{"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":"b9f79120-9386-5e05-b423-6c57501ec2b7","task_key":"precalculus--test--b9f79120-9386-5e05-b423-6c57501ec2b7","task_revision_id":"1","upstream_id":"","short_description":"Let $\\mathbf{a}$ and $\\mathbf{b}$ be vectors such that","config":"precalculus","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"Let $\\\\mathbf{a}$ and $\\\\mathbf{b}$ be vectors such that\\n\\\\[\\\\mathbf{v} = \\\\operatorname{proj}_{\\\\mathbf{a}} \\\\mathbf{v} + \\\\operatorname{proj}_{\\\\mathbf{b}} \\\\mathbf{v}\\\\]for all vectors $\\\\mathbf{v}.$  Enter all possible values of $\\\\mathbf{a} \\\\cdot \\\\mathbf{b},$ separated by commas.\",\"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":[]}