{"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":"29e3618b-58ef-565a-b36d-61f70c5464a2","task_key":"precalculus--test--29e3618b-58ef-565a-b36d-61f70c5464a2","task_revision_id":"1","upstream_id":"","short_description":"Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be unit vectors, such that the…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"Let $\\\\mathbf{a},$ $\\\\mathbf{b},$ and $\\\\mathbf{c}$ be unit vectors, such that the angle between $\\\\mathbf{a}$ and $\\\\mathbf{b}$ is $\\\\arccos \\\\frac{1}{5},$ the angle between $\\\\mathbf{a}$ and $\\\\mathbf{c}$ is $\\\\arccos \\\\frac{1}{6},$ and the angle between $\\\\mathbf{b}$ and $\\\\mathbf{c}$ is $60^\\\\circ.$\\n\\nLet $P$ be the plane which contains the origin, $\\\\mathbf{b},$ and $\\\\mathbf{c}.$  Then the projection of $\\\\mathbf{a}$ onto $P$ can be expressed in the form\\n\\\\[p \\\\mathbf{b} + q \\\\mathbf{c}\\\\]for some scalars $p$ and $q.$  Enter the ordered pair $(p,q).$\",\"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":[]}