{"id":41273,"date":"2016-11-02T11:37:20","date_gmt":"2016-11-02T08:37:20","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles2016\/6-lix-2016\/"},"modified":"2019-06-18T21:58:29","modified_gmt":"2019-06-18T18:58:29","slug":"6-lix-2016","status":"publish","type":"post","link":"https:\/\/strategies.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/","title":{"rendered":"Mathematics and Informatics, Number 6\/2016, Volume 59"},"content":{"rendered":"<p><strong><a id=\"top\"><\/a>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>EDUCATIONAL MATTERS \/ \u041d\u0410\u0423\u0427\u041d\u041e-\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0418 \u0421\u0422\u0410\u0422\u0418\u0418<\/strong><br \/><a href=\"#art01\">\u041f\u0435\u0440\u0432\u044b\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u044b\u0439 \u0441\u0435\u0442\u0435\u0432\u043e\u0439 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u0441\u043a\u0438\u0439 \u043f\u0440\u043e\u0435\u043a\u0442 \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 MITE <\/a><br \/><a href=\"#art01\">[First International Net Research Project of Secondary Students in the Frames of MITE] <\/a><br \/>\/ \u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u041c\u0430\u0440\u0438\u043d\u0430 \u0411\u0435\u043b\u043e\u0440\u0443\u043a\u043e\u0432\u0430, \u0420\u043e\u0437\u0430 \u0410\u0442\u0430\u043c\u0443\u0440\u0430\u0442\u043e\u0432\u0430, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432&nbsp;\/ Maria Shabanova, Marina Belorukova, Roza Atamuratova, Veselin Nenkov &ndash; \u0441\u0442\u0440. 567<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_MariaShabanova_i_avt.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art02\">\u041d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u044b \u0440\u0430\u0432\u043d\u043e\u0431\u0435\u0434\u0440\u0435\u043d\u043d\u044b\u043c\u0438 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430\u043c\u0438 <\/a><br \/><a href=\"#art02\">[Some Trajectories Determined by Isosceles Triangles] <\/a><br \/>\/ \u041a\u0441\u0435\u043d\u0438\u044f \u0413\u043e\u0440\u0441\u043a\u0430\u044f, \u0414\u0430\u0440\u044c\u044f \u041a\u043e\u043f\u0442\u0435\u0432\u0430, \u0414\u0430\u043d\u0438\u0438\u043b \u041c\u0438\u043a\u0443\u0440\u043e\u0432, \u0415\u0440\u043a\u0435\u043d \u041c\u0443\u0434\u0435\u0431\u0430\u0435\u0432, \u041a\u0430\u0437\u0431\u0435\u043a \u041c\u0443\u0445\u0430\u043c\u0431\u0435\u0442\u043e\u0432, \u0410\u0434\u0438\u043b\u0431\u0435\u043a \u0422\u0435\u043c\u0438\u0440\u0445\u0430\u043d\u043e\u0432, \u041b\u0438\u043b\u0438 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u0418\u0440\u0438\u043d\u0430 \u0425\u0440\u0438\u0441\u0442\u043e\u0432\u0430, \u0420\u0430\u0434\u0438\u043d\u0430 \u0418\u0432\u0430\u043d\u043e\u0432\u0430 \/ Kseniya Gorskaya, Daria Kopteva, Daniil Mikurov, Yerkin Mudebaev, Kazbek Mukhambetov, Adilbek Temirkhanov, Lilly Stefanova, Irina Hristova, Radina Ivanova &ndash; \u0441\u0442\u0440. 572<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_GorskayaKseniya_i_avt.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art03\">\u0414\u043e\u043c\u0430\u0448\u043d\u0430\u0442\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 &ndash; \u043d\u0435\u0434\u0435\u043b\u0438\u043c\u0430 \u0447\u0430\u0441\u0442 \u043e\u0442 \u0443\u0447\u0435\u0431\u043d\u0438\u044f \u043f\u0440\u043e\u0446\u0435\u0441 <\/a><br \/><a href=\"#art03\">[The Homework &ndash; an Integral Part of the Educational Process] <\/a><br \/>\/ \u0415\u043b\u0435\u043d\u0430 \u0420\u0430\u0434\u043e\u0432\u0430\u043d\u043e\u0432\u0430 \/ Elena Radovanova &ndash; \u0441\u0442\u0440. 589<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_ElenaRadovanova.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art04\">\u041d\u044f\u043a\u043e\u0438 \u0433\u0440\u0443\u043f\u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0442\u043c\u0438\u0447\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u0415xcel <\/a><br \/><a href=\"#art04\">[Groups of Logarithmic Equations by the Help of Excel] <\/a><br \/>\/ \u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432 \/ Diana Stefanova, Plamen Penev &ndash; \u0441\u0442\u0440. 599<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_Stefanova_Penev.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>EDUCATIONAL TECHNOLOGIES \/ \u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u0422\u0415\u041b\u041d\u0418 \u0422\u0415\u0425\u041d\u041e\u041b\u041e\u0413\u0418\u0418<\/strong><br \/><a href=\"#art05\">\u041f\u0441\u0435\u0432\u0434\u043e\u0446\u0435\u043d\u0442\u044a\u0440 \u0438 \u043e\u0440\u0442\u043e\u0446\u0435\u043d\u0442\u044a\u0440 &ndash; \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \u0432 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430 <\/a><br \/><a href=\"#art05\">[Pseudocenter and Orthocenter &ndash; Notable Points in the Quadrilateral] <\/a><br \/>\/ \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432, \u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432, \u0425\u0430\u0438\u043c \u0425\u0430\u0438\u043c\u043e\u0432 \/ Veselin Nenkov, Stanislav Stefanov, Haim Haimov &ndash; \u0441\u0442\u0440. 614<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_Nenkov_Stefanov_Haimov.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art06\">\u041d\u044f\u043a\u043e\u0438 \u043d\u043e\u0432\u0438 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u043c\u0435\u0436\u0434\u0443 \u0441\u0440\u0435\u0434\u043d\u0438\u0442\u0435 \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u043e, \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u043e, \u0445\u0430\u0440\u043c\u043e\u043d\u0438\u0447\u043d\u043e \u0438 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u043e <\/a><br \/><a href=\"#art06\">[Some New Inequalities \u0430mong the Arithmetic, Geometric, Harmonic and Quadratic Means] <\/a><br \/>\/ \u0422\u043e\u0434\u043e\u0440 \u041c\u0438\u0442\u0435\u0432 \/ Todor Mitev &ndash; \u0441\u0442\u0440. 626<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_TodorMitev.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art07\">Fuzzy Logic <\/a><br \/>\/ Reinhard Magenreuter &ndash; \u0441\u0442\u0440. 657<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_ReinhardMagenreuter.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"#art08\">Genetic Algorithm <\/a> <br \/>\/ Reinhard Magenreuter &ndash; \u0441\u0442\u0440. 665<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_ReinhardMagenreuter_1.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>CONTEST PROBLEMS \/ \u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong><br \/>Contest Problems of this Issue <br \/>[Contest Problems of this Issue] &ndash; \u0441\u0442\u0440. 673<\/p>\n<p>&nbsp;<\/p>\n<p>\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u0431\u0440\u043e\u0439 1, 2016 <br \/>[Solutions of the Contest Problems from Issue 1, 2016] &ndash; \u0441\u0442\u0440. 675<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/azbuki.bg_dmdocuments_Matematika06_16_ContestProblems.pdf\" class=\"doclink\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" alt=\"icon\" border=\"0\" width=\"16\" height=\"16\" \/>&nbsp;Open the full text<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>ANNUAL CONTENTS \/ \u0413\u041e\u0414\u0418\u0428\u041d\u041e \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415<\/strong> &ndash; \u0441\u0442\u0440. 678<\/p>\n<p>&nbsp;<\/p>\n<p><strong>READ IN THE LATEST ISSUES OF &bdquo;AZBUKI&ldquo; JOURNALS <\/strong><br \/><strong>\/ \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 &bdquo;\u0410\u0417\u0411\u0423\u041a\u0418&ldquo; \u0427\u0415\u0422\u0415\u0422\u0415<\/strong> &ndash; \u0441\u0442\u0440. 683<\/p>\n<p>&nbsp;<\/p>\n<p><strong>GUIDE FOR AUTHORS \/ \u0423\u041a\u0410\u0417\u0410\u041d\u0418\u042f \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415<\/strong> &ndash; \u0441\u0442\u0440. 687<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a id=\"art01\"><\/a>\u041f\u0415\u0420\u0412\u042b\u0419 \u041c\u0415\u0416\u0414\u0423\u041d\u0410\u0420\u041e\u0414\u041d\u042b\u0419 \u0421\u0415\u0422\u0415\u0412\u041e\u0419 \u0418\u0421\u0421\u041b\u0415\u0414\u041e\u0412\u0410\u0422\u0415\u041b\u042c\u0421\u041a\u0418\u0419 \u041f\u0420\u041e\u0415\u041a\u0422 \u0423\u0427\u0410\u0429\u0418\u0425\u0421\u042f \u0412 \u0420\u0410\u041c\u041a\u0410\u0425 MITE<\/strong> <br \/>\u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u0421\u0435\u0432\u0435\u0440\u043d\u044b\u0439 (\u0410\u0440\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439) \u0444\u0435\u0434\u0435\u0440\u0430\u0442\u0438\u0432\u043d\u044b\u0439 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u0438\u043c\u0435\u043d\u0438 \u041c.\u0412. \u041b\u043e\u043c\u043e\u043d\u043e\u0441\u043e\u0432\u0430 &ndash; \u0410\u0440\u0445\u0430\u043d\u0433\u0435\u043b\u044c\u0441\u043a, \u0420\u043e\u0441\u0441\u0438\u044f<br \/>\u041c\u0430\u0440\u0438\u043d\u0430 \u0411\u0435\u043b\u043e\u0440\u0443\u043a\u043e\u0432\u0430, \u041c\u0443\u043d\u0438\u0446\u0438\u043f\u0430\u043b\u044c\u043d\u043e\u0435 \u0431\u044e\u0434\u0436\u0435\u0442\u043d\u043e\u0435 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0443\u0447\u0440\u0435\u0436\u0434\u0435\u043d\u0438\u0435 &bdquo;\u0421\u0440\u0435\u0434\u043d\u044f\u044f \u0448\u043a\u043e\u043b\u0430 \u2116 8&rdquo; &ndash; \u0410\u0440\u0445\u0430\u043d\u0433\u0435\u043b\u044c\u0441\u043a, \u0420\u043e\u0441\u0441\u0438\u044f<br \/>\u0420\u043e\u0437\u0430 \u0410\u0442\u0430\u043c\u0443\u0440\u0430\u0442\u043e\u0432\u0430, \u041e\u0431\u043b\u0430\u0441\u0442\u043d\u0430\u044f \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u0430\u044f \u0448\u043a\u043e\u043b\u0430-\u0438\u043d\u0442\u0435\u0440\u043d\u0430\u0442 \u0434\u043b\u044f \u043e\u0434\u0430\u0440\u0435\u043d\u043d\u044b\u0445 \u0434\u0435\u0442\u0435\u0439 \u0441 \u0443\u0433\u043b\u0443\u0431\u043b\u0435\u043d\u043d\u044b\u043c \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u0435\u043c \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u043c\u0435\u0442\u043e\u0432 &ndash; \u0410\u043a\u0442\u0430\u0443, \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d<br \/>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432, \u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u043a\u043e\u043b\u043b\u0435\u0434\u0436 &ndash; \u041b\u043e\u0432\u0435\u0447, \u0411\u043e\u043b\u0433\u0430\u0440\u0438\u044f<\/p>\n<p>&nbsp;<\/p>\n<p>\u0410\u043d\u043d\u043e\u0442\u0430\u0446\u0438\u044f. \u041e\u0434\u043d\u043e\u0439 \u0438\u0437 \u0437\u0430\u0434\u0430\u0447 \u041c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0435\u043a\u0442\u0430 &bdquo;\u041c\u0435\u0442\u043e\u0434\u0438\u043a\u0438 \u0438 \u0438\u043d&not;\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0435 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0432 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0438 (MITE)&rdquo; \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0441\u043e\u0437\u0434\u0430\u043d\u0438\u0435 \u0443\u0441\u043b\u043e\u0432\u0438\u0439 \u0434\u043b\u044f \u0432\u044b\u044f\u0432\u043b\u0435\u043d\u0438\u044f \u0438 \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u044f \u043c\u043e\u043b\u043e\u0434\u044b\u0445 \u0442\u0430\u043b\u0430\u043d\u0442\u043e\u0432. \u041e\u0434\u043d\u043e\u0439 \u0438\u0437 \u0437\u0430\u0440\u0435\u043a\u043e\u043c\u0435\u043d\u0434\u043e\u0432\u0430\u0432\u0448\u0438\u0445 \u0441\u0435\u0431\u044f \u0444\u043e\u0440\u043c \u0440\u0430\u0431\u043e\u0442\u044b \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u041c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u044b\u0439 \u043a\u043e\u043d\u043a\u0443\u0440\u0441 &ldquo;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u043f\u0440\u043e\u0435\u043a\u0442\u0438&not;\u0440\u043e\u0432\u0430\u043d\u0438\u0435&rdquo;. \u041e\u043d \u043f\u043e\u043b\u044c\u0437\u0443\u0435\u0442\u0441\u044f \u0431\u043e\u043b\u044c\u0448\u043e\u0439 \u043f\u043e\u043f\u0443\u043b\u044f\u0440\u043d\u043e\u0441\u0442\u044c\u044e \u0441\u0440\u0435\u0434\u0438 \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0411\u043e\u043b\u0433\u0430\u0440\u0438\u0438, \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0430 \u0438 \u0420\u043e\u0441\u0441\u0438\u0438. \u0412 2016 \u0433\u043e\u0434\u0443 \u044d\u0442\u043e\u0442 \u043a\u043e\u043d\u043a\u0443\u0440\u0441 \u043f\u0440\u0430\u0437\u0434\u043d\u043e\u0432\u0430\u043b \u0441\u0432\u043e\u0439 \u0434\u0435\u0441\u044f\u0442\u0438\u043b\u0435\u0442\u043d\u0438\u0439 \u044e\u0431\u0438\u043b\u0435\u0439. \u0412 \u044d\u0442\u043e\u043c \u0433\u043e\u0434\u0443 \u0432\u043f\u0435\u0440\u0432\u044b\u0435 \u0432 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u0443 \u043a\u043e\u043d\u043a\u0443\u0440\u0441\u0430 \u0431\u044b\u043b\u0430 \u0432\u043a\u043b\u044e\u0447\u0435\u043d\u0430 \u043d\u043e\u043c\u0438\u043d\u0430&not;\u0446\u0438\u044f &ldquo;\u0421\u0435\u0442\u0435\u0432\u044b\u0435 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u0441\u043a\u0438\u0435 \u043f\u0440\u043e\u0435\u043a\u0442\u044b&rdquo;. \u041f\u0435\u0440\u0432\u043e\u0435 \u043c\u0435\u0441\u0442\u043e \u0432 \u044d\u0442\u043e\u0439 \u043d\u043e\u043c\u0438\u043d\u0430\u0446\u0438\u0438 \u0437\u0430\u043d\u044f\u043b \u043f\u0440\u043e\u0435\u043a\u0442 &ldquo;\u0413\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0439 Scrabble \u0432 \u043e\u0431\u043b\u0430\u043a\u0430\u0445&rdquo; \u043f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043b\u0435\u043d\u043d\u044b\u0439 \u043e\u0431\u044a\u0435\u0434\u0438&not;\u043d\u0435\u043d\u043d\u043e\u0439 \u043a\u043e\u043c\u0430\u043d\u0434\u043e\u0439 \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0438\u0437 \u0442\u0440\u0435\u0445 \u0441\u0442\u0440\u0430\u043d: \u0411\u043e\u043b\u0433\u0430\u0440\u0438\u0439, \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0430 \u0438 \u0420\u043e\u0441\u0441\u0438\u0438. \u0414\u0430\u043d\u043d\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f \u043f\u043e\u0434\u0432\u043e\u0434\u0438\u0442 \u0438\u0442\u043e\u0433\u0438 \u0440\u0430\u0431\u043e\u0442\u044b \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0438 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u0435\u0442 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0443 \u043e\u0440&not;\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u0438 \u0438\u0445 \u0441\u043e\u0432\u043c\u0435\u0441\u0442\u043d\u043e\u0439 \u0440\u0430\u0431\u043e\u0442\u044b \u043f\u043e \u043f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0435 \u043f\u0440\u043e\u0435\u043a\u0442\u0430.<br \/>Keywords: geometry, secondary education, students&rsquo; research, international network collaboration, dynamical geometry software, cloud service.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>FIRST INTERNATIONAL NET RESEARCH PROJECT OF SECONDARY STUDENTS IN THE FRAMES OF MITE<\/strong><\/p>\n<p>Abstract .One of the goals of the International project &ldquo;Methodology and Information Technologies in Education (MITE)&rdquo; is to create conditions for identification and support of young talents. A proven effective form of activity is the International Competition &ldquo;Mathematics and projecting&rdquo;, which is quite popular in Bulgaria, Kazakhstan and Russia. In 2016 the competition celebrated its 10-th anniversary. This year for the first time the competition program included the nomination &ldquo;Network Research Projects&rdquo;. The project &ldquo;Geometry Scrabble in Cloud&rdquo; was awarded first prize in this nomination. It was prepared by an international group of students from the three countries: Bulgaria, Kazakhstan and Russia. The present paper summarizes the students&rsquo; work and presents the organizational methodology of the communication during the preparation of the project.<\/p>\n<p>&nbsp;<\/p>\n<p>Prof. Maria Shabanova, DSc.<br \/>Northern Arctic Federal University &ldquo;M.V. Lomonosov&rdquo;<br \/>163 002 Arkhangelsk, Russia<\/p>\n<p>Ms. Marina Belorukova, teacher<br \/>Public Secondary school \u21168<br \/>163 002 Arkhangelsk, Russia<\/p>\n<p>Ms. Roza Atamuratova, teacher <br \/>Regional Special Boarding School for Gifted Children with Profound Studying of Different Subjects<br \/>130 000 Aktau, Kazakhstan<\/p>\n<p>Dr. Veselin Nenkov, Assoc. Prof. <br \/>Technical College, Lovech,<br \/>Lovech, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a id=\"art02\"><\/a>\u041d\u0415\u041a\u041e\u0422\u041e\u0420\u042b\u0415 \u0422\u0420\u0410\u0415\u041a\u0422\u041e\u0420\u0418\u0418, \u041a\u041e\u0422\u041e\u0420\u042b\u0415 \u041e\u041f\u0420\u0415\u0414\u0415\u041b\u0415\u041d\u042b \u0420\u0410\u0412\u041d\u041e\u0411\u0415\u0414\u0420\u0415\u041d\u041d\u042b\u041c\u0418 \u0422\u0420\u0415\u0423\u0413\u041e\u041b\u042c\u041d\u0418\u041a\u0410\u041c\u0418<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u041a\u0441\u0435\u043d\u0438\u044f \u0413\u043e\u0440\u0441\u043a\u0430\u044f, \u0414\u0430\u0440\u044c\u044f \u041a\u043e\u043f\u0442\u0435\u0432\u0430, \u0414\u0430\u043d\u0438\u0438\u043b \u041c\u0438\u043a\u0443\u0440\u043e\u0432<br \/>\u041c\u0443\u043d\u0438\u0446\u0438\u043f\u0430\u043b\u044c\u043d\u043e\u0435 \u0431\u044e\u0434\u0436\u0435\u0442\u043d\u043e\u0435 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0443\u0447\u0440\u0435\u0436\u0434\u0435\u043d\u0438\u0435 &ldquo;\u0421\u0440\u0435\u0434\u043d\u044f\u044f \u0448\u043a\u043e\u043b\u0430 \u2116 8&rdquo; &ndash; \u0410\u0440\u0445\u0430\u043d\u0433\u0435\u043b\u044c\u0441\u043a, \u0420\u043e\u0441\u0441\u0438\u044f<br \/>\u0415\u0440\u043a\u0435\u043d \u041c\u0443\u0434\u0435\u0431\u0430\u0435\u0432, \u041a\u0430\u0437\u0431\u0435\u043a \u041c\u0443\u0445\u0430\u043c\u0431\u0435\u0442\u043e\u0432, \u0410\u0434\u0438\u043b\u0431\u0435\u043a \u0422\u0435\u043c\u0438\u0440\u0445\u0430\u043d\u043e\u0432<br \/>\u041e\u0431\u043b\u0430\u0441\u0442\u043d\u0430\u044f \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u0438\u0437\u0438\u0440\u043e\u0432\u0430\u043d\u043d\u0430\u044f \u0448\u043a\u043e\u043b\u0430-\u0438\u043d\u0442\u0435\u0440\u043d\u0430\u0442 \u0434\u043b\u044f \u043e\u0434\u0430\u0440\u0435\u043d\u043d\u044b\u0445 \u0434\u0435\u0442\u0435\u0439 \u0441 \u0443\u0433\u043b\u0443\u0431\u043b\u0435\u043d\u043d\u044b\u043c \u0438\u0437\u0443\u0447\u0435\u043d\u0438\u0435\u043c \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u044b\u0445 \u043f\u0440\u0435\u0434\u043c\u0435\u0442\u043e\u0432 &ndash; \u0410\u043a\u0442\u0430\u0443, \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d<br \/>\u041b\u0438\u043b\u0438 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u0418\u0440\u0438\u043d\u0430 \u0425\u0440\u0438\u0441\u0442\u043e\u0432\u0430, \u0420\u0430\u0434\u0438\u043d\u0430 \u0418\u0432\u0430\u043d\u043e\u0432\u0430 <br \/>\u041f\u0440\u0438\u0440\u043e\u0434\u043e-\u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0433\u0438\u043c\u043d\u0430\u0437\u0438\u044f &ndash; \u041b\u043e\u0432\u0435\u0447, \u0411\u043e\u043b\u0433\u0430\u0440\u0438\u044f<\/p>\n<p>&nbsp;<\/p>\n<p>\u0410\u043d\u043d\u043e\u0442\u0430\u0446\u0438\u044f. \u0412 \u0441\u0442\u0430\u0442\u044c\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u044b \u043e\u0441\u043d\u043e\u0432\u043d\u044b\u0435 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0440\u0430\u0431\u043e\u0442\u044b \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0439 \u0433\u0440\u0443\u043f\u043f\u044b \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u0441\u043a\u043e\u0433\u043e \u043f\u0440\u043e\u0435\u043a\u0442\u0430 &bdquo;\u0413\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0439 Scrabble \u0432 \u043e\u0431\u043b\u0430\u043a\u0430\u0445&ldquo;. \u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0435 \u0441\u043e\u0441\u0442\u043e\u044f\u043b\u043e \u0432 \u043f\u043e\u0441\u0442\u0430\u043d\u043e\u0432\u043a\u0435 \u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u0438 \u0437\u0430\u0434\u0430\u0447, \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u044f\u044e\u0449\u0438\u0445 \u0441\u043e\u0431\u043e\u0439 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u0438 \u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0435 \u0430\u043d\u0430\u043b\u043e\u0433\u0438 \u0441\u0442\u0430\u0440\u0442\u043e\u0432\u043e\u0439 \u0437\u0430\u0434\u0430\u0447\u0438. \u0412 \u0445\u043e\u0434\u0435 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u0441\u044f \u0430\u043f\u043f\u0430\u0440\u0430\u0442 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438, \u0430 \u0442\u0430\u043a\u0436\u0435 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u043d\u044b\u0435 \u043f\u0440\u043e\u0434\u0443\u043a\u0442\u044b, \u0442\u0430\u043a\u0438\u0435 \u043a\u0430\u043a GeoGebra, Geometer&rsquo;s Sketchpad, Maple. \u0412 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u0435 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u044b \u0437\u0430\u0434\u0430\u0447\u0438, \u043f\u0440\u0438\u0432\u043e\u0434\u044f\u0449\u0438\u0435 \u043a \u043f\u043e\u044f\u0432\u043b\u0435\u043d\u0438\u044e \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u043c\u0435\u0441\u0442 \u0442\u043e\u0447\u0435\u043a, \u043a\u043e\u0442\u043e\u0440\u044b\u043c\u0438 \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f: \u043f\u0440\u044f\u043c\u0430\u044f \u042d\u0439\u043b\u0435\u0440\u0430, \u043a\u0440\u0438\u0432\u044b\u0435 \u0432\u0442\u043e\u0440\u043e\u0433\u043e \u043f\u043e\u0440\u044f\u0434\u043a\u0430, \u0430 \u0442\u0430\u043a\u0436\u0435 \u043a\u0440\u0438\u0432\u044b\u0435 \u0431\u043e\u043b\u0435\u0435 \u0432\u044b\u0441\u043e\u043a\u0438\u0445 \u043f\u043e\u0440\u044f\u0434\u043a\u043e\u0432.<br \/>Keywords: locus, Euler line, Euler curve, curve of second degree<\/p>\n<p>&nbsp;<\/p>\n<p><strong>SOME TRAJECTORIES DETERMINED BY ISOSCELES TRIANGLES<\/strong><\/p>\n<p>Abstract. The paper presents the main results of the work of an international group of students in the frames of the research project &ldquo;Geometry Scrabble in Cloud&rdquo;. The study consisted in creating and solving problems that were generalizations and natural analogues of a starting problem. It applied analytical geometry methods and was software supported by GeoGebra, Geometer&rsquo;s Sketchpad and Maple. As a result various problems were created leading to loci like Euler line, curves of second degree and also curves of higher order.<\/p>\n<p>&nbsp;<\/p>\n<p>Ms. Gorskaya Kseniya, 2)Ms. Kopteva Daria, 3)Mr. Mikurov Daniil<br \/>Municipal High School \u2116 8<br \/>163060 Arkhangelsk, Russia<br \/>E-mail: 1)<a href=\"mailto:gorksuvas@mail.ru\">gorksuvas@mail.ru<\/a>; 2)<a href=\"mailto:kopteva-8@yandex.ru\">kopteva-8@yandex.ru<\/a>; 3)<a href=\"mailto:hygermik@gmail.com\">hygermik@gmail.com<\/a><\/p>\n<p>Mr. Yerkin Amirzhanovich Mudebaev, Mr. Adilbek Timurovich Temirkhanov, Mr. Kazbek Batyrbekovich Mukhambetov<br \/>Regional Specialized Boarding School for Gifted Children with In-depth Study of Various Subjects<br \/>Aktau, Kazahstan<\/p>\n<p>Ms. Irina Hristova, Ms. Radina Ivanova, Ms. Lilly Stefanova <br \/>Mathematics High School &ndash; Lovech<br \/>5500 Lovech, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: justify;\"><strong><a id=\"art03\"><\/a>\u0414\u041e\u041c\u0410\u0428\u041d\u0410\u0422\u0410 \u0420\u0410\u0411\u041e\u0422\u0410 &ndash; \u041d\u0415\u0414\u0415\u041b\u0418\u041c\u0410 \u0427\u0410\u0421\u0422 \u041e\u0422 \u0423\u0427\u0415\u0411\u041d\u0418\u042f \u041f\u0420\u041e\u0426\u0415\u0421<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u0415\u043b\u0435\u043d\u0430 \u0420\u0430\u0434\u043e\u0432\u0430\u043d\u043e\u0432\u0430 <br \/>\u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 &ndash; \u0421\u043e\u0444\u0438\u044f<\/p>\n<p>&nbsp;<\/p>\n<p>\u0420\u0435\u0437\u044e\u043c\u0435. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0438\u0442\u0435 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u043d\u0435\u043c\u0441\u043a\u0438 \u0435\u0437\u0438\u043a \u0432 \u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438\u044f \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 &ndash; \u0421\u043e\u0444\u0438\u044f. \u0410\u043a\u0446\u0435\u043d\u0442\u0438\u0440\u0430 \u0441\u0435 \u043d\u0430 \u0434\u043e\u043c\u0430\u0448&not;\u043d\u0430\u0442\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 \u043a\u0430\u0442\u043e \u043d\u0435\u0434\u0435\u043b\u0438\u043c\u0430 \u0447\u0430\u0441\u0442 \u043e\u0442 \u0443\u0447\u0435\u0431\u043d\u0438\u044f \u043f\u0440\u043e\u0446\u0435\u0441. \u041e\u0431\u043e\u0441\u043d\u043e\u0432\u0430\u043d\u0430 \u0435 \u043d\u0435\u043e\u0431\u0445\u043e\u0434\u0438&not;\u043c\u043e\u0441\u0442\u0442\u0430 \u043e\u0442 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0430\u0442\u0430 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0430\u0442\u0438\u043a\u0430: \u043e\u0431\u0443\u0447\u0430\u0432\u0430\u043d\u0438\u044f\u0442 \u0442\u0440\u044f\u0431\u0432\u0430 \u0434\u0430 \u043c\u043e\u0436\u0435 \u0434\u0430 \u043f\u043e\u043b\u0437\u0432\u0430 \u043b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u0430 \u043d\u0430 \u043d\u0435\u043c\u0441\u043a\u0438 \u0435\u0437\u0438\u043a, \u0434\u0430 \u0441\u0435 \u0438\u0437\u0440\u0430\u0437\u044f\u0432\u0430 \u043f\u0440\u0430\u0432\u0438\u043b\u043d\u043e \u043f\u0438\u0441\u043c\u0435&not;\u043d\u043e \u0438 \u0433\u043e\u0432\u043e\u0440\u0438\u043c\u043e \u0438 \u0434\u0430 \u0443\u043c\u0435\u0435 \u0434\u0430 \u043e\u0441\u044a\u0449\u0435\u0441\u0442\u0432\u044f\u0432\u0430 \u0442\u0440\u0430\u043d\u0441\u0444\u0435\u0440 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0437\u043d\u0430\u043d\u0438\u044f; \u0432 \u0441\u044a\u0449\u043e\u0442\u043e \u0432\u0440\u0435\u043c\u0435 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043f\u0440\u0435\u0434 \u043e\u0431\u0443\u0447\u0430\u0432\u0430\u0449\u0438\u044f \u0441\u0430 \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0432\u044a\u0437&not;\u043d\u0438\u043a\u0432\u0430\u0449\u0438\u0442\u0435 \u0435\u0437\u0438\u043a\u043e\u0432\u0438, \u043c\u0435\u0442\u043e\u0434\u043e\u043b\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438 \u0438 \u043f\u0440\u043e\u0446\u0435\u0441\u0443\u0430\u043b\u043d\u0438 \u043f\u0440\u043e\u0431\u043b\u0435\u043c\u0438 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u043d\u0430 \u043d\u0435\u043c\u0441\u043a\u0438 \u0435\u0437\u0438\u043a. <br \/>Keywords: homework, educational system, learning, methodology of teaching<\/p>\n<p>&nbsp;<\/p>\n<p><strong>THE HOMEWORK &ndash; AN INTEGRAL PART OF THE EDUCATIONAL PROCESS<\/strong><\/p>\n<p>Abstract. The paper is dedicted to the problems of the mathematical education in German taking place in the Technical University, Sofia. It focuses on the homework as an integral part of the learning process. The need of considering the suggested problematics is justified: the lesrner should be able to use literaure in German, to express him\/herself correctly in writing and orally, and should be capable in transfering mathematical knowledge; at the same time the tasks which the instructor faces are connected with the solution of linguistic, methodological and procedural problems appearing in the mathematical education in German.<\/p>\n<p>&nbsp;<\/p>\n<p>Elena Radovanova, Assist. Prof. <br \/>Faculty of Mathematics and Informatics <br \/>Technical University <br \/>Sofia, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a id=\"art04\"><\/a>\u041d\u042f\u041a\u041e\u0418 \u0413\u0420\u0423\u041f\u0418 \u041b\u041e\u0413\u0410\u0420\u0418\u0422\u041c\u0418\u0427\u041d\u0418 \u0423\u0420\u0410\u0412\u041d\u0415\u041d\u0418\u042f \u0421 \u041f\u041e\u041c\u041e\u0429\u0422\u0410 \u041d\u0410 EXCEL<\/strong><\/p>\n<p>\u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432<\/p>\n<p>&nbsp;<\/p>\n<p>\u0420\u0435\u0437\u044e\u043c\u0435. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0438 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043d\u044f\u043a\u043e\u0438 \u0433\u0440\u0443\u043f\u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0442\u043c\u0438\u0447\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0441 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430\u0442\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0430 Excel. \u0426\u0435\u043b&not;\u0442\u0430 \u0435 \u0443\u0441\u044a\u0432\u044a\u0440\u0448\u0435\u043d\u0441\u0442\u0432\u0430\u043d\u0435 \u043d\u0430 \u0443\u043c\u0435\u043d\u0438\u044f\u0442\u0430 \u0438 \u043f\u043e\u0432\u0438\u0448\u0430\u0432\u0430\u043d\u0435 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u0430 \u043d\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u0438\u043c \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u0441\u0440\u0435\u0434\u043d\u043e\u0442\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435. <br \/>Keywords: problem solving, log equations, number, table, macros<\/p>\n<p>&nbsp;<\/p>\n<p><strong>GROUPS OF LOGARITHMIC EQUATIONS BY THE HELP OF EXCEL<\/strong><\/p>\n<p>Abstract. The paper proposes possibilities to solve some groups of logarithmic equations by the help of the computer program Excel. The goal is to improve the capacities and to increase the interest of students studying mathematics in the secondary school.<\/p>\n<p>Dr. Diana Stefanova <br \/>Teacher in Mathematics, Asenovgrad<br \/>Mr. Plamen Penev <br \/>Teacher in Informatics, Shumen<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a id=\"art05\"><\/a>\u041f\u0421\u0415\u0412\u0414\u041e\u0426\u0415\u041d\u0422\u042a\u0420 \u0418 \u041e\u0420\u0422\u041e\u0426\u0415\u041d\u0422\u042a\u0420 &ndash; \u0417\u0410\u0411\u0415\u041b\u0415\u0416\u0418\u0422\u0415\u041b\u041d\u0418 \u0422\u041e\u0427\u041a\u0418 \u0412 \u0427\u0415\u0422\u0418\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a\u0410<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 <br \/>\u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u043a\u043e\u043b\u0435\u0436 &ndash; \u041b\u043e\u0432\u0435\u0447<br \/>\u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432, \u0425\u0430\u0438\u043c \u0425\u0430\u0438\u043c\u043e\u0432 <br \/>\u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 &ndash; \u0421\u043e\u0444\u0438\u044f<\/p>\n<p>&nbsp;<\/p>\n<p>\u0420\u0435\u0437\u044e\u043c\u0435. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0438 \u043d\u044f\u043a\u043e\u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430 \u0434\u0432\u0435 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \u0432 \u0438\u0437\u043f\u044a\u043a\u043d\u0430\u043b\u0438\u044f \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a, \u043a\u043e\u0438\u0442\u043e \u0441\u0430 \u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u0438 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u043d\u0430 \u0446\u0435\u043d\u0442\u044a\u0440\u0430 \u043d\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0430\u0442\u0430 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442 \u0438 \u043e\u0440\u0442\u043e\u0446\u0435\u043d\u0442\u044a\u0440\u0430 \u043d\u0430 \u0432\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a.<br \/>Keywords: quadrilateral, circumcircle, circumcenter, orthocenter, Euler circle<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>PSEUDOCENTER AND ORTHOCENTER &ndash; NOTABLE POINTS IN THE QUADRILATERAL<\/strong><\/p>\n<p>Abstract. The paper describes some properties of two notable points in the convex quadrilateral, which naturally generalize the circum-center and the orthocenter of the inscribed quadrilateral.<\/p>\n<p>Dr. Veselin Nenkov, Assoc. Prof. <br \/>Technical College &ndash; Lovech <br \/>Mr. Stanislav Stefanov, PhD student Mr. Haim Haimov, Researcher<br \/>Technical University &ndash; Sofia<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a id=\"art06\"><\/a>\u041d\u042f\u041a\u041e\u0418 \u041d\u041e\u0412\u0418 \u041d\u0415\u0420\u0410\u0412\u0415\u041d\u0421\u0422\u0412\u0410 \u041c\u0415\u0416\u0414\u0423 \u0421\u0420\u0415\u0414\u041d\u0418\u0422\u0415\u0410\u0420\u0418\u0422\u041c\u0415\u0422\u0418\u0427\u041d\u041e, \u0413\u0415\u041e\u041c\u0415\u0422\u0420\u0418\u0427\u041d\u041e, \u0425\u0410\u0420\u041c\u041e\u041d\u0418\u0427\u041d\u041e \u0418 \u041a\u0412\u0410\u0414\u0420\u0410\u0422\u0418\u0427\u041d\u041e<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>\u0422\u043e\u0434\u043e\u0440 \u041c\u0438\u0442\u0435\u0432 <br \/>\u0420\u0443\u0441\u0435\u043d\u0441\u043a\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 &bdquo;\u0410\u043d\u0433\u0435\u043b \u041a\u044a\u043d\u0447\u0435\u0432&ldquo;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Abstract.<\/strong> \u041d\u0435\u043a\u0430 <img decoding=\"async\" loading=\"lazy\" width=\"74\" height=\"25\" class=\"size-full wp-image-41266 size-full wp-image-41267 size-full wp-image-41268 size-full wp-image-41269 size-full wp-image-41270 size-full wp-image-41271 size-full wp-image-41272 size-full wp-image-41266 size-full wp-image-41267 size-full wp-image-41268 size-full wp-image-41269 size-full wp-image-41270 size-full wp-image-41271 size-full wp-image-41272\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_1.jpg\" alt=\"\" \/>&nbsp;\u0441\u0430 \u0441\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u043e \u0430\u0440\u0438\u0442\u043c\u0435\u0442\u0438\u0447\u043d\u043e\u0442\u043e, \u0445\u0430\u0440\u043c\u043e\u043d\u0438\u0447\u043d\u043e\u0442\u043e \u0438 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0447\u043d\u043e\u0442\u043e \u0441\u0440\u0435\u0434\u043d\u0438 \u0437\u0430 \u043f\u043e\u043b\u043e\u0436\u0438\u0442\u0435\u043b\u043d\u0438\u0442\u0435 \u0440\u0435\u0430\u043b\u043d\u0438 \u0447\u0438\u0441\u043b\u0430 <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_2.jpg\" alt=\"\" \/>. \u0412 \u0442\u0430\u0437\u0438 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0430 \u0434\u043e\u043a\u0430\u0437\u0430\u043d\u0438 \u0441\u043b\u0435\u0434\u043d\u0438\u0442\u0435 \u0442\u0435\u043e\u0440\u0435\u043c\u0438:<\/p>\n<p><strong>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 1<\/strong>. \u0417\u0430<em><strong> n = 3<\/strong><\/em> \u0435 \u0438\u0437\u043f\u044a\u043b\u043d\u0435\u043d\u043e<br \/> <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_3.jpg\" alt=\"\" \/>and <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_4.jpg\" alt=\"\" \/>.<br \/><strong>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 2<\/strong>. \u0417\u0430 \u0435 \u0438\u0437\u043f\u044a\u043b\u043d\u0435\u043d\u043e<br \/> <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_5.jpg\" alt=\"\" \/>and <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_6.jpg\" alt=\"\" \/>.<br \/><strong>\u0422\u0435\u043e\u0440\u0435\u043c\u0430 3.<\/strong> \u0417\u0430<em><strong> n = 5<\/strong> <\/em>\u0435 \u0438\u0437\u043f\u044a\u043b\u043d\u0435\u043d\u043e <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_7.jpg\" alt=\"\" \/>.<br \/>\u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0438 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u043d\u0435\u0440\u0435\u0448\u0435\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438.<br \/>Keywords: ineaquality, arithmetic mean, geometric mean, harmonic mean, quadratic mean<\/p>\n<p>&nbsp;<\/p>\n<p><strong>SOME NEW INEQUALITIES AMONG THE ARITHMETIC, GEOMETRIC, HARMONIC AND QUADRATIC MEANS <\/strong><\/p>\n<p><strong>Abstract.<\/strong> Let <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_1.jpg\" alt=\"\" \/>&nbsp;be the arithmetic, harmonic and quadratic means respectivity for the positive real numbers <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_2.jpg\" alt=\"\" \/>. In this article we prove the following theorems:<\/p>\n<p><strong>Theorem 1<\/strong>. For&nbsp;<em><strong>n = 3<\/strong><\/em> we have <br \/> <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_3.jpg\" alt=\"\" \/>and <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_4.jpg\" alt=\"\" \/>.<\/p>\n<p><strong>Theotrem 2.<\/strong> For <em><strong>n = 4<\/strong> <\/em>we have<br \/> <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_5.jpg\" alt=\"\" \/>and <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_6.jpg\" alt=\"\" \/>.<\/p>\n<p><strong>Theorem 3.&nbsp;<\/strong>For <em><strong>n = 5<\/strong>&nbsp;<\/em>we have <img decoding=\"async\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2016\/12\/images_Matematika06_16_pic_7.jpg\" alt=\"\" \/>.<br \/>Some open problems are proposed too.<\/p>\n<p>&nbsp;<\/p>\n<p>Dr. Todor Mitev<br \/>Mathematics Department Faculty of Natural Sciences<br \/>University of Ruse<br width=\"74\" height=\"25\" \/>Ruse, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a id=\"art07\"><\/a>FUZZY LOGIC<\/strong><br \/>Reinhard Magenreuter <br \/>University of Finance, Business and Entrepreneurship &ndash; Sofia<\/p>\n<p>&nbsp;<\/p>\n<p>Abstract. There are four backbones to analyze time-series in general and fore&not;cast time-series for financial markets: Chaos-theory, Fuzzy logic, Neural networks and Genetic algorithms. The first one is considered in (Magenreuter, 2016a), while te third one in (Magenreuter, 2016b). The present paper is dedicated to the second one analyzing posibilities for its application. Some results are discussed in promis&not;ing outcomes. <br \/>Keywords: financial market, time-series, fuzzy logic, forecasting<\/p>\n<p>&nbsp;<\/p>\n<p>Mr. Reinhard Magenreuter, PhD student<br \/>University of Finance, Business and Entrepreneurship<br \/>1618 Sofia, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a id=\"art08\"><\/a>GENETIC ALGORITHM<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Reinhard Magenreuter <br \/>University of Finance, Business and Entrepreneurship &ndash; Sofia<\/p>\n<p>&nbsp;<\/p>\n<p>Abstract. There are four backbones to analyze time-series in general and forecast time-series for financial markets: Chaos-theory, Fuzzy logic, Neural networks and Genetic algorithms. The first one is considered in (Magenreuter, 2016a), while the second one &ndash; in (Magenreuter, 2016c) and the third one &ndash; in (Magenreuter, 2016b). The present paper is dedicated to the fourth backbone including promising outcomes in financial markts. <br \/>Keywords: chromosome, reproduction, genetic algorithm, neural network, financial market, time-series, forecasting<\/p>\n<p>&nbsp;<\/p>\n<p>Mr. Reinhard Magenreuter, PhD student<br \/>University of Finance, Business and Entrepreneurship<br \/>1618 Sofia, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>","protected":false},"excerpt":{"rendered":"<p>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 &nbsp; EDUCATIONAL MATTERS \/ \u041d\u0410\u0423\u0427\u041d\u041e-\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0418 \u0421\u0422\u0410\u0422\u0418\u0418\u041f\u0435\u0440\u0432\u044b\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u044b\u0439 \u0441\u0435\u0442\u0435\u0432\u043e\u0439 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u0441\u043a\u0438\u0439 \u043f\u0440\u043e\u0435\u043a\u0442 \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 MITE [First International Net Research Project of Secondary Students in the Frames of MITE] \/ \u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u041c\u0430\u0440\u0438\u043d\u0430 \u0411\u0435\u043b\u043e\u0440\u0443\u043a\u043e\u0432\u0430, \u0420\u043e\u0437\u0430 \u0410\u0442\u0430\u043c\u0443\u0440\u0430\u0442\u043e\u0432\u0430, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432&nbsp;\/ Maria Shabanova, Marina Belorukova, Roza Atamuratova, Veselin Nenkov &ndash; \u0441\u0442\u0440. 567&nbsp;\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 &nbsp; \u041d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438, [&hellip;]<\/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":[3290],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>&quot;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&quot;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016 - \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:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"&quot;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&quot;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016 - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 &nbsp; EDUCATIONAL MATTERS \/ \u041d\u0410\u0423\u0427\u041d\u041e-\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0418 \u0421\u0422\u0410\u0422\u0418\u0418\u041f\u0435\u0440\u0432\u044b\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u044b\u0439 \u0441\u0435\u0442\u0435\u0432\u043e\u0439 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u0441\u043a\u0438\u0439 \u043f\u0440\u043e\u0435\u043a\u0442 \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 MITE [First International Net Research Project of Secondary Students in the Frames of MITE] \/ \u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u041c\u0430\u0440\u0438\u043d\u0430 \u0411\u0435\u043b\u043e\u0440\u0443\u043a\u043e\u0432\u0430, \u0420\u043e\u0437\u0430 \u0410\u0442\u0430\u043c\u0443\u0440\u0430\u0442\u043e\u0432\u0430, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432&nbsp;\/ Maria Shabanova, Marina Belorukova, Roza Atamuratova, Veselin Nenkov &ndash; \u0441\u0442\u0440. 567&nbsp;\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 &nbsp; \u041d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438, [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/\" \/>\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=\"2016-11-02T08:37:20+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-06-18T18:58:29+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:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/\"},\"author\":{\"name\":\"\",\"@id\":\"\"},\"headline\":\"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016\",\"datePublished\":\"2016-11-02T08:37:20+00:00\",\"dateModified\":\"2019-06-18T18:58:29+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/\"},\"wordCount\":1866,\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/#organization\"},\"articleSection\":[\"\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\", 2016 \u0433.\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/\",\"url\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/\",\"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 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016 - \u0410\u0437-\u0431\u0443\u043a\u0438\",\"isPartOf\":{\"@id\":\"https:\/\/azbuki.bg\/#website\"},\"datePublished\":\"2016-11-02T08:37:20+00:00\",\"dateModified\":\"2019-06-18T18:58:29+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathinfo.azbuki.bg\/en\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/azbuki.bg\/#website\",\"url\":\"https:\/\/azbuki.bg\/\",\"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\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/azbuki.bg\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/azbuki.bg\/#organization\",\"name\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\",\"url\":\"https:\/\/azbuki.bg\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/azbuki.bg\/#\/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\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/Azbuki55\/\",\"https:\/\/www.instagram.com\/nion_azbuki\/\"]},{\"@type\":\"Person\",\"@id\":\"\",\"url\":\"https:\/\/strategies.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 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016 - \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:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/","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 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016 - \u0410\u0437-\u0431\u0443\u043a\u0438","og_description":"CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 &nbsp; EDUCATIONAL MATTERS \/ \u041d\u0410\u0423\u0427\u041d\u041e-\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0418 \u0421\u0422\u0410\u0422\u0418\u0418\u041f\u0435\u0440\u0432\u044b\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u044b\u0439 \u0441\u0435\u0442\u0435\u0432\u043e\u0439 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u0441\u043a\u0438\u0439 \u043f\u0440\u043e\u0435\u043a\u0442 \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 MITE [First International Net Research Project of Secondary Students in the Frames of MITE] \/ \u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u041c\u0430\u0440\u0438\u043d\u0430 \u0411\u0435\u043b\u043e\u0440\u0443\u043a\u043e\u0432\u0430, \u0420\u043e\u0437\u0430 \u0410\u0442\u0430\u043c\u0443\u0440\u0430\u0442\u043e\u0432\u0430, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432&nbsp;\/ Maria Shabanova, Marina Belorukova, Roza Atamuratova, Veselin Nenkov &ndash; \u0441\u0442\u0440. 567&nbsp;\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 &nbsp; \u041d\u0435\u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u0438, [&hellip;]","og_url":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2016-11-02T08:37:20+00:00","article_modified_time":"2019-06-18T18:58:29+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:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/#article","isPartOf":{"@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/"},"author":{"name":"","@id":""},"headline":"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016","datePublished":"2016-11-02T08:37:20+00:00","dateModified":"2019-06-18T18:58:29+00:00","mainEntityOfPage":{"@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/"},"wordCount":1866,"publisher":{"@id":"https:\/\/azbuki.bg\/#organization"},"articleSection":["\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", 2016 \u0433."],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/","url":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/","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 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016 - \u0410\u0437-\u0431\u0443\u043a\u0438","isPartOf":{"@id":"https:\/\/azbuki.bg\/#website"},"datePublished":"2016-11-02T08:37:20+00:00","dateModified":"2019-06-18T18:58:29+00:00","breadcrumb":{"@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016\/6-lix-2016\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathinfo.azbuki.bg\/en\/"},{"@type":"ListItem","position":2,"name":"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LIX, 2016"}]},{"@type":"WebSite","@id":"https:\/\/azbuki.bg\/#website","url":"https:\/\/azbuki.bg\/","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\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/azbuki.bg\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/azbuki.bg\/#organization","name":"\u0410\u0437-\u0431\u0443\u043a\u0438","url":"https:\/\/azbuki.bg\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/azbuki.bg\/#\/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\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/Azbuki55\/","https:\/\/www.instagram.com\/nion_azbuki\/"]},{"@type":"Person","@id":"","url":"https:\/\/strategies.azbuki.bg\/en\/author\/"}]}},"_links":{"self":[{"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/41273"}],"collection":[{"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/comments?post=41273"}],"version-history":[{"count":4,"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/41273\/revisions"}],"predecessor-version":[{"id":59693,"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/41273\/revisions\/59693"}],"wp:attachment":[{"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/media?parent=41273"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/categories?post=41273"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/strategies.azbuki.bg\/en\/wp-json\/wp\/v2\/tags?post=41273"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}