{"id":42773,"date":"2018-06-27T08:49:13","date_gmt":"2018-06-27T05:49:13","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles2016-3\/3-lxi-2018\/"},"modified":"2019-06-24T17:33:14","modified_gmt":"2019-06-24T14:33:14","slug":"3-lxi-2018","status":"publish","type":"post","link":"https:\/\/newspaper.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/","title":{"rendered":"Mathematics and Informatics, Number 3\/2018, Volume 61"},"content":{"rendered":"<p><strong><a name=\"top\"><\/a>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"text1\"><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><\/p>\n<p><a href=\"#art01\">\u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u043b\u0438\u043d\u0435\u0435\u043d \u0442\u0440\u0435\u043d\u0434 \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f <\/a><br \/><a href=\"#art01\">[Using a Trendline to Solve Rational Equations] <\/a><br \/>\/ <em>\u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432 \/ <\/em><em>Plamen Penev<\/em> \u2013 \u0441\u0442\u0440. 223<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art02\">\u0421\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0438\u0433\u0440\u0438 \u0432 \u0447\u0430\u0441\u043e\u0432\u0435\u0442\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0447\u0440\u0435\u0437 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u043e\u0440 \u043d\u0430 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u0438 \u0447\u0438\u0441\u043b\u0430 <\/a><br \/><a href=\"#art02\">[Creating of Games in the Informatics Classes by Using a Generator of Random Numbers] <\/a><br \/>\/ <em>\u0415\u043c\u0438\u043b\u0438\u044f \u041d\u0438\u043a\u043e\u043b\u043e\u0432\u0430, \u0414\u0430\u043d\u0438\u0435\u043b\u0430 \u0422\u0443\u043f\u0430\u0440\u043e\u0432\u0430<\/em><em> \/ Emilia Nikolova, Daniela Tuparova<\/em> \u2013 \u0441\u0442\u0440. 232<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art03\">\u041a\u0430\u043a \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u044a\u0442 \u0440\u0435\u0448\u0430\u0432\u0430 \u0441\u0443\u0434\u043e\u043a\u0443 \u2013 \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043c\u043e\u0434\u0435\u043b \u043d\u0430 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u044a\u043c\u0430 <\/a><br \/><a href=\"#art03\">[How Does the Computer Solve Sudoku \u2013 A Mathematical Model of the Algorithm] <\/a><br \/>\/ <em>\u041a\u0440\u0430\u0441\u0438\u043c\u0438\u0440 \u0419\u043e\u0440\u0434\u0436\u0435\u0432<\/em><em> \/ Krasimir Yordzhev<\/em> \u2013 \u0441\u0442\u0440. 259<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"text1\">\u00a0<\/p>\n<p class=\"text1\"><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><\/p>\n<p><a href=\"#art04\">\u041f\u043e\u0441\u0442\u0438\u0433\u0430\u043d\u0435 \u043d\u0430 \u0442\u0432\u043e\u0440\u0447\u0435\u0441\u043a\u0438 \u0446\u0435\u043b\u0438 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 VI. \u043a\u043b\u0430\u0441 \u0440\u0430\u0437\u0434\u0435\u043b \u201e\u0441\u0442\u0435\u043f\u0435\u043d\u0443\u0432\u0430\u043d\u0435\u201d <\/a><br \/><a href=\"#art04\">[Achieving Creative Goals in Mathematics Education in 6. Grade Section \u201cExponents\u201d] <\/a><br \/>\/ <em>\u0421\u0435\u0432\u0434\u0430\u043b\u0438\u043d\u0430 \u0413\u0435\u043e\u0440\u0433\u0438\u0435\u0432\u0430 \/ <\/em><em>Sevdalina Georgie<\/em><em>va <\/em>\u2013 \u0441\u0442\u0440. 265<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art05\">\u041f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u043e\u0442 \u0442\u0440\u0435\u0442\u0430 \u0441\u0442\u0435\u043f\u0435\u043d \u0441 \u043a\u043e\u043b\u0438\u043d\u0435\u0430\u0440\u043d\u0438 \u043a\u043e\u0440\u0435\u043d\u0438 <\/a><br \/><a href=\"#art05\">[Polynomials of Third Degree with Collinear Roots] <\/a><br \/>\/ <em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \/ Sava Grozdev, Veselin Nenkov <\/em>\u2013 \u0441\u0442\u0440. 277<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art06\">\u041f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u043e\u0442 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0430 \u0441\u0442\u0435\u043f\u0435\u043d \u0441 \u043a\u043e\u0440\u0435\u043d\u0438 \u0432\u044a\u0432 \u0432\u044a\u0440\u0445\u043e\u0432\u0435\u0442\u0435 \u043d\u0430 \u0443\u0441\u043f\u043e\u0440\u0435\u0434\u043d\u0438\u043a <\/a><br \/><a href=\"#art06\">[Polynomials of Fourth Degree with Roots in the Vertices of a Parallelogram] <\/a><br \/>\/ <em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \/ Sava Grozdev, Veselin Nenkov <\/em>\u2013 page 283<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art07\">\u0427\u0435\u0442\u0438\u0440\u0438\u0434\u0435\u0441\u0435\u0442 \u0438 \u043f\u0435\u0442\u0430 \u043d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0441\u043a\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 <\/a><br \/><a href=\"#art07\">[Forty Fifth National Olympiad in Mathematics for University Students] <\/a><br \/>\/ <em>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432, \u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432\u0430 \u0421\u0442\u043e\u0438\u043b\u043e\u0432\u0430, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/em> \/ <em>Sava Grozdev, Rosen Nikolaev, <\/em><em>Stanislava <\/em><em>Stoilova, Veselin Nenkov <\/em>\u2013 \u0441\u0442\u0440. 294<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><br \/>\u00a0<\/p>\n<p><a href=\"#art08\">Two New Phd Dissertations in Methodology of Informatics Teaching <\/a><br \/>\/ <em>Petya <\/em><em>Asenova <\/em>\u2013 page 299<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"text1\">\u00a0<\/p>\n<p class=\"text1\"><strong>CONTEST PROBLEMS \/ \u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong><\/p>\n<p class=\"text1\">Contest Problems of this Issue <br \/>[Contest Problems of this Issue] \u2013 \u0441\u0442\u0440. 302<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"text1\">\u00a0<\/p>\n<p class=\"text1\">\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u0431\u0440\u043e\u0439 4, 2017 <br \/>[Solutions of the Contest Problems from Issue 4, 2017] \u2013 \u0441\u0442\u0440. 303<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"text1\">\u00a0<\/p>\n<p class=\"BasicParagraph\"><strong>READ IN THE LATEST ISSUES OF \u201eAZBUKI\u201c 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 \u201e\u0410\u0417 \u0411\u0423\u041a\u0418\u201c \u0427\u0415\u0422\u0415\u0422\u0415<\/strong> \u2013 \u0441\u0442\u0440. 307<\/p>\n<p class=\"BasicParagraph\">\u00a0<\/p>\n<p class=\"Pa14\"><strong>GUIDE FOR AUTHORS \/ \u0423\u041a\u0410\u0417\u0410\u041d\u0418\u042f \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415<\/strong> \u2013 \u0441\u0442\u0440. 309<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p><strong><a name=\"art01\"><\/a>\u0418\u0417\u041f\u041e\u041b\u0417\u0412\u0410\u041d\u0415 \u041d\u0410 \u041b\u0418\u041d\u0415\u0415\u041d \u0422\u0420\u0415\u041d\u0414 \u041f\u0420\u0418 \u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415 \u041d\u0410 \u0420\u0410\u0426\u0418\u041e\u041d\u0410\u041b\u041d\u0418 \u0423\u0420\u0410\u0412\u041d\u0415\u041d\u0418\u042f<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>Plamen Penev<\/p>\n<p class=\"Default\">\u00a0<\/p>\n<p class=\"Pa52\"><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u201e\u0420\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043b\u0438\u043d\u0435\u0439\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u0441 Excel\u201c \u043e\u0442 \u0431\u0440. 2\/2018 \u0433. \u043d\u0430 \u0441\u043f. \u201e\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u201c \u0438 \u043f\u043e \u0441\u044a\u0449\u0435\u0441\u0442\u0432\u043e \u0435 \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435 \u043d\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0432 \u043d\u0435\u044f \u043c\u043e\u0434\u0435\u043b. \u0412\u044a\u0432\u0435\u0436\u0434\u0430\u0442 \u0441\u0435 \u043d\u043e\u0432\u0438 \u0442\u0435\u0445\u043d\u0438\u043a\u0438 \u0437\u0430 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435 \u043d\u0430 \u0434\u0435\u0444\u0438\u043d\u0438\u0446\u0438\u043e\u043d\u043d\u0438 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430.<br \/><em>Keywords:<\/em> problem solving; equation; numbers; tables; trendline; macros<\/p>\n<p>\u00a0<\/p>\n<p><strong>USING A TRENDLINE TO SOLVE RATIONAL EQUATIONS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The present article shows development of a model for solving linear equations with Excel. New technique is introduced for the examination of domains.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Mr. Plamen Penev<br \/>Teacher in Informatics<br \/>Shumen, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u00a0\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p><strong><a name=\"art02\"><\/a>\u0421\u042a\u0417\u0414\u0410\u0412\u0410\u041d\u0415 \u041d\u0410 \u0418\u0413\u0420\u0418 \u0412 \u0427\u0410\u0421\u041e\u0412\u0415\u0422\u0415 \u041f\u041e \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0422\u0418\u041a\u0410 \u0427\u0420\u0415\u0417 \u0418\u0417\u041f\u041e\u041b\u0417\u0412\u0410\u041d\u0415 \u041d\u0410 \u0413\u0415\u041d\u0415\u0420\u0410\u0422\u041e\u0420 \u041d\u0410 \u0421\u041b\u0423\u0427\u0410\u0419\u041d\u0418 \u0427\u0418\u0421\u041b\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>\u0415\u043c\u0438\u043b\u0438\u044f \u041d\u0438\u043a\u043e\u043b\u043e\u0432\u0430, \u0414\u0430\u043d\u0438\u0435\u043b\u0430 \u0422\u0443\u043f\u0430\u0440\u043e\u0432\u0430<br \/>South-West University \u201eNeofit Rilski\u201c\nBlagoevgrad, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\"><strong>Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0434\u0430\u0432\u0430\u0442 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0438\u0434\u0435\u0438 \u0437\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u043e\u00ad\u0440\u0430 \u043d\u0430 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u0438 \u0447\u0438\u0441\u043b\u0430 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435. \u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d \u0435 \u043d\u0430\u0431\u043e\u0440 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0432\u0438\u0437\u0443\u0430\u043b\u043d\u043e \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0431\u0430\u0437\u0430\u0442\u0430 \u043d\u0430 \u0435\u0437\u0438\u043a\u0430 C#, \u043a\u044a\u043c \u0432\u0441\u044f\u043a\u0430 \u043e\u0442 \u043a\u043e\u0438\u0442\u043e \u0441\u0430 \u0438\u0437\u043b\u043e\u0436\u0435\u043d\u0438 \u0446\u0435\u043b\u0438\u0442\u0435 \u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0435\u0442\u0430\u043f\u0438 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u045d. \u0417\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u0438\u043b\u044e\u0441\u0442\u0440\u0438\u0440\u0430\u0442 \u0438 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442 \u0437\u0430 \u0441\u044a\u0447\u0435\u0442\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043f\u0440\u043e\u0446\u0435\u0441\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435 \u0441 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442\u0442\u0430 \u0437\u0430 \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0438\u0433\u0440\u0438. \u041f\u043e\u0441\u043e\u0447\u0435\u043d\u0430\u0442\u0430 \u0433\u0440\u0443\u043f\u0430 \u0441\u044a\u0434\u044a\u0440\u0436\u0430 10 \u0437\u0430\u0434\u0430\u0447\u0438, \u043e\u0442 \u043a\u043e\u0438\u0442\u043e 7 \u0441\u0430 \u0430\u0432\u0442\u043e\u0440\u0441\u043a\u0438, \u0430 3 \u0441\u0435 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0438 \u043a\u043b\u0430\u0441\u0438\u0447\u0435\u0441\u043a\u0438 \u0438\u0433\u0440\u0438. \u041a\u044a\u043c \u0432\u0441\u044f\u043a\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u0441\u0430 \u043f\u043e\u0441\u043e\u0447\u0435\u043d\u0438 \u0446\u0435\u043b\u0442\u0430 \u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0435\u0442\u0430\u043f\u0438 \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u045d.<br \/><em>Keywords:<\/em> computer science education; high school; problem solving in programming; game; event driven programming; random numbers generator<\/p>\n<p>\u00a0<\/p>\n<p><strong>CREATING OF GAMES IN THE INFORMATICS CLASSES BY USING A GENERATOR OF RANDOM NUMBERS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract. <\/strong>Methodological ideas are given in the paper for using the random numbers generator in programming education. A set of problems is presented in connection with visual programming education on the base of the C# language. The aims and the main stages are exposed for the solution of each problem. Also, the problems illustrate a possibility to combine the process of programming education with the creation of games. The set under consideration contains 10 problems, including 7 ones which are original and 3 ones are well-known classic games. The aim and the main stages of the solution are presented for each problem.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Ms. Emilia Nikolova, PhD student, Dr. Daniela Tuparova, Assoc. Prof.<br \/>Department of Mathematics<br \/>South-West University \u201cNeofit Rilski\u201d<br \/>Blagoevgrad, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u00a0\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p><strong><a name=\"art03\"><\/a>\u041a\u0410\u041a \u041a\u041e\u041c\u041f\u042e\u0422\u042a\u0420\u042a\u0422 \u0420\u0415\u0428\u0410\u0412\u0410 \u0421\u0423\u0414\u041e\u041a\u0423 \u2013 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u0427\u0415\u0421\u041a\u0418 \u041c\u041e\u0414\u0415\u041b \u041d\u0410 \u0410\u041b\u0413\u041e\u0420\u0418\u0422\u042a\u041c\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">\u041a\u0440\u0430\u0441\u0438\u043c\u0438\u0440 \u0419\u043e\u0440\u0434\u0436\u0435\u0432<br \/>South-West University \u201eNeofit Rilski\u201c\nBlagoevgrad, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\"><strong>Abstract.<\/strong> \u0412 \u0440\u0430\u0431\u043e\u0442\u0430\u0442\u0430 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u043d\u044f\u043a\u043e\u0438 \u0430\u0441\u043f\u0435\u043a\u0442\u0438 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435. \u041d\u0430\u0431\u043b\u044f\u0433\u0430 \u0441\u0435 \u043d\u0430 \u0437\u0430\u043d\u0438\u043c\u0430\u0442\u0435\u043b\u043d\u0438\u044f \u043c\u043e\u043c\u0435\u043d\u0442 \u043f\u0440\u0438 \u043f\u043e\u0434\u0431\u043e\u0440\u0430 \u043d\u0430 \u043f\u043e\u0434\u0445\u043e\u0434\u044f\u0449\u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u0438 \u0437\u0430 \u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0430\u0446\u0438\u044f \u043d\u0430 \u043e\u0442\u0434\u0435\u043b\u043d\u0438\u0442\u0435 \u0435\u0437\u0438\u043a\u043e\u0432\u0438 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u0438 \u0438 \u0441\u0442\u0440\u0443\u043a\u0442\u0443\u0440\u0438 \u043e\u0442 \u0434\u0430\u043d\u043d\u0438. \u0422\u0430\u043a\u044a\u0432 \u043f\u0440\u0438\u043c\u0435\u0440 \u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438\u044f\u0442 \u0430\u043b\u0433\u043e\u0440\u0438\u0442\u044a\u043c \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0448\u0438\u0440\u043e\u043a\u043e \u0440\u0430\u0437\u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0435\u043d\u0430\u0442\u0430 \u0432 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u043e \u0432\u0440\u0435\u043c\u0435 \u0433\u043b\u0430\u0432\u043e\u0431\u043b\u044a\u0441\u043a\u0430\u043d\u0438\u0446\u0430 \u0441\u0443\u0434\u043e\u043a\u0443. \u0422\u043e\u0432\u0430 \u0435 \u043d\u0430\u043f\u0440\u0430\u0432\u0435\u043d\u043e \u0432\u044a\u0432 \u0432\u0440\u044a\u0437\u043a\u0430 \u0441 \u043f\u043e\u043d\u044f\u0442\u0438\u0435\u0442\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0438 \u043d\u0435\u0433\u043e\u0432\u043e\u0442\u043e \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u0432 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435\u0442\u043e.<br \/><em>Keywords:<\/em> education in programming; data structures; set; Sudoku matrix; Sudoku puzzle; mathematical model of an algorithm<\/p>\n<p>\u00a0<\/p>\n<p><strong>HOW DOES THE COMPUTER SOLVE SUDOKU \u2013 A MATHEMATICAL MODEL OF THE ALGORITHM<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>\u00a0<strong>Abstract.<\/strong> Some aspects of programming education are examined in this work. The emphasis is on the entertainment value. The most appropriate examples are chosen to demonstrate the different language constructions and data structures. Such an example is the demonstrated algorithm for solving the widespread nowadays sudoku puzzle. This is made because of the connection with the term set and putting it into practice in the programming.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Assoc. Prof. Krasimir Yordzhev, DSc.<br \/>Faculty of Natural Sciences<br \/>South-West University \u201cNeofit Rilski\u201d<br \/>Blagoevgrad, Bulgaria<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p><a name=\"art04\"><\/a>\u00a0<strong>\u041f\u041e\u0421\u0422\u0418\u0413\u0410\u041d\u0415 \u041d\u0410 \u0422\u0412\u041e\u0420\u0427\u0415\u0421\u041a\u0418 \u0426\u0415\u041b\u0418 \u0412 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415\u0422\u041e \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410 \u0412 VI. \u041a\u041b\u0410\u0421 \u0420\u0410\u0417\u0414\u0415\u041b \u201e\u0421\u0422\u0415\u041f\u0415\u041d\u0423\u0412\u0410\u041d\u0415\u201d<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Sevdalina Georgieva<br \/>Department for Information Qualification and Lifelong Learning \u2013 Varna\nKonstantin Preslavsky University of Shumen<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\"><strong>Abstract.<\/strong> \u0420\u0430\u0437\u0432\u0438\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0442\u0435 \u043a\u043e\u043c\u043f\u0435\u0442\u0435\u043d\u0442\u043d\u043e\u0441\u0442\u0438 \u0432 \u043f\u0440\u043e\u0446\u0435\u0441\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0441\u0435 \u043e\u043a\u0430\u0437\u0432\u0430 \u043d\u0435\u043b\u0435\u043a\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u0437\u0430 \u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438\u0442\u0435 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u0438\u0441\u0442\u0438, \u0430\u043a\u043e \u043d\u0435 \u0441\u0435 \u043d\u0430\u0441\u043e\u0447\u0438 \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435\u0442\u043e \u043a\u044a\u043c \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0430\u043a\u0442\u0438\u0432\u043d\u0430 \u0442\u0432\u043e\u0440\u0447\u0435\u0441\u043a\u0430 \u043b\u0438\u0447\u043d\u043e\u0441\u0442 \u0443 \u0443\u0447\u0435\u043d\u0438\u043a\u0430. \u041d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430 \u043f\u0440\u0435\u0434\u043b\u0430\u0433\u0430 \u0430\u043d\u0430\u043b\u0438\u0437 \u043d\u0430 \u0446\u0435\u043b\u0435\u043f\u043e\u043b\u0430\u0433\u0430\u043d\u0435\u0442\u043e \u0432 \u0443\u0440\u043e\u043a\u0430 \u043e\u0442 \u0433\u043b\u0435\u0434\u043d\u0430 \u0442\u043e\u0447\u043a\u0430 \u043d\u0430 \u0442\u0432\u043e\u0440\u0447\u0435\u0441\u043a\u0438\u0442\u0435 \u0446\u0435\u043b\u0438 \u0438 \u0434\u0435\u043c\u043e\u043d\u0441\u0442\u0440\u0438\u0440\u0430 \u043f\u0440\u0438\u043b\u0430\u0433\u0430\u043d\u0435\u0442\u043e \u0438\u043c \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 VI \u043a\u043b\u0430\u0441, \u0440\u0430\u0437\u0434\u0435\u043b \u201e\u0421\u0442\u0435\u043f\u0435\u043d\u0443\u0432\u0430\u043d\u0435\u201c.<br \/><em>Keywords:<\/em> creative goals; learning problem; mathematics education<\/p>\n<p>\u00a0<\/p>\n<p><strong>ACHIEVING CREATIVE GOALS IN MATHEMATICS EDUCATION IN 6. GRADE SECTION \u201cEXPONENTS\u201d<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The development of mathematical competences in the process of learning proves to be a difficult task for pedagogical specialists if not enough attention is paid to the formation of an active and creative behavior of students. The present work provides an analysis of the class aims in terms of the creative purposes and demonstrates their application in mathematics education in 6th grade, Section \u201cExponents\u201d.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Ms. Sevdalina Georgieva, Assist. Prof.<br \/>Department for Information, Qualification and Continuous Education \u2013 Varna<br \/>University of Shumen<br \/>Shumen, Bulgaria<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p><strong><a name=\"art05\"><\/a>\u041f\u041e\u041b\u0418\u041d\u041e\u041c\u0418 \u041e\u0422 \u0422\u0420\u0415\u0422\u0410 \u0421\u0422\u0415\u041f\u0415\u041d \u0421 \u041a\u041e\u041b\u0418\u041d\u0415\u0410\u0420\u041d\u0418 \u041a\u041e\u0420\u0415\u041d\u0418<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0438\u0441\u0448\u0435 \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u043f\u043e \u0437\u0430\u0441\u0442\u0440\u0430\u0445\u043e\u0432\u0430\u043d\u0435 \u0438 \u0444\u0438\u043d\u0430\u043d\u0441\u0438 \u2013 \u0421\u043e\u0444\u0438\u044f<br \/>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p class=\"Default\">\u00a0<\/p>\n<p class=\"Pa18\"><strong>Abstract.<\/strong> \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0430 \u0435 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0430 \u0432\u0440\u044a\u0437\u043a\u0430 \u043c\u0435\u0436\u0434\u0443 \u043f\u043e\u043b\u0438\u043d\u043e\u043c \u0441 \u043a\u043e\u043b\u0438\u043d\u0435\u0430\u0440\u043d\u0438 \u043a\u043e\u0440\u0435\u043d\u0438 \u0438 \u043a\u043e\u0440\u0435\u043d\u0438\u0442\u0435 \u043d\u0430 \u043d\u0435\u0433\u043e\u0432\u0430\u0442\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430 \u043f\u043e\u0441\u0440\u0435\u0434\u0441\u0442\u0432\u043e\u043c \u0435\u0434\u043d\u0430 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u043d\u0430 \u0435\u043b\u0438\u043f\u0441\u0430, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0430 \u043e\u0442 \u043a\u043e\u0440\u0435\u043d\u0438\u0442\u0435 \u043d\u0430 \u043f\u043e\u043b\u0438\u043d\u043e\u043c\u0430.<br \/><em>Keywords:<\/em> polynomial of third degree; derivative; equilateral triangle; ellipse; circle; vertices; centroid<\/p>\n<p>\u00a0<\/p>\n<p><strong>POLYNOMIALS OF THIRD DEGREE WITH COLLINEAR ROOTS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>Abstract. It is considered a geometric relation between a polynomial with collinear roots and the roots of its derivative by means of a special ellipse, which is generated by the roots of the polynomial.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Prof. Sava Grozdev, DSc.<br \/>University of Finance, Business and Entrepreneurship<br \/>Sofia, Bulgaria<\/p>\n<p class=\"Default\">\u00a0<\/p>\n<p class=\"Default\">Dr. Veselin Nenkov, Assoc. Prof.<br \/>Beli Osam, Bulgaria<\/p>\n<p class=\"Default\" style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p class=\"Default\" style=\"text-align: right;\">\u00a0<\/p>\n<p class=\"Default\"><strong><a name=\"art06\"><\/a>\u041f\u041e\u041b\u0418\u041d\u041e\u041c\u0418 \u041e\u0422 \u0427\u0415\u0422\u0412\u042a\u0420\u0422\u0410 \u0421\u0422\u0415\u041f\u0415\u041d \u0421 \u041a\u041e\u0420\u0415\u041d\u0418 \u0412\u042a\u0412 \u0412\u042a\u0420\u0425\u041e\u0412\u0415\u0422\u0415 \u041d\u0410 \u0423\u0421\u041f\u041e\u0420\u0415\u0414\u041d\u0418\u041a<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0438\u0441\u0448\u0435 \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u043f\u043e \u0437\u0430\u0441\u0442\u0440\u0430\u0445\u043e\u0432\u0430\u043d\u0435 \u0438 \u0444\u0438\u043d\u0430\u043d\u0441\u0438 \u2013 \u0421\u043e\u0444\u0438\u044f<br \/>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\"><strong>Abstract.<\/strong> \u0418\u0437\u0432\u0435\u0434\u0435\u043d\u0430 \u0435 \u0435\u0434\u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0430 \u0432\u0440\u044a\u0437\u043a\u0430 \u043c\u0435\u0436\u0434\u0443 \u043a\u043e\u0440\u0435\u043d\u0438\u0442\u0435 \u043d\u0430 \u043f\u043e\u043b\u0438\u043d\u043e\u043c \u043e\u0442 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0430 \u0441\u0442\u0435\u043f\u0435\u043d \u043d\u0430 \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u0430 \u043f\u0440\u043e\u043c\u0435\u043d\u043b\u0438\u0432\u0430 \u0441 \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u0438 \u043a\u043e\u0435\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u0438 \u0438 \u043a\u043e\u0440\u0435\u043d\u0438\u0442\u0435 \u043d\u0430 \u043d\u0435\u0433\u043e\u0432\u0430\u0442\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430. \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u0441\u0430 \u043d\u044f\u043a\u043e\u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u043d\u0430 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u0438\u0442\u0435 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438 \u0437\u0430 \u043f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u043d\u0430 \u0440\u0435\u0430\u043b\u043d\u0430 \u043f\u0440\u043e\u043c\u0435\u043d\u043b\u0438\u0432\u0430 \u0441 \u0440\u0435\u0430\u043b\u043d\u0438 \u043a\u043e\u0435\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u0438. \u041d\u0430\u043a\u0440\u0430\u044f \u0435 \u043e\u043f\u0438\u0441\u0430\u043d\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0430 \u0440\u0435\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f \u043d\u0430 \u043f\u043e\u043b\u0438\u043d\u043e\u043c\u0438\u0442\u0435 \u043d\u0430 \u0440\u0435\u0430\u043b\u043d\u0430 \u043f\u0440\u043e\u043c\u0435\u043d\u043b\u0438\u0432\u0430.<br \/><em>Keywords:<\/em> polynomial; roots of polynomial; parallelogram; rhombus; rectangle; ellipse; focus; centre<\/p>\n<p>\u00a0<\/p>\n<p><strong>POLYNOMIALS OF FOURTH DEGREE WITH ROOTS IN THE VERTICES OF A PARALLELOGRAM<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> It is deduced a geometric relation between the roots of a complex polynomial of fourth degree with a complex variable and the roots of its derivatives. Some applications of the obtained results are presented considering real polynomials with a real variable. Finally, a computer realization of the real polynomials is described.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Prof. Sava Grozdev, DSc.<br \/>University of Finance, Business and Entrepreneurship<br \/>Sofia, Bulgaria<\/p>\n<p class=\"Default\">\u00a0<\/p>\n<p class=\"Pa17\">Dr. Veselin Nenkov, Assoc. Prof.<br \/>Beli Osam, Bulgaria<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p><strong><a name=\"art07\"><\/a>\u0427\u0415\u0422\u0418\u0420\u0418\u0414\u0415\u0421\u0415\u0422 \u0418 \u041f\u0415\u0422\u0410 \u041d\u0410\u0426\u0418\u041e\u041d\u0410\u041b\u041d\u0410 \u0421\u0422\u0423\u0414\u0415\u041d\u0422\u0421\u041a\u0410 \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u0410 \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa27\">\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0438\u0441\u0448\u0435 \u0443\u0447\u0438\u043b\u0438\u0449\u0435 \u043f\u043e \u0437\u0430\u0441\u0442\u0440\u0430\u0445\u043e\u0432\u0430\u043d\u0435 \u0438 \u0444\u0438\u043d\u0430\u043d\u0441\u0438 \u2013 \u0421\u043e\u0444\u0438\u044f<br \/>\u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432, \u0418\u043a\u043e\u043d\u043e\u043c\u0438\u0447\u0435\u0441\u043a\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u2013 \u0412\u0430\u0440\u043d\u0430<br \/>\u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432\u0430 \u0421\u0442\u043e\u0438\u043b\u043e\u0432\u0430, \u0423\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u043f\u043e \u0430\u0440\u0445\u0438\u0442\u0435\u043a\u0442\u0443\u0440\u0430, \u0441\u0442\u0440\u043e\u0438\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0438 \u0433\u0435\u043e\u0434\u0435\u0437\u0438\u044f \u2013 \u0421\u043e\u0444\u0438\u044f<br \/>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\"><strong>Abstract.\u00a0<\/strong>\u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u0442\u0435\u043c\u0430\u0442\u0430 \u0437\u0430 \u0433\u0440\u0443\u043f\u0430 \u0412 \u043d\u0430 \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430\u0442\u0430 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0441\u043a\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u0440\u0435\u0437 2018 \u0433. \u0412 \u0442\u0430\u0437\u0438 \u0433\u0440\u0443\u043f\u0430 \u0441\u0435 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u0432\u0430\u0442 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u0438, \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u0449\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u043f\u044a\u0440\u0432\u0438 \u043a\u0443\u0440\u0441 \u043d\u0430 \u0432\u0438\u0441\u0448\u0438\u0442\u0435 \u0443\u0447\u0438\u043b\u0438\u0449\u0430 \u0432 \u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f \u0441 \u0445\u043e\u0440\u0430\u0440\u0438\u0443\u043c, \u043d\u0435 \u043f\u043e-\u0433\u043e\u043b\u044f\u043c \u043e\u0442 240 \u0447\u0430\u0441\u0430. \u0412\u043a\u043b\u044e\u0447\u0435\u043d\u0438 \u0441\u0430 \u0438 \u043a\u0440\u0438\u0442\u0435\u0440\u0438\u0438\u0442\u0435 \u0437\u0430 \u043e\u0446\u0435\u043d\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435.<br \/><em>Keywords:<\/em> Olympiad; university student; problem solving; assessment<\/p>\n<p>\u00a0<\/p>\n<p><strong>FORTY FIFTH NATIONAL OLYMPIAD IN MATHEMATICS FOR UNIVERSITY STUDENTS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> Methodological solutions are proposed of the problems from the paper for Group B of the Forty fifth national Olympiad in Mathematics for University students in 2018. The contestants in this group are students who study Mathematics no more than 240 classes during the first year in the Bulgarian Universities.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Prof. Sava Grozdev, DSc.<br \/>University of Finance, Business and Entrepreneurship<br \/>Sofia, Bulgaria<\/p>\n<p class=\"Default\">\u00a0<\/p>\n<p class=\"Pa17\">Prof. Dr. Rosen Nikolaev<br \/>University of Economics \u2013 Varna (Bulgaria)<\/p>\n<p class=\"Default\">\u00a0<\/p>\n<p class=\"Pa17\">Dr. Stanislava Stoilova, Assoc. Prof.<br \/>University of Architecture, Civil Engineering and Geodesy<br \/>Sofia, Bulgaria<\/p>\n<p class=\"Default\">\u00a0<\/p>\n<p class=\"Pa17\">Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n<p class=\"Pa27\">Beli Osam, Bulgaria<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p>\u00a0<a name=\"art08\"><\/a><strong>TWO NEW PHD DISSERTATIONS IN METHODOLOGY OF INFORMATICS TEACHING<\/strong><\/p>\n<p>\u00a0\u00a0<\/p>\n<p class=\"Pa18\"><strong>Abstract.<\/strong> Annotations of two new dissertations of PhD students are presented. Both of them are in the domain of methodology of Informatics education and are defended successfully in Plovdiv University \u201cP. Hilendarski\u201d.<br \/><em>Keywords:<\/em> PhD student; dissertation; methodology; Informatics; education; teaching<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Dr. Petya Asenova, Assoc. Prof.<br \/>New Bulgarian University<br \/>Sofia, Bulgaria<\/p>\n<p style=\"text-align: right;\" align=\"left\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\" align=\"left\">\u00a0<\/p>","protected":false},"excerpt":{"rendered":"<p>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \u00a0 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 \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u043b\u0438\u043d\u0435\u0435\u043d \u0442\u0440\u0435\u043d\u0434 \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f [Using a Trendline to Solve Rational Equations] \/ \u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432 \/ Plamen Penev \u2013 \u0441\u0442\u0440. 223 \u00a0 \u0421\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0438\u0433\u0440\u0438 \u0432 \u0447\u0430\u0441\u043e\u0432\u0435\u0442\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0447\u0440\u0435\u0437 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u043e\u0440 \u043d\u0430 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u0438 \u0447\u0438\u0441\u043b\u0430 [Creating of Games in the Informatics [&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":[3614],"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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018 - \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:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/\" \/>\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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018 - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \u00a0 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 \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u043b\u0438\u043d\u0435\u0435\u043d \u0442\u0440\u0435\u043d\u0434 \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f [Using a Trendline to Solve Rational Equations] \/ \u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432 \/ Plamen Penev \u2013 \u0441\u0442\u0440. 223 \u00a0 \u0421\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0438\u0433\u0440\u0438 \u0432 \u0447\u0430\u0441\u043e\u0432\u0435\u0442\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0447\u0440\u0435\u0437 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u043e\u0440 \u043d\u0430 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u0438 \u0447\u0438\u0441\u043b\u0430 [Creating of Games in the Informatics [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/\" \/>\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=\"2018-06-27T05:49:13+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-06-24T14:33:14+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/\"},\"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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018\",\"datePublished\":\"2018-06-27T05:49:13+00:00\",\"dateModified\":\"2019-06-24T14:33:14+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/\"},\"wordCount\":1784,\"publisher\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#organization\"},\"articleSection\":[\"\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\", 2018 \u0433.\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/\",\"url\":\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/\",\"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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018 - \u0410\u0437-\u0431\u0443\u043a\u0438\",\"isPartOf\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#website\"},\"datePublished\":\"2018-06-27T05:49:13+00:00\",\"dateModified\":\"2019-06-24T14:33:14+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/strategies.azbuki.bg\/\"},{\"@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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#website\",\"url\":\"https:\/\/vocedu.azbuki.bg\/en\/\",\"name\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\",\"description\":\"\u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e \u0438\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0437\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u043d\u0430\u0443\u043a\u0430\",\"publisher\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/vocedu.azbuki.bg\/en\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#organization\",\"name\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\",\"url\":\"https:\/\/vocedu.azbuki.bg\/en\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png\",\"contentUrl\":\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png\",\"width\":250,\"height\":70,\"caption\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\"},\"image\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/Azbuki55\/\",\"https:\/\/www.instagram.com\/nion_azbuki\/\"]},{\"@type\":\"Person\",\"@id\":\"\",\"url\":\"https:\/\/newspaper.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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018 - \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:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/","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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018 - \u0410\u0437-\u0431\u0443\u043a\u0438","og_description":"CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \u00a0 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 \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u043b\u0438\u043d\u0435\u0435\u043d \u0442\u0440\u0435\u043d\u0434 \u043f\u0440\u0438 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0440\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f [Using a Trendline to Solve Rational Equations] \/ \u041f\u043b\u0430\u043c\u0435\u043d \u041f\u0435\u043d\u0435\u0432 \/ Plamen Penev \u2013 \u0441\u0442\u0440. 223 \u00a0 \u0421\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0438\u0433\u0440\u0438 \u0432 \u0447\u0430\u0441\u043e\u0432\u0435\u0442\u0435 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0447\u0440\u0435\u0437 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0435\u043d\u0435\u0440\u0430\u0442\u043e\u0440 \u043d\u0430 \u0441\u043b\u0443\u0447\u0430\u0439\u043d\u0438 \u0447\u0438\u0441\u043b\u0430 [Creating of Games in the Informatics [&hellip;]","og_url":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2018-06-27T05:49:13+00:00","article_modified_time":"2019-06-24T14:33:14+00:00","og_image":[{"url":"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png"}],"twitter_card":"summary_large_image","twitter_misc":{"Written by":"","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/#article","isPartOf":{"@id":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/"},"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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018","datePublished":"2018-06-27T05:49:13+00:00","dateModified":"2019-06-24T14:33:14+00:00","mainEntityOfPage":{"@id":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/"},"wordCount":1784,"publisher":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#organization"},"articleSection":["\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", 2018 \u0433."],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/","url":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/","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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018 - \u0410\u0437-\u0431\u0443\u043a\u0438","isPartOf":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#website"},"datePublished":"2018-06-27T05:49:13+00:00","dateModified":"2019-06-24T14:33:14+00:00","breadcrumb":{"@id":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/philosophy.azbuki.bg\/en\/matematics\/matharticles2016-3\/3-lxi-2018\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/strategies.azbuki.bg\/"},{"@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 3, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018"}]},{"@type":"WebSite","@id":"https:\/\/vocedu.azbuki.bg\/en\/#website","url":"https:\/\/vocedu.azbuki.bg\/en\/","name":"\u0410\u0437-\u0431\u0443\u043a\u0438","description":"\u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e \u0438\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0437\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u043d\u0430\u0443\u043a\u0430","publisher":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/vocedu.azbuki.bg\/en\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/vocedu.azbuki.bg\/en\/#organization","name":"\u0410\u0437-\u0431\u0443\u043a\u0438","url":"https:\/\/vocedu.azbuki.bg\/en\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/","url":"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png","contentUrl":"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png","width":250,"height":70,"caption":"\u0410\u0437-\u0431\u0443\u043a\u0438"},"image":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/Azbuki55\/","https:\/\/www.instagram.com\/nion_azbuki\/"]},{"@type":"Person","@id":"","url":"https:\/\/newspaper.azbuki.bg\/en\/author\/"}]}},"_links":{"self":[{"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/42773"}],"collection":[{"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/comments?post=42773"}],"version-history":[{"count":11,"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/42773\/revisions"}],"predecessor-version":[{"id":60523,"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/42773\/revisions\/60523"}],"wp:attachment":[{"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/media?parent=42773"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/categories?post=42773"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/newspaper.azbuki.bg\/en\/wp-json\/wp\/v2\/tags?post=42773"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}