{"id":43041,"date":"2018-10-16T13:33:36","date_gmt":"2018-10-16T10:33:36","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles2016-3\/5-lxi-2018\/"},"modified":"2019-06-24T17:33:14","modified_gmt":"2019-06-24T14:33:14","slug":"5-lxi-2018","status":"publish","type":"post","link":"https:\/\/newspaper.azbuki.bg\/en\/matematics\/matharticles2016-3\/5-lxi-2018\/","title":{"rendered":"Mathematics and Informatics, Number 5\/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><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\u0442\u043e\u0433\u0438 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0432\u0442\u043e\u0440\u043e\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0439 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u044b \u043f\u043e \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432\u043e\u0439 \u0438 \u0430\u043a\u0442\u0443\u0430\u0440\u043d\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0441\u0440\u0435\u0434\u0438 \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u043e\u0432 \u0438 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u043e\u0432 <\/a><br \/><a href=\"#art01\">[Results of the Second International Olympiad in Financial and Actuarial Mathematics for School and 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, \u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u041b\u0430\u0440\u0438\u0441\u0430 \u0424\u043e\u0440\u043a\u0443\u043d\u043e\u0432\u0430, \u041d\u0438\u043d\u0430 \u041f\u0430\u0442\u0440\u043e\u043d\u043e\u0432\u0430 \/ Sava Grozdev, Rosen Nikolaev, Maria Shabanova, Larisa Forkunova, Nina Patronova<\/em> \u2013 page 423<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\">Learning and Assessment Based on Gamified E-Course in Moodle <\/a><br \/><a href=\"#art02\">[\u041e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0438 \u043e\u0446\u0435\u043d\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043e\u0441\u043d\u043e\u0432\u0430\u0442\u0430 \u043d\u0430 \u0438\u0433\u0440\u043e\u0432\u0438\u0437\u0438\u0440\u0430\u043d \u0435-\u043a\u0443\u0440\u0441 \u0432 moodle] <\/a><br \/>\/ <em>\u041c\u0430\u0440\u0438\u044f \u0413\u0430\u0447\u043a\u043e\u0432\u0430, \u041c\u0430\u0440\u0442\u0438\u043d \u0422\u0430\u043a\u0435\u0432, \u0415\u043b\u0435\u043d\u0430 \u0421\u043e\u043c\u043e\u0432\u0430 \/ Mariya Gachkova, Martin Takev, Elena Somova<\/em> \u2013 page 444<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\">\u041f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u043d\u0438, \u043b\u043e\u0433\u0430\u0440\u0438\u0442\u043c\u0438\u0447\u043d\u0438 \u0438 \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f (IV \u0447\u0430\u0441\u0442) <\/a><br \/><a href=\"#art03\">[Exponential, Logarithmic and Trigonometric Functions in Transcendental Equations (Part Iv)] <\/a><br \/><em>\/<\/em> <em>\u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430 \/ Diana Stefanova<\/em> \u2013 \u0441\u0442\u0440. 455<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=\"#art04\">\u0423\u043b\u0438\u0442\u043a\u0430 \u041f\u0430\u0441\u043a\u0430\u043b\u044f <\/a><br \/><a href=\"#art04\">[Pascal\u2019s Limacon]<\/a> <br \/>\/ <em>\u0414\u0430\u0440\u044c\u044f \u041a\u043e\u043f\u0442\u0435\u0432\u0430, \u041a\u0441\u0435\u043d\u0438\u044f \u0413\u043e\u0440\u0441\u043a\u0430\u044f \/ Daria Kopteva, \u041aseniya Gorskaya<\/em> \u2013 \u0441\u0442\u0440. 465<br \/><a href=\"https:\/\/newspaper.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>\u00a0<\/strong><\/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><a href=\"#art05\">\u041a\u043e\u043d\u0441\u0442\u0440\u0443\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0435\u0434\u0438\u043d \u0432\u0438\u0434 \u0430\u043b\u0433\u0435\u0431\u0440\u0438\u0447\u043d\u0438 \u043a\u0440\u0438\u0432\u0438 \u0432 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a <\/a><br \/><a href=\"#art05\">[Construction of a Type of Algebraic Curves in the Plane of a Triangle] <\/a><br \/><em>\/ \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \/ Veselin Nenkov<\/em> \u2013 \u0441\u0442\u0440. 481<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 \u0441 \u043a\u043e\u0440\u0435\u043d\u0438 \u0432 \u0442\u0440\u0438 \u043a\u043e\u043b\u0438\u043d\u0435\u0430\u0440\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 <\/a><br \/><a href=\"#art06\">[Polynomials with Roots in Three Collinear Points] <\/a><br \/>\/ <em>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \/ Veselin Nenkov<\/em> \u2013 page 498<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\">\u041a\u043e\u043c\u0431\u0438\u043d\u0430\u0442\u043e\u0440\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a <\/a><br \/><a href=\"#art07\">[Combinatorial Problems Related to Triangles] <\/a><br \/>\/ <em>\u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432, \u0422\u0430\u043d\u043a\u0430 \u041c\u0438\u043b\u043a\u043e\u0432\u0430, \u041a\u0430\u0442\u044f \u0427\u0430\u043b\u044a\u043a\u043e\u0432\u0430 \/ Rosen Nikolaev, Tanka Milkova, Katya Chalakova<\/em> \u2013 \u0441\u0442\u0440. 506<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><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. 518<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>\u0420\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435 \u043e\u0442 \u0431\u0440\u043e\u0439 6, 2017 <br \/>[Solutions of the Contest Problems from issue 6, 2017] \u2013 \u0441\u0442\u0440. 519<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 class=\"BasicParagraph\"><strong>READ IN THE LATEST ISSUES OF \u201eAZBUKI\u201c JOURNALS \/ \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 p. 522<\/p>\n<p class=\"Pa14\">\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. 524<\/p>\n<p class=\"BasicParagraph\" style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p class=\"BasicParagraph\" style=\"text-align: right;\">\u00a0<\/p>\n<p><strong><a name=\"art01\"><\/a>\u0418\u0422\u041e\u0413\u0418 \u041f\u0420\u041e\u0412\u0415\u0414\u0415\u041d\u0418\u042f \u0412\u0422\u041e\u0420\u041e\u0419 \u041c\u0415\u0416\u0414\u0423\u041d\u0410\u0420\u041e\u0414\u041d\u041e\u0419 \u041e\u041b\u0418\u041c\u041f\u0418\u0410\u0414\u042b \u041f\u041e \u0424\u0418\u041d\u0410\u041d\u0421\u041e\u0412\u041e\u0419 \u0418 \u0410\u041a\u0422\u0423\u0410\u0420\u041d\u041e\u0419 \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0415 \u0421\u0420\u0415\u0414\u0418 \u0428\u041a\u041e\u041b\u042c\u041d\u0418\u041a\u041e\u0412 \u0418 \u0421\u0422\u0423\u0414\u0415\u041d\u0422\u041e\u0412<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p>\u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u044b\u0441\u0448\u0430\u044f \u0448\u043a\u043e\u043b\u0430 \u0441\u0442\u0440\u0430\u0445\u043e\u0432\u0430\u043d\u0438\u044f \u0438 \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432 \u2013 \u0421\u043e\u0444\u0438\u044f<\/p>\n<p>\u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432, \u042d\u043a\u043e\u043d\u043e\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u2013 \u0412\u0430\u0440\u043d\u0430<\/p>\n<p>\u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u0413\u0410\u041e\u0423 \u0414\u041f\u041e \u201e\u041c\u043e\u0441\u043a\u043e\u0432\u0441\u043a\u0438\u0439 \u0446\u0435\u043d\u0442\u0440 \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u044f \u043a\u0430\u0434\u0440\u043e\u0432\u043e\u0433\u043e \u043f\u043e\u0442\u0435\u043d\u0446\u0438\u0430\u043b\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f\u201d \u2013 \u041c\u043e\u0441\u043a\u0432\u0430<\/p>\n<p>\u041b\u0430\u0440\u0438\u0441\u0430 \u0424\u043e\u0440\u043a\u0443\u043d\u043e\u0432\u0430, \u0413\u0411\u041e\u0423 \u0412\u041e \u041c\u041e \u201e\u0410\u043a\u0430\u0434\u0435\u043c\u0438\u044f \u0441\u043e\u0446\u0438\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u0443\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f\u201c \u2013 \u041c\u043e\u0441\u043a\u0432\u0430 (\u0420\u043e\u0441\u0441\u0438\u044f)<\/p>\n<p>\u041d\u0438\u043d\u0430 \u041f\u0430\u0442\u0440\u043e\u043d\u043e\u0432\u0430 \u2013 \u0424\u0413\u0410\u041e\u0423 \u0412\u041e \u201e\u0421\u0435\u0432\u0435\u0440\u043d\u044b\u0439 (\u0410\u0440\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0439) \u0424\u0435\u0434\u0435\u0440\u0430\u043b\u044c\u043d\u044b\u0439 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u2013 \u0410\u0440\u0445\u0430\u043d\u0433\u0435\u043b\u044c\u0441\u043a<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>Abstract.<\/strong> \u0414\u0430\u043d\u043d\u0430\u044f \u0441\u0442\u0430\u0442\u044c\u044f \u043f\u043e\u0434\u0432\u043e\u0434\u0438\u0442 \u0438\u0442\u043e\u0433\u0438 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0412\u0442\u043e\u0440\u043e\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0439 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u044b \u043f\u043e \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432\u043e\u0439 \u0438 \u0430\u043a\u0442\u0443\u0430\u0440\u043d\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435, \u0443\u0447\u0440\u0435\u0434\u0438\u0442\u0435\u043b\u044f\u043c\u0438 \u043a\u043e\u0442\u043e\u0440\u043e\u0439 \u044f\u0432\u043b\u044f\u044e\u0442\u0441\u044f \u0434\u0432\u0430 \u0431\u043e\u043b\u0433\u0430\u0440\u0441\u043a\u0438\u0445 \u0432\u0443\u0437\u0430 \u042d\u043a\u043e\u043d\u043e\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 &#8211; \u0412\u0430\u0440\u043d\u0430 \u0438 \u0412\u044b\u0441\u0448\u0430\u044f \u0448\u043a\u043e\u043b\u0430 \u0441\u0442\u0440\u0430\u0445\u043e\u0432\u0430\u043d\u0438\u044f \u0438 \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432 &#8211; \u0421\u043e\u0444\u0438\u044f. \u041e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u0440\u043e\u0445\u043e\u0434\u0438\u0442 \u0432 \u0447\u0435\u0442\u044b\u0440\u0435\u0445 \u0432\u043e\u0437\u0440\u0430\u0441\u0442\u043d\u044b\u0445 \u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u044f\u0445: \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u0438 &#8211; \u043c\u043b\u0430\u0434\u0448\u0430\u044f \u0433\u0440\u0443\u043f\u043f\u0430 (\u0443\u0447\u0430\u0449\u0438\u0435\u0441\u044f 5-6 \u043a\u043b\u0430\u0441\u0441\u043e\u0432, 11-12 \u043b\u0435\u0442), \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u0438 &#8211; \u0441\u0440\u0435\u0434\u043d\u044f\u044f (\u0443\u0447\u0430\u0449\u0438\u0435\u0441\u044f 7-9 \u043a\u043b\u0430\u0441\u0441\u043e\u0432, 13-15 \u043b\u0435\u0442), \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u0438 &#8211; \u0441\u0442\u0430\u0440\u0448\u0430\u044f \u0433\u0440\u0443\u043f\u043f\u0430 (\u0443\u0447\u0430\u0449\u0438\u0435\u0441\u044f \u0441\u0442\u0430\u0440\u0448\u0438\u0445 \u043a\u043b\u0430\u0441\u0441\u043e\u0432 \u043e\u0431\u0449\u0435\u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0448\u043a\u043e\u043b\u044b \u0438 \u0443\u0447\u0440\u0435\u0436\u0434\u0435\u043d\u0438\u0439 \u0441\u0440\u0435\u0434\u043d\u0435\u0433\u043e \u043f\u0440\u043e\u0444\u0435\u0441\u0441\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u043e\u0433\u043e \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u044f, 16-18 \u043b\u0435\u0442), \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u044b \u0438 \u0432\u0437\u0440\u043e\u0441\u043b\u044b\u0435 (\u043a\u0430\u0442\u0435\u0433\u043e\u0440\u0438\u044f 18+). \u041c\u0438\u0441\u0441\u0438\u044f \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u044b &#8211; \u043f\u0440\u0438\u0432\u043b\u0435\u0447\u044c \u0432\u043d\u0438\u043c\u0430\u043d\u0438\u0435 \u043a \u0432\u043e\u043f\u0440\u043e\u0441\u0443 \u043e \u0437\u043d\u0430\u0447\u0438\u043c\u043e\u0441\u0442\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0445 \u0437\u043d\u0430\u043d\u0438\u0439 \u0434\u043b\u044f \u043f\u043e\u0432\u044b\u0448\u0435\u043d\u0438\u044f \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432\u043e\u0439 \u0433\u0440\u0430\u043c\u043e\u0442\u043d\u043e\u0441\u0442\u0438 \u043d\u0430\u0441\u0435\u043b\u0435\u043d\u0438\u044f.<\/p>\n<p><em>Keywords:<\/em> financial literacy; Olympiad; mathematical methods<\/p>\n<p>\u00a0<\/p>\n<p><strong>RESULTS OF THE SECOND INTERNATIONAL OLYMPIAD IN FINANCIAL AND ACTUARIAL MATHEMATICS FOR SCHOOL AND UNIVERSITY STUDENTS<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>Abstract. <\/strong>The present paper sums up the results of the Second international Olympiad in financial and actuarial mathematics established by two Bulgarian universities: Economical university-Varna and Higher School of Insurance and Finance-Sofia. The participants were divided into four groups: the junior group \u2013 school students 11 \u2013 12 years old, 5 \u2013 6 grades; the middle group \u2013 school students 13 \u2013 15 years old, 7 \u2013 9 grades; the higher group \u2013 school students 16 \u2013 18 years old, 10 \u2013 11 (12) grades and professional college students; the senior group \u2013 University students and citizens, 18+ years old. The Olympiad mission is to draw attention to the importance issue of mathematical knowledge for improving the financial literacy of the population.<\/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>\u00a0<\/p>\n<p class=\"Pa18\">Dr. Rosen Nikolaev, Prof.<\/p>\n<p class=\"Pa19\">University of Economics \u2013 Varna<\/p>\n<p class=\"Pa19\">Varna, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\">Dr. Nina Patronova, Assoc. Prof.<\/p>\n<p class=\"Pa19\">Northern (Arctic) Federal University<\/p>\n<p>Arkhangelsk, Russian Federation<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\">Dr. Larisa Forkunova, Assoc. Prof.<\/p>\n<p class=\"Pa19\">Academy of Public Administration<\/p>\n<p>Moscow, Russian Federation<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\">Prof. Maria Shabanova, DSc.<\/p>\n<p class=\"Pa19\">Moscow Center for Human Resource Development<\/p>\n<p>Moscow, Russian Federation<\/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=\"art02\"><\/a>LEARNING AND ASSESSMENT BASED ON GAMIFIED E-COURSE IN MOODLE<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa18\">Mariya Gachkova, Martin Takev, Elena Somova<\/p>\n<p class=\"Pa19\">University \u201cPaisii Hilendarski\u201d \u2013 Plovdiv (Bulgaria)<\/p>\n<p class=\"Pa20\">\u00a0<\/p>\n<p class=\"Pa20\"><strong>Abstract.<\/strong> The paper presents game-like elements suitable for the realization of a gamified e-course and their representation as components in the e-learning environment Moodle. An example of the design of a gamified e-learning course in Moodle is shown. The paper contains comparison charts for the assessment of the course participants. Feedback of the effectiveness of certain aspects of the gamified e-course is collected by a survey.<\/p>\n<p><em>Keywords:<\/em> gamification; e-learning course design; e-learning<\/p>\n<p>\u00a0<\/p>\n<p><strong>\u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415 \u0418 \u041e\u0426\u0415\u041d\u042f\u0412\u0410\u041d\u0415 \u041d\u0410 \u041e\u0421\u041d\u041e\u0412\u0410\u0422\u0410 \u041d\u0410 \u0418\u0413\u0420\u041e\u0412\u0418\u0417\u0418\u0420\u0410\u041d \u0415-\u041a\u0423\u0420\u0421 \u0412 MOODLE<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f \u0438\u0433\u0440\u043e\u0432\u0438\u0442\u0435 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0438, \u043f\u043e\u0434\u0445\u043e\u0434\u044f\u0449\u0438 \u0437\u0430 \u0440\u0435\u0430\u043b\u0438\u0437\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0438\u0433\u0440\u043e\u0432\u0438\u0437\u0438\u0440\u0430\u043d \u0435-\u043a\u0443\u0440\u0441, \u0438 \u0442\u044f\u0445\u043d\u043e\u0442\u043e \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u044f\u043d\u0435 \u043a\u0430\u0442\u043e \u043a\u043e\u043c\u043f\u043e\u043d\u0435\u043d\u0442\u0438 \u0432 \u0441\u0440\u0435\u0434\u0430\u0442\u0430 \u0437\u0430 \u0435-\u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 Moodle. \u041f\u043e\u043a\u0430\u0437\u0430\u043d \u0435 \u043f\u0440\u0438\u043c\u0435\u0440 \u0437\u0430 \u0434\u0438\u0437\u0430\u0439\u043d \u043d\u0430 \u0438\u0433\u0440\u043e\u0432\u0438\u0437\u0438\u0440\u0430\u043d \u0435-\u043a\u0443\u0440\u0441 \u0432 Moodle. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u044a\u0434\u044a\u0440\u0436\u0430 \u0434\u0438\u0430\u0433\u0440\u0430\u043c\u0438 \u0437\u0430 \u0441\u0440\u0430\u0432\u043d\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043e\u0446\u0435\u043d\u044f\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u0446\u0438\u0442\u0435 \u0432 \u043a\u0443\u0440\u0441\u0430. \u0427\u0440\u0435\u0437 \u043f\u0440\u043e\u0443\u0447\u0432\u0430\u043d\u0435 \u0435 \u0441\u044a\u0431\u0440\u0430\u043d\u0430 \u043e\u0431\u0440\u0430\u0442\u043d\u0430 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u044f \u0437\u0430 \u0435\u0444\u0435\u043a\u0442\u0438\u0432\u043d\u043e\u0441\u0442\u0442\u0430 \u043d\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438 \u0430\u0441\u043f\u0435\u043a\u0442\u0438 \u043e\u0442 \u0438\u0433\u0440\u043e\u0432\u0438\u0437\u0438\u0440\u0430\u043d\u0438\u044f \u0435-\u043a\u0443\u0440\u0441.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\">Ms. Mariya Gachkova, PhD student<\/p>\n<p>Mr. Martin Takev, PhD student<\/p>\n<p>Dr. Elena Somova, Assoc. Prof.<\/p>\n<p class=\"Pa19\">Faculty of Mathematics and Informatics<\/p>\n<p class=\"Pa19\">University of Plovdiv \u201cPaisii Hilendarski\u201d<\/p>\n<p class=\"Pa19\">Plovdiv, 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=\"art03\"><\/a>\u041f\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\u041d\u0418, \u041b\u041e\u0413\u0410\u0420\u0418\u0422\u041c\u0418\u0427\u041d\u0418 \u0418 \u0422\u0420\u0418\u0413\u041e\u041d\u041e\u041c\u0415\u0422\u0420\u0418\u0427\u041d\u0418 \u0424\u0423\u041d\u041a\u0426\u0418\u0418 \u0412 \u0422\u0420\u0410\u041d\u0421\u0426\u0415\u041d\u0414\u0415\u041d\u0422\u041d\u0418 \u0423\u0420\u0410\u0412\u041d\u0415\u041d\u0418\u042f (IV \u0427\u0410\u0421\u0422)<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa18\">Diana Stefanova<\/p>\n<p class=\"Pa19\">\u041e\u0423 \u201e\u041d. \u0419. \u0412\u0430\u043f\u0446\u0430\u0440\u043e\u0432\u201c \u2013 \u0410\u0441\u0435\u043d\u043e\u0432\u0433\u0440\u0430\u0434<\/p>\n<p class=\"Pa20\"><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430 \u043d\u0430 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u0438\u0442\u0435 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f. \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u043a\u043e\u043c\u0431\u0438\u043d\u0430\u0446\u0438\u0438 \u043e\u0442 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\u043d\u0438, \u043b\u043e\u0433\u0430\u0440\u0438\u0442\u043c\u0438\u0447\u043d\u0438 \u0438 \u0442\u0440\u0438\u0433\u043e\u043d\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438 \u0432 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u043e \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u0435. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u043f\u043e\u0434\u0445\u043e\u0434\u0438 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u0438\u043c.<\/p>\n<p><em>Keywords:<\/em> transcendental equation; problem solving<\/p>\n<p>\u00a0<\/p>\n<p><strong>EXPONENTIAL, LOGARITHMIC AND TRIGONOMETRIC FUNCTIONS IN TRANSCENDENTAL EQUATIONS (PART IV)<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa20\"><strong>Abstract.<\/strong> The paper is dedicated to transcendental equations. Presently, combinations of exponential, logarithmic and trigonometric functions in a transcendental equation are considered. Approaches to solve them are proposed.<\/p>\n<p class=\"Pa18\">\u00a0<\/p>\n<p class=\"Pa18\">Dr. Diana Stefanova<\/p>\n<p class=\"Pa19\">Teacher in Mathematics<\/p>\n<p>Asenovgrad, 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>\u0423\u041b\u0418\u0422\u041a\u0410 \u041f\u0410\u0421\u041a\u0410\u041b\u042f<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa18\">\u0414\u0430\u0440\u044c\u044f \u041a\u043e\u043f\u0442\u0435\u0432\u0430, \u041a\u0441\u0435\u043d\u0438\u044f \u0413\u043e\u0440\u0441\u043a\u0430\u044f<\/p>\n<p class=\"Pa19\">\u041c\u0443\u043d\u0438\u0446\u0438\u043f\u0430\u043b\u044c\u043d\u043e\u0435 \u0431\u044e\u0434\u0436\u0435\u0442\u043d\u043e\u0435 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0435 \u0443\u0447\u0440\u0435\u0436\u0434\u0435\u043d\u0438\u0435 \u201e\u0421\u0440\u0435\u0434\u043d\u044f\u044f \u0448\u043a\u043e\u043b\u0430 \u2116 8\u201c \u2013 \u0410\u0440\u0445\u0430\u043d\u0433\u0435\u043b\u044c\u0441\u043a (\u0420\u043e\u0441\u0441\u0438\u044f)<\/p>\n<p class=\"Pa20\">\u00a0<\/p>\n<p class=\"Pa20\"><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, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 \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 \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u043a\u0430\u043c\u0438 \u0438\u0437 \u0420\u043e\u0441\u0441\u0438\u0438. \u041f\u0440\u043e\u0435\u043a\u0442 \u0431\u044b\u043b \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0438\u0437\u0430\u0446\u0438\u044e \u0438 \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u044f, \u0441\u0438\u043b\u0430\u043c\u0438 \u0443\u0447\u0430\u0449\u0438\u0445\u0441\u044f, \u0437\u043d\u0430\u043d\u0438\u0439 \u043e \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. \u0412\u0437\u0430\u0438\u043c\u043e\u0434\u0435\u0439\u0441\u0442\u0432\u0438\u0435 \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u043a\u043e\u0432 \u043f\u0440\u043e\u0435\u043a\u0442\u043e\u0432 \u043e\u0441\u0443\u0449\u0435\u0441\u0442\u0432\u043b\u044f\u043b\u043e\u0441\u044c \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u043e\u0431\u043b\u0430\u0447\u043d\u044b\u0445 \u0441\u0435\u0440\u0432\u0438\u0441\u043e\u0432 Google. \u0418\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u0435 \u043f\u0440\u043e\u0432\u043e\u0434\u0438\u043b\u043e\u0441\u044c \u043c\u0435\u0442\u043e\u0434\u0430\u043c\u0438 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0438 \u0441 \u0438\u0441\u043f\u043e\u043b\u044c\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u043c \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u043d\u043e\u0433\u043e \u043f\u0440\u043e\u0434\u0443\u043a\u0442\u0430 GeoGebra.<\/p>\n<p><em>Keywords:<\/em> circle; curve; equation; GeoGebra; Pascal\u2019s limacon<\/p>\n<p>\u00a0<\/p>\n<p><strong>PASCAL\u2019S LIMACON<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>\u0410bstract.<\/strong> The paper presents results of the Russian participants in the net research project \u201cEncyclopedia of notable plane figures: we work by ourselves\u201d. The project aimed at systematization and development of knowledge on notable plane curves by secondary students. The net interaction between the participants was realizes using Google Cloud Service. The investigation was carried out by analytical geometry methods applying the software product GeoGebra.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\">Ms. Kopteva Daria, student<\/p>\n<p>Ms. Gorskaya Kseniya, student<\/p>\n<p class=\"Pa19\">Public Secondary School \u21168<\/p>\n<p>Arkhangelsk, Russia<\/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>\u041a\u041e\u041d\u0421\u0422\u0420\u0423\u0418\u0420\u0410\u041d\u0415 \u041d\u0410 \u0415\u0414\u0418\u041d \u0412\u0418\u0414 \u0410\u041b\u0413\u0415\u0411\u0420\u0418\u0427\u041d\u0418 \u041a\u0420\u0418\u0412\u0418 \u0412 \u0420\u0410\u0412\u041d\u0418\u041d\u0410\u0422\u0410 \u041d\u0410 \u0422\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p>\u0412\u0438\u0441\u0448\u0435 \u0432\u043e\u0435\u043d\u043d\u043e\u043c\u043e\u0440\u0441\u043a\u043e \u0443\u0447\u0438\u043b\u0438\u0449\u0435 &#8222;\u041d. \u0419. \u0412\u0430\u043f\u0446\u0430\u0440\u043e\u0432&#8220;<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0412\u044a\u0432 \u0432\u0440\u044a\u0437\u043a\u0430 \u0441 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0435\u0434\u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430 \u0437\u0430 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u044f\u043d\u0435 \u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u043e \u043c\u044f\u0441\u0442\u043e \u0441\u0430 \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u0438 \u0434\u0432\u0430 \u043a\u043b\u0430\u0441\u0430 \u043a\u0440\u0438\u0432\u0438 \u043e\u0442 \u0441\u0442\u0435\u043f\u0435\u043d \u00a0\u0432 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u0434\u0430\u0434\u0435\u043d \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a. \u041e\u043f\u0438\u0441\u0430\u043d\u0438 \u0441\u0430 \u043e\u0431\u0449\u0438\u0442\u0435 \u0442\u043e\u0447\u043a\u0438 \u043d\u0430 \u0442\u0435\u0437\u0438 \u043a\u0440\u0438\u0432\u0438 \u0438 \u0434\u0432\u0430 \u043f\u0440\u0430\u0432\u0438\u043b\u043d\u0438 \u043c\u043d\u043e\u0433\u043e\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438 \u043e\u0442 \u043f\u0440\u0435\u0441\u0435\u0447\u043d\u0438\u0442\u0435 \u0442\u043e\u0447\u043a\u0438 \u043d\u0430 \u043a\u0440\u0438\u0432\u0438\u0442\u0435 \u0438 \u043e\u043f\u0438\u0441\u0430\u043d\u0430\u0442\u0430 \u043e\u043a\u043e\u043b\u043e \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442.<\/p>\n<p><em>Keywords:<\/em> triangle; circle; hyperbola; regular polygon; curve of ()-th degree<\/p>\n<p>\u00a0<\/p>\n<p><strong>CONSTRUCTION OF A TYPE OF ALGEBRAIC CURVES IN THE PLANE OF A TRIANGLE<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>Abstract:<\/strong> Two classes of curves of \u00a0degree \u00a0are obtained in the plane of a given triangle in connection with the generalization of a problem on locus determination. What are described are the common points of those curves and two regular polygons determined by the points of intersection of the curves and the circumcircle of the triangle.<\/p>\n<p class=\"Pa18\">Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n<p class=\"Pa19\">\u201cNikola Vaptsarov\u201d Naval Academy<\/p>\n<p>Varna, 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=\"art06\"><\/a><strong>\u041f\u041e\u041b\u0418\u041d\u041e\u041c\u0418 \u0421 \u041a\u041e\u0420\u0415\u041d\u0418 \u0412 \u0422\u0420\u0418 \u041a\u041e\u041b\u0418\u041d\u0415\u0410\u0420\u041d\u0418 \u0422\u041e\u0427\u041a\u0418<\/strong><\/p>\n<p><strong style=\"font-size: 12.16px;\">\u00a0<\/strong><\/p>\n<p class=\"Pa18\">\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432<\/p>\n<p class=\"Pa19\">Nikola Vaptsarov Naval Academy\nVarna, Bulgaria<\/p>\n<p class=\"Pa20\">\u00a0<\/p>\n<p class=\"Pa20\"><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 \u043f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u0441 \u043a\u043e\u0440\u0435\u043d\u0438 \u0432 \u0442\u0440\u0438 \u043a\u043e\u043b\u0438\u043d\u0435\u0430\u0440\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \u0438 \u043a\u043e\u0440\u0435\u043d\u0438\u0442\u0435 \u043d\u0430 \u0441\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u0438\u0442\u0435 \u0438\u043c \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u0438 \u043f\u043e\u0441\u0440\u0435\u0434\u0441\u0442\u0432\u043e\u043c \u0435\u0434\u043d\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u043e\u0442 \u0435\u043b\u0438\u043f\u0441\u0438, \u043f\u043e\u0440\u043e\u0434\u0435\u043d\u0438 \u043e\u0442 \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; derivative; ellipse; circle; vertices; centroid<\/p>\n<p>\u00a0<\/p>\n<p><strong>POLYNOMIALS WITH ROOTS IN THREE COLLINEAR POINTS<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa20\"><strong>Abstract.<\/strong> A geometric relation is considered between polynomials with roots in three collinear points and the roots of their corresponding derivatives by means of a set of ellipses generated by the polynomial roots.<\/p>\n<p class=\"Pa18\">\u00a0<\/p>\n<p class=\"Pa18\">Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n<p class=\"Pa19\">\u201cNikola Vaptsarov\u201d Naval Academy<\/p>\n<p>Varna, 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=\"art07\"><\/a><strong>\u041a\u041e\u041c\u0411\u0418\u041d\u0410\u0422\u041e\u0420\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418, \u0421\u0412\u042a\u0420\u0417\u0410\u041d\u0418 \u0421 \u0422\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa18\">Rosen Nikolaev, Tanka Milkova<\/p>\n<p class=\"Pa18\">University of Economics \u2013 Varna<\/p>\n<p>\u041a\u0430\u0442\u044f \u0427\u0430\u043b\u044a\u043a\u043e\u0432\u0430<\/p>\n<p>\u041f\u041c\u0413 \u201e\u0418\u0432\u0430\u043d \u0412\u0430\u0437\u043e\u0432\u201c \u2013 \u0414\u0438\u043c\u0438\u0442\u0440\u043e\u0432\u0433\u0440\u0430\u0434<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa20\"><strong>Abstract<\/strong><strong>.<\/strong> \u0412 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u043d\u0435\u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e\u0442\u043e \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430 \u0437\u0430 \u0446\u0435\u043b\u043e\u0447\u0438\u0441\u043b\u0435\u043d\u0438 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438. \u041d\u0430\u043c\u0435\u0440\u0435\u043d\u0438 \u0441\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0438 \u0437\u0430 \u0431\u0440\u043e\u044f \u0438\u043c \u0432 \u0441\u043b\u0443\u0447\u0430\u0438\u0442\u0435 \u043d\u0430 \u0440\u0430\u0432\u043d\u043e\u0441\u0442\u0440\u0430\u043d\u043d\u0438, \u0440\u0430\u0432\u043d\u043e\u0431\u0435\u0434\u0440\u0435\u043d\u0438 \u0438 \u0440\u0430\u0437\u043d\u043e\u0441\u0442\u0440\u0430\u043d\u043d\u0438 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438. \u0424\u043e\u0440\u043c\u0443\u043b\u0438\u0442\u0435 \u0434\u0430\u0432\u0430\u0442 \u0432\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442 \u0437\u0430 \u0431\u044a\u0440\u0437\u043e \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438 \u0432 \u0440\u0435\u0434\u043e\u0432\u043d\u0438\u0442\u0435 \u0447\u0430\u0441\u043e\u0432\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0432 \u0443\u0447\u0438\u043b\u0438\u0449\u0435, \u0430 \u0442\u0430\u043a\u0430 \u0441\u044a\u0449\u043e \u0438 \u043d\u0430 \u043d\u0438\u0432\u043e \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f.<\/p>\n<p><em>Keywords:<\/em> triangle inequality; integer triangle; counting; combinatorics<\/p>\n<p>\u00a0<\/p>\n<p><strong>COMBINATORIAL PROBLEMS RELATED TO TRIANGLES<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p><strong>Abstract<\/strong><strong>.<\/strong> The present paper considers an application of the triangle inequality to integer triangles. Formulae are found for their number in the cases of equilateral, isosceles and scalene triangles. The formulae give opportunities for quick problem solving in regular math classes at school as well as at the level of competitions.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa18\">Dr. Rosen Nikolaev, Prof.<\/p>\n<p class=\"Pa18\">Dr. Tanka Milkova, Assoc. Prof.<\/p>\n<p class=\"Pa19\">University of Economics \u2013 Varna<\/p>\n<p class=\"Pa19\">Varna, Bulgaria<\/p>\n<p class=\"Pa18\">Ms. Katya Chalakova<\/p>\n<p class=\"Pa19\">Mathematics High School \u2013 Dimitrovgrad<\/p>\n<p>Dimitrovgrad, Bulgaria<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: 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 \u0418\u0442\u043e\u0433\u0438 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0432\u0442\u043e\u0440\u043e\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0439 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u044b \u043f\u043e \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432\u043e\u0439 \u0438 \u0430\u043a\u0442\u0443\u0430\u0440\u043d\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0441\u0440\u0435\u0434\u0438 \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u043e\u0432 \u0438 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u043e\u0432 [Results of the Second International Olympiad in Financial and Actuarial Mathematics for School and University Students] \/ \u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432, \u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u041b\u0430\u0440\u0438\u0441\u0430 \u0424\u043e\u0440\u043a\u0443\u043d\u043e\u0432\u0430, \u041d\u0438\u043d\u0430 \u041f\u0430\u0442\u0440\u043e\u043d\u043e\u0432\u0430 \/ Sava Grozdev, Rosen Nikolaev, [&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 5, \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:\/\/foreignlanguages.azbuki.bg\/en\/matematics\/matharticles2016-3\/5-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 5, \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\u0442\u043e\u0433\u0438 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0432\u0442\u043e\u0440\u043e\u0439 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u043e\u0439 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u044b \u043f\u043e \u0444\u0438\u043d\u0430\u043d\u0441\u043e\u0432\u043e\u0439 \u0438 \u0430\u043a\u0442\u0443\u0430\u0440\u043d\u043e\u0439 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435 \u0441\u0440\u0435\u0434\u0438 \u0448\u043a\u043e\u043b\u044c\u043d\u0438\u043a\u043e\u0432 \u0438 \u0441\u0442\u0443\u0434\u0435\u043d\u0442\u043e\u0432 [Results of the Second International Olympiad in Financial and Actuarial Mathematics for School and University Students] \/ \u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0420\u043e\u0441\u0435\u043d \u041d\u0438\u043a\u043e\u043b\u0430\u0435\u0432, \u041c\u0430\u0440\u0438\u044f \u0428\u0430\u0431\u0430\u043d\u043e\u0432\u0430, \u041b\u0430\u0440\u0438\u0441\u0430 \u0424\u043e\u0440\u043a\u0443\u043d\u043e\u0432\u0430, \u041d\u0438\u043d\u0430 \u041f\u0430\u0442\u0440\u043e\u043d\u043e\u0432\u0430 \/ Sava Grozdev, Rosen Nikolaev, [&hellip;]\" \/>\n<meta property=\"og:url\" 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