{"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":"64b5d63b-b976-51e7-96d4-f2f5f9695f8e","task_key":"intermediate~5falgebra--test--64b5d63b-b976-51e7-96d4-f2f5f9695f8e","task_revision_id":"1","upstream_id":"","short_description":"Jonathon is given the following problem: Let $a$ and $b$ be positive real…","config":"intermediate_algebra","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"Jonathon is given the following problem: Let $a$ and $b$ be positive real numbers such that $a + b = 1.$  Find the minimum value of\\n\\\\[\\\\left( a + \\\\frac{1}{a} \\\\right)^2 + \\\\left( b + \\\\frac{1}{b} \\\\right)^2.\\\\]Jonathon reasons as follows: By AM-GM,\\n\\\\[a + \\\\frac{1}{a} \\\\ge 2,\\\\]and\\n\\\\[b + \\\\frac{1}{b} \\\\ge 2,\\\\]so\\n\\\\[\\\\left( a + \\\\frac{1}{a} \\\\right)^2 + \\\\left( b + \\\\frac{1}{b} \\\\right)^2 \\\\ge 2^2 + 2^2 = 8.\\\\]He concludes that the minimum value is 8, but this answer is incorrect.  What is the minimum value?\\n\\nAs a bonus, determine where Jonathon went wrong.\",\"type\":\"Intermediate Algebra\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}