{"id":38068,"date":"2012-04-20T18:35:03","date_gmt":"2012-04-20T15:35:03","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/math2\/"},"modified":"2019-06-24T17:33:15","modified_gmt":"2019-06-24T14:33:15","slug":"math2","status":"publish","type":"post","link":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles\/math2\/","title":{"rendered":"Mathematics and Informatics, Number 2\/2012, Volume 55"},"content":{"rendered":"
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\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS<\/strong><\/p>\n

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19th \"Minko Balkanski\" Competition in Mathematics and Physics - p. 103<\/p>\n

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List of Mathematics teachers who defended their first and second professional qualification degrees in 2011 - p. 104<\/p>\n

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SEEMOUS Olympiad for University Students
[\u041e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430\u0442\u0430 SEEMOUS \u0437\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438]<\/a> \/
\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev \/ \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov<\/em> - p. 106<\/p>\n

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EUROMATH Scientific Conference 
[\u041d\u0430\u0443\u0447\u043d\u0430 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u044f EUROMATH]<\/a> \/
<\/a>\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev \/ \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov <\/em>- p. 113<\/p>\n

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Five Ways to Solve a Problem for a Triangle <\/a>
[\u041f\u0435\u0442 \u043d\u0430\u0447\u0438\u043d\u0430 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0435\u0434\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u0437\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a]<\/a> \/
\u0428. \u0410\u0440\u0441\u043b\u0430\u043d\u0430\u0433\u0438\u0447 \/ \u0160. Arslanagi\u0107 \/ \u0414. \u041c\u0438\u043b\u043e\u0448\u0435\u0432\u0438\u0447 \/ D. Milo\u0161evi\u0107 <\/em>- p. 121<\/p>\n

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\u0420\u0435\u0444\u043b\u0435\u043a\u0441\u0438\u044f \u0432 \u0434\u0435\u0439\u043d\u043e\u0441\u0442\u0442\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438
[Refl ection on the Activity of Solving Problems]<\/a> \/
\u0414. \u0413\u044a\u043b\u044a\u0431\u043e\u0432\u0430 \/ D. Galabova \/ \u041c. \u0411\u0435\u0439\u043a\u043e\u0432 \/ M. Bejkov<\/em> - p. 131<\/p>\n

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\u041f\u044a\u0442\u0435\u0448\u0435\u0441\u0442\u0432\u0438\u0435 \u0432 \u0441\u0432\u0435\u0442\u0430 \u043d\u0430 \u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0442\u043e\u0440\u0438\u043a\u0430\u0442\u0430
[A Journey in the World of Combinatorics]<\/a> \/
\u0420. \u041a\u0435\u0440\u0447\u0435\u0432\u0430 \/ R. Kercheva \/ \u0420. \u0418\u0432\u0430\u043d\u043e\u0432\u0430 \/ R. Ivanova<\/em> - p. 142<\/p>\n

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\u0414\u043e\u043f\u0438\u0440\u0430\u0442\u0435\u043b\u043d\u0438 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442\u0438, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u043e\u0442 \u0438\u043d\u0446\u0438\u0434\u0435\u043d\u0442\u043d\u0438 \u0442\u043e\u0447\u043a\u0430 \u0438 \u043f\u0440\u0430\u0432\u0430
[Tangent Circles, Generated by an Incident Point and a Line]<\/a> \/
\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev \/ \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov<\/em> - p. 151<\/p>\n

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\n

CONTEST PROBLEMS<\/strong>
The column is led by Dr. Svetlozar Doychev and Dr. Veselin Nenkov
<\/em>
contest problems - p. 161<\/p>\n

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\u041f\u043e\u043b\u0437\u043e\u0442\u0432\u043e\u0440\u043d\u0430 \u043f\u0440\u043e\u043c\u044f\u043d\u0430 \/Beneficial change
\u0418. \u0421\u0442\u0430\u0440\u0438\u0431\u0440\u0430\u0442\u043e\u0432 \/ Y. Staribratov \u2013 p. 164<\/em><\/p>\n

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\u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415 \/ GUIDE FOR AUTHORS<\/strong> – \u0441\u0442\u0440. 188<\/p>\n

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<\/a>SEEMOUS OLYMPIAD FOR UNIVERSITY STUDENTS<\/strong><\/p>\n

Sava Grozdev, Veselin Nenkov<\/strong><\/p>\n


Abstract<\/strong>. The paper considers the results of a recent International Mathematical Olympiad for University students. Solutions of the problems with methodological and heuristic details are proposed.<\/p>\n

Keywords<\/em>: Olympiad, problem solving, University student, result.<\/p>\n

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\u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u0410\u0422\u0410 SEEMOUS \u0417\u0410 \u0421\u0422\u0423\u0414\u0415\u041d\u0422\u0418<\/strong><\/p>\n


Abstract<\/strong>. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438\u0442\u0435 \u043e\u0442 \u043d\u0435\u043e\u0442\u0434\u0430\u0432\u043d\u0430\u0448\u043d\u0430 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0441\u043a\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430\u0442\u0430 \u0441 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0438 \u0435\u0432\u0440\u0438\u0441\u0442\u0438\u0447\u043d\u0438 \u0434\u0435\u0442\u0430\u0439\u043b\u0438.<\/p>\n

Sava Grozdev<\/strong>
Professor, Doctor of Mathematics, Doctor of Pedagogical Sciences<\/em>
Institute of Mathematics and Informatics \u2013 BAS<\/em><\/p>\n

Veselin Nenkov<\/strong>
doctor of mathematics<\/em>
Technical College - Lovech<\/em><\/p>\n

 <\/p>\n

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

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<\/a>EUROMATH SCIENTIFIC CONFERENCE<\/p>\n

Sava Grozdev, Veselin Nenkov<\/strong><\/p>\n

Abstract<\/strong>. The paper is dedicated to a recent International Scientific Conference for secondary school students. Main parts of the Programme are described and one of the Workshops is presented with its methodological aspects.<\/p>\n

Keywords<\/em>: science, innovator, researcher, fame-lab, math-factor.<\/p>\n

\u041d\u0410\u0423\u0427\u041d\u0410 \u041a\u041e\u041d\u0424\u0415\u0420\u0415\u041d\u0426\u0418\u042f EUROMATH<\/strong><\/p>\n


Abstract<\/strong>. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043d\u0435\u043e\u0442\u0434\u0430\u0432\u043d\u0430\u0448\u043d\u0430\u0442\u0430 \u041c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0430 \u043d\u0430\u0443\u0447\u043d\u0430 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u044f \u0437\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438. \u041e\u043f\u0438\u0441\u0430\u043d\u0438 \u0441\u0430 \u0433\u043b\u0430\u0432\u043d\u0438\u0442\u0435 \u0447\u0430\u0441\u0442\u0438 \u043d\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430\u0442\u0430 \u0438 \u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d \u0435\u0434\u0438\u043d \u043e\u0442 \u0443\u044a\u0440\u043a\u0448\u043e\u043f\u0438\u0442\u0435 \u0441 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438\u0442\u0435 \u043c\u0443 \u0430\u0441\u043f\u0435\u043a\u0442\u0438.<\/p>\n

\n

Sava Grozdev
<\/strong>Professor, Doctor of Mathematics, Doctor of Pedagogical Sciences
<\/em>Institute of Mathematics and Informatics \u2013 BAS<\/em><\/p>\n

\n

Veselin Nenkov<\/strong>
doctor of mathematics<\/em>
Technical College - Lovech<\/em><\/p>\n

 <\/p>\n

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

\n

<\/a>FIVE WAYS TO SOLVE A PROBLEM FOR A TRIANGLE<\/p>\n

\u0160efket Arslanagi\u0107, Dragoljub Milo\u0161evi\u0107<\/strong><\/p>\n

Abstract<\/strong>. A triangle with the property 3\u03b1 + 2\u03b2 =1800 is considered in the
paper. Five different ways are proposed to prove that a2<\/sup> + bc = c2<\/sup> . The paper is of methodological character.<\/p>\n

Keywords<\/em>: triangle, problem solving, trigonometry, similarity.<\/p>\n

\u041f\u0415\u0422 \u041d\u0410\u0427\u0418\u041d\u0410 \u0417\u0410 \u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415 \u041d\u0410 \u0415\u0414\u041d\u0410 \u0417\u0410\u0414\u0410\u0427\u0410 \u0417\u0410 \u0422\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a<\/p>\n

\u0428\u0435\u0444\u043a\u0435\u0442 \u0410\u0440\u0441\u043b\u0430\u043d\u0430\u0433\u0438\u0447, \u0414\u0440\u0430\u0433\u043e\u043b\u044e\u0431 \u041c\u0438\u043b\u043e\u0448\u0435\u0432\u0438\u0447
<\/strong><\/p>\n

Abstract<\/strong>. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a \u0441\u044a\u0441 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u043e\u0442\u043e 3\u03b1 + 2\u03b2 =1800. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u043f\u0435\u0442 \u043d\u0430\u0447\u0438\u043d\u0430 \u0437\u0430 \u0434\u043e\u043a\u0430\u0437\u0432\u0430\u043d\u0435, \u0447\u0435 a2<\/sup> + bc = c2<\/sup> . \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u0441 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440.<\/p>\n

\u0160efket Arslanagi\u0107<\/strong>
Professor, Doctor in Mathematics<\/em>
Faculty of Mathematics and Natural Sciences<\/em>
University of Sarajevo<\/em><\/p>\n

Dragoljub Milo\u0161evi\u0107<\/strong>
Teacher in Mathematics<\/em>
Gornji Milanovac<\/em>
Serbia<\/em><\/p>\n

 <\/p>\n

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

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<\/a>\u041f\u0420\u041e\u041f\u041e\u0420\u0426\u0418\u0418<\/p>\n

\u0412\u0430\u043b\u044f \u0413\u0435\u043e\u0440\u0433\u0438\u0435\u0432\u0430<\/strong>
\u0421\u041e\u0423 \u201e\u0421\u0432\u0435\u0442\u0438 \u041f\u0430\u0442\u0440\u0438\u0430\u0440\u0445 \u0415\u0432\u0442\u0438\u043c\u0438\u0439\u201d, \u0433\u0440. \u041f\u043b\u043e\u0432\u0434\u0438\u0432<\/em><\/p>\n

Abstract<\/strong>. \u201e\u041f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u044f\u0442 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440 \u0438 \u0437\u043d\u0430\u0447\u0438\u043c\u043e\u0441\u0442 \u043d\u0430 \u0437\u043d\u0430\u043d\u0438\u044f\u0442\u0430 \u043e\u0442 \u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u041f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u0438\u201d \u0435 \u0431\u0435\u0437\u0441\u043f\u043e\u0440\u0435\u043d. \u0422\u0435\u0437\u0438 \u0437\u043d\u0430\u043d\u0438\u044f \u0441\u0430 \u0447\u0430\u0441\u0442 \u043e\u0442 \u043e\u043d\u0430\u0437\u0438 \u043e\u0431\u0449\u0430\u0442\u0430 \u043a\u0443\u043b\u0442\u0443\u0440\u0430 \u043d\u0430 \u0441\u044a\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u0438\u044f \u0447\u043e\u0432\u0435\u043a, \u0431\u0435\u0437 \u043a\u043e\u044f\u0442\u043e \u0442\u043e\u0439 \u043d\u0435 \u043c\u043e\u0436\u0435 \u0434\u0430 \u0438\u0437\u0432\u043b\u0438\u0447\u0430 \u0438\u043b\u0438 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u0430 \u043f\u043e \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u043d\u0430\u0447\u0438\u043d\u0438 \u0438\u043b\u0438 \u0441 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430\u201d (\u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0443\u043a\u0430\u0437\u0430\u043d\u0438\u044f, \u0434\u0430\u0434\u0435\u043d\u0438 \u043a\u044a\u043c \u0443\u0447\u0435\u0431\u043d\u043e\u0442\u043e \u0441\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u0437\u0430 VI \u043a\u043b\u0430\u0441). \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d \u0443\u0440\u043e\u043a \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d \u043d\u0430 28.03.2011 \u0433\u043e\u0434\u0438\u043d\u0430 \u0432 \u0421\u041e\u0423 \u201d\u0421\u0432. \u041f\u0430\u0442\u0440\u0438\u0430\u0440\u0445 \u0415\u0432\u0442\u0438\u043c\u0438\u0439\u201d, \u0433\u0440\u0430\u0434 \u041f\u043b\u043e\u0432\u0434\u0438\u0432, \u0441 \u0448\u0435\u0441\u0442\u0438 \u201e\u0430\u201d \u043a\u043b\u0430\u0441 \u0441\u0435 \u043f\u0440\u043e\u0432\u0435\u0436\u0434\u0430 \u043e\u0442\u043a\u0440\u0438\u0442 \u043e\u0431\u043e\u0431\u0449\u0438\u0442\u0435\u043b\u0435\u043d \u0443\u0440\u043e\u043a \u043d\u0430 \u0442\u0435\u043c\u0430 \u201e\u041f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u0438\u201d.<\/p>\n

Keywords<\/em>: ratio, lesson plan, interdisciplinary approach<\/p>\n

\n

PROPORTIONS<\/strong><\/p>\n

Abstract<\/strong>. \u201ePractical nature and knowledge significance of the topic of \u201eProportions\u201d are unquestionable. This knowledge is part of one\u2018s contemporary general culture which is essential in the extraction and presentation of information obtained by different ways and manners.\u201d (methological instructions to the 6th grade curriculum). This article presents a 6th grade open class on 28.03.2011 at \u201eSt. Patriarch Evtimii\u201d High School in the city of Plovdiv generalizing the topic of \u201eProportions\u201d.<\/p>\n

Valya Georgieva<\/strong>
Mathematics teacher<\/em>
\u201cSt. Patriarch Evtimii\u201d High School<\/em><\/p>\n

 <\/p>\n

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

\n

<\/a>\u0420\u0415\u0424\u041b\u0415\u041a\u0421\u0418\u042f\u0422\u0410 \u0412 \u0414\u0415\u0419\u041d\u041e\u0421\u0422\u0422\u0410 \u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415 \u041d\u0410 \u0417\u0410\u0414\u0410\u0427\u0418(\u0412\u042a\u0420\u0425\u0423 \u041f\u0420\u0418\u041c\u0415\u0420\u0418 \u041e\u0422 9 \u041a\u041b\u0410\u0421)<\/p>\n

\u0414\u0430\u0440\u0438\u043d\u0430 \u0413\u044a\u043b\u044a\u0431\u043e\u0432\u0430, \u041c\u0438\u043d\u0447\u043e \u0411\u0435\u0439\u043a\u043e\u0432<\/strong><\/p>\n

Abstract<\/strong>. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u0442\u0435\u043c\u0438 \u043e\u0442 \u0443\u0447\u0435\u0431\u043d\u043e\u0442\u043e \u0441\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u0437\u0430 9 \u043a\u043b\u0430\u0441. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d \u0435 \u0440\u0435\u0444\u043b\u0435\u043a\u0441\u0438\u0432\u0435\u043d \u043f\u043e\u0434\u0445\u043e\u0434, \u0437\u0430 \u043a\u043e\u0439\u0442\u043e \u0441\u0435 \u0442\u0432\u044a\u0440\u0434\u0438, \u0447\u0435 \u0435 \u0435\u0444\u0435\u043a\u0442\u0438\u0432\u0435\u043d \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0441\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u0422\u0432\u044a\u0440\u0434\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0435 \u0435 \u043f\u043e\u0434\u043a\u0440\u0435\u043f\u0435\u043d\u043e \u0441 \u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438 \u0435\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442.<\/p>\n

Keywords<\/em>: reflection, problem solving, 9th grade curriculum<\/p>\n

\n

REFLECTION ON THE ACTIVITY OF SOLVING PROBLEMS<\/strong><\/p>\n

Abstract<\/strong>. Several topics from the 9th grade curriculum are considered in the paper. An approach by refl ection is proposed and the assertion is that this approach is effective in a corresponding problem solving. The assertion is not supported by a case study.<\/p>\n

Darinka Galabova<\/strong>
Doctor<\/em>
St. Kiril and St. Methodius University of Veliko Turnovo<\/em><\/p>\n

Mincho Bejkov<\/strong>
\u201eEmiliyan Stanev\u201d High School<\/em><\/p>\n

 <\/p>\n

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

\n

<\/a>\u041f\u042a\u0422\u0415\u0428\u0415\u0421\u0422\u0412\u0418\u0415 \u0412 \u0421\u0412\u0415\u0422\u0410 \u041d\u0410 \u041a\u041e\u041c\u0411\u0418\u041d\u0410\u0422\u041e\u0420\u0418\u041a\u0410\u0422\u0410<\/p>\n

\u0420\u043e\u0441\u0438\u0446\u0430 \u041a\u0435\u0440\u0447\u0435\u0432\u0430, \u0420\u0443\u043c\u044f\u043d\u0430 \u0418\u0432\u0430\u043d\u043e\u0432\u0430<\/strong><\/p>\n

Abstract<\/strong>. \u0412 \u0418\u043d\u0434\u0438\u044f \u043d\u0430 \u0433\u043b\u0430\u0432\u043d\u0438\u044f \u0432\u0445\u043e\u0434 \u043d\u0430 \u0435\u0434\u043d\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u0441\u0442\u043e\u0438 \u043d\u0430\u0434\u043f\u0438\u0441: \u201e\u0421\u0442\u0440\u0435\u043c\u0438 \u0441\u0435, \u0434\u0435\u0439\u0441\u0442\u0432\u0430\u0439, \u043f\u043e\u0441\u0442\u0438\u0433\u0430\u0439!\u201d. \u041d\u0430\u0448\u0438\u044f\u0442 \u043c\u0430\u043b\u044a\u043a, \u043d\u043e \u0430\u043c\u0431\u0438\u0446\u0438\u043e\u0437\u0435\u043d \u0435\u043a\u0438\u043f \u0440\u0435\u0448\u0438 \u0434\u0430 \u043f\u043e\u0433\u043b\u0435\u0434\u043d\u0435 \u043f\u043e \u0434\u0440\u0443\u0433 \u043d\u0430\u0447\u0438\u043d \u043d\u0430 \u0443\u0447\u0435\u0431\u043d\u043e\u0442\u043e \u0441\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430, \u043a\u0430\u0442\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0438 \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b \u043e\u0442 \u0440\u0430\u0437\u0434\u0435\u043b \u041a\u043e\u043c\u0431\u0438\u043d\u0430\u0442\u043e\u0440\u0438\u043a\u0430\u201d \u0432 \u043a\u0440\u0430\u044f \u043d\u0430 10 \u043a\u043b\u0430\u0441 \u2013 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u0435\u043d, \u0437\u0430\u0431\u0430\u0432\u0435\u043d \u0438 \u0434\u043e\u0440\u0438 \u0432\u044a\u043b\u043d\u0443\u0432\u0430\u0449. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u044a\u0434\u044a\u0440\u0436\u0430 \u043f\u044a\u0440\u0432\u0430\u0442\u0430 \u0447\u0430\u0441\u0442 \u043e\u0442 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430\u0442\u0430 \u043f\u0440\u0435\u0437\u0435\u043d\u0442\u0430\u0446\u0438\u044f \u201e\u041c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u0438 \u0441\u044a\u0435\u0434\u0438\u043d\u0435\u043d\u0438\u044f\u201d, \u043a\u043b\u0430\u0441\u0438\u0440\u0430\u043d\u0430 \u043d\u0430 \u0432\u0442\u043e\u0440\u043e \u043c\u044f\u0441\u0442\u043e \u0432 \u043d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438\u044f \u0435\u0442\u0430\u043f \u043d\u0430 \u041c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0438 \u043a\u043e\u043d\u043a\u0443\u0440\u0441 \u0437\u0430 \u0443\u0447\u0438\u0442\u0435\u043b\u0438 \u201e\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u043f\u0440\u043e\u0435\u043a\u0442\u0438\u0440\u0430\u043d\u0435\u201d. \u0412 \u043d\u0435\u044f \u0441\u043b\u0435\u0434 \u043a\u0440\u0430\u0442\u043a\u0430 \u0438\u0441\u0442\u043e\u0440\u0438\u0447\u0435\u0441\u043a\u0430 \u0441\u043f\u0440\u0430\u0432\u043a\u0430 \u0441\u0430 \u0440\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u043f\u043e\u043d\u044f\u0442\u0438\u044f\u0442\u0430 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u0438 \u0441\u044a\u0435\u0434\u0438\u043d\u0435\u043d\u0438\u044f, \u043f\u043e\u0434\u043a\u0440\u0435\u043f\u0435\u043d\u0438 \u0441 \u043f\u043e\u0434\u0445\u043e\u0434\u044f\u0449\u0438 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u0438. \u0421\u043b\u0435\u0434\u0432\u0430\u0442 \u043f\u0440\u0430\u0432\u0438\u043b\u0430\u0442\u0430 \u0437\u0430 \u0441\u044a\u0431\u0438\u0440\u0430\u043d\u0435 \u0438 \u0443\u043c\u043d\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u044a\u0435\u0434\u0438\u043d\u0435\u043d\u0438\u044f, \u043a\u0430\u043a\u0442\u043e \u0438 \u0440\u0430\u0437\u043d\u043e\u043e\u0431\u0440\u0430\u0437\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438: \u0437\u0430 \u0448\u0430\u0445\u043c\u0430\u0442\u043d\u0438 \u0434\u044a\u0441\u043a\u0438, \u0437\u0430 \u043f\u0440\u0438\u043a\u0430\u0437\u043d\u0438 \u0433\u0435\u0440\u043e\u0438, \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u0438 \u043f\u043e\u0449\u0438, \u0445\u0430\u043a\u0435\u0440\u0438, \u043f\u0430\u0440\u043e\u043b\u0438, \u0441\u0435\u0439\u0444\u043e\u0432\u0435, \u0446\u0432\u0435\u0442\u044f \u0438 \u0434\u0440. \u0412\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u0442\u0435 \u0441\u043b\u0430\u0439\u0434\u043e\u0432\u0435 \u0441\u0430 \u0441\u0430\u043c\u043e \u043c\u0430\u043b\u043a\u0430 \u0447\u0430\u0441\u0442 \u043e\u0442 \u043f\u0440\u0435\u0437\u0435\u043d\u0442\u0430\u0446\u0438\u044f\u0442\u0430. \u0422\u0435 \u0441\u0435 \u0432\u044a\u0432\u0435\u0436\u0434\u0430\u0442 \u043f\u043e\u0441\u0442\u0435\u043f\u0435\u043d\u043d\u043e, \u0441\u0442\u044a\u043f\u043a\u0430 \u043f\u043e \u0441\u0442\u044a\u043f\u043a\u0430, \u043a\u043e\u0435\u0442\u043e \u043e\u0441\u0438\u0433\u0443\u0440\u044f\u0432\u0430 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e\u0442\u043e \u0443\u0447\u0435\u0431\u043d\u043e \u0432\u0440\u0435\u043c\u0435 \u0437\u0430 \u043f\u043e-\u043f\u044a\u043b\u043d\u043e \u043e\u0441\u043c\u0438\u0441\u043b\u044f\u043d\u0435 \u0438 \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u043d\u043e\u0432\u0438\u0442\u0435 \u0437\u043d\u0430\u043d\u0438\u044f, \u043a\u0430\u043a\u0442\u043e \u0438 \u0437\u0430 \u0430\u043d\u0430\u043b\u0438\u0437\u0438 \u0438 \u0440\u0430\u0437\u0441\u044a\u0436\u0434\u0435\u043d\u0438\u044f \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438. \u0426\u044f\u043b\u0430\u0442\u0430 \u043f\u0440\u0435\u0437\u0435\u043d\u0442\u0430\u0446\u0438\u044f \u0441\u0435 \u043d\u0430\u043c\u0438\u0440\u0430 \u043d\u0430 \u0441\u0430\u0439\u0442\u0430 \u043d\u0430 \u0413\u041f\u0427\u0415 \u201e\u0415\u043a\u0437\u0430\u0440\u0445 \u0419\u043e\u0441\u0438\u0444\u201d, \u0433\u0440. \u0420\u0430\u0437\u0433\u0440\u0430\u0434.<\/p>\n

Keywords<\/em>: Combinatorics, set, compound, multiplication rule.<\/p>\n

\n

A JOURNEY IN THE WORD OF COMBINATORICS<\/strong><\/p>\n

Abstract<\/strong>. A sign on the front gate of a school in India reads: \u201eStrive for it and do it.\u201d Our small but ambitious team has decided to look at the Combinatorics section of the athematics textbook for 10th grade from another angle in order to present it in an interesting, amusing, and exciting way. The paper contains the first part of the PowerPoint presentation of \u201eA Journey in the World of Combinatorics\u201d, ranked second in the National stage of the International Competition for teachers \u201eMathematics and Projecting\u201d. After a short historical review, the notions of a set and a combination are considered under the support of suitable and interesting examples. The rules for summation and multiplication of combinations follow as well as various tasks: for chess games of chess, fairy characters, e-mails, hackers, passwords, safes, fl owers, etc. The included slides constitute a small part of the presentation. They are introduced successively, step by step, which assures the necessary teaching time for a more profound and meaningful understanding of new knowledge, for analysis and reasoning in problem solving. The whole presentation is on the website of the \u201eEkzarh Jossiff\u201d High School for Foreign Languages, Razgrad.<\/p>\n

Rositsa Kercheva<\/strong>
Teacher in Matematics<\/em>
\u201cEkzarh Jossiff\u201d High School for Foreign Languages, Razgrad<\/em><\/p>\n

Rumyana Ivanova<\/strong>
Teacher in Matematics<\/em>
\u201cEkzarh Jossiff\u201d High School for Foreign Languages, Razgrad<\/em><\/p>\n

 <\/p>\n

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

\n

<\/a>\u0414\u041e\u041f\u0418\u0420\u0410\u0422\u0415\u041b\u041d\u0418 \u041e\u041a\u0420\u042a\u0416\u041d\u041e\u0421\u0422\u0418, \u041f\u041e\u0420\u041e\u0414\u0415\u041d\u0418 \u041e\u0422 \u0418\u041d\u0426\u0418\u0414\u0415\u041d\u0422\u041d\u0418 \u0422\u041e\u0427\u041a\u0410 \u0418 \u041f\u0420\u0410\u0412\u0410<\/p>\n

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

Abstract<\/strong>. \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u0441\u0430 \u0442\u0432\u044a\u0440\u0434\u0435\u043d\u0438\u044f, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0435\u0434\u043d\u0430 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u041c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0430\u0442\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043f\u0440\u0435\u0437 2011 \u0433. \u041e\u0431\u044a\u0440\u043d\u0430\u0442\u043e \u0435 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u043d\u043e \u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u043d\u0430 \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u0430\u0442\u0430, \u043a\u0430\u0442\u043e \u0437\u0430 \u043e\u043f\u0440\u043e\u0441\u0442\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0438\u0437\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u044f\u0442\u0430 \u0441\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0438 \u0438\u043d\u0432\u0435\u0440\u0441\u0438\u044f \u0438 \u043f\u043e\u0434\u0445\u043e\u0434\u044f\u0449\u0430 \u0442\u043e\u0447\u043a\u0430 \u043d\u0430 \u041c\u0438\u043a\u0435\u043b.<\/p>\n

Keywords<\/em>: THE GEOMETER\u2019S SKETCHPAD (GSP), triangle, circle, tangent point, Miquel point.<\/p>\n

\n

TANGENT CIRCLES, GENERATED BY AN INCIDENT POINT AND A LINE<\/strong><\/p>\n

Abstract<\/strong>. Some assertions are considered in connection with the generalization of a problem from the International Mathematical Olympiad in 2011. Special notice is taken of the proofs. Inversion and a suitable Miquel point are used to simplify the computing process.<\/p>\n

Sava Grozdev<\/strong>
Professor, Doctor in Mathematics, DSc in Pedagogy<\/em>
Institute of Mathematics and Informatics \u2013 BAS<\/em>

Veselin Nenkov<\/strong>
Doctor in Mathematics<\/em>
Technical College Lovec<\/em>h<\/p>\n

 <\/p>\n

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

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

\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS 19-\u0442\u0438 \u041a\u043e\u043d\u043a\u0443\u0440\u0441 \u201e\u041c\u0438\u043d\u043a\u043e \u0411\u0430\u043b\u043a\u0430\u043d\u0441\u043a\u0438\u201d \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0444\u0438\u0437\u0438\u043a\u0430 – \u0441\u0442\u0440. 103 \u0423\u0447\u0438\u0442\u0435\u043b\u0438 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430, \u0437\u0430\u0449\u0438\u0442\u0438\u043b\u0438 \u043f\u044a\u0440\u0432\u0430 \u0438 \u0432\u0442\u043e\u0440\u0430 \u043f\u0440\u043e\u0444\u0435\u0441\u0438\u043e\u043d\u0430\u043b\u043d\u043e-\u043a\u0432\u0430\u043b\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u043e\u043d\u043d\u0430 \u0441\u0442\u0435\u043f\u0435\u043d \u043f\u0440\u0435\u0437 2011 \u0433. – \u0441\u0442\u0440. 104 SEEMOUS Olympiad for University Students [\u041e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430\u0442\u0430 SEEMOUS \u0437\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438] \/\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev \/ \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov – \u0441\u0442\u0440. 106 EUROMATH Scientific Conference [\u041d\u0430\u0443\u0447\u043d\u0430 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u044f […]<\/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 2, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \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:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/\" \/>\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 2, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS 19-\u0442\u0438 \u041a\u043e\u043d\u043a\u0443\u0440\u0441 \u201e\u041c\u0438\u043d\u043a\u043e \u0411\u0430\u043b\u043a\u0430\u043d\u0441\u043a\u0438\u201d \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0444\u0438\u0437\u0438\u043a\u0430 – \u0441\u0442\u0440. 103 \u0423\u0447\u0438\u0442\u0435\u043b\u0438 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430, \u0437\u0430\u0449\u0438\u0442\u0438\u043b\u0438 \u043f\u044a\u0440\u0432\u0430 \u0438 \u0432\u0442\u043e\u0440\u0430 \u043f\u0440\u043e\u0444\u0435\u0441\u0438\u043e\u043d\u0430\u043b\u043d\u043e-\u043a\u0432\u0430\u043b\u0438\u0444\u0438\u043a\u0430\u0446\u0438\u043e\u043d\u043d\u0430 \u0441\u0442\u0435\u043f\u0435\u043d \u043f\u0440\u0435\u0437 2011 \u0433. – \u0441\u0442\u0440. 104 SEEMOUS Olympiad for University Students [\u041e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430\u0442\u0430 SEEMOUS \u0437\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438] \/\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev \/ \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov – \u0441\u0442\u0440. 106 EUROMATH Scientific Conference [\u041d\u0430\u0443\u0447\u043d\u0430 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u044f […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/\" \/>\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=\"2012-04-20T15:35:03+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-06-24T14:33:15+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:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/\"},\"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 2, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012\",\"datePublished\":\"2012-04-20T15:35:03+00:00\",\"dateModified\":\"2019-06-24T14:33:15+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/\"},\"wordCount\":1699,\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/#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:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/\",\"url\":\"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/\",\"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 2, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - 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Grozdev \/ \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov – \u0441\u0442\u0440. 106 EUROMATH Scientific Conference [\u041d\u0430\u0443\u0447\u043d\u0430 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u044f […]","og_url":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2012-04-20T15:35:03+00:00","article_modified_time":"2019-06-24T14:33:15+00:00","og_image":[{"url":"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png"}],"twitter_card":"summary_large_image","twitter_misc":{"Written by":"","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/#article","isPartOf":{"@id":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/"},"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 2, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012","datePublished":"2012-04-20T15:35:03+00:00","dateModified":"2019-06-24T14:33:15+00:00","mainEntityOfPage":{"@id":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/"},"wordCount":1699,"publisher":{"@id":"https:\/\/azbuki.bg\/#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:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/","url":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/math2\/","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 2, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - 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