{"id":551,"date":"2026-03-24T20:42:51","date_gmt":"2026-03-24T12:42:51","guid":{"rendered":"https:\/\/barzov.com\/?p=551"},"modified":"2026-04-26T11:30:13","modified_gmt":"2026-04-26T03:30:13","slug":"math-articles","status":"publish","type":"post","link":"https:\/\/barzov.com\/index.php\/2026\/03\/24\/math-articles\/","title":{"rendered":"Math articles"},"content":{"rendered":"\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"401\" height=\"374\" src=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/blue_dino.png\" alt=\"\" class=\"wp-image-662\" style=\"aspect-ratio:1.072203869749882;width:213px;height:auto\" srcset=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/blue_dino.png 401w, https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/blue_dino-300x280.png 300w\" sizes=\"auto, (max-width: 401px) 100vw, 401px\" \/><\/figure>\n\n\n\n<p>Here are a few articles from my younger days, newly polished and translated into English.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Sums of sets <\/h2>\n\n\n\n<p>This article is based on a problem from number theory. We present several generalizations and corollaries. The paper studies residues modulo a prime, sets of such residues, and a special type of summing sets. The examples are taken from past mathematical competitions.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/sums_of_sets.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Sums of Sets.\"><\/object><a id=\"wp-block-file--media-ba5ae0ab-868f-4dcb-b150-3c343d9cdade\" href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/sums_of_sets.pdf\">Sums of Sets<\/a><a href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/sums_of_sets.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-ba5ae0ab-868f-4dcb-b150-3c343d9cdade\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Similar triangles inscribed in one another<\/h2>\n\n\n\n<p>This is an article I wrote in high school with the guidance of my teacher Nikolay Nikolov.<\/p>\n\n\n\n<p>The use of complex numbers in geometry is often regarded as less<br>elegant than purely geometric methods. Yet, despite its lack of aesthetics, a bit of algebra can lead to surprisingly interesting geometric results. In this article, we examine families of triangles inscribed in one another.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/similar_triangles.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Similar triangles inscribed in one another.\"><\/object><a id=\"wp-block-file--media-5b4c5d3f-ecd4-4f3d-a76e-a7c20ab4b018\" href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/similar_triangles.pdf\">Similar triangles inscribed in one another<\/a><a href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/similar_triangles.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-5b4c5d3f-ecd4-4f3d-a76e-a7c20ab4b018\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Use of polynomials for arithmetical and combinatorial problems<\/h2>\n\n\n\n<p>Sometimes number theory and combinatorics problems can be easily translated into algebra problems by introducing suitable polynomials. The present note considers such applications in connection with some problems from various mathematical competitions and olympiads.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/polynomials.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Polynomials.\"><\/object><a id=\"wp-block-file--media-9fb38734-e36e-456c-8f18-041ccb23d579\" href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/polynomials.pdf\">Polynomials<\/a><a href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/polynomials.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-9fb38734-e36e-456c-8f18-041ccb23d579\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Certain applications of the roots of unity<\/h2>\n\n\n\n<p>The objective of this article is to illustrate the use of the roots of unity for solving problems from various mathematical areas, like combinatorics, algebra, number theory. The selected examples are either original or from national and international mathematical competitions.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/roots_of_unity.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Roots of unity.\"><\/object><a id=\"wp-block-file--media-fb68b7ab-e2b6-4ec7-b950-7e102fe9b84c\" href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/roots_of_unity.pdf\">Roots of unity<\/a><a href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/roots_of_unity.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-fb68b7ab-e2b6-4ec7-b950-7e102fe9b84c\">Download<\/a><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">On the distribution of fractional parts<\/h2>\n\n\n\n<p>A short study of the fractional parts of an arithmetic progression.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/fractional_parts.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Fractional parts.\"><\/object><a id=\"wp-block-file--media-13252008-7840-456f-b457-f45c01ed006b\" href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/fractional_parts.pdf\">Fractional parts<\/a><a href=\"https:\/\/barzov.com\/wp-content\/uploads\/2026\/03\/fractional_parts.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-13252008-7840-456f-b457-f45c01ed006b\">Download<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Here are a few articles from my younger days, newly polished and translated into English. Sums of sets This article is based on a problem from number theory. We present several generalizations and corollaries. The paper studies residues modulo a prime, sets of such residues, and a special type of summing sets. The examples are [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-551","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"_links":{"self":[{"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/posts\/551","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/comments?post=551"}],"version-history":[{"count":26,"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/posts\/551\/revisions"}],"predecessor-version":[{"id":664,"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/posts\/551\/revisions\/664"}],"wp:attachment":[{"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/media?parent=551"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/categories?post=551"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/barzov.com\/index.php\/wp-json\/wp\/v2\/tags?post=551"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}