{"id":38384,"date":"2013-10-16T13:16:45","date_gmt":"2013-10-16T10:16:45","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles\/mathinfo052013\/"},"modified":"2019-06-24T17:33:15","modified_gmt":"2019-06-24T14:33:15","slug":"mathinfo052013","status":"publish","type":"post","link":"https:\/\/newspaper.azbuki.bg\/en\/matematics\/matharticles\/mathinfo052013\/","title":{"rendered":"Mathematics and Informatics, Number 5\/2013, Volume 56"},"content":{"rendered":"<p><strong><a name=\"top\">&nbsp;<\/a>\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS<\/strong><\/p>\n<p><a href=\"#art01\">\u041f\u0435\u0440\u0432\u044b\u0439 \u0442\u0432\u043e\u0440\u0447\u0435\u0441\u043a\u0438\u0439 \u043a\u043e\u043d\u043a\u0443\u0440\u0441 \u0443\u0447\u0438\u0442\u0435\u043b\u0435\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0430<br \/>Creative Competition for Mathematics Teachers of Kazakhstan<\/a><br \/><i>\u041a\u0430\u0439\u0440\u043e\u0448 \u041c\u0430\u043a\u0438\u0448\u0435\u0432, Kairosh Makishev<\/i> \u2013 page 409<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><br \/><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art03\">\u0412\u0435\u0440\u0438\u0433\u0438 \u043d\u0430 \u041c\u0430\u0440\u043a\u043e\u0432 \u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435\u0442\u043e \u0438\u043c \u0432 Google<br \/>Markov Chains and their Application in Google<\/a><br \/><i>\u0415\u0432\u0433\u0435\u043d\u0438\u044f \u0421\u0442\u043e\u0438\u043c\u0435\u043d\u043e\u0432\u0430, Eugenia Stoimenova <\/i>\u2013 \u0441\u0442\u0440. 426<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/em><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art06\">\u0421\u0438\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u0435\u043d \u043f\u043e\u0434\u0445\u043e\u0434 \u043a\u044a\u043c \u043f\u0440\u043e\u0446\u0435\u0441\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438<br \/>Synergetic Approach to the Training Process in Information Technology<\/a><br \/><i>\u0415\u0432\u0433\u0435\u043d\u0438\u044f \u0413\u043e\u0440\u0430\u043d\u043e\u0432\u0430, Evgenia Goranova <\/i>\u2013 page 463<br \/><em><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/em><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art07\">Boosting Efficiency of Project-oriented Teaching and Learning through Classroom Management and Online Testing<br \/>\u041f\u043e\u0432\u0438\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0435\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0441\u0442\u0442\u0430 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0442\u043d\u043e-\u043e\u0440\u0438\u0435\u043d\u0442\u0438\u0440\u0430\u043d\u043e\u0442\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0447\u0440\u0435\u0437 \u043c\u0435\u043d\u0438\u0434\u0436\u043c\u044a\u043d\u0442 \u043d\u0430 \u043a\u043b\u0430\u0441\u0430 \u0438 \u043e\u043d\u043b\u0430\u0439\u043d \u0442\u0435\u0441\u0442\u043e\u0432\u043e \u0438\u0437\u043f\u0438\u0442\u0432\u0430\u043d\u0435<\/a><br \/><i>Ivan Shotlekov, \u0418\u0432\u0430\u043d \u0428\u043e\u0442\u043b\u0435\u043a\u043e\u0432 \/ Vanya Ivanova, \u0412\u0430\u043d\u044f \u0418\u0432\u0430\u043d\u043e\u0432\u0430 \/ Kirina Boykova, \u041a\u0438\u0440\u0438\u043d\u0430 \u0411\u043e\u0439\u043a\u043e\u0432\u0430 <\/i>\u2013 \u0441\u0442\u0440. 476<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418\/ CONTEST PROBLEMS<br \/><\/strong><i>\u0420\u0443\u0431\u0440\u0438\u043a\u0430\u0442\u0430 \u0441\u0435 \u0432\u043e\u0434\u0438 \u043e\u0442 \u0434-\u0440 \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \u0438 \u0434-\u0440 \u0416\u0438\u0432\u043a\u043e \u0416\u0435\u043b\u0435\u0432<\/i><\/p>\n<p>Contest Problems of this Issue<br \/>Contest problems of the issu\u0435 \u2013 \u0441\u0442\u0440. 489<\/p>\n<p>\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u043e\u0442 \u0431\u0440\u043e\u0439 6, 2012 <br \/>Solutions of the contest problems from issue 6, 2012 \u2013 \u0441\u0442\u0440. 490<\/p>\n<p><strong><br \/>\u0412 \u041d\u041e\u0412\u0418\u0422\u0415 \u0411\u0420\u041e\u0415\u0412\u0415 \u041d\u0410 \u0421\u041f\u0418\u0421\u0410\u041d\u0418\u042f\u0422\u0410 \u041d\u0410 \u0418\u0417\u0414\u0410\u0422\u0415\u041b\u0421\u0422\u0412\u041e \u201e\u0410\u0417 \u0411\u0423\u041a\u0418\u201d \u0427\u0415\u0422\u0415\u0422\u0415 \/ <\/strong><br \/><strong>READ IN THE LATEST ISSUES OF \u201eAZ BUKI\u201d JOURNALS <\/strong>\u2013 \u0441\u0442\u0440. 493<\/p>\n<p><strong><br \/>\u0423\u041a\u0410\u0417\u0410\u041d\u0418\u0415 \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415 \/ GUIDE FOR AUTHORS<\/strong> \u2013 \u0441\u0442\u0440. 495<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art01\">&nbsp;<\/a>\u041f\u0415\u0420\u0412\u042cI\u0419 \u0422\u0412\u041e\u0420\u0427\u0415\u0421\u041a\u0418\u0419 \u041a\u041e\u041d\u041a\u0423\u0420\u0421 <\/strong><strong>\u0423\u0427\u0418\u0422\u0415\u041b\u0415\u0419 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0418 \u041a\u0410\u0417\u0410\u0425\u0421\u0422\u0410\u041d\u0410<\/p>\n<p><\/strong><strong>\u041a\u0430\u0439\u0440\u043e\u0448 \u041c\u0430\u043a\u0438\u0448\u0435\u0432<\/p>\n<p><\/strong><strong>Abstract. <\/strong>\u041f\u0435\u0440\u0432\u044b\u0439 \u0441\u044a\u0435\u0437\u0434 \u0443\u0447\u0438\u0442\u0435\u043b\u0435\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0430, \u0441\u043e\u0441\u0442\u043e\u044f\u0432\u0448\u0438\u0439\u0441\u044f 10\u201312 \u043c\u0430\u044f 2011 \u0433\u043e\u0434\u0430, \u043f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u0438\u043b \u0441\u043e\u0437\u0434\u0430\u0442\u044c \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0441\u043a\u0443\u044e \u0410\u0441\u0441\u043e\u0446\u0438\u0430\u0446\u0438\u044e \u0443\u0447\u0438\u0442\u0435\u043b\u0435\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438. \u0410\u0441\u0441\u043e\u0446\u0438\u0430\u0446\u0438\u044f \u0431\u044b\u043b\u0430 \u0441\u043e\u0437\u0434\u0430\u043d\u0430 \u0438 \u043e\u0444\u0438\u0446\u0438\u0430\u043b\u044c\u043d\u043e \u0437\u0430\u0440\u0435\u0433\u0438\u0441\u0442\u0440\u0438\u0440\u043e\u0432\u0430\u043d\u0430 6 \u0438\u044e\u043b\u044f 2012 \u0433\u043e\u0434\u0430. &nbsp; \u0421\u043e\u0433\u043b\u0430\u0441\u043d\u043e \u041f\u043b\u0430\u043d\u0443 \u043c\u0435\u0440\u043e\u043f\u0440\u0438\u044f\u0442\u0438\u0439 \u043d\u0430 2013 \u0433\u043e\u0434 \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0441\u043a\u0430\u044f \u0410\u0441\u0441\u043e\u0446\u0438\u0430\u0446\u0438\u044f \u0443\u0447\u0438\u0442\u0435\u043b\u0435\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u043f\u0440\u043e\u0432\u043e\u0434\u0438\u043b\u0430 \u041f\u0435\u0440\u0432\u044b\u0439 \u0442\u0432\u043e\u0440\u0447\u0435\u0441\u043a\u0438\u0439 \u043a\u043e\u043d\u043a\u0443\u0440\u0441 \u0443\u0447\u0438\u0442\u0435\u043b\u0435\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438. \u0412 \u0434\u0430\u043d\u043d\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u043e\u0431\u0441\u0443\u0436\u0434\u0430\u044e\u0442\u0441\u044f \u0437\u0430\u0434\u0430\u0447\u0438\u044f \u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u0430.<\/p>\n<p><i><br \/>Keywords<\/i><strong>: <\/strong>creativecompetition, problem solving, teacher, achievement, methodology<\/p>\n<p><strong>FIRST CREATIVE COMPETITION&nbsp;<\/strong><strong>FOR MATHEMATICS TEACHERS OF KAZAKHSTAN<\/p>\n<p><\/strong><strong style=\"text-align: right;\">Kairosh Makishev<\/p>\n<p><\/strong><strong style=\"text-align: right;\">Abstract. <\/strong><span style=\"text-align: right;\">The First Congress of Mathematics Teachers of Kazakhstan was held on 10\u201312 May 2011. One of the decisions was to create the Kazakhstan Association of Mathematics Teachers, which was registered officially on 6 July 2012. According to the Event Plan the First creative competition for Mathematics teachers was organized. The present paper considers the problems and the solutions of the competition. &nbsp;<\/span><\/p>\n<p><strong><br \/>Kairosh Makishev<br \/><\/strong>Doctor in Physics-Mathematics<br \/>Director RSPhMSBS<br \/>Republic of Kazakhstan<\/p>\n<hr \/>\n<p><strong><a name=\"art02\">&nbsp;<\/a>\u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u0427\u0415\u0421\u041a\u0418\u0415 \u0420\u0415\u0413\u0410\u0422\u042cI<\/p>\n<p><\/strong><strong>\u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440 \u0411\u043b\u0438\u043d\u043a\u043e\u0432<\/p>\n<p><\/strong><strong>Abstract. <\/strong>\u0421\u0444\u0435\u0440 \u0440\u0435\u0430\u043b\u044c\u043d\u043e\u0439 \u0447\u0435\u043b\u043e\u0432\u0435\u0447\u0435\u0441\u043a\u043e\u0439 \u0434\u0435\u044f\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438 \u0442\u0440\u0435\u0431\u0443\u0435\u0442 \u0443\u0441\u043f\u0435\u0448\u043d\u043e\u0439 \u0440\u0430\u0431\u043e\u0442\u044b \u201c\u0432 \u043a\u043e\u043c\u0430\u043d\u0434\u0435\u201d: \u0441\u043e\u0432\u043c\u0435\u0441\u0442\u043d\u043e\u0433\u043e \u043f\u043b\u0430\u043d\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0440\u0430\u0431\u043e\u0442\u044b; \u0440\u0430\u0437\u0434\u0435\u043b\u0435\u043d\u0438\u044f \u0442\u0440\u0443\u0434\u0430, \u043e\u0431\u0441\u0443\u0436\u0434\u0435\u043d\u0438\u044f \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u043d\u044b\u0445 \u043c\u043e\u043c\u0435\u043d\u0442\u043e\u0432, \u0438 \u0442\u043e\u043c\u0443 \u043f\u043e\u0434\u043e\u0431\u043d\u043e\u0435. \u0420\u044f\u0434 \u043a\u043e\u043c\u0430\u043d\u0434\u043d\u044b\u0445 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0441\u043e\u0440\u0435\u0432\u043d\u043e\u0432\u0430\u043d\u0438\u0439 \u0434\u043b\u044f \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u043e\u0432 \u0434\u0430\u0435\u0442 \u0432\u043e\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u044c \u0441\u043c\u043e\u0434\u0435\u043b\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u044d\u0442\u0443 \u0434\u0435\u044f\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u044c. \u0412 \u0434\u0430\u043d\u043d\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044f \u043f\u0440\u0438\u043c\u0435\u0440 \u0440\u0443\u0441\u0441\u043a\u043e\u0433\u043e \u043a\u043e\u043c\u0430\u043d\u0434\u043d\u043e\u0433\u043e \u0441\u043e\u0440\u0435\u0432\u043d\u043e\u0432\u0430\u043d\u0438\u044f \u2013 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0440\u0435\u0433\u0430\u0442\u044b. &nbsp;<\/p>\n<p><i><br \/>Keywords:<\/i> team competition, round, mathematical problem, problem solving, solution<\/p>\n<p><strong><br \/>MATHEMATICAL REGATTAS<br \/><\/strong><strong><br \/>Alexander Blinkov<\/strong><\/p>\n<p><strong>Abstract. <\/strong>The domain of real human activity requires a successive \u201cteam\u201d work: joint planning of actions; division of labor; discussion of problematic moments and so forth. Various team mathematical competitions give possibilities to model such activities. The present paper considers an example of a Russian team competition \u2013 the Mathematical regattas.&nbsp;<\/p>\n<p><strong>Alexander Blinkov<br \/><\/strong>Doctor in Physics-Mathematics<br \/>Assoc. Professor<br \/>Vice-Director<br \/>Moscow Center for Continuous<br \/>Mathematical Education<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art03\">&nbsp;<\/a>\u0412\u0415\u0420\u0418\u0413\u0418 \u041d\u0410 \u041c\u0410\u0420\u041a\u041e\u0412 <\/strong><strong>\u0418 \u041f\u0420\u0418\u041b\u041e\u0416\u0415\u041d\u0418\u0415\u0422\u041e \u0418\u041c \u0412 GOOGLE<\/strong><\/p>\n<p><strong>\u0415\u0432\u0433\u0435\u043d\u0438\u044f \u0421\u0442\u043e\u0438\u043c\u0435\u043d\u043e\u0432\u0430<\/p>\n<p><\/strong><strong>Abstract.<\/strong> \u0412 \u0442\u0430\u0437\u0438 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u043f\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0444\u0438\u0437\u0438\u0447\u0435\u0441\u043a\u0438 \u0441\u0438\u0441\u0442\u0435\u043c\u0438, \u043a\u043e\u0438\u0442\u043e \u0441\u0435 \u0440\u0430\u0437\u0432\u0438\u0432\u0430\u0442 \u0432\u044a\u0432 \u0432\u0440\u0435\u043c\u0435\u0442\u043e. \u041c\u043e\u0434\u0435\u043b\u0438\u0442\u0435 \u0441\u0430 \u0441\u0442\u043e\u0445\u0430\u0441\u0442\u0438\u0447\u043d\u0438 \u0438 \u0441\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f\u0442 \u0441 \u043c\u0430\u0440\u043a\u043e\u0432\u0441\u043a\u0438 \u0432\u0435\u0440\u0438\u0433\u0438 \u0441 \u043a\u0440\u0430\u0435\u043d \u0431\u0440\u043e\u0439 \u0441\u044a\u0441\u0442\u043e\u044f\u043d\u0438\u044f \u0438 \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u0438 \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 \u0437\u0430 \u043f\u0440\u0435\u0445\u043e\u0434. \u041c\u0430\u0442\u0435\u0440\u0438\u0430\u043b\u044a\u0442 \u0435 \u0434\u043e\u0441\u0442\u044a\u043f\u0435\u043d \u0437\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u043e\u0442 \u0441\u0440\u0435\u0434\u043d\u0438\u044f \u043a\u0443\u0440\u0441. \u0411\u0430\u0437\u043e\u0432\u0438\u0442\u0435 \u0437\u043d\u0430\u043d\u0438\u044f \u043f\u043e \u0432\u0435\u0440\u043e\u044f\u0442\u043d\u043e\u0441\u0442\u0438 \u043e\u0442 \u0441\u0440\u0435\u0434\u043d\u0438\u044f \u043a\u0443\u0440\u0441 \u0441\u044a\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043e \u043f\u043e\u043c\u0430\u0433\u0430\u0442 \u0437\u0430 \u0440\u0430\u0437\u0431\u0438\u0440\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0438\u0434\u0435\u0438\u0442\u0435. \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u0442 \u0441\u0435 \u043e\u0449\u0435 \u0430\u043b\u0433\u0435\u0431\u0440\u0438\u0447\u043d\u0438 \u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f \u0441 \u0432\u0435\u043a\u0442\u043e\u0440\u0438 \u0438 \u043c\u0430\u0442\u0440\u0438\u0446\u0438, \u043a\u043e\u0438\u0442\u043e \u043d\u0435 \u0437\u0430\u0442\u0440\u0443\u0434\u043d\u044f\u0432\u0430\u0442 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0441 \u043f\u043e\u0432\u0438\u0448\u0435\u043d\u0438 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u0438 \u043a\u044a\u043c \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u0442\u0430. \u0422\u0435\u043a\u0441\u0442\u044a\u0442 \u0441\u0435 \u043e\u0441\u043d\u043e\u0432\u0430\u0432\u0430 \u043d\u0430 \u043b\u0435\u043a\u0446\u0438\u0438, \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438 \u043e\u0442 \u0430\u0432\u0442\u043e\u0440\u0430 \u043d\u0430 \u041b\u044f\u0442\u043d\u0430\u0442\u0430 \u0438\u0437\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u0441\u043a\u0430 \u0448\u043a\u043e\u043b\u0430 \u043d\u0430 \u0423\u0447\u0418\u041c\u0418 2013.<\/p>\n<p><i><br \/>Keywords<\/i><strong>: <\/strong>Markov chain, web page ranking, <i>PageRank<\/p>\n<p><\/i><strong>MARKOV CHAINS AND THEIR APPLICATION IN GOOGLE<\/strong><\/p>\n<p><strong><br \/>Abstract.<\/strong> In this paper we study the behaviour of physical systems that evolves over time. The model is stochastic and is described by Markov chain with finite number of steps and constant transition probabilities. The text is suitable for high school students. Common school knowledge in probability helps the understanding of the ideas. Simple matrix algebra is used which is not difficult for students with enhanced interest in informatics and mathematics. The paper is based on the author\u2019s lectures presented at the Summer school of High School Student Institute, Varna, Bulgaria, in 2013.<\/p>\n<p><strong>Eugenia Stoimenova<br \/><\/strong>Professor, DSc in Mathematics<br \/>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art04\">&nbsp;<\/a>\u041d\u042f\u041a\u041e\u0418 \u041f\u0420\u0418\u041b\u041e\u0416\u0415\u041d\u0418\u042f&nbsp;<\/strong><strong>\u041d\u0410 \u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u041d\u0410\u0422\u0410 \u041f\u0420\u041e\u0413\u0420\u0410\u041c\u0410 \u201e\u041e\u0422\u041a\u0420\u0418\u0412\u0410\u0422\u0415\u041b\u201c<\/p>\n<p><\/strong><strong>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432,&nbsp; \u0414\u0435\u043a\u043e \u0414\u0435\u043a\u043e\u0432<\/p>\n<p><\/strong><strong>Abstract. <\/strong>\u041a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430\u0442\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c, \u043a\u043e\u044f\u0442\u043e \u0435 \u0432 \u043f\u0440\u043e\u0446\u0435\u0441 \u043d\u0430 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0432\u0430\u043d\u0435 \u043e\u0442 \u0430\u0432\u0442\u043e\u0440\u0438\u0442\u0435 \u043d\u0430 \u0442\u0430\u0437\u0438 \u0441\u0442\u0430\u0442\u0438\u044f, \u0435 \u043f\u0440\u0435\u0434\u043d\u0430\u0437\u043d\u0430\u0447\u0435\u043d\u0430 \u0434\u0430 \u043e\u0442\u043a\u0440\u0438\u0432\u0430 \u043d\u043e\u0432\u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u0438 \u0432 \u0415\u0432\u043a\u043b\u0438\u0434\u043e\u0432\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f. \u0412 \u0442\u0430\u0437\u0438 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0430 \u0434\u0430\u0434\u0435\u043d\u0438 \u043d\u044f\u043a\u043e\u0438 \u0442\u0435\u043e\u0440\u0435\u043c\u0438, \u0441\u044a\u0437\u0434\u0430\u0434\u0435\u043d\u0438 \u043e\u0442 \u201e\u041e\u0442\u043a\u0440\u0438\u0432\u0430\u0442\u0435\u043b\u201c, \u0438 \u0435 \u043f\u043e\u0441\u043e\u0447\u0435\u043d\u043e \u043a\u0430\u043a \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0431\u044a\u0434\u0430\u0442 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0438.<\/p>\n<p><i>Keywords<\/i>: computer-generated mathematics, Euclidean geometry, Discoverer, mathematical olympiad<\/p>\n<p><strong>SOME APPLICATIONS&nbsp;<\/strong><strong>OF THE COMPUTER PROGRAM \u201eDISCOVERER\u201c<\/strong><\/p>\n<p><strong>Abstract.<\/strong> The aim of the computer program \u201eDiscoverer\u201c, which is in process of development by the authors of this article, is to discover new theorems in Euclidean Geometry. The authors present some theorems, generated by \u201eDiscoverer\u201c and show how these theorems could be used.<\/p>\n<p><strong>Sava Grozdev<br \/><\/strong>Professor, Doctor in Mathematics, DSc in Pedagogy<br \/>Institute of Mathematics and Informatics \u2013 BAS<\/p>\n<p><strong>Deko Dekov<br \/><\/strong>Associated Professor, PhD in Mathematics<br \/>Stara Zagora<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art05\">&nbsp;<\/a>\u0415\u0414\u041d\u041e \u0418\u041d\u0422\u0415\u0420\u0415\u0421\u041d\u041e \u0421\u0412\u041e\u0419\u0421\u0422\u0412\u041e&nbsp;<\/strong><strong>\u0417\u0410 \u041d\u042f\u041a\u041e\u0418 \u041a\u041b\u0410\u0421\u041e\u0412\u0415 \u0420\u0415\u0410\u041b\u041d\u0418 \u0424\u0423\u041d\u041a\u0426\u0418\u0418<br \/><\/strong><\/p>\n<p><strong>\u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432<br \/><\/strong><strong><br \/>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u043e\u0442\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u043d\u0435 \u043d\u0430 \u043d\u044f\u043a\u043e\u0438 \u043a\u043b\u0430\u0441\u043e\u0432\u0435 \u0440\u0435\u0430\u043b\u043d\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438, \u0437\u0430 \u043a\u043e\u0438\u0442\u043e \u0440\u0430\u0437\u0441\u0442\u043e\u044f\u043d\u0438\u0435\u0442\u043e \u043c\u0435\u0436\u0434\u0443 \u0430\u0431\u0441\u0446\u0438\u0441\u0438\u0442\u0435 \u043d\u0430 \u043a\u043e\u0438 \u0434\u0430 \u0435 \u0434\u0432\u0435 \u0442\u043e\u0447\u043a\u0438 \u043e\u0442 \u0433\u0440\u0430\u0444\u0438\u043a\u0438\u0442\u0435 \u0438\u043c \u0435 \u0440\u0430\u0432\u043d\u043e \u043d\u0430 \u0440\u0430\u0437\u0441\u0442\u043e\u044f\u043d\u0438\u0435\u0442\u043e \u043c\u0435\u0436\u0434\u0443 \u043f\u0440\u0435\u0441\u0435\u0447\u043d\u0438\u0442\u0435 \u0442\u043e\u0447\u043a\u0438 \u043d\u0430 \u0434\u043e\u043f\u0438\u0440\u0430\u0442\u0435\u043b\u043d\u0438\u0442\u0435 \u043a\u044a\u043c \u0433\u0440\u0430\u0444\u0438\u043a\u0438\u0442\u0435 \u043f\u0440\u0435\u0437 \u0434\u0432\u0435\u0442\u0435 \u0442\u043e\u0447\u043a\u0438 \u0438 \u0430\u0431\u0441\u0446\u0438\u0441\u043d\u0430\u0442\u0430 \u043e\u0441.<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">&nbsp;<\/a><\/p>\n<p><i>Keywords<\/i>: function, exponent, graphics, tangent, distance &nbsp;&nbsp;<\/p>\n<p><strong>AN INTERESTING PROPERTY <\/strong><strong>FOR SOME CLASSES OF REAL FUNCTIONS<\/strong><\/p>\n<p><strong><br \/>Abstract.<\/strong> The paper is dedicated to the methodological determination of some classes of real functions, for which the distance between the abscises of any two points from their graphics is equal to the distance between the intersection points of the tangents to the graphics and the abscise axis.&nbsp;<\/p>\n<p><strong>Rosen Nikolaev<br \/><\/strong>PhD, Associated Professor<br \/>Head of Department<br \/>Economical University \u2013 Varna<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art06\">&nbsp;<\/a>\u0421\u0418\u041d\u0415\u0420\u0413\u0415\u0422\u0418\u0427\u0415\u041d \u041f\u041e\u0414\u0425\u041e\u0414 \u041a\u042a\u041c \u041f\u0420\u041e\u0426\u0415\u0421\u0410 \u041d\u0410 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415&nbsp;<\/strong><strong>\u041f\u041e \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0426\u0418\u041e\u041d\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418<\/p>\n<p><\/strong><strong>\u0415\u0432\u0433\u0435\u043d\u0438\u044f \u0413\u043e\u0440\u0430\u043d\u043e\u0432\u0430<\/p>\n<p><\/strong><strong>Abstract. <\/strong>\u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u043e\u0442\u043e \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435 \u0441\u0435 \u043f\u0440\u0438\u043b\u0430\u0433\u0430 \u0441\u0438\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u0435\u043d \u043f\u043e\u0434\u0445\u043e\u0434, \u0437\u0430\u0449\u043e\u0442\u043e \u043f\u0440\u043e\u0446\u0435\u0441\u044a\u0442 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0435 \u043d\u0435\u043b\u0438\u043d\u0435\u0439\u043d\u0430, \u043e\u0442\u0432\u043e\u0440\u0435\u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430, \u043f\u043e\u0434\u043b\u043e\u0436\u0435\u043d\u0430 \u043d\u0430 \u0432\u044a\u0442\u0440\u0435\u0448\u043d\u0438 \u0438 \u0432\u044a\u043d\u0448\u043d\u0438 \u0432\u044a\u0437\u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f. \u0421\u0438\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u043d\u0438\u044f\u0442 \u043f\u043e\u0434\u0445\u043e\u0434 \u043e\u0431\u044f\u0441\u043d\u044f\u0432\u0430 \u0432\u0437\u0430\u0438\u043c\u043d\u0438\u0442\u0435 \u0432\u0440\u044a\u0437\u043a\u0438 \u043c\u0435\u0436\u0434\u0443 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0438\u0442\u0435 \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430; \u043e\u0442\u043a\u0440\u043e\u044f\u0432\u0430 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u0438\u0442\u0435 \u043f\u043e\u0441\u043e\u043a\u0438 \u0437\u0430 \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435\u0442\u043e \u0438\u043c;&nbsp; \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u044f \u043d\u0430\u0447\u0438\u043d \u0437\u0430 \u0443\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430 \u043f\u0440\u043e\u0446\u0435\u0441\u0438\u0442\u0435 \u0432 \u043f\u0435\u0440\u0438\u043e\u0434 \u043d\u0430 \u043a\u0440\u0438\u0437\u0430 \u043a\u044a\u043c \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435 \u0432 \u0431\u043b\u0430\u0433\u043e\u043f\u0440\u0438\u044f\u0442\u043d\u0430 \u043f\u043e\u0441\u043e\u043a\u0430; \u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u0430 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u043d\u0438 \u0430\u0442\u0440\u0430\u043a\u0442\u043e\u0440\u0438 \u043a\u0430\u0442\u043e \u0446\u0435\u043b \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e. \u041d\u0430\u0439-\u0447\u0443\u0432\u0441\u0442\u0432\u0438\u0442\u0435\u043b\u043d\u0438\u044f\u0442 \u0443\u043f\u0440\u0430\u0432\u043b\u044f\u0432\u0430\u0449 \u043f\u0430\u0440\u0430\u043c\u0435\u0442\u044a\u0440 (\u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u0435\u043d \u0430\u0442\u0440\u0430\u043a\u0442\u043e\u0440) \u0435 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430\u0442\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438, \u043a\u043e\u044f\u0442\u043e \u0441\u0435 \u0434\u0435\u0444\u0438\u043d\u0438\u0440\u0430 \u043d\u0430 \u043c\u0430\u043a\u0440\u043e\u0440\u0430\u0432\u043d\u0438\u0449\u0435. \u0422\u044f \u043c\u043e\u0436\u0435 \u0434\u0430 \u043e\u0442\u0440\u0430\u0437\u0438 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440\u0438\u0441\u0442\u0438\u043a\u0438\u0442\u0435 \u043d\u0430 \u0434\u0438\u0433\u0438\u0442\u0430\u043b\u043d\u043e\u0442\u043e \u043f\u043e\u043a\u043e\u043b\u0435\u043d\u0438\u0435. \u0421\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u043e\u0442 \u0437\u043d\u0430\u043d\u0438\u044f, \u0443\u043c\u0435\u043d\u0438\u044f \u0438 \u043a\u043e\u043c\u043f\u0435\u0442\u0435\u043d\u0442\u043d\u043e\u0441\u0442\u0438, \u043a\u0430\u0442\u043e \u0430\u0442\u0440\u0430\u043a\u0442\u043e\u0440\u0435\u043d \u0431\u0430\u0441\u0435\u0439\u043d, \u0441\u0435 \u0434\u0435\u0444\u0438\u043d\u0438\u0440\u0430 \u043d\u0430 \u043c\u0438\u043a\u0440\u043e\u0440\u0430\u0432\u043d\u0438\u0449\u0435. \u041f\u043e\u0441\u043b\u0435\u0434\u043d\u0430\u0442\u0430 \u043c\u043e\u0436\u0435 \u0434\u0430 \u0441\u0435 \u0440\u0435\u0430\u043b\u0438\u0437\u0438\u0440\u0430 \u0447\u0440\u0435\u0437 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0432 \u043c\u0443\u043b\u0442\u0438\u043c\u0435\u0434\u0438\u0439\u043d\u0430 \u0441\u0440\u0435\u0434\u0430, \u0438\u0437\u0433\u0440\u0430\u0434\u0435\u043d\u0430 \u043d\u0430 \u0431\u0430\u0437\u0430\u0442\u0430 \u043d\u0430 \u0443\u0447\u0435\u0431\u043d\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430.<\/p>\n<p><span class=\"s1\"><i><br \/>Keywords<\/i>: <\/span>synergetic approach, process of information technology training, attractor at macro level, methodology for training of information technology<\/p>\n<p><strong>SYNERGETIC APPROACH TO THE TRAINING PROCESS&nbsp;<\/strong><strong>IN INFORMATION TECHNOLOGY<\/p>\n<p><\/strong><strong>Abstract.<\/strong> Synergetic approach is applied in this study, because the training process of Information Technology is non-linear, open system, subjected to internal and external influences. Synergetics explains the existing interrelationships among the elements of the system; highlights possible directions for their development; provides a way to manage the process in a crisis for the development in a favorable direction; determines structural attractors as aim of training. The most sensitive control parameter (structural attractor) is<i> the<\/i> <i>methodology for the Information Technology training<\/i>, which is defined at macro level. It can reflect the characteristics of the digital generation. The system of knowledge, skills and competences as an attractor pool is determined at micro level. It can be realized through learning in a multimedia environment based on the curriculum.<\/p>\n<p><strong>Goranova<br \/><\/strong>Assistant Professor<br \/>University of Rousse&nbsp;<br \/>Silistra Branch<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art07\">&nbsp;<\/a>BOOSTING EFFICIENCY OF PROJECT-ORIENTED TEACHING&nbsp;<\/strong><strong>AND LEARNING THROUGH CLASSROOM MANAGEMENT&nbsp;<\/strong><strong>AND ONLINE TESTING<\/p>\n<p><\/strong><strong>Ivan Shotlekov, Vanya Ivanova, Kirina Boykova<\/p>\n<p><\/strong><strong>Abstract. <\/strong>A set of classroom management and testing policies was elaborated and, after two years of piloting, it was further developed and administered to 247 students at the Faculty of Mathematics and Informatics, Plovdiv University \u201cPaisiy Hilendarski\u201d. The main purpose of this effort was to enhance the efficiency of project-oriented teaching and learning. The findings confirmed the positive outcome of this approach and the potential to be successfully applied to traditional classroom settings. &nbsp;<\/p>\n<p><i>Keywords<\/i>: classroom management, project-oriented learning, The Common European framework of reference for languages: learning, teaching, assessment, testing<\/p>\n<p><strong>\u041f\u041e\u0412\u0418\u0428\u0410\u0412\u0410\u041d\u0415 \u0415\u0424\u0415\u041a\u0422\u0418\u0412\u041d\u041e\u0421\u0422\u0422\u0410 \u041d\u0410 \u041f\u0420\u041e\u0415\u041a\u0422\u041d\u041e-\u041e\u0420\u0418\u0415\u041d\u0422\u0418\u0420\u0410\u041d\u041e\u0422\u041e \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415 \u0427\u0420\u0415\u0417 \u041c\u0415\u041d\u0418\u0414\u0416\u041c\u042a\u041d\u0422 \u041d\u0410 \u041a\u041b\u0410\u0421\u0410 \u0418 \u041e\u041d\u041b\u0410\u0419\u041d \u0422\u0415\u0421\u0422\u041e\u0412\u041e \u0418\u0417\u041f\u0418\u0422\u0412\u0410\u041d\u0415<\/p>\n<p><\/strong><strong>\u0418\u0432\u0430\u043d \u0428\u043e\u0442\u043b\u0435\u043a\u043e\u0432, \u0412\u0430\u043d\u044f \u0418\u0432\u0430\u043d\u043e\u0432\u0430, \u041a\u0438\u0440\u0438\u043d\u0430 \u0411\u043e\u0439\u043a\u043e\u0432\u0430<\/strong><\/p>\n<p class=\"p6\"><strong><br \/>Abstract. <\/strong>\u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0435\u043d \u0435 \u043d\u0430\u0431\u043e\u0440 \u043e\u0442 \u043f\u043e\u043b\u0438\u0442\u0438\u043a\u0438 \u0437\u0430 \u0443\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435 \u043d\u0430 \u043a\u043b\u0430\u0441\u0430 \u0438 \u0442\u0435\u0441\u0442\u043e\u0432\u043e \u0438\u0437\u043f\u0438\u0442\u0432\u0430\u043d\u0435, \u043a\u043e\u0438\u0442\u043e \u0441\u043b\u0435\u0434 \u0434\u0432\u0443\u0433\u043e\u0434\u0438\u0448\u043d\u0430 \u0430\u043f\u0440\u043e\u0431\u0430\u0446\u0438\u044f \u0431\u044f\u0445\u0430 \u0434\u043e\u0440\u0430\u0437\u0432\u0438\u0442\u0438 \u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438 \u043f\u0440\u0438 247 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438 \u0432\u044a\u0432 \u0424\u0430\u043a\u0443\u043b\u0442\u0435\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 \u043d\u0430 \u041f\u043b\u043e\u0432\u0434\u0438\u0432\u0441\u043a\u0438\u044f \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 &#8222;\u041f\u0430\u0438\u0441\u0438\u0439 \u0425\u0438\u043b\u0435\u043d\u0434\u0430\u0440\u0441\u043a\u0438&#8220;. \u041e\u0441\u043d\u043e\u0432\u043d\u0430\u0442\u0430 \u0446\u0435\u043b \u043d\u0430 \u0442\u0435\u0437\u0438 \u0443\u0441\u0438\u043b\u0438\u044f \u0435 \u0434\u0430 \u0441\u0435 \u043f\u043e\u0432\u0438\u0448\u0438 \u0435\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0441\u0442\u0442\u0430 \u043d\u0430 \u043f\u0440\u043e\u0435\u043a\u0442\u043d\u043e-\u043e\u0440\u0438\u0435\u043d\u0442\u0438\u0440\u0430\u043d\u043e\u0442\u043e \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435. \u0420\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438\u0442\u0435 \u043f\u043e\u0442\u0432\u044a\u0440\u0434\u0438\u0445\u0430 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u043d\u0438\u044f \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442 \u043e\u0442 \u0442\u043e\u0437\u0438 \u043f\u043e\u0434\u0445\u043e\u0434 \u0438 \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u0430 \u0434\u0430 \u0441\u0435 \u043f\u0440\u0438\u043b\u0430\u0433\u0430 \u0443\u0441\u043f\u0435\u0448\u043d\u043e \u0438 \u0432 \u043a\u043e\u043d\u0442\u0435\u043a\u0441\u0442\u0430 \u043d\u0430 \u0442\u0440\u0430\u0434\u0438\u0446\u0438\u043e\u043d\u043d\u0438\u0442\u0435 \u0444\u043e\u0440\u043c\u0438 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435.<\/p>\n<p><strong><br \/>Ivan Shotlekov <br \/><\/strong>PhD in Methodology<br \/>FMI, Plovdiv University \u201cPaissiy Hilendarski\u201d<\/p>\n<p><strong>Vanya Ivanova&nbsp;<br \/><\/strong>FMI, Plovdiv University \u201cPaissiy Hilendarski\u201d<\/p>\n<p><strong>Kirina Boykova<br \/><\/strong>FMI, Plovdiv University \u201cPaissiy Hilendarski\u201d<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>","protected":false},"excerpt":{"rendered":"<p>&nbsp;\u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \/ CONTENTS \u041f\u0435\u0440\u0432\u044b\u0439 \u0442\u0432\u043e\u0440\u0447\u0435\u0441\u043a\u0438\u0439 \u043a\u043e\u043d\u043a\u0443\u0440\u0441 \u0443\u0447\u0438\u0442\u0435\u043b\u0435\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0438 \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0430Creative Competition for Mathematics Teachers of Kazakhstan\u041a\u0430\u0439\u0440\u043e\u0448 \u041c\u0430\u043a\u0438\u0448\u0435\u0432, Kairosh Makishev \u2013 \u0441\u0442\u0440. 409 &nbsp; \u0412\u0435\u0440\u0438\u0433\u0438 \u043d\u0430 \u041c\u0430\u0440\u043a\u043e\u0432 \u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435\u0442\u043e \u0438\u043c \u0432 GoogleMarkov Chains and their Application in Google\u0415\u0432\u0433\u0435\u043d\u0438\u044f \u0421\u0442\u043e\u0438\u043c\u0435\u043d\u043e\u0432\u0430, Eugenia Stoimenova \u2013 \u0441\u0442\u0440. 426 &nbsp; \u0421\u0438\u043d\u0435\u0440\u0433\u0435\u0442\u0438\u0447\u0435\u043d \u043f\u043e\u0434\u0445\u043e\u0434 \u043a\u044a\u043c \u043f\u0440\u043e\u0446\u0435\u0441\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438Synergetic Approach to the 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