{"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":"75732aab-e66b-5c49-87a1-bffda352735f","task_key":"precalculus--test--75732aab-e66b-5c49-87a1-bffda352735f","task_revision_id":"1","upstream_id":"","short_description":"Let $\\alpha,$ $\\beta,$ and $\\gamma$ be three real numbers. Suppose that","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"Let $\\\\alpha,$ $\\\\beta,$ and $\\\\gamma$ be three real numbers.  Suppose that\\n\\\\begin{align*}\\n\\\\cos \\\\alpha + \\\\cos \\\\beta + \\\\cos \\\\gamma &= 1, \\\\\\\\\\n\\\\sin \\\\alpha + \\\\sin \\\\beta + \\\\sin \\\\gamma &= 1.\\n\\\\end{align*}Then the minimum value of $\\\\cos \\\\alpha$ can be expressed in the form $-\\\\frac{p + \\\\sqrt{q}}{r},$ where $p,$ $q,$ and $r$ are positive integers, and $q$ is not divisible by the square of a prime.  Find $p + q + r.$\",\"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":[]}