{"id":38631,"date":"2014-02-18T21:59:00","date_gmt":"2014-02-18T18:59:00","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/mathinfocontent012014\/"},"modified":"2019-06-18T21:59:00","modified_gmt":"2019-06-18T18:59:00","slug":"mathinfocontent012014","status":"publish","type":"post","link":"https:\/\/foreignlanguages.azbuki.bg\/en\/matematics\/matharticles\/mathinfocontent012014\/","title":{"rendered":"Mathematics and Informatics, Number 1\/2014, Volume 57"},"content":{"rendered":"

 <\/a>CONTENTS<\/strong><\/p>\n

EDUCATIONAL TECHNOLOGIES<\/strong>
 <\/a>
<\/strong>\u0418\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u2013  \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438
Integrating Software Applications \u2013 Some Exemplary Problems<\/a>
\u0410\u043d\u0433\u0435\u043b \u0410\u043d\u0433\u0435\u043b\u043e\u0432 \/ Angel Angelov<\/em> – p. 10<\/p>\n

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\"icon\" Full Text<\/a> <\/p>\n

\u0415\u0432\u0440\u0438\u0441\u0442\u0438\u043a\u0430 \u0441 Excel
Heuristics by Excel<\/a>
\u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432 \/ Plamen Penev<\/em> – p. 18<\/p>\n

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\u041a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u043e \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0430\u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430: \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0432\u0430\u043d\u0435 \u043d\u0430 \u0442\u0435\u043c\u0430 \u043e\u0442 \u0435\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f
Computer-generated mathematics: Elaboration of a Topic from Euclidean Geometry<\/a>
\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432 \/ Sava Grozdev, Deko Dekov<\/em> – p. 34<\/p>\n

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

\u0422\u0435\u043e\u0440\u0435\u0442\u0438\u043a\u043e-\u043f\u0440\u0438\u043b\u043e\u0436\u043d\u0438 \u043e\u0441\u043d\u043e\u0432\u0438 \u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e-\u0430\u043b\u0433\u0435\u0431\u0440\u0438\u0447\u043d\u043e \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f
Theoretical-practical Base of Vector- algebraic Modeling in Geometry Education<\/a>
\u0418\u0440\u0438\u043d\u0430 \u0412\u0443\u0442\u043e\u0432\u0430 \/ Irina Voutova<\/em> – p. 43<\/p>\n

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\u0415\u0434\u043d\u043e \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u0435 \u043d\u0430 \u0442\u0435\u043e\u0440\u0435\u043c\u0430\u0442\u0430 \u043d\u0430 \u043f\u0438\u0442\u0430\u0433\u043e\u0440 \u0432 \u0438\u0437\u0432\u044a\u043d\u043a\u043b\u0430\u0441\u043d\u0430\u0442\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430
\u0410 Generalization of the Pythagorean Theorem in Extracurricular Classes of Mathematics<\/a>
\u0420\u0443\u043c\u044f\u043d\u0430 \u041d\u0435\u0441\u0442\u043e\u0440\u043e\u0432\u0430 \/ Rumjana Nestorova<\/em> – p. 59<\/p>\n

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EDUCATIONAL RESEARCH<\/strong><\/p>\n

Some inequalities In the triangle<\/a>
\u0160efket Arslanagi\u0107<\/em> – p. 64<\/p>\n

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 \"icon\" Full Text<\/a><\/p>\n

\u041d\u044f\u043a\u043e\u043b\u043a\u043e \u0445\u043e\u043c\u043e\u0442\u0435\u0442\u0438\u0447\u043d\u043e \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430 \u0444\u043e\u0439\u0435\u0440\u0431\u0430\u0445\u043e\u0432\u0438\u0442\u0435 \u043a\u043e\u043d\u0444\u0438\u0433\u0443\u0440\u0430\u0446\u0438\u0438
Several Homothetically Generated Properties Of The Feuerbach Configurations<\/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, \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432 \/ Sava Grozdev, Veselin Nenkov, Deko Dekov<\/em> – p. 68<\/p>\n

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COMPETITIONS AND OLYMPIADS<\/p>\n

<\/strong>\u0415\u0434\u0438\u043d\u0430\u0434\u0435\u0441\u0435\u0442\u0430 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u043e \u043b\u0438\u043d\u0433\u0432\u0438\u0441\u0442\u0438\u043a\u0430
Eleventh International Linguistics Olympiad<\/a>
\u0418\u0432\u0430\u043d \u0414\u0435\u0440\u0436\u0430\u043d\u0441\u043a\u0438 \/ Ivan Derzhanski<\/em> – p. 73<\/p>\n

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OUR TEACHERS<\/strong> <\/p>\n

\u0415\u0434\u0438\u043d \u043a\u043b\u0430\u0441\u0438\u0447\u0435\u0441\u043a\u0438 \u0443\u0447\u0435\u0431\u043d\u0438\u043a \u043f\u043e \u0430\u043b\u0433\u0435\u0431\u0440\u0430
A Classical Textbook in Algebra<\/a>
\u0414\u043e\u043d\u043a\u0430 \u041f\u0430\u0448\u043a\u0443\u043b\u0435\u0432\u0430 \/ Donka Pashkuleva<\/em>  – p. 79<\/p>\n

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 \"icon\" Full Text<\/a><\/p>\n

\u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 \/  CONTEST PROBLEMS<\/strong>
Contest Problems of this Issue<\/span>
[Contest Problems of the Issue] – p. 84<\/p>\n

\"icon\" Full text<\/a><\/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 2, 2013<\/span> 
[Solutions of the Contest Problems from issue 2, 2013] – p. 86<\/p>\n

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\"icon\" Full Text<\/a><\/p>\n

\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>
READ IN THE LATEST ISSUES OF \u201eAZ BUKI\u201c JOURNALS<\/strong> – p. 89<\/p>\n

\u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415 \/ GUIDE FOR AUTHORS<\/strong> – p. 91<\/p>\n

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 <\/a>\u0418\u041d\u0422\u0415\u0413\u0420\u0418\u0420\u0410\u041d\u0415 \u041d\u0410 \u041f\u0420\u0418\u041b\u041e\u0416\u0415\u041d\u0418\u042f \u2013  \u041f\u0420\u0418\u041c\u0415\u0420\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong><\/p>\n

Angel Angelov<\/strong>
Sofia University<\/em><\/p>\n

Abstract.<\/strong> \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0435 \u043f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430, \u0440\u0435\u0448\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u043a\u043e\u044f\u0442\u043e \u0435 \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u043e \u0441 \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0430\u043d\u043e \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u043f\u0440\u0438\u043b\u043e\u0436\u043d\u0438 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438, \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0438 \u0432  \u0443\u0447\u0438\u043b\u0438\u0449\u043d\u0438\u044f \u043a\u0443\u0440\u0441 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438. \u041f\u043e\u0441\u043e\u0447\u0435\u043d\u0438 \u0441\u0430 \u0448\u0438\u0440\u043e\u043a \u043a\u0440\u044a\u0433 \u043e\u0442 \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u0438 \u043d\u0430 \u0443\u0447\u0435\u0431\u043d\u043e\u0442\u043e \u0441\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 (\u0432 \u0442\u0435\u043e\u0440\u0435\u0442\u0438\u0447\u0435\u043d \u0438 \u043c\u0430\u043d\u0438\u043f\u0443\u043b\u0430\u0442\u0438\u0432\u0435\u043d \u043f\u043b\u0430\u043d), \u043a\u043e\u0438\u0442\u043e \u0441\u0430 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u0438 \u0437\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0442\u0430\u0437\u0438 \u0437\u0430\u0434\u0430\u0447\u0430. \u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0430 \u0435 \u0438 \u0438\u0434\u0435\u044f \u0437\u0430 \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u043f\u043e\u0434\u043e\u0431\u043d\u0438, \u043f\u0440\u0438 \u0442\u043e\u0432\u0430 \u0433\u0440\u0430\u0434\u0438\u0440\u0430\u043d\u0438 \u043f\u043e \u0442\u0440\u0443\u0434\u043d\u043e\u0441\u0442 \u0437\u0430\u0434\u0430\u0447\u0438. \u0417\u0430\u0434\u0430\u0447\u0438 \u0441 \u043f\u043e\u0434\u043e\u0431\u043d\u0438 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043d\u0438 \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u0438 \u0441\u0430 \u0435\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e \u0434\u0438\u0434\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u043e \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u043e \u0432 \u0443\u0440\u043e\u0447\u043d\u0430\u0442\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 \u043f\u043e \u043c\u043e\u0434\u0443\u043b \u201e\u0418\u043d- \u0442\u0435\u0433\u0440\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0434\u0435\u0439\u043d\u043e\u0441\u0442\u0438\u201c, \u0430 \u0442\u0430\u043a\u0430 \u0441\u044a\u0449\u043e \u0438 \u0437\u0430 \u0434\u0438\u0430\u0433\u043d\u043e\u0441\u0442\u0438\u0446\u0438\u0440\u0430\u043d\u0435 \u0438 \u043e\u0446\u0435\u043d\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438\u0442\u0435 \u043e\u0442 \u0446\u044f\u043b\u043e\u0441\u0442\u043d\u0438\u044f \u0446\u0438\u043a\u044a\u043b \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438.<\/p>\n

Keywords:<\/em> word processing, integrated activities,  integrated document, didactical problems<\/p>\n

INTEGRATING SOFTWARE APPLICATIONS \u2013 SOME EXEMPLARY PROBLEMS<\/strong><\/p>\n

Abstract<\/strong>. The present article deals with a problem, the solution of which is related to the integrated use of the basic software applications studied within the high school course of Information Technologies (IT). A large group of components of the school curriculum necessary for solving the problem are considered (both in theoretical and manipulation aspects). An idea on generating similar problems ordered by level of dif- ficulty is presented as well. Problems embracing structural components of the kind are an efficient didactic tool in a class setting when working on the module \u201cIntegrating of activities\u201d. Furthermore, such problems can be used for diagnosing and evaluating the results of the whole IT learning cycle. <\/p>\n

Dr. Angel Angelov,<\/strong>
Assoc. Prof. Faculty of Mathematics and Informatics
University of Sofia<\/p>\n

top<\/a><\/p>\n

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 <\/a>\u0415\u0412\u0420\u0418\u0421\u0422\u0418\u041a\u0410 \u0421 EXCEL<\/strong><\/p>\n

Plamen Penev<\/strong>
\u0421\u041e\u0423 \u201e\u041f\u0430\u043d\u0430\u0439\u043e\u0442 \u0412\u043e\u043b\u043e\u0432\u201c
<\/em>
Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u043e\u043f\u0438\u0441\u0432\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0448\u0438\u0440\u043e\u043a \u043a\u0440\u044a\u0433 \u043e\u0442 \u0447\u0438\u0441\u043b\u043e\u0432\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0441 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u0430 \u0442\u0430\u0431\u043b\u0438\u0446\u0430. \u0427\u0440\u0435\u0437 \u0440\u0430\u0437\u0448\u0438\u0440\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0435\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430\u043b\u043d\u0430\u0442\u0430 \u043e\u0431\u043b\u0430\u0441\u0442 \u043d\u0430 \u043d\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043c\u0430\u0442\u0430 \u043f\u0440\u043e\u043c\u0435\u043d\u043b\u0438\u0432\u0430 \u0441\u0435 \u043f\u0440\u043e\u0432\u0435\u0436\u0434\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u0435\u043d \u0430\u043d\u0430\u043b\u0438\u0437 \u0437\u0430 \u043b\u043e\u043a\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d\u0435 \u0438 \u043d\u0430\u043c\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u043a\u043e\u0440\u0435\u043d. \u0412 \u0441\u043b\u0443\u0447\u0430\u0438\u0442\u0435, \u043a\u043e\u0433\u0430\u0442\u043e \u0442\u0430\u0431\u043b\u0438\u0447\u043d\u0438\u044f\u0442 \u043c\u0435\u0442\u043e\u0434 \u0441\u0435 \u043e\u043a\u0430\u0437\u0432\u0430 \u043d\u0435\u0434\u043e\u0441\u0442\u0430\u0442\u044a\u0447\u0435\u043d, \u0441\u0435 \u0434\u043e\u0431\u0430\u0432\u044f\u0442 \u043c\u0430\u043a\u0440\u043e\u0441\u0438, \u0430\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u0437\u0438\u0440\u0430\u0449\u0438 \u0438\u0437\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f\u0442\u0430. \u0412\u043e\u0434\u0435\u0449\u0430 \u0440\u043e\u043b\u044f \u0441\u0435 \u043e\u0442\u0440\u0435\u0436\u0434\u0430 \u043d\u0430 \u0433\u0440\u0430\u0444\u0438\u0447\u043d\u0438\u044f \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b. <\/p>\n

Keywords:<\/em> problem solving, equation, numbers, tables, macros<\/p>\n


HEURISTICS BY EXCEL<\/strong><\/p>\n

Abstract<\/strong>. The paper considers methodology for a large scale mathematical problem solving by Electronic Table. A consecutive analysis is produced for localization and argumentation of the solutions by augmenting the experimentation domain. When the table method is insufficient some macros are added, which makes computing automatic. The leading role is prescribed for graphical material.  <\/p>\n

Mr. Plamen Penev<\/strong>
Teacher in Informatics
\u201ePanayot Volov\u201c School<\/p>\n

top<\/a><\/p>\n

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 <\/a>\u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u041e \u0413\u0415\u041d\u0415\u0420\u0418\u0420\u0410\u041d\u0410 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410: \u0420\u0410\u0417\u0420\u0410\u0411\u041e\u0422\u0412\u0410\u041d\u0415 \u041d\u0410 \u0422\u0415\u041c\u0410 \u041e\u0422 \u0415\u0412\u041a\u041b\u0418\u0414\u041e\u0412\u0410\u0422\u0410 \u0413\u0415\u041e\u041c\u0415\u0422\u0420\u0418\u042f<\/strong><\/p>\n

\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432<\/strong>
Bulgarian Academy of Sciences (Bulgaria)<\/em><\/p>\n

Abstract.<\/strong> \u0418\u043b\u044e\u0441\u0442\u0440\u0438\u0440\u0430\u043d\u043e \u0435 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0442\u0435\u043c\u0430 \u043e\u0442 \u0415\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430\u0442\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c. \u041a\u0430\u0442\u043e \u0438\u0437\u0445\u043e\u0434\u0435\u043d \u043f\u0443\u043d\u043a\u0442 \u0441\u0435 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430 \u043e\u0442\u0434\u0435\u043b\u043d\u0430 \u0442\u0435\u043e\u0440\u0435\u043c\u0430 \u0438\u043b\u0438 \u0433\u0440\u0443\u043f\u0430 \u043e\u0442 \u0441\u0445\u043e\u0434\u043d\u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u0438. \u0412 \u0442\u0430\u0437\u0438 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0430 \u0434\u0430\u0434\u0435\u043d\u0438 283 \u0442\u0435\u043e\u0440\u0435\u043c\u0438, \u043e\u0442\u043a\u0440\u0438\u0442\u0438 \u043e\u0442 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430\u0442\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c. \u0410\u0432\u0442\u043e\u0440\u0438\u0442\u0435 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430\u0442, \u0447\u0435 \u0435\u0434\u043d\u0430 \u0447\u0430\u0441\u0442 \u043e\u0442 \u0442\u044f\u0445 \u0441\u0430 \u043d\u043e\u0432\u0438. \u0422\u0435\u043e\u0440\u0435\u043c\u0438\u0442\u0435, \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438 \u043e\u0442 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c, \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0431\u044a\u0434\u0430\u0442 \u043f\u0440\u0435\u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u0430\u043d\u0438 \u043a\u0430\u0442\u043e \u0437\u0430\u0434\u0430\u0447\u0438, \u0432 \u043e\u0431\u0449\u0438\u044f \u0441\u043b\u0443\u0447\u0430\u0439 \u043f\u043e \u043f\u043e\u0432\u0435\u0447\u0435 \u043e\u0442 \u0435\u0434\u0438\u043d \u043d\u0430\u0447\u0438\u043d. <\/p>\n

Keywords:<\/em> computer-generated mathematics, Euclidean geometry, Discoverer, mathematical olympiad<\/p>\n

COMPUTER-GENERATED MATHEMATICS: ELABORATION OF A TOPIC FROM EUCLIDEAN GEOMETRY<\/strong><\/p>\n

Abstract<\/strong>. In the paper an elaboration of a topic from Euclidean Geometry by the computer program \u2018Discoverer\u2019 is shown. A theorem or a group of similar theorems are used as starting points. The paper presents 283 theorems which are discovered by the computer program \u2018Discoverer\u2019. The authors presume that some of them are new. The theorems produced by \u2018Discoverer\u2019 could be reformulated as problems, in more than one way in the general case.
 <\/p>\n

Prof. Sava Grozdev,<\/em><\/strong>
DSc in Pedagogy Institute of Mathematics and Informatics \u2013 BAS <\/p>\n

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

top<\/a><\/p>\n

\n

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

\u0418\u0440\u0438\u043d\u0430 \u0412\u0443\u0442\u043e\u0432\u0430 <\/strong>
Sofia University<\/em><\/p>\n

Abstract.<\/strong> \u0420\u0430\u0431\u043e\u0442\u0430\u0442\u0430 \u0441\u0435 \u0441\u044a\u0441\u0442\u043e\u0438 \u043e\u0442 \u0434\u0432\u0435 \u0447\u0430\u0441\u0442\u0438 \u2013 \u0442\u0435\u043e\u0440\u0435\u0442\u0438\u0447\u043d\u0430 \u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u043d\u0430. \u0412 \u0442\u0435\u043e\u0440\u0435\u0442\u0438\u0447\u043d\u0430\u0442\u0430 \u0447\u0430\u0441\u0442 \u0435 \u043d\u0430\u043f\u0440\u0430\u0432\u0435\u043d\u0430 \u0441\u0435\u043b\u0435\u043a\u0446\u0438\u044f \u0438 \u043a\u043b\u0430\u0441\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u044f \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u043f\u043e\u043d\u044f\u0442\u0438\u044f (\u043e\u0431\u0435\u043a\u0442\u0438  \u0438 \u0440\u0435\u043b\u0430\u0446\u0438\u0438) \u0438 \u0442\u0432\u044a\u0440\u0434\u0435\u043d\u0438\u044f (\u0430\u043a\u0441\u0438\u043e\u043c\u0438 \u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u0438) \u043e\u0442 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0430\u0442\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u043d\u0430 \u0412\u0430\u0439\u043b \u0437\u0430 \u0446\u0435\u043b\u0438\u0442\u0435 \u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e-\u0430\u043b\u0433\u0435\u0431\u0440\u0438\u0447\u043d\u043e\u0442\u043e \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435 \u0432 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0430\u0440\u043d\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f. \u0412 \u043f\u0440\u0438\u043b\u043e\u0436\u043d\u0430\u0442\u0430 \u0447\u0430\u0441\u0442 \u0441\u0430 \u043f\u043e\u043c\u0435\u0441\u0442\u0435\u043d\u0438 \u0432 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d \u0440\u0435\u0434 \u0438 \u0441\u0430 \u0440\u0435\u0448\u0435\u043d\u0438 \u0441 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430  \u0448\u0435\u0441\u0442 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u043a\u043e\u0438\u0442\u043e \u043e\u0441\u0432\u0435\u043d \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u0435\u043d\u0438 \u0438\u043b\u044e\u0441\u0442\u0440\u0438\u0440\u0430\u0442 \u0438 \u0435\u0432\u0440\u0438\u0441\u0442\u0438\u0447\u043d\u0438 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e-\u0430\u043b\u0433\u0435\u0431\u0440\u0438\u0447\u043d\u043e\u0442\u043e \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f.<\/p>\n

Keywords:<\/em> vector algebra, modeling, education, geometry, heuristics<\/p>\n

THEORETICAL-PRACTICAL BASE OF VECTOR-ALGEBRAIC MODELING IN GEOMETRY EDUCATION<\/strong><\/p>\n

Abstract<\/strong>. The paper consists of two parts \u2013 theoretical and applied. In the theoretical part a selection and classification of the basic concepts (objects and relations) and statements (axioms and theorems) of Veil\u2019s geometric system has been made, with the help of which a vector-algebraic modeling of concepts and statements from the Euclidean geometry could be established. Six geometric problems have been chosen and presented in the applied part. The problems are solved, using vector means. It is shown that in geometry education vector-algebraic modeling can be used not only for proving statements but as a heuristic possibility as well. <\/p>\n

Irina Voutova, Assist. Prof.<\/strong>
Faculty of Mathematics and Informatics
Sofia University \u201eSt. Kliment Ohridski\u201c  <\/p>\n

top<\/a><\/p>\n

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 <\/a>\u0415\u0414\u041d\u041e \u041e\u0411\u041e\u0411\u0429\u0415\u041d\u0418\u0415 \u041d\u0410 \u0422\u0415\u041e\u0420\u0415\u041c\u0410\u0422\u0410 \u041d\u0410 \u041f\u0418\u0422\u0410\u0413\u041e\u0420 \u0412 \u0418\u0417\u0412\u042a\u041d\u041a\u041b\u0410\u0421\u041d\u0410\u0422\u0410 \u0420\u0410\u0411\u041e\u0422\u0410 \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n

\u0420\u0443\u043c\u044f\u043d\u0430 \u041d\u0435\u0441\u0442\u043e\u0440\u043e\u0432\u0430<\/strong>
\u0420\u0435\u0433\u0438\u043e\u043d\u0430\u043b\u0435\u043d \u0438\u043d\u0441\u043f\u0435\u043a\u0442\u043e\u0440\u0430\u0442 \u043f\u043e \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u0442\u043e – \u0412\u0440\u0430\u0446\u0430<\/em><\/p>\n

Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u0442\u0435\u043e\u0440\u0435\u043c\u0430\u0442\u0430 \u043d\u0430 \u041f\u0438\u0442\u0430\u0433\u043e\u0440, \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u0435\u0442\u043e \u0439 \u0437\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u0435\u043d \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a \u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f. \u041f\u043e\u0434\u0445\u043e\u0434\u044f\u0449\u0430 \u0435 \u0437\u0430 \u0438\u0437\u044f\u0432\u0435\u043d\u0438 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043f\u043e  \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 7. \u0438 8. \u043a\u043b\u0430\u0441.<\/p>\n

Keywords:<\/em> Pythagorean theorem, generalization, proof, application<\/p>\n


A GENERALIZATION OF THE PYTHAGORAEN THEOREM IN EXTRACURRICULAR CLASSES OF MATHEMATICS<\/strong><\/p>\n

Abstract<\/strong>. The paper considers the Pythagorean theorem, its generalization for arbitrary triangle and exemplary applications. It is suitable for outstanding students in the 7th and 8th grade. <\/p>\n

Mrs. Rumjana Nestorova <\/strong>
Senior expert in Mathematics<\/p>\n

top<\/a><\/p>\n

\n

 <\/a>SOME INEQUALITIES IN THE TRIANGLE <\/strong><\/p>\n

\u0160efket Arslanagi\u0107<\/strong>
University of Sarajevo<\/em><\/p>\n

Abstract<\/strong>. The paper considers some inequalities in the triangle connecting the semi- perimeter, the circumradius and the inradius. Keywords: triangle, semi-perimeter, circumradius, inradius, inequality<\/p>\n

Keywords:<\/em> triangle, semi-perimeter, circumradius, inradius, inequality<\/p>\n

top<\/a><\/p>\n

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 <\/a>\u041d\u042f\u041a\u041e\u041b\u041a\u041e \u0425\u041e\u041c\u041e\u0422\u0415\u0422\u0418\u0427\u041d\u041e \u041f\u041e\u0420\u041e\u0414\u0415\u041d\u0418 \u0421\u0412\u041e\u0419\u0421\u0422\u0412\u0410 \u041d\u0410 \u0424\u041e\u0419\u0415\u0420\u0411\u0410\u0425\u041e\u0412\u0418\u0422\u0415 \u041a\u041e\u041d\u0424\u0418\u0413\u0423\u0420\u0410\u0426\u0418\u0418<\/strong>

1\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, <\/strong>
2\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432, <\/strong>
\u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432<\/strong>
1\u0411\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0430 \u0430\u043a\u0430\u0434\u0435\u043c\u0438\u044f \u043d\u0430 \u043d\u0430\u0443\u043a\u0438\u0442\u0435, <\/em>
2\u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u043a\u043e\u043b\u0435\u0436 \u2013 \u041b\u043e\u0432\u0435\u0447<\/em><\/p>\n

Abstract.<\/strong> \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u043d\u0430 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043e\u0442 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f\u0442\u0430 \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u0432\u043f\u0438\u0441\u0430\u043d\u0438\u0442\u0435 \u0438 \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u0442\u0435 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442\u0438,. \u0422\u0435\u0437\u0438 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u0441\u0430 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430 \u0435\u0434\u043d\u0430 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u043d\u0430 \u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0446\u0438\u044f \u043e\u0442 \u0446\u0435\u043d\u0442\u0440\u0430\u043b\u043d\u0438 \u043a\u043e\u043d\u0438\u0447\u043d\u0438 \u0441\u0435\u0447\u0435\u043d\u0438\u044f \u0432 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430, \u043d\u0430\u0440\u0435\u0447\u0435\u043d\u0430 \u0432 (\u041d\u0435\u043d\u043a\u043e\u0432, 2010) \u0424\u043e\u0439\u0435\u0440\u0431\u0430\u0445\u043e\u0432\u0430 \u043a\u043e\u043d\u0444\u0438\u0433\u0443\u0440\u0430\u0446\u0438\u044f.<\/p>\n

Keywords:<\/em> triangle, homothety, Feuerbach configuration, Euler curve<\/p>\n


SEVERAL HOMOTHETICALLY GENERATED PROPERTIES OF THE FEUERBACH CONFIGURATIONS<\/strong><\/p>\n

Abstract<\/strong>. The present paper considers generalizations of some known properties from the geometry of the triangle, which are connected with the circum-circles and the in-circles. The generalizations are several properties of a special combination of central conics, which is called to be Feuerbach configuration in (\u041d\u0435\u043d\u043a\u043e\u0432, 2010).<\/p>\n

Prof. Sava Grozdev,<\/strong>
DSc in Pedagogy Institute of Mathematics and Informatics \u2013 BAS<\/p>\n

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

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

top<\/a><\/p>\n

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 <\/a>\u0415\u0414\u0418\u041d\u0410\u0414\u0415\u0421\u0415\u0422\u0410 \u041c\u0415\u0416\u0414\u0423\u041d\u0410\u0420\u041e\u0414\u041d\u0410 \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u0410 <\/strong>\u041f\u041e \u041b\u0418\u041d\u0413\u0412\u0418\u0421\u0422\u0418\u041a\u0410<\/p>\n

Ivan Derzhanski<\/strong>
\u0411\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0430 \u0430\u043a\u0430\u0434\u0435\u043c\u044f \u043d\u0430 \u043d\u0430\u0443\u043a\u0438\u0442\u0435<\/em><\/p>\n

Abstract<\/strong>. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0430\u0442\u0430 \u043f\u0440\u0435\u0437 2013 \u0433. \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u043e \u043b\u0438\u043d\u0433\u0432\u0438\u0441\u0442\u0438\u043a\u0430. \u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0441\u0430 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438\u0442\u0435 \u043d\u0430 \u0431\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0438\u0442\u0435 \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u0446\u0438, \u0447\u0430\u0441\u0442 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u0438 \u0442\u0435\u0445\u043d\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f.<\/p>\n

Keywords:<\/em> linguistics, Olympiad, problem solving<\/p>\n


ELEVENTH INTERNATIONAL LINGUISTICS OLYMPIAD<\/strong><\/p>\n

Abstract.<\/strong> The paper is dedicated to the International Linguistics Olympiad in 2013. The results of the Bulgarian participants, some of the tasks and their solutions are presented. <\/p>\n

Dr. Ivan Derzhanski, Assoc. Prof.<\/strong>
Institute of Mathematics and Informatics \u2013  BAS<\/p>\n

top<\/a><\/p>\n

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 <\/a>\u0415\u0414\u0418\u041d \u041a\u041b\u0410\u0421\u0418\u0427\u0415\u0421\u041a\u0418 \u0423\u0427\u0415\u0411\u041d\u0418\u041a \u041f\u041e \u0410\u041b\u0413\u0415\u0411\u0420\u0410 <\/p>\n

\u0414\u043e\u043d\u043a\u0430 \u041f\u0430\u0448\u043a\u0443\u043b\u0435\u0432\u0430<\/strong>
Bulgarian Academy of Sciences (Bulgaria)<\/em><\/p>\n

Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f \u0443\u0447\u0435\u0431\u043d\u0438\u043a \u043f\u043e \u0430\u043b\u0433\u0435\u0431\u0440\u0430 \u043d\u0430 \u0438\u0437\u0442\u044a\u043a\u043d\u0430\u0442\u0438\u0442\u0435 \u0431\u044a\u043b\u0433\u0430\u0440\u0441\u043a\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0446\u0438 \u0430\u043a\u0430\u0434e\u043c\u0438\u043a \u041b\u044e\u0431\u043e\u043c\u0438\u0440 \u0418\u043b\u0438\u0435\u0432 \u0438 \u043f\u0440\u043e\u0444\u0435\u0441\u043e\u0440 \u0421\u043f\u0430\u0441 \u041c\u0430\u043d\u043e\u043b\u043e\u0432, \u043a\u043e\u0439\u0442\u043e \u043d\u0435 \u0435 \u0437\u0430\u0433\u0443\u0431\u0438\u043b \u0430\u043a\u0442\u0443\u0430\u043b\u043d\u043e\u0441\u0442 \u0438 \u0434\u043d\u0435\u0441. <\/p>\n

Keywords:<\/em> algebra, textbook, polynomial, function, arithmetic progression<\/p>\n


A CLASSICAL TEXTBOOK IN ALGEBRA <\/strong><\/p>\n

Abstract.<\/strong> The paper presents a textbook in Algebra by the outstanding Bulgarian mathematicians Academician Ljubomir Iliev and Professor Spas Manolov, which still is on modern lines.<\/p>\n

Dr. Donka Pashkuleva, Assoc. Prof.<\/strong>
Institute of Mathematics and Informatics \u2013 BAS<\/p>\n

top<\/a><\/p>\n

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 <\/p>","protected":false},"excerpt":{"rendered":"

 CONTENTS EDUCATIONAL TECHNOLOGIES \u0418\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u2013  \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438Integrating Software Applications \u2013 Some Exemplary Problems\u0410\u043d\u0433\u0435\u043b \u0410\u043d\u0433\u0435\u043b\u043e\u0432 \/ Angel Angelov – p. 10    Full Text  \u0415\u0432\u0440\u0438\u0441\u0442\u0438\u043a\u0430 \u0441 Excel Heuristics by Excel\u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432 \/ Plamen Penev – p. 18     Full Text \u041a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u043e \u0433\u0435\u043d\u0435\u0440\u0438\u0440\u0430\u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430: \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0432\u0430\u043d\u0435 \u043d\u0430 \u0442\u0435\u043c\u0430 \u043e\u0442 \u0435\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f Computer-generated mathematics: Elaboration of a Topic […]<\/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 1, \u0433\u043e\u0434\u0438\u043d\u0430 LVII, 2014 - 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