{"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":"148c437d-bc86-5c6b-b915-1b8a0e2210be","task_key":"precalculus--test--148c437d-bc86-5c6b-b915-1b8a0e2210be","task_revision_id":"1","upstream_id":"","short_description":"In triangle $ABC,$ $a = 8,$ $b = 7,$ and $c = 5.$ Let $H$ be the orthocenter.","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"In triangle $ABC,$ $a = 8,$ $b = 7,$ and $c = 5.$  Let $H$ be the orthocenter.\\n\\n[asy]\\nunitsize(0.6 cm);\\n\\npair A, B, C, D, E, F, H;\\n\\nB = (0,0);\\nC = (8,0);\\nA = intersectionpoint(arc(B,5,0,180),arc(C,7,0,180));\\nH = orthocenter(A,B,C);\\nD = (A + reflect(B,C)*(A))/2;\\nE = (B + reflect(C,A)*(B))/2;\\nF = (C + reflect(A,B)*(C))/2;\\n\\ndraw(A--B--C--cycle);\\ndraw(A--D,dashed);\\ndraw(B--E,dashed);\\ndraw(C--F,dashed);\\n\\nlabel(\\\"$A$\\\", A, N);\\nlabel(\\\"$B$\\\", B, SW);\\nlabel(\\\"$C$\\\", C, SE);\\nlabel(\\\"$H$\\\", H, SE, UnFill);\\n\\ndot(H);\\n[/asy]\\n\\nThen\\n\\\\[\\\\overrightarrow{H} = x \\\\overrightarrow{A} + y \\\\overrightarrow{B} + z \\\\overrightarrow{C},\\\\]where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$  Enter the ordered triple $(x,y,z).$\",\"type\":\"Precalculus\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}