{"id":38067,"date":"2012-02-20T18:31:06","date_gmt":"2012-02-20T15:31:06","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/math\/"},"modified":"2019-06-18T21:59:02","modified_gmt":"2019-06-18T18:59:02","slug":"math-2","status":"publish","type":"post","link":"https:\/\/mathinfo.azbuki.bg\/en\/matematics\/matharticles\/math-2\/","title":{"rendered":"Mathematics and Informatics, Number 1\/2012, Volume 55"},"content":{"rendered":"

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

<\/p>\n

\u041a\u042a\u041c \u0427\u0418\u0422\u0410\u0422\u0415\u041b\u042f \/ \u0415DITORIAL
<\/strong>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ Sava Grozdev<\/em> \u2013 p. 7<\/p>\n

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Academician Blagovest Sendov - 80th anniversary - p. 9<\/p>\n

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\u0421\u0440\u0430\u0432\u043d\u0438\u0442\u0435\u043b\u0435\u043d \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 8-\u0430 \u0436\u0430\u0443\u0442\u0438\u043a\u043e\u0432\u0441\u043a\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430
[Comparative Analysis of the 8th Zhautykov Olympiad] <\/a>\/ 
\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ Sava Grozdev \/ \u041a\u0430\u0439\u0440\u043e\u0448 \u041c\u0430\u043a\u0438\u0448\u0435\u0432 \/ Kairosh Makishev<\/em> – \u0441\u0442\u0440. 12<\/p>\n

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\u0418\u0433\u043b\u0430 \u043d\u0430 \u0431\u044e\u0444\u043e\u043d \u0438 \u043f\u0440\u0435\u0441\u043c\u044f\u0442\u0430\u043d\u0435 \u043d\u0430 \u0447\u0438\u0441\u043b\u043e\u0442\u043e 
[Buffon\u2019s Needle and Estimation of the \u03c0 Number] <\/a>\/ 
\u0415\u0432\u0433\u0435\u043d\u0438\u044f \u0421\u0442\u043e\u0438\u043c\u0435\u043d\u043e\u0432\u0430 \/ Evgenia Stoimenova –<\/em> \u0441\u0442\u0440. 23<\/p>\n

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\u0417\u0430 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e\u0442\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 
[On the e-Learning] \/
<\/a>\u0414\u0430\u043d\u0438\u0435\u043b\u0430 \u0414\u0443\u0440\u0435\u0432\u0430 (\u0422\u0443\u043f\u0430\u0440\u043e\u0432\u0430) \/ Daniela Dureva (Tuparova) –<\/em> \u0441\u0442\u0440. 30<\/p>\n

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\u0423\u0441\u043b\u043e\u0432\u0438\u044f \u0437\u0430 \u043f\u0440\u0438\u0434\u043e\u0431\u0438\u0432\u0430\u043d\u0435 \u043d\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\u0438 \u0441\u0442\u0435\u043f\u0435\u043d\u0438 
[Requirements to obtain Profession-qualification Degrees]<\/a> \/ 
\u0413\u0430\u043b\u044f \u041a\u043e\u0436\u0443\u0445\u0430\u0440\u043e\u0432\u0430 \/ Galia Kozhuharova – <\/em>\u0441\u0442\u0440. 34<\/p>\n

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International competitions for project development<\/span> – <\/strong>p. 38<\/span><\/p>\n

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\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430\u0442\u0430 e \u0437\u0430\u0431\u0430\u0432\u043d\u0430 
[Mathematics is amusing] \/
<\/a>\u0412\u0435\u0441\u0435\u043b\u0438\u043d\u0430 \u0412\u044a\u043b\u043a\u0430\u043d\u043e\u0432\u0430 \/ Veselina Valkanova – <\/em>\u0441\u0442\u0440. 42<\/p>\n

\"icon\" Open the full text<\/a><\/p>\n


\u0421\u0440\u0430\u0432\u043d\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0438\u0437\u0440\u0430\u0437\u0438 \u0441 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043d\u0438 \u043a\u043e\u0440\u0435\u043d\u0438 
[Comparison of Expressions with Square Roots]
<\/a>\u0413\u0438\u043d\u043a\u0430 \u0411\u0438\u0437\u043e\u0432\u0430 \/ Ginka Bizova \/ \u0412\u0430\u043d\u044f \u041b\u0430\u043b\u0435\u0432\u0430 \/ Vania Lal\u0435va –<\/em> \u0441\u0442\u0440. 47<\/p>\n

\n

\u041f\u0440\u0435\u0431\u0440\u043e\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u0435\u0441\u0435\u0447\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \u043d\u0430 \u043e\u0442\u0441\u0435\u0447\u043a\u0438 \u0438 \u043f\u0440\u0430\u0432\u0438
<\/span>[Counting of Intersection Points of Segments and Lines] \/
\u042e\u043b\u0438\u044f \u041a\u0440\u044a\u0441\u0442\u0435\u0432\u0430 \/ Julia Krasteva –<\/em> p. <\/em>54<\/p>\n

\"icon\" <\/a><\/em>Open the full text<\/a><\/p>\n

\u0414\u0432\u0435 \u0434\u0432\u043e\u0439\u043a\u0438 \u0442\u043e\u0447\u043a\u0438, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u043e\u0442 \u0430\u0441\u043e\u0446\u0438\u0438\u0440\u0430\u043d\u0438 \u0441\u043f\u0440\u044f\u043c\u043e \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a \u0446\u0435\u043d\u0442\u0440\u0430\u043b\u043d\u0438 \u043a\u043e\u043d\u0438\u0447\u043d\u0438 \u0441\u0435\u0447\u0435\u043d\u0438\u044f
[Two Pairs of Points, Generated by Central Conics with Respect to a Triangle] \/ 
\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ Sava Grozdev \/ \u0412\u0432\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \/ Veselin Nenkov – <\/em>\u0441\u0442\u0440. 60<\/p>\n

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Contest Problems<\/strong>
The column is led by Dr. Svetlozar Doychev and Dr. Veselin Nenkov - <\/em>p. 84<\/p>\n

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National Gymnasium of Natural Sciences and Mathematics \"Academician Lyubomir Chakalov\"
sample topic for internal profiling exam in mathematics -<\/em> p.88<\/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> p. 92<\/p>\n

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<\/div>\n

<\/a>\u0421\u0420\u0410\u0412\u041d\u0418\u0422\u0415\u041b\u0415\u041d \u0410\u041d\u0410\u041b\u0418\u0417 \u041d\u0410 8-\u0430 \u0416\u0410\u0423\u0422\u0418\u041a\u041e\u0412\u0421\u041a\u0410 \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u0410<\/strong><\/p>\n

\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u041a\u0430\u0439\u0440\u043e\u0448 \u041c\u0430\u043a\u0438\u0448\u0435\u0432<\/strong><\/p>\n

Abstract. <\/strong>\u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u0442 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438\u0442\u0435 \u043e\u0442 8-\u043c\u0430 \u0416\u0430\u0443\u0442\u0438\u043a\u043e\u0432\u0441\u043a\u0430 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430, \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0430 \u0432 \u0410\u043b\u043c\u0430\u0442\u0438, \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d, 15\u201321 \u044f\u043d\u0443\u0430\u0440\u0438 2012 \u0433. \u0414\u0430\u0434\u0435\u043d\u0438 \u0441\u0430 \u0431\u0430\u043b\u043e\u0432\u0435\u0442\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u0442\u0440\u0438\u0442\u0435 \u043d\u0430\u0439-\u0432\u0438\u0441\u043e\u043a\u043e \u043a\u043b\u0430\u0441\u0438\u0440\u0430\u043d\u0438 \u0443\u0447\u0438\u043b\u0438\u0449\u0430 \u043e\u0442 \u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f, \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d \u0438 \u0420\u0443\u0441\u0438\u044f. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \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 \u0437\u0430 \u0448\u0435\u0441\u0442\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 \u0442\u0435\u043c\u0430\u0442\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430.<\/span><\/p>\n


Keywords:<\/em> Olympiad, problem solving, achievement, methodology, heuristics<\/span><\/p>\n

<\/p>\n

COMPARATIVE ANALYSIS<\/strong> OF THE 8TH ZHAUTYKOV OLYMPIAD<\/strong><\/p>\n

Abstract.<\/strong> The paper analyses the results of the 8th Zhautykov International Olympiad, held in Almaty, Kazkhstan, January 15\u201321, 2012. The scores in Mathematics and Informatics of the three highly ranked schools from Bulgaria, Kazakhstan and Russia are listed. Solutions with methodological and heuristic details are proposed for the six problems from the Mathematics Competition paper.<\/p>\n

Sava Grozdev
<\/strong>Institute of Mathematics and informatics \u2013 BAS<\/em><\/p>\n


<\/em>Kairosh Makishev
<\/strong>Director RSPhMSBS<\/em><\/p>\n

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

\n
<\/div>\n

<\/a>\u0418\u0413\u041b\u0410 \u041d\u0410 \u0411\u042e\u0424\u041e\u041d \u0418 \u041f\u0420\u0415\u0421\u041c\u042f\u0422\u0410\u041d\u0415 \u041d\u0410 \u0427\u0418\u0421\u041b\u041e\u0422\u041e <\/strong><\/p>\n

\u0415\u0432\u0433\u0435\u043d\u0438\u044f \u0421\u0442\u043e\u0438\u043c\u0435\u043d\u043e\u0432\u0430<\/strong><\/p>\n


Abstract. <\/strong>\u041f\u0440\u0435\u0437 1777 \u0433. \u0444\u0440\u0435\u043d\u0441\u043a\u0438\u044f\u0442 \u0443\u0447\u0435\u043d \u0414\u0436\u043e\u0440\u0434\u0436-\u041b\u0443\u0438 \u041b\u0435\u043a\u043b\u0435\u0440\u043a, \u041a\u043e\u043d\u0442 \u043d\u0430 \u0411\u044e\u0444\u043e\u043d, \u043f\u043e\u0441\u0442\u0430\u0432\u044f \u0438 \u0440\u0435\u0448\u0430\u0432\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430 \u0437\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u043d\u0435 \u043d\u0430 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0442\u0430 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u043e \u0445\u0432\u044a\u0440\u043b\u0435\u043d\u0430 \u0438\u0433\u043b\u0430 \u0441 \u0434\u044a\u043b\u0436\u0438\u043d\u0430 <\/span>l<\/em> \u0434\u0430 \u043f\u0440\u0435\u0441\u0435\u0447\u0435 \u043f\u0440\u0430\u0432\u0430 \u043e\u0442 \u043c\u0440\u0435\u0436\u0430 \u043e\u0442 \u0443\u0441\u043f\u043e\u0440\u0435\u0434\u043d\u0438 \u043f\u0440\u0430\u0432\u0438 \u043d\u0430 \u0440\u0430\u0437\u0441\u0442\u043e\u044f\u043d\u0438\u0435 <\/span>b<\/em> \u0435\u0434\u043d\u0430 \u043e\u0442 \u0434\u0440\u0443\u0433\u0430. \u0422\u0430\u0437\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442 \u0441\u0435 \u043e\u043a\u0430\u0437\u0432\u0430, \u0447\u0435 \u0435 \u0440\u0430\u0432\u043d\u0430 \u043d\u0430 <\/span>. \u0420\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u044a\u0442 \u0435 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u0435\u043d \u0438 \u0438\u0437\u043d\u0435\u043d\u0430\u0434\u0432\u0430\u0449. \u0422\u043e\u0439 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f \u0434\u043e\u0431\u0440\u0435 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0430\u0442\u0430 \u043a\u043e\u043d\u0441\u0442\u0430\u043d\u0442\u0430 <\/span>\u043a\u0430\u0442\u043e \u0433\u0440\u0430\u043d\u0438\u0446\u0430 \u043d\u0430 \u043c\u043d\u043e\u0433\u043e \u0433\u043e\u043b\u044f\u043c \u0431\u0440\u043e\u0439 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u0438 \u0445\u0432\u044a\u0440\u043b\u044f\u043d\u0438\u044f. \u0412 \u0442\u0430\u0437\u0438 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0435 \u043f\u0440\u0430\u0432\u0438 \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0435\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u0430 \u043d\u0430 \u0411\u044e\u0444\u043e\u043d.<\/span><\/p>\n


Keywords:<\/em> probability, number <\/span>, Buffon<\/span><\/p>\n

<\/p>\n

BUFFON’S NEEDLE AND ESTIMATION OF THE <\/strong>NUMBER<\/strong><\/strong><\/p>\n


<\/strong>Abstract.<\/strong> In 1777 the French scientist Georges Louis Leclerc, Comte de Buffon, asked and solved the following problem: what is the probability that a needle of length l <\/em>when dropped randomly onto a net of parallel cracks, distance b<\/em> apart, lands on a crack?
The answer, leads to a remarkable and surprising result estimating the value of the well known constant as a limit of a great number of random drops. The aim of the present paper is to analyze Buffon\u2019s experiment.<\/p>\n

Evgenia Stoimenova
<\/strong>Institute of Mathematics and informatics \u2013 BAS<\/em><\/p>\n

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

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<\/div>\n

<\/a>\u0417\u0410 \u0415\u041b\u0415\u041a\u0422\u0420\u041e\u041d\u041d\u041e\u0422\u041e \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415<\/strong><\/p>\n


<\/strong>\u0414\u0430\u043d\u0438\u0435\u043b\u0430 \u0414\u0443\u0440\u0435\u0432\u0430 (\u0422\u0443\u043f\u0430\u0440\u043e\u0432\u0430)<\/strong><\/p>\n


Abstract. <\/strong>\u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0431\u0435\u043b\u0435\u0436\u043a\u0430 \u0441\u0435 \u0430\u043d\u043e\u043d\u0441\u0438\u0440\u0430 \u043d\u0430\u043c\u0435\u0440\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0441\u043f. \u201c\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u201d \u0434\u0430 \u043f\u0443\u0431\u043b\u0438\u043a\u0443\u0432\u0430 \u043c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u0438 \u0437\u0430 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e\u0442\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435. \u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0441\u0430 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043a\u0438 \u043d\u0430 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e\u0442\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435.<\/p>\n

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Keywords:<\/em> internet, computer, software, e-learning, web-learning, blendedlearning<\/p>\n

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ON THE E-LEARNING<\/strong><\/p>\n


Abstract.<\/strong> The present note announces the intention of Mathematics and Informatics<\/em>Journal to publish materials on e-learning. Basic characteristics of e-learning are presented<\/p>\n

\n

Daniela Dureva (Tuparova)
<\/strong>Department \u201cInformatics\u201d<\/em>
SWU \u201cNeofit Rilski\u201d, Blagoevgrad<\/em><\/p>\n

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

\n
<\/div>\n

<\/a>\u0423\u0421\u041b\u041e\u0412\u0418\u042f \u0417\u0410 \u041f\u0420\u0418\u0414\u041e\u0411\u0418\u0412\u0410\u041d\u0415 \u041d\u0410 \u041f\u0420\u041e\u0424\u0415\u0421\u0418\u041e\u041d\u0410\u041b\u041d\u041e-\u041a\u0412\u0410\u041b\u0418\u0424\u0418\u041a\u0410\u0426\u0418\u041e\u041d\u041d\u0418 \u0421\u0422\u0415\u041f\u0415\u041d\u0418 <\/strong><\/p>\n


<\/strong>\u0413\u0430\u043b\u044f \u041a\u043e\u0436\u0443\u0445\u0430\u0440\u043e\u0432\u0430<\/strong><\/p>\n


Abstract. <\/strong>\u041d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0431\u0435\u043b\u0435\u0436\u043a\u0430 \u0441\u044a\u0434\u044a\u0440\u0436\u0430 \u043e\u0441\u043d\u043e\u0432\u043d\u0430 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u0437\u0430 \u043f\u0440\u0438\u0434\u043e\u0431\u0438\u0432\u0430\u043d\u0435 \u043d\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\u0438 \u0441\u0442\u0435\u043f\u0435\u043d\u0438. \u0410\u043d\u043e\u043d\u0441\u0438\u0440\u0430 \u0441\u0435 \u043d\u0430\u043c\u0435\u0440\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0441\u043f. \u201c\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u201d \u0434\u0430 \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u044f \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442 \u0437\u0430 \u0441\u043f\u043e\u0434\u0435\u043b\u044f\u043d\u0435 \u043d\u0430 \u0432\u044a\u0437\u0433\u043b\u0435\u0434\u0438 \u0432\u044a\u0432 \u0432\u0440\u044a\u0437\u043a\u0430 \u0441 \u043a\u0430\u0440\u0438\u0435\u0440\u043d\u043e\u0442\u043e \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435 \u043d\u0430 \u0443\u0447\u0438\u0442\u0435\u043b\u0438\u0442\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430.<\/p>\n

Keywords:<\/em> teacher, career development, profession-qualigication degree<\/p>\n

REQUIREMENTS TO OBTAIN PROFESSION-QUALIFICATION DEGREES<\/strong><\/p>\n


Abstract.<\/strong>The present note considers basic information about the requirements for the profession-qualification degrees. It announces the intention of Mathematics and Informatics Journal to share opinions on the career development of teachers in Mathematics and Informatics.<\/p>\n

Galia Kozhuharova
<\/strong>IPKU, Stara Zagora<\/em><\/p>\n

 <\/strong><\/p>\n

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

\n
<\/div>\n

<\/a>\u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410\u0422\u0410 E \u0417\u0410\u0411\u0410\u0412\u041d\u0410<\/strong><\/p>\n


<\/strong>\u0412\u0435\u0441\u0435\u043b\u0438\u043d\u0430 \u0412\u044a\u043b\u043a\u0430\u043d\u043e\u0432\u0430<\/strong><\/p>\n


Abstract. <\/strong><\/strong>\u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u043e \u0435 \u0437\u0430\u043d\u044f\u0442\u0438\u0435 \u0441 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043e\u0442 5 \u043a\u043b\u0430\u0441 \u0432\u044a\u0440\u0445\u0443 \u0435\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0437\u0430 \u043e\u0431\u0438\u043a\u043e\u043b\u043a\u0430 \u0438 \u043b\u0438\u0446\u0435 \u043d\u0430 \u043f\u0440\u0430\u0432\u043e\u044a\u0433\u044a\u043b\u043d\u0438\u043a. \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0430 \u0435 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u043d\u0430 \u0441\u0440\u0435\u0434\u0430 GeoGebra.<\/p>\n

<\/strong>Keywords: <\/em>rectangle, square, perimeter, area, extremal problem, dynamicenvironment<\/p>\n

MATHEMATICS IS AMUSING<\/strong><\/p>\n


Abstract.<\/strong>A school hour with 5th grade students is presented on extremal problems connected with perimeters and areas of rectangles. The dynamic environment of GeoGebra is used.<\/p>\n

Veselina Valkanova
<\/strong>\u201cSt. Sofronii Vrachanski\u201d school<\/em><\/p>\n

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

\n
<\/div>\n

<\/a>\u0421\u0420\u0410\u0412\u041d\u042f\u0412\u0410\u041d\u0415 \u041d\u0410 \u0418\u0417\u0420\u0410\u0417\u0418 \u0421 \u041a\u0412\u0410\u0414\u0420\u0410\u0422\u041d\u0418 \u041a\u041e\u0420\u0415\u041d\u0418<\/strong><\/p>\n


<\/strong>\u0413\u0438\u043d\u043a\u0430 \u0411\u0438\u0437\u043e\u0432\u0430, \u0412\u0430\u043d\u044f \u041b\u0430\u043b\u0435\u0432\u0430<\/strong><\/p>\n


Abstract. <\/strong>\u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u043e \u0435 \u0437\u0430\u043d\u044f\u0442\u0438\u0435 \u0441 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043e\u0442 5 \u043a\u043b\u0430\u0441 \u0432\u044a\u0440\u0445\u0443 \u0435\u043a\u0441\u0442\u0440\u0435\u043c\u0430\u043b\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0437\u0430 \u043e\u0431\u0438\u043a\u043e\u043b\u043a\u0430 \u0438 \u043b\u0438\u0446\u0435 \u043d\u0430 \u043f\u0440\u0430\u0432\u043e\u044a\u0433\u044a\u043b\u043d\u0438\u043a. \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0430 \u0435 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u043d\u0430 \u0441\u0440\u0435\u0434\u0430 GeoGebra.<\/p>\n

Keywords: <\/em>rectangle, square, perimeter, area, extremal problem, dynamicenvironment<\/p>\n

COMPARISON OF EXPRESSIONS WITH SQUARE ROOTS<\/strong><\/p>\n


Abstract.<\/strong> the present note considers applications of the properties and the operations with square roots in comparison of irrational expressions.<\/p>\n

Ginka Bizova 
<\/strong>Faculty of Mathematics,
<\/em>Plovdiv University \u201eP. Hilendarski\u201c<\/em><\/p>\n

Vania Laleva<\/strong>
National School of Commerce, Plovdiv<\/em><\/p>\n

\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/span><\/p>","protected":false},"excerpt":{"rendered":"

\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS \u041a\u042a\u041c \u0427\u0418\u0422\u0410\u0422\u0415\u041b\u042f \/ \u0415DITORIAL\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ Sava Grozdev – \u0441\u0442\u0440. 7  \u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u0410\u043a\u0430\u0434\u0435\u043c\u0438\u043a \u0411\u043b\u0430\u0433\u043e\u0432\u0435\u0441\u0442 \u0421\u0435\u043d\u0434\u043e\u0432 \u2013 80-\u0433\u043e\u0434\u0438\u0448\u0435\u043d \u044e\u0431\u0438\u043b\u0435\u0439 – \u0441\u0442\u0440. 9  \u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u0421\u0440\u0430\u0432\u043d\u0438\u0442\u0435\u043b\u0435\u043d \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 8-\u0430 \u0436\u0430\u0443\u0442\u0438\u043a\u043e\u0432\u0441\u043a\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430[Comparative Analysis of the 8th Zhautykov Olympiad] \/ \u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ Sava Grozdev \/ \u041a\u0430\u0439\u0440\u043e\u0448 \u041c\u0430\u043a\u0438\u0448\u0435\u0432 \/ Kairosh Makishev – \u0441\u0442\u0440. 12 […]<\/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 LV, 2012 - 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