{"id":42957,"date":"2018-09-05T12:45:08","date_gmt":"2018-09-05T09:45:08","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles2016-3\/4-lxi-2018\/"},"modified":"2019-06-24T17:33:14","modified_gmt":"2019-06-24T14:33:14","slug":"4-lxi-2018","status":"publish","type":"post","link":"https:\/\/science.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-lxi-2018\/","title":{"rendered":"Mathematics and Informatics, Number 4\/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><strong>\u00a0<\/strong><\/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><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art01\">\u0417\u0430 \u043f\u0440\u043e\u0441\u0442\u0438\u0442\u0435 \u0447\u0438\u0441\u043b\u0430 <\/a><br \/><a href=\"#art01\">[About the Prime Numbers] <\/a><br \/>\/ <em>Sava Grozdev, Veselin Nenkov<\/em><em> \/ Sava Grozdev, Veselin Nenkov\u00a0 <\/em>\u2013 \u0441\u0442\u0440. 327<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art02\">\u0418\u043d\u0446\u0435\u043d\u0442\u044a\u0440 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a <\/a><br \/><a href=\"#art02\">[Incenter of a Quadrilateral] <\/a><br \/>\/ <em>\u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432<\/em><em> \/ Stanoslav Stefanov <\/em>\u2013 \u0441\u0442\u0440. 338<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art03\">\u041f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u0441 \u043a\u0440\u0430\u0442\u043d\u0438 \u043a\u043e\u0440\u0435\u043d\u0438 \u0432\u044a\u0432 \u0432\u044a\u0440\u0445\u043e\u0432\u0435\u0442\u0435 \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a <\/a><br \/><a href=\"#art03\">[Polynomials with Multiple Roots in the Vertices of a Triangle] <\/a><br \/>\/ <em>Sava Grozdev, Veselin Nenkov<\/em><em> \/ Sava Grozdev, Veselin Nenkov <\/em>\u2013 page 352<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/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><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art04\">\u042d\u043f\u0438\u0446\u0438\u043a\u043b\u043e\u0438\u0434\u0430 <\/a><br \/><a href=\"#art04\">[Epicycloid] <\/a><br \/>\/ <em>\u0410\u0441\u043a\u0430\u0440 \u0418\u043d\u043a\u0430\u0440, \u0421\u0430\u0440\u0441\u0435\u043c\u0431\u0430\u0435\u0432\u0430<\/em><em>\u041a\u0430\u043c\u0438\u043b\u0430<\/em><em> \/ Askar Inkar, Sarsenbayeva Kamila <\/em>\u2013 \u0441\u0442\u0440. 360<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\"><a href=\"#art05\">\u0413\u0438\u043f\u043e\u0446\u0438\u043a\u043b\u043e\u0438\u0434\u0430 <\/a><br \/><a href=\"#art05\">[Hypocycloid] <\/a><br \/>\/ <em>\u0411\u043e\u0440\u0438\u0441\u043b\u0430\u0432 \u0411\u043e\u0440\u0438\u0441\u043e\u0432, \u0414\u0435\u044f\u043d \u0414\u0438\u043c\u0438\u0442\u0440\u043e\u0432, \u0418\u0432\u0430\u043d \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432, \u041d\u0438\u043a\u043e\u043b\u0430\u0439 \u041d\u0438\u043d\u043e\u0432, \u0422\u0435\u043e\u0434\u043e\u0440 \u0425\u0440\u0438\u0441\u0442\u043e\u0432 \/ Borislav Borisov, Deyan Dimitrov, Ivan Stefanov, Nikolay Ninov, <\/em><em>Teodor <\/em><em>Hristov <\/em>\u2013 \u0441\u0442\u0440. 368<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"Pa17\">\u00a0<\/p>\n<p><a href=\"#art06\">\u041c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u0430 \u043e\u0442 \u0442\u043e\u0447\u043a\u0438, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u043e\u0442 \u0434\u0432\u043e\u0439\u043a\u0438 \u0440\u0430\u0432\u043d\u043e\u0431\u0435\u0434\u0440\u0435\u043d\u0438 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438 \u0441\u044a\u0441 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u043d\u043e \u0440\u0430\u0437\u043f\u043e\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0438\u0442\u0435 <\/a><br \/><a href=\"#art06\">[Sets of Points, Generated by a Pair of Isosceles Triangles with a Special Location of their Bases] <\/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 \/ <\/em><em>Sava<\/em><em>Grozdev<\/em><em>, <\/em><em>Veselin<\/em><em>Nenkov<\/em>\u2013 \u0441\u0442\u0440. 378<br \/><a href=\"https:\/\/science.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 \u043d\u043e\u0432\u0438 \u0434\u043e\u043a\u0442\u043e\u0440\u0441\u043a\u0438 \u0434\u0438\u0441\u0435\u0440\u0442\u0430\u0446\u0438\u0438 \u043f\u043e \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 <\/a><br \/><a href=\"#art07\">[Four New PhD Dissertations In Methodology of Mathematics and Informatics Teaching]<\/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 \/ <\/em><em>Sava<\/em><em>Grozdev<\/em><em>, <\/em><em>Veselin<\/em><em>Nenkov<\/em>\u2013 page 396<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><strong>CONTEST PROBLEMS \/ \u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong><\/p>\n<p>Contest Problems of this Issue <br \/>[Contest Problems of Issue] \u2013 \u0441\u0442\u0440. 403<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/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 5, 2017 <br \/>[Solutions of the Contest Problems from issue 5, 2017] \u2013 \u0441\u0442\u0440. 404<br \/><a href=\"https:\/\/science.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\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. 411<\/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 page 413<\/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<\/p>\n<p><strong><a name=\"art01\"><\/a>\u0417\u0410 \u041f\u0420\u041e\u0421\u0422\u0418\u0422\u0415 \u0427\u0418\u0421\u041b\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, <br \/>University of Finance, Business and Entrepreneurship<\/p>\n<p class=\"Pa17\">\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u0441\u0430 \u043d\u044f\u043a\u043e\u0438 \u043e\u0442 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0445\u0438\u043f\u043e\u0442\u0435\u0437\u0438 \u0437\u0430 \u043f\u0440\u043e\u0441\u0442\u0438\u0442\u0435 \u0447\u0438\u0441\u043b\u0430, \u043a\u043e\u0438\u0442\u043e \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u0442 \u0442\u0435\u0445\u043d\u0438 \u0432\u0430\u0436\u043d\u0438 \u0437\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430\u0442\u0430 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430.<\/p>\n<p><em>Keywords:<\/em> prime number; perfect number; Mersenne prime number; twin prime numbers; hypothesis<\/p>\n<p>\u00a0<\/p>\n<p><strong>ABOUT THE PRIME NUMBERS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> Some of the basic hypotheses about the prime numbers are considered, which determine their important properties for Mathematics.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Prof. Sava Grozdev, DSc.<\/p>\n<p class=\"Pa27\">University of Finance, Business and Entrepreneurship<\/p>\n<p class=\"Pa27\">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>\u00a0<\/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<\/p>\n<p><strong><a name=\"art02\"><\/a>\u0418\u041d\u0426\u0415\u041d\u0422\u042a\u0420 \u041d\u0410 \u0427\u0415\u0422\u0418\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>\u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432<\/p>\n<p>Technical university \u2013 Sofia<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0412 \u0442\u0430\u0437\u0438 \u043f\u0443\u0431\u043b\u0438\u043a\u0430\u0446\u0438\u044f \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430\u0442\u0430 \u043d\u0430 \u0435\u0434\u043d\u0430 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0430 \u0442\u043e\u0447\u043a\u0430 \u0432 \u0438\u0437\u043f\u044a\u043a\u043d\u0430\u043b \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a, \u043a\u043e\u044f\u0442\u043e \u0435 \u0435\u0441\u0442\u0435\u0441\u0442\u0432\u0435\u043d\u043e \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u0435 \u043d\u0430 \u0446\u0435\u043d\u0442\u044a\u0440\u0430 \u043d\u0430 \u0432\u043f\u0438\u0441\u0430\u043d\u0430\u0442\u0430 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442 \u043d\u0430 \u043e\u043f\u0438\u0441\u0430\u043d\u0438\u044f \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a. \u0418\u0437\u0441\u043b\u0435\u0434\u0432\u0430 \u0441\u0435 \u0432\u0440\u044a\u0437\u043a\u0430\u0442\u0430 \u043d\u0430 \u0442\u0430\u0437\u0438 \u0442\u043e\u0447\u043a\u0430 \u0441 \u0435\u0434\u043d\u043e \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0430\u043b\u043d\u043e \u043f\u0440\u0435\u043e\u0431\u0440\u0430\u0437\u0443\u0432\u0430\u043d\u0438\u0435 \u0432 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430, \u043a\u0430\u043a\u0442\u043e \u0438 \u0441 \u0432\u0435\u0447\u0435 \u0438\u0437\u0443\u0447\u0435\u043d\u0430 \u0434\u0440\u0443\u0433\u0430 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0430 \u0442\u043e\u0447\u043a\u0430 \u0432 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430.<\/p>\n<p><em>Keywords:<\/em> quadrilateral; notable point; transformation<\/p>\n<p>\u00a0<\/p>\n<p><strong>INCENTER OF A QUADRILATERAL<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The paper describes some properties of a notable point in a convex quadrilateral, which naturally generalize the incenter of the outscribed quadrilateral. What are considered are the connections of this point with a transformation in the plane of the quadrilateral and the connections with another notable point.<\/p>\n<p>\u00a0<\/p>\n<p>Mr. Stanislav Stefanov, PhD student<\/p>\n<p>Technical University Sofia<\/p>\n<p>Sofia, 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<\/p>\n<p><strong><a name=\"art03\"><\/a>\u041f\u041e\u041b\u0418\u041d\u041e\u041c\u0418 \u0421 \u041a\u0420\u0410\u0422\u041d\u0418 \u041a\u041e\u0420\u0415\u041d\u0418 \u0412\u042a\u0412 \u0412\u042a\u0420\u0425\u041e\u0412\u0415\u0422\u0415 \u041d\u0410 \u0422\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/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<\/p>\n<p class=\"Pa17\">\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\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 \u0438 \u0442\u0435\u0437\u0438 \u043d\u0430 \u043d\u0435\u0433\u043e\u0432\u0430\u0442\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0430, \u043a\u043e\u0433\u0430\u0442\u043e \u0432\u0441\u0438\u0447\u043a\u0438 \u043a\u043e\u0440\u0435\u043d\u0438 \u043d\u0430 \u043f\u043e\u043b\u0438\u043d\u043e\u043c\u0430 \u0441\u0435 \u043d\u0430\u043c\u0438\u0440\u0430\u0442 \u0432 \u0442\u0440\u0438 \u043d\u0435\u043a\u043e\u043b\u0438\u043d\u0435\u0430\u0440\u043d\u0438 \u0442\u043e\u0447\u043a\u0438. \u0422\u0430\u0437\u0438 \u0432\u0440\u044a\u0437\u043a\u0430 \u0441\u0435 \u043e\u0441\u044a\u0449\u0435\u0441\u0442\u0432\u044f\u0432\u0430 \u0447\u0440\u0435\u0437 \u0444\u043e\u043a\u0443\u0441\u0438\u0442\u0435 \u043d\u0430 \u0441\u043f\u0435\u0446\u0438\u0430\u043b\u043d\u0430 \u0435\u043b\u0438\u043f\u0441\u0430, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0430 \u043e\u0442 \u043a\u0440\u0430\u0442\u043d\u043e\u0441\u0442\u0438\u0442\u0435 \u043d\u0430 \u043a\u043e\u0440\u0435\u043d\u0438\u0442\u0435 \u043d\u0430 \u043f\u043e\u043b\u0438\u043d\u043e\u043c\u0430.<\/p>\n<p><em>Keywords:<\/em> polynomial; root; triangle; ellipse; focus<\/p>\n<p>\u00a0<\/p>\n<p><strong>POLYNOMIALS WITH MULTIPLE ROOTS IN THE VERTICES OF A TRIANGLE<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>Abstract.<\/strong> It is considered a geometric relation between the roots of a polynomial and the roots of its derivative, when all roots of the polynomial are in three noncollinear points. This relation is realized through the focuses of a special ellipse, which is determined by the multiplicities of the polynomial roots.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Prof. Sava Grozdev, DSc.<\/p>\n<p class=\"Pa27\">University of Finance, Business and Entrepreneurship<\/p>\n<p class=\"Pa27\">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><strong><a name=\"art04\"><\/a>\u042d\u041f\u0418\u0426\u0418\u041a\u041b\u041e\u0418\u0414\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>\u0418\u043d\u043a\u0430\u0440 \u0410\u0441\u043a\u0430\u0440<\/p>\n<p>\u041a\u0430\u043c\u0438\u043b\u0430 \u0421\u0430\u0440\u0441\u0435\u043c\u0431\u0430\u0435\u0432\u0430<\/p>\n<p>\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 \u2013 \u0433. \u0410\u043a\u0442\u0430\u0443<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0412 \u0434\u0430\u043d\u043d\u043e\u0439 \u0441\u0442\u0430\u0442\u044c\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u044b \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f \u044d\u043f\u0438\u0446\u0438\u043a\u043b\u043e\u0438\u0434\u044b, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u0443\u0447\u0430\u0449\u0438\u043c\u0438\u0441\u044f \u0438\u0437 \u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d\u0430 \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0433\u043e \u0441\u0435\u0442\u0435\u0432\u043e\u0433\u043e \u043f\u0440\u043e\u0435\u043a\u0442\u0430 \u201e\u042d\u043d\u0446\u0438\u043a\u043b\u043e\u043f\u0435\u0434\u0438\u044f \u0437\u0430\u043c\u0435\u0447\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u043f\u043b\u043e\u0441\u043a\u0438\u0445 \u043a\u0440\u0438\u0432\u044b\u0445: \u043f\u0438\u0448\u0435\u043c \u0441\u0430\u043c\u0438\u201c. \u0412 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f\u0445 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u0438\u0441\u044c \u043c\u0435\u0442\u043e\u0434\u044b \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 \u0438 \u043a\u043e\u043c\u043f\u044c\u044e\u0442\u0435\u0440\u043d\u044b\u0435 \u044d\u043a\u0441\u043f\u0435\u0440\u0438\u043c\u0435\u043d\u0442\u044b, \u043a\u043e\u0442\u043e\u0440\u044b\u0435 \u043f\u0440\u043e\u0432\u043e\u0434\u0438\u043b\u0438\u0441\u044c \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 GeoGebra \u0438\u043b\u0438 The Geometer\u2019s Sketchpad. \u0414\u043b\u044f \u043e\u0440\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u0438 \u0441\u0435\u0442\u0435\u0432\u043e\u0433\u043e \u0432\u0437\u0430\u0438\u043c\u043e\u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u043a\u043e\u0432 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u0438\u0441\u044c \u043e\u0431\u043b\u0430\u0447\u043d\u044b\u0435 \u0441\u0435\u0440\u0432\u0438\u0441\u044b Google.<\/p>\n<p><em>Keywords:<\/em> circle; curve; trajectory; \u0435\u0440\u0456cycloid<\/p>\n<p>\u00a0<\/p>\n<p><strong>EPICYCLOID<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>\u0410bstract.<\/strong> The paper presents results on the epicycloid of Kazakhstan students in the frames of the net research project \u201cEncyclopedia of notable plane figures: we work by ourselves\u201d. What are used in the investigation are analytical geometry methods and experiments by computer applying the software products GeoGebra and The Geometer\u2019s Sketchpad. The Google Cloud Service is used in the organization of the interaction between the participants.<\/p>\n<p>\u00a0<\/p>\n<p>Ms. Inkar Askar, Student<\/p>\n<p>Ms. Kamila Sarsenbayeva, Student<\/p>\n<p>Regional Special Boarding School for Gifted Children with Profound Studying of Different Subjects<\/p>\n<p>Aktau, Kazakhstan<\/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=\"art05\"><\/a><strong>\u0413\u0418\u041f\u041e\u0426\u0418\u041a\u041b\u041e\u0418\u0414\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>\u0411\u043e\u0440\u0438\u0441\u043b\u0430\u0432 \u0411\u043e\u0440\u0438\u0441\u043e\u0432, \u0414\u0435\u044f\u043d \u0414\u0438\u043c\u0438\u0442\u0440\u043e\u0432, \u0418\u0432\u0430\u043d \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432, \u041d\u0438\u043a\u043e\u043b\u0430\u0439 \u041d\u0438\u043d\u043e\u0432, \u0422\u0435\u043e\u0434\u043e\u0440 \u0425\u0440\u0438\u0441\u0442\u043e\u0432<\/p>\n<p>\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 \u2013 \u041b\u043e\u0432\u0435\u0447<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u044c\u0435 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u043b\u0435\u043d\u044b \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u0440\u0430\u0431\u043e\u0442\u044b \u0411\u043e\u043b\u0433\u0430\u0440\u0441\u043a\u043e\u0439 \u043f\u043e\u0434\u043a\u043e\u043c\u0430\u043d\u0434\u044b \u2013 \u0447\u0430\u0441\u0442\u0438 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0439 \u043a\u043e\u043c\u0430\u043d\u0434\u044b \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f. \u042d\u0442\u0430 \u043a\u043e\u043c\u0430\u043d\u0434\u0430 \u0431\u044b\u043b\u0430 \u0441\u043e\u0437\u0434\u0430\u043d\u0430 \u0434\u043b\u044f \u0440\u0435\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u0438 \u0441\u0435\u0442\u0435\u0432\u043e\u0433\u043e \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 \u201e\u042d\u043d\u0446\u0438\u043a\u043b\u043e\u043f\u0435\u0434\u0438\u044f \u0437\u0430\u043c\u0435\u0447\u0430\u0442\u0435\u043b\u044c\u043d\u044b\u0445 \u043f\u043b\u043e\u0441\u043a\u0438\u0445 \u043a\u0440\u0438\u0432\u044b\u0445: \u043f\u0438\u0448\u0435\u043c \u0441\u0430\u043c\u0438\u201c. \u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0435 \u043f\u0440\u043e\u0432\u043e\u0434\u0438\u043b\u043e\u0441\u044c \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u043d\u044b\u0445 \u043f\u0440\u043e\u0434\u0443\u043a\u0442\u043e\u0432 GeoGebra, Geometer\u2019s Sketchpad \u0438 Maple. \u0414\u043b\u044f \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u044c\u0441\u0442\u0432\u0430 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0445 \u0433\u0438\u043f\u043e\u0442\u0435\u0437 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u0441\u044f \u043c\u0435\u0442\u043e\u0434 \u043a\u043e\u043c\u043f\u043b\u0435\u043a\u0441\u043d\u044b\u0445 \u0447\u0438\u0441\u0435\u043b. \u0414\u043b\u044f \u043e\u0440\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u0438 \u0441\u0435\u0442\u0435\u0432\u043e\u0433\u043e \u0432\u0437\u0430\u0438\u043c\u043e\u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u043a\u043e\u0432 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043b\u0438\u0441\u044c \u043e\u0431\u043b\u0430\u0447\u043d\u044b\u0435 \u0441\u0435\u0440\u0432\u0438\u0441\u044b Google.<\/p>\n<p><em>Keywords:<\/em> circle; curve; trajectory; hypocycloid<\/p>\n<p>\u00a0<\/p>\n<p><strong>HYPOCYCLOID<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The paper presents results of the Bulgarian sub-team \u2013 a part of an international team of students. The team was created for the realization of the net research project \u201cEncyclopedia of notable plane figures: we work by ourselves\u201d. The investigation is carried out using the software products GeoGebra, Geometer\u2019s Sketchpad and Maple. The method of the complex numbers is applied in the proofs. The organization of the net interaction is based on Google Cloud Service.<\/p>\n<p>\u00a0<\/p>\n<p>Mr. Borislav Borisov, Student, Mr. Deyan Dimitrov, Student, Mr. Ivan Stefanov, Student<\/p>\n<p>Mr. Nikolay Ninov, Student, Mr. Teodor Hristov, Student<\/p>\n<p>Mathematics High School \u2013 Lovech<\/p>\n<p>Lovech, 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<\/p>\n<p><strong><a name=\"art06\"><\/a>\u041c\u041d\u041e\u0416\u0415\u0421\u0422\u0412\u0410 \u041e\u0422 \u0422\u041e\u0427\u041a\u0418, \u041f\u041e\u0420\u041e\u0414\u0415\u041d\u0418 \u041e\u0422 \u0414\u0412\u041e\u0419\u041a\u0418 \u0420\u0410\u0412\u041d\u041e\u0411\u0415\u0414\u0420\u0415\u041d\u0418 \u0422\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u0426\u0418 \u0421\u042a\u0421 \u0421\u041f\u0415\u0426\u0418\u0410\u041b\u041d\u041e \u0420\u0410\u0417\u041f\u041e\u041b\u041e\u0416\u0415\u041d\u0418\u0415 \u041d\u0410 \u041e\u0421\u041d\u041e\u0412\u0418\u0422\u0415<\/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<\/p>\n<p class=\"Pa17\">\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430: B \u0435 \u0442\u043e\u0447\u043a\u0430 \u043e\u0442 \u043e\u0442\u0441\u0435\u0447\u043a\u0430\u0442\u0430 AC, \u0430 X \u0438 Y \u0441\u0430 \u0442\u043e\u0447\u043a\u0438, \u043b\u0435\u0436\u0430\u0449\u0438 \u0432 \u0435\u0434\u043d\u0430 \u0438 \u0441\u044a\u0449\u0430 \u043f\u043e\u043b\u0443\u0440\u0430\u0432\u043d\u0438\u043d\u0430 \u043e\u0442\u043d\u043e\u0441\u043d\u043e \u0441\u0442\u0440\u0430\u043d\u0430\u0442\u0430 AC \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a ABC, \u0442\u0430\u043a\u0430 \u0447\u0435 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438\u0442\u0435 ABX \u0438 BCY \u0441\u0430 \u0440\u0430\u0432\u043d\u043e\u0441\u0442\u0440\u0430\u043d\u043d\u0438. \u0414\u0430 \u0441\u0435 \u0434\u043e\u043a\u0430\u0436\u0435 \u0438\u043b\u0438 \u043e\u043f\u0440\u043e\u0432\u0435\u0440\u0433\u0430\u0435 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e\u0442\u043e AY = CX.<\/p>\n<p><em>Keywords:<\/em> triangle; co-ordinate system; curve of second degree; polarity<\/p>\n<p>\u00a0<\/p>\n<p><strong>SETS OF POINTS, GENERATED BY A PAIR OF ISOSCELES TRIANGLES WITH A SPECIAL LOCATION OF THEIR BASES<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>Abstract.<\/strong> The paper is dedicated to some generalizations of the problem: B is a point on the segment AC, while X and Y are points in one and the same half-plane with respect to the side AC of the triangle ABC. When the triangles ABX and BCY are equilateral, prove or disprove the equality AY = CX.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Prof. Sava Grozdev, DSc.<\/p>\n<p class=\"Pa27\">University of Finance, Business and Entrepreneurship<\/p>\n<p class=\"Pa27\">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><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong><a name=\"art07\"><\/a>\u0427\u0415\u0422\u0418\u0420\u0418 \u041d\u041e\u0412\u0418 \u0414\u041e\u041a\u0422\u041e\u0420\u0421\u041a\u0418 \u0414\u0418\u0421\u0415\u0420\u0422\u0410\u0426\u0418\u0418 \u041f\u041e \u041c\u0415\u0422\u041e\u0414\u0418\u041a\u0410 \u041d\u0410 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415\u0422\u041e \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410 \u0418 \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0422\u0418\u041a\u0410<\/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<\/p>\n<p class=\"Pa17\">\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0441\u0430 \u0430\u043d\u043e\u0442\u0430\u0446\u0438\u0438 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438 \u043d\u043e\u0432\u0438 \u0434\u043e\u043a\u0442\u043e\u0440\u0441\u043a\u0438 \u0434\u0438\u0441\u0435\u0440\u0442\u0430\u0446\u0438\u0438. \u0422\u0440\u0438 \u043e\u0442 \u0442\u044f\u0445 \u0441\u0430 \u0432 \u043e\u0431\u043b\u0430\u0441\u0442\u0442\u0430 \u043d\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430\u0442\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0441\u0430 \u0443\u0441\u043f\u0435\u0448\u043d\u043e \u0437\u0430\u0449\u0438\u0442\u0435\u043d\u0438 \u0432 \u041f\u043b\u043e\u0432\u0434\u0438\u0432\u0441\u043a\u0438\u044f \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u201e\u041f. \u0425\u0438\u043b\u0435\u043d\u0434\u0430\u0440\u0441\u043a\u0438\u201c. \u0427\u0435\u0442\u0432\u044a\u0440\u0442\u0430\u0442\u0430 \u0435 \u0432 \u043e\u0431\u043b\u0430\u0441\u0442\u0442\u0430 \u043d\u0430 \u043c\u0435\u0442\u043e\u0434\u0438\u043a\u0430\u0442\u0430 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0435 \u0443\u0441\u043f\u0435\u0448\u043d\u043e \u0437\u0430\u0449\u0438\u0442\u0435\u043d\u0430 \u0432 \u042e\u0433\u043e\u0437\u0430\u043f\u0430\u0434\u043d\u0438\u044f \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u201e\u041d. \u0420\u0438\u043b\u0441\u043a\u0438\u201c.<\/p>\n<p><em>Keywords:<\/em> PhD student; dissertation; methodology; Mathematics; Informatics; education; teaching<\/p>\n<p>\u00a0<\/p>\n<p><strong>FOUR NEW PHD DISSERTATIONS IN METHODOLOGY OF MATHEMATICS AND INFORMATICS TEACHING<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>Abstract.<\/strong> Annotations of four new dissertations of PhD students are presented. Three of them are in the domain of methodology of Informatics education and are defended successfully in Plovdiv University \u201cP. Hilendarski\u201d. The forth one is in the domain of methodology of Mathematics education and is defended successfully in Southwest University \u201cN. Rilski\u201d.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa17\">Prof. Sava Grozdev, DSc.<\/p>\n<p class=\"Pa27\">University of Finance, Business and Entrepreneurship<\/p>\n<p class=\"Pa27\">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>","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 \u00a0 \u0417\u0430 \u043f\u0440\u043e\u0441\u0442\u0438\u0442\u0435 \u0447\u0438\u0441\u043b\u0430 [About the Prime Numbers] \/ \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\u00a0 \u2013 \u0441\u0442\u0440. 327 \u00a0 \u0418\u043d\u0446\u0435\u043d\u0442\u044a\u0440 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a [Incenter of a Quadrilateral] \/ \u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432 \/ Stanoslav Stefanov \u2013 \u0441\u0442\u0440. 338 \u00a0 \u041f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u0441 \u043a\u0440\u0430\u0442\u043d\u0438 \u043a\u043e\u0440\u0435\u043d\u0438 \u0432\u044a\u0432 \u0432\u044a\u0440\u0445\u043e\u0432\u0435\u0442\u0435 \u043d\u0430 [&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 4, \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:\/\/press.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-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 4, \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 \u00a0 \u0417\u0430 \u043f\u0440\u043e\u0441\u0442\u0438\u0442\u0435 \u0447\u0438\u0441\u043b\u0430 [About the Prime Numbers] \/ \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\u00a0 \u2013 \u0441\u0442\u0440. 327 \u00a0 \u0418\u043d\u0446\u0435\u043d\u0442\u044a\u0440 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a [Incenter of a Quadrilateral] \/ \u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432 \/ Stanoslav Stefanov \u2013 \u0441\u0442\u0440. 338 \u00a0 \u041f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u0441 \u043a\u0440\u0430\u0442\u043d\u0438 \u043a\u043e\u0440\u0435\u043d\u0438 \u0432\u044a\u0432 \u0432\u044a\u0440\u0445\u043e\u0432\u0435\u0442\u0435 \u043d\u0430 [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/press.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-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=\"2018-09-05T09:45:08+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:\/\/press.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-lxi-2018\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/press.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-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 4, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018\",\"datePublished\":\"2018-09-05T09:45:08+00:00\",\"dateModified\":\"2019-06-24T14:33:14+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/press.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-lxi-2018\/\"},\"wordCount\":1523,\"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:\/\/press.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-lxi-2018\/\",\"url\":\"https:\/\/press.azbuki.bg\/en\/matematics\/matharticles2016-3\/4-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 4, \u0433\u043e\u0434\u0438\u043d\u0430 LXI, 2018 - 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