{"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":"a2f57358-2c33-504f-993a-a8419b735655","task_key":"precalculus--test--a2f57358-2c33-504f-993a-a8419b735655","task_revision_id":"1","upstream_id":"","short_description":"Below is the graph of $y = a \\sin (bx + c) + d$ for some positive constants $a,$…","config":"precalculus","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"Below is the graph of $y = a \\\\sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$  Find the smallest possible value of $c.$\\n\\n[asy]import TrigMacros;\\n\\nsize(400);\\n\\nreal f(real x)\\n{\\n\\treturn 2*sin(3*x + pi) + 1;\\n}\\n\\ndraw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);\\ntrig_axes(-3*pi,3*pi,-4,4,pi/2,1);\\nlayer();\\nrm_trig_labels(-5,5, 2);\\n\\nlabel(\\\"$1$\\\", (0,1), E);\\nlabel(\\\"$2$\\\", (0,2), E);\\nlabel(\\\"$3$\\\", (0,3), E);\\nlabel(\\\"$-1$\\\", (0,-1), E);\\nlabel(\\\"$-2$\\\", (0,-2), E);\\nlabel(\\\"$-3$\\\", (0,-3), E);\\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":[]}