{"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":"70d73ffe-9bf3-5ef2-abce-08fe6a37f21a","task_key":"precalculus--test--70d73ffe-9bf3-5ef2-abce-08fe6a37f21a","task_revision_id":"1","upstream_id":"","short_description":"Let $A,$ $B,$ $C,$ and $D$ be points on a circle. For each pair of points, we…","config":"precalculus","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"Let $A,$ $B,$ $C,$ and $D$ be points on a circle.  For each pair of points, we draw the line passing through the midpoint of those two points that is perpendicular to the line formed by the other two points.  For example, we draw the line passing through the midpoint of $\\\\overline{AB}$ that is perpendicular to line $CD.$  In this way, we draw a total of $\\\\binom{4}{2} = 6$ lines.\\n\\n[asy]\\nunitsize(2.5 cm);\\n\\npair A, B, C, D, P;\\npair[] M;\\n\\nA = dir(118);\\nB = dir(62);\\nC = dir(323);\\nD = dir(165);\\nM[12] = (A + B)/2;\\nM[13] = (A + C)/2;\\nM[14] = (A + D)/2;\\nM[23] = (B + C)/2;\\nM[24] = (B + D)/2;\\nM[34] = (C + D)/2;\\nP = (A + B + C + D)/2;\\n\\ndraw(Circle((0,0),1),red);\\ndraw(A--B,green);\\ndraw(A--C,green);\\ndraw(A--D,green);\\ndraw(B--C,green);\\ndraw(B--D,green);\\ndraw(C--D,green);\\ndraw(interp(M[12],P,-2)--interp(M[12],P,6),dashed);\\ndraw(interp(M[13],P,-2)--interp(M[13],P,2),dashed);\\ndraw(interp(M[14],P,-0.5)--interp(M[14],P,2.5),dashed);\\ndraw(interp(M[23],P,-0.7)--interp(M[23],P,1.5),dashed);\\ndraw(interp(M[24],P,-3.3)--interp(M[24],P,4),dashed);\\ndraw(interp(M[34],P,-0.3)--interp(M[34],P,1.6),dashed);\\n\\ndot(\\\"$A$\\\", A, A);\\ndot(\\\"$B$\\\", B, B);\\ndot(\\\"$C$\\\", C, C);\\ndot(\\\"$D$\\\", D, D);\\ndot(M[12]);\\ndot(M[13]);\\ndot(M[14]);\\ndot(M[23]);\\ndot(M[24]);\\ndot(M[34]);\\n[/asy]\\n\\nIt turns out that all six lines will always pass through the same point, say $P.$  There exist constants $a,$ $b,$ $c,$ and $d$ such that $\\\\overrightarrow{P}$ can always be expressed in the form\\n\\\\[\\\\overrightarrow{P} = a \\\\overrightarrow{A} + b \\\\overrightarrow{B} + c \\\\overrightarrow{C} + d \\\\overrightarrow{D}.\\\\]Find $a + b + c + d.$\",\"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":[]}