 <\/p>\n

COMPOSITE ARITHMETIC PROBLEMS. STRUCTURE-TECHNOLOGICAL MODEL AND MZ-MAP OF A PROBLEM. WORD PROBLEMS<\/strong><\/p>\n

Abstract.<\/strong> The paper is focused on composite arithmetic problems in primary school mathematics. The authors follow the concept that the composite arithmetic problem is a composition of connected elementary arithmetic problems. With regard to this the notions of a structure and a problem-component of a composite arithmetic problem are introduced. The dialectic unity of the formulation and the solution of an arithmetic problem reflects on the concept of structural-technological model of a problem in primary school mathematics. Aiming at a graphical representation of the model when studying the problem, the authors suggest a didactical approach, which is connected with the construction of a MZ-map of the problem. In several examples of composite arithmetic problems the explicative role in comprehension of the MZ-map is shown together with its heuristic role in searching an idea for the problem solution. The paper is a continuation of the research of arithmetic problems.<\/p>\n

 <\/p>\n

Prof. Dr. Zdravko Lalchev<\/strong><\/p>\n

Faculty of Preschool and Primary Education<\/p>\n

University of Sofia<\/p>\n

69A, Shipchenski prohod Blvd.<\/p>\n

1574 Sofia, Bulgaria<\/p>\n

 <\/p>\n

Prof. Dr. Margarita Varbanova<\/strong><\/p>\n

Faculty of Mathematics and Informatics<\/p>\n

University of VelikoTarnovo<\/p>\n

3A, Arh. G. Kozarev Blvd.<\/p>\n

5000 Veliko Tarnovo, Bulgaria<\/p>\n

 <\/p>\n

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 <\/a>\u0415\u0414\u0418\u041d \u0423\u0420\u041e\u041a \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410 \u0412 \u0427\u0415\u0422\u0412\u042a\u0420\u0422\u0418 \u041a\u041b\u0410\u0421<\/strong><\/p>\n

Diana Stefanova<\/strong><\/p>\n

\u041e\u0423 \u201e\u041d. \u0412\u0430\u043f\u0446\u0430\u0440\u043e\u0432\u201c \u2013 \u0410\u0441\u0435\u043d\u043e\u0432\u0433\u0440\u0430\u0434<\/p>\n

 <\/p>\n

Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u0441\u043f\u043e\u0434\u0435\u043b\u0435\u043d \u043e\u043f\u0438\u0442 \u043e\u0442\u043d\u043e\u0441\u043d\u043e \u0443\u0440\u043e\u043a \u0441 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043d\u0430 \u0442\u0435\u043c\u0430 \u201e\u041b\u0438\u0446\u0435 \u043d\u0430 \u043f\u0440\u0430\u0432\u043e\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u201c. \u0418\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0438 \u0441\u0430 \u043e\u0440\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u044f\u0442\u0430 \u0438 \u043f\u0440\u043e\u0432\u0435\u0436\u0434\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0443\u0440\u043e\u043a\u0430 \u0441 \u0446\u0435\u043b \u0430\u043a\u0442\u0438\u0432\u0438\u0437\u0438\u0440\u0430\u043d\u0435 \u0438 \u043c\u043e\u0442\u0438\u0432\u0438\u0440\u0430\u043d\u0435 \u0434\u0435\u0439\u043d\u043e\u0441\u0442\u0442\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435 \u0437\u0430 \u0444\u043e\u0440\u043c\u0438-\u0440\u0430\u043d\u0435 \u043d\u0430 \u0442\u0440\u0430\u0439\u043d\u0438 \u0437\u043d\u0430\u043d\u0438\u044f.<\/p>\n

Keywords: activity, rectangle, length, width, circumference, surface<\/p>\n

 <\/p>\n

A MATHEMATICAL CLASS IN FOURTH GRADE<\/strong><\/p>\n

Abstract.<\/strong> The paper shares an experience concerning a class on \u201cRectangle area\u201d. The organization and the realization of the class are examined aiming at activation and motivation of pupils\u2019 activity in the formation of durable knowledge.<\/p>\n

 <\/p>\n

Dr. Diana Stefanova<\/strong><\/p>\n

Teacher in Mathematics<\/p>\n

Primary School \u201cNikola Vaptzarov\u201d<\/p>\n

116, Tsar Ivan Asen II Str.<\/p>\n

4230 Asenovgrad, Bulgaria<\/p>\n

 <\/p>\n

\n

 <\/p>\n

 <\/a>\u0417\u041b\u0410\u0422\u041d\u041e\u0422\u041e \u0421\u0415\u0427\u0415\u041d\u0418\u0415 \u0418 \u0427\u0418\u0421\u041b\u0410\u0422\u0410 \u041d\u0410 \u0424\u0418\u0411\u041e\u041d\u0410\u0427\u0418 \u2013 \u0421\u0418\u041d\u0415\u0420\u0413\u0415\u0422\u0418\u0427\u041d\u0418 \u0412\u0420\u042a\u0417\u041a\u0418 \u041c\u0415\u0416\u0414\u0423 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410, \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0422\u0418\u041a\u0410 \u0418 \u041c\u0423\u0417\u0418\u041a\u0410<\/strong><\/p>\n

1)<\/sup>\u041a\u0430\u043c\u0435\u043b\u0438\u044f \u041a\u043e\u043b\u0435\u0432\u0430, 2)<\/sup>\u0412\u0430\u043b\u0435\u043d\u0442\u0438\u043d \u0411\u0430\u043a\u043e\u0435\u0432<\/p>\n

1)<\/sup>\u041d\u0412\u0423 \u201e\u0412\u0430\u0441\u0438\u043b \u041b\u0435\u0432\u0441\u043a\u0438\u201c<\/p>\n

2)<\/sup>\u0412\u0422\u0423 \u201e\u0421\u0432. \u041a\u0438\u0440\u0438\u043b \u0438 \u041c\u0435\u0442\u043e\u0434\u0438\u0439\u201c<\/p>\n

 <\/p>\n

\u0420\u0435\u0437\u044e\u043c\u0435. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u0438 \u0441\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0437\u0430 \u0437\u043b\u0430\u0442\u043d\u043e\u0442\u043e \u0441\u0435\u0447\u0435\u043d\u0438\u0435 \u0438 \u0447\u0438\u0441\u043b\u0430\u0442\u0430 \u043d\u0430 \u0424\u0438\u0431\u043e\u043d\u0430\u0447\u0438, \u043a\u0430\u043a\u0442\u043e \u0438 \u043c\u0435\u0436\u0434\u0443\u043f\u0440\u0435\u0434\u043c\u0435\u0442\u043d\u0438\u0442\u0435 \u0438\u043c \u0432\u0440\u044a\u0437\u043a\u0438 \u0441 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u0442\u0430 \u0438 \u0441 \u043c\u0443\u0437\u0438\u043a\u0430\u0442\u0430 \u0432 \u0441\u0438\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u0435\u043d \u0430\u0441\u043f\u0435\u043a\u0442.<\/p>\n

Keywords: golden section, fibonacci numbers, recurrence relation, fibonacci tree,<\/p>\n

music, harmony.<\/p>\n

 <\/p>\n

THE GOLDEN SECTION AND THE FIBONACCI NUMBERS \u2212 SYNERGETIC RELATIONSHIP BETWEEN MATHEMATICS, INFORMATICS AND MUSIC<\/strong><\/p>\n

 <\/p>\n

Abstract.<\/strong> This paper represents the basic information about the golden section and the Fibonacci numbers as well as their interdisciplinary connections with informatics and music in a synergetic aspect.<\/p>\n

Dr. Kamelia Koleva, Senior Lecturer<\/p>\n

Land Forces Faculty<\/p>\n

National Military University \u201cVassil Levski\u201d<\/p>\n

76 Bulgaria str.<\/p>\n

5006 Veliko Turnovo, Bulgaria<\/p>\n

 <\/p>\n

Dr. Valentin Bakoev, Assoc. Prof.<\/p>\n

Faculty of Mathematics and Informatics<\/p>\n

University of Veliko Turnovo \u201cSt. Cyril and St. Methodius\u201d<\/p>\n

2 Teodosi Turnovski str.<\/p>\n

5003 Veliko Turnovo, Bulgaria<\/p>\n

 <\/p>\n

\n

 <\/p>\n

 <\/a>\u0410\u041d\u0410\u041b\u0418\u0417 \u041d\u0410 \u0417\u0410\u0414\u0410\u0427\u0418\u0422\u0415 \u041e\u0422 \u041e\u0411\u041b\u0410\u0421\u0422\u041d\u0418\u042f \u041a\u0420\u042a\u0413 \u041d\u0410 \u041c\u0415\u0416\u0414\u0423\u041d\u0410\u0420\u041e\u0414\u041d\u041e\u0422\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u0427\u0415\u0421\u041a\u041e \u0421\u042a\u0421\u0422\u0415\u0417\u0410\u041d\u0418\u0415 \u201e\u0415\u0412\u0420\u041e\u041f\u0415\u0419\u0421\u041a\u041e \u041a\u0415\u041d\u0413\u0423\u0420\u0423\u201c \u0417\u0410 \u0421\u0422\u0423\u0414\u0415\u041d\u0422\u0418<\/strong><\/p>\n

1)<\/sup>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432, 2)<\/sup>\u0419\u043e\u0440\u0434\u0430\u043d \u041f\u0435\u0442\u043a\u043e\u0432<\/strong><\/p>\n

1)<\/sup>\u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u043a\u043e\u043b\u0435\u0436 \u2013 \u041b\u043e\u0432\u0435\u0447<\/p>\n

2)<\/sup>University of Economics \u2013 Varna<\/p>\n

Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u0442\u0435\u043b\u043d\u0430\u0442\u0430 \u0442\u0435\u043c\u0430 \u0437\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438 \u043e\u0442 \u043e\u0431\u043b\u0430\u0441\u0442\u043d\u0438\u044f \u043a\u0440\u044a\u0433 \u043d\u0430 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0442\u043e \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u0435 \u201e\u0415\u0432\u0440\u043e\u043f\u0435\u0439\u0441\u043a\u043e \u043a\u0435\u043d\u0433\u0443\u0440\u0443\u201c \u043f\u0440\u0435\u0437 2015 \u0433. \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u0441\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u0438 \u0435 \u043d\u0430\u043f\u0440\u0430\u0432\u0435\u043d \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u043f\u043e\u0441\u0442\u0438\u0433\u043d\u0430\u0442\u0438\u0442\u0435 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438.<\/p>\n

Keywords: mathematics competition, problem solving, university student<\/p>\n

 <\/p>\n

ANALYSIS OF THE PAPER FOR THE REGIONAL ROUND OF THE INTERNATIONAL MATHEMATICAL COMPETITION FOR UNIVERSITY STUDENTS \u201cEUROPEAN KANGAROO\u201d<\/strong><\/p>\n

Abstract.<\/strong> The paper proposes the competition paper for university students from the regional round of the International mathematical competition \u201cEuropean K\u0430ngar\u043e\u043e\u201d in 2015. Methodological solutions of the problems are considered and an analysis of the results is proposed.<\/p>\n

Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n

Technical College Lovech<\/p>\n

31, Sajko Saev Str.<\/p>\n

Lovech, Bulgaria<\/p>\n

 <\/p>\n

Dr. Yordan Petkov, Chief Assist. Prof.<\/p>\n

University of Economics \u2013 Varna<\/p>\n

77, Kniaz Boris I Blvd.<\/p>\n

9002 Varna, Bulgaria<\/p>\n

 <\/p>\n

\n

 <\/p>\n

 <\/a>\u0414\u0412\u0415 \u0421\u0418\u041c\u0415\u0422\u0420\u0418\u0427\u041d\u0418 \u041f\u041e\u041b\u042f\u0420\u0418 \u0418 \u0414\u0412\u0410 \u0425\u0410\u0420\u041c\u041e\u041d\u0418\u0427\u041d\u041e \u0421\u041f\u0420\u0415\u0413\u041d\u0410\u0422\u0418 \u041f\u041e\u041b\u042e\u0421\u0410<\/strong><\/p>\n

 <\/p>\n

1)<\/strong><\/sup>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, 2)<\/sup>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/strong><\/p>\n

1)<\/sup>VUZF University<\/p>\n

2)<\/sup>\u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u043a\u043e\u043b\u0435\u0436 \u2013 \u041b\u043e\u0432\u0435\u0447<\/p>\n

 <\/p>\n

Abstract.<\/strong> \u041e\u043f\u0438\u0441\u0432\u0430\u0442 \u0441\u0435 \u043d\u044f\u043a\u043e\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430 \u043f\u043e\u043b\u044e\u0441\u0438 \u0438 \u043f\u043e\u043b\u044f\u0440\u0438, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u043e\u0442 \u0447\u0435\u0432\u0438\u0430\u043d\u0438 \u0432 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u0434\u0430\u0434\u0435\u043d \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a.<\/p>\n

Keywords: triangle, conic, Euler curve, GSP<\/p>\n

 <\/p>\n

TWO SYMMETRIC POLAR LINES AND TWO HARMONICALLY CONJUGATED POLES<\/strong><\/p>\n

 <\/p>\n

Abstract.<\/strong> Some properties are described of poles and polar lines, generated by cevians in the plane of a given triangle.<\/p>\n

 <\/p>\n

Prof. Sava Grozdev, DSc<\/strong><\/p>\n

University of Finance, Business and Entrepreneurship<\/p>\n

1, Gusla Str.<\/p>\n

1618 Sofia, Bulgaria<\/p>\n

 <\/p>\n

Dr. Veselin Nenkov, Assoc. Prof.<\/strong><\/p>\n

Technical College Lovech<\/p>\n

31, Sajko Saev Str.<\/p>\n

Lovech, Bulgaria<\/p>\n

 <\/p>\n

\n

 <\/p>\n

 <\/a>COMPUTER-DISCOVERED MATHEMATICS: CEVIAN CORNER PRODUCTS<\/strong><\/p>\n

 <\/p>\n

1)<\/sup>Sava Grozdev, Deko Dekov<\/strong><\/p>\n

1)<\/sup>University of Finance Business and Entrepreneurship<\/p>\n

 <\/p>\n

Abstract.<\/strong> By using the computer program \u201cDiscoverer\u201d, we give theorems about cevian corner products.<\/p>\n

 <\/p>\n

Keywords: cevian corner product, triangle geometry, remarkable point, computer-discovered mathematics, \u0435uclidean geometry, discoverer.<\/p>\n

 <\/p>\n

Prof. Sava Grozdev, DSc<\/strong><\/p>\n

University of Finance, Business and Entrepreneurship<\/p>\n

1, Gusla Str.<\/p>\n

1618 Sofia, Bulgaria<\/p>\n

 <\/p>\n

Dr. Deko Dekov, Assoc. Prof.<\/strong><\/p>\n

81, Zahari Knjazheski Str.<\/p>\n

6000 Stara Zagora, Bulgaria<\/p>\n

\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n

","protected":false},"excerpt":{"rendered":"

 CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415   EUCATIONAL MATTERS\/\u0412\u042a\u041f\u0420\u041e\u0421\u0418 \u041d\u0410 \u041f\u0420\u0415\u041f\u041e\u0414\u0410\u0412\u0410\u041d\u0415\u0422\u041e \u0421\u044a\u0441\u0442\u0430\u0432\u043d\u0438 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u0421\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043d\u043e-\u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0447\u0435\u043d \u043c\u043e\u0434\u0435\u043b \u0438 MZ-\u043a\u0430\u0440\u0442\u0430 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430. \u0422\u0435\u043a\u0441\u0442\u043e\u0432\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 [Composite Arithmetic Problems. Structure-Technological Model and MZ-map of a Problem. Word Problems] \/\u0417\u0434\u0440\u0430\u0432\u043a\u043e \u041b\u0430\u043b\u0447\u0435\u0432, \u041c\u0430\u0440\u0433\u0430\u0440\u0438\u0442\u0430 \u0412\u044a\u0440\u0431\u0430\u043d\u043e\u0432\u0430 \/ Zdravko Lalchev, Margarita Varbanova – 343   \u0415\u0434\u0438\u043d \u0443\u0440\u043e\u043a \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0438 \u043a\u043b\u0430\u0441 [A Mathematical Class in Fourth Grade] […]<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[2882],"tags":[],"yoast_head":"\n"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LVIII, 2015 - \u0410\u0437-\u0431\u0443\u043a\u0438<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\""\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LVIII, 2015 - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\" CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415   EUCATIONAL MATTERS\/\u0412\u042a\u041f\u0420\u041e\u0421\u0418 \u041d\u0410 \u041f\u0420\u0415\u041f\u041e\u0414\u0410\u0412\u0410\u041d\u0415\u0422\u041e \u0421\u044a\u0441\u0442\u0430\u0432\u043d\u0438 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u0421\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043d\u043e-\u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0447\u0435\u043d \u043c\u043e\u0434\u0435\u043b \u0438 MZ-\u043a\u0430\u0440\u0442\u0430 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430. \u0422\u0435\u043a\u0441\u0442\u043e\u0432\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 [Composite Arithmetic Problems. Structure-Technological Model and MZ-map of a Problem. Word Problems] \/\u0417\u0434\u0440\u0430\u0432\u043a\u043e \u041b\u0430\u043b\u0447\u0435\u0432, \u041c\u0430\u0440\u0433\u0430\u0440\u0438\u0442\u0430 \u0412\u044a\u0440\u0431\u0430\u043d\u043e\u0432\u0430 \/ Zdravko Lalchev, Margarita Varbanova – 343   \u0415\u0434\u0438\u043d \u0443\u0440\u043e\u043a \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0438 \u043a\u043b\u0430\u0441 [A Mathematical Class in Fourth Grade] […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/\" \/>\n<meta property=\"og:site_name\" content=\"\u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/Azbuki55\/\" \/>\n<meta property=\"article:published_time\" content=\"2015-08-04T17:59:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-06-24T14:33:14+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/\"},\"author\":{\"name\":\"\",\"@id\":\"\"},\"headline\":\"„\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430“, \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LVIII, 2015\",\"datePublished\":\"2015-08-04T17:59:00+00:00\",\"dateModified\":\"2019-06-24T14:33:14+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/\"},\"wordCount\":1462,\"publisher\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#organization\"},\"articleSection\":[\"\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\"\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/\",\"url\":\"https:\/\/vocedu.azbuki.bg\/matematics\/matharticles\/mathinfocontent042015\/\",\"name\":\"\\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LVIII, 2015 - 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Structure-Technological Model and MZ-map of a Problem. 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