{"id":42281,"date":"2017-11-15T09:22:12","date_gmt":"2017-11-15T06:22:12","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles2016-2\/6-lx-2017\/"},"modified":"2019-06-24T17:33:14","modified_gmt":"2019-06-24T14:33:14","slug":"6-lx-2017","status":"publish","type":"post","link":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/","title":{"rendered":"Mathematics and Informatics, Number 6\/2017, Volume 60"},"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\">A Survey of MathematicsDiscovered by Computers. Part 2. <\/a><br \/>\/ <em>Sava Grozdev,Hiroshi Okumura, Deko Dekov<\/em> \u2013 \u0441\u0442\u0440.\/page 543<br \/><a href=\"https:\/\/bel.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art02\">\u0422\u0440\u0438 \u0438\u043d\u0432\u0430\u0440\u0438\u0430\u043d\u0442\u044b \u0432 \u043e\u0434\u043d\u0443 \u0437\u0430\u0434\u0430\u0447\u0443 <\/a><br \/><a href=\"#art02\">[Three Invariants in One Problem]<\/a><br \/>\/ <em>\u041a\u0441\u0435\u043d\u0438\u044f<\/em><em>\u0413\u043e\u0440\u0441\u043a\u0430\u044f<\/em><em>, <\/em><em>\u0414\u0430\u0440\u044c\u044f<\/em><em>\u041a\u043e\u043f\u0442\u0435\u0432\u0430<\/em><em>,<\/em><em>\u0410\u0441\u0445\u0430\u0442 \u0415\u0440\u043c\u0435\u043a\u0431\u0430\u0435\u0432<\/em><em>, <\/em><em>\u0410\u0440\u043c\u0430\u043d<\/em><em>\u0416\u0435\u0442\u0438\u0440\u0443<\/em><em>, <\/em><em>\u0410\u0437\u0430\u0442<\/em><em>\u0411\u0435\u0440\u043c\u0443\u0445\u0430\u043c\u0435\u0434\u043e\u0432<\/em><em>, <\/em><em>\u0421\u0430\u043b\u0442\u0430\u043d\u0430\u0442<\/em><em>\u041a\u043e\u0448\u0435\u0440<\/em><em>,<\/em><em>\u041b\u0438\u043b\u0438<\/em><em>\u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430<\/em><em>, <\/em><em>\u0418\u0440\u0438\u043d\u0430<\/em><em>Hristova<\/em><em>, <\/em><em>\u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u0430<\/em><em>\u0419\u043e\u0432\u043a\u043e\u0432\u0430<\/em><em> \/ Kseniya Gorskaya, Daria Kopteva, Ashat Ermekbaev, Arman Zhetiru, Azat Bermuhamedov, Saltanat Kosher, Lilly Stefanova, Irina Hristova, Alexandra Yovkova<\/em> \u2013 \u0441\u0442\u0440.\/page 551<br \/><a href=\"https:\/\/bel.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art03\">Games with Modified Dice <\/a><br \/>\/ <em>AldiyarZhumashov<\/em> \u2013 \u0441\u0442\u0440.\/page 565<br \/><a href=\"https:\/\/bel.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/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=\"#art04\">\u0415\u043b\u0435\u043c\u0435\u043d\u0442\u0430\u0440\u043d\u0438 \u0442\u043e\u0447\u043a\u043e\u0432\u0438 \u043a\u043e\u043d\u0444\u0438\u0433\u0443\u0440\u0430\u0446\u0438\u0438. \u0414\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0435\u043d \u043f\u0440\u0438\u043d\u0446\u0438\u043f. \u0438\u043d\u0432\u0430\u0440\u0438\u0430\u043d\u0442\u0438 <\/a><br \/><a href=\"#art04\">[Basic Configurations of Points. Diagonal Principle. Invariants]<\/a><strong> <br \/>\/ <\/strong><em>\u0417\u0434\u0440\u0430\u0432\u043a\u043e \u041b\u0430\u043b\u0447\u0435\u0432, \u0418\u0440\u0438\u043d\u0430 \u0412\u0443\u0442\u043e\u0432\u0430 \/ Zdravko Lalchev, Irina Voutova<\/em> \u2013 \u0441\u0442\u0440.\/page 577<br \/><a href=\"https:\/\/bel.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art05\">\u041b\u043e\u0433\u0430\u0440\u0438\u0442\u043c\u0438\u0447\u043d\u0438 \u0438 \u043f\u043e\u043a\u0430\u0437\u0430\u0442\u0435\u043b\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 (III \u0447\u0430\u0441\u0442) <\/a><br \/><a href=\"#art05\">[Logarithmic and Exponential Functions in Transcendental Equations (Part III)] <\/a><br \/>\/ <em>\u0414\u0438\u0430\u043d\u0430 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430 \/ <\/em><em>Diana<\/em><em>Stefanova<\/em> \u2013 \u0441\u0442\u0440.\/page 601<br \/><a href=\"https:\/\/bel.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art06\">Some Numerical Sequences Concerning Square Roots (Part Two) <\/a><br \/>\/ <em>Rosen Nikolaev, Tanka Milkova, Yordan Petkov<\/em> \u2013 \u0441\u0442\u0440.\/page 616<br \/><a href=\"https:\/\/bel.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art07\">\u0417\u0430\u043d\u0438\u043c\u0430\u0442\u0435\u043b\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438 \u043f\u043e \u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u041a\u0430\u0440\u0442\u0438\u043d\u043d\u0430 \u0433\u0430\u043b\u0435\u0440\u0438\u044f\u201c <\/a><br \/><a href=\"#art07\">[Entertaining Problems on the Topic \u201ePicture Gallery\u201c]<\/a><br \/>\/ <em>\u041c\u0438\u0440\u043e\u0441\u043b\u0430\u0432 \u0421\u0442\u043e\u0438\u043c\u0438\u0440\u043e\u0432<\/em><em>, <\/em><em>\u0418\u0440\u0438\u043d\u0430 \u0412\u0443\u0442\u043e\u0432\u0430<\/em><em> \/ <\/em><em>MiroslavStoimirov<\/em><strong><em>, <\/em><\/strong><em>IrinaVoutova<\/em> \u2013 \u0441\u0442\u0440.\/page 626<br \/><a href=\"https:\/\/bel.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 this Issue] \u2013 \u0441\u0442\u0440.\/page 641<br \/><a href=\"https:\/\/bel.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 1, 2017 <br \/>[Solutions of the Contest Problems from Issue 1, 2017] \u2013 \u0441\u0442\u0440.\/page 642<br \/><a href=\"https:\/\/bel.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><strong>ANNUAL CONTENTS \/ \u0413\u041e\u0414\u0418\u0428\u041d\u041e \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415\u00a0<\/strong>\u00a0\u2013 \u0441\u0442\u0440.\/page 645<\/p>\n<p>\u00a0<\/p>\n<p><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.\/page 650<\/p>\n<p>\u00a0<\/p>\n<p><strong>GUIDE FOR AUTHORS \/ \u0423\u041a\u0410\u0417\u0410\u041d\u0418\u042f \u0417\u0410 \u0410\u0412\u0422\u041e\u0420\u0418\u0422\u0415<\/strong> \u2013 \u0441\u0442\u0440.\/page 652<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p><strong><a name=\"art01\"><\/a>A SURVEY OF MATHEMATICS DISCOVERED BY COMPUTERS. PART 2.<\/strong><\/p>\n<p class=\"avtor\" align=\"left\">\u00a0<\/p>\n<p>1)Sava Grozdev, 2)Hiroshi Okumura, 3)Deko Dekov<br \/>1) University of Finance, Business and Entrepreneurship \u2013 Sofia (Bulgaria)<br \/>2) Independent researcher \u2013 Japan<br \/>3) Independent researcher \u2013 Bulgaria<\/p>\n<p class=\"avtor\" align=\"left\">\u00a0<\/p>\n<p class=\"anotacia\" align=\"left\"><strong>Abstract.<\/strong> This is the second part of the survey on mathematics discovered by computers. The first part is published in 2015 in the International Journal of Computer Discovered Mathematics. The second part of the survey contains articles published in 2015 \u2013 2017.<\/p>\n<p class=\"anotacia\" align=\"left\"><em>Keywords: <\/em>computer discovered mathematics; learning through inquiry; learning through discovery; Euclidean geometry; problems for students<\/p>\n<p>\u00a0<\/p>\n<p>Prof. Sava Grozdev, DSc.<\/p>\n<p>University of Finance, Business and Entrepreneurship<\/p>\n<p>1618 Sofia, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p>Prof. Hiroshi Okumura, PhD<\/p>\n<p>Maebashi Gunma, 371-0123 Japan<\/p>\n<p>\u00a0<\/p>\n<p>Dr. Deko Dekov, Assoc.Prof.<\/p>\n<p>6000 Stara Zagora, 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><strong><a name=\"art02\"><\/a>\u0422\u0420\u0418 \u0418\u041d\u0412\u0410\u0420\u0418\u0410\u041d\u0422\u042b \u0412 \u041e\u0414\u041d\u0423 \u0417\u0410\u0414\u0410\u0427\u0423<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>\u041a\u0441\u0435\u043d\u0438\u044f \u0413\u043e\u0440\u0441\u043a\u0430\u044f, \u0414\u0430\u0440\u044c\u044f \u041a\u043e\u043f\u0442\u0435\u0432\u0430 &#8211; \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>\u0410\u0441\u0445\u0430\u0442 \u0415\u0440\u043c\u0435\u043a\u0431\u0430\u0435\u0432, \u0410\u0440\u043c\u0430\u043d \u0416\u0435\u0442\u0438\u0440\u0443, \u0410\u0437\u0430\u0442 \u0411\u0435\u0440\u043c\u0443\u0445\u0430\u043c\u0435\u0434\u043e\u0432, \u0421\u0430\u043b\u0442\u0430\u043d\u0430\u0442 \u041a\u043e\u0448\u0435\u0440 &#8211; \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 \u0410\u043a\u0442\u0430\u0443 (\u041a\u0430\u0437\u0430\u0445\u0441\u0442\u0430\u043d)<\/p>\n<p>\u041b\u0438\u043b\u0438 \u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u0418\u0440\u0438\u043d\u0430 \u0425\u0440\u0438\u0441\u0442\u043e\u0432\u0430, \u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u0430 \u0419\u043e\u0432\u043a\u043e\u0432\u0430 &#8211; \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 (\u0411\u043e\u043b\u0433\u0430\u0440\u0438\u044f)<\/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 \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, \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u043d\u044b\u0435 \u0432 \u0440\u0430\u043c\u043a\u0430\u0445 \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\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u043c\u043e\u0437\u0430\u0438\u043a\u0430\u201c. \u042d\u0442\u0438 \u0440\u0435\u0437\u0443\u043b\u044c\u0442\u0430\u0442\u044b \u043f\u043e\u043b\u0443\u0447\u0435\u043d\u044b \u0432 \u0445\u043e\u0434\u0435 \u0438\u0441\u0441\u043b\u0435\u0434\u043e\u0432\u0430\u043d\u0438\u044f, \u043a\u043e\u0442\u043e\u0440\u043e\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 \u0443\u0442\u0432\u0435\u0440\u0436\u0434\u0435\u043d\u0438\u0439 \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.<\/p>\n<p><em>Keywords:<\/em> triangle; circle; locus; curve of second degree; complex number<\/p>\n<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n<p><strong>THREE INVARIANTS IN ONE PROBLEM<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The paper presents results of the work of an international group of students in the frames of the realization of the research net project \u201cMathematical mosaic\u201d. The results are obtained during investigations applying software products GeoGebra, Geometer\u2019s Sketchpad and Maple. The method of the complex numbers is used for the proofs of the assertions.<\/p>\n<p>\u00a0<\/p>\n<p>Ms. Gorskaya Kseniya, Ms. Kopteva Daria<\/p>\n<p>163060 Arkhangelsk, Russia<\/p>\n<p>\u00a0<\/p>\n<p>Mr. Ermekbaev Ashat<\/p>\n<p>130200 Zhanaozen, Mangystau, Kazakhstan<\/p>\n<p>\u00a0<\/p>\n<p>Mr. Zhetiru Arman, Mr. Bermuhamedov Azat, Mr. Kosher Saltanat<\/p>\n<p>130000 Aktau, Mangystau, Kazakhstan<\/p>\n<p>\u00a0<\/p>\n<p>Ms. Lilly Stefanova, Ms. Irina Hristova, Ms. Alexandra Yovkova<\/p>\n<p>5500 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><strong><a name=\"art03\"><\/a>GAMES WITH MODIFIED DICE<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The paper studies variations of a standard six-sided dice. In particular, we consider dice with other integer numbers written on the faces, but with the same sum of those numbers. What is examined is how any two of those modified dice play against one another in a random toss. Several assertions are proved.<\/p>\n<p><em>Keywords:<\/em> dice; face; toss; modified dice; random<\/p>\n<p>\u00a0<\/p>\n<p>Mr. Aldiyar Zhumashov<\/p>\n<p>Haileybury Almaty International School<\/p>\n<p>Almaty 050040, 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><strong><a name=\"art04\"><\/a>\u0415\u041b\u0415\u041c\u0415\u041d\u0422\u0410\u0420\u041d\u0418 \u0422\u041e\u0427\u041a\u041e\u0412\u0418 \u041a\u041e\u041d\u0424\u0418\u0413\u0423\u0420\u0410\u0426\u0418\u0418. \u0414\u0418\u0410\u0413\u041e\u041d\u0410\u041b\u0415\u041d \u041f\u0420\u0418\u041d\u0426\u0418\u041f. \u0418\u041d\u0412\u0410\u0420\u0418\u0410\u041d\u0422\u0418<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>Zdravko Lalchev, Irina Voutova<\/p>\n<p>Sofia University<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u041f\u043e\u043d\u044f\u0442\u0438\u044f\u0442\u0430 \u043b\u0438\u0446\u0435 \u043d\u0430 \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a \u0438 \u043e\u0431\u0435\u043c \u043d\u0430 \u0442\u0435\u0442\u0440\u0430\u0435\u0434\u044a\u0440 \u0441\u0430 \u0438\u043d\u0442\u0435\u0440\u043f\u0440\u0435\u0442\u0438\u0440\u0430\u043d\u0438 \u043a\u0430\u0442\u043e \u0438\u043d\u0432\u0430\u0440\u0438\u0430\u043d\u0442\u0438 \u043d\u0430 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0430\u0440\u043d\u0438 \u0442\u043e\u0447\u043a\u043e\u0432\u0438 \u043a\u043e\u043d\u0444\u0438\u0433\u0443\u0440\u0430\u0446\u0438\u0438 \u0432 \u0434\u0432\u0443\u043c\u0435\u0440\u043d\u043e \u0438 \u0442\u0440\u0438\u043c\u0435\u0440\u043d\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e \u0438 \u0435 \u043d\u0430\u043c\u0435\u0440\u0435\u043d \u0442\u0435\u0445e\u043d \u0430\u043d\u0430\u043b\u043e\u0433 \u0432 \u0447\u0435\u0442\u0438\u0440\u0438\u043c\u0435\u0440\u043d\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e. \u0418\u0434\u0435\u044f\u0442\u0430 \u0437\u0430 \u0438\u043d\u0432\u0430\u0440\u0438\u0430\u043d\u0442 \u043d\u0430 \u0435\u043b\u0435\u043c\u0435\u043d\u0442\u0430\u0440\u043d\u0430 \u0442\u043e\u0447\u043a\u043e\u0432\u0430 \u043a\u043e\u043d\u0444\u0438\u0433\u0443\u0440\u0430\u0446\u0438\u044f \u0435 \u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0430 \u0438 \u0432 n-\u043c\u0435\u0440\u043d\u043e \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u043e, \u043a\u0430\u0442\u043e \u0435 \u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u0430\u043d \u043e\u0431\u0449 \u043f\u0440\u0438\u043d\u0446\u0438\u043f \u0437\u0430 \u043d\u0435\u0439\u043d\u043e\u0442\u043e \u0440\u0430\u0437\u0432\u0438\u0442\u0438\u0435. \u0418\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0438 \u0441\u0430 \u043c\u0435\u0442\u043e\u0434\u0438 \u043d\u0430 \u0430\u043d\u0430\u043b\u0438\u0442\u0438\u0447\u043d\u0430\u0442\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u044f (\u0432\u0435\u043a\u0442\u043e\u0440\u043d\u043e-\u0430\u043b\u0433\u0435\u0431-\u0440\u0438\u0447\u043d\u043e \u043c\u043e\u0434\u0435\u043b\u0438\u0440\u0430\u043d\u0435, \u0430\u0444\u0438\u043d\u043d\u0430 \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430, \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0438 \u043d\u0430 \u0442\u043e\u0447\u043a\u0430, \u043a\u043e\u043e\u0440\u0434\u0438\u043d\u0430\u0442\u0438 \u043d\u0430 \u0432\u0435\u043a\u0442\u043e\u0440) \u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430 \u043d\u0430 \u043b\u0438\u043d\u0435\u0439\u043d\u0430\u0442\u0430 \u0430\u043b\u0433\u0435\u0431\u0440\u0430 (\u0434\u0435\u0442\u0435\u0440\u043c\u0438\u043d\u043d\u0430\u0442\u0438).<\/p>\n<p><em>Keywords:<\/em> triangle; area; tetrahedron; volume; invariant; vector-algebraic modeling; affine co-ordinates; diagonal principle<\/p>\n<p>\u00a0<\/p>\n<p><strong>BASIC CONFIGURATIONS OF POINTS. DIAGONAL PRINCIPLE. INVARIANTS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The concepts of area of a triangle and volume of a tetrahedron are interpreted as invariants of basic configurations of points in two and three-dimensional spaces and their analogues are found in four-dimensional space. The idea of the invariant of a basic configuration of points is continued for n-dimensional space and a general principle for its development is formulated. Methods of analytical geometry (vector-algebraic modeling, affine co-ordinates, point co-ordinates, vector co-ordinates) and instruments of linear algebra (determinants) are used.<\/p>\n<p>\u00a0<\/p>\n<p>Prof. Dr. Zdravko Lalchev<\/p>\n<p>Faculty of Preschool and Primary Education<\/p>\n<p>University of Sofia<\/p>\n<p>1574 Sofia, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p>Dr. Irina Voutova, Assist. Prof.<\/p>\n<p>Faculty of Mathematics and Informatics<\/p>\n<p>University of Sofia<\/p>\n<p>1164 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><strong><a name=\"art05\"><\/a>\u041b\u041e\u0413\u0410\u0420\u0418\u0422\u041c\u0418\u0427\u041d\u0418 \u0418 \u041f\u041e\u041a\u0410\u0417\u0410\u0422\u0415\u041b\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 (<\/strong><strong>III<\/strong><strong> \u0427\u0410\u0421\u0422)<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>Diana Stefanova<\/p>\n<p>\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>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u043f\u044a\u0440\u0432\u0438\u0442\u0435 \u0447\u0430\u0441\u0442\u0438 \u043f\u043e \u0442\u0435\u043c\u0430\u0442\u0430 \u0437\u0430 \u0442\u0440\u0430\u043d\u0441\u0446\u0435\u043d\u0434\u0435\u043d\u0442\u043d\u0438 \u0443\u0440\u0430\u0432\u043d\u0435\u043d\u0438\u044f \u043e\u0442 \u0441\u044a\u0449\u0438\u044f \u0430\u0432\u0442\u043e\u0440 \u0432 \u0431\u0440\u043e\u0439 1\/2017 \u0433. \u0438 \u0431\u0440\u043e\u0439 4\/2017 \u0433. \u043d\u0430 \u0441\u043f. \u201e\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\u201c, \u043a\u0430\u0442\u043e \u0441\u0435\u0433\u0430 \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 \u0438 \u043b\u043e\u0433\u0430\u0440\u0438\u0442\u043c\u0438\u0447\u043d\u0438 \u0444\u0443\u043d\u043a\u0446\u0438\u0438. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d\u0438 \u0441\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \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; logarithmic function; exponential function; problem solving<\/p>\n<p>\u00a0<\/p>\n<p><strong>LOGARITHMIC AND EXPONENTIAL FUNCTIONS IN TRANSCENDENTAL EQUATIONS (PART III)<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The paper is a continuation of the first parts on the topic of transcendental equations by the same author in issues 1\/2017 and 4\/2017 of the \u201cMathematics and Informatics\u201d journal. Presently, combinations of exponential and logarithmic functions are considered. Different approaches for the solutions are proposed.<\/p>\n<p>\u00a0<\/p>\n<p>Dr. Diana Stefanova<\/p>\n<p>4230 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=\"art06\"><\/a>SOME NUMERICAL SEQUENCES CONCERNING SQUARE ROOTS (PART two)<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The present article is a continuation of the homonymous publication in the journal \u201cMathematics and Informatics\u201d, issue 5, 2017. The theoretical considerations in it cover all cases when calculating the value of a finite and infinite number of nested square radicals.<\/p>\n<p><em>Keywords:<\/em> infinite sequence; nested radical; convergence; mathematical induction<\/p>\n<p>\u00a0<\/p>\n<p>Dr. Rosen Nikolaev, Assoc. Prof.; Dr. Tanka Milkova, Assoc. Prof.; Dr. Jordan Petkov, Assist. Prof.<\/p>\n<p>University of Economics \u2013 Varna<\/p>\n<p>9002 Varna, 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=\"art07\"><\/a>\u0417\u0410\u041d\u0418\u041c\u0410\u0422\u0415\u041b\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418 \u041f\u041e \u0422\u0415\u041c\u0410\u0422\u0410 \u201e\u041a\u0410\u0420\u0422\u0418\u041d\u041d\u0410 \u0413\u0410\u041b\u0415\u0420\u0418\u042f\u201c<\/strong><\/p>\n<p>\u00a0<\/p>\n<p>\u041c\u0438\u0440\u043e\u0441\u043b\u0430\u0432 \u0421\u0442\u043e\u0438\u043c\u0438\u0440\u043e\u0432 &#8211; \u0412\u0435\u043b\u0438\u043a\u043e\u0442\u044a\u0440\u043d\u043e\u0432\u0441\u043a\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u201e\u0421\u0432. \u0441\u0432. \u041a\u0438\u0440\u0438\u043b \u0438 \u041c\u0435\u0442\u043e\u0434\u0438\u0439\u201c<\/p>\n<p>\u0418\u0440\u0438\u043d\u0430 \u0412\u0443\u0442\u043e\u0432\u0430 &#8211; \u0421\u043e\u0444\u0438\u0439\u0441\u043a\u0438 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442 \u201e\u0421\u0432. \u041a\u043b\u0438\u043c\u0435\u043d\u0442 \u041e\u0445\u0440\u0438\u0434\u0441\u043a\u0438\u201c<\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> \u041f\u0440\u0435\u0434\u043c\u0435\u0442 \u043d\u0430 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0435 \u0438\u043d\u0434\u0443\u043a\u0442\u0438\u0432\u043d\u0438\u044f\u0442 \u043f\u043e\u0434\u0445\u043e\u0434 \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u043d\u0438\u043c\u0430\u0442\u0435\u043b\u043d\u0438 \u0437\u0430\u0434\u0430\u0447\u0438, \u0441\u0432\u044a\u0440\u0437\u0430\u043d\u0438 \u0441 \u0434\u0432\u0438\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0441\u0432\u0435\u0442\u043b\u0438\u043d\u0435\u043d \u043b\u044a\u0447 \u0432 \u043f\u0440\u0430\u0432\u043e\u044a\u0433\u044a\u043b\u043d\u0430 \u0433\u0430\u043b\u0435\u0440\u0438\u044f \u0441 \u043e\u0433\u043b\u0435\u0434\u0430\u043b\u043d\u0438 \u0441\u0442\u0435\u043d\u0438, \u0432 \u0442\u0440\u0438 \u043e\u0442 \u044a\u0433\u043b\u0438\u0442\u0435 \u043d\u0430 \u043a\u043e\u044f\u0442\u043e \u0441\u0430 \u043f\u043e\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u043a\u0430\u0440\u0442\u0438\u043d\u0438. \u0420\u0430\u0437\u0433\u043b\u0435\u0434\u0430\u043d\u0438 \u0441\u0430 \u0440\u0435\u0434\u0438\u0446\u0430 \u0441\u043b\u0443\u0447\u0430\u0438 \u0438 \u0441\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u0440\u0435\u0448\u0435\u043d\u0438\u044f \u043d\u0430 \u0442\u0440\u0438 \u0433\u0440\u0443\u043f\u0438 \u0437\u0430\u0434\u0430\u0447\u0438. \u0412 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442 \u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u043f\u043e\u0441\u0442\u0440\u043e\u0435\u043d\u0438\u044f \u0441\u0430 \u0441\u044a\u0431\u0440\u0430\u043d\u0438 \u0434\u0430\u043d\u043d\u0438 (\u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0438 \u0441\u0430 \u0433\u0430\u043b\u0435\u0440\u0438\u0438 \u0441 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u0440\u0430\u0437\u043c\u0435\u0440\u0438), \u0443\u0441\u0442\u0430\u043d\u043e\u0432\u0435\u043d\u0438 \u0441\u0430 \u0437\u0430\u043a\u043e\u043d\u043e\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438 \u0438 \u0441\u0430 \u0444\u043e\u0440\u043c\u0443\u043b\u0438\u0440\u0430\u043d\u0438 \u0438\u0437\u0432\u043e\u0434\u0438 (\u0445\u0438\u043f\u043e\u0442\u0435\u0437\u0438). \u041d\u0430 \u043e\u0441\u043d\u043e\u0432\u0430 \u043d\u0430 \u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0442\u0435 \u0437\u0430\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u044f \u043c\u043e\u0433\u0430\u0442 \u0434\u0430 \u0431\u044a\u0434\u0430\u0442 \u043f\u0440\u043e\u0433\u043d\u043e\u0437\u0438\u0440\u0430\u043d\u0438 \u0442\u0440\u0430\u0435\u043a\u0442\u043e\u0440\u0438\u044f\u0442\u0430 \u043d\u0430 \u0441\u0432\u0435\u0442\u043b\u0438\u043d\u043d\u0438\u044f \u043b\u044a\u0447, \u0431\u0440\u043e\u044f\u0442 \u043d\u0430 \u043d\u0435\u0433\u043e\u0432\u0438\u0442\u0435 \u043e\u0442\u0440\u0430\u0436\u0435\u043d\u0438\u044f \u0438 \u043d\u043e\u043c\u0435\u0440\u044a\u0442 \u043d\u0430 \u043e\u0441\u0432\u0435\u0442\u0435\u043d\u0430\u0442\u0430 \u043a\u0430\u0440\u0442\u0438\u043d\u0430. \u0412 \u043c\u0435\u0442\u043e\u0434\u043e\u043b\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u043e \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u0435 \u0438\u0437\u0441\u043b\u0435\u0434\u0432\u0430\u043d\u0435\u0442\u043e \u0435 \u043f\u0440\u043e\u0434\u044a\u043b\u0436\u0435\u043d\u0438\u0435 \u043d\u0430 \u0433\u0435\u043e\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438\u044f \u043c\u0435\u0442\u043e\u0434 \u043d\u0430 \u041f\u0435\u0440\u0435\u043b\u043c\u0430\u043d \u0437\u0430 \u0440\u0435\u0448\u0430\u0432\u0430\u043d\u0435 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0430\u0442\u0430 \u043d\u0430 \u041f\u043e\u0430\u0441\u043e\u043d.<\/p>\n<p><em>Keywords:<\/em> picture; gallery; ray; reflection; rectangle; co-ordinate net; incomplete induction<\/p>\n<p>\u00a0<\/p>\n<p><strong>ENTERTAINING PROBLEMS ON THE TOPIC \u201ePICTURE GALLERY\u201c<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract.<\/strong> The topic of the article is the inductive approach to solve entertaining problems, connected with the motion of a light ray in a rectangular gallery with mirror walls and pictures in three of its corners. Several cases are considered and geometric solutions of three groups of problems are presented. Data is collected as a result of geometric constructions (galleries with different dimensions are examined), regularities are established and conclusions (hypotheses) are formulated. What are prognosticated on the base of the derived conclusions are the trajectory of the light ray, the number of its reflections and the number of the illuminated picture. In methodological aspect the study is a continuation of the Perelman\u2019s geometric method for solving the Poisson\u2019s problem.<\/p>\n<p>\u00a0<\/p>\n<p>Mr. Miroslav Stoimirov, PhD student<\/p>\n<p>Faculty of Mathematics and Informatics<\/p>\n<p>University of Veliko Tarnovo<\/p>\n<p>5000 Veliko Tarnovo, Bulgaria<\/p>\n<p>\u00a0<\/p>\n<p>Dr. Irina Voutova, Assist. Prof.<\/p>\n<p>Faculty of Mathematics and Informatics<\/p>\n<p>University of Sofia<\/p>\n<p>1164 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>","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 A Survey of MathematicsDiscovered by Computers. Part 2. \/ Sava Grozdev,Hiroshi Okumura, Deko Dekov \u2013 \u0441\u0442\u0440.\/page 543 \u00a0 \u0422\u0440\u0438 \u0438\u043d\u0432\u0430\u0440\u0438\u0430\u043d\u0442\u044b \u0432 \u043e\u0434\u043d\u0443 \u0437\u0430\u0434\u0430\u0447\u0443 [Three Invariants in One Problem]\/ \u041a\u0441\u0435\u043d\u0438\u044f\u0413\u043e\u0440\u0441\u043a\u0430\u044f, \u0414\u0430\u0440\u044c\u044f\u041a\u043e\u043f\u0442\u0435\u0432\u0430,\u0410\u0441\u0445\u0430\u0442 \u0415\u0440\u043c\u0435\u043a\u0431\u0430\u0435\u0432, \u0410\u0440\u043c\u0430\u043d\u0416\u0435\u0442\u0438\u0440\u0443, \u0410\u0437\u0430\u0442\u0411\u0435\u0440\u043c\u0443\u0445\u0430\u043c\u0435\u0434\u043e\u0432, \u0421\u0430\u043b\u0442\u0430\u043d\u0430\u0442\u041a\u043e\u0448\u0435\u0440,\u041b\u0438\u043b\u0438\u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u0418\u0440\u0438\u043d\u0430\u0425\u0440\u0438\u0441\u0442\u043e\u0432\u0430, \u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u0430\u0419\u043e\u0432\u043a\u043e\u0432\u0430 \/ Kseniya Gorskaya, Daria Kopteva, Ashat Ermekbaev, Arman Zhetiru, Azat [&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":[3486],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>&quot;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&quot;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LX, 2017 - \u0410\u0437-\u0431\u0443\u043a\u0438<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"&quot;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&quot;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LX, 2017 - \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 A Survey of MathematicsDiscovered by Computers. Part 2. \/ Sava Grozdev,Hiroshi Okumura, Deko Dekov \u2013 \u0441\u0442\u0440.\/page 543 \u00a0 \u0422\u0440\u0438 \u0438\u043d\u0432\u0430\u0440\u0438\u0430\u043d\u0442\u044b \u0432 \u043e\u0434\u043d\u0443 \u0437\u0430\u0434\u0430\u0447\u0443 [Three Invariants in One Problem]\/ \u041a\u0441\u0435\u043d\u0438\u044f\u0413\u043e\u0440\u0441\u043a\u0430\u044f, \u0414\u0430\u0440\u044c\u044f\u041a\u043e\u043f\u0442\u0435\u0432\u0430,\u0410\u0441\u0445\u0430\u0442 \u0415\u0440\u043c\u0435\u043a\u0431\u0430\u0435\u0432, \u0410\u0440\u043c\u0430\u043d\u0416\u0435\u0442\u0438\u0440\u0443, \u0410\u0437\u0430\u0442\u0411\u0435\u0440\u043c\u0443\u0445\u0430\u043c\u0435\u0434\u043e\u0432, \u0421\u0430\u043b\u0442\u0430\u043d\u0430\u0442\u041a\u043e\u0448\u0435\u0440,\u041b\u0438\u043b\u0438\u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u0418\u0440\u0438\u043d\u0430\u0425\u0440\u0438\u0441\u0442\u043e\u0432\u0430, \u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u0430\u0419\u043e\u0432\u043a\u043e\u0432\u0430 \/ Kseniya Gorskaya, Daria Kopteva, Ashat Ermekbaev, Arman Zhetiru, Azat [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/\" \/>\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=\"2017-11-15T06:22:12+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:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/\"},\"author\":{\"name\":\"\",\"@id\":\"\"},\"headline\":\"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LX, 2017\",\"datePublished\":\"2017-11-15T06:22:12+00:00\",\"dateModified\":\"2019-06-24T14:33:14+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/\"},\"wordCount\":1593,\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/#organization\"},\"articleSection\":[\"\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\", 2017 \u0433.\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/\",\"url\":\"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/\",\"name\":\"\\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\", \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LX, 2017 - 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Part 2. \/ Sava Grozdev,Hiroshi Okumura, Deko Dekov \u2013 \u0441\u0442\u0440.\/page 543 \u00a0 \u0422\u0440\u0438 \u0438\u043d\u0432\u0430\u0440\u0438\u0430\u043d\u0442\u044b \u0432 \u043e\u0434\u043d\u0443 \u0437\u0430\u0434\u0430\u0447\u0443 [Three Invariants in One Problem]\/ \u041a\u0441\u0435\u043d\u0438\u044f\u0413\u043e\u0440\u0441\u043a\u0430\u044f, \u0414\u0430\u0440\u044c\u044f\u041a\u043e\u043f\u0442\u0435\u0432\u0430,\u0410\u0441\u0445\u0430\u0442 \u0415\u0440\u043c\u0435\u043a\u0431\u0430\u0435\u0432, \u0410\u0440\u043c\u0430\u043d\u0416\u0435\u0442\u0438\u0440\u0443, \u0410\u0437\u0430\u0442\u0411\u0435\u0440\u043c\u0443\u0445\u0430\u043c\u0435\u0434\u043e\u0432, \u0421\u0430\u043b\u0442\u0430\u043d\u0430\u0442\u041a\u043e\u0448\u0435\u0440,\u041b\u0438\u043b\u0438\u0421\u0442\u0435\u0444\u0430\u043d\u043e\u0432\u0430, \u0418\u0440\u0438\u043d\u0430\u0425\u0440\u0438\u0441\u0442\u043e\u0432\u0430, \u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u0430\u0419\u043e\u0432\u043a\u043e\u0432\u0430 \/ Kseniya Gorskaya, Daria Kopteva, Ashat Ermekbaev, Arman Zhetiru, Azat [&hellip;]","og_url":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2017-11-15T06:22:12+00:00","article_modified_time":"2019-06-24T14:33:14+00:00","og_image":[{"url":"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png"}],"twitter_card":"summary_large_image","twitter_misc":{"Written by":"","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/#article","isPartOf":{"@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/"},"author":{"name":"","@id":""},"headline":"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LX, 2017","datePublished":"2017-11-15T06:22:12+00:00","dateModified":"2019-06-24T14:33:14+00:00","mainEntityOfPage":{"@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/"},"wordCount":1593,"publisher":{"@id":"https:\/\/azbuki.bg\/#organization"},"articleSection":["\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", 2017 \u0433."],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/","url":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-2\/6-lx-2017\/","name":"\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", \u043a\u043d\u0438\u0436\u043a\u0430 6, \u0433\u043e\u0434\u0438\u043d\u0430 LX, 2017 - 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