{"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":"963c3834-9814-5802-a741-e09b784400c1","task_key":"intermediate~5falgebra--test--963c3834-9814-5802-a741-e09b784400c1","task_revision_id":"1","upstream_id":"","short_description":"Let $f(m,1) = f(1,n) = 1$ for $m \\geq 1, n \\geq 1,$ and let $f(m,n) = f(m-1,n) +…","config":"intermediate_algebra","split":"test","body":"{\"level\":\"Level 3\",\"problem\":\"Let $f(m,1) = f(1,n) = 1$ for $m \\\\geq 1, n \\\\geq 1,$ and let $f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1)$ for $m > 1$ and $n > 1.$  Also, let\\n\\n$$S(k) = \\\\sum_{a+b=k} f(a,b), \\\\text{ for } a \\\\geq 1, b \\\\geq 1.$$Note: The summation notation means to sum over all positive integers $a,b$ such that $a+b=k.$\\n\\nGiven that\\n\\n$$S(k+2) = pS(k+1) + qS(k) \\\\text{ for all } k \\\\geq 2,$$for some constants $p$ and $q$, find $pq.$\",\"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":[]}