{"id":42623,"date":"2018-04-25T13:00:37","date_gmt":"2018-04-25T10:00:37","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles2016-3\/2-lxi-2018\/"},"modified":"2019-06-24T17:33:14","modified_gmt":"2019-06-24T14:33:14","slug":"2-lxi-2018","status":"publish","type":"post","link":"https:\/\/foreignlanguages.azbuki.bg\/en\/matematics\/matharticles2016-3\/2-lxi-2018\/","title":{"rendered":"Mathematics and Informatics, Number 2\/2018, Volume 61"},"content":{"rendered":"<p class=\"text1\"><strong><a name=\"top\"><\/a>CONTENTS \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415<\/strong><\/p>\n<p class=\"text1\">\u00a0<\/p>\n<p class=\"text1\"><strong>EDITORIAL \/ \u041a\u042a\u041c \u0427\u0418\u0422\u0410\u0422\u0415\u041b\u042f<\/strong> \u2013 \u0441\u0442\u0440. 111<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"text1\" align=\"left\">\u00a0<\/p>\n<p class=\"text1\" align=\"left\"><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 align=\"left\"><a href=\"#art01\">A Possibility to Teach and Learn Mathematics by Theatre Technology <\/a><br \/><a href=\"#art01\">[\u0412\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442 \u0437\u0430 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435 \u0438 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0447\u0440\u0435\u0437 \u0442\u0435\u0430\u0442\u044a\u0440] <\/a><br \/>\/ <em>Sava Grozdev<\/em><em>,<\/em><em> Veselin Nenkov \/ <\/em><em>Sava Grozdev, Veselin Nenkov<\/em> \u2013 \u0441\u0442\u0440. 113<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p align=\"left\">\u00a0<\/p>\n<p align=\"left\"><a href=\"#art02\">\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043f\u0440\u043e\u0435\u043a\u0442\u0438 \u0432 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u043d\u0438 \u0441\u0440\u0435\u0434\u0438 <\/a><br \/><a href=\"#art02\">[Mathematical Projects in Dynamic Environment] <\/a><br \/>\/ \u0410\u043d\u0433\u0435\u043b \u0413\u0443\u0448\u0435\u0432 \/ Angel Gushev \u2013 \u0441\u0442\u0440. 125<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p align=\"left\">\u00a0<\/p>\n<p align=\"left\"><a href=\"#art03\">\u041f\u0440\u0430\u043a\u0442\u0438\u043a\u043e\u043f\u0440\u0438\u043b\u043e\u0436\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0432 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435\u0442\u043e \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 <\/a><br \/><a href=\"#art03\">[Practice-Applicable Problems in Mathematics Education] <\/a><br \/>\/ <em>\u0420\u0443\u043c\u044f\u043d\u0430 \u041d\u0435\u0441\u0442\u043e\u0440\u043e\u0432\u0430<\/em><em> \/ <\/em><em>Rumyana Nestorova<\/em>\u2013 \u0441\u0442\u0440. 144<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p align=\"left\">\u00a0<\/p>\n<p><a href=\"#art04\">\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 <\/a><br \/><a href=\"#art04\">[Solving Linear Equations with Excel] <\/a><br \/>\/ <em>Plamen Penev<\/em><em> \/ Plamen Penev <\/em>\u2013 \u0441\u0442\u0440. 154<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p align=\"left\"><a href=\"#art05\">Two Interesting Inequalities for Acute Triangles <\/a><br \/>\/ <em>\u0160efket Arslanagi\u0107, Amar Ba\u0161i\u0107<\/em>\u2013 \u0441\u0442\u0440. 164<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p align=\"left\">\u00a0<\/p>\n<p><a href=\"#art06\">Computer Discovered Mathematics: An Alternative Construction of Malfatti Squares <\/a><br \/><a href=\"#art06\">[M\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430, \u043e\u0442\u043a\u0440\u0438\u0442\u0430 \u043e\u0442 \u043a\u043e\u043c\u043f\u044e\u0442\u0440\u0438: \u0410\u043b\u0442\u0435\u0440\u043d\u0430\u0442\u0438\u0432\u043d\u0430 \u043a\u043e\u043d\u0441\u0442\u0440\u0443\u043a\u0446\u0438\u044f \u043d\u0430 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u0438\u0442\u0435 \u043d\u0430 M\u0430\u043b\u0444\u0430\u0442\u0438] <\/a><br \/>\/ <em>Sava Grozdev, Hiroshi Okumura, Deko Dekov <\/em>\u2013 \u0441\u0442\u0440. 171<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"text1\" align=\"left\">\u00a0<\/p>\n<p class=\"text1\" align=\"left\"><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 align=\"left\"><a href=\"#art07\">\u041f\u0435\u0440\u0444\u0435\u043a\u0442\u043d\u0430 \u0438\u0437\u043e\u0433\u043e\u043d\u0430\u043b\u043d\u043e\u0441\u0442 \u0432 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a <\/a><br \/><a href=\"#art07\">[Perfect Isogonality in Quadrilateral] <\/a><br \/>\/ <em>\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432, \u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432, \u0425\u0430\u0438\u043c \u0425\u0430\u0438\u043c\u043e\u0432<\/em><em> \/ <\/em><em>Veselin Nenkov<\/em><em>, <\/em><em>Stanislav Stefanov<\/em><em>, <\/em><em>Haim Haimov<\/em>\u2013 \u0441\u0442\u0440. 175<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p align=\"left\">\u00a0<\/p>\n<p align=\"left\"><a href=\"#art08\">\u0414\u0432\u0435 \u0442\u0440\u0430\u043d\u0441\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u0438 \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 \u0438 \u0442\u044f\u0445\u043d\u043e\u0442\u043e \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435 <\/a><br \/><a href=\"#art08\">[Two Transformations in the Plane of a Triangle and Their Application] <\/a><br \/>\/ <em>\u0414\u0438\u043c\u0438\u0442\u044a\u0440\u00a0<\/em><em>\u041e\u043f\u0430\u0440\u043b\u0430\u043a\u043e\u0432<\/em><em> \/ <\/em><em>Dimitar Oparlakov<\/em>\u2013 \u0441\u0442\u0440. 190<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p align=\"left\">\u00a0<\/p>\n<p class=\"Pa9\"><a href=\"#art09\">\u041d\u044f\u043a\u043e\u0438 \u0442\u0438\u043f\u043e\u0432\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u0441\u044a\u0441 \u0441\u0438\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u0447\u0438\u0441\u043b\u0430 <\/a><br \/><a href=\"#art09\">[Some Types of Problems with Symmetric Numbers] <\/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, \u0420\u0430\u0434\u0430\u043d \u041c\u0438\u0440\u044f\u043d\u043e\u0432 \/ <\/em><em>Rosen Nikolaev<\/em><em>, <\/em><em>Tanka Milkova<\/em><em>, <\/em><em>Radan Mir<\/em><em>ya<\/em><em>nov<\/em>\u2013 \u0441\u0442\u0440. 200<br \/><a href=\"https:\/\/foreignlanguages.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p class=\"text1\" align=\"left\">\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. 206<br \/><a href=\"https:\/\/foreignlanguages.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 3, 2017 <br \/>[Solutions of the Contest Problems from Issue 3, 2017] \u2013 \u0441\u0442\u0440. 207<br \/><a href=\"https:\/\/foreignlanguages.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 <br \/>\/ \u0412 \u041d\u041e\u0412\u0418\u0422\u0415 \u0411\u0420\u041e\u0415\u0412\u0415 \u041d\u0410 \u0421\u041f\u0418\u0421\u0410\u041d\u0418\u042f\u0422\u0410 \u041d\u0410 \u0418\u0417\u0414\u0410\u0422\u0415\u041b\u0421\u0422\u0412\u041e \u201e\u0410\u0417 \u0411\u0423\u041a\u0418\u201c \u0427\u0415\u0422\u0415\u0422\u0415<\/strong> \u2013 \u0441\u0442\u0440. 210<\/p>\n<p class=\"BasicParagraph\">\u00a0<\/p>\n<p class=\"Pa14\"><strong>GUIDE FOR AUTHORS <br \/>\/ \u0423\u041a\u0410\u0417\u0410\u041d\u0418\u042f \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415<\/strong> \u2013 \u0441\u0442\u0440. 212<\/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><span style=\"font-size: 12.16px;\"><a name=\"art01\"><\/a>A POSSIBILITY TO TEACH AND LEARN MATHEMATICS BY THEATRE TECHNOLOGY<\/span><\/strong><\/p>\n<p align=\"left\"><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa25\"><strong>Abstract. <\/strong>In the recent years new approaches were created to teach and learn Mathematics. One of them is by using theatre plays with mathematical content. The present paper proposes a scenario of such a play based on linear Diophantine equations.<\/p>\n<p align=\"left\"><em>Keywords: <\/em>linear Diophantine equation; Euclid; Euler; theatre; Little Red Riding Hood; intelligence<\/p>\n<p align=\"left\">\u00a0<\/p>\n<p>Prof. Sava Grozdev, DSc.<\/p>\n<p>University of Finance, Business and Entrepreneurship<\/p>\n<p>Sofia, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n<p class=\"Pa23\">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=\"art02\"><\/a><strong>\u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u0427\u0415\u0421\u041a\u0418 \u041f\u0420\u041e\u0415\u041a\u0422\u0418 \u0412 \u0414\u0418\u041d\u0410\u041c\u0418\u0427\u041d\u0418 \u0421\u0420\u0415\u0414\u0418<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">\u0410\u043d\u0433\u0435\u043b \u0413\u0443\u0448\u0435\u0432<\/p>\n<p>\u201cVasil Drumev\u201d High school of Mathematics and Natural Science \u2013 Veliko Tarnovo (Bulgaria)<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> \u0426\u0435\u043b\u0442\u0430 \u043d\u0430 \u043d\u0430\u0441\u0442\u043e\u044f\u0449\u0430\u0442\u0430 \u0441\u0442\u0430\u0442\u0438\u044f \u0435 \u0434\u0430 \u0441\u0435 \u043f\u043e\u043a\u0430\u0436\u0435, \u0447\u0435 \u0437\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u0435\u043d \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0441\u043e\u0444\u0442\u0443\u0435\u0440 \u043d\u044f\u043c\u0430 \u0432\u044a\u0437\u0440\u0430\u0441\u0442\u043e\u0432\u0438 \u0433\u0440\u0430\u043d\u0438\u0446\u0438. \u041d\u044f\u043c\u0430 \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u044f \u0438 \u0432 \u043d\u0430\u0447\u0438\u043d\u0438\u0442\u0435 \u043d\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435\u0442\u043e \u043c\u0443 \u0438 \u0434\u044a\u043b\u0431\u043e\u0447\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u043f\u043e\u0441\u0442\u0438\u0433\u043d\u0430\u0442\u0438\u0442\u0435 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438. \u0415\u0434\u0438\u043d\u0441\u0442\u0432\u0435\u043d\u043e\u0442\u043e \u043e\u0433\u0440\u0430\u043d\u0438\u0447\u0435\u043d\u0438\u0435 \u0441\u0435 \u043a\u0440\u0438\u0435 \u0432 \u0441\u043f\u043e\u0441\u043e\u0431\u043d\u043e\u0441\u0442\u0438\u0442\u0435 \u0438 \u0432\u044a\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0435\u0442\u043e \u043d\u0430 \u0442\u0435\u0437\u0438, \u043a\u043e\u0438\u0442\u043e \u0433\u043e \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u0442.<\/p>\n<p><em>Keywords: <\/em>dynamic mathematical software; ways of use; depth of achievements<\/p>\n<p>\u00a0<\/p>\n<p><strong>MATHEMATICAL PROJECTS IN DYNAMIC ENVIRONMENT<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><span style=\"font-size: 12.16px;\"><strong>Abstract.<\/strong> The purpose of this article is to show that there are no age limits for the use of dynamic mathematical software. There are no limitations on how to use it and the depth of the results. The only limitation lies in the capabilities and imagination of those who use it.<\/span><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Mr. Angel Gushev<\/p>\n<p class=\"Pa23\">Principal<\/p>\n<p class=\"Pa23\">High School of Mathematics and Natural Science \u201cVasil Drumev\u201d<\/p>\n<p>Veliko Tarnovo, 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\u0420\u0410\u041a\u0422\u0418\u041a\u041e\u041f\u0420\u0418\u041b\u041e\u0416\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 \u0412 \u041e\u0411\u0423\u0427\u0415\u041d\u0418\u0415\u0422\u041e \u041f\u041e \u041c\u0410\u0422\u0415\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">\u0420\u0443\u043c\u044f\u043d\u0430 \u041d\u0435\u0441\u0442\u043e\u0440\u043e\u0432\u0430<\/p>\n<p class=\"Pa22\">\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0430 \u0440\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u043d\u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u043d\u0430 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0438 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u0437\u043d\u0430\u043d\u0438\u044f \u0432 \u0433\u0438\u043c\u043d\u0430\u0437\u0438\u0430\u043b\u0435\u043d \u0435\u0442\u0430\u043f \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043f\u043e \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438\u0442\u0435 \u0437\u0430\u0434\u0430\u0447\u0438 \u0441\u0430 \u043f\u043e \u0444\u043e\u0440\u043c\u0430\u0442\u0430 \u043d\u0430 PISA \u0438 \u0444\u043e\u0440\u043c\u0438\u0440\u0430\u0442 \u0443\u043c\u0435\u043d\u0438\u044f \u0437\u0430 \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435 \u043d\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u0441\u0438\u0442\u0443\u0430\u0446\u0438\u0438.<\/p>\n<p><em>Keywords:<\/em> problem; problem solving; practice-applicable problem; PISA<\/p>\n<p>\u00a0<\/p>\n<p><strong>PRACTICE-APPLICABLE PROBLEMS IN MATHEMATICS EDUCATION<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> Exemplary applications are considered in the paper in connection with the mathematical curriculum of high school mathematics education. The proposed problems are in the format of PISA and form skills for modeling different situations.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Dr. Rumyana Nestorova<\/p>\n<p class=\"Pa23\">Teacher in Mathematics<\/p>\n<p>Vratsa, 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: left;\"><strong style=\"font-size: 12.16px;\"><a name=\"art04\"><\/a>\u0420\u0415\u0428\u0410\u0412\u0410\u041d\u0415 \u041d\u0410 \u041b\u0418\u041d\u0415\u0419\u041d\u0418 \u0423\u0420\u0410\u0412\u041d\u0415\u041d\u0418\u042f \u0421 EXCEL<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa22\">Plamen Penev<\/p>\n<p class=\"Pa22\">\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d \u043c\u043e\u0434\u0435\u043b \u0437\u0430 \u0440\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 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 Excel. \u041a\u043e\u0440\u0435\u043d\u0438\u0442\u0435 \u043d\u0430 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f\u0442\u0430 \u0441\u0435 \u0438\u0437\u0447\u0438\u0441\u043b\u044f\u0432\u0430\u0442 \u0432\u044a\u0437 \u043e\u0441\u043d\u043e\u0432\u0430 \u043d\u0430 \u043a\u043e\u0435\u0444\u0438\u0446\u0438\u0435\u043d\u0442\u0438, \u0432\u0437\u0435\u0442\u0438 \u043e\u0442 \u043b\u0438\u043d\u0438\u044f\u0442\u0430 \u043d\u0430 \u0442\u0440\u0435\u043d\u0434\u0430. \u0421 \u043d\u0435\u0441\u043b\u043e\u0436\u043d\u0438 \u0430\u0432\u0442\u043e\u043c\u0430\u0442\u0438\u0437\u0430\u0446\u0438\u0438 \u0441\u0435 \u0432\u044a\u0432\u0435\u0436\u0434\u0430 \u0440\u0430\u0431\u043e\u0442\u0430 \u0441 \u043e\u0431\u0438\u043a\u043d\u043e\u0432\u0435\u043d\u0438 \u0434\u0440\u043e\u0431\u0438. \u0421\u043f\u043e\u0434\u0435\u043b\u044f \u0441\u0435 \u043f\u0440\u0430\u043a\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043e\u043f\u0438\u0442 \u043e\u0442 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043c\u043e\u0434\u0435\u043b\u0430.<\/p>\n<p><em>Keywords:<\/em> problem solving; equation; numbers; tables; trendline; macros<\/p>\n<p>\u00a0<\/p>\n<p><strong>SOLVING LINEAR EQUATIONS WITH EXCEL<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> A model is proposed in the paper to solve linear equations with the help of Excel. The solutions of the equations are found on the base of coefficients from the trend line. Work with ordinary fractions is introduced by not complicated automations. A practical experience of the model application is shared.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Mr. Plamen Penev<\/p>\n<p class=\"Pa23\">Teacher in Informatics<\/p>\n<p>Shumen, Bulgaria<\/p>\n<p style=\"text-align: right;\">\u00a0\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>TWO INTERESTING INEQUALITIES FOR ACUTE TRIANGLES<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> The paper considers the proofs of two interesting geometric inequalities for acute triangles.<\/p>\n<p><em>Keywords:<\/em> acute triangle; inequality; orthocentre of a triangle<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Prof. \u0160efket Arslanagi\u0107, DSc.<\/p>\n<p class=\"Pa22\">Mr. Amar Ba\u0161i\u0107<\/p>\n<p class=\"Pa23\">Faculty of Mathematics and Natural Sciences<\/p>\n<p class=\"Pa23\">University of Sarajevo<\/p>\n<p>Sarajevo, Bosnia and Herzegovina<\/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><strong><a name=\"art06\"><\/a>COMPUTER DISCOVERED MATHEMATICS: AN ALTERNATIVE CONSTRUCTION OF MALFATTI SQUARES<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> In comparison with (Grozdev, Okumura. &amp; Dekov, 2018) an alternative construction of Malfatti squares is presented.<\/p>\n<p><em>Keywords:<\/em> Malfatti square; geometric construction; Euclidean geometry; computer discovered mathematics; \u201cDiscoverer\u201d<\/p>\n<p>\u00a0<\/p>\n<p>Prof. Sava Grozdev, DSc.<\/p>\n<p>University of Finance, Business and Entrepreneurship<\/p>\n<p>Sofia, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p>Prof. Dr. Hiroshi Okumura<\/p>\n<p>Maebashi Gunma, Japan<\/p>\n<p>\u00a0<\/p>\n<p>Dr. Deko Dekov, Assoc. Prof.<\/p>\n<p>Stara Zagora, 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<a name=\"art07\"><\/a><\/span><strong>\u041f\u0415\u0420\u0424\u0415\u041a\u0422\u041d\u0410 \u0418\u0417\u041e\u0413\u041e\u041d\u0410\u041b\u041d\u041e\u0421\u0422 \u0412 \u0427\u0415\u0422\u0418\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=\"Pa22\">\u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432, \u0425\u0430\u0438\u043c \u0425\u0430\u0438\u043c\u043e\u0432<\/p>\n<p class=\"Pa23\">\u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432, \u0422\u0435\u0445\u043d\u0438\u0447\u0435\u0441\u043a\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u2013 \u0421\u043e\u0444\u0438\u044f<\/p>\n<p class=\"Pa23\">\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u043e \u0435 \u0435\u0434\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\u043e\u0435\u0442\u043e \u043f\u043e\u0437\u0432\u043e\u043b\u044f\u0432\u0430 \u0434\u0430 \u0441\u0435 \u043f\u043e\u043b\u0443\u0447\u0430\u0442 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f \u043d\u0430 \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0438\u0442\u0435 \u043f\u0440\u0430\u0432\u0438 \u043d\u0430 \u0413\u0430\u0443\u0441 \u0438 \u0413\u0430\u0443\u0441-\u0411\u043e\u0434\u0435\u043d\u043c\u0438\u043b\u0435\u0440 \u0437\u0430 \u043f\u044a\u043b\u043d\u0438\u044f \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a.<\/p>\n<p><em>Keywords:\u00a0<\/em>quadrilateral; notable point; line; circle<\/p>\n<p>\u00a0<\/p>\n<p><strong>PERFECT ISOGONALITY IN QUADRILATERAL<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> It is considered a transformation in the plane of a quadrilateral, allowing to obtain generalizations of the well-known lines of Gauss and Gauss- Bodenmiller for the complete quadrilateral.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n<p class=\"Pa23\">Beli Osam, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Mr. Stanislav Stefanov, PhD student<\/p>\n<p class=\"Pa23\">Technical University\u2013 Sofia<\/p>\n<p class=\"Pa23\">Sofia, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Mr. Haim Haimov, Researcher<\/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><strong><a name=\"art08\"><\/a>\u0414\u0412\u0415 \u0422\u0420\u0410\u041d\u0421\u0424\u041e\u0420\u041c\u0410\u0426\u0418\u0418 \u0412 \u0420\u0410\u0412\u041d\u0418\u041d\u0410\u0422\u0410 \u041d\u0410 \u0427\u0415\u0422\u0418\u0420\u0418\u042a\u0413\u042a\u041b\u041d\u0418\u041a \u0418 \u0422\u042f\u0425\u041d\u041e\u0422\u041e \u041f\u0420\u0418\u041b\u041e\u0416\u0415\u041d\u0418\u0415<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">\u0414\u0438\u043c\u0438\u0442\u044a\u0440 \u041e\u043f\u0430\u0440\u043b\u0430\u043a\u043e\u0432, \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430 \u0433\u0438\u043c\u043d\u0430\u0437\u0438\u044f \u2013 \u0412\u0430\u0440\u043d\u0430<\/p>\n<p class=\"Pa22\">\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u0443\u0447\u0435\u043d\u0438\u0447\u0435\u0441\u043a\u0430 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0430 \u043f\u043e\u0434 \u0440\u044a\u043a\u043e\u0432\u043e\u0434\u0441\u0442\u0432\u043e\u0442\u043e \u043d\u0430 \u0430\u0441. \u0421\u0442\u0430\u043d\u0438\u0441\u043b\u0430\u0432 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432. \u0420\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u0435\u043d\u0430\u0442\u0430 \u0442\u0435\u043c\u0430 \u0443\u0447\u0430\u0441\u0442\u0432\u0430 \u0432 \u043c\u0435\u0436\u0434\u0443\u043d\u0430\u0440\u043e\u0434\u043d\u0438\u044f \u043a\u043e\u043d\u043a\u0443\u0440\u0441 \u201e\u041c\u0435\u0442\u043e\u0434\u043e\u043b\u043e\u0433\u0438\u044f \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438 \u0432 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435\u0442\u043e\u201c \u043f\u0440\u0435\u0437 2018 \u0433. \u0412 \u043d\u0435\u044f \u0441\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0438\u0437\u0438\u0440\u0430\u043d\u0438 \u0438 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u043d\u044f\u043a\u043e\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u0438 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438 \u043e\u0442 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f\u0442\u0430 \u043d\u0430 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u0434\u0432\u0435 \u043d\u043e\u0432\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0430\u043b\u043d\u0438 \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u044f \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, \u0438\u043c\u0430\u0449\u0438 \u0432\u0430\u0436\u043d\u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u0432 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f\u0442\u0430 \u043d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u044f. \u041e\u0441\u043d\u043e\u0432\u043d\u043e\u0442\u043e \u043e\u0442 \u0442\u0435\u0437\u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u044f \u0441\u0435 \u0441\u044a\u0441\u0442\u043e\u0438 \u0432 \u043e\u0431\u043e\u0431\u0449\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0442\u0435\u043e\u0440\u0435\u043c\u0438\u0442\u0435 \u043d\u0430 \u041c\u0438\u043a\u0435\u043b, \u0413\u0430\u0443\u0441 \u0438 \u0413\u0430\u0443\u0441-\u0411\u043e\u0434\u0435\u043d\u043c\u0438\u043b\u0435\u0440 \u0437\u0430 \u043f\u044a\u043b\u043d\u0438\u044f \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a, \u043a\u0430\u043a\u0442\u043e \u0438 \u043d\u0430 \u0441\u044a\u043f\u044a\u0442\u0441\u0442\u0432\u0430\u0449\u0438 \u0433\u0438 \u0442\u0432\u044a\u0440\u0434\u0435\u043d\u0438\u044f. \u0412\u044a\u043f\u0440\u043e\u0441\u043d\u0438\u0442\u0435 \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0441\u043b\u0443\u0436\u0430\u0442 \u0438 \u0437\u0430 \u0434\u043e\u043a\u0430\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u0432\u0440\u044a\u0437\u043a\u0438 \u043c\u0435\u0436\u0434\u0443 \u043d\u043e\u0432\u043e\u043e\u0442\u043a\u0440\u0438\u0442\u0438 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \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 \u0437\u0430 \u0438\u0437\u0432\u0435\u0436\u0434\u0430\u043d\u0435\u0442\u043e \u043e\u0442 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430 \u0435\u0434\u043d\u0438 \u043e\u0442 \u0442\u044f\u0445 \u0438 \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430 \u043d\u0430 \u0434\u0440\u0443\u0433\u0438, \u043e\u0431\u0440\u0430\u0437\u0438 \u043d\u0430 \u043f\u044a\u0440\u0432\u0438\u0442\u0435 \u043f\u0440\u0438 \u0438\u0437\u043e\u0431\u0440\u0430\u0437\u044f\u0432\u0430\u043d\u0435\u0442\u043e. \u041c\u0430\u043a\u0430\u0440 \u0438 \u0441 \u0440\u0430\u0437\u043d\u043e\u0440\u043e\u0434\u043d\u0438 \u0434\u0435\u0444\u0438\u043d\u0438\u0446\u0438\u0438, \u0434\u0432\u0435\u0442\u0435 \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0441\u0430 \u0442\u044f\u0441\u043d\u043e \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u043f\u043e\u043c\u0435\u0436\u0434\u0443 \u0441\u0438. \u0415\u0434\u043d\u043e\u0442\u043e \u043e\u0442 \u0442\u044f\u0445 \u0441\u0435 \u044f\u0432\u044f\u0432\u0430 \u0440\u0430\u0437\u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0435\u043d\u0438\u0435 \u043d\u0430 \u0434\u0440\u0443\u0433\u043e\u0442\u043e \u043e\u0442 \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u043e \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u043e\u0442 \u0442\u043e\u0447\u043a\u0438 \u0432 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u0434\u0430\u0434\u0435\u043d\u0438\u044f \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a \u0432 \u0446\u044f\u043b\u0430\u0442\u0430 \u0440\u0430\u0432\u043d\u0438\u043d\u0430.<\/p>\n<p><em>Keywords:<\/em> quadrilateral; remarkable point; line; circle<\/p>\n<p>\u00a0<\/p>\n<p><strong>TWO TRANSFORMATIONS IN THE PLANE OF A TRIANGLE AND THEIR APPLICATION<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The paper is an elaboration of a high school student under the supervision of Assist. Prof. Stanislav Stefanov. It was awarded first prize in the National round of the International Competition \u201cMethodology and information technologies in Education in February, 2018. Some basic results from the geometry of the quadrilateral are systemized and presented in connection with two new transformations in the plane of a quadrilateral that have important applications in the geometry of the quadrilateral. The main of the applications are generalizations of the theorems of Miquel, Gauss and Gauss-Bedenmiller for the complete quadrilateral and the accompanying assertions. The transformations in question serve also in proving geometric relations among the newly found notable points in the plane of a quadrilateral and in deducing properties of some of them and properties of others that are images of the first ones after being transformed. None the less, that they have heterogeneous definitions, the two transformations are connected narrowly. One of them turns to be a diffusion of the other from a determined set of points in the plane of a given quadrilateral to the whole plane.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Mr. Dimitar Oparlakov<\/p>\n<p class=\"Pa23\">Student<\/p>\n<p class=\"Pa23\">Mathematics High School \u2013 Varna<\/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><strong>\u00a0<a name=\"art09\"><\/a>\u041d\u042f\u041a\u041e\u0418 \u0422\u0418\u041f\u041e\u0412\u0415 \u0417\u0410\u0414\u0410\u0427\u0418 \u0421\u042a\u0421 \u0421\u0418\u041c\u0415\u0422\u0420\u0418\u0427\u041d\u0418 \u0427\u0418\u0421\u041b\u0410<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">\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, \u0420\u0430\u0434\u0430\u043d \u041c\u0438\u0440\u044f\u043d\u043e\u0432<\/p>\n<p class=\"Pa23\">University of Economics \u2013 Varna<\/p>\n<p class=\"Pa23\">\u00a0<\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0435 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430\u0442 \u0442. \u043d\u0430\u0440. \u201e\u0441\u0438\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438\u201c \u0447\u0438\u0441\u043b\u0430 \u0438 \u0430\u0432\u0442\u043e\u0440\u0438\u0442\u0435 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u0442 \u0447\u0430\u0441\u0442 \u043e\u0442 \u0440\u0430\u0437\u043d\u043e\u043e\u0431\u0440\u0430\u0437\u043d\u0438\u0442\u0435 \u0438\u043c \u0441\u0432\u043e\u0439\u0441\u0442\u0432\u0430. \u041f\u043e\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0441\u0430 \u0440\u0435\u0434\u0438\u0446\u0430 \u0438\u043d\u0442\u0435\u0440\u0435\u0441\u043d\u0438 \u0432\u044a\u043f\u0440\u043e\u0441\u0438 \u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u0437\u0430 \u0432\u044a\u043f\u0440\u043e\u0441\u043d\u0438\u0442\u0435 \u0447\u0438\u0441\u043b\u0430, \u043a\u0430\u0442\u043e \u0441\u0430 \u0440\u0435\u0448\u0435\u043d\u0438 \u043d\u044f\u043a\u043e\u043b\u043a\u043e \u043a\u043e\u043d\u043a\u0440\u0435\u0442\u043d\u0438 \u043f\u0440\u0438\u043c\u0435\u0440\u0430. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0438 \u0441\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u0438 \u0442\u0435\u0445\u043d\u0438 \u043e\u0431\u043e\u0431\u0449\u0435\u043d\u0438\u044f. \u0418\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u0442\u043e \u043c\u043e\u0436\u0435 \u0434\u0430 \u0431\u044a\u0434\u0435 \u043e\u0441\u043d\u043e\u0432\u0430 \u0437\u0430 \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u043d\u043e\u0436\u0435\u0441\u0442\u0432\u043e \u0434\u0440\u0443\u0433\u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u043a\u043e\u0438\u0442\u043e \u0434\u0430 \u0431\u044a\u0434\u0430\u0442 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0438 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0438 \u0438 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f, \u043a\u0430\u043a\u0442\u043e \u0438 \u043f\u0440\u0438 \u043f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0430\u0442\u0430 \u0437\u0430 \u0442\u0430\u043a\u0438\u0432\u0430.<\/p>\n<p><em>Keywords:<\/em> symmetric number; digit; problem; problem solving<\/p>\n<p>\u00a0<\/p>\n<p><strong>SOME TYPES OF PROBLEMS WITH SYMMETRIC NUMBERS<\/strong><\/p>\n<p><span style=\"font-size: 12.16px;\">\u00a0<\/span><\/p>\n<p class=\"Pa25\"><strong>Abstract.<\/strong> The so called \u201csymmetric\u201d numbers are considered in the paper and the authors study some of their various properties. Several interesting questions and problems for numbers of that kind are formulated and some concrete examples are solved. Corresponding generalizations of them are proposed. The investigation could be a base for creating many other problems, which could be used in mathematical Olympiads and competitions, as well as for the preparation for them.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa22\">Prof. Dr. Rosen Nikolaev<\/p>\n<p class=\"Pa22\">Dr. Tanka Milkova, Assoc. Prof.<\/p>\n<p class=\"Pa22\">Dr. Radan Miryanov, Assist. Prof.<\/p>\n<p class=\"Pa23\">University of Economics \u2013 Varna<\/p>\n<p>Varna, 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 EDITORIAL \/ \u041a\u042a\u041c \u0427\u0418\u0422\u0410\u0422\u0415\u041b\u042f \u2013 \u0441\u0442\u0440. 111 \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 A Possibility to Teach and Learn Mathematics by Theatre Technology [\u0412\u044a\u0437\u043c\u043e\u0436\u043d\u043e\u0441\u0442 \u0437\u0430 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435 \u0438 \u0438\u0437\u0443\u0447\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0447\u0440\u0435\u0437 \u0442\u0435\u0430\u0442\u044a\u0440] \/ Sava Grozdev, Veselin Nenkov \/ \u0421\u0430\u0432\u0430 \u0413\u0440\u043e\u0437\u0434\u0435\u0432, \u0412\u0435\u0441\u0435\u043b\u0438\u043d \u041d\u0435\u043d\u043a\u043e\u0432 \u2013 \u0441\u0442\u0440. 113 \u00a0 \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043f\u0440\u043e\u0435\u043a\u0442\u0438 \u0432 \u0434\u0438\u043d\u0430\u043c\u0438\u0447\u043d\u0438 \u0441\u0440\u0435\u0434\u0438 [&hellip;]<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[3614],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - 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