{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/math","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"a56d770f-4148-567e-834a-d208cb18a722","task_key":"geometry--test--a56d770f-4148-567e-834a-d208cb18a722","task_revision_id":"1","upstream_id":"","short_description":"Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ as shown…","config":"geometry","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ as shown with $\\\\overline{ED}\\\\perp\\\\overline{BC}.$ What is the ratio of the area of $\\\\triangle DEF$ to the area of $\\\\triangle\\nABC$? [asy]\\nimport olympiad; import geometry; size(100); defaultpen(linewidth(0.8));\\ndraw(dir(90)--dir(210)--(dir(-30))--cycle);\\npair[] inters = intersectionpoints(Circle(origin,1/sqrt(3)),dir(90)--dir(210)--(dir(-30))--cycle);\\ninters[5] = dir(210) + ((dir(-30) - dir(210))/3);\\ndraw(inters[0]--inters[2]--inters[5]--cycle);\\ndraw(rightanglemark(inters[2],inters[5],dir(210),4));\\ndot(\\\"$A$\\\",dir(90),N);\\ndot(\\\"$C$\\\",dir(210),W);\\ndot(\\\"$B$\\\",dir(-30),E);\\ndot(\\\"$F$\\\",inters[0],E);\\ndot(\\\"$E$\\\",inters[2],W);\\ndot(\\\"$D$\\\",inters[5],S);\\n[/asy]\",\"type\":\"Geometry\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}