{"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":"8a6aa8e6-0703-5c7c-996f-3873faf74b42","task_key":"geometry--test--8a6aa8e6-0703-5c7c-996f-3873faf74b42","task_revision_id":"1","upstream_id":"","short_description":"Circles with centers at $O$ and $P$ have radii 2 and 4, respectively, and are…","config":"geometry","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"Circles with centers at $O$ and $P$ have radii 2 and 4, respectively, and are externally tangent. Points $A$ and $B$ on the circle with center $O$ and points  $C$ and $D$ on the circle with center $P$ are such that  $\\\\overline{AD}$ and $\\\\overline{BC}$ are common external tangents to the circles. What is the area of the concave hexagon $AOBCPD$?\\n\\n[asy]\\npair A,B,C,D,H,O,P;\\nO = (0,0);\\nP = (6,0);\\nH = (-6,0);\\nA = intersectionpoint(arc((A + H)/2, abs(A -  H)/2, 0, 180), arc(O, 2, 0, 180));\\nD = intersectionpoint(arc((A + P)/2, abs(A -  P)/2, 0, 180), arc(P, 4, 0, 180));\\nB = reflect(O,P)*(A);\\nC = reflect(O,P)*(D);\\ndraw(O--P,linewidth(0.7));\\ndraw(Circle(O,2),linewidth(0.7));\\ndraw(Circle(P,4),linewidth(0.7));\\ndraw(interp(A,D,-0.2)--interp(A,D,1.2),linewidth(0.7));\\ndraw(interp(B,C,-0.2)--interp(B,C,1.2),linewidth(0.7));\\ndraw(A--O--B,linewidth(0.7));\\ndraw(D--P--C,linewidth(0.7));\\nlabel(\\\"$O$\\\",O,SE);\\nlabel(\\\"$P$\\\",P,E);\\nlabel(\\\"$D$\\\",D,NW);\\nlabel(\\\"$A$\\\",A,NW);\\nlabel(\\\"$B$\\\",B,S);\\nlabel(\\\"$C$\\\",C,S);\\nlabel(\\\"2\\\",(-0.45,0.95),E);\\nlabel(\\\"6\\\",(3,0),N);\\nlabel(\\\"4\\\",(5.55,1.95),E);\\n\\n[/asy]\",\"type\":\"Geometry\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}