{"id":4790,"date":"2026-03-14T12:56:13","date_gmt":"2026-03-14T16:56:13","guid":{"rendered":"https:\/\/josmfs.net\/wordpress\/?p=4790"},"modified":"2026-04-02T13:12:21","modified_gmt":"2026-04-02T17:12:21","slug":"language-students-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2026\/03\/14\/language-students-problem\/","title":{"rendered":"Language Students Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-4787\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/03\/Language-Students-Prob-Fig2.jpg\" alt=\"\" width=\"201\" height=\"178\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/03\/Language-Students-Prob-Fig2.jpg 400w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/03\/Language-Students-Prob-Fig2-300x266.jpg 300w\" sizes=\"auto, (max-width: 201px) 100vw, 201px\" \/>This is a problem from the 2001 American Invitational Mathematics Exam (AIME).<\/p>\n<p>\u201cEach of the 2000 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let <em>m<\/em> be the smallest number of students who could study both languages, and let <em>M<\/em> be the largest number of students who could study both languages. Find <em>M \u2013 m<\/em>.\u201d<\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/03\/Language-Students-Prob-Answer.jpg\">Answer<\/a>.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/03\/Language-Students-Problem-260311.pdf\">Language Students Problem<\/a> for a solution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a problem from the 2001 American Invitational Mathematics Exam (AIME). \u201cEach of the 2000 students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between 80 percent and 85 percent of the school population, and the number who study French is between 30 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[238,133],"class_list":["post-4790","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-aime","tag-algebra"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4790","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=4790"}],"version-history":[{"count":1,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4790\/revisions"}],"predecessor-version":[{"id":4791,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4790\/revisions\/4791"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=4790"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=4790"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=4790"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}