{"id":43399,"date":"2019-03-08T12:19:33","date_gmt":"2019-03-08T09:19:33","guid":{"rendered":"https:\/\/new.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/"},"modified":"2019-06-18T21:58:26","modified_gmt":"2019-06-18T18:58:26","slug":"1-lxii-2019","status":"publish","type":"post","link":"https:\/\/bel.azbuki.bg\/en\/matematics\/matharticles2016-4\/1-lxii-2019\/","title":{"rendered":"Mathematics and Informatics, Number 1\/2019, Volume 62"},"content":{"rendered":"<p class=\"text1\"><a name=\"top\"><\/a><strong>CONTENTS of issue 1\/2019 \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \u043d\u0430 \u0431\u0440\u043e\u0439 1\/2019<\/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 p. 7<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_Editorial.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open 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><a href=\"#art01\">\u0410\u0440\u0445\u0438\u0432\u0438\u0442\u0435 \u0433\u043e\u0432\u043e\u0440\u044f\u0442: \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f\u0442\u0430 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 <\/a><br \/><a href=\"#art01\">[The Archives Speak: National Competitions in Informatics] \/ <\/a><br \/><em>\u041f\u0430\u0432\u0435\u043b \u0410\u0437\u044a\u043b\u043e\u0432 \/ Pavel Azalov <\/em>\u2013 p. 11<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_PavelAzalov.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art02\">\u041c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043a\u0443\u043b\u044c\u0442\u0443\u0440\u044b \u043f\u0435\u0434\u0430\u0433\u043e\u0433\u043e\u0432 \u2013 \u043f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u043e\u0432 \u0432 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0441\u0440\u0435\u0434\u0435 <\/a><br \/><a href=\"#art02\">[A Methodical System for the Formation of Information Culture of Teachers \u2013 Psychologists in Information Educational Environment] \/ <\/a><br \/><em>\u0413\u0435\u043d\u043d\u0430\u0434\u0438\u0439 \u041a\u0438\u0441\u0435\u043b\u0435\u0432, \u0410\u043b\u044c\u0431\u0438\u043d\u0430 \u0427\u0435\u0440\u0432\u043e\u0432\u0430 \/ Gennady Kiselev, Albina Chervova <\/em>\u2013 p. 32<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_Kiselev_Chervova.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art03\">Increasing the Digital Competences of Students \/ <\/a><br \/><em>Lyubka Slavova, Kosta Garov <\/em>\u2013 p. 43<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_Slavova_Garov.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p><em>\u00a0<\/em><\/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><a href=\"#art04\">\u0421\u0438\u0441\u0442\u0435\u043c\u0430 \u043e\u0442 \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=\"#art04\">[System of Tasks in Mathematics Education] \/ <\/a><br \/><em>\u041f\u0435\u0442\u044f \u0410\u0441\u0435\u043d\u043e\u0432\u0430, \u041c\u0430\u0440\u0438\u043d \u041c\u0430\u0440\u0438\u043d\u043e\u0432 \/ Petya Asenova, Marin Marinov <\/em>\u2013 p. 53<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_Asenova_Marinov.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art05\">Euler- Gergonne&#8217; s Theorem and Its Applications \/ <\/a><br \/><em>\u0160efket Arslanagi\u0107 <\/em>\u2013 p. 72<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_%C5%A0efketArslanagi%C4%87.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art06\">\u0415\u0434\u043d\u043e \u0442\u0432\u044a\u0440\u0434\u0435\u043d\u0438\u0435 \u0437\u0430 \u043a\u043e\u043d\u043a\u0443\u0440\u0435\u043d\u0442\u043d\u043e\u0441\u0442 \u043d\u0430 \u043f\u0435\u0434\u0430\u043b\u043d\u0438 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u0442\u043e\u0447\u043a\u0430 \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 <\/a><br \/><a href=\"#art06\">[A Statement for Concurrence of Pedal Circles of a Point in the Plane of a Quadrilateral] \/ <\/a><br \/><em>\u0425\u0430\u0438\u043c \u0425\u0430\u0438\u043c\u043e\u0432 \/ Haim Haimov <\/em>\u2013 p. 77<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_HaimHaimov.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art07\">Polynomials of Fourth Degree with Colinear Central Symmetric Roots <\/a><br \/><a href=\"#art07\">[\u041f\u043e\u043b\u0438\u043d\u043e\u043c\u0438 \u043e\u0442 \u0447\u0435\u0442\u0432\u044a\u0440\u0442\u0430 \u0441\u0442\u0435\u043f\u0435\u043d \u0441 \u043a\u043e\u043b\u0438\u043d\u0435\u0430\u0440\u043d\u0438 \u0446\u0435\u043d\u0442\u0440\u0430\u043b\u043d\u043e \u0441\u0438\u043c\u0435\u0442\u0440\u0438\u0447\u043d\u0438 \u043a\u043e\u0440\u0435\u043d\u0438] <\/a><em><a href=\"#art07\">\/ <\/a><br \/>Sava Grozdev, Veselin Nenkov <\/em>\u2013 p. 91<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_GrozdevNenkov.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p>\u00a0<\/p>\n<p><a href=\"#art08\">Problem 6. From IMO\u20192018 <\/a><br \/><a href=\"#art08\">[\u0417\u0430\u0434\u0430\u0447\u0430 6. \u041e\u0442 IMO\u20192018] \/ <\/a><br \/><em>Sava Grozdev, Veselin Nenkov <\/em>\u2013 p. 97<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_GrozdevNenkov_2.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p class=\"text1\" align=\"left\">\u00a0<\/p>\n<p class=\"text1\" align=\"left\"><strong>CONTEST PROBLEMS \/ \u041a\u041e\u041d\u041a\u0423\u0420\u0421\u041d\u0418 \u0417\u0410\u0414\u0410\u0427\u0418<\/strong><\/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 2, 2018 <br \/>[Solutions of the Contest Problems from Issue 2, 2018] \u2013 p.108<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_ContestProblems_reshenia.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open the full text<\/a><\/p>\n<p class=\"text1\">\u00a0<\/p>\n<p class=\"text1\">Contest Problems of this Issue <br \/>[Contest Problems of this Issue] \u2013 p.113<br \/><a href=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/03\/azbuki.bg_dmdocuments_Math_Info_1_19_ContestProblems.pdf\"><img decoding=\"async\" loading=\"lazy\" class=\"size-full wp-image-38022\" src=\"https:\/\/azbuki.bg\/wp-content\/uploads\/2012\/08\/components_com_docman_themes_default_images_icons_16x16_pdf.png\" border=\"0\" alt=\"icon\" width=\"16\" height=\"16\" \/>\u00a0Open 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 p. 115<\/p>\n<p style=\"text-align: right;\"><a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p><strong><a name=\"art01\"><\/a>\u0410\u0420\u0425\u0418\u0412\u0418\u0422\u0415 \u0413\u041e\u0412\u041e\u0420\u042f\u0422: \u041d\u0410\u0426\u0418\u041e\u041d\u0410\u041b\u041d\u0418 \u0421\u042a\u0421\u0422\u0415\u0417\u0410\u041d\u0418\u042f\u0422\u0410 \u041f\u041e \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0422\u0418\u041a\u0410<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p><strong>Pavel Azalov<\/strong><\/p>\n<p><em>Penn State, Hazleton, U.S.A.<\/em><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>Abstract<\/strong>. \u0412 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0441\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u043e\u0441\u043d\u043e\u0432\u043d\u0438\u0442\u0435 \u0441\u044a\u0431\u0438\u0442\u0438\u044f, \u0434\u043e\u043f\u0440\u0438\u043d\u0435\u0441\u043b\u0438 \u0437\u0430 \u0438\u043d\u0438\u0446\u0438\u0438\u0440\u0430\u043d\u0435\u0442\u043e \u0438 \u0443\u0442\u0432\u044a\u0440\u0436\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u043f\u044a\u0440\u0432\u0438\u0442\u0435 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 \u0437\u0430 \u0443\u0447\u0435\u043d\u0438\u0446\u0438 \u0432 \u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f. \u041f\u043e\u0434\u0440\u043e\u0431\u043d\u043e \u0435 \u043e\u043f\u0438\u0441\u0430\u043d\u0430 \u043f\u0440\u0435\u0434\u0438\u0441\u0442\u043e\u0440\u0438\u044f\u0442\u0430 \u043d\u0430 \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0430\u0442\u0430 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0430 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 (\u041d\u041e\u0418). \u0422\u044f \u0432\u043a\u043b\u044e\u0447\u0432\u0430 \u043f\u044a\u0440\u0432\u0438\u0442\u0435 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f\u0442\u0430 \u043f\u043e \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435 \u0443 \u043d\u0430\u0441. \u0422\u0435 \u0441\u0430 \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0438 \u0432 \u0428\u0443\u043c\u0435\u043d, \u041f\u043b\u043e\u0432\u0434\u0438\u0432 \u0438 \u041f\u0430\u0437\u0430\u0440\u0434\u0436\u0438\u043a \u0441\u044a\u043e\u0442\u0432\u0435\u0442\u043d\u043e \u043f\u0440\u0435\u0437 1982, 1983 \u0438 1984 \u0433\u043e\u0434\u0438\u043d\u0430 \u0438 \u0441\u0430 \u0438\u043d\u0438\u0446\u0438\u0438\u0440\u0430\u043d\u0438 \u043e\u0442 \u0430\u0432\u0442\u043e\u0440\u0430 \u043d\u0430 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430. \u0412 \u0442\u044f\u0445 \u0443\u0447\u0435\u043d\u0438\u0446\u0438\u0442\u0435 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u0442 \u043a\u043e\u043c\u043f\u044e\u0442\u0440\u0438 \u043e\u0442 \u0415\u0434\u0438\u043d\u043d\u0430\u0442\u0430 \u0441\u0435\u0440\u0438\u044f \u0438 \u0424\u041e\u0420\u0422\u0420\u0410\u041d \u043a\u0430\u0442\u043e \u043e\u0441\u043d\u043e\u0432\u0435\u043d \u0435\u0437\u0438\u043a \u0437\u0430 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u0438\u0440\u0430\u043d\u0435. \u041f\u043e\u0434\u0440\u043e\u0431\u043d\u043e \u0435 \u043e\u043f\u0438\u0441\u0430\u043d\u0430 \u041f\u044a\u0440\u0432\u0430\u0442\u0430 \u041d\u041e\u0418, \u043f\u0440\u043e\u0432\u0435\u0434\u0435\u043d\u0430 \u043f\u0440\u0435\u0437 1985 \u0433. \u041f\u043e-\u043d\u0430\u043a\u0440\u0430\u0442\u043a\u043e \u0441\u0430 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438 \u043d\u044f\u043a\u043e\u0438 \u043e\u0442 \u0441\u043b\u0435\u0434\u0432\u0430\u0449\u0438\u0442\u0435 \u043e\u043b\u0438\u043c\u043f\u0438\u0430\u0434\u0438. \u0414\u0430\u0434\u0435\u043d\u0438 \u0441\u0430 \u0441\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0437\u0430 \u0417\u0438\u043c\u043d\u0438\u0442\u0435 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438 \u043f\u0440\u0430\u0437\u043d\u0438\u0446\u0438 \u0438 \u0443\u0447\u0435\u043d\u0438\u0447\u0435\u0441\u043a\u0430\u0442\u0430 \u0441\u0435\u043a\u0446\u0438\u044f \u043d\u0430 \u041f\u0440\u043e\u043b\u0435\u0442\u043d\u0438\u0442\u0435 \u043a\u043e\u043d\u0444\u0435\u0440\u0435\u043d\u0446\u0438\u0438 \u043d\u0430 \u0421\u044a\u044e\u0437\u0430 \u043d\u0430 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0446\u0438\u0442\u0435 \u0432 \u0411\u044a\u043b\u0433\u0430\u0440\u0438\u044f. \u041f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0430 \u0435 \u0438 \u043a\u0440\u0430\u0442\u043a\u0430 \u0438\u0441\u0442\u043e\u0440\u0438\u044f \u043e\u0442 \u0441\u044a\u0437\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u041f\u0440\u043e\u043b\u0435\u0442\u043d\u0438\u044f \u0442\u0443\u0440\u043d\u0438\u0440 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430. \u041f\u043e\u0440\u0430\u0434\u0438 \u0434\u043e\u043a\u0443\u043c\u0435\u043d\u0442\u0430\u043b\u043d\u0438\u044f \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440 \u043d\u0430 \u0441\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0432 \u043d\u0435\u044f \u0441\u0430 \u0446\u0438\u0442\u0438\u0440\u0430\u043d\u0438 \u043c\u043d\u043e\u0433\u043e \u0441\u0442\u0430\u0442\u0438\u0438, \u0438\u043c\u0430 \u0438\u0437\u0432\u0430\u0434\u043a\u0438 \u043e\u0442 \u0442\u0435\u043a\u0441\u0442\u043e\u0432\u0435 \u043d\u0430 \u043f\u0443\u0431\u043b\u0438\u043a\u0430\u0446\u0438\u0438, \u043a\u0430\u043a\u0442\u043e \u0438 \u0441\u043d\u0438\u043c\u043a\u0438, \u0441 \u043a\u043e\u0438\u0442\u043e \u0442\u043e\u0447\u043d\u043e \u0441\u0435 \u0434\u043e\u043a\u0443\u043c\u0435\u043d\u0442\u0438\u0440\u0430\u0442 \u0444\u0430\u043a\u0442\u0438\u0442\u0435 \u0432 \u043f\u0440\u0435\u0434\u0441\u0442\u0430\u0432\u0435\u043d\u0438\u0442\u0435 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430.<\/p>\n<p><em>Keywords<\/em>: programming contests; \u043elympiad in informatics; national olympiad in informatics; NOI; national spring tournament in informatics<\/p>\n<p>\u00a0<\/p>\n<p><strong>THE ARCHIVES SPEAK: NATIONAL COMPETITIONS IN INFORMATICS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract<\/strong>. The article discusses the key events leading to the ideation, initiation, and establishment of the first programming contests for high school students in Bulgaria. The competitions preceding the First National Olympiad in Informatics (NOI) are presented in detail. More specifically, they include the programming contests, initiated by the author and held in Shoumen, Plovdiv and Pazardjik in 1982, 1983, and 1984 respectively. At the time, the main programming language used was FORTRAN for IBM\/360 compatible computers. Next, the first NOI is discussed at length. Some of the subsequent NOIs are also outlined. The paper also includes facts about the Winter Mathematics Competitions and the high school student competitions held during the Spring Conferences of the Union of Bulgarian Mathematicians. A brief historical data about the Spring Tournament in Informatics is also presented. Due to its factual nature, the article contains an extensive list of references, including text excerpts, citations, pictures, and other relevant materials, supporting the historical facts related to the competitions in informatics.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\">Dr. Pavel Azalov<\/p>\n<p class=\"Pa22\">Professor Emeritus of Computer Science<\/p>\n<p>Penn State, U.S.A.<\/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>\u041c\u0415\u0422\u041e\u0414\u0418\u0427\u0415\u0421\u041a\u0410\u042f \u0421\u0418\u0421\u0422\u0415\u041c\u0410 \u0424\u041e\u0420\u041c\u0418\u0420\u041e\u0412\u0410\u041d\u0418\u042f \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0426\u0418\u041e\u041d\u041d\u041e\u0419 \u041a\u0423\u041b\u042c\u0422\u0423\u0420\u042b \u041f\u0415\u0414\u0410\u0413\u041e\u0413\u041e\u0412 \u2013 \u041f\u0421\u0418\u0425\u041e\u041b\u041e\u0413\u041e\u0412 \u0412 \u0418\u041d\u0424\u041e\u0420\u041c\u0410\u0426\u0418\u041e\u041d\u041d\u041e\u0419 \u041e\u0411\u0420\u0410\u0417\u041e\u0412\u0410\u0422\u0415\u041b\u042c\u041d\u041e\u0419 \u0421\u0420\u0415\u0414\u0415<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\"><strong>\u041a\u0438\u0441\u0435\u043b\u0435\u0432, \u0413\u0435\u043d\u043d\u0430\u0434\u0438\u0439 \u041c\u0438\u0445\u0430\u0439\u043b\u043e\u0432\u0438\u0447,<\/strong> <br \/><em>\u0410\u041d\u041e \u0412\u041e \u00ab\u041c\u043e\u0441\u043a\u043e\u0432\u0441\u043a\u0438\u0439 \u0440\u0435\u0433\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b\u0439 \u0441\u043e\u0446\u0438\u0430\u043b\u044c\u043d\u043e-\u044d\u043a\u043e\u043d\u043e\u043c\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0438\u043d\u0441\u0442\u0438\u0442\u0443\u0442\u00bb (\u0420\u043e\u0441\u0441\u0438\u044f)<\/em><\/p>\n<p><strong>\u0427\u0435\u0440\u0432\u043e\u0432\u0430, \u0410\u043b\u044c\u0431\u0438\u043d\u0430 \u0410\u043b\u0435\u043a\u0441\u0430\u043d\u0434\u0440\u043e\u0432\u043d\u0430,<\/strong> <br \/><em>\u0428\u0443\u0439\u0441\u043a\u0438\u0439 \u0444\u0438\u043b\u0438\u0430\u043b \u0424\u0413\u0411\u041e\u0423 \u0412\u041e \u00ab\u0418\u0432\u0430\u043d\u043e\u0432\u0441\u043a\u0438\u0439 \u0433\u043e\u0441\u0443\u0434\u0430\u0440\u0441\u0442\u0432\u0435\u043d\u043d\u044b\u0439 \u0443\u043d\u0438\u0432\u0435\u0440\u0441\u0438\u0442\u0435\u0442\u00bb (\u0420\u043e\u0441\u0441\u0438\u044f)<\/em><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>\u0410\u043d\u043d\u043e\u0442\u0430\u0446\u0438\u044f<\/strong>. \u041f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0435 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0445 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0439 \u0432 \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e\u043c \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u043c \u043f\u0440\u043e\u0441\u0442\u0440\u0430\u043d\u0441\u0442\u0432\u0435 \u043f\u0440\u043e\u0434\u043e\u043b\u0436\u0430\u0435\u0442 \u0432\u044b\u0437\u044b\u0432\u0430\u0442\u044c \u043f\u043e\u0432\u044b\u0448\u0435\u043d\u043d\u044b\u0439 \u0438\u043d\u0442\u0435\u0440\u0435\u0441 \u043a\u0430\u043a \u043e\u0442\u0435\u0447\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e\u0439, \u0442\u0430\u043a \u0438 \u0437\u0430\u0440\u0443\u0431\u0435\u0436\u043d\u043e\u0439 \u043f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u043e-\u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u043d\u0430\u0443\u043a\u0438. \u042d\u0442\u043e \u0432 \u043f\u0435\u0440\u0432\u0443\u044e \u043e\u0447\u0435\u0440\u0435\u0434\u044c \u0441\u0432\u044f\u0437\u0430\u043d\u043e \u0441\u043e \u0441\u0442\u0440\u0435\u043c\u043b\u0435\u043d\u0438\u0435\u043c \u0443\u0447\u0430\u0441\u0442\u043d\u0438\u043a\u043e\u0432 \u043f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u043e-\u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u043e\u0433\u043e \u043f\u0440\u043e\u0446\u0435\u0441\u0441\u0430 \u0441\u043e\u0432\u0435\u0440\u0448\u0435\u043d\u0441\u0442\u0432\u043e\u0432\u0430\u0442\u044c \u0441\u0432\u043e\u0438 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0438. \u0424\u0440\u0430\u0433\u043c\u0435\u043d\u0442\u0430\u0440\u043d\u043e\u0441\u0442\u044c \u0441\u0443\u0449\u0435\u0441\u0442\u0432\u0443\u044e\u0449\u0435\u0439 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0438 \u0432 \u0432\u0443\u0437\u0435, \u043d\u0430\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u043d\u043e\u0439 \u043f\u0440\u0435\u0438\u043c\u0443\u0449\u0435\u0441\u0442\u0432\u0435\u043d\u043d\u043e \u043d\u0430 \u0438\u043d\u0441\u0442\u0440\u0443\u043c\u0435\u043d\u0442\u0430\u043b\u044c\u043d\u043e-\u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438\u0439 \u0443\u0440\u043e\u0432\u0435\u043d\u044c \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u0434\u0435\u044f\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438, \u0435\u0435 \u0440\u0435\u043f\u0440\u043e\u0434\u0443\u043a\u0442\u0438\u0432\u043d\u044b\u0439 \u0445\u0430\u0440\u0430\u043a\u0442\u0435\u0440 \u043d\u0435 \u0441\u043f\u043e\u0441\u043e\u0431\u0441\u0442\u0432\u0443\u0435\u0442 \u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044e \u043c\u0438\u0440\u043e\u0432\u043e\u0437\u0437\u0440\u0435\u043d\u0447\u0435\u0441\u043a\u0438\u0445, \u0446\u0435\u043d\u043d\u043e\u0441\u0442\u043d\u043e-\u0441\u043c\u044b\u0441\u043b\u043e\u0432\u044b\u0445 \u0430\u0441\u043f\u0435\u043a\u0442\u043e\u0432 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u0434\u0435\u044f\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438 \u0431\u0443\u00ad\u0434\u0443\u0449\u0438\u0445 \u043f\u0435\u0434\u0430\u0433\u043e\u0433\u043e\u0432-\u043f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u043e\u0432. \u0412 \u0441\u0442\u0430\u0442\u044c\u0435 \u0440\u0430\u0441\u0441\u043c\u0430\u0442\u0440\u0438\u0432\u0430\u044e\u0442\u0441\u044f \u043e\u0441\u043e\u0431\u0435\u043d\u043d\u043e\u0441\u0442\u0438 \u0440\u0430\u0437\u0440\u0430\u0431\u043e\u0442\u043a\u0438 \u0438 \u0440\u0435\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u0438 \u043c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0441\u0438\u0441\u0442\u0435\u043c\u044b \u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043a\u0443\u043b\u044c\u0442\u0443\u0440\u044b \u043f\u0435\u0434\u0430\u0433\u043e\u0433\u043e\u0432 \u2013 \u043f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u043e\u0432 \u0432 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e\u0439 \u0441\u0440\u0435\u0434\u0435 \u0432\u0443\u0437\u0430. \u0410\u0432\u0442\u043e\u0440\u044b \u043f\u044b\u0442\u0430\u044e\u0442\u0441\u044f \u0441\u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u0442\u044c \u0438 \u043f\u043e\u0432\u044b\u0441\u0438\u0442\u044c \u0443\u0440\u043e\u0432\u0435\u043d\u044c \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043a\u0443\u043b\u044c\u0442\u0443\u0440\u044b \u0438\u043d\u0442\u0435\u0433\u0440\u0438\u0440\u0443\u044f \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0435 \u0438 \u043f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u043e-\u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u0434\u0438\u0441\u0446\u0438\u043f\u043b\u0438\u043d\u044b \u043f\u0440\u0438 \u043f\u043e\u0441\u0442\u043e\u044f\u043d\u043d\u043e\u043c \u043f\u0440\u0438\u043c\u0435\u043d\u0435\u043d\u0438\u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0445 \u0438 \u043a\u043e\u043c\u043c\u0443\u043d\u0438\u043a\u0430\u0446\u0438\u043e\u043d\u043d\u044b\u0445 \u0442\u0435\u0445\u043d\u043e\u043b\u043e\u0433\u0438\u0439 \u043d\u0430 \u0432\u0441\u0435\u0445 \u0432\u0438\u0434\u0430\u0445 \u0438 \u0444\u043e\u0440\u043c\u0430\u0445 \u043f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u043e-\u043f\u0435\u0434\u0430\u0433\u043e\u0433\u0438\u0447\u0435\u0441\u043a\u043e\u0439 \u0434\u0435\u044f\u0442\u0435\u043b\u044c\u043d\u043e\u0441\u0442\u0438.<\/p>\n<p><em>Keywords: <\/em>information culture; information technologies in professional activity; information educational environment; methodical system<\/p>\n<p>\u00a0<\/p>\n<p><strong>A METHODICAL SYSTEM FOR THE FORMATION OF INFORMATION CULTURE OF TEACHERS \u2013 PSYCHOLOGISTS IN INFORMATION EDUCATIONAL ENVIRONMENT<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract. <\/strong>The use of information technologies in the modern educational environment continues to cause increased interest in both domestic and foreign psychological and pedagogical science. This is primarily due to the desire of participants in the psychological and pedagogical process to improve their technologies. The fragmentary nature of the existing information training in universities, aimed mainly at instrumental and technological level of information activity, its reproductive nature does not contribute to the formation of ideological, value-semantic aspects of information activities of future teachers-psychologists. The article discusses the features of the development and implementation of a methodical system for the formation of information culture of teachers \u2013 psychologists in the information educational environment in universities. The authors try to form and improve the level of information culture by integrating information, psychological and pedagogical disciplines with the constant use of information and communication technologies in all types and forms of psychological and pedagogical activity.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\">Prof. Dr. Gennady Kiselev<\/p>\n<p class=\"Pa22\">Moscow Regional Social-Economic Institute<\/p>\n<p>Vidnoe, Russia<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\">Prof. Dr. Albina Chervova<\/p>\n<p class=\"Pa22\">Shuisky Branch of Ivanovo State University<\/p>\n<p>Shuya, Ivanovo region, 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=\"art03\"><\/a>INCREASING THE DIGITAL COMPETENCES OF STUDENTS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\"><strong>Lyubka Slavova, Kosta Garov<\/strong><\/p>\n<p><em>Plovdiv University \u201cPaisii Hilendarski\u201d (Bulgaria)<\/em><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>Abstract. <\/strong>The present paper examines the digital competences of secondary school students in Bulgaria in the light of the National External Evaluation of the 10th grade that has been held during the last years by the Ministry of Education and Science. There are ways to raise these competences and to improve the practical experience using the results of a specific experiment in a school in the city of Plovdiv. The main results are discussed in the paper.<\/p>\n<p><em>Keywords: <\/em>digital competence; national external evaluation; information literacy; information technology; education<\/p>\n<p>\u00a0<\/p>\n<p>Ms. Lyubka Slavova, PhD student<\/p>\n<p>Prof. Dr. Kosta Garov<\/p>\n<p>Faculty of Mathematics and Informatics<\/p>\n<p>Plovdiv University \u201cPaisii Hilendarski\u201d<\/p>\n<p>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>\u00a0<a name=\"art04\"><\/a><strong>\u0421\u0418\u0421\u0422\u0415\u041c\u0410 \u041e\u0422 \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=\"Pa21\"><strong>\u041f\u0435\u0442\u044f \u0410\u0441\u0435\u043d\u043e\u0432\u0430, \u041c\u0430\u0440\u0438\u043d \u041c\u0430\u0440\u0438\u043d\u043e\u0432<\/strong><\/p>\n<p><em>New Bulgarian University<\/em><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>Abstract<\/strong>. \u0421\u0442\u0430\u0442\u0438\u044f\u0442\u0430 \u0440\u0430\u0437\u0433\u043b\u0435\u0436\u0434\u0430 \u043f\u043e\u043d\u044f\u0442\u0438\u0435\u0442\u043e \u201e\u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438\u201c \u043a\u0430\u0442\u043e \u043c\u0435\u0442\u043e\u0434 \u0437\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435, \u043a\u043e\u0435\u0442\u043e \u043e\u0437\u043d\u0430\u0447\u0430\u0432\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u0447\u0440\u0435\u0437 \u0437\u0430\u0434\u0430\u0447\u0438. \u0422\u044f \u043f\u043e\u043a\u0440\u0438\u0432\u0430 \u0432\u0441\u0438\u0447\u043a\u0438 \u0435\u0442\u0430\u043f\u0438 \u043d\u0430 \u0443\u0441\u0432\u043e\u044f\u0432\u0430\u043d\u0435 \u043d\u0430 \u043d\u043e\u0432\u043e\u0442\u043e \u0437\u043d\u0430\u043d\u0438\u0435 \u2013 \u0432\u044a\u0432\u0435\u0436\u0434\u0430\u043d\u0435, \u0437\u0430\u0442\u0432\u044a\u0440\u0436\u0434\u0430\u0432\u0430\u043d\u0435 \u0438 \u043f\u0440\u0438\u043b\u043e\u0436\u0435\u043d\u0438\u0435. \u041f\u0440\u0435\u0434\u043b\u043e\u0436\u0435\u043d \u0435 \u043c\u043e\u0434\u0435\u043b \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438 \u043d\u0430 \u0440\u0430\u0437\u043b\u0438\u0447\u043d\u0438 \u0440\u0430\u0432\u043d\u0438\u0449\u0430. \u041f\u0440\u043e\u0435\u043a\u0442\u0438\u0440\u0430\u043d\u0435\u0442\u043e \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u0435 \u0439\u0435\u0440\u0430\u0440\u0445\u0438\u0447\u043d\u043e \u0438 \u0432\u043a\u043b\u044e\u0447\u0432\u0430: \u0446\u0435\u043b; \u043e\u0447\u0430\u043a\u0432\u0430\u043d\u0438 \u0440\u0435\u0437\u0443\u043b\u0442\u0430\u0442\u0438; \u043f\u043e\u0434\u0431\u043e\u0440 \u043d\u0430 \u0437\u0430\u0434\u0430\u0447\u0438\u0442\u0435; \u043e\u0440\u0433\u0430\u043d\u0438\u0437\u0430\u0446\u0438\u044f, \u043c\u0435\u0442\u043e\u0434\u0438 \u0438 \u0441\u0440\u0435\u0434\u0441\u0442\u0432\u0430 \u043f\u0440\u0438 \u043f\u0440\u0435\u043f\u043e\u0434\u0430\u0432\u0430\u043d\u0435\u0442\u043e. \u0420\u0435\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f\u0442\u0430 \u043d\u0430 \u0441\u0438\u0441\u0442\u0435\u043c\u0430\u0442\u0430 \u0435 \u0438\u043b\u044e\u0441\u0442\u0440\u0438\u0440\u0430\u043d\u0430 \u0432\u044a\u0440\u0445\u0443 \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u043e\u0442 \u0437\u0430\u0434\u0430\u0447\u0438 \u043f\u044a\u0440\u0432\u043e \u0440\u0430\u0432\u043d\u0438\u0449\u0435 \u043f\u043e \u0442\u0435\u043c\u0430\u0442\u0430 \u201e\u0421\u0435\u0447\u0435\u043d\u0438\u0435 \u043d\u0430 \u043c\u043d\u043e\u0433\u043e\u0441\u0442\u0435\u043d \u0441 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u201c \u0437\u0430 \u0441\u0432\u043e\u0431\u043e\u0434\u043d\u043e\u0438\u0437\u0431\u0438\u0440\u0430\u0435\u043c\u0430 \u043f\u043e\u0434\u0433\u043e\u0442\u043e\u0432\u043a\u0430 \u0432 XI \u043a\u043b\u0430\u0441. \u041c\u0435\u0442\u043e\u0434\u0438\u043a\u0430\u0442\u0430 \u043f\u0440\u0435\u0434\u043f\u043e\u043b\u0430\u0433\u0430 \u0438\u0437\u043f\u043e\u043b\u0437\u0432\u0430\u043d\u0435 \u043d\u0430 \u043a\u043e\u043c\u043f\u044e\u0442\u044a\u0440\u043d\u0438 \u0441\u0438\u0441\u0442\u0435\u043c\u0438 \u0437\u0430 \u0432\u0438\u0437\u0443\u0430\u043b\u0438\u0437\u0430\u0446\u0438\u044f \u0438 \u0430\u043d\u0438\u043c\u0430\u0446\u0438\u044f.<\/p>\n<p><em>Keywords<\/em>: education in Mathematics; intersections; system of tasks; computer visualization and animation<\/p>\n<p>\u00a0<\/p>\n<p><strong>SYSTEM OF TASKS IN MATHEMATICS EDUCATION<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract<\/strong>. This paper presents a view to the notion System of tasks as a peda-gogical approach, which means education trough tasks. It covers all steps in the teaching process \u2013 introduction of a new knowledge, training and application. The multilevel model of the system of tasks is proposed here. Its design is based on hierarchy: objectives and planned achievements; selection of the tasks; organiza-tion, teaching technics and tools. The implementation of the system of tasks is il-lustrated for the first level on the Solid geometry topic on Intersections for upper secondary school, elective form of education. The teaching technics proposed are strongly based on using computer software as a tool for visualization and animation.<\/p>\n<p>\u00a0<\/p>\n<p>Dr. Petya Asenova, Assoc. Prof.<\/p>\n<p>Prof. Dr. Marin Marinov<\/p>\n<p>Department of Computer Science<\/p>\n<p>New Bulgarian University<\/p>\n<p>Sofia, Bulgaria<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p>\u00a0<\/p>\n<p><strong><a name=\"art05\"><\/a>EULER- GERGONNE&#8217; S THEOREM AND ITS APPLICATIONS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\"><strong>\u0160efket Arslanagi\u0107<\/strong><\/p>\n<p><em>University of Sarajevo (Bosnia and Herzegovina)<\/em><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>Abstract<\/strong>. The present paper considers a proof of the Euler-Gergonne\u2019s theorem and its application to two assertions.<\/p>\n<p><em>Keywords<\/em>: triangle; circumcentre; orthocentre; incentre; centroid; radius of the circumcircle; radius of the incircle of a triangle<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\">Prof. \u0160efket Arslanagi\u0107, DSc.<\/p>\n<p class=\"Pa22\">Faculty of Mathematics and Natural Sciences<\/p>\n<p class=\"Pa22\">University of Sarajevo<\/p>\n<p>Sarajevo, Bosnia and Herzegovina<\/p>\n<p style=\"text-align: right;\">\u00a0<a href=\"#top\">\u043d\u0430\u0433\u043e\u0440\u0435<\/a><\/p>\n<hr \/>\n<p style=\"text-align: right;\">\u00a0<\/p>\n<p>\u00a0<\/p>\n<p><strong><a name=\"art06\"><\/a>\u0415\u0414\u041d\u041e \u0422\u0412\u042a\u0420\u0414\u0415\u041d\u0418\u0415 \u0417\u0410 \u041a\u041e\u041d\u041a\u0423\u0420\u0415\u041d\u0422\u041d\u041e\u0421\u0422 \u041d\u0410 \u041f\u0415\u0414\u0410\u041b\u041d\u0418 \u041e\u041a\u0420\u042a\u0416\u041d\u041e\u0421\u0422\u0418 \u041d\u0410 \u0422\u041e\u0427\u041a\u0410 \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<\/strong><\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<p class=\"Pa21\"><strong>\u0425\u0430\u0438\u043c \u0425\u0430\u0438\u043c\u043e\u0432<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>Abstract. <\/strong>\u0412 \u0442\u0430\u0437\u0438 \u043f\u0443\u0431\u043b\u0438\u043a\u0430\u0446\u0438\u044f \u0441\u0435 \u0434\u043e\u043a\u0430\u0437\u0432\u0430, \u0447\u0435 \u043f\u0435\u0434\u0430\u043b\u043d\u0438\u0442\u0435 \u043e\u043a\u0440\u044a\u0436\u043d\u043e\u0441\u0442\u0438 \u043d\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u043d\u0430 \u0442\u043e\u0447\u043a\u0430 \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 \u043e\u0442\u043d\u043e\u0441\u043d\u043e \u0442\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u0446\u0438\u0442\u0435, \u043e\u043f\u0440\u0435\u0434\u0435\u043b\u0435\u043d\u0438 \u043e\u0442 \u0441\u0442\u0440\u0430\u043d\u0438\u0442\u0435 \u0438 \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0438\u0442\u0435 \u043d\u0430 \u043f\u043e\u0441\u043b\u0435\u0434\u043d\u0438\u044f, \u0441\u0435 \u043f\u0440\u0435\u0441\u0438\u0447\u0430\u0442 \u0432 \u0435\u0434\u043d\u0430 \u0442\u043e\u0447\u043a\u0430. \u0421 \u043f\u043e\u043c\u043e\u0449\u0442\u0430 \u043d\u0430 \u0442\u043e\u0432\u0430 \u0442\u0432\u044a\u0440\u0434\u0435\u043d\u0438\u0435 \u0441\u0435 \u0434\u0435\u0444\u0438\u043d\u0438\u0440\u0430 \u0435\u0434\u043d\u043e \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0435 \u0432 \u0440\u0430\u0432\u043d\u0438\u043d\u0430\u0442\u0430 \u043d\u0430 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u043b\u0435\u043d \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a. \u041d\u0430\u043c\u0438\u0440\u0430\u0442 \u0441\u0435 \u043e\u0431\u0440\u0430\u0437\u0438\u0442\u0435 \u043d\u0430 \u043d\u044f\u043a\u043e\u0438 \u0437\u0430\u0431\u0435\u043b\u0435\u0436\u0438\u0442\u0435\u043b\u043d\u0438 \u0442\u043e\u0447\u043a\u0438 \u0432 \u0447\u0435\u0442\u0438\u0440\u0438\u044a\u0433\u044a\u043b\u043d\u0438\u043a\u0430. \u041e\u043f\u0438\u0441\u0432\u0430\u0442 \u0441\u0435 \u0432\u0440\u044a\u0437\u043a\u0438\u0442\u0435 \u043d\u0430 \u0438\u0437\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0435\u0442\u043e \u0441 \u0434\u0440\u0443\u0433\u0438 \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.<\/p>\n<p><em>Keywords: <\/em>quadrilateral; pedalcircles; transformations; remarkablepoints<\/p>\n<p>\u00a0<\/p>\n<p><strong>A STATEMENT FOR CONCURRENCE OF PEDAL CIRCLES OF A POINT IN THE PLANE OF A QUADRILATERAL<\/strong><\/p>\n<p>\u00a0<\/p>\n<p><strong>Abstract<em>. <\/em><\/strong>It is proved in the present paper that the pedal circles of an arbitrary point in the plane of a quadrilateral, determined by its sides and diagonals, are concurrent. With the help of this statement a transformation is defined in the plane of an arbitrary quadrilateral. The images of some notable points in the quadrilateral are found. The connection of this transformation with other transformations in the plane of the quadrilateral are described.<\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\">Mr. Haim Haimov, Resercher<\/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<\/p>\n<p><strong><a name=\"art07\"><\/a>POLYNOMIALS OF FOURTH DEGREE WITH COLINEAR CENTRAL SYMMETRIC ROOTS<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\"><strong>Sava Grozdev,<\/strong> <br \/><em>University of Finance, Business and Entrepreneurship \u2013 Sofia (Bulgaria)<\/em><\/p>\n<p><strong>Veselin Nenkov,<\/strong> <br \/><em>Nikola Vaptsarov Naval Academy \u2013 Varna (Bulgaria)<\/em><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>Abstract. <\/strong>A geometric relation is considered between a polynomial of fourth degree with symmetrically located roots on a line and the roots of its derivative. A special ellipse is used for the purpose, which is generated by the roots of the polynomial.<\/p>\n<p><em>Keywords: <\/em>polynomial of fourth degree; derivative; symmetric points; equilateral triangle; square; center; ellipse; circle; vertices<\/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 class=\"Pa21\">Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n<p class=\"Pa22\">Nikola Vaptsarov 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><strong><a name=\"art08\"><\/a>PROBLEM 6. FROM IMO\u20192018<\/strong><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa21\"><strong>Sava Grozdev,<\/strong> <br \/><em>University of Finance, Business and Entrepreneurship \u2013 Sofia (Bulgaria)<\/em><\/p>\n<p><strong>Veselin Nenkov,<\/strong> <br \/><em>Nikola Vaptsarov Naval Academy \u2013 Varna (Bulgaria)<\/em><\/p>\n<p>\u00a0<\/p>\n<p class=\"Pa24\"><strong>Abstract. <\/strong>The International Mathematical Olympiad is one of the respectable events and one of the most long-lived international educational and scientific competitions. It is the largest, oldest and most prestigious scientific Olympiad for high school students. The 59<sup>th<\/sup> edition of the event took place in Cluj-Napoca, Romania, 3 \u2013 14 July 2018. The present paper is dedicated to the sixth problem on the Olympiad paper. A detailed analysis of the problem is proposed in a methodological way, which will be useful for students and teachers in the preparatory process for future participations in mathematical competitions.<\/p>\n<p><em>Keywords<\/em>: Olympiad; problem solving<\/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 class=\"Pa21\">Dr. Veselin Nenkov, Assoc. Prof.<\/p>\n<p class=\"Pa22\">Nikola Vaptsarov 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>","protected":false},"excerpt":{"rendered":"<p>CONTENTS of issue 1\/2019 \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \u043d\u0430 \u0431\u0440\u043e\u0439 1\/2019 \u00a0 EDITORIAL \/ \u041a\u042a\u041c \u0427\u0418\u0422\u0410\u0422\u0415\u041b\u042f \u2013 \u0441\u0442\u0440. 7\u00a0\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \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 \u0410\u0440\u0445\u0438\u0432\u0438\u0442\u0435 \u0433\u043e\u0432\u043e\u0440\u044f\u0442: \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f\u0442\u0430 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 [The Archives Speak: National Competitions in Informatics] \/ \u041f\u0430\u0432\u0435\u043b \u0410\u0437\u044a\u043b\u043e\u0432 \/ Pavel Azalov \u2013 \u0441\u0442\u0440. 11\u00a0\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u00a0 \u041c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043a\u0443\u043b\u044c\u0442\u0443\u0440\u044b [&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":[3732],"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 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019 - \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\/matematics\/matharticles2016-4\/1-lxii-2019\/\" \/>\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 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019 - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"CONTENTS of issue 1\/2019 \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \u043d\u0430 \u0431\u0440\u043e\u0439 1\/2019 \u00a0 EDITORIAL \/ \u041a\u042a\u041c \u0427\u0418\u0422\u0410\u0422\u0415\u041b\u042f \u2013 \u0441\u0442\u0440. 7\u00a0\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \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 \u0410\u0440\u0445\u0438\u0432\u0438\u0442\u0435 \u0433\u043e\u0432\u043e\u0440\u044f\u0442: \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f\u0442\u0430 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 [The Archives Speak: National Competitions in Informatics] \/ \u041f\u0430\u0432\u0435\u043b \u0410\u0437\u044a\u043b\u043e\u0432 \/ Pavel Azalov \u2013 \u0441\u0442\u0440. 11\u00a0\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u00a0 \u041c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043a\u0443\u043b\u044c\u0442\u0443\u0440\u044b [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/\" \/>\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=\"2019-03-08T09:19:33+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-06-18T18:58:26+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=\"6 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/\"},\"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 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019\",\"datePublished\":\"2019-03-08T09:19:33+00:00\",\"dateModified\":\"2019-06-18T18:58:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/\"},\"wordCount\":1853,\"publisher\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#organization\"},\"articleSection\":[\"\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \\\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\\\", 2019 \u0433.\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/\",\"url\":\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/\",\"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 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019 - \u0410\u0437-\u0431\u0443\u043a\u0438\",\"isPartOf\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#website\"},\"datePublished\":\"2019-03-08T09:19:33+00:00\",\"dateModified\":\"2019-06-18T18:58:26+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/strategies.azbuki.bg\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#website\",\"url\":\"https:\/\/vocedu.azbuki.bg\/en\/\",\"name\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\",\"description\":\"\u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e \u0438\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0437\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u043d\u0430\u0443\u043a\u0430\",\"publisher\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/vocedu.azbuki.bg\/en\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#organization\",\"name\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\",\"url\":\"https:\/\/vocedu.azbuki.bg\/en\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png\",\"contentUrl\":\"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png\",\"width\":250,\"height\":70,\"caption\":\"\u0410\u0437-\u0431\u0443\u043a\u0438\"},\"image\":{\"@id\":\"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/Azbuki55\/\",\"https:\/\/www.instagram.com\/nion_azbuki\/\"]},{\"@type\":\"Person\",\"@id\":\"\",\"url\":\"https:\/\/bel.azbuki.bg\/en\/author\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", \u043a\u043d\u0438\u0436\u043a\u0430 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019 - \u0410\u0437-\u0431\u0443\u043a\u0438","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/","og_locale":"en_US","og_type":"article","og_title":"\"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", \u043a\u043d\u0438\u0436\u043a\u0430 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019 - \u0410\u0437-\u0431\u0443\u043a\u0438","og_description":"CONTENTS of issue 1\/2019 \/ \u0421\u042a\u0414\u042a\u0420\u0416\u0410\u041d\u0418\u0415 \u043d\u0430 \u0431\u0440\u043e\u0439 1\/2019 \u00a0 EDITORIAL \/ \u041a\u042a\u041c \u0427\u0418\u0422\u0410\u0422\u0415\u041b\u042f \u2013 \u0441\u0442\u0440. 7\u00a0\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \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 \u0410\u0440\u0445\u0438\u0432\u0438\u0442\u0435 \u0433\u043e\u0432\u043e\u0440\u044f\u0442: \u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u0438 \u0441\u044a\u0441\u0442\u0435\u0437\u0430\u043d\u0438\u044f\u0442\u0430 \u043f\u043e \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430 [The Archives Speak: National Competitions in Informatics] \/ \u041f\u0430\u0432\u0435\u043b \u0410\u0437\u044a\u043b\u043e\u0432 \/ Pavel Azalov \u2013 \u0441\u0442\u0440. 11\u00a0\u041e\u0442\u0432\u043e\u0440\u0438 \u043f\u044a\u043b\u043d\u0438\u044f \u0442\u0435\u043a\u0441\u0442 \u00a0 \u041c\u0435\u0442\u043e\u0434\u0438\u0447\u0435\u0441\u043a\u0430\u044f \u0441\u0438\u0441\u0442\u0435\u043c\u0430 \u0444\u043e\u0440\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u044f \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0446\u0438\u043e\u043d\u043d\u043e\u0439 \u043a\u0443\u043b\u044c\u0442\u0443\u0440\u044b [&hellip;]","og_url":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2019-03-08T09:19:33+00:00","article_modified_time":"2019-06-18T18:58:26+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":"6 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/#article","isPartOf":{"@id":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/"},"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 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019","datePublished":"2019-03-08T09:19:33+00:00","dateModified":"2019-06-18T18:58:26+00:00","mainEntityOfPage":{"@id":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/"},"wordCount":1853,"publisher":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#organization"},"articleSection":["\u0421\u044a\u0434\u044a\u0440\u0436\u0430\u043d\u0438\u0435 \u043d\u0430 \u0441\u043f. \"\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430\", 2019 \u0433."],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/","url":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/","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 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019 - \u0410\u0437-\u0431\u0443\u043a\u0438","isPartOf":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#website"},"datePublished":"2019-03-08T09:19:33+00:00","dateModified":"2019-06-18T18:58:26+00:00","breadcrumb":{"@id":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/bel.azbuki.bg\/matematics\/matharticles2016-4\/1-lxii-2019\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/strategies.azbuki.bg\/"},{"@type":"ListItem","position":2,"name":"&#8222;\u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0430 \u0438 \u0438\u043d\u0444\u043e\u0440\u043c\u0430\u0442\u0438\u043a\u0430&#8220;, \u043a\u043d\u0438\u0436\u043a\u0430 1, \u0433\u043e\u0434\u0438\u043d\u0430 LXII, 2019"}]},{"@type":"WebSite","@id":"https:\/\/vocedu.azbuki.bg\/en\/#website","url":"https:\/\/vocedu.azbuki.bg\/en\/","name":"\u0410\u0437-\u0431\u0443\u043a\u0438","description":"\u041d\u0430\u0446\u0438\u043e\u043d\u0430\u043b\u043d\u043e \u0438\u0437\u0434\u0430\u0442\u0435\u043b\u0441\u0442\u0432\u043e \u0437\u0430 \u043e\u0431\u0440\u0430\u0437\u043e\u0432\u0430\u043d\u0438\u0435 \u0438 \u043d\u0430\u0443\u043a\u0430","publisher":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/vocedu.azbuki.bg\/en\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/vocedu.azbuki.bg\/en\/#organization","name":"\u0410\u0437-\u0431\u0443\u043a\u0438","url":"https:\/\/vocedu.azbuki.bg\/en\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/","url":"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png","contentUrl":"https:\/\/azbuki.bg\/wp-content\/uploads\/2019\/04\/logo-azbuki-e1580374651845.png","width":250,"height":70,"caption":"\u0410\u0437-\u0431\u0443\u043a\u0438"},"image":{"@id":"https:\/\/vocedu.azbuki.bg\/en\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/Azbuki55\/","https:\/\/www.instagram.com\/nion_azbuki\/"]},{"@type":"Person","@id":"","url":"https:\/\/bel.azbuki.bg\/en\/author\/"}]}},"_links":{"self":[{"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/43399"}],"collection":[{"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/comments?post=43399"}],"version-history":[{"count":5,"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/43399\/revisions"}],"predecessor-version":[{"id":59583,"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/posts\/43399\/revisions\/59583"}],"wp:attachment":[{"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/media?parent=43399"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/categories?post=43399"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bel.azbuki.bg\/en\/wp-json\/wp\/v2\/tags?post=43399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}