{"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":"fa885011-feef-55c2-8d19-577ece5a7c8b","task_key":"precalculus--test--fa885011-feef-55c2-8d19-577ece5a7c8b","task_revision_id":"1","upstream_id":"","short_description":"Let $\\mathbf{a} = \\begin{pmatrix} 7 \\\\ - 1 \\\\ 4 \\end{pmatrix}$ and $\\mathbf{b} =…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"Let $\\\\mathbf{a} = \\\\begin{pmatrix} 7 \\\\\\\\ - 1 \\\\\\\\ 4 \\\\end{pmatrix}$ and $\\\\mathbf{b} = \\\\begin{pmatrix} 3 \\\\\\\\ 1 \\\\\\\\ 2 \\\\end{pmatrix}.$  Find the vector $\\\\mathbf{c}$ so that $\\\\mathbf{a},$ $\\\\mathbf{b},$ and $\\\\mathbf{c}$ are collinear, and $\\\\mathbf{b}$ bisects the angle between $\\\\mathbf{a}$ and $\\\\mathbf{c}.$\\n\\n[asy]\\nunitsize(0.5 cm);\\n\\npair A, B, C, O;\\n\\nA = (-2,5);\\nB = (1,3);\\nO = (0,0);\\nC = extension(O, reflect(O,B)*(A), A, B);\\n\\ndraw(O--A,Arrow(6));\\ndraw(O--B,Arrow(6));\\ndraw(O--C,Arrow(6));\\ndraw(interp(A,C,-0.1)--interp(A,C,1.1),dashed);\\n\\nlabel(\\\"$\\\\mathbf{a}$\\\", A, NE);\\nlabel(\\\"$\\\\mathbf{b}$\\\", B, NE);\\nlabel(\\\"$\\\\mathbf{c}$\\\", C, NE);\\n[/asy]\",\"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":[]}