{"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":"53603063-6407-5c28-9565-c4583d2f0bae","task_key":"precalculus--test--53603063-6407-5c28-9565-c4583d2f0bae","task_revision_id":"1","upstream_id":"","short_description":"Three unit circles are drawn so they are mutually tangent, as shown below. A…","config":"precalculus","split":"test","body":"{\"level\":\"Level 5\",\"problem\":\"Three unit circles are drawn so they are mutually tangent, as shown below.  A blue circle that is externally tangent to all three unit circles is drawn.  Finally, three red circles are drawn, so that each red circle is externally tangent to two unit circles and externally tangent to the blue circle.  Then the radius of each red circle can be expressed in the form\\n\\\\[\\\\frac{a - b \\\\sqrt{c}}{d},\\\\]where $a,$ $b,$ $c,$ and $d$ are positive integers, when simplified.  Find $a + b + c + d$.\\n\\n[asy]\\nunitsize(2 cm);\\n\\npair A, B, C, D, E, F, O;\\nreal s = 2/sqrt(3) - 1, r = (9 - 4*sqrt(3))/33;\\n\\nA = 2/sqrt(3)*dir(150);\\nB = 2/sqrt(3)*dir(30);\\nC = 2/sqrt(3)*dir(270);\\nO = (0,0);\\nD = (r + s)*dir(330);\\nE = (r + s)*dir(210);\\nF = (r + s)*dir(90);\\n\\nfilldraw(Circle(O,s),blue);\\nfilldraw(Circle(D,r),red);\\nfilldraw(Circle(E,r),red);\\nfilldraw(Circle(F,r),red);\\ndraw(Circle(A,1));\\ndraw(Circle(B,1));\\ndraw(Circle(C,1));\\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":[]}