{"id":38121,"date":"2012-08-12T10:38:54","date_gmt":"2012-08-12T07:38:54","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012\/"},"modified":"2019-06-24T17:33:15","modified_gmt":"2019-06-24T14:33:15","slug":"matematika-i-informatika-knizhka-4-godina-lv-2012-2","status":"publish","type":"post","link":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/","title":{"rendered":"Mathematics and Informatics, Number 4\/2012, Volume 55"},"content":{"rendered":"

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

<\/strong>\u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u043d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430\u0442\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<\/a>
[Analysis of the Problems from the National Olympiad in Mathematics for University Students] \/<\/a> 
\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov<\/em> \u2013 p. 295<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0434\u0432\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 M\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<\/a>
[Analysis of Two Problems from the International Mathematics Olympiad] \/ <\/a>
\u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev<\/em> \u2013 p. 304<\/p>\n

Log in to read the full text<\/a><\/p>\n

Ubiquitous Archimedean Circles \/ <\/a>
Hiroshi Okumura<\/em> \u2013 p. 308<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u0421\u043e\u0444\u0442\u0443\u0435\u0440\u043d\u043e \u043e\u0431\u043d\u043e\u0432\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0437\u0430 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0441 \u043e\u0442\u0432\u043e\u0440\u0435\u043d \u043a\u043e\u0434<\/a>
[Managing Upgrades of Open Source Virtual Learning Environment] \/<\/a>
\u0412. \u0414\u0436\u0430\u043c\u0431\u0430\u0437\u043e\u0432 \/ V. Dzhambazov<\/em> \u2013 p. 312<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u0418\u0437\u043f\u0438\u0442\u044a\u0442 \u0437\u0430 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\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\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 <\/a>
[The Exam for the Fourth Professional Quali\ufb01 cation Degree in Mathematics] \/ <\/a>
\u0413. \u041a\u043e\u0436\u0443\u0445\u0430\u0440\u043e\u0432\u0430 \/ G. Kozhuharova<\/em> \u2013 p. 319<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u041f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0430 \u043d\u0430 \u0443\u0447\u0438\u0442\u0435\u043b\u0438\u0442\u0435 \u0437\u0430 \u043f\u0440\u0438\u0434\u043e\u0431\u0438\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0435\u0442\u0430 \u0438 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\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 <\/a>
\u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 <\/a>
[Preparation of Teachers for Obtaining Fifth and Fourth Professional Degrees in Informatics and Information Technologies] \/ <\/a>
\u0414. \u0411\u0440\u044a\u043d\u0435\u043a\u043e\u0432\u0430 \/ D. Branekova<\/em> \u2013 p. 328<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u0414\u0432\u0435 \u043d\u0430\u0443\u0447\u043d\u043e-\u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0438\u043d\u0438\u0446\u0438\u0430\u0442\u0438\u0432\u0438 <\/a>
[Two Education Research Initiatives] \/<\/a>
\u0411. \u041b\u0430\u0437\u0430\u0440\u043e\u0432 \/ B. Lazarov<\/em> \u2013 p. 348<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 \/ CONTEST PROBLEMS
<\/strong>The column is led by Dr. Svetlozar Doychev and Dr. Veselin Nenkov<\/em><\/p>\n

Contest problems \u2013 p. 352
Solutions of the contest problems from issue 4, 2011 - p. 353
Solutions of the contest problems from issue 5, 2011 - p. 458<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0442\u0443\u0440\u043d\u0438\u0440 \u201e\u0420\u0443\u043c\u0435\u043d \u0413\u0440\u043e\u0437\u0434\u0430\u043d\u043e\u0432\u201c \u043d\u0430 10 \u0433\u043e\u0434\u0438\u043d\u0438 (\u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0438\u0435 \u043e\u0442 \u0431\u0440\u043e\u0439 3) <\/a>
[Mathematical Tournament \u201cRumen Grozdanov\u201d – XTH Edition] \/<\/a>
\u0418. \u0421\u0442\u0430\u0440\u0438\u0431\u0440\u0430\u0442\u043e\u0432 \/ Y. Staribratov<\/em> \u2013 p. 361<\/p>\n

Log in to read the full text<\/a><\/p>\n

\u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415 \/ GUIDE FOR AUTHORS <\/strong>\u2013 p. 380<\/p>\n

Log in to read the full text<\/a><\/p>\n

 <\/p>\n

\n

<\/a>\u0410\u041d\u0410\u041b\u0418\u0417 \u041d\u0410 \u0417\u0410\u0414\u0410\u0427\u0418\u0422\u0415 \u041e\u0422 \u041d\u0410\u0426\u0418\u041e\u041d\u0410\u041b\u041d\u0410\u0422\u0410 \u0421\u0422\u0423\u0414\u0415\u041d\u0422\u0421\u041a\u0410 \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u0410 \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n

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

Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u0442\u0435\u043b\u043d\u0438\u0442\u0435 \u0442\u0435\u043c\u0438 \u043e\u0442 \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430\u0442\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 \u043f\u0440\u0435\u0437 2012 \u0433. \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u0441\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u0437\u0430 \u0442\u0440\u0438\u0442\u0435 \u0433\u0440\u0443\u043f\u0438 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438 \u0432 \u0437\u0430\u0432\u0438\u0441\u0438\u043c\u043e\u0441\u0442 \u043e\u0442 \u043f\u0440\u043e\u0444\u0438\u043b\u0430 \u043d\u0430 \u0432\u0438\u0441\u0448\u0435\u0442\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435, \u0432 \u043a\u043e\u0435\u0442\u043e \u0443\u0447\u0430\u0442.<\/p>\n

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

ANALYSIS OF THE PROBLEMS FROM THE NATIONAL OLYMPIAD IN MATHEMATICS FOR UNIVERSITY STUDENTS<\/strong><\/p>\n


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


Abstract.<\/strong> The paper proposes the competition papers of the National Olympiad in Mathematics for University students in 2012. Methodological solutions of the problems are considered for the three groups of students corresponding to the pro\ufb01 le of the Higher school
where they study.<\/p>\n

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

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

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

\n

<\/a>\u0410\u041d\u0410\u041b\u0418\u0417 \u041d\u0410 \u0414\u0412\u0415 \u0417\u0410\u0414\u0410\u0427\u0418 \u041e\u0422 \u041c\u0415\u0416\u0414\u0423\u041d\u0410\u0420\u041e\u0414\u041d\u0410\u0422\u0410 \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u0410 \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n


Sava Grozdev<\/strong><\/p>\n


Abstract.<\/strong> \u0423\u0441\u043f\u0435\u0448\u043d\u043e\u0442\u043e \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u044a\u0440\u0432\u0438\u0442\u0435 \u0434\u0432\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 \u0442\u0435\u043c\u0430\u0442\u0430 \u043d\u0430 \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 2012 \u0433. \u0435 \u0431\u0438\u043b\u043e \u0434\u043e\u0441\u0442\u0430\u0442\u044a\u0447\u043d\u043e \u0437\u0430 \u043f\u043e\u043b\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0431\u0440\u043e\u043d\u0437\u043e\u0432 \u043c\u0435\u0434\u0430\u043b. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u0442\u044f\u0445\u043d\u043e\u0442\u043e \u0430\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d\u0435. \u041e\u0441\u0442\u0430\u043d\u0430\u043b\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u0441\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0441 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0442\u0430 \u0438\u043c.<\/p>\n

Keywords:<\/em> Olympiad, problem solving, orthocent\u0435r, derivative.<\/p>\n

ANALYSIS OF TWO PROBLEMS FROM THE INTERNATIONAL MATHEMATICAL OLYMPIAD<\/strong><\/p>\n


Sava Grozdev<\/strong><\/p>\n


Abstract.<\/strong> Successful solutions of the \ufb01 rst two problems from the paper of the International Mathematical Olympiad in 2012 were enough for a bronze medal. The paper is dedicated to the analysis of these problems. The other problems are stated only.<\/p>\n

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

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

\n

<\/a>UBIQUITOUS ARCHIMEDEAN CIRCLES <\/strong><\/p>\n


Hiroshi Okumura<\/strong><\/p>\n


Abstract.<\/strong> Ubiquitous Archimedean circles of the arbelos arising from a result of the old Japanese geometry are considered. It is shown that they cover the plane.<\/p>\n

Keywords:<\/em> arbelos, Archimedean circle, paralleogram, similarity, ubiquitous circle.<\/p>\n

\u041f\u041e\u041a\u0420\u0418\u0412\u0410\u0429\u0418 \u0420\u0410\u0412\u041d\u0418\u041d\u0410\u0422\u0410 \u0410\u0420\u0425\u0418\u041c\u0415\u0414\u041e\u0412\u0418 \u041e\u041a\u0420\u042a\u0416\u041d\u041e\u0421\u0422\u0418<\/strong><\/p>\n


\u0425\u0438\u0440\u043e\u0448\u0438 \u041e\u043a\u0443\u043c\u0443\u0440\u0430<\/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 \u0435\u0434\u0438\u043d \u043a\u043b\u0430\u0441 \u0410\u0440\u0445\u0438\u043c\u0435\u0434\u043e\u0432\u0438 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u0430\u0440\u0431\u0435\u043b\u043e\u0441\u0430 (\u043e\u0431\u0443\u0449\u0430\u0440\u0441\u043a\u0438 \u043d\u043e\u0436), \u043a\u043e\u0439\u0442\u043e \u0441\u0435 \u043f\u043e\u044f\u0432\u044f\u0432\u0430 \u0432\u044a\u0437 \u043e\u0441\u043d\u043e\u0432\u0430 \u043d\u0430 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442 \u043e\u0442 \u0441\u0442\u0430\u0440\u043e\u044f\u043f\u043e\u043d\u0441\u043a\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f. \u041f\u043e\u043a\u0430\u0437\u0430\u043d\u043e \u0435, \u0447\u0435 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442\u0438\u0442\u0435 \u043e\u0442 \u0442\u043e\u0437\u0438 \u043a\u043b\u0430\u0441 \u043f\u043e\u043a\u0440\u0438\u0432\u0430\u0442 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430. <\/p>\n

Hiroshi Okumura<\/strong>
Professor<\/em>
251 Moo 15 Ban Kesorn, Tambol Sila <\/em>
Amphur Muang Khonkaen 40000<\/em>
Thailand<\/em><\/p>\n

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

\n

<\/a>\u0421\u041e\u0424\u0422\u0423\u0415\u0420\u041d\u041e \u041e\u0411\u041e\u041d\u041e\u0412\u042f\u0412\u0410\u041d\u0415 \u041d\u0410 \u0421\u0418\u0421\u0422\u0415\u041c\u0410 \u0417\u0410 \u0415\u041b\u0415\u041a\u0422\u0420\u041e\u041d\u041d\u041e \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415 \u0421 \u041e\u0422\u0412\u041e\u0420\u0415\u041d \u041a\u041e\u0414<\/strong><\/p>\n


\u0412\u0435\u043d\u0446\u0438\u0441\u043b\u0430\u0432 \u0414\u0436\u0430\u043c\u0431\u0430\u0437\u043e\u0432<\/strong><\/p>\n


Abstract.<\/strong> \u0421\u0432\u043e\u0435\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u0442\u043e \u0432\u044a\u0432\u0435\u0436\u0434\u0430\u043d\u0435 \u0432 \u0435\u043a\u0441\u043f\u043b\u043e\u0430\u0442\u0430\u0446\u0438\u044f \u043d\u0430 \u043f\u043e\u0440\u0435\u0434\u043d\u0430\u0442\u0430 \u043e\u0431\u043d\u043e\u0432\u0435\u043d\u0430 \u0432\u0435\u0440\u0441\u0438\u044f \u043d\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0430\u0442\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0437\u0430 \u0435\u043b\u0435\u043a\u0442\u0440\u043e\u043d\u043d\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0435 \u0432\u0430\u0436\u043d\u043e \u0438 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438\u043c\u043e. \u041d\u043e \u043d\u0430 \u043f\u0440\u0430\u043a\u0442\u0438\u043a\u0430 \u0441\u0435 \u043e\u043a\u0430\u0437\u0432\u0430, \u0447\u0435 \u0435 \u043f\u043e\u0447\u0442\u0438 \u043d\u0435\u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e \u0434\u0430 \u0441\u0435 \u0440\u0435\u0430\u043b\u0438\u0437\u0438\u0440\u0430 \u0430\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u0437\u0438\u0440\u0430\u043d\u043e \u0438 \u043d\u0435\u0437\u0430\u0431\u0430\u0432\u043d\u043e \u0442\u0430\u043a\u0430, \u043a\u0430\u043a\u0442\u043e
\u0435 \u043f\u0440\u0435\u043f\u043e\u0440\u044a\u0447\u0438\u0442\u0435\u043b\u043d\u043e. \u041f\u0440\u043e\u0431\u043b\u0435\u043c\u0438\u0442\u0435, \u043e\u0442 \u0435\u0434\u043d\u0430 \u0441\u0442\u0440\u0430\u043d\u0430, \u0441\u0430 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0447\u043d\u0438, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u043f\u0435\u0440\u0441\u043e\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d\u0438\u0442\u0435 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u0430\u0442\u0430 \u0438\u043d\u0441\u0442\u0430\u043b\u0430\u0446\u0438\u044f, \u043e\u0442 \u0434\u0440\u0443\u0433\u0430 \u2013 \u043f\u0440\u043e\u043c\u044f\u043d\u0430\u0442\u0430 \u043d\u0430 \u0438\u043d\u0442\u0435\u0440\u0444\u0435\u0439\u0441\u0430, \u043d\u0430 \u0438\u0434\u0435\u043e\u043b\u043e\u0433\u0438\u044f\u0442\u0430 \u0438 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430\u0442\u0430 \u043e\u0431\u044a\u0440\u043a\u0432\u0430 \u043f\u043e\u0442\u0440\u0435\u0431\u0438\u0442\u0435\u043b\u0438\u0442\u0435 \u0438 \u0433\u0438 \u043f\u043e\u0441\u0442\u0430\u0432\u044f \u0432 \u0441\u0442\u0440\u0435\u0441\u043e\u0432\u0430 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u044f.
\u0412\u044a\u0432\u0435\u0436\u0434\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043d\u043e\u0432\u0430 \u0432\u0435\u0440\u0441\u0438\u044f \u0442\u0440\u044f\u0431\u0432\u0430 \u0434\u0430 \u0441\u0442\u0430\u0432\u0430 \u043f\u043b\u0430\u043d\u043e\u0432\u043e, \u043f\u0440\u0438 \u043e\u0442\u0447\u0438\u0442\u0430\u043d\u0435 \u043d\u0430 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0444\u0430\u043a\u0442\u043e\u0440\u0438 \u2013 \u043e\u0442 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0447\u043d\u0438, \u043f\u0440\u0435\u0437 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438, \u0434\u043e \u043e\u0440\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0438 \u0430\u0434\u043c\u0438\u043d\u0438\u0441\u0442\u0440\u0430\u0442\u0438\u0432\u043d\u0438.<\/p>\n

Keywords<\/em>:e-Learning, VLE, Moodle, upgrade<\/p>\n

MANAGING UPGRADES OF OPEN SOURCE <\/span>VIRTUAL LEARNING ENVIRONMENT<\/strong><\/p>\n

Abstract.<\/strong> The timely upgrade to the next updated version of an eLearning system is important and necessary. But in practice it appears that it is almost impossible to realize it automatically and immediately, as recommended. Problems on the one hand are related with customization of the installation, of another \u2013 the change of interface, ideology and methodology confuse consumers and puts them in a stressful situation. A new version should be planned, taking into account many factors – from technologycal, through methodical, to organizational and administrative ones.\u0002<\/p>\n

Vencislav Dzhambazov<\/strong>
Assistant Professor, PhD<\/em>
\u201cInformatics\u201d Department<\/em>
New Bulgarian University<\/em><\/p>\n

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

\n

<\/a>\u0418\u0417\u041f\u0418\u0422\u042a\u0422 \u0417\u0410 \u0427\u0415\u0422\u0412\u042a\u0420\u0422\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\u0410 \u0421\u0422\u0415\u041f\u0415\u041d \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n


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


Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u044a\u0434\u044a\u0440\u0436\u0430 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u0437\u0430 \u0443\u0441\u043b\u043e\u0432\u0438\u044f\u0442\u0430, \u0440\u0435\u0434\u0430 \u0438 \u043e\u0440\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u044f\u0442\u0430 \u043d\u0430 \u0438\u0437\u043f\u0438\u0442\u0430 \u0437\u0430 \u043f\u0440\u0438\u0434\u043e\u0431\u0438\u0432\u0430\u043d\u0435 \u043d\u0430 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\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 \u043d\u0430 \u0443\u0447\u0438\u0442\u0435\u043b\u0438\u0442\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u0438\u0437\u043f\u0438\u0442\u0430 \u043f\u0440\u0435\u0437 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0442\u0435 \u0447\u0435\u0442\u0438\u0440\u0438 \u0433\u043e\u0434\u0438\u043d\u0438 \u0438 \u0430\u043d\u0430\u043b\u0438\u0437
\u0441 \u043a\u0440\u0430\u0442\u043a\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 2011 \u0433.<\/p>\n

Keywords:<\/em> degree, professional quali\ufb01 cation degree, problem.<\/p>\n

THE EXAM FOR THE FOURTH PROFESSIONAL QUALIFICATION DEGREE IN MATHEMATICS<\/strong><\/p>\n

Galia Kozhuhrova<\/strong><\/p>\n

Abstract.<\/strong> The paper contains information about the requirements, the regulations and the organization of the exam to acquire fourth professional quali\ufb01 cation degree by Mathematics teachers. The last four years exam papers are given and an analysis with short solutions of the 2011 problems is proposed. <\/p>\n

Galya Kozhuharova<\/strong>
Associated Professor, PhD<\/em>
IPKU, Trakia University<\/em>
Stara Zagora<\/em><\/p>\n

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

 <\/p>\n

\n

<\/a>\u041f\u041e\u0414\u0413\u041e\u0422\u041e\u0412\u041a\u0410 \u041d\u0410 \u0423\u0427\u0418\u0422\u0415\u041b\u0418\u0422\u0415 \u0417\u0410 \u041f\u0420\u0418\u0414\u041e\u0411\u0418\u0412\u0410\u041d\u0415 <\/strong>\u041d\u0410 \u041f\u0415\u0422\u0410 \u0418 \u0427\u0415\u0422\u0412\u042a\u0420\u0422\u0410 \u041f\u0420\u041e\u0424\u0415\u0421\u0418\u041e\u041d\u0410\u041b\u041d\u041e-<\/strong>\u041a\u0412\u0410\u041b\u0418\u0424\u0418\u041a\u0410\u0426\u0418\u041e\u041d\u041d\u0410 \u0421\u0422\u0415\u041f\u0415\u041d
\u041f\u041e \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0422\u0418\u041a\u0410<\/strong> \u0418 \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0426\u0418\u041e\u041d\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418<\/strong><\/p>\n


\u0414\u0438\u043c\u0438\u0442\u0440\u0438\u043d\u0430 \u0411\u0440\u044a\u043d\u0435\u043a\u043e\u0432\u0430<\/strong><\/p>\n


Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u044a\u0434\u044a\u0440\u0436\u0430 \u043d\u0430\u0441\u043e\u043a\u0438 \u0432 \u043f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0430\u0442\u0430 \u043d\u0430 \u0443\u0447\u0438\u0442\u0435\u043b\u0438\u0442\u0435 \u0437\u0430 \u0443\u0441\u043f\u0435\u0448\u043d\u043e \u043f\u043e\u043b\u0430\u0433\u0430\u043d\u0435 \u043d\u0430 \u0438\u0437\u043f\u0438\u0442\u0438\u0442\u0435 \u0437\u0430 \u043f\u0440\u0438\u0434\u043e\u0431\u0438\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0435\u0442\u0430 \u0438 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\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 \u0441\u0442\u0435\u043f\u0435\u043d \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0432 \u0414\u0418\u041f\u041a\u0423 \u2013 \u0421\u0442\u0430\u0440\u0430 \u0417\u0430\u0433\u043e\u0440\u0430. \u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0441\u0430 \u0442\u0435\u043c\u0438\u0442\u0435, \u0438\u0437\u0442\u0435\u0433\u043b\u0435\u043d\u0438 \u043d\u0430 \u043b\u043e\u0442\u0430\u0440\u0438\u0435\u043d \u043f\u0440\u0438\u043d\u0446\u0438\u043f \u043e\u0442 \u0438\u0437\u043f\u0438\u0442\u0430 \u0437\u0430 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0430 \u0441\u0442\u0435\u043f\u0435\u043d \u043e\u0442 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u0442\u0435 \u0442\u0440\u0438 \u0433\u043e\u0434\u0438\u043d\u0438. \u0410\u043d\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d\u0438 \u0441\u0430 \u043f\u0440\u043e\u043f\u0443\u0441\u043a\u0438\u0442\u0435 \u0438 \u0447\u0435\u0441\u0442\u043e \u0434\u043e\u043f\u0443\u0441\u043a\u0430\u043d\u0438\u0442\u0435 \u0433\u0440\u0435\u0448\u043a\u0438 \u043d\u0430 \u043a\u0430\u043d\u0434\u0438\u0434\u0430\u0442\u0438\u0442\u0435.<\/p>\n

Keywords:<\/em> Professional Qualification Degree, Teachers, Education, Informatics, Information Technologies<\/p>\n

PREPARATION OF TEACHERS FOR OBTAINING FIFTH AND FOURTH PROFESSIONAL DEGREES IN INFORMATICS 
<\/strong>AND INFORMATION TECHNOLOGIES<\/strong><\/p>\n


Abstract.<\/strong> This article provides guidance in the preparation of teachers for passing the exams for obtaining \ufb01 fth and fourth professional degrees for informatics and information technologies from the Department for Information and In-Service Teacher Training \u2013 Stara Zagora. Below are the topics for the written examinations in the past three years, which drawn on a lottery basis. Common mistakes and gaps were analyzed.<\/p>\n

Dimitrina Branekova,<\/strong>
Assistant Professor, PhD<\/em>
Department for Information and In-Service Teacher Training,<\/em>
Trakia University, Stara Zagora, Bulgaria<\/em><\/p>\n

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

\n

<\/a>\u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u0418\u0422\u0415 \u0415\u0412\u0420\u0418\u0421\u0422\u0418\u041a\u0418 \u0418 \u0412\u042a\u0417\u041c\u041e\u0416\u041d\u041e\u0421\u0422\u0418\u0422\u0415 \u0418\u041c \u0417\u0410 \u0418\u0417\u041f\u041e\u041b\u0417\u0412\u0410\u041d\u0415 \u0412 \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u041d\u0410\u0422\u0410 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n


\u0416\u0438\u0432\u043a\u043e \u0416\u0435\u043b\u0435\u0432<\/strong><\/p>\n


Abstract.<\/strong> \u0418\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0435\u0432\u0440\u0438\u0441\u0442\u0438\u0447\u043d\u0438 \u043f\u043e\u0445\u0432\u0430\u0442\u0438 \u0435 \u043a\u043b\u044e\u0447\u043e\u0432\u043e \u0437\u0432\u0435\u043d\u043e \u043f\u0440\u0438 \u0444\u043e\u0440\u043c\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0443\u043c\u0435\u043d\u0438\u044f \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u0415\u0434\u043d\u0430 \u043e\u0442 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0446\u0435\u043b\u0438 \u043d\u0430 \u0441\u044a\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u0430\u0442\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0435 \u0434\u0430 \u0441\u0435 \u0441\u044a\u0437\u0434\u0430\u0434\u0435 \u043f\u0440\u043e\u0444\u0435\u0441\u0438\u043e\u043d\u0430\u043b\u043d\u0430 \u043d\u0430\u0433\u043b\u0430\u0441\u0430 \u0438 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0443\u043c\u0435\u043d\u0438\u044f \u0437\u0430 \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435 \u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u043d\u0438 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u0438 \u0441\u044a\u0441 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430\u0442\u0430 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430\u0442\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u0442\u0430. \u0411\u0435\u0437\u0441\u043f\u043e\u0440\u043d\u043e \u0435\u0434\u043d\u043e \u043e\u0442 \u043d\u0430\u0439-\u0432\u0430\u0436\u043d\u0438\u0442\u0435 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u043d\u0430 \u0441\u044a\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u0438\u0442\u0435 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0438 \u0441\u043e\u0444\u0442\u0443\u0435\u0440 \u0435 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0447\u0440\u0435\u0437 \u043f\u043e-\u0434\u043e\u0431\u0440\u0430 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u043c\u0438\u0437\u0430\u0446\u0438\u044f \u043d\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u0442\u0438\u043f\u043e\u0432\u0435 \u0435\u0432\u0440\u0438\u0441\u0442\u0438\u0447\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 (\u0432 \u0442\u043e\u0432\u0430 \u0447\u0438\u0441\u043b\u043e \u0438 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u043d\u0438) \u043d\u0430 \u0431\u0430\u0437\u0430\u0442\u0430 \u043d\u0430 \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u0438 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043d\u043e \u0431\u0430\u0437\u0438\u0440\u0430\u043d\u0438 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0438 \u0435\u0432\u0440\u0438\u0441\u0442\u0438\u043a\u0438.<\/p>\n

Keywords:<\/em> Olympiad, problem, problem solving, heuristics, computer.<\/p>\n

COMPUTER HEURISTICS AND THEIR US\u0415 <\/span>IN THE OLYMPIAD MATHEMATICS<\/strong><\/p>\n


Abstract.<\/strong> The study of heuristics is a crucial element in the formation of skills for problem solving in Mathematics. One of the main tasks of the contemporary methodology in Mathematics is to establish a professional ground and practical abilities to modeling and solving of problem situations by the means of Mathematics and Informatics. No doubt that one of the most important applications of the contemporary technologies and software is to Mathematics Education by a better algorithmization of different types of heuristics problems (Olympiad problems included) on the base of concrete and programming-based computer heuristics.<\/p>\n

\u0002Zhivko Zhelev
<\/strong>Assistant Professor, PhD
<\/em>Faculty of Economics
<\/em>Trakia University
<\/em>Stara Zagora, Bulgaria<\/em><\/p>\n

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

 <\/p>\n

\n

<\/a>\u0414\u0412\u0415 \u041d\u0410\u0423\u0427\u041d\u041e-\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0418 \u0418\u041d\u0418\u0426\u0418\u0410\u0422\u0418\u0412\u0418<\/strong><\/p>\n


Borislav Lazarov<\/strong><\/p>\n


Abstract.<\/strong> \u0418\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0430\u0442\u0430 \u0431\u0435\u043b\u0435\u0436\u043a\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0438\u0442\u0435 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u0438 \u0432 \u0440\u0430\u043c\u043a\u0438\u0442\u0435 \u043d\u0430 \u041c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0438\u044f \u043f\u0440\u043e\u0435\u043a\u0442 MITE \u0438 \u043d\u0430 \u043d\u0430\u0443\u0447\u043d\u0430\u0442\u0430 \u0434\u0435\u0439\u043d\u043e\u0441\u0442, \u043e\u0441\u044a\u0449\u0435\u0441\u0442\u0432\u044f\u0432\u0430\u043d\u0430 \u043e\u0442 \u0441\u0435\u043c\u0438\u043d\u0430\u0440\u0430 \u201c\u0414\u0438\u0434\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u043e \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435\u201d \u043f\u0440\u0438 \u0418\u043d\u0441\u0442\u0438\u0442\u0443\u0442\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u2013 \u0411\u0410\u041d \u043f\u0440\u0435\u0437 \u0430\u043a\u0430\u0434\u0435\u043c\u0438\u0447\u043d\u0430\u0442\u0430 2010\/2011 \u0433.<\/p>\n

Keywords:<\/em> competition, scienti\ufb01 c project, mathematical methodology, Didactics, didactical modeling.<\/p>\n

TWO EDUCATION RESEARCH INITIATIVES<\/strong><\/p>\n


Borislav Lazarov<\/strong><\/p>\n


Abstract.<\/strong> The present information note considers the International Competitions in the frames of the International Project MITE and the scienti\ufb01 c activities of the Workshop \u201cDidactical modeling\u201d of the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, during the 2010\/2011 academic year.<\/p>\n

Borislav Lazarov<\/strong>
Associate Professor, Doctor in Pedagogy<\/em>
Institute of Mathematics and Informatics \u2013 BAS<\/em><\/p>\n

 <\/p>\n

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

 <\/p>\n

 <\/p>\n

\n

<\/a>\u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u0427\u0415\u0421\u041a\u0418 \u0422\u0423\u0420\u041d\u0418\u0420 \u201e\u0420\u0423\u041c\u0415\u041d \u0413\u0420\u041e\u0417\u0414\u0410\u041d\u041e\u0412\u201c \u041d\u0410 10 \u0413\u041e\u0414\u0418\u041d\u0418
<\/strong>(\u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0438\u0435 \u043e\u0442 \u0431\u0440\u043e\u0439 3)<\/strong><\/p>\n


Ivailo Staribratov<\/strong><\/p>\n

\u0412 \u041f\u043b\u043e\u0432\u0434\u0438\u0432 \u043d\u0430 19.02.2012 \u0441\u0435 \u043f\u0440\u043e\u0432\u0435\u0434\u0435 \u0437\u0430 \u0434\u0435\u0441\u0435\u0442\u0438 \u043f\u044a\u0442 \u0422\u0443\u0440\u043d\u0438\u0440 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u0438\u043c\u0435\u0442\u043e \u043d\u0430 \u0435\u0434\u0438\u043d \u0437\u043d\u0430\u0447\u0438\u043c \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a – \u0420\u0443\u043c\u0435\u043d \u0413\u0440\u043e\u0437\u0434\u0430\u043d\u043e\u0432. \u041f\u0440\u0435\u0437 90-\u0442\u0435 \u0433\u043e\u0434\u0438\u043d\u0438 \u043d\u0430 \u043c\u0438\u043d\u0430\u043b\u0438\u044f \u0432\u0435\u043a \u0442\u043e\u0439 \u043f\u043e\u0434\u0433\u043e\u0442\u0432\u0438 \u043d\u0430\u0434 \u0441\u0442\u043e \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0437\u0430 \u0443\u0447\u0430\u0441\u0442\u0438\u0435 \u0432 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0438 \u0438 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f. \u0422\u0432\u043e\u0440\u0447\u0435\u0441\u0442\u0432\u043e\u0442\u043e \u043c\u0443 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438 \u0435 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u043e. \u0410\u0432\u0442\u043e\u0440 \u0435 \u043d\u0430 \u0433\u043e\u043b\u044f\u043c \u0431\u0440\u043e\u0439 \u0441\u0442\u0430\u0442\u0438\u0438 \u0438 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u043a\u043e\u0438\u0442\u043e \u0441\u0430 \u043f\u0443\u0431\u043b\u0438\u043a\u0443\u0432\u0430\u043d\u0438 \u0432 \u0441\u043f. \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430. \u0417\u0430 \u0437\u0430\u0441\u043b\u0443\u0433\u0438\u0442\u0435 \u0441\u0438 \u0435 \u043d\u0430\u0433\u0440\u0430\u0434\u0435\u043d \u0441 \u043e\u0440\u0434\u0435\u043d \u201e\u041a\u0438\u0440\u0438\u043b \u0438 \u041c\u0435\u0442\u043e\u0434\u0438\u0439\u201c. \u041e\u0442 2003 \u0433\u043e\u0434\u0438\u043d\u0430 \u0432 \u041e\u041c\u0413 \u201e\u0410\u043a\u0430\u0434. \u041a\u0438\u0440\u0438\u043b \u041f\u043e\u043f\u043e\u0432\u201c – \u041f\u043b\u043e\u0432\u0434\u0438\u0432 \u0441\u0435 \u043f\u0440\u043e\u0432\u0435\u0436\u0434\u0430 \u0442\u0443\u0440\u043d\u0438\u0440 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u043d\u0435\u0433\u043e\u0432\u043e \u0438\u043c\u0435. \u0422\u0430\u0437\u0438 \u0433\u043e\u0434\u0438\u043d\u0430 \u0432\u0437\u0435\u0445\u0430 \u0443\u0447\u0430\u0441\u0442\u0438\u0435 488 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043e\u0442 \u0433\u0440\u0430\u0434\u043e\u0432\u0435\u0442\u0435 \u0421\u043e\u0444\u0438\u044f, \u041f\u043b\u043e\u0432\u0434\u0438\u0432, \u0412\u0430\u0440\u043d\u0430, \u041a\u0430\u0440\u043b\u043e\u0432\u043e, \u0410\u0441\u0435\u043d\u043e\u0432\u0433\u0440\u0430\u0434, \u0421\u044a\u0435\u0434\u0438\u043d\u0435\u043d\u0438\u0435 \u0438 \u0434\u0440\u0443\u0433\u0438 \u0431\u043b\u0438\u0437\u043a\u0438 \u0434\u043e \u041f\u043b\u043e\u0432\u0434\u0438\u0432 \u043d\u0430\u0441\u0435\u043b\u0435\u043d\u0438 \u043c\u0435\u0441\u0442\u0430.<\/p>\n

MATHEMATICAL TOURNAMENT RUMEN GROZDANOV – XTH EDITION<\/strong><\/p>\n

Mathematical tournament Rumen Grozdanov was held on 19th February 2012 for the tenth time. The tournament is named after a signi\ufb01 cant mathematician who worked at Academician Kiril Popov Model High School of Mathematics in Plovdiv. He prepared more than hundred
of students to participate in national and international contests and competitons in 90s of the XXth century. His works of mathematical tasks and problems is remarkable. He wrote numerous of articles and he is an author of performing tasks published in the magazine Mathematics.
For his merits he was awarded the medal \u201cCyril and Methodius\u201d. Academician Kiril Popov Model High School of Mathematics in Plovdiv is founder and it has hosted the tournament in mathematics in his name from its beginning in 2003. This year the tournament was attended by over 488 students from So\ufb01 a, Plovdiv, Varna, Karlovo, Asenovgrad and other small towns near Plovdiv.<\/p>\n

Ivaylo Staribratov<\/strong>
Doctor<\/em>
\u201cAcad. Kiril Popov\u201d High School of Mathematic<\/em><\/p>\n

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

 <\/p>\n

\n

 <\/p>\n

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

\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS \u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u043d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430\u0442\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[Analysis of the Problems from the National Olympiad in Mathematics for University Students] \/ \u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov – \u0441\u0442\u0440. 295 \u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0434\u0432\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 M\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[Analysis of Two Problems from the International Mathematics Olympiad] \/ […]<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[2882],"tags":[],"yoast_head":"\n"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 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:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\""\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 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 \u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u043d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430\u0442\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[Analysis of the Problems from the National Olympiad in Mathematics for University Students] \/ \u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov – \u0441\u0442\u0440. 295 \u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0434\u0432\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 M\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[Analysis of Two Problems from the International Mathematics Olympiad] \/ […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/\" \/>\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=\"2012-08-12T07:38:54+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=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/\"},\"author\":{\"name\":\"\",\"@id\":\"\"},\"headline\":\"„\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430“, \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012\",\"datePublished\":\"2012-08-12T07:38:54+00:00\",\"dateModified\":\"2019-06-24T14:33:15+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/\"},\"wordCount\":2107,\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/en\/#organization\"},\"articleSection\":[\"\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\"\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/\",\"url\":\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/\",\"name\":\"\\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \u0410\u0437-\u0431\u0443\u043a\u0438\",\"isPartOf\":{\"@id\":\"https:\/\/azbuki.bg\/en\/#website\"},\"datePublished\":\"2012-08-12T07:38:54+00:00\",\"dateModified\":\"2019-06-24T14:33:15+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/history.azbuki.bg\/en\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"„\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430“, \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/azbuki.bg\/en\/#website\",\"url\":\"https:\/\/azbuki.bg\/en\/\",\"name\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\",\"description\":\"\u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e \u0438\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0437\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u043d\u0430\u0443\u043a\u0430\",\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/en\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/azbuki.bg\/en\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/azbuki.bg\/en\/#organization\",\"name\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\",\"url\":\"https:\/\/azbuki.bg\/en\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/azbuki.bg\/en\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png\",\"contentUrl\":\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png\",\"width\":250,\"height\":70,\"caption\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\"},\"image\":{\"@id\":\"https:\/\/azbuki.bg\/en\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/Azbuki55\/\",\"https:\/\/www.instagram.com\/nion_azbuki\/\"]},{\"@type\":\"Person\",\"@id\":\"\",\"url\":\"https:\/\/science.azbuki.bg\/en\/author\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \u0410\u0437-\u0431\u0443\u043a\u0438","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/","og_locale":"en_US","og_type":"article","og_title":"\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \u0410\u0437-\u0431\u0443\u043a\u0438","og_description":"\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS \u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u043d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430\u0442\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[Analysis of the Problems from the National Olympiad in Mathematics for University Students] \/ \u0421. \u0413\u0440\u043e\u0437\u0434\u0435\u0432 \/ S. Grozdev, \u0412. \u041d\u0435\u043d\u043a\u043e\u0432 \/ V. Nenkov – \u0441\u0442\u0440. 295 \u0410\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0434\u0432\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 M\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[Analysis of Two Problems from the International Mathematics Olympiad] \/ […]","og_url":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2012-08-12T07:38:54+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":"6 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/#article","isPartOf":{"@id":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/"},"author":{"name":"","@id":""},"headline":"„\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430“, \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012","datePublished":"2012-08-12T07:38:54+00:00","dateModified":"2019-06-24T14:33:15+00:00","mainEntityOfPage":{"@id":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/"},"wordCount":2107,"publisher":{"@id":"https:\/\/azbuki.bg\/en\/#organization"},"articleSection":["\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\""],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/","url":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/","name":"\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012 - \u0410\u0437-\u0431\u0443\u043a\u0438","isPartOf":{"@id":"https:\/\/azbuki.bg\/en\/#website"},"datePublished":"2012-08-12T07:38:54+00:00","dateModified":"2019-06-24T14:33:15+00:00","breadcrumb":{"@id":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/press.azbuki.bg\/matematics\/matharticles\/matematika-i-informatika-knizhka-4-godina-lv-2012-2\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/history.azbuki.bg\/en\/"},{"@type":"ListItem","position":2,"name":"„\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430“, \u043a\u043d\u0438\u0436\u043a\u0430 4, \u0433\u043e\u0434\u0438\u043d\u0430 LV, 2012"}]},{"@type":"WebSite","@id":"https:\/\/azbuki.bg\/en\/#website","url":"https:\/\/azbuki.bg\/en\/","name":"\u0410\u0437-\u0431\u0443\u043a\u0438","description":"\u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e \u0438\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0437\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u043d\u0430\u0443\u043a\u0430","publisher":{"@id":"https:\/\/azbuki.bg\/en\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/azbuki.bg\/en\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/azbuki.bg\/en\/#organization","name":"\u0410\u0437-\u0431\u0443\u043a\u0438","url":"https:\/\/azbuki.bg\/en\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/azbuki.bg\/en\/#\/schema\/logo\/image\/","url":"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png","contentUrl":"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png","width":250,"height":70,"caption":"\u0410\u0437-\u0431\u0443\u043a\u0438"},"image":{"@id":"https:\/\/azbuki.bg\/en\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/Azbuki55\/","https:\/\/www.instagram.com\/nion_azbuki\/"]},{"@type":"Person","@id":"","url":"https:\/\/science.azbuki.bg\/en\/author\/"}]}},"_links":{"self":[{"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/38121"}],"collection":[{"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/comments?post=38121"}],"version-history":[{"count":11,"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/38121\/revisions"}],"predecessor-version":[{"id":60547,"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/38121\/revisions\/60547"}],"wp:attachment":[{"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/media?parent=38121"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/categories?post=38121"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/science.azbuki.bg\/en\/wp-json\/wp\/v2\/tags?post=38121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}