diff --git "a/GeoGramBench.json" "b/GeoGramBench.json" --- "a/GeoGramBench.json" +++ "b/GeoGramBench.json" @@ -5,7 +5,8 @@ "geo_code": "[asy]\nsize(175);\ndefaultpen(linewidth(0.8));\npair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;\ndraw(A--B--C--cycle^^X--Y--Z--cycle);\nlabel(\" $A$ \",A,N);\nlabel(\" $B$ \",B,S);\nlabel(\" $C$ \",C,E);\nlabel(\" $X$ \",X,W);\nlabel(\" $Y$ \",Y,S);\nlabel(\" $Z$ \",Z,NE);[/asy]", "answer": "$122$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 2, @@ -13,7 +14,8 @@ "geo_code": "[asy]size(200);defaultpen(linewidth(0.7)+fontsize(10));\npair O=origin, D=dir(195), A=dir(150), B=dir(30), C=B+1*dir(0);\ndraw(O--A--C--D);\ndot(A^^B^^C^^D^^O);\npair point=O;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $O$ \", O, dir(285));\nlabel(\" $x$ \", O+0.1*dir(172.5), dir(172.5));\nlabel(\" $y$ \", C+0.4*dir(187.5), dir(187.5));\ndraw(Circle(O,1));\n[/asy]", "answer": "$3$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 3, @@ -21,7 +23,8 @@ "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair A=origin, B=(14,0), C=(10,6);\ndraw(A--B--C--cycle);\nlabel(\" $A$ \", A, SW);\nlabel(\" $B$ \", B, SE);\nlabel(\" $C$ \", C, N);\nlabel(\" $a$ \", B--C, dir(B--C)*dir(-90));\nlabel(\" $b$ \", A--C, dir(C--A)*dir(-90));\nlabel(\" $c$ \", A--B, dir(A--B)*dir(-90));\n[/asy]", "answer": "$35$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 4, @@ -29,7 +32,8 @@ "geo_code": "[asy]unitsize(8mm);\ndefaultpen(linewidth(.8pt));\ndotfactor=4;\n\ndraw(Circle((2,0),2));\ndraw(Circle((0,4),4));\nclip(scale(4)*unitsquare);\ndraw(scale(4)*unitsquare);\nfilldraw(Circle((2,0),0.07));\nlabel(\" $A$ \",(0,4),NW);\nlabel(\" $B$ \",(4,4),NE);\nlabel(\" $C$ \",(4,0),SE);\nlabel(\" $D$ \",(0,0),SW);\nlabel(\" $M$ \",(2,0),S);\nlabel(\" $P$ \",(3.6,1.4),N);[/asy]", "answer": "$\\frac{16}{5}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 5, @@ -37,7 +41,8 @@ "geo_code": "[asy]import olympiad;\nunitsize(2cm);\ndefaultpen(fontsize(8pt)+linewidth(.8pt));\nlabelmargin=0.2;\ndotfactor=3;\n\npair O=(0,0);\npair A=(1,0);\npair B=(1,1.5);\npair D=bisectorpoint(A,B,O);\npair C=extension(B,D,O,A);\n\ndraw(Circle(O,1));\ndraw(O--A--B--cycle);\ndraw(B--C);\n\nlabel(\" $O$ \",O,SW);\ndot(O);\nlabel(\" $\\theta$ \",(0.1,0.05),ENE);\n\ndot(C);\nlabel(\" $C$ \",C,S);\n\ndot(A);\nlabel(\" $A$ \",A,E);\n\ndot(B);\nlabel(\" $B$ \",B,E);[/asy]", "answer": "$\\frac{2}{3}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 6, @@ -45,7 +50,8 @@ "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A);\nreal r=degrees(B);\ndraw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270));\npair point=incenter(A,B,C);\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $M$ \", M, dir(point--M));\nlabel(\" $N$ \", N, dir(point--N));\n\nlabel(\" $12$ \", (6,0), S);\nlabel(\" $5$ \", (12,3.5), E);[/asy]", "answer": "$4$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 7, @@ -53,7 +59,8 @@ "geo_code": "[asy]\nsize(200);\ndefaultpen(linewidth(0.8)+fontsize(10.6));\npair A = (0,sqrt(850));\npair B = (0,0);\npair C = (sqrt(850),0);\npair D = (sqrt(850),sqrt(850));\ndraw(A--B--C--D--cycle);\ndotfactor = 3;\ndot(\" $A$ \",A,dir(135));\ndot(\" $B$ \",B,dir(215));\ndot(\" $C$ \",C,dir(305));\ndot(\" $D$ \",D,dir(45));\npair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850));\npair F = ((2sqrt(850)+sqrt(306)+7)/6,0);\ndot(\" $H$ \",H,dir(90));\ndot(\" $F$ \",F,dir(270));\ndraw(H--F);\npair E = (0,(sqrt(850)-6)/2);\npair G = (sqrt(850),(sqrt(850)+sqrt(100))/2);\ndot(\" $E$ \",E,dir(180));\ndot(\" $G$ \",G,dir(0));\ndraw(E--G);\npair P = extension(H,F,E,G);\ndot(\" $P$ \",P,dir(60));\nlabel(\" $w$ \", (H+E)/2,fontsize(15));\nlabel(\" $x$ \", (E+F)/2,fontsize(15));\nlabel(\" $y$ \", (G+F)/2,fontsize(15));\nlabel(\" $z$ \", (H+G)/2,fontsize(15));\nlabel(\" $w:x:y:z=269:275:405:411$ \",(sqrt(850)/2,-4.5),fontsize(11));\n[/asy]", "answer": "$850$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 8, @@ -61,7 +68,8 @@ "geo_code": "[asy]\ndraw((4, 8)--(0, 0)--(14, 0)--cycle);\ndraw((2.54,0)--(2.54+5.1,0)--(2.54+5.1,5.1)--(2.54,5.1)--cycle);\nlabel(\" $X$ \",(4,8),N);\nlabel(\" $Y$ \",(0,0),W);\nlabel(\" $Z$ \",(14,0),E);\n[/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 9, @@ -69,7 +77,8 @@ "geo_code": "[asy]\npair S, D;\nD = 1.27;\nS = 2.55;\ndraw((2, 4)--(0, 0)--(7, 0)--cycle);\ndraw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);\nlabel(\" $X$ \",(2,4),N);\nlabel(\" $Y$ \",(0,0),W);\nlabel(\" $Z$ \",(7,0),E);\n[/asy]", "answer": "$6$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 10, @@ -77,7 +86,8 @@ "geo_code": "[asy]\npair S, D;\nD = 1.27;\nS = 2.55;\ndraw((2, 4)--(0, 0)--(7, 0)--cycle);\ndraw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);\nlabel(\" $X$ \",(2,4),N);\nlabel(\" $Y$ \",(0,0),W);\nlabel(\" $Z$ \",(7,0),E);\n[/asy]", "answer": "$4034$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 11, @@ -85,7 +95,8 @@ "geo_code": "[asy]\nunitsize(0.3 cm);\n\npair A, B, P, Q, R, S;\nreal r = (3 + sqrt(17))/2;\n\nA = (-1,0);\nB = (1,0);\nP = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));\nR = -P;\nQ = (r + 2,0);\nS = (-r - 2,0);\n\ndraw(Circle(A,1));\ndraw(Circle(B,1));\ndraw(Circle(P,r));\ndraw(Circle(Q,r));\ndraw(Circle(R,r));\ndraw(Circle(S,r));\n\nlabel(\" $A$ \", A);\nlabel(\" $B$ \", B);\nlabel(\" $P$ \", P);\nlabel(\" $Q$ \", Q);\nlabel(\" $R$ \", R);\nlabel(\" $S$ \", S);\n[/asy]", "answer": "$2$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 12, @@ -93,7 +104,8 @@ "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);\ndraw(C--A--D^^B--O^^Circle(O,1));\npair point=O;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $O$ \", O, dir(305));\n\nlabel(\" $5$ \", B--O, dir(O--B)*dir(90));\nlabel(\" $60^\\circ$ \", dir(185), dir(185));\nlabel(\" $60^\\circ$ \", B+0.05*dir(-25), dir(-25));[/asy]", "answer": "$5$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 13, @@ -101,7 +113,8 @@ "geo_code": "[asy]\nunitsize(0.3 cm);\npair F, H, M, O;\nF = (0,0);\nH = (0,5);\nO = (11,5);\nM = (11,0);\ndraw(H--O--M--F--cycle);\nlabel(\" $F$ \", F, SW);\nlabel(\" $H$ \", H, NW);\nlabel(\" $M$ \", M, SE);\nlabel(\" $O$ \", O, NE);\n[/asy]", "answer": "$28$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 14, @@ -109,7 +122,8 @@ "geo_code": "[asy]defaultpen(linewidth(.9pt));\n\nreal r = 2 + sqrt(2);\npair A = dir(45)*(r + sqrt(2));\npair B = dir(135)*(r + sqrt(2));\npair C = dir(-135)*(r + sqrt(2));\npair D = dir(-45)*(r + sqrt(2));\n\ndraw(Circle(origin,sqrt(2)));\ndraw(Circle(A,r));draw(Circle(B,r));draw(Circle(C,r));draw(Circle(D,r));\n\ndraw(A--(dir(45)*r + A));\ndraw(B--(dir(45)*r + B));\ndraw(C--(dir(45)*r + C));\ndraw(D--(dir(45)*r + D));\ndraw(origin--(dir(25)*sqrt(2)));\n\nlabel(\" $r$ \",midpoint(A--(dir(45)*r + A)), SE);\nlabel(\" $r$ \",midpoint(B--(dir(45)*r + B)), SE);\nlabel(\" $r$ \",midpoint(C--(dir(45)*r + C)), SE);\nlabel(\" $r$ \",midpoint(D--(dir(45)*r + D)), SE);\nlabel(\" $1$ \",origin,W);[/asy]", "answer": "$1 + \\sqrt{2}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 15, @@ -117,7 +131,8 @@ "geo_code": "[asy]unitsize(1mm);\ndefaultpen(linewidth(.8pt));\nfilldraw((0,0)--(8,0)--(8,1)--(0,1)--cycle,grey,black);\nfilldraw((0,5)--(8,5)--(8,6)--(0,6)--cycle,grey,black);\ndraw((0,1)--(0,5));\ndraw((8,1)--(8,5));[/asy]", "answer": "$2.7$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 16, @@ -125,7 +140,8 @@ "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2cm);\npair A = origin;\npair M = (1,0);\npair C = (2,0);\npair P = (2,0.5);\npair B = (2,1);\npair Q = (1,0.5);\n\ndraw(A--B--C--cycle);\ndraw(M--Q--P);\n\nlabel(\" $A$ \",A,SW);\nlabel(\" $M$ \",M,S);\nlabel(\" $C$ \",C,SE);\nlabel(\" $P$ \",P,E);\nlabel(\" $B$ \",B,NE);\nlabel(\" $N$ \",Q,NW);[/asy]", "answer": "$\\frac{109}{12}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 17, @@ -133,7 +149,8 @@ "geo_code": "[asy]size(200);\ndefaultpen(linewidth(0.7)+fontsize(10));\npair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F);\ndraw(D--G--H--A--cycle);\ndraw(Circle(O, abs(O-C)));\nlabel(\" $A$ \", A, NW);\nlabel(\" $B$ \", B, NW);\nlabel(\" $C$ \", C, NE);\nlabel(\" $D$ \", D, SW);\nlabel(\" $E$ \", E, SE);\nlabel(\" $F$ \", F, SW);\n\nlabel(\"4\", (2,0.85), N);\nlabel(\"3\", D--E, S);\nlabel(\"5\", (6.5,0.85), N);\n[/asy]", "answer": "$7$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 18, @@ -141,7 +158,8 @@ "geo_code": "[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\nreal r = (sqrt(133)-9)/2;\ndraw(circle(origin,1)^^circle(origin,4));\nfor(int i=0;i<=2;i=i+1)\n{\nfilldraw(circle(dir(90 + i*120)*(4-r),r),gray);\n}\nfor(int j=0;j<=2;j=j+1)\n{\nfilldraw(circle(dir(30+j*120)*(1+r),r),darkgray);\n}\n[/asy]", "answer": "$126$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 19, @@ -149,7 +167,8 @@ "geo_code": "[asy]\npair A,B,C,D,EE,F,G;\nA = (0,0);\nB = (9,4);\nC = (21,0);\nD = (13,-12);\nEE = (4,-16);\nF = (13/2,-6);\nG = (8,0);\n\ndraw(A--C--EE--B--D--cycle);\n\nlabel(\" $A$ \",A,W);\nlabel(\" $B$ \",B,N);\nlabel(\" $C$ \",C,E);\nlabel(\" $D$ \",D,SE);\nlabel(\" $E$ \",EE,SW);\nlabel(\" $F$ \",F,WSW);\nlabel(\" $G$ \",G,NW);\n[/asy]", "answer": "$80$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "angle" }, { "index": 20, @@ -157,7 +176,8 @@ "geo_code": "[asy]\ndefaultpen(linewidth(0.8));\npen blu = rgb(0,112,191);\nreal r=sqrt(3);\nfill((8,0)--(0,8r)--(-8,0)--cycle, blu);\nfill(origin--(4,4r)--(-4,4r)--cycle, white);\nfill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);\nfill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]", "answer": "$\\frac{\\sqrt{3}}{5}$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 21, @@ -165,7 +185,8 @@ "geo_code": "[asy]unitsize(2cm);\ndefaultpen(fontsize(8)+linewidth(0.8));\npair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);\npair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0);\ndraw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A);\nlabel(\" $A$ \",A,SE);\nlabel(\" $B$ \",B,SW);\nlabel(\" $C$ \",C,NW);\nlabel(\" $D$ \",D,NE);\nlabel(\" $K$ \",K,NNW);\nlabel(\" $L$ \",L,E);\nlabel(\" $M$ \",M,S);\nlabel(\" $N$ \",N,W);[/asy]", "answer": "$32 + 16\\sqrt{3}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 22, @@ -173,7 +194,8 @@ "geo_code": "[asy] \nsize(150);\nfilldraw((0,0)--(0,12)--(24,-60/7)--cycle, lightgray);\nfilldraw((14,0)--(14,5)--(0,12)--cycle, gray);\ndraw((0,0)--(24,0)--(0,12)--cycle);\ndraw((0,0)--(24,0)--(24,-60/7)--cycle);\ndraw((0,12)--(24,-60/7));\ndraw((14,5)--(14,0));\ndot((0,0));\ndot((0,12));\ndot((14,5));\ndot((24,0));\ndot((14,0));\ndot((24,-60/7));\nlabel(\" $14$ \", (7,0), S);\nlabel(\" $10$ \", (19,0), S);\ndraw((0,2/3)--(2/3,2/3)--(2/3,0));\ndraw((14,2/3)--(14+2/3,2/3)--(14+2/3,0));\ndraw((24-2/3,0)--(24-2/3,-2/3)--(24,-2/3));\n[/asy]", "answer": "$144$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 23, @@ -181,7 +203,8 @@ "geo_code": "[asy]\nimport three; size(2inch);\ncurrentprojection=orthographic(4,2,2);\ndraw((0,0,0)--(0,0,3),dashed);\ndraw((0,0,0)--(0,4,0),dashed);\ndraw((0,0,0)--(5,0,0),dashed);\ndraw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3));\ndraw((0,4,3)--(5,4,3)--(5,4,0));\nlabel(\"3\",(5,0,3)--(5,0,0),W);\nlabel(\"4\",(5,0,0)--(5,4,0),S);\nlabel(\"5\",(5,4,0)--(0,4,0),SE);\n[/asy]", "answer": "$10$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "volume" }, { "index": 24, @@ -189,7 +212,8 @@ "geo_code": "[asy]\ndraw((0,0)--(0,1)--(1,1)--(1,0)--cycle);\ndraw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));\ndraw((1,0)--(1.5,0.5)--(1.5,1.5));\ndraw((0.5,1.5)--(1,2)--(1.5,2));\ndraw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));\ndraw((1.5,3.5)--(2.5,3.5));\ndraw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));\ndraw((3,4)--(3,3)--(2.5,2.5));\ndraw((3,3)--(4,3)--(4,2)--(3.5,1.5));\ndraw((4,3)--(3.5,2.5));\ndraw((2.5,.5)--(3,1)--(3,1.5));[/asy]", "answer": "$26$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 25, @@ -197,7 +221,8 @@ "geo_code": "[asy]\ndraw(circle((0,0),10));\ndraw((0,10)--(0,0)--(10,0)--(0,10));\ndraw((0,3)--(4,0));\nlabel(\"O\",(0,0),SW);\nlabel(\"C\",(0,3),W);\nlabel(\"A\",(0,10),N);\nlabel(\"D\",(4,0),S);\nlabel(\"B\",(10,0),E);\n[/asy]", "answer": "$44$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 26, @@ -205,7 +230,8 @@ "geo_code": "[asy]\n\npair A = (0,3);\npair B = (0,0);\npair C = (3,0);\npair D = (0,1.5);\npair E = (0.35,0);\npair F = (1.2,1.8);\npair J = (0.17,0);\npair Y = (0.17,0.75);\n\npair Z = (1.6,0.2);\ndraw(A--B);\ndraw(B--C);\ndraw(C--A);\ndraw(D--F--Z--E--D);\ndraw(\" $O$ \", B, dir(180));\ndraw(\" $B$ \", A, dir(45));\ndraw(\" $A$ \", C, dir(45));\ndraw(\" $Q$ \", E, dir(45));\ndraw(\" $P$ \", D, dir(45));\ndraw(\" $R$ \", Z, dir(45));\ndraw(\" $S$ \", F, dir(45));\ndraw(\" $a$ \", Y, dir(210));\ndraw(\" $b$ \", J, dir(100));\n[/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 27, @@ -213,7 +239,8 @@ "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2.5cm);\n\nreal m = 0;\nreal b = 0;\n\npair O = origin;\npair X = (-1,0);\npair Y = (1,0);\npair Q = midpoint(O--X);\npair A = (Q.x, -1*sqrt(3)/2);\npair B = (Q.x, -1*A.y);\npair M = (Q.x + sqrt(3)/2,0);\n\nm = (B.y - M.y)/(B.x - M.x);\nb = (B.y - m*B.x);\n\npair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));\n\nm = (A.y - M.y)/(A.x - M.x);\nb = (A.y - m*A.x);\n\npair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));\n\ndraw(Circle(O,1));\ndraw(Arc(Q,sqrt(3)/2,-90,90));\ndraw(A--B);\ndraw(X--Y);\ndraw(B--D);\ndraw(A--C);\ndraw(A--D);\ndot(O);dot(M);\n\nlabel(\" $B$ \",B,NW);\nlabel(\" $C$ \",C,NE);\nlabel(\" $Y$ \",Y,E);\nlabel(\" $D$ \",D,SE);\nlabel(\" $A$ \",A,SW);\nlabel(\" $X$ \",X,W);\nlabel(\" $Q$ \",Q,SW);\nlabel(\" $O$ \",O,SW);\nlabel(\" $M$ \",M,NE+2N);[/asy]", "answer": "$\\sqrt{2}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 28, @@ -221,7 +248,8 @@ "geo_code": "[asy]unitsize(2cm);\ndefaultpen(linewidth(.8pt));\n\npair A = (0,0);\npair C = (2,0);\npair B = dir(57.5)*2;\npair E = waypoint(C--A,0.25);\npair D = waypoint(C--B,0.25);\npair T = intersectionpoint(D--A,E--B);\n\nlabel(\" $B$ \",B,NW);label(\" $A$ \",A,SW);label(\" $C$ \",C,SE);label(\" $D$ \",D,NE);label(\" $E$ \",E,S);label(\" $T$ \",T,2*W+N);\n\ndraw(A--B--C--cycle);\ndraw(A--D);\ndraw(B--E);[/asy]", "answer": "$\\frac{4}{11}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 29, @@ -229,7 +257,8 @@ "geo_code": "[asy]\nunitsize (2 cm);\n\npair A, B, C, D, Bp, Cp, Dp, P;\n\nA = (0,0);\nB = (-1,0);\nC = (-1,1);\nD = (0,1);\nBp = rotate(-30)*(B);\nCp = rotate(-30)*(C);\nDp = rotate(-30)*(D);\nP = extension(C, D, Bp, Cp);\n\nfill(A--Bp--P--D--cycle, gray(0.8));\ndraw(A--B--C--D--cycle);\ndraw(A--Bp--Cp--Dp--cycle);\n\nlabel(\" $30^\\circ$ \", (-0.5,0.1), fontsize(10));\n[/asy]", "answer": "$48\\sqrt{3}$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 30, @@ -237,7 +266,8 @@ "geo_code": "[asy] unitsize(7); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E; A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10); draw(A--B--C--cycle); draw(D--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,S); label(\"$E$\",E,NE); draw(rightanglemark(B,D,E,30)); [/asy]", "answer": "$58\\frac{1}{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 31, @@ -245,7 +275,8 @@ "geo_code": "[asy] unitsize(5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); pair O=(0,3/8); draw((-2/3,9/16)--(2/3,9/16)); draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); draw(Circle(O,3/16)); draw((-2/3,0)--(2/3,0)); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$\\frac{3}{8}$\",O); draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); label(\"$\\frac{1}{2}$\",(.5,.25)); draw((.5,.33)--(.5,.5),EndArrow(3)); draw((.5,.17)--(.5,0),EndArrow(3)); label(\"$x$\",midpoint((.5,.5)--(.5,9/16))); draw((.5,5/8)--(.5,9/16),EndArrow(3)); label(\"$60^{\\circ}$\",(0.01,0.12)); dot(A); dot(B); dot(C);[/asy]", "answer": "$\\frac{1}{16}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 32, @@ -253,7 +284,8 @@ "geo_code": "[asy] size(6cm); pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7); draw(A--B--C--D--cycle); draw(F--C--E--B); label(\"$A$\", A, SW); label(\"$B$\", B, S); label(\"$C$\", C, N); label(\"$D$\", D, NW); label(\"$E$\", E, SE); label(\"$F$\", F, W); [/asy]", "answer": "$12$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 33, @@ -261,7 +293,8 @@ "geo_code": "[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label(\"D\",(0,1),NW);label(\"E\",(.4,1),N);label(\"C\",(1,1),NE); label(\"P\",(0,.6),W);label(\"M\",(.25,.55),E);label(\"Q\",(1,.2),E); label(\"A\",(0,0),SW);label(\"B\",(1,0),SE); [/asy]", "answer": "$\\frac{5}{19}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 34, @@ -269,7 +302,8 @@ "geo_code": "[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label(\"A\", (0,0), S); label(\"B\", (5,0), S); label(\"C\", (6,3), NE); label(\"D\", (1,3), NW); label(\"P\", (10,0), E); label(\"Q\", (-5,0), W); label(\"M\", (.5,1.5), NW); label(\"N\", (5.65, 1.5), NE); label(\"O\", (3.4,1.75));[/asy]", "answer": "$\\frac{9}{8}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 35, @@ -277,7 +311,8 @@ "geo_code": "[asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label(\"A\", (0,0), S); label(\"B\", (12,0), S); label(\"C\", (14,7.75), E); label(\"E\", (8.75,4.84), N); label(\"F\", (5,0), S); label(\"M\", (13,3.875), E); label(\"G\", (7,1)); [/asy]", "answer": "$\\frac{3}{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 36, @@ -285,7 +320,8 @@ "geo_code": "[asy] size(120); path c = Circle((0, 0), 1); pair A = dir(20), B = dir(130), C = dir(240), D = dir(330); draw(c); pair F = 3(A-B) + B; pair G = 3(D-C) + C; pair E = intersectionpoints(B--F, C--G)[0]; draw(B--E--C--A); label(\"$A$\", A, NE); label(\"$B$\", B, NW); label(\"$C$\", C, SW); label(\"$D$\", D, SE); label(\"$E$\", E, E); [/asy]", "answer": "$15$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 37, @@ -293,7 +329,8 @@ "geo_code": "[asy] size(120); real t = 2/sqrt(3); real x = 1 + sqrt(3); pair A = t*dir(90), D = x*A; pair B = t*dir(210), E = x*B; pair C = t*dir(330), F = x*C; draw(D--E--F--cycle); draw(Circle(A, 1)); draw(Circle(B, 1)); draw(Circle(C, 1)); [/asy]", "answer": "$18+18\\sqrt{3}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 38, @@ -301,7 +338,8 @@ "geo_code": "[asy] size(100); real a=4, b=3; // import cse5; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP(\"A\",A)--MP(\"B\",B,N)--MP(\"C\",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP(\"R\",IP(CP((X+C)/2,C),B--C),NW)--MP(\"Q\",IP(CP((X+C)/2,C),A--C+(0.1,0)))); [/asy]", "answer": "$4.8$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 39, @@ -309,7 +347,8 @@ "geo_code": "[asy]\nimport cse5;pathpen=black;pointpen=black;\nsize(2inch);\npair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162);\nD(MP(\"A\",A,A)--MP(\"B\",B,B)--MP(\"C\",C,C)--MP(\"D\",D,D)--MP(\"E\",E,E)--cycle,linewidth(1.5));\nD(A--C--E--B--D--cycle);\npair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A);\nD(MP(\"F\",F,dir(126))--MP(\"I\",I,dir(270))--MP(\"G\",G,dir(54))--MP(\"J\",J,dir(198))--MP(\"H\",H,dir(342))--cycle);\n[/asy]", "answer": "$1 + \\sqrt{5}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 40, @@ -317,7 +356,8 @@ "geo_code": "[asy]\nimport graph; size(7cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -4.3, xmax = 14.52, ymin = -8.3, ymax = 6.3; \ndraw((0,1)--(0,0)--(1,0)--(1,1)--cycle); \ndraw((1,1)--(1,0)--(1.87,-0.5)--(2.73,0)--(2.73,1)--(1.87,1.5)--cycle); \ndraw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); \ndraw((0,0)--(1,0)--(1.5,-0.87)--(1,-1.73)--(0,-1.73)--(-0.5,-0.87)--cycle); \ndraw((0,1)--(0,0)--(-0.87,-0.5)--(-1.73,0)--(-1.73,1)--(-0.87,1.5)--cycle); \n\ndraw((0,1)--(0,0)); \ndraw((0,0)--(1,0)); \ndraw((1,0)--(1,1)); \ndraw((1,1)--(0,1)); \ndraw((1,1)--(1,0)); \ndraw((1,0)--(1.87,-0.5)); \ndraw((1.87,-0.5)--(2.73,0)); \ndraw((2.73,0)--(2.73,1)); \ndraw((2.73,1)--(1.87,1.5)); \ndraw((1.87,1.5)--(1,1)); \ndraw((0,1)--(1,1)); \ndraw((1,1)--(1.5,1.87)); \ndraw((1.5,1.87)--(1,2.73)); \ndraw((1,2.73)--(0,2.73)); \ndraw((0,2.73)--(-0.5,1.87)); \ndraw((-0.5,1.87)--(0,1)); \n\ndot((1.87,-0.5),dotstyle); \nlabel(\" $C_1$ \", (1.72,-0.1), NE * labelscalefactor); \ndot((1.87,1.5),dotstyle); \nlabel(\" $B_2$ \", (1.76,1.04), NE * labelscalefactor); \ndot((1.5,1.87),dotstyle); \nlabel(\" $B_1$ \", (0.96,1.8), NE * labelscalefactor); \ndot((-0.5,1.87),dotstyle); \nlabel(\" $A_2$ \", (-0.26,1.78), NE * labelscalefactor); \ndot((-0.87,1.5),dotstyle); \nlabel(\" $A_1$ \", (-0.96,1.08), NE * labelscalefactor); \ndot((-0.87,-0.5),dotstyle); \nlabel(\" $D_2$ \", (-1.02,-0.18), NE * labelscalefactor); \ndot((-0.5,-0.87),dotstyle); \nlabel(\" $D_1$ \", (-0.22,-0.96), NE * labelscalefactor); \ndot((1.5,-0.87),dotstyle); \nlabel(\" $C_2$ \", (0.9,-0.94), NE * labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n[/asy]", "answer": "$67$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 41, @@ -325,7 +365,8 @@ "geo_code": "[asy]\nunitsize(1 cm);\npair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C);\ndraw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8));\nlabel(\" $A$ \",A,.5A); label(\" $B$ \",B,.5B); label(\" $C$ \",C,.5C); label(\" $E$ \",E,.5E); label(\" $D$ \",D,dir(-60));\n[/asy]", "answer": "$8$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 42, @@ -333,7 +374,8 @@ "geo_code": "[asy]\ndraw((0,0)--(0,1)--(2,1)--(2,0)--cycle^^(.5,1)--(.5,2)--(1.5,2)--(1.5,1)--(.5,2)^^(.5,1)--(1.5,2)^^(1,2)--(1,0));\n[/asy]", "answer": "$\\frac{5\\sqrt{17}}{16} $", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 43, @@ -341,7 +383,8 @@ "geo_code": "[asy]\nsize(5cm);\npen dps = fontsize(10);\ndefaultpen(dps);\npair A,B,C,D,E,F,G,H,I;\nG=origin;\nH=(4,0);\nI=(2,2*sqrt(3));\nF=(1.5,3*sqrt(3)/2);\nC=F+(1,0);\nB=F-(1,0);\nD=C+(2,0);\nA=F+(0,sqrt(3));\nE=C+(0.5,3*sqrt(3)/2);\ndraw(A--H--G--E--D--B--cycle);\nlabel(\" $A$ \",A,N*.5);\nlabel(\" $B$ \",B,S*.5);\nlabel(\" $C$ \",C,SW*.5);\nlabel(\" $D$ \",D,S*.5);\nlabel(\" $E$ \",E,N*.5);\nlabel(\" $F$ \",F,SE*.5);\nlabel(\" $G$ \",G,S*.5);\nlabel(\" $H$ \",H,S*.5);\nlabel(\" $I$ \",I,N*2);\n[/asy]", "answer": "$114$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 44, @@ -349,7 +392,8 @@ "geo_code": "[asy]\ndraw((3,-13)--(21.5,-5)--(19,-18)--(9,-18)--(10,-6)--(23,-14.5)--cycle);\nlabel(\"A\",(3,-13),W);label(\"C\",(21.5,-5),N);label(\"E\",(19,-18),E);label(\"F\",(9,-18),W);label(\"B\",(10,-6),N);label(\"D\",(23,-14.5),E);\n[/asy]", "answer": "$4$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 45, @@ -357,7 +401,8 @@ "geo_code": "[asy]\nsize(150);\npair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25);\ndraw(circle(A,1)^^D--B);\ndraw(circle(E,.5)^^circle( F ,.25));\nlabel(\" $A$ \", D, W);\nlabel(\" $K$ \", A, S);\nlabel(\" $B$ \", B, dir(0));\nlabel(\" $L$ \", E, N);\nlabel(\" $M$ \",shift(-.05,.05)*F);\n[/asy]", "answer": "$16$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 46, @@ -365,7 +410,8 @@ "geo_code": "[asy]\nimport graph; size(4.26cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -1.52, xmax = 2.74, ymin = -2.18, ymax = 6.72;\ndraw((0,1)--(2,1)--(2,3)--(0,3)--cycle); \ndraw((0,3)--(2,3)--(1,4.73)--cycle); \ndraw((0,1)--(2,1)); \ndraw((2,1)--(2,3)); \ndraw((2,3)--(0,3)); \ndraw((0,3)--(0,1)); \ndraw((0,3)--(2,3)); \ndraw((2,3)--(1,4.73)); \ndraw((1,4.73)--(0,3)); \ndraw(circle((0,3), 1.44)); \nlabel(\" $C$ \",(-0.4,3.14),SE*labelscalefactor); \nlabel(\" $A$ \",(2.1,3.1),SE*labelscalefactor); \nlabel(\" $B$ \",(0.86,5.18),SE*labelscalefactor); \nlabel(\" $D$ \",(-0.28,0.88),SE*labelscalefactor); \nlabel(\" $E$ \",(2.1,0.8),SE*labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]", "answer": "$20$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 47, @@ -373,7 +419,8 @@ "geo_code": "[asy]\ndraw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0));\nlabel(\"A\", (0,0), S);\nlabel(\"B\", (2,0), S);\nlabel(\"C\", (2,2), N);\nlabel(\"D\", (0,2), N);\nlabel(\"M\", (1,0), S);\nlabel(\"N\", (2,1), E);\nlabel(\"O\", (1.2, .8));\n[/asy]", "answer": "$\\frac{2}{3}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 48, @@ -381,7 +428,8 @@ "geo_code": "[asy]\nimport graph; size(5.75cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); \npen dotstyle = black; \nreal xmin = -2, xmax = 21, ymin = -2, ymax = 16; \ndraw((0,0)--(20,0)); \ndraw((13.48,14.62)--(7,0)); \ndraw((0,0)--(15.93,9.12)); \ndraw((13.48,14.62)--(20,0)); \ndraw((13.48,14.62)--(0,0)); \nlabel(\"6\",(15.16,12.72),SE*labelscalefactor); \nlabel(\"10\",(18.56,5.1),SE*labelscalefactor); \nlabel(\"7\",(3.26,-0.6),SE*labelscalefactor); \nlabel(\"13\",(13.18,-0.71),SE*labelscalefactor); \nlabel(\"20\",(5.07,8.33),SE*labelscalefactor); \ndot((0,0),dotstyle); \nlabel(\" $B$ \", (-1.23,-1.48), NE * labelscalefactor); \ndot((20,0),dotstyle); \nlabel(\" $C$ \", (19.71,-1.59), NE * labelscalefactor); \ndot((7,0),dotstyle); \nlabel(\" $D$ \", (6.77,-1.64), NE * labelscalefactor); \ndot((13.48,14.62),dotstyle); \nlabel(\" $A$ \", (12.36,14.91), NE * labelscalefactor); \ndot((15.93,9.12),dotstyle); \nlabel(\" $E$ \", (16.42,9.21), NE * labelscalefactor); \ndot((9.38,5.37),dotstyle); \nlabel(\" $F$ \", (9.68,4.5), NE * labelscalefactor); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n[/asy]", "answer": "$\\frac{14}{15}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 49, @@ -389,7 +437,8 @@ "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=sqrt(22);\npair B=origin, A=16*dir(60), C=(16,0), D=(10-r,0), E=(10+r,0), F=C+1*dir(120), G=C+14*dir(120), H=13*dir(60), J=6*dir(60), O=circumcenter(G,H,J);\ndot(A^^B^^C^^D^^E^^F^^G^^H^^J);\ndraw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7));\nlabel(\" $A$ \", A, N);\nlabel(\" $B$ \", B, dir(210));\nlabel(\" $C$ \", C, dir(330));\nlabel(\" $D$ \", D, SW);\nlabel(\" $E$ \", E, SE);\nlabel(\" $F$ \", F, dir(170));\nlabel(\" $G$ \", G, dir(250));\nlabel(\" $H$ \", H, SE);\nlabel(\" $J$ \", J, dir(0));\nlabel(\"2\", A--G, dir(30));\nlabel(\"13\", F--G, dir(180+30));\nlabel(\"1\", F--C, dir(30));\nlabel(\"7\", H--J, dir(-30));[/asy]", "answer": "$2\\sqrt{22}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 50, @@ -397,7 +446,8 @@ "geo_code": "[asy]\n unitsize(40);\n pair A = dir(90);\n pair B = dir(210);\n pair C = dir(330);\n dot(A); \n dot(B);\n dot(C);\n draw(B -- C);\n label(\" $A$ \", A, N);\n label(\" $B$ \", B, W);\n label(\" $C$ \", C, E);\n[/asy]", "answer": "$56\\pi+9\\sqrt{3}$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 51, @@ -405,7 +455,8 @@ "geo_code": "[asy]\npair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P;\nA = origin;\nB = (0,3);\nC = 3*dir(150);\nD = (0,1);\nE = (0,2);\nF = C+2*dir(30);\nG = C+dir(30);\nH = 2*dir(150);\nI = dir(150);\nJ = (1,1);\nK = J+dir(30);\nL = (1,2);\nM = F+dir(120);\nN = G+dir(120);\nO = H+dir(240);\nP = I+dir(240);\ndraw(A--B--C--cycle);\ndraw(D--E--F--G--H--I--cycle);\ndraw(D--E--L--J--cycle);\ndraw(F--G--N--M--cycle);\ndraw(H--I--P--O--cycle);\ndraw(J--K--L--cycle);\n[/asy]", "answer": "$\\frac{5}{3\\sqrt{2}}$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "volume" }, { "index": 52, @@ -413,7 +464,8 @@ "geo_code": "[asy]\nimport cse5;\nsize(200);\npair O=(2, 0), B=(0, 0), C=(4, 0), A=(1, 3), D, E;\nD=MP(\"D\",D(IP(D(CP(O,B)),D(MP(\"A\",D(A),N)--MP(\"B\",D(B),W)))),NW);\nE=MP(\"E\",D(IP(CP(O,B),D(MP(\"C\",D(C),NE)--A),1)),NE);\nD(B--C);\n[/asy]", "answer": "$7.18$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 53, @@ -421,7 +473,8 @@ "geo_code": "[asy]\nsize(150);\ndotfactor=4;\ndraw(circle((0,0),4));\ndraw(circle((10,-6),3));\npair O,A,P,Q;\nO = (0,0);\nA = (10,-6);\nP = (-.55, -4.12);\nQ = (10.7, -2.86);\ndot(\" $O$ \", O, NE);\ndot(\" $O'$ \", A, SW);\ndot(\" $P$ \", P, SW);\ndot(\" $Q$ \", Q, NE);\ndraw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle);\ndraw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle);\ndraw(P--Q--cycle);\n[/asy]", "answer": "$6$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 54, @@ -429,7 +482,8 @@ "geo_code": "[asy]\ndraw((0,0)--(1,0)--(1,1)--(0,1)--cycle);\ndraw((0,0)--(1,0.5));\ndraw((1,0)--(0.5,1));\ndraw((1,1)--(0,0.5));\ndraw((0,1)--(0.5,0));\nlabel(\" $A$ \",(0.21,0.6),N);\nlabel(\" $B$ \",(0.4,0.21),W);\nlabel(\" $C$ \",(0.8,0.4),S);\nlabel(\" $D$ \",(0.6,0.8),E);\n[/asy]", "answer": "$\\frac{1}{5}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 55, @@ -437,7 +491,8 @@ "geo_code": "[asy]unitsize(7);\n draw((-7, 0) -- (12, 0));\n draw((-7, 12) -- (12, 12));\n real r = 169 / 48;\n draw(circle((0, r), r));\n draw(circle((5, 12 - r), r));\n pair A = (0, 0);\n pair B = (5, 12);\n dot(A);\n dot(B);\n label(\" $A$ \", A, plain.S);\n label(\" $B$ \", B, plain.N);\n label(\" $\\ell$ \", (12, 0), plain.E);\n label(\" $m$ \", (12, 12), plain.E);[/asy]", "answer": "$\\frac{169}{48}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 56, @@ -445,7 +500,8 @@ "geo_code": "[asy]\nimport olympiad;\n\tsize(80);\ndefaultpen(linewidth(0.8));\ndraw((0,0)--(3,0)--(0,4.25)--(0,0)^^(0,3)--(4.25,0)--(3,0)^^rightanglemark((0,3),(0,0),(3,0),10));\npair P = intersectionpoint((3,0)--(0,4.25),(0,3)--(4.25,0));\ndraw(anglemark((4.25,0),P,(0,4.25),10));\nlabel(\" $\\alpha$ \",P,2 * NE);\n[/asy]", "answer": "$170$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "angle" }, { "index": 57, @@ -453,7 +509,8 @@ "geo_code": "[asy]defaultpen(linewidth(0.7));pointpen=black; pathpen=black;\nsize(7cm);\npair A,B,C,D,E,F,G,H,W,X,Y,Z;\nA=(0,2); B=(1,2); C=(2,2); D=(3,2);\nH=(0,0); G=(1,0); F=(2,0); E=(3,0);\nD('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,NE); D('F',F,NE); D('G',G,NW); D('H',H,NW);\nD(A--F); D(B--E); D(D--G); D(C--H);\nZ=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H);\nD('W',W,N); D('X',X,plain.E); D('Y',Y,S); D('Z',Z,plain.W);\nD(A--D--E--H--cycle);[/asy]", "answer": "$\\frac{1}{2}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 58, @@ -461,7 +518,8 @@ "geo_code": "[asy]size(2inch);\npair O,A,B,C,D,E;\nB=(0,0);\nO=(2,0);\nC=(4,0);\nD=(.333,1.333);\nA=(.75,2.67);\nE=(1.8,2);\ndraw(Arc(O,2,0,360));\ndraw(B--C--A--B);\nlabel(\" $A$ \",A,N);\nlabel(\" $B$ \",B,W);\nlabel(\" $C$ \",C,E);\nlabel(\" $D$ \",D,W);\nlabel(\" $E$ \",E,N);\nlabel(\"Figure not drawn to scale\",(2,-2.5),S);\n[/asy]", "answer": "$26$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 59, @@ -469,7 +527,8 @@ "geo_code": "[asy]\nsize(200);\ndefaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28);\ndraw(P--Q--R--S--cycle);\npicture p = new picture;\ndraw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5));\ndraw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3));\ntransform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5);\nadd(t*p);\nlabel(\" $P$ \",P,SW); label(\" $Q$ \",Q,NW); label(\" $R$ \",R,NE); label(\" $S$ \",S,SE); label(\" $A$ \",t*(0,-3),W); label(\" $B$ \",B,N); label(\" $C$ \",t*(3,0),plain.ENE); label(\" $D$ \",t*(3,-2),NE); label(\" $E$ \",t*(5,-2),plain.E); label(\" $F$ \",t*(5,-3),plain.SW); label(\" $G$ \",t*(3,-3),(0.81,-1.3)); label(\" $H$ \",t*(3,-5),plain.S); label(\" $I$ \",t*(2,-5),NW); label(\" $J$ \",t*(2,-3),SW);[/asy]", "answer": "$338$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 60, @@ -477,7 +536,8 @@ "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2cm);\n\npair O = origin;\npair C = (-1,0);\npair D = (1,0);\npair E = (0,1);\npair A = dir(-135);\npair B = dir(-60);\n\ndraw(Circle(O,1));\ndraw(C--E--D--cycle);\ndraw(A--O--B--cycle);\n\nlabel(\" $A$ \",A,SW);\nlabel(\" $C$ \",C,W);\nlabel(\" $E$ \",E,N);\nlabel(\" $D$ \",D,NE);\nlabel(\" $B$ \",B,SE);\nlabel(\" $O$ \",O,N);[/asy]", "answer": "$2$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 61, @@ -485,7 +545,8 @@ "geo_code": "[asy]unitsize(5cm);\ndefaultpen(linewidth(.8pt)+fontsize(8pt));\ndotfactor=3;\n\npair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);\npair O=(0,3/8);\n\ndraw((-2/3,9/16)--(2/3,9/16));\ndraw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2));\ndraw(Circle(O,3/16));\ndraw((-2/3,0)--(2/3,0));\n\nlabel(\" $A$ \",A,SW);\nlabel(\" $B$ \",B,SE);\nlabel(\" $C$ \",C,N);\nlabel(\" $\\frac{3}{8}$ \",O);\ndraw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3));\ndraw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3));\nlabel(\" $\\frac{1}{2}$ \",(.5,.25));\ndraw((.5,.33)--(.5,.5),EndArrow(3));\ndraw((.5,.17)--(.5,0),EndArrow(3));\nlabel(\" $x$ \",midpoint((.5,.5)--(.5,9/16)));\ndraw((.5,5/8)--(.5,9/16),EndArrow(3));\nlabel(\" $60^{\\circ}$ \",(0.01,0.12));\ndot(A);\ndot(B);\ndot(C);[/asy]", "answer": "$\\frac{1}{16}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 62, @@ -493,7 +554,8 @@ "geo_code": "[asy]\nsize(100); defaultpen(linewidth(0.7)+fontsize(10)); \npair D2(pair P) {\ndot(P,linewidth(3)); return P;\n}\npair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];\ndraw(A--B--C--D--cycle);\ndraw(F--A--H); draw(E--F); draw(G--H); \nlabel(\" $A$ \",D2(A),NW);\nlabel(\" $B$ \",D2(B),SW);\nlabel(\" $C$ \",D2(C),SE);\nlabel(\" $D$ \",D2(D),NE);\nlabel(\" $E$ \",D2(E),plain.N);\nlabel(\" $F$ \",D2(F),S);\nlabel(\" $G$ \",D2(G),W);\nlabel(\" $H$ \",D2(H),plain.E);\nlabel(\" $P$ \",D2(P),SE);\n[/asy]", "answer": "$45$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "angle" }, { "index": 63, @@ -501,7 +563,8 @@ "geo_code": "[asy]\nmarkscalefactor=0.15;\nsize(8cm);\npair A = (0,0);\npair B = (17,0);\npair E = (0,17);\npair D = (17,17);\npair F = (-120/17,225/17);\npair C = (17+120/17, 64/17);\ndraw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle); \nlabel(\" $A$ \", A, S);\nlabel(\" $B$ \", B, S);\nlabel(\" $C$ \", C, dir(0));\nlabel(\" $D$ \", D, N);\nlabel(\" $E$ \", E, N);\nlabel(\" $F$ \", F, W);\nlabel(\" $8$ \", (F+E)/2, NW);\nlabel(\" $15$ \", (F+A)/2, SW);\nlabel(\" $8$ \", (C+B)/2, SE);\nlabel(\" $15$ \", (D+C)/2, NE);\ndraw(rightanglemark(E,F,A));\ndraw(rightanglemark(D,C,B));\n[/asy]", "answer": "$25$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 64, @@ -509,7 +572,8 @@ "geo_code": "[asy]\nsize(300);\ndefaultpen(linewidth(0.8));\ndraw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));\nlabel(\" $A$ \",origin,SW);\nlabel(\" $B$ \",(0,4),dir(160));\nlabel(\" $C$ \",(3,0),S);\nlabel(\" $D$ \",(6,8),dir(80));\nlabel(\" $E$ \",(30,-4),E);[/asy]", "answer": "$5393$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 65, @@ -517,7 +581,8 @@ "geo_code": "[asy]\nunitsize(90);\npair A = dir(0);\npair B = dir(120);\npair C = dir(240);\ndraw(A -- B -- C -- cycle);\npair D = (2*A + B)/3;\npair E = (A + 2*B)/3;\npair F = (2*B + C)/3;\npair G = (B + 2*C)/3;\npair H = (2*C + A)/3;\npair I = (C + 2*A)/3;\ndraw(E -- F);\ndraw(G -- H);\ndraw(I -- D);\ndraw(D -- G);\ndraw(E -- H);\ndraw(F -- I);\npair O = (0, 0);\nreal r = 1/sqrt(3);\ndraw(circle(O, r));\nfill(O -- D -- E -- cycle, gray);\nfill(O -- F -- G -- cycle, gray);\nfill(O -- H -- I -- cycle, gray);\nfill(arc(O, r, -30, 30) -- cycle, gray);\nfill(arc(0, r, 90, 150) -- cycle, gray);\nfill(arc(0, r, 210, 270) -- cycle, gray);\nlabel(\" $A$ \", A, A);\nlabel(\" $B$ \", B, B);\nlabel(\" $C$ \", C, C);\nlabel(\" $D$ \", D, unit(D));\nlabel(\" $E$ \", E, unit(E));\nlabel(\" $F$ \", F, unit(F));\nlabel(\" $G$ \", G, unit(G));\nlabel(\" $H$ \", H, unit(H));\nlabel(\" $I$ \", I, unit(I));\nlabel(\" $O$ \", O, C);\n[/asy]", "answer": "$\\frac{2\\sqrt{3}}{27}$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 66, @@ -525,7 +590,8 @@ "geo_code": "[asy]draw((75,0)--(0,0)--(0,50)--(75,50)--(75,0)--(55,0)--(55,20)--(100,20)--(100,0)--cycle);\ndraw((55,5)--(60,5)--(60,0));\ndraw((75,5)--(80,5)--(80,0));\nlabel(\"A\",(0,50),NW);\nlabel(\"B\",(0,0),SW);\nlabel(\"C\",(75,0),SE);\nlabel(\"D\",(75,50),NE);\nlabel(\"E\",(55,20),NW);\nlabel(\"F\",(55,0),SW);\nlabel(\"G\",(100,0),SE);\nlabel(\"H\",(100,20),NE);\nlabel(\"K\",(75,20),NE);[/asy]", "answer": "$6.25$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 67, @@ -533,7 +599,8 @@ "geo_code": "[asy]\nsize(250);\n defaultpen (linewidth (0.7) + fontsize (10));\n\t\t draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);\n\t\t label(\" $A$ \", (0,23), NW);\n\t\t label(\" $B$ \", (23, 23), NE);\n\t\t label(\" $C$ \", (23,0), SE);\n\t\t label(\" $D$ \", (0,0), SW);\n\t\t draw((0,6)--(23,6));\n\t\t draw((0,19)--(23,19));\n\t\t draw((5,0)--(5,23));\n\t\t draw((12,0)--(12,23));\n\t\t label(\"13\", (17/2, 21));\n\t\t label(\"111\",(35/2,25/2));\n\t\t label(\"37\",(17/2,3));\n\t\t label(\"123\",(2.5,12.5));[/asy]", "answer": "$180$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 68, @@ -541,7 +608,8 @@ "geo_code": "[asy]\nimport graph; size(3.2cm); \nreal labelscalefactor = 0.5; \npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);\ndraw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle); \ndraw((-1,3)--(-1,2)); \ndraw((-1,2)--(-0.13,1.5)); \ndraw((-0.13,1.5)--(0.73,2)); \ndraw((0.73,2)--(0.73,3)); \ndraw((0.73,3)--(-0.13,3.5)); \ndraw((-0.13,3.5)--(-1,3)); \ndraw((-1,3.5)--(0.73,3.5)); \ndraw((0.73,3.5)--(0.73,1.5)); \ndraw((-1,1.5)--(0.73,1.5)); \ndraw((-1,3.5)--(-1,1.5)); \nlabel(\" $ A $ \",(-1.4,3.9),SE*labelscalefactor); \nlabel(\" $ B $ \",(-1.4,3.28),SE*labelscalefactor); \nlabel(\" $ C $ \",(-1.4,2.29),SE*labelscalefactor); \nlabel(\" $ D $ \",(-1.4,1.45),SE*labelscalefactor); \nlabel(\" $ E $ \",(-0.3,1.4),SE*labelscalefactor); \nlabel(\" $ F $ \",(0.8,1.45),SE*labelscalefactor); \nlabel(\" $ G $ \",(0.8,2.24),SE*labelscalefactor); \nlabel(\" $ H $ \",(0.8,3.26),SE*labelscalefactor); \nlabel(\" $ I $ \",(0.8,3.9),SE*labelscalefactor); \nlabel(\" $ J $ \",(-0.25,3.9),SE*labelscalefactor); [/asy]", "answer": "$55$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 69, @@ -549,7 +617,8 @@ "geo_code": "[asy]\nsize(200);\nimport three; defaultpen(linewidth(0.7)+fontsize(10)); currentprojection = orthographic(0,4,2.5); \n// 1.15 x-scale distortion factor \ntriple A = (0,0,0), B = (75^.5/1.15,-125^.5,0), C = (-75^.5/1.15,-125^.5,0), D = (A+B)/2 + (0,0,abs((B-A)/2)), E = (A+C)/2 + (0,0,abs((C-A)/2)), F = (C+B)/2 + (0,0,abs((B-C)/2)); \ndraw(D--E--F--cycle); draw(B--A--C); \nreal r = 1.38; draw(B--(r*B+C)/(1+r)^^(B+r*C)/(1+r)--C,linetype(\"4 4\")); draw((B+r*C)/(1+r)--(r*B+C)/(1+r));\ndraw(arc((A+B)/2,A,D)); draw(arc((A+B)/2,D,B)); draw(arc((A+C)/2,E,A)); draw(arc((A+C)/2,E,C)); draw(arc((C+B)/2,F,B)); draw(arc((C+B)/2,F,C)); \nlabel(\" $A$ \",A,S); label(\" $B$ \",B,W); label(\" $C$ \",C,plain.E);\nlabel(\" $D$ \",D,SW); label(\" $E$ \",E,SE); label(\" $F$ \",F,N);[/asy]", "answer": "$24$", "category": "Global Abstract Integration", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 70, @@ -557,7 +626,8 @@ "geo_code": "[asy]\nunitsize(5);\npair A = (-9 sqrt(3), 0);\npair B = (9 sqrt(3), 0);\npair C = (-18 sqrt(3), 0);\npair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3);\npair E = (2 sqrt(3), 4 sqrt(6));\npair F = (7 sqrt(3), 5 sqrt(6));\npair G = (12 sqrt(3), 6 sqrt(6));\nreal r = 9sqrt(3);\ndraw(circle(A, r));\ndraw(circle(B, r));\ndraw(circle((B + C) / 2, 3r / 2));\ndraw(C -- D);\ndraw(\" $6$ \", E -- D);\ndraw(E -- F);\ndraw(\" $9$ \", F -- G);\ndot(A);\ndot(B);\nlabel(\" $A$ \", A, plain.E);\nlabel(\" $B$ \", B, plain.E);\nlabel(\" $C$ \", C, W);\nlabel(\" $D$ \", D, dir(160));\nlabel(\" $E$ \", E, S);\nlabel(\" $F$ \", F, SSW);\nlabel(\" $G$ \", G, N);\n[/asy]", "answer": "$9\\sqrt{19}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 71, @@ -565,7 +635,8 @@ "geo_code": "[asy]\nfor (int a = 0; a < 7; ++a)\n{\nfor (int b = 0; b < 8; ++b)\n{\ndot((a,b));\n}\n}\ndraw((3,0)--(0,5)--(3,7)--(6,5)--cycle);[/asy]", "answer": "$189$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 72, @@ -573,7 +644,8 @@ "geo_code": "[asy]\nsize(4cm);\ndraw(circle((0,0),3));\ndraw(circle((12,0),9));\ndraw(3*dir(120)--(12,0)+9*dir(120));\ndraw(3*dir(240)--(12,0)+9*dir(240));\n[/asy]", "answer": "$29$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 73, @@ -581,7 +653,8 @@ "geo_code": "[asy]defaultpen(linewidth(.8pt));\nunitsize(2cm);\n\npair A = origin;\npair B = (2.25,0);\npair C = (2,1);\npair D = (1,1);\npair E = waypoint(A--D,0.25);\npair F = waypoint(B--C,0.25);\n\ndraw(A--B--C--D--cycle);\ndraw(E--F);\n\nlabel(\"6\",midpoint(A--D),NW);\nlabel(\"3\",midpoint(C--D),N);\nlabel(\"4\",midpoint(C--B),NE);\nlabel(\"9\",midpoint(A--B),S);[/asy]", "answer": "$4$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 74, @@ -589,7 +662,8 @@ "geo_code": "[asy]\nsize(200);\npair O=(0,0);\nreal R=10, r=4.7;\ndraw(arc(O,R,0,180)--cycle);\npair P=(sqrt((R-r)^2-r^2),r),Q;\ndraw(circle(P,r));\nreal a=0,b=r,c;\nfor(int k=0;k<20;++k)\n{\nc=(a+b)/2;\nQ=(-sqrt((R-c)^2-c^2),c);\nif(abs(P-Q)>c+r) a=c; else b=c;\n}\ndraw(circle(Q,c));[/asy]", "answer": "$361$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 75, @@ -597,7 +671,8 @@ "geo_code": "[asy]\npair A,C,T,R;\nC = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = (3/2 - sqrt(5)/2,1.175570);\ndraw(C--A--T--cycle);\ndraw(T--R);\nlabel(\" $A$ \",A,N);\nlabel(\" $T$ \",T,SE);\nlabel(\" $C$ \",C,SW);\nlabel(\" $R$ \",R,NW);\n[/asy]", "answer": "$72$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "angle" }, { "index": 76, @@ -605,7 +680,8 @@ "geo_code": "[asy]\ndraw((0,0)--(8,0)--(4,12)--cycle);\ndraw((8,0)--(1.6,4.8));\nlabel(\"A\", (4,12), N);\nlabel(\"B\", (0,0), W);\nlabel(\"C\", (8,0), E);\nlabel(\"P\", (1.6,4.8), NW);\ndot((0,0));\ndot((4,12));\ndot((8,0));\ndot((1.6,4.8));\n[/asy]", "answer": "$36$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "angle" }, { "index": 77, @@ -613,7 +689,8 @@ "geo_code": "[asy]\nunitsize(75);\npathpen = black; pointpen=black;\npair A = MP(\"A\", D((0,0)), dir(200));\npair B = MP(\"B\", D((2,0)), dir(-20));\npair C = MP(\"C\", D((1/2,1)), dir(100));\npair D = MP(\"D\", D(midpoint(B--C)), dir(30));\npair E = MP(\"E\", D(midpoint(A--B)), dir(-90));\npair P = MP(\"P\", D(IP(A--D, C--E)), dir(150)*2.013);\ndraw(A--B--C--cycle);\ndraw(A--D--E--C);\n[/asy]", "answer": "$13.5$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 78, @@ -621,7 +698,8 @@ "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=100.8933946;\npair A=sqrt(7)*dir(r), B=origin, C=(3,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);\ndraw(A--B--C--A--D^^B--E^^C--F);\npair point=G;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $E$ \", E, dir(point--E));\nlabel(\" $F$ \", F, dir(point--F));\nlabel(\" $G$ \", G, dir(20));\nlabel(\"1\", B--G, dir(150));\nlabel(\"1\", D--G, dir(30));\nlabel(\"1\", B--D, dir(270));[/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 79, @@ -629,7 +707,8 @@ "geo_code": "[asy]\nsize(4inch,2inch);\ndraw((0,0)--(31,0)--(16,8)--(6,8)--cycle);\ndraw((11,8)--(11,0), linetype(\"8 4\"));\ndraw((11,1)--(12,1)--(12,0));\nlabel(\" $A$ \", (0,0), SW);\nlabel(\" $D$ \", (31,0), SE);\nlabel(\" $B$ \", (6,8), NW);\nlabel(\" $C$ \", (16,8), NE);\nlabel(\"10\", (3,5), W);\nlabel(\"8\", (11,4), E);\nlabel(\"17\", (22.5,5), E);[/asy]", "answer": "$10$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 80, @@ -637,7 +716,8 @@ "geo_code": "[asy]\npair A, B, C, X, Y;\nA = origin;\nX = dir(30);\nY = X + dir(0);\nB = Y + dir(60);\nC = B + dir(330);\ndraw(A--B--C--cycle);\ndraw(X--Y--B);\nlabel(\" $A$ \",A,W);\nlabel(\" $B$ \",B,N);\nlabel(\" $C$ \",C,E);\nlabel(\" $X$ \",X,NW);\nlabel(\" $Y$ \",Y,SE);\n[/asy]", "answer": "$15$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "angle" }, { "index": 81, @@ -645,7 +725,8 @@ "geo_code": "[asy]\nunitsize (1 cm);\n\npair A, B, C, D, E, F, S;\n\nA = (0,0);\nB = (5,0);\nC = (1,4);\nS = (14*A + 15*B + 6*C)/35;\nD = extension(A,S,B,C);\nE = extension(B,S,C,A);\nF = extension(C,S,A,B);\n\ndraw(A--B--C--cycle);\ndraw(A--D);\ndraw(B--E);\ndraw(C--F);\n\ndot(\" $A$ \", A, SW);\ndot(\" $B$ \", B, SE);\ndot(\" $C$ \", C, N);\ndot(\" $D$ \", D, NE);\ndot(\" $E$ \", E, W);\ndot(\" $F$ \", F, dir(270));\ndot(\" $S$ \", S, NE);\n[/asy]", "answer": "$2.889$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 82, @@ -653,7 +734,8 @@ "geo_code": "[asy]\nmarkscalefactor=0.15;\nsize(8cm);\npair A = (0,0);\npair B = (17,0);\npair E = (0,17);\npair D = (17,17);\npair F = (-120/17,225/17);\npair C = (17+120/17, 64/17);\ndraw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle); \nlabel(\" $A$ \", A, S);\nlabel(\" $B$ \", B, S);\nlabel(\" $C$ \", C, dir(0));\nlabel(\" $D$ \", D, N);\nlabel(\" $E$ \", E, N);\nlabel(\" $F$ \", F, W);\nlabel(\" $8$ \", (F+E)/2, NW);\nlabel(\" $15$ \", (F+A)/2, SW);\nlabel(\" $8$ \", (C+B)/2, SE);\nlabel(\" $15$ \", (D+C)/2, NE);\ndraw(rightanglemark(E,F,A));\ndraw(rightanglemark(D,C,B));\n[/asy]", "answer": "$23\\sqrt{2}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 83, @@ -661,7 +743,8 @@ "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=100.8933946;\npair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);\ndraw(A--B--C--A--D^^B--E^^C--F);\npair point=G;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $E$ \", E, dir(point--E));\nlabel(\" $F$ \", F, dir(point--F));\nlabel(\" $G$ \", G, dir(20));\nlabel(\"1\", B--G, dir(150));\nlabel(\"1\", D--G, dir(30));\nlabel(\"1\", B--D, dir(270));[/asy]", "answer": "$\\sqrt{7}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 84, @@ -669,7 +752,8 @@ "geo_code": "[asy]\nsize(200);\ndefaultpen(fontsize(10));\nreal r=100.8933946;\npair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);\ndraw(A--B--C--A--D^^B--E^^C--F);\npair point=G;\nlabel(\" $A$ \", A, dir(point--A));\nlabel(\" $B$ \", B, dir(point--B));\nlabel(\" $C$ \", C, dir(point--C));\nlabel(\" $D$ \", D, dir(point--D));\nlabel(\" $E$ \", E, dir(point--E));\nlabel(\" $F$ \", F, dir(point--F));\nlabel(\" $G$ \", G, dir(20));\nlabel(\"1\", B--G, dir(150));\nlabel(\"1\", D--G, dir(30));\nlabel(\"1\", B--D, dir(270));[/asy]", "answer": "$\\sqrt{13}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 85, @@ -677,7 +761,8 @@ "geo_code": "[asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP(\"$A$\",A)); D(MP(\"$B$\",B)); D(MP(\"$C$\",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP(\"$B'$\",BB[0],NW);MP(\"$C'$\",CC[1],NE); [/asy]", "answer": "$1+\\sqrt{3(r^2-1)}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 86, @@ -685,7 +770,8 @@ "geo_code": "[asy] size(200); import cse5; pathpen=black; anglefontpen=black; pointpen=black; anglepen=black; dotfactor=3; pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE; EE=(B+C)/2; D(MP(\"$A$\",A,W)--MP(\"$B$\",B,N)--MP(\"$C$\",C,E)--cycle); D(MP(\"$E$\",EE,N)--MP(\"$D$\",D,S)); D(D);D(EE); MA(\"80^\\circ\",8,D,EE,C,0.1); MA(\"20^\\circ\",8,EE,C,D,0.3,2,shift(1,3)*C); draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow()); MA(\"100^\\circ\",8,A,B,C,0.1,0); MA(\"60^\\circ\",8,C,A,B,0.1,0); [/asy]", "answer": "$\\frac{\\sqrt{3}}{8}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 87, @@ -693,7 +779,8 @@ "geo_code": "[asy] defaultpen(linewidth(.8pt)); pair A = (0,11); pair B = (2,0); pair D = (4,0); pair E = (7,0); pair C = (13,0); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,S); label(\"$E$\",E,S); label(\"$2$\",midpoint(B--D),N); label(\"$3$\",midpoint(D--E),NW); label(\"$6$\",midpoint(E--C),NW); draw(A--B--C--cycle); draw(A--D); draw(A--E); [/asy]", "answer": "$2\\sqrt{10}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 88, @@ -701,7 +788,8 @@ "geo_code": "[asy] draw((1,9)--(6,9)--(6,0)--(2,0)--(2,4)--(1,4)--cycle); label(\"A\",(1,9),NW); label(\"B\",(6,9),NE); label(\"C\",(6,0),SE); label(\"D\",(2,0),SW); label(\"E\",(2,4),NE); label(\"F\",(1,4),SW); label(\"6\",(3,9),N); label(\"9\",(6,4.5),E); label(\"4\",(4,0),S); label(\"5\",(1,6.5),W); [/asy]", "answer": "$46$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 89, @@ -709,7 +797,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label(\"A\", A, NE); label(\"B\", B, NE); label(\"C\", C, dir(0)); label(\"D\", D, dir(180)); label(\"$L$\", (x/2,0), SW); label(\"$L^\\prime$\", C, SW); label(\"1\", D--F, NW); label(\"1\", F--E, SE); label(\"1\", E--B, SE); clip(W--X--Y--Z--cycle);[/asy]", "answer": "$4.2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 90, @@ -717,7 +806,8 @@ "geo_code": "[asy] size(4cm); draw((0,0)--(0,6)--(7,6)--(7,3)--(2.7,3)--(2.7,0)--cycle); label(\"$6$\",(0,3),W); label(\"$8$\",(4,6),N); draw((0.5,0)--(0.5,0.5)--(0,0.5)); draw((0.5,6)--(0.5,5.5)--(0,5.5)); draw((6.5,6)--(6.5,5.5)--(7,5.5)); draw((6.5,3)--(6.5,3.5)--(7,3.5)); draw((2.2,0)--(2.2,0.5)--(2.7,0.5)); draw((2.7,2.5)--(3.2,2.5)--(3.2,3)) [/asy]", "answer": "$28$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 91, @@ -725,7 +815,8 @@ "geo_code": "[asy] size(4cm);draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(.5 mm)); label(\"2\",(8,2),E); label(\"4\",(4,0),S); [/asy]", "answer": "$8$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 92, @@ -733,7 +824,8 @@ "geo_code": "[asy] draw((0,0)--(18,0)--(18,12)--(0,12)--cycle); draw((0,6)--(18,6)); for(int a=6; a<12; ++a) { draw((1.5a,0)--(1.5a,6)); } fill((15,0)--(18,0)--(18,6)--(15,6)--cycle,black); label(\"0\",(0,0),W); label(\"9\",(9,0),S); label(\"18\",(18,0),S); label(\"6\",(0,6),W); label(\"12\",(0,12),W); [/asy]", "answer": "$18$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 93, @@ -741,7 +833,8 @@ "geo_code": "[asy] pair A,B,C,D; A=(0,4); B=(3,4); C=(3,0); D=origin; draw(circle(D,5)); fill((0,5)..(1.5,4.7697)..B--A--cycle,black); fill(B..(4,3)..(5,0)--C--cycle,black); draw((0,5)--D--(5,0)); label(\"A\",A,NW); label(\"B\",B,NE); label(\"C\",C,S); label(\"D\",D,SW); [/asy]", "answer": "$\\frac{25}{4}\\pi - 12$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 94, @@ -749,7 +842,8 @@ "geo_code": "[asy] size((220)); draw((0,0)--(20,0)--(7,6)--cycle); draw((6,6)--(10,-1)); label(\"A\", (0,0), W); label(\"B\", (20,0), E); label(\"C\", (7,6), NE); label(\"D\", (9.5,-1), W); label(\"E\", (5.9, 6.1), SW); label(\"$45^{\\circ}$\", (2.5,.5)); label(\"$60^{\\circ}$\", (7.8,.5)); label(\"$30^{\\circ}$\", (16.5,.5)); [/asy]", "answer": "$\\sqrt[4]{12}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 95, @@ -757,7 +851,8 @@ "geo_code": "[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label(\"O\",(-2.5,0),W); label(\"A\",A,W); label(\"B\",B,S); label(\"C\",C,W); label(\"D\",D,E); label(\"0\",(-1.8,0),W); label(\"20\",(-1.7,.5),NW); label(\"160\",(1.6,.5),NE); label(\"180\",(1.7,0),E); [/asy]", "answer": "$50$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 96, @@ -765,7 +860,8 @@ "geo_code": "[asy] fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); label(\"2\",(0,.9),W); label(\"3\",(7.3,4.5),N); draw((0,-3.3)--(0,-5.3),linewidth(1)); draw((0,-4.3)--(3.7,-4.3),linewidth(1)); label(\"10\",(4.7,-3.7),S); draw((5.7,-4.3)--(10,-4.3),linewidth(1)); draw((10,-3.3)--(10,-5.3),linewidth(1)); draw((11,4.5)--(13,4.5),linewidth(1)); draw((12,4.5)--(12,2),linewidth(1)); label(\"10\",(11.3,1),E); draw((12,0)--(12,-3.5),linewidth(1)); draw((11,-3.5)--(13,-3.5),linewidth(1)); [/asy]", "answer": "$44$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 97, @@ -773,7 +869,8 @@ "geo_code": "[asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label(\"A\",A,SW); label(\"B\",B,NW); label(\"C\",C,NE); label(\"D\",D,SE); label(\"E\",E,S); label(\"$10$\",(9,8),N); label(\"$6$\",(7,0),S); label(\"$8$\",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy]", "answer": "$4$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 98, @@ -781,7 +878,8 @@ "geo_code": "[asy] unitsize(24); draw((0,0)--(1,0)--(1,3)--(0,3)--cycle); draw((1,0)--(1+9*sqrt(3)/2,9/2)--(1+9*sqrt(3)/2,15/2)--(1+5*sqrt(3)/2,11/2)--(1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),4)--(1+2*sqrt(3),5)--(1,3)); draw((0,3)--(2*sqrt(3),5)--(1+2*sqrt(3),5)); draw((1+9*sqrt(3)/2,15/2)--(9*sqrt(3)/2,15/2)--(5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,5)); draw((1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),9/2)); draw((1+5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,11/2)); label(\"$1'$\",(.5,0),S); label(\"$3'$\",(1,1.5),E); label(\"$9'$\",(1+9*sqrt(3)/4,9/4),S); label(\"$1'$\",(1+9*sqrt(3)/4,17/4),S); label(\"$1'$\",(1+5*sqrt(3)/2,5),E);label(\"$1'$\",(1/2+5*sqrt(3)/2,11/2),S); [/asy]", "answer": "$0$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 99, @@ -789,7 +887,8 @@ "geo_code": "[asy] fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray); fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white); fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white); fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white); fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white); draw(circle((3,3),3)); draw(circle((3,-3),3)); draw(circle((-3,-3),3)); draw(circle((-3,3),3)); draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle); [/asy]", "answer": "$36-9\\pi$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 100, @@ -797,7 +896,8 @@ "geo_code": "[asy] fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill((0,0)--arc((0,0),1,180,360)--cycle,white); draw((1,0)--arc((1,0),2,180,225)--(1,0),black+linewidth(1)); draw((-1,0)--arc((-1,0),2,315,360)--(-1,0),black+linewidth(1)); draw((0,0)--arc((0,0),1,180,360)--(0,0),black+linewidth(1)); draw(arc((0,-1),2-sqrt(2),225,315),black+linewidth(1)); draw((0,0)--(0,-1),black+linewidth(1)); MP(\"C\",(0,0),N);MP(\"A\",(-1,0),N);MP(\"B\",(1,0),N); MP(\"D\",(0,-.8),NW);MP(\"E\",(1-sqrt(2),-sqrt(2)),SW);MP(\"F\",(-1+sqrt(2),-sqrt(2)),SE); [/asy]", "answer": "$2\\pi-\\pi \\sqrt{2}-1$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 101, @@ -805,7 +905,8 @@ "geo_code": "[asy] draw((-1,0)--(1,0)--(1+sqrt(2),sqrt(2))--(0,sqrt(2)+sqrt(13-2*sqrt(2)))--(-1-sqrt(2),sqrt(2))--cycle,black+linewidth(.75)); MP(\"A\",(-1,0),SW);MP(\"B\",(1,0),SE);MP(\"C\",(1+sqrt(2),sqrt(2)),E);MP(\"D\",(0,sqrt(2)+sqrt(13-2*sqrt(2))),N);MP(\"E\",(-1-sqrt(2),sqrt(2)),W); dot((-1,0));dot((1,0));dot((1+sqrt(2),sqrt(2)));dot((-1-sqrt(2),sqrt(2)));dot((0,sqrt(2)+sqrt(13-2*sqrt(2)))); [/asy]", "answer": "$28\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 102, @@ -813,7 +914,8 @@ "geo_code": "[asy] draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75)); draw((0,-1)--(0,1), black+linewidth(.75)); draw((-1,0)--(1,0), black+linewidth(.75)); draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75)); draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75)); draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75)); draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75)); [/asy]", "answer": "$\\frac{\\sqrt{3}+1}{4}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 103, @@ -821,7 +923,8 @@ "geo_code": "[asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label(\"6\", (1.5,6)); label(\"?\", (1.5,2.5)); label(\"14\", (6.5,6)); label(\"35\", (6.5,2.5)); [/asy]", "answer": "$70$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 104, @@ -829,7 +932,8 @@ "geo_code": "[asy] Label l; l.p=fontsize(6); xaxis(\"$x$\",0,6,Ticks(l,1.0,0.5),EndArrow); yaxis(\"$y$\",0,4,Ticks(l,1.0,0.5),EndArrow); draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy]", "answer": "$\\frac{9}{7}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 105, @@ -837,7 +941,8 @@ "geo_code": "[asy] draw(Circle((0,0), 13)); draw((-13,0)--(12,5)--(12,-5)--cycle); dot((-13,0)); dot((12,5)); dot((12,-5)); label(\"A\", (-13,0), W); label(\"B\", (12,5), NE); label(\"C\", (12,-5), SE); [/asy]", "answer": "$\\2\\sin\\left(\\frac{1}{4}\\right)$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 106, @@ -845,7 +950,8 @@ "geo_code": "[asy] import graph; size(5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label(\"$11$\",(1,1.63),W); dot((1,3),ds); label(\"$A$\",(1,3),N); dot((0,0),ds); label(\"$B$\",(0,0),SW); dot((2,0),ds); label(\"$C$\",(2,0),SE); dot((1,0),ds); label(\"$M$\",(1,0),S); dot((1,0.7),ds); label(\"$D$\",(1,0.7),NE); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]", "answer": "$11+\\sqrt{605}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 107, @@ -853,7 +959,8 @@ "geo_code": "[asy] pair O,P,Q,R,T; O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0); draw((-5,0)--(3,0)); draw((0,-1)--(0,3)); draw(P--Q--R); draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8)); draw((-0.2,2.8)--(0,3)--(0.2,2.8)); draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2)); draw((2.8,-0.2)--(3,0)--(2.8,0.2)); draw(Q--T); label(\"$O$\",O,SW); label(\"$P$\",P,S); label(\"$Q$\",Q,NE); label(\"$R$\",R,W); label(\"$T$\",T,S); [/asy]", "answer": "$-6$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 108, @@ -861,7 +968,8 @@ "geo_code": "[asy] size(120); import three; currentprojection=orthographic(1, 4/5, 1/3); draw(box(O, (4,4,3))); triple A=(0,4,3), B=(0,0,0) , C=(4,4,0), D=(0,4,0); draw(A--B--C--cycle, linewidth(0.9)); label(\"$A$\", A, NE); label(\"$B$\", B, NW); label(\"$C$\", C, S); label(\"$D$\", D, E); label(\"$4$\", (4,2,0), SW); label(\"$4$\", (2,4,0), SE); label(\"$3$\", (0, 4, 1.5), E); [/asy]", "answer": "$\\frac{12}{\\sqrt{34}}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 109, @@ -869,7 +977,8 @@ "geo_code": "[asy] unitsize(8); fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black); fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white); fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black); fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white); fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black); fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white); draw((0,6)--(0,0)--(6,0)); [/asy]", "answer": "$\\frac{5}{12}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 110, @@ -877,7 +986,8 @@ "geo_code": "[asy] pair H,I,J,K,L; H = (0,0); I = 10*dir(70); J = I + 10*dir(290); K = J + 5*dir(110); L = J + 5*dir(0); draw(H--I--J--cycle); draw(K--L--J); draw(arc((0,0),dir(70),(1,0),CW)); label(\"$70^\\circ$\",dir(30),NE); draw(arc(I,I+dir(250),I+dir(290),CCW)); label(\"$40^\\circ$\",I+1.25*dir(270),S); label(\"$1$\",J+0.25*dir(162.5),NW); label(\"$2$\",J+0.25*dir(17.5),NE); label(\"$3$\",L+dir(162.5),WNW); label(\"$4$\",K+dir(-52.5),SE); [/asy]", "answer": "$35$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 111, @@ -885,7 +995,8 @@ "geo_code": "[asy] draw((2.7,3.99)--(0,3)--(0,0)); draw((3.7,3.99)--(1,3)--(1,0)); draw((4.7,3.99)--(2,3)--(2,0)); draw((5.7,3.99)--(3,3)--(3,0)); draw((0,0)--(3,0)--(5.7,0.99)); draw((0,1)--(3,1)--(5.7,1.99)); draw((0,2)--(3,2)--(5.7,2.99)); draw((0,3)--(3,3)--(5.7,3.99)); draw((0,3)--(3,3)--(3,0)); draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33)); draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66)); draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99)); [/asy]", "answer": "$216$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 112, @@ -893,7 +1004,8 @@ "geo_code": "[asy] pair A,B,C,D,EE; A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0); fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray); draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)); draw(circle((5,0),5)); dot(A); dot(B); dot(C); dot(D); dot(EE); label(\"$A$\",A,W); label(\"$B$\",B,N); label(\"$C$\",C,E); label(\"$D$\",D,N); label(\"$E$\",EE,W); [/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 113, @@ -901,7 +1013,8 @@ "geo_code": "[asy] size(6cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0)); label(\"$\\textbf{A}$\",foot((0,2),(0,3),(1,2)),SW); label(\"$\\textbf{B}$\",midpoint((0,1)--(1,2))); label(\"$\\textbf{C}$\",midpoint((-1,0)--(0,1))); label(\"$\\textbf{D}$\",midpoint((0,0)--(1,1))); label(\"$\\textbf{E}$\",midpoint((1,0)--(2,1)),NW); label(\"$\\textbf{F}$\",midpoint((0,-1)--(1,0)),NW); label(\"$\\textbf{G}$\",midpoint((0,-1)--(1,0)),2SE);[/asy]", "answer": "$\\frac{20}{3}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "volume" }, { "index": 114, @@ -909,7 +1022,8 @@ "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (2,0); pair C = (3,1); pair P = (1,2.25); pair D = intersectionpoint(P--B,C--A); dot(A);dot(B);dot(C);dot(P);dot(D); label(\"$A$\",A,SW);label(\"$B$\",B,SE);label(\"$C$\",C,N);label(\"$D$\",D,NE + N);label(\"$P$\",P,N); draw(A--B--P--cycle); draw(A--C--B--cycle);[/asy]", "answer": "$20$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 115, @@ -917,7 +1031,8 @@ "geo_code": "[asy] unitsize(12); draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle); draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW)); draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW)); draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW)); label(rotate(30)*\"$40^\\circ$\",(2,-8.9),ENE); label(\"$100^\\circ$\",(21/3,-2/3),SE); label(\"$110^\\circ$\",(900/83,-317/83),NNW); label(\"$A$\",(0,0),NW); [/asy]", "answer": "$30$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 116, @@ -925,7 +1040,8 @@ "geo_code": "[asy] label(\"A\", (0,0), W); label(\"B\", (64,0), E); label(\"C\", (32, 32*sqrt(3)), N); draw(arc((0,0),64,0,60)); draw(arc((64,0),64,120,180)); draw((0,0)--(64,0)); draw(circle((32, 24), 24)); [/asy]", "answer": "$27$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 117, @@ -933,7 +1049,8 @@ "geo_code": "[asy] fill((0,2)--(1,3)--(2,3)--(2,4)--(3,5)--(4,4)--(4,3)--(5,3)--(6,2)--(5,1)--(4,1)--(4,0)--(2,0)--(2,1)--(1,1)--cycle, mediumgrey); fill((7,1)--(6,2)--(7,3)--(8,3)--(8,4)--(9,5)--(10,4)--(7,0)--cycle, mediumgrey); fill((3,5)--(2,6)--(2,7)--(1,7)--(0,8)--(1,9)--(2,9)--(2,10)--(3,11)--(4,10)--(4,9)--(5,9)--(6,8)--(5,7)--(4,7)--(4,6)--cycle, mediumgrey); fill((6,8)--(7,9)--(8,9)--(8,10)--(9,11)--(10,10)--(10,9)--(11,9)--(11,7)--(10,7)--(10,6)--(9,5)--(8,6)--(8,7)--(7,7)--cycle, mediumgrey); draw((0,0)--(0,11)--(11,11)); for ( int x = 1; x < 11; ++x ) { draw((x,11)--(x,0), linetype(\"4 4\")); } for ( int y = 1; y < 11; ++y ) { draw((0,y)--(11,y), linetype(\"4 4\")); } clip((0,0)--(0,11)--(11,11)--(11,5)--(4,1)--cycle);[/asy]", "answer": "$\\frac{5}{9}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 118, @@ -941,7 +1058,8 @@ "geo_code": "[asy] import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P); draw(A--C--B--cycle); draw(B--D); dot(P); label(\"A\",A,S); label(\"D\",D,S); label(\"C\",C,S); label(\"P\",P,NE); label(\"B\",B,N);[/asy]", "answer": "$\\frac{\\sqrt3}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 119, @@ -949,7 +1067,8 @@ "geo_code": "[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy]", "answer": "$22\\frac{1}{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 120, @@ -957,7 +1076,8 @@ "geo_code": "[asy]size(3inch, 1.5inch); pair a=(0,0), b=(18,24), c=(68,24), d=(75,0), f=(68,0), e=(18,0); draw(a--b--c--d--cycle); draw(b--e); draw(shift(0,2)*e--shift(2,2)*e--shift(2,0)*e); label(\"30\", (9,12), W); label(\"50\", (43,24), N); label(\"25\", (71.5, 12), E); label(\"24\", (18, 12), E); label(\"$A$\", a, SW); label(\"$B$\", b, N); label(\"$C$\", c, N); label(\"$D$\", d, SE); label(\"$E$\", e, S);[/asy]", "answer": "$1500$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 121, @@ -965,7 +1085,8 @@ "geo_code": "[asy] size(7cm); pathpen = linewidth(0.7); D(CR((0,0),10)); D(CR((0,0),9.5)); D(CR((0,-18.5),9.5)); D(CR((0,-18.5),9)); MP(\"$\\vdots$\",(0,-31),(0,0)); D(CR((0,-39),3)); D(CR((0,-39),2.5)); D(CR((0,-43.5),2.5)); D(CR((0,-43.5),2)); D(CR((0,-47),2)); D(CR((0,-47),1.5)); D(CR((0,-49.5),1.5)); D(CR((0,-49.5),1.0)); D((12,-10)--(12,10)); MP('20',(12,0),E); D((12,-51)--(12,-48)); MP('3',(12,-49.5),E);[/asy]", "answer": "$346$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 122, @@ -973,7 +1094,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,0)--(2*i+1,6)--(2*i,6)--cycle, mediumgray); } pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10); draw(B--A--C); fill((7.3,0)--(7.8,0)--(7.8,6)--(7.3,6)--cycle, white); clip(B--A--C--cycle); for(int i=0; i<9; i=i+1) { draw((i,1)--(i,6)); } label(\"$\\mathcal{A}$\", A+0.2*dir(-17), S); label(\"$\\mathcal{B}$\", A+2.3*dir(-17), S); label(\"$\\mathcal{C}$\", A+4.4*dir(-17), S); label(\"$\\mathcal{D}$\", A+6.5*dir(-17), S);[/asy]", "answer": "$408$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 123, @@ -981,7 +1103,8 @@ "geo_code": "[asy] size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label(\"$\\mathcal{T}$\",(2.1,-1.6)); label(\"$\\mathcal{P}$\",(0,-1),NE); label(\"$\\mathcal{Q}$\",(4.2,-1),NW); label(\"$\\mathcal{R}$\",(0,-2.2),SE); label(\"$\\mathcal{S}$\",(4.2,-2.2),SW); [/asy]", "answer": "$89$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 124, @@ -989,7 +1112,8 @@ "geo_code": "[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } [/asy]", "answer": "$24$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 125, @@ -997,7 +1121,8 @@ "geo_code": "[asy] size((150)); draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle); draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle); draw((20,0)--(-20,0)); draw((0,20)--(0,-20)); draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow); draw((-18,1)--(-12, 1), EndArrow); draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow); draw((1,-9)--(1,9), EndArrow); draw((0,12)..(8.3, 8.3)..(12,0), EndArrow); draw((12,-1)--(18,-1), EndArrow); label(\"$A$\", (0,20), N); label(\"$K$\", (20,0), E); [/asy]", "answer": "$10\\pi+20$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 126, @@ -1005,7 +1130,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.65)); real d=90-63.43494882; draw(ellipse((origin), 2, 4)); fill((0,4)--(0,-4)--(-8,-4)--(-8,4)--cycle, white); draw(ellipse((-4,0), 2, 4)); draw((0,4)--(-4,4)); draw((0,-4)--(-4,-4)); draw(shift(-2,0)*rotate(-d-5)*ellipse(origin, 1.82, 4.56), linetype(\"10 10\")); draw((-4,4)--(-8,4), dashed); draw((-4,-4)--(-8,-4), dashed); draw((-4,4.3)--(-4,5)); draw((0,4.3)--(0,5)); draw((-7,4)--(-7,-4), Arrows(5)); draw((-4,4.7)--(0,4.7), Arrows(5)); label(\"$8$ cm\", (-7,0), W); label(\"$6$ cm\", (-2,4.7), N);[/asy]", "answer": "$80\\pi$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 127, @@ -1013,7 +1139,8 @@ "geo_code": "[asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label(\"\\(x\\)\",(-1.55,2.1),E); label(\"\\(1\\)\",(-0.5,3.8),S);[/asy]", "answer": "$\\frac{3\\sqrt{7}-\\sqrt{3}}{2}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 128, @@ -1021,7 +1148,8 @@ "geo_code": "[asy] size(250); defaultpen(linewidth(0.55)); pair A=(-6,0), B=origin, C=(0,6), D=(0,12); pair ac=C+2.828*dir(45), ca=A+2.828*dir(225), ad=D+2.828*dir(A--D), da=A+2.828*dir(D--A), ab=(2.828,0), ba=(-6-2.828, 0); fill(A--C--D--cycle, gray); draw(ba--ab); draw(ac--ca); draw(ad--da); draw((0,-1)--(0,15)); draw((1/3, -1)--(1/3, 15)); int i; for(i=1; i<15; i=i+1) { draw((-1/10, i)--(13/30, i)); } label(\"$A$\", A, SE); label(\"$B$\", B, SE); label(\"$C$\", C, SE); label(\"$D$\", D, SE); label(\"$3$\", (1/3,3), E); label(\"$3$\", (1/3,9), E); label(\"$3$\", (-3,0), S); label(\"Main\", (-3,0), N); label(rotate(45)*\"Aspen\", A--C, SE); label(rotate(63.43494882)*\"Brown\", A--D, NW); [/asy]", "answer": "$9$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 129, @@ -1029,7 +1157,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy]", "answer": "$23$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 130, @@ -1037,7 +1166,8 @@ "geo_code": "[asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(2,8/15,7/15); int t=0; void f(real x) { path3 r=(t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--cycle; path3 f=(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)--cycle; path3 u=(t,1,x)--(t+1,1,x)--(t+1,0,x)--(t,0,x)--cycle; draw(surface(r), white, nolight); draw(surface(f), white, nolight); draw(surface(u), white, nolight); draw((t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--(t,1,x)--(t,0,x)--(t+1,0,x)--(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)); t=t+1; } f(d); f(1/2); f(1/3); f(1/17); label(\"D\", (1/2, 1, 0), SE); label(\"A\", (1+1/2, 1, 0), SE); label(\"B\", (2+1/2, 1, 0), SE); label(\"C\", (3+1/2, 1, 0), SE);[/asy][asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(1,8/15,7/15); draw(unitcube, white, thick(), nolight); void f(real x) { draw((0,1,x)--(1,1,x)--(1,0,x)); } f(d); f(1/6); f(1/2); label(\"A\", (1,0,3/4), W); label(\"B\", (1,0,1/3), W); label(\"C\", (1,0,1/6-d/4), W); label(\"D\", (1,0,d/2), W); label(\"1/2\", (1,1,3/4), E); label(\"1/3\", (1,1,1/3), E); label(\"1/17\", (0,1,1/6-d/4), E);[/asy]", "answer": "$11$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 131, @@ -1045,7 +1175,8 @@ "geo_code": "[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); draw((1,1)--(-1,0)); pair P=foot((1,-1),(1,1),(-1,0)); draw((1,-1)--P); draw(rightanglemark((-1,0),P,(1,-1),4)); label(\"$M$\",(-1,0),W); label(\"$C$\",(-0.1,-0.3)); label(\"$A$\",(-0.4,0.7)); label(\"$B$\",(0.7,0.4)); [/asy]", "answer": "$\\frac{32}{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 132, @@ -1053,7 +1184,8 @@ "geo_code": "[asy] import graph; size(9cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.381056062031275, xmax = 15.020004395092375, ymin = -4.051697595316909, ymax = 10.663513514111651; /* image dimensions */ draw((0.,0.)--(4.714285714285714,7.666518779999279)--(7.,0.)--cycle); /* draw figures */ draw((0.,0.)--(4.714285714285714,7.666518779999279)); draw((4.714285714285714,7.666518779999279)--(7.,0.)); draw((7.,0.)--(0.,0.)); label(\"7\",(3.2916797119724284,-0.07831656949355523),SE*labelscalefactor); label(\"9\",(2.0037562070503783,4.196493361737088),SE*labelscalefactor); label(\"8\",(6.114150371695219,3.785453945272603),SE*labelscalefactor); draw((0.,0.)--(6.428571428571427,1.9166296949998194)); draw((7.,0.)--(2.2,3.5777087639996634)); draw((4.714285714285714,7.666518779999279)--(3.7058823529411766,0.)); /* dots and labels */ dot((0.,0.),dotstyle); label(\"$A$\", (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor); dot((7.,0.),dotstyle); label(\"$B$\", (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor); dot((3.7058823529411766,0.),dotstyle); label(\"$E$\", (3.8123296394941084,0.16830708038513573), NE * labelscalefactor); dot((4.714285714285714,7.666518779999279),dotstyle); label(\"$C$\", (4.579603216894479,7.895848109917452), NE * labelscalefactor); dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle); label(\"$D$\", (2.1407693458718726,3.127790878929427), NE * labelscalefactor); dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle); label(\"$H$\", (6.004539860638023,1.9494778850645704), NE * labelscalefactor); dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle); label(\"$Q$\", (4.935837377830365,1.7302568629501784), NE * labelscalefactor); dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle); label(\"$P$\", (3.538303361851119,1.2370095631927964), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]", "answer": "$\\frac{8}{27}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 133, @@ -1061,7 +1193,8 @@ "geo_code": "[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,NE); label(\"$E$\",E,NW); label(\"$F$\",F,1.5*N); [/asy]", "answer": "$\\frac{2}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 134, @@ -1069,7 +1202,8 @@ "geo_code": "[asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$E$\",E,dir(90)); label(\"$F$\",F,NE); label(\"$G$\",G,NE); label(\"$H$\",H,W); label(\"$J$\",J,S); label(\"$K$\",K,SE); label(\"$L$\",L,SE); label(\"$M$\",M,dir(90)); label(\"$N$\",N,dir(180)); [/asy]", "answer": "$99$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 135, @@ -1077,7 +1211,8 @@ "geo_code": "[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]", "answer": "$\\frac{143}{14}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 136, @@ -1085,7 +1220,8 @@ "geo_code": "[asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP(\"x\",(3.5,0),S)--(0,0)--MP(\"\\frac{3}{2}\",(0,3/2),W)--MP(\"y\",(0,3.5),W)); path P=(0,0)--MP(\"3\",(3,0),S)..(3*dir(45))..MP(\"3\",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy]", "answer": "$\\frac{9\\pi}{8}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 137, @@ -1093,7 +1229,8 @@ "geo_code": "[asy]size(170);defaultpen(linewidth(0.9)+fontsize(13pt));draw(unitcircle^^circle((0,1.5),0.5)); path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle; for(int i=1;i<=12;i=i+1){draw(0.9*dir(90-30*i)--dir(90-30*i));label(\"$\"+(string) i+\"$\",0.78*dir(90-30*i));} dot(origin);draw(shift((0,1.87))*arrow);draw(arc(origin,1.5,68,30),EndArrow(size=12));[/asy]", "answer": "$4$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 138, @@ -1101,7 +1238,8 @@ "geo_code": "[asy] pair A,B,C,D,E,F,G,H,O,X; A=dir(45); B=dir(90); C=dir(135); D=dir(180); E=dir(-135); F=dir(-90); G=dir(-45); H=dir(0); O=(0,0); X=midpoint(A--B); fill(X--B--C--D--E--O--cycle,rgb(0.75,0.75,0.75)); draw(A--B--C--D--E--F--G--H--cycle); dot(\"$A$\",A,dir(45)); dot(\"$B$\",B,dir(90)); dot(\"$C$\",C,dir(135)); dot(\"$D$\",D,dir(180)); dot(\"$E$\",E,dir(-135)); dot(\"$F$\",F,dir(-90)); dot(\"$G$\",G,dir(-45)); dot(\"$H$\",H,dir(0)); dot(\"$X$\",X,dir(135/2)); dot(\"$O$\",O,dir(0)); draw(E--O--X); [/asy]", "answer": "$\\frac{7}{16}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 139, @@ -1109,7 +1247,8 @@ "geo_code": "[asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(\" \",A,NW); dot(\" \",B,NW); dot(\" \",C,NW); dot(\" \",D,NW); dot(\" \",E,NW); dot(\" \",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(\" \",R,NW); dot(\" \",S,NW); dot(\" \",T,NW); dot(\" \",X,NW); dot(\" \",Y,NW); dot(\" \",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy]", "answer": "$12\\sqrt{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 140, @@ -1117,7 +1256,8 @@ "geo_code": "[asy] pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2); draw(A--B--C--D--cycle); draw(K--H--G--J--cycle); draw(F--E); label(\"$A$\",A,SE); label(\"$B$\",B,SW); label(\"$C$\",C,NW); label(\"$D$\",D,NE); label(\"$E$\",E,S); label(\"$F$\",F,N); label(\"$G$\",G,E); label(\"$H$\",H,N); label(\"$J$\",J,S); label(\"$K$\",K,W); [/asy]", "answer": "$2\\sqrt{3}-2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 141, @@ -1125,7 +1265,8 @@ "geo_code": "[asy] draw((0,0)--(-5,8.66025404)--(0, 17.3205081)--(10, 17.3205081)--(15,8.66025404)--(10, 0)--(0, 0)); draw((30,0)--(25,8.66025404)--(30, 17.3205081)--(40, 17.3205081)--(45, 8.66025404)--(40, 0)--(30, 0)); draw((30,0)--(25,-8.66025404)--(30, -17.3205081)--(40, -17.3205081)--(45, -8.66025404)--(40, 0)--(30, 0)); draw((0,0)--(-5, -8.66025404)--(0, -17.3205081)--(10, -17.3205081)--(15, -8.66025404)--(10, 0)--(0, 0)); draw((15,8.66025404)--(10, 17.3205081)--(15, 25.9807621)--(25, 25.9807621)--(30, 17.3205081)--(25, 8.66025404)--(15, 8.66025404)); draw((15,-8.66025404)--(10, -17.3205081)--(15, -25.9807621)--(25, -25.9807621)--(30, -17.3205081)--(25, -8.66025404)--(15, -8.66025404)); label(\"A\", (0,0), W); label(\"B\", (30, 17.3205081), NE); label(\"C\", (30, -17.3205081), SE); draw((0,0)--(30, 17.3205081)--(30, -17.3205081)--(0, 0)); //(Diagram Creds-DivideBy0) [/asy]", "answer": "$12\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 142, @@ -1133,7 +1274,8 @@ "geo_code": "[asy]\nimport graph; size(9.115122858763474cm); \nreal labelscalefactor = 0.5; /* changes label-to-point distance */\npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ \npen dotstyle = black; /* point style */ \nreal xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273; /* image dimensions */\n\n /* draw figures */\ndraw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); \ndraw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); \ndraw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); \ndraw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); \ndraw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); \ndraw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); \ndraw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); \ndraw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); \nlabel(\"$A$\",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); \nlabel(\"$B$\",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); \nlabel(\"$C$\",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); \n /* dots and labels */\ndot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); \ndot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); \ndot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); \ndot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); \ndot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); \ndot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); \ndot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n\n[/asy]", "answer": "$72$", "category": "Global Abstract Integration", - "source": "olympiads" + "source": "olympiads", + "problem_type": "angle" }, { "index": 143, @@ -1141,7 +1283,8 @@ "geo_code": "[asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(point--D)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$G$\", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label(\"10\", A--F, dir(90)*dir(A--F)); label(\"27\", F--C, dir(90)*dir(F--C)); label(\"3\", (0,10), W); label(\"9\", (0,4), W); [/asy]", "answer": "$147$", "category": "Local Relation Composition", - "source": "AIME-83-24" + "source": "AIME-83-24", + "problem_type": "area" }, { "index": 144, @@ -1149,7 +1292,8 @@ "geo_code": "[asy]\npair A, B, C, X, Y;\nA = origin;\nX = dir(30);\nY = X + dir(0);\nB = Y + dir(60);\nC = B + dir(330);\ndraw(A--B--C--cycle);\ndraw(X--Y--B);\nlabel(\"$A$\",A,W);\nlabel(\"$B$\",B,N);\nlabel(\"$C$\",C,E);\nlabel(\"$X$\",X,NW);\nlabel(\"$Y$\",Y,SE);\n[/asy]", "answer": "$45$", "category": "Local Relation Composition", - "source": "olympiads" + "source": "olympiads", + "problem_type": "angle" }, { "index": 145, @@ -1157,7 +1301,8 @@ "geo_code": "[asy]\nimport graph; size(9.115122858763474cm); \nreal labelscalefactor = 0.5; /* changes label-to-point distance */\npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ \npen dotstyle = black; /* point style */ \nreal xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273; /* image dimensions */\n\n /* draw figures */\ndraw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); \ndraw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); \ndraw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); \ndraw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); \ndraw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); \ndraw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); \ndraw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); \ndraw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); \nlabel(\"$A$\",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); \nlabel(\"$B$\",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); \nlabel(\"$C$\",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); \n /* dots and labels */\ndot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); \ndot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); \ndot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); \ndot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); \ndot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); \ndot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); \ndot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n /* end of picture */\n[/asy]", "answer": "$36$", "category": "Local Relation Composition", - "source": "olympiads" + "source": "olympiads", + "problem_type": "angle" }, { "index": 146, @@ -1165,7 +1310,8 @@ "geo_code": "[asy] import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((0,0)--(6,0),linewidth(1.2pt)); draw((0,0)--(0,1),linewidth(1.2pt)); draw((0,1)--(1,1),linewidth(1.2pt)); draw((1,1)--(1,2),linewidth(1.2pt)); draw((1,2)--(5,2),linewidth(1.2pt)); draw((5,2)--(5,1),linewidth(1.2pt)); draw((5,1)--(6,1),linewidth(1.2pt)); draw((6,1)--(6,0),linewidth(1.2pt)); draw((1,1)--(5,1),linewidth(1.2pt)); draw((1,1)--(1,0),linewidth(1.2pt)); draw((2,2)--(2,0),linewidth(1.2pt)); draw((3,2)--(3,0),linewidth(1.2pt)); draw((4,2)--(4,0),linewidth(1.2pt)); draw((5,1)--(5,0),linewidth(1.2pt)); draw((0,0)--(5,1.5),linewidth(1.2pt)); dot((0,0),ds); label(\"$P$\", (-0.23,-0.26),NE*lsf); dot((0,1),ds); dot((1,1),ds); dot((1,2),ds); dot((5,2),ds); label(\"$X$\", (5.14,2.02),NE*lsf); dot((5,1),ds); label(\"$Y$\", (5.12,1.14),NE*lsf); dot((6,1),ds); dot((6,0),ds); dot((1,0),ds); dot((2,0),ds); dot((3,0),ds); dot((4,0),ds); dot((5,0),ds); dot((2,2),ds); dot((3,2),ds); dot((4,2),ds); dot((5,1.5),ds); label(\"$Q$\", (5.14,1.51),NE*lsf); clip((-4.19,-5.52)--(-4.19,6.5)--(10.08,6.5)--(10.08,-5.52)--cycle); [/asy]", "answer": "$\\frac{2}{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 147, @@ -1173,7 +1319,8 @@ "geo_code": "[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]", "answer": "$\\frac{375}{4}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "volume" }, { "index": 148, @@ -1181,7 +1328,8 @@ "geo_code": "[asy] import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label(\"$ P(-1,1) $\",(-2.57,2.17),SE*lsf); label(\"$ Q(1,1) $\",(1.55,2.21),SE*lsf); label(\"$ R(-1,-1) $\",(-2.72,-1.45),SE*lsf); label(\"$S(1,-1)$\",(1.59,-1.49),SE*lsf); dot((0,0),ds); label(\"$O$\",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]", "answer": "$\\pi$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 149, @@ -1189,7 +1337,8 @@ "geo_code": "[asy] unitsize(2mm); defaultpen(linewidth(.8pt)); fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0)); [/asy]", "answer": "$\\frac{1}{6}$", "category": "Local Relation Composition", - "source": "olympiads" + "source": "olympiads", + "problem_type": "ratio" }, { "index": 150, @@ -1197,7 +1346,8 @@ "geo_code": "[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label(\"$C_4$\", D); label(\"$C_1$\", (-1.375, 0)); label(\"$C_2$\", (1.375,0)); label(\"$\\frac{1}{2}$\", (0, -.125)); label(\"$C_3$\", (-0.4, -0.4)); label(\"$1$\", (-.85, 0.70)); label(\"$1$\", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]", "answer": "$\\frac{28}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 151, @@ -1205,7 +1355,8 @@ "geo_code": "[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label(\"$\\theta$\", (-41.5,-10.5)); label(\"$\\theta$\", (-13,10.5)); label(\"$\\theta$\", (15.5,-10.5)); label(\"$\\theta$\", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label(\"$D$\",(-58,15)); label(\"$A$\",(-58,-15)); label(\"$C$\",(58,15)); label(\"$B$\",(58,-15)); label(\"$S$\",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label(\"$P$\",(-22.5,23)); label(\"$Q$\",(5,-23)); label(\"$R$\",(32.5,23)); [/asy]", "answer": "$60$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 152, @@ -1213,7 +1364,8 @@ "geo_code": "[asy] unitsize(1cm); draw(scale(3)*polygon(6)); filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray); filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray); filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray); filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray); filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray); filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray); [/asy]", "answer": "$18\\sqrt{3}}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 153, @@ -1221,7 +1373,8 @@ "geo_code": "[asy] size(12cm); real h = 2.5; // height real g=4; //c2c space real s = 0.65; //Xcord of Hline real adj = 0.08; //adjust line diffs pair A,B,C; B=(0,h); C=(1,0); A=-conj(C); pair PONE=(s,h*(1-s)); //Endpoint of Hline ONE pair PTWO=(s+adj,h*(1-s-adj)); //Endpoint of Hline ONE path LONE=PONE--(-conj(PONE)); //Hline ONE path LTWO=PTWO--(-conj(PTWO)); path T=A--B--C--cycle; //Triangle fill (shift(g,0)*(LTWO--B--cycle),mediumgrey); fill (LONE--A--C--cycle,mediumgrey); draw(LONE); draw(T); label(\"$A$\",A,SW); label(\"$B$\",B,N); label(\"$C$\",C,SE); draw(shift(g,0)*LTWO); draw(shift(g,0)*T); label(\"$A$\",shift(g,0)*A,SW); label(\"$B$\",shift(g,0)*B,N); label(\"$C$\",shift(g,0)*C,SE); draw(B--shift(g,0)*B,dashed); draw(C--shift(g,0)*A,dashed); draw((g/2,0)--(g/2,h),dashed); draw((0,h*(1-s))--B,dashed); draw((g,h*(1-s-adj))--(g,0),dashed); label(\"$5$\", midpoint((g,h*(1-s-adj))--(g,0)),UnFill); label(\"$h$\", midpoint((g/2,0)--(g/2,h)),UnFill); label(\"$11$\", midpoint((0,h*(1-s))--B),UnFill); [/asy]", "answer": "$14.6$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 154, @@ -1229,7 +1382,8 @@ "geo_code": "[asy] import graph; size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*\"$Q$\", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*\"$?$\", (8.5,2.05)); pair g = (-8,0); real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); pair DA,DB,DC,CD,O; DA = (4*sqrt(3),0); DB = (2*sqrt(3),6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label(\"$7$\",trf*DC); label(\"$Q$\",trf*DC+DA-DB); label(\"$5$\",trf*DC-DB); label(\"$3$\",trf*DC-DA-DB); label(\"$6$\",trf*CD); label(\"$4$\",trf*CD-DA); label(\"$2$\",trf*CD-DA-DB); label(\"$1$\",trf*CD-2DA); [/asy]", "answer": "$6$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 155, @@ -1237,7 +1391,8 @@ "geo_code": "[asy] size(6cm); draw(circle((3,3),3)); filldraw(circle((2,3),2),lightgrey); filldraw(circle((3,3),1),white); filldraw(circle((1,3),1),white); filldraw(circle((5.5,3),0.5),lightgrey); filldraw(circle((4.5,4.5),0.5),lightgrey); filldraw(circle((4.5,1.5),0.5),lightgrey); int i, j; for(i=0; i<7; i=i+1) { draw((0,i)--(6,i), dashed+grey); draw((i,0)--(i,6), dashed+grey); } [/asy]", "answer": "$\\frac{25}{36}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 156, @@ -1245,7 +1400,8 @@ "geo_code": "[asy] import three; size(225); currentprojection= orthographic(camera=(-5.52541796301147,-2.61548797564715,1.6545450372312), up=(0.00247902062334861,0.000877141782387748,0.00966536329192992), target=(0,0,0), zoom=0.570588560870951); currentpen = black+1.5bp; triple A = O; triple M = (X+Y)/2; triple B = (-1/2,-1/2,1/sqrt(2)); triple C = (-1,0,sqrt(2)); triple D = (0,-1,sqrt(2)); transform3 rho = rotate(90,M,M+Z); //arrays of vertices for the lower level (the square), the middle level, //and the interleaves vertices of the upper level (the octagon) triple[] lVs = {A}; triple[] mVs = {B}; triple[] uVsl = {C}; triple[] uVsr = {D}; for(int i = 0; i < 3; ++i){ lVs.push(rho*lVs[i]); mVs.push(rho*mVs[i]); uVsl.push(rho*uVsl[i]); uVsr.push(rho*uVsr[i]); } lVs.cyclic = true; uVsl.cyclic = true; for(int i : new int[] {0,1,2,3}){ draw(uVsl[i]--uVsr[i]); draw(uVsr[i]--uVsl[i+1]); } draw(lVs[0]--lVs[1]^^lVs[0]--lVs[3]); for(int i : new int[] {0,1,3}){ draw(lVs[0]--lVs[i]); draw(lVs[i]--mVs[i]); draw(mVs[i]--uVsl[i]); } for(int i : new int[] {0,3}){ draw(mVs[i]--uVsr[i]); } for(int i : new int[] {1,3}) draw(lVs[2]--lVs[i],dashed); draw(lVs[2]--mVs[2],dashed); draw(mVs[2]--uVsl[2]^^mVs[2]--uVsr[2],dashed); draw(mVs[1]--uVsr[1],dashed); //Comment two lines below to remove red edges //draw(lVs[1]--lVs[3],red+2bp); //draw(uVsl[0]--uVsr[0],red+2bp); [/asy]", "answer": "$63$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 157, @@ -1253,7 +1409,8 @@ "geo_code": "[asy] pair A,B; size(8cm); A=(0,0); B=(480,0); draw((0,0)--(480,0),linetype(\"3 4\")); filldraw(circle((8,0),8),black); draw((0,0)..(100,-100)..(200,0)); draw((200,0)..(260,60)..(320,0)); draw((320,0)..(400,-80)..(480,0)); draw((100,0)--(150,-50sqrt(3)),Arrow(size=4)); draw((260,0)--(290,30sqrt(3)),Arrow(size=4)); draw((400,0)--(440,-40sqrt(3)),Arrow(size=4)); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$R_1$\", (100,-40), W); label(\"$R_2$\", (260,40), SW); label(\"$R_3$\", (400,-40), W);[/asy]", "answer": "$296\\pi$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 158, @@ -1261,7 +1418,8 @@ "geo_code": "[asy] unitsize(8mm); defaultpen(linewidth(.8pt)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((0,3)--(0,4)--(1,4)--(1,3)--cycle); draw((1,3)--(1,4)--(2,4)--(2,3)--cycle); draw((2,3)--(2,4)--(3,4)--(3,3)--cycle); draw((3,3)--(3,4)--(4,4)--(4,3)--cycle); [/asy]", "answer": "$\\frac{3}{4}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 159, @@ -1269,7 +1427,8 @@ "geo_code": "[asy]\nimport graph; size(9.115122858763474cm); \nreal labelscalefactor = 0.5; /* changes label-to-point distance */\npen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ \npen dotstyle = black; /* point style */ \nreal xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273; /* image dimensions */\n\n /* draw figures */\ndraw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); \ndraw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); \ndraw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); \ndraw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); \ndraw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); \ndraw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); \ndraw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); \ndraw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); \ndraw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); \ndraw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); \ndraw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); \ndraw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); \nlabel(\"$A$\",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); \nlabel(\"$B$\",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); \nlabel(\"$C$\",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); \n /* dots and labels */\ndot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); \ndot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); \ndot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); \ndot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); \ndot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); \ndot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); \ndot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); \nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); \n\n[/asy]", "answer": "$72$", "category": "Primitive Recognition", - "source": "olympiads" + "source": "olympiads", + "problem_type": "angle" }, { "index": 160, @@ -1277,7 +1436,8 @@ "geo_code": "[asy] size(250); defaultpen(linewidth(0.8)); pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0); draw(A--B--E--D--cycle^^C--D); draw(rightanglemark(D,C,E,30)); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,S); label(\"$D$\",D,N); label(\"$E$\",E,S); label(\"$5$\",A/2,W); label(\"$6$\",(A+D)/2,N); [/asy]", "answer": "$\\sqrt{601}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 161, @@ -1285,7 +1445,8 @@ "geo_code": "[asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,gray,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]", "answer": "$\\frac{2}{\\pi-1}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 162, @@ -1293,7 +1454,8 @@ "geo_code": "[asy] usepackage(\"mathptmx\"); size(275); defaultpen(linewidth(0.8)); real r = 2, s = 2.5, theta = 14; pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta); pair N = (B+G)/2, J = N + s/2 * dir(180+theta); pair E = F + r * dir(- 45 - theta/2), D = I+E-F; pair H = J + r * dir(135 + theta/2), A = B+H-J; draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B)); draw(J--B--G^^C--F--I,linetype (\"4 4\")); dot(\"$A$\",A,N); dot(\"$B$\",B,1.2*N); dot(\"$C$\",C,N); dot(\"$D$\",D,dir(0)); dot(\"$E$\",E,S); dot(\"$F$\",F,1.5*dir(-100)); dot(\"$G$\",G,S); dot(\"$H$\",H,W); dot(\"$I$\",I,NE); dot(\"$J$\",J,1.5*S); [/asy]", "answer": "$192$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 163, @@ -1301,7 +1463,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.5)); size(5cm); defaultpen(fontsize(14pt)); label(\"$\\textbf{Math}$\", (2.1,3.7)--(3.9,3.7)); label(\"$\\textbf{Team}$\", (2.1,3)--(3.9,3)); filldraw((1,2)--(2,1)--(3,2)--(4,1)--(5,2)--(4,3)--(5,4)--(4,5)--(3,4)--(2,5)--(1,4)--(2,3)--(1,2)--cycle, mediumgray*0.5 + lightgray*0.5); draw((0,0)--(6,0), gray); draw((0,1)--(6,1), gray); draw((0,2)--(6,2), gray); draw((0,3)--(6,3), gray); draw((0,4)--(6,4), gray); draw((0,5)--(6,5), gray); draw((0,6)--(6,6), gray); draw((0,0)--(0,6), gray); draw((1,0)--(1,6), gray); draw((2,0)--(2,6), gray); draw((3,0)--(3,6), gray); draw((4,0)--(4,6), gray); draw((5,0)--(5,6), gray); draw((6,0)--(6,6), gray); [/asy]", "answer": "$12\\sqrt{2}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 164, @@ -1309,7 +1472,8 @@ "geo_code": "[asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8); pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label(\"${\"+string(i)+\"}$\", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label(\"${\"+string(i)+\"}$\", (0,i), 2*W); } } label(\"$0$\", O, 2*SW); draw(O--X+(0.35,0), black+1.5, EndArrow(10)); draw(O--Y+(0,0.35), black+1.5, EndArrow(10)); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]", "answer": "$6$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 165, @@ -1317,7 +1481,8 @@ "geo_code": "[asy] size(10cm); usepackage(\"mathptmx\"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label(\"2\",C--Y,S); label(\"1\",Y--X,S); label(\"3\",X--A,S); label(\"$A$\",A,2*E); label(\"$B$\",B,2*N); label(\"$C$\",C,2*W); label(\"$D$\",D,2*S); label(\"$Y$\",Y,2*sqrt(2)*NE); label(\"$X$\",X,2*N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy]", "answer": "$\\frac{6\\sqrt{35}}{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 166, @@ -1325,7 +1490,8 @@ "geo_code": "[asy] size(10cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label(\"$C$\",C,2*S); label(\"$D$\",D,2*S); label(\"$E$\",E,2*S); label(\"$F$\",F,2*dir(0)); label(\"$A$\",A,2*N); label(\"$B$\",B,2*W); [/asy]", "answer": "$24\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 167, @@ -1333,7 +1499,8 @@ "geo_code": "[asy] size(6cm); pair A = (0,10); label(\"$A$\", A, N); pair B = (0,0); label(\"$B$\", B, S); pair C = (10,0); label(\"$C$\", C, S); pair D = (10,10); label(\"$D$\", D, SW); pair EE = (15,11.8); label(\"$E$\", EE, N); pair F = (3,10); label(\"$F$\", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label(\"$110^\\circ$\", (15,9), SW); [/asy]", "answer": "$10$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 168, @@ -1341,7 +1508,8 @@ "geo_code": "[asy] size(170); defaultpen(linewidth(0.6)+fontsize(10)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot(\"$A$\",A,S); dot(\"$B$\",B,S); dot(\"$C$\",C,N); dot(\"$D$\",D,N); dot(\"$Q$\",Q,S); dot(\"$P$\",P,W); dot(\"$R$\",R,1.3*S); label(\"$7$\",(P+R)/2,NE); label(\"$6$\",(R+B)/2,NE); [/asy]", "answer": "$117$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 169, @@ -1349,7 +1517,8 @@ "geo_code": "[asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label(\"$A$\",A,dir(90)); label(\"$B$\",B,dir(225)); label(\"$C$\",C,dir(-45)); label(\"$D$\",D,dir(180)); label(\"$E$\",E,dir(-45)); label(\"$F$\",F,dir(225)); label(\"$G$\",G,dir(0)); label(\"$\\ell$\",midpoint(E--F),dir(90)); [/asy]", "answer": "$504$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 170, @@ -1357,7 +1526,8 @@ "geo_code": "[asy] unitsize(100); pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); draw(A--B--C--D--cycle, black); draw(A--P, black); draw(B--P, black); draw(C--P, black); draw(D--P, black); label(\"$A$\",A,(-1,0)); label(\"$B$\",B,(1,0)); label(\"$C$\",C,(1,-0)); label(\"$D$\",D,(-1,0)); label(\"$2$\",E,(0,0)); label(\"$3$\",F,(0,0)); label(\"$4$\",G,(0,0)); label(\"$5$\",H,(0,0)); dot(A^^B^^C^^D^^P); [/asy]", "answer": "$2-\\sqrt{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 171, @@ -1365,7 +1535,8 @@ "geo_code": "[asy] pair A=(-2.4638,4.10658); pair B=(-4,2.6567453480756127); pair C=(-3.47132,0.6335248637894945); pair D=(-1.464483379039766,0.6335248637894945); pair E=(-0.956630463955801,2.6567453480756127); pair F=(-2,2); pair G=(-3,2); draw(A--B--C--D--E--A); draw(A--F--A--G); draw(B--F--C); draw(E--G--D); label(\"A\",A,N); label(\"B\",B,W); label(\"C\",C,S); label(\"D\",D,S); label(\"E\",E,dir(0)); dot(A^^B^^C^^D^^E^^F^^G); [/asy]", "answer": "$\\sqrt{12}+\\sqrt{11}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 172, @@ -1373,7 +1544,8 @@ "geo_code": "[asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); pair B = (0,0); pair C = (1,0); pair A = rotate(60,B)*C; pair E = rotate(270,A)*B; pair D = rotate(270,E)*A; pair F = rotate(90,A)*C; pair G = rotate(90,F)*A; pair I = rotate(270,B)*C; pair H = rotate(270,I)*B; draw(A--B--C--cycle); draw(A--E--D--B); draw(A--F--G--C); draw(B--I--H--C); draw(E--F); draw(D--I); draw(I--H); draw(H--G); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,W); label(\"$E$\",E,W); label(\"$F$\",F,E); label(\"$G$\",G,E); label(\"$H$\",H,SE); label(\"$I$\",I,SW); [/asy]", "answer": "$27+9\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 173, @@ -1381,7 +1553,8 @@ "geo_code": "[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(1,0); pair C=(0.8,-0.4); draw(A--(2,0)); draw((0,-1)--(2,-1)); draw((0,-2)--(1,-2)); draw(A--(0,-2)); draw(B--(1,-2)); draw((2,0)--(2,-1)); draw(A--(2,-1)); draw(B--(0,-2)); pair[] ps={A,B,C}; dot(ps); label(\"$A$\",A,N); label(\"$B$\",B,N); label(\"$C$\",C,W); [/asy]", "answer": "$5$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 174, @@ -1389,7 +1562,8 @@ "geo_code": "[asy] size(200); defaultpen(linewidth(1)); pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60); pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A; draw(A--B--C--D--E--F--cycle); draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype(\"2 2\")); dot(\"$A$\",A,W,linewidth(4)); dot(\"$B$\",B,dir(0),linewidth(4)); dot(\"$C$\",C,dir(0),linewidth(4)); dot(\"$D$\",D,dir(20),linewidth(4)); dot(\"$E$\",E,dir(100),linewidth(4)); dot(\"$F$\",F,W,linewidth(4)); dot(\"$X$\",X,dir(0),linewidth(4)); dot(\"$Y$\",Y,N,linewidth(4)); dot(\"$Z$\",Z,W,linewidth(4)); [/asy]", "answer": "$116\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 175, @@ -1397,7 +1571,8 @@ "geo_code": "[asy] size(350); defaultpen(linewidth(0.8)); real h1 = 10, r = 3.1, s=0.75; pair P = (r,h1), Q = (-r,h1), Pp = s * P, Qp = s * Q; path e = ellipse((0,h1),r,0.9), ep = ellipse((0,h1*s),r*s,0.9); draw(ellipse(origin,r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill(origin--Pp--Qp--cycle,gray(0.8)); draw((-r,h1)--(0,0)--(r,h1)^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype(\"4 4\")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(0,Qp.y),Arrows(size=8)); draw(origin--(0,12),linetype(\"4 4\")); draw(origin--(r*(s-0.1),0)); label(\"$3$\",(-0.9,h1*s),N,fontsize(10)); real h2 = 7.5, r = 6, s=0.6, d = 14; pair P = (d+r-0.05,h2-0.15), Q = (d-r+0.05,h2-0.15), Pp = s * P + (1-s)*(d,0), Qp = s * Q + (1-s)*(d,0); path e = ellipse((d,h2),r,1), ep = ellipse((d,h2*s+0.09),r*s,1); draw(ellipse((d,0),r*(s-0.1),0.8)); fill(ep,gray(0.8)); fill((d,0)--Pp--Qp--cycle,gray(0.8)); draw(P--(d,0)--Q^^e); draw(subpath(ep,0,reltime(ep,0.5)),linetype(\"4 4\")); draw(subpath(ep,reltime(ep,0.5),reltime(ep,1))); draw(Qp--(d,Qp.y),Arrows(size=8)); draw((d,0)--(d,10),linetype(\"4 4\")); draw((d,0)--(d+r*(s-0.1),0)); label(\"$6$\",(d-r/4,h2*s-0.06),N,fontsize(10)); [/asy]", "answer": "$\\frac{1}{4}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 176, @@ -1405,7 +1580,8 @@ "geo_code": "[asy] pair A=(0,1); pair CC=(0.666666666666,1); pair D=(1,1); pair F=(1,0.440062); pair C=(1,0); pair B=(0,0); pair G=(0,0.22005); pair H=(-0.13,0.41); pair E=(0,0.5); dot(A^^CC^^D^^C^^B^^E); draw(E--A--D--F); draw(G--B--C--F, dashed); fill(E--CC--F--G--H--E--CC--cycle, gray); draw(E--CC--F--G--H--E--CC); label(\"A\",A,NW); label(\"B\",B,SW); label(\"C\",C,SE); label(\"D\",D,NE); label(\"E\",E,NW); label(\"C'\",CC,N); label(\"F\",F,NE); [/asy]", "answer": "$12$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 177, @@ -1413,7 +1589,8 @@ "geo_code": "[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy]", "answer": "$7$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 178, @@ -1421,7 +1598,8 @@ "geo_code": "[asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw((3,0)--(3,1)--(0,1)); draw((3,1)--(3,2)--(5,2)); draw((3,2)--(2,2)--(2,1)--(2,3)); label(\"$R_1$\",(3/2,1/2)); label(\"$S_3$\",(4,1)); label(\"$S_2$\",(5/2,3/2)); label(\"$S_1$\",(1,2)); label(\"$R_2$\",(7/2,5/2)); [/asy]", "answer": "$2604$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 179, @@ -1429,7 +1607,8 @@ "geo_code": "[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label(\"$A$\", A, W*r); label(\"$B$\", B, S*r); label(\"$C$\", C, S*r); label(\"$D$\", D, E*r); label(\"$E$\", EE, E*r); label(\"$F$\", F, N*r); label(\"$G$\", G, N*r); label(\"$H$\", H, W*r); label(\"$J$\", J, W*r); [/asy]", "answer": "$184$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 180, @@ -1437,7 +1616,8 @@ "geo_code": "[asy] size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label(\"$A$\",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype(\"2.5 2.5\")+linewidth(.5)); draw((3,0)--(-3,0),linetype(\"2.5 2.5\")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label(\"$A$\",(6.5,0),NW); dot((6.5,0)); [/asy]", "answer": "$50$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 181, @@ -1445,7 +1625,8 @@ "geo_code": "[asy] unitsize(8mm); for (int i=0; i<7; ++i) { draw((i,0)--(i,7),gray); draw((0,i+1)--(7,i+1),gray); } draw((1,3)--(2,4)--(2,5)--(3,6)--(4,5)--(5,5)--(6,4)--(5,3)--(5,2)--(4,1)--(3,2)--(2,2)--cycle,black+2bp); [/asy]", "answer": "$117$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 182, @@ -1453,7 +1634,8 @@ "geo_code": "[asy]draw((1,1.732)--(2,3.464)--(3,1.732)); draw(arc((0,0),(2,0),(1,1.732))); draw(arc((4,0),(3,1.732),(2,0))); label(\"$U$\", (2,3.464), N); label(\"$S$\", (1,1.732), W); label(\"$T$\", (3,1.732), E); label(\"$R$\", (2,0), S);[/asy]", "answer": "$\\sqrt{3}-\\frac{\\pi}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 183, @@ -1461,7 +1643,8 @@ "geo_code": "[asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0)); label(\"$B$\", (0, 0), SW); label(\"$A$\", (12, 0), ESE); label(\"$C$\", (2.4, 3.6), SE); label(\"$D$\", (0, 5), N);[/asy]", "answer": "$96$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 184, @@ -1469,7 +1652,8 @@ "geo_code": "[asy]draw((0,0)--(4,0)--(0,3)--(0,0)); label(\"$A$\", (0,0), SW); label(\"$B$\", (4,0), ESE); label(\"$C$\", (0, 3), N); label(\"$3$\", (0, 1.5), W); label(\"$4$\", (2, 0), S); label(\"$5$\", (2, 1.5), NE);[/asy]", "answer": "$\\frac{18}{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 185, @@ -1477,7 +1661,8 @@ "geo_code": "[asy] size(300); defaultpen(linewidth(0.8)); pair A=(-1,0),C=(1,0),B=dir(40),D=origin; draw(A--B--C--A); draw(D--B); dot(\"$A$\", A, SW); dot(\"$B$\", B, NE); dot(\"$C$\", C, SE); dot(\"$D$\", D, S); label(\"$70^\\circ$\",C,2*dir(180-35));[/asy]", "answer": "$2$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 186, @@ -1485,7 +1670,8 @@ "geo_code": "[asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,-1)--(5.5,-1));[/asy]", "answer": "$\\frac{\\pi}{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 187, @@ -1493,7 +1679,8 @@ "geo_code": "[asy] import graph; size(6cm); pen dps = linewidth(0.7) + fontsize(8); defaultpen(dps); pair B = (0,0); pair C = (1,0); pair A = rotate(60,B)*C; pair E = rotate(270,A)*B; pair D = rotate(270,E)*A; pair F = rotate(90,A)*C; pair G = rotate(90,F)*A; pair I = rotate(270,B)*C; pair H = rotate(270,I)*B; draw(A--B--C--cycle); draw(A--E--D--B); draw(A--F--G--C); draw(B--I--H--C); draw(E--F); draw(D--I); draw(I--H); draw(H--G); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,W); label(\"$E$\",E,W); label(\"$F$\",F,E); label(\"$G$\",G,E); label(\"$H$\",H,SE); label(\"$I$\",I,SW); [/asy]", "answer": "$\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 188, @@ -1501,7 +1688,8 @@ "geo_code": "[asy] dotfactor = 3; size(10cm); dot((0, 10)); label(\"$X$\", (0,10),W,fontsize(8pt)); dot((6,2)); label(\"$Y$\", (6,2),E,fontsize(8pt)); draw((0, 0)--(0, 10)--(1, 10)--(1, 9)--(2, 9)--(2, 7)--(3, 7)--(3,4)--(4, 4)--(4, 0)--cycle); draw((0,9)--(1, 9)--(1.5, 9.5)--(1.5, 10.5)--(0.5, 10.5)--(0, 10)); draw((1, 10)--(1.5,10.5)); draw((1.5, 10)--(3,10)--(3,8)--(2,7)--(0,7)); draw((2,9)--(3,10)); draw((3,8.5)--(4.5,8.5)--(4.5,5.5)--(3,4)--(0,4)); draw((3,7)--(4.5,8.5)); draw((4.5,6)--(6,6)--(6,2)--(4,0)); draw((4,4)--(6,6)); label(\"$1$\", (1,9.5), W,fontsize(8pt)); label(\"$2$\", (2,8), W,fontsize(8pt)); label(\"$3$\", (3,5.5), W,fontsize(8pt)); label(\"$4$\", (4,2), W,fontsize(8pt)); [/asy]", "answer": "$\\frac{6\\sqrt{33}}{5}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 189, @@ -1509,7 +1697,8 @@ "geo_code": "[asy] draw((0,0)--(-5,8.66025404)--(0, 17.3205081)--(10, 17.3205081)--(15,8.66025404)--(10, 0)--(0, 0)); draw((30,0)--(25,8.66025404)--(30, 17.3205081)--(40, 17.3205081)--(45, 8.66025404)--(40, 0)--(30, 0)); draw((30,0)--(25,-8.66025404)--(30, -17.3205081)--(40, -17.3205081)--(45, -8.66025404)--(40, 0)--(30, 0)); draw((0,0)--(-5, -8.66025404)--(0, -17.3205081)--(10, -17.3205081)--(15, -8.66025404)--(10, 0)--(0, 0)); draw((15,8.66025404)--(10, 17.3205081)--(15, 25.9807621)--(25, 25.9807621)--(30, 17.3205081)--(25, 8.66025404)--(15, 8.66025404)); draw((15,-8.66025404)--(10, -17.3205081)--(15, -25.9807621)--(25, -25.9807621)--(30, -17.3205081)--(25, -8.66025404)--(15, -8.66025404)); label(\"A\", (0,0), W); label(\"B\", (30, 17.3205081), NE); label(\"C\", (30, -17.3205081), SE); draw((0,0)--(30, 17.3205081)--(30, -17.3205081)--(0, 0)); //(Diagram Creds-DivideBy0) [/asy]", "answer": "$6\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 190, @@ -1517,7 +1706,8 @@ "geo_code": "[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill((6,0)--(8,0)--(8,26)--(6,26)--cycle,gray); fill((12,0)--(14,0)--(14,26)--(12,26)--cycle,gray); fill((18,0)--(20,0)--(20,26)--(18,26)--cycle,gray); fill((24,0)--(26,0)--(26,26)--(24,26)--cycle,gray); fill((0,0)--(26,0)--(26,2)--(0,2)--cycle,gray); fill((0,12)--(26,12)--(26,14)--(0,14)--cycle,gray); fill((0,24)--(26,24)--(26,26)--(0,26)--cycle,gray); [/asy]", "answer": "$676$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 191, @@ -1525,7 +1715,8 @@ "geo_code": "[asy] real r = 3;\npair A = (r, -r); \npair B = (3*r, -r);\npair C = (6, -8.2);\nreal width = 4*r;\nreal top = 0;\nreal bottom = C.y - r;\ndraw(box((0,top),(width,bottom)), linewidth(0.8bp));\ndraw(circle(A, r), blue);\ndraw(circle(B, r), blue);\ndraw(circle(C, r), blue);\n\ndraw(A--(A+(0,r)), red+linewidth(1));\nlabel(\"$1,\\mathrm{cm}$\", A+(0,r/2), W, red);\ndraw(B--(B+(0,r)), red+linewidth(1));\nlabel(\"$1,\\mathrm{cm}$\", B+(0,r/2), E, red);\ndraw(C--(C+(0,-r)), red+linewidth(1));\nlabel(\"$1,\\mathrm{cm}$\", C+(0,-r/2), E, red);\ndot(midpoint(A--B));\ndot(midpoint(A--C));\ndot(midpoint(B--C));\ndot(A); label(\"$A$\", A, W);\ndot(B); label(\"$B$\", B, E);\ndot(C); label(\"$C$\", C, S);[/asy]", "answer": "$8+4\\sqrt{3}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 192, @@ -1533,7 +1724,8 @@ "geo_code": "[asy] import three; currentprojection=orthographic(1/2,-1,1/2); /* three - currentprojection, orthographic */ draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle); draw((0,0,0)--(0,0,1)); draw((0,1,0)--(0,1,1)); draw((1,1,0)--(1,1,1)); draw((1,0,0)--(1,0,1)); draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle); label(\"$D$\",(0,0,0),S); label(\"$A$\",(0,0,1),N); label(\"$H$\",(0,1,0),S); label(\"$E$\",(0,1,1),N); label(\"$C$\",(1,0,0),S); label(\"$B$\",(1,0,1),N); label(\"$G$\",(1,1,0),S); label(\"$F$\",(1,1,1),N); [/asy]", "answer": "$16$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 193, @@ -1541,7 +1733,8 @@ "geo_code": "[asy] draw((1,0)--(1,5),linewidth(.5)); draw((2,0)--(2,5),linewidth(.5)); draw((3,0)--(3,5),linewidth(.5)); draw((4,0)--(4,5),linewidth(.5)); draw((5,0)--(5,5),linewidth(.5)); draw((6,0)--(6,5),linewidth(.5)); draw((0,1)--(6,1),linewidth(.5)); draw((0,2)--(6,2),linewidth(.5)); draw((0,3)--(6,3),linewidth(.5)); draw((0,4)--(6,4),linewidth(.5)); draw((0,5)--(6,5),linewidth(.5)); draw((0,0)--(0,6),EndArrow); draw((0,0)--(7,0),EndArrow); draw((1,3)--(4,4)--(5,1)--cycle); label(\"$y$\",(0,6),W); label(\"$x$\",(7,0),S); label(\"$A$\",(1,3),dir(210)); label(\"$B$\",(5,1),SE); label(\"$C$\",(4,4),dir(100)); [/asy]", "answer": "$6$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 194, @@ -1549,7 +1742,8 @@ "geo_code": "[asy] draw((-4,6*sqrt(2))--(4,6*sqrt(2))); draw((-4,-6*sqrt(2))--(4,-6*sqrt(2))); draw((-8,0)--(-4,6*sqrt(2))); draw((-8,0)--(-4,-6*sqrt(2))); draw((4,6*sqrt(2))--(8,0)); draw((8,0)--(4,-6*sqrt(2))); draw((-4,6*sqrt(2))--(4,6*sqrt(2))--(4,8+6*sqrt(2))--(-4,8+6*sqrt(2))--cycle); draw((-8,0)--(-4,-6*sqrt(2))--(-4-6*sqrt(2),-4-6*sqrt(2))--(-8-6*sqrt(2),-4)--cycle); label(\"$I$\",(-4,8+6*sqrt(2)),dir(100)); label(\"$J$\",(4,8+6*sqrt(2)),dir(80)); label(\"$A$\",(-4,6*sqrt(2)),dir(280)); label(\"$B$\",(4,6*sqrt(2)),dir(250)); label(\"$C$\",(8,0),W); label(\"$D$\",(4,-6*sqrt(2)),NW); label(\"$E$\",(-4,-6*sqrt(2)),NE); label(\"$F$\",(-8,0),E); draw((4,8+6*sqrt(2))--(4,6*sqrt(2))--(4+4*sqrt(3),4+6*sqrt(2))--cycle); label(\"$K$\",(4+4*sqrt(3),4+6*sqrt(2)),E); draw((4+4*sqrt(3),4+6*sqrt(2))--(8,0),dashed); label(\"$H$\",(-4-6*sqrt(2),-4-6*sqrt(2)),S); label(\"$G$\",(-8-6*sqrt(2),-4),W); label(\"$32$\",(-10,-8),N); label(\"$18$\",(0,6*sqrt(2)+2),N); [/asy]", "answer": "$\\frac{3}{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 195, @@ -1557,7 +1751,8 @@ "geo_code": "[asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP(\"x\",(3.5,0),S)--(0,0)--MP(\"\\frac{3}{2}\",(0,3/2),W)--MP(\"y\",(0,3.5),W)); path P=(0,0)--MP(\"3\",(3,0),S)..(3*dir(45))..MP(\"3\",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy]", "answer": "$\\frac{9\\pi}{4}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 196, @@ -1565,7 +1760,8 @@ "geo_code": "[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy]", "answer": "$\\frac{143}{14}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 197, @@ -1573,15 +1769,17 @@ "geo_code": "[asy] pair A,B,C,D,E,F,G,H,J,K,L,M,N; B=(0,0); real m=7*sqrt(55)/5; J=(m,0); C=(7*m/2,0); A=(0,7*m/2); D=(7*m/2,7*m/2); E=(A+D)/2; H=(0,2m); N=(0,2m+3*sqrt(55)/2); G=foot(H,E,C); F=foot(J,E,C); draw(A--B--C--D--cycle); draw(C--E); draw(G--H--J--F); pair X=foot(N,E,C); M=extension(N,X,A,D); K=foot(N,H,G); L=foot(M,H,G); draw(K--N--M--L); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$E$\",E,dir(90)); label(\"$F$\",F,NE); label(\"$G$\",G,NE); label(\"$H$\",H,W); label(\"$J$\",J,S); label(\"$K$\",K,SE); label(\"$L$\",L,SE); label(\"$M$\",M,dir(90)); label(\"$N$\",N,dir(180)); [/asy]", "answer": "$\\frac{49}{9}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 198, - "problem": "In rectangle $ABCD$, point $M$ is the midpoint of $\\overline{AD}$. The area of $\\triangle AMC$ is \\$12$, and $\\frac{AD}{AB} = \\frac{3}{2}$. Find the length of side $AD$. [asy]size(4cm);draw((0,2)--(0,0)--(3,0)--(3,4)--(0,4)--(0,2)--(3,4)--(0,0));label(\"$A$\", (0,0), SW); label(\"$B$\", (3, 0), SE); label(\"$C$\", (3,4), NE); label(\"$D$\", (0, 4), NW); label(\"$M$\", (0, 2), W);", - "geo_code": "", + "problem": "In rectangle $ABCD$, point $M$ is the midpoint of $\\overline{AD}$. The area of $\\triangle AMC$ is \\$12$, and $\\frac{AD}{AB} = \\frac{3}{2}$. Find the length of side $AD$.", + "geo_code": "[asy]size(4cm);draw((0,2)--(0,0)--(3,0)--(3,4)--(0,4)--(0,2)--(3,4)--(0,0));label(\"$A$\", (0,0), SW); label(\"$B$\", (3, 0), SE); label(\"$C$\", (3,4), NE); label(\"$D$\", (0, 4), NW); label(\"$M$\", (0, 2), W); [/asy]", "answer": "$8$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 199, @@ -1589,7 +1787,8 @@ "geo_code": "[asy] draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0)); draw((3,0)--(1,4)--(0,0)); fill((0,0)--(1,4)--(1.5,3)--cycle, black); fill((3,0)--(2,4)--(1.5,3)--cycle, black); label(\"$A$\",(3.05,4.2)); label(\"$B$\",(2,4.2)); label(\"$C$\",(1,4.2)); label(\"$D$\",(0,4.2)); label(\"$E$\", (0,-0.2)); label(\"$F$\", (3,-0.2)); label(\"$1$\", (0.5, 4), N); label(\"$1$\", (1.5, 4), N); label(\"$1$\", (2.5, 4), N); label(\"$4$\", (3.2, 2), E); [/asy]", "answer": "$9$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 200, @@ -1597,7 +1796,8 @@ "geo_code": "[asy] size(6cm); defaultpen(fontsize(9pt)); path rectangle(pair X, pair Y){ return X--(X.x,Y.y)--Y--(Y.x,X.y)--cycle; } filldraw(rectangle((0,0),(7,5)),gray(0.5)); filldraw(rectangle((1,1),(6,4)),gray(0.75)); filldraw(rectangle((2,2),(5,3)),white); label(\"$1$\",(0.5,2.5)); draw((0.3,2.5)--(0,2.5),EndArrow(TeXHead)); draw((0.7,2.5)--(1,2.5),EndArrow(TeXHead)); label(\"$1$\",(1.5,2.5)); draw((1.3,2.5)--(1,2.5),EndArrow(TeXHead)); draw((1.7,2.5)--(2,2.5),EndArrow(TeXHead)); label(\"$1$\",(4.5,2.5)); draw((4.5,2.7)--(4.5,3),EndArrow(TeXHead)); draw((4.5,2.3)--(4.5,2),EndArrow(TeXHead)); label(\"$1$\",(4.1,1.5)); draw((4.1,1.7)--(4.1,2),EndArrow(TeXHead)); draw((4.1,1.3)--(4.1,1),EndArrow(TeXHead)); label(\"$1$\",(3.7,0.5)); draw((3.7,0.7)--(3.7,1),EndArrow(TeXHead)); draw((3.7,0.3)--(3.7,0),EndArrow(TeXHead)); [/asy]", "answer": "$6$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 201, @@ -1605,7 +1805,8 @@ "geo_code": "[asy] draw(circle((0,0),3)); draw(circle((0,0),1)); draw(circle((1,sqrt(3)),1)); draw(circle((-1,sqrt(3)),1)); draw(circle((-1,-sqrt(3)),1)); draw(circle((1,-sqrt(3)),1)); draw(circle((2,0),1)); draw(circle((-2,0),1)); [/asy]", "answer": "$2\\sqrt{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 202, @@ -1613,7 +1814,8 @@ "geo_code": "[asy]pair A1=(2,0),A2=(4,4); pair B1=(0,4),B2=(5,1); pair C1=(5,0),C2=(0,4); draw(A1--A2); draw(B1--B2); draw(C1--C2); draw((0,0)--B1--(5,4)--C1--cycle); dot((20/7,12/7)); dot((3.07692307692,2.15384615384)); label(\"$Q$\",(3.07692307692,2.15384615384),N); label(\"$P$\",(20/7,12/7),W); label(\"$A$\",(0,4), NW); label(\"$B$\",(5,4), NE); label(\"$C$\",(5,0),SE); label(\"$D$\",(0,0),SW); label(\"$F$\",(2,0),S); label(\"$G$\",(5,1),E); label(\"$E$\",(4,4),N);[/asy]", "answer": "$\\frac{10}{91}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 203, @@ -1621,7 +1823,8 @@ "geo_code": "[asy] pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); draw(A--B--C--A--D^^B--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,NE); label(\"$E$\",E,NW); label(\"$F$\",F,1.5*N); [/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 204, @@ -1629,7 +1832,8 @@ "geo_code": "[asy] import graph; size(9cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.381056062031275, xmax = 15.020004395092375, ymin = -4.051697595316909, ymax = 10.663513514111651; /* image dimensions */ draw((0.,0.)--(4.714285714285714,7.666518779999279)--(7.,0.)--cycle); /* draw figures */ draw((0.,0.)--(4.714285714285714,7.666518779999279)); draw((4.714285714285714,7.666518779999279)--(7.,0.)); draw((7.,0.)--(0.,0.)); label(\"7\",(3.2916797119724284,-0.07831656949355523),SE*labelscalefactor); label(\"9\",(2.0037562070503783,4.196493361737088),SE*labelscalefactor); label(\"8\",(6.114150371695219,3.785453945272603),SE*labelscalefactor); draw((0.,0.)--(6.428571428571427,1.9166296949998194)); draw((7.,0.)--(2.2,3.5777087639996634)); draw((4.714285714285714,7.666518779999279)--(3.7058823529411766,0.)); /* dots and labels */ dot((0.,0.),dotstyle); label(\"$A$\", (-0.2432592696221352,-0.5715638692509372), NE * labelscalefactor); dot((7.,0.),dotstyle); label(\"$B$\", (7.0458397156813835,-0.48935598595804014), NE * labelscalefactor); dot((3.7058823529411766,0.),dotstyle); label(\"$E$\", (3.8123296394941084,0.16830708038513573), NE * labelscalefactor); dot((4.714285714285714,7.666518779999279),dotstyle); label(\"$C$\", (4.579603216894479,7.895848109917452), NE * labelscalefactor); dot((2.2,3.5777087639996634),linewidth(3.pt) + dotstyle); label(\"$D$\", (2.1407693458718726,3.127790878929427), NE * labelscalefactor); dot((6.428571428571427,1.9166296949998194),linewidth(3.pt) + dotstyle); label(\"$H$\", (6.004539860638023,1.9494778850645704), NE * labelscalefactor); dot((5.,1.49071198499986),linewidth(3.pt) + dotstyle); label(\"$Q$\", (4.935837377830365,1.7302568629501784), NE * labelscalefactor); dot((3.857142857142857,1.1499778169998918),linewidth(3.pt) + dotstyle); label(\"$P$\", (3.538303361851119,1.2370095631927964), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]", "answer": "$\\frac{8}{15}\\sqrt{5}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 205, @@ -1637,7 +1841,8 @@ "geo_code": "[asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0)); label(\"$B$\", (0, 0), SW); label(\"$A$\", (12, 0), ESE); label(\"$C$\", (2.4, 3.6), SE); label(\"$D$\", (0, 5), N);[/asy]", "answer": "$4$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 206, @@ -1645,7 +1850,8 @@ "geo_code": "[asy] draw((0,0)--(0,10)--(15,10)--(15,0)--cycle); fill((0,0)--(0,10)--(15,10)--(15,0)--cycle, lightgray); draw((1,1)--(1,3)--(7,3)--(7,1)--cycle); fill((1,1)--(1,3)--(7,3)--(7,1)--cycle, white); draw((1,4)--(1,6)--(7,6)--(7,4)--cycle); fill((1,4)--(1,6)--(7,6)--(7,4)--cycle, white); draw((1,7)--(1,9)--(7,9)--(7,7)--cycle); fill((1,7)--(1,9)--(7,9)--(7,7)--cycle, white); draw((8,1)--(8,3)--(14,3)--(14,1)--cycle); fill((8,1)--(8,3)--(14,3)--(14,1)--cycle, white); draw((8,4)--(8,6)--(14,6)--(14,4)--cycle); fill((8,4)--(8,6)--(14,6)--(14,4)--cycle, white); draw((8,7)--(8,9)--(14,9)--(14,7)--cycle); fill((8,7)--(8,9)--(14,9)--(14,7)--cycle, white); defaultpen(fontsize(8, lineskip=1)); label(\"2\", (1.2, 2)); label(\"6\", (4, 1.2)); defaultpen(linewidth(.2)); draw((0,8)--(1,8), arrow=Arrows); draw((7,8)--(8,8), arrow=Arrows); draw((14,8)--(15,8), arrow=Arrows); draw((11,0)--(11,1), arrow=Arrows); draw((11,3)--(11,4), arrow=Arrows); draw((11,6)--(11,7), arrow=Arrows); label(\"1\", (.5,7.8)); label(\"1\", (7.5,7.8)); label(\"1\", (14.5,7.8)); label(\"1\", (10.8,.5)); label(\"1\", (10.8,3.5)); label(\"1\", (10.8,6.5)); [/asy]", "answer": "$78$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 207, @@ -1653,7 +1859,8 @@ "geo_code": "[asy] size(5cm); draw((0,0)--(6,0)--(6,6)--(0,6)--cycle); draw((0,6)--(6,0)); draw((3,0)--(6,6)); label(\"$A$\",(0,6),NW); label(\"$B$\",(6,6),NE); label(\"$C$\",(6,0),SE); label(\"$D$\",(0,0),SW); label(\"$E$\",(3,0),S); label(\"$F$\",(4,2),E); [/asy]", "answer": "$6\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 208, @@ -1661,7 +1868,8 @@ "geo_code": "[asy] size(6cm); pair A,B,C,D,EE,F,G,H,I,J; C = (0,0); B = (-1,1); D = (2,0.5); A = B+D; G = (0,2); F = B+G; H = G+D; EE = G+B+D; I = (D+H)/2; J = (B+F)/2; filldraw(C--I--EE--J--cycle,lightgray,black); draw(C--D--H--EE--F--B--cycle); draw(G--F--G--C--G--H); draw(A--B,dashed); draw(A--EE,dashed); draw(A--D,dashed); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(I); dot(J); label(\"$A$\",A,E); label(\"$B$\",B,W); label(\"$C$\",C,S); label(\"$D$\",D,E); label(\"$E$\",EE,N); label(\"$F$\",F,W); label(\"$G$\",G,N); label(\"$H$\",H,E); label(\"$I$\",I,E); label(\"$J$\",J,W); [/asy]", "answer": "$\\frac{3}{2}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 209, @@ -1669,7 +1877,8 @@ "geo_code": "[asy] size(4cm);draw((0,0)--(18,0)); draw(arc((9,0),9,0,180)); filldraw(arc((1,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((3,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((5,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((7,0),1,0,180)--cycle,gray(0.8)); label(\"...\",(9,0.5)); filldraw(arc((11,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((13,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((15,0),1,0,180)--cycle,gray(0.8)); filldraw(arc((17,0),1,0,180)--cycle,gray(0.8)); [/asy]", "answer": "$38$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 210, @@ -1677,7 +1886,8 @@ "geo_code": "[asy] size(250); defaultpen(fontsize(10pt)); pair A =origin; pair B = (4.75,0); pair E1=(0,3); pair F = (4.75,3); pair G = (5.95,4.2); pair C = (5.95,1.2); pair D = (1.2,1.2); pair H= (1.2,4.2); pair M = ((4.75+5.95)/2,3.6); draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H); draw(B--C); draw(F--G); draw(A--D--H--C--D,dashed); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,E); label(\"$D$\",D,W); label(\"$E$\",E1,W); label(\"$F$\",F,SW); label(\"$G$\",G,NE); label(\"$H$\",H,NW); label(\"$M$\",M,N); dot(A); dot(B); dot(E1); dot(F); dot(G); dot(C); dot(D); dot(H); dot(M); label(\"3\",A/2+B/2,S); label(\"2\",C/2+G/2,E); label(\"1\",C/2+B/2,SE); [/asy]", "answer": "$2$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "volume" }, { "index": 211, @@ -1685,7 +1895,8 @@ "geo_code": "[asy] size(160); pair A, B, C, D, F; A = origin; B = (4,0); C = (0,3); D = (2/7,2/7); F = foot(D,B,C); fill(A--(2/7,0)--D--(0,2/7)--cycle, lightgray); draw(A--B--C--cycle); draw((2/7,0)--D--(0,2/7)); label(\"$4$\", midpoint(A--B), N); label(\"$3$\", midpoint(A--C), E); label(\"$2$\", midpoint(D--F), SE); label(\"$S$\", midpoint(A--D)); draw(D--F, dashed); [/asy]", "answer": "$\\frac{145}{147}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 212, @@ -1693,7 +1904,8 @@ "geo_code": "[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label(\"$A$\", A, W*r); label(\"$B$\", B, S*r); label(\"$C$\", C, S*r); label(\"$D$\", D, E*r); label(\"$E$\", EE, E*r); label(\"$F$\", F, N*r); label(\"$G$\", G, N*r); label(\"$H$\", H, W*r); label(\"$J$\", J, W*r); [/asy]", "answer": "$184$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 213, @@ -1701,7 +1913,8 @@ "geo_code": "[asy] draw((0,0)--(3,0)); draw((0,0)--(0,2)); draw((0,2)--(3,2)); draw((3,2)--(3,0)); dot((0,0)); dot((0,2)); dot((3,0)); dot((3,2)); draw((2,0)--(2,2)); draw((0,1)--(2,1)); label(\"A\",(0,0),S); label(\"B\",(3,0),S); label(\"C\",(3,2),N); label(\"D\",(0,2),N); [/asy]", "answer": "$50$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 214, @@ -1709,7 +1922,8 @@ "geo_code": "[asy] draw((0,0)--(0,8)); draw((0,8)--(8,8)); draw((8,8)--(8,0)); draw((8,0)--(0,0)); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((0,6)); dot((0,7)); dot((0,8)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((1,6)); dot((1,7)); dot((1,8)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((2,6)); dot((2,7)); dot((2,8)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((3,6)); dot((3,7)); dot((3,8)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); dot((4,6)); dot((4,7)); dot((4,8)); dot((5,0)); dot((5,1)); dot((5,2)); dot((5,3)); dot((5,4)); dot((5,5)); dot((5,6)); dot((5,7)); dot((5,8)); dot((6,0)); dot((6,1)); dot((6,2)); dot((6,3)); dot((6,4)); dot((6,5)); dot((6,6)); dot((6,7)); dot((6,8)); dot((7,0)); dot((7,1)); dot((7,2)); dot((7,3)); dot((7,4)); dot((7,5)); dot((7,6)); dot((7,7)); dot((7,8)); dot((8,0)); dot((8,1)); dot((8,2)); dot((8,3)); dot((8,4)); dot((8,5)); dot((8,6)); dot((8,7)); dot((8,8)); label(\"P\",(4,4),NE); [/asy]", "answer": "$\\frac{2}{5}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 215, @@ -1717,7 +1931,8 @@ "geo_code": "[asy] unitsize(2cm); pair A,B,C,DD,EE,FF; B = (0,0); C = (3,0); A = (1.2,1.7); DD = (2/3)*A+(1/3)*C; EE = (B+DD)/2; FF = intersectionpoint(B--C,A--A+2*(EE-A)); draw(A--B--C--cycle); draw(A--FF); draw(B--DD);dot(A); label(\"$A$\",A,N); dot(B); label(\"$B$\", B,SW);dot(C); label(\"$C$\",C,SE); dot(DD); label(\"$D$\",DD,NE); dot(EE); label(\"$E$\",EE,NW); dot(FF); label(\"$F$\",FF,S); [/asy]", "answer": "$12$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 216, @@ -1725,7 +1940,8 @@ "geo_code": "[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0)); [/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 217, @@ -1733,7 +1949,8 @@ "geo_code": "[asy] import graph; size(8.016233639805293cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.001920114613276, xmax = 4.014313525192017, ymin = -2.552570341575814, ymax = 5.6249093771911145; /* image dimensions */ draw((-1.6742337260757447,-1.)--(-1.6742337260757445,-0.6742337260757447)--(-2.,-0.6742337260757447)--(-2.,-1.)--cycle, linewidth(2.)); draw((-1.7696484586262846,2.7696484586262846)--(-1.5392969172525692,3.)--(-1.7696484586262846,3.2303515413737154)--(-2.,3.)--cycle, linewidth(2.)); /* draw figures */ draw((-2.,3.)--(-2.,-1.), linewidth(2.)); draw((-2.,-1.)--(2.,-1.), linewidth(2.)); draw((2.,-1.)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(-2.,3.), linewidth(2.)); draw((-0.6404058554606791,4.3595941445393205)--(2.,-1.), linewidth(2.)); label(\"$D$\",(-0.9382446143428628,4.887784444795223),SE*labelscalefactor,fontsize(14)); label(\"$A$\",(1.9411496528285788,-1.0783204767840298),SE*labelscalefactor,fontsize(14)); label(\"$B$\",(-2.5046350956841272,-0.9861798602345433),SE*labelscalefactor,fontsize(14)); label(\"$C$\",(-2.5737405580962416,3.5747806589650395),SE*labelscalefactor,fontsize(14)); label(\"$1$\",(-2.665881174645728,1.2712652452278765),SE*labelscalefactor,fontsize(14)); label(\"$1$\",(-0.3393306067712029,-1.3547423264324894),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((-2.,3.),linewidth(4.pt) + dotstyle); dot((-2.,-1.),linewidth(4.pt) + dotstyle); dot((2.,-1.),linewidth(4.pt) + dotstyle); dot((-0.6404058554606791,4.3595941445393205),linewidth(4.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]", "answer": "$\\frac{7}{9}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 218, @@ -1741,7 +1958,8 @@ "geo_code": "[asy] draw(arc((0,0),34,180,0)); draw((-34,0)--(34,0)); fill((-16,0)--(-16,30)--(16,30)--(16,0)--cycle, 1.5*grey); draw((-16,0)--(-16,30)--(16,30)--(16,0)--cycle); dot(\"$A$\",(16,0), 1.25*S); dot(\"$B$\",(16,30), 1.25*N); dot(\"$C$\",(-16,30), 1.25*N); dot(\"$D$\",(-16,0), 1.25*S); dot(\"$E$\",(34,0), 1.25*S); dot(\"$F$\",(-34,0), 1.25*S); label(\"$16$\",(0,0),N); label(\"$9$\",(25,0),N); label(\"$9$\",(-25,0),N); [/asy]", "answer": "$15$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 219, @@ -1749,7 +1967,8 @@ "geo_code": "[asy] draw((0,0)--(13,0)--(13,13)--(0,13)--cycle); filldraw((1,1)--(4,1)--(4,4)--(1,4)--cycle, mediumgray); filldraw((1,5)--(4,5)--(4,8)--(1,8)--cycle, mediumgray); filldraw((1,9)--(4,9)--(4,12)--(1,12)--cycle, mediumgray); filldraw((5,1)--(8,1)--(8,4)--(5,4)--cycle, mediumgray); filldraw((5,5)--(8,5)--(8,8)--(5,8)--cycle, mediumgray); filldraw((5,9)--(8,9)--(8,12)--(5,12)--cycle, mediumgray); filldraw((9,1)--(12,1)--(12,4)--(9,4)--cycle, mediumgray); filldraw((9,5)--(12,5)--(12,8)--(9,8)--cycle, mediumgray); filldraw((12,12)--(12,9)--(9,9)--(9,12)--cycle, mediumgray); [/asy]", "answer": "$\\frac{6}{25}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 220, @@ -1757,7 +1976,8 @@ "geo_code": "[asy] draw((-4,0)--(4,0)--(0,12)--cycle); draw((-2,6)--(4,0)); draw((2,6)--(-4,0)); label(\"M\", (-4,0), W); label(\"C\", (4,0), E); label(\"A\", (0, 12), N); label(\"V\", (2, 6), NE); label(\"U\", (-2, 6), NW); label(\"P\", (0, 3.6), S); [/asy]", "answer": "$96$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 221, @@ -1765,7 +1985,8 @@ "geo_code": "[asy] pair A=(0,1); pair CC=(0.666666666666,1); pair D=(1,1); pair F=(1,0.440062); pair C=(1,0); pair B=(0,0); pair G=(0,0.22005); pair H=(-0.13,0.41); pair E=(0,0.5); dot(A^^CC^^D^^C^^B^^E); draw(E--A--D--F); draw(G--B--C--F, dashed); fill(E--CC--F--G--H--E--CC--cycle, gray); draw(E--CC--F--G--H--E--CC); label(\"A\",A,NW); label(\"B\",B,SW); label(\"C\",C,SE); label(\"D\",D,NE); label(\"E\",E,NW); label(\"C'\",CC,N); label(\"F\",F,NE); [/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 222, @@ -1773,7 +1994,8 @@ "geo_code": "[asy] unitsize(100); pair A=(-1, 0), B=(1, 0), C=(0.3, 0.9), D=(-0.3, 0.9), P=(0.2, 0.5), E=(0.1, 0.75), F=(0.4, 0.5), G=(0.15, 0.2), H=(-0.3, 0.5); draw(A--B--C--D--cycle, black); draw(A--P, black); draw(B--P, black); draw(C--P, black); draw(D--P, black); label(\"$A$\",A,(-1,0)); label(\"$B$\",B,(1,0)); label(\"$C$\",C,(1,-0)); label(\"$D$\",D,(-1,0)); label(\"$2$\",E,(0,0)); label(\"$3$\",F,(0,0)); label(\"$4$\",G,(0,0)); label(\"$5$\",H,(0,0)); dot(A^^B^^C^^D^^P); [/asy]", "answer": "$2+\\sqrt{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 223, @@ -1781,7 +2003,8 @@ "geo_code": "[asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label(\"$A$\",A,dir(90)); label(\"$B$\",B,dir(225)); label(\"$C$\",C,dir(-45)); label(\"$D$\",D,dir(180)); label(\"$E$\",E,dir(-45)); label(\"$F$\",F,dir(225)); label(\"$G$\",G,dir(0)); label(\"$\\ell$\",midpoint(E--F),dir(90)); [/asy]", "answer": "$336$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 224, @@ -1789,7 +2012,8 @@ "geo_code": "[asy] size(170); defaultpen(linewidth(0.6)+fontsize(10)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot(\"$A$\",A,S); dot(\"$B$\",B,S); dot(\"$C$\",C,N); dot(\"$D$\",D,N); dot(\"$Q$\",Q,S); dot(\"$P$\",P,W); dot(\"$R$\",R,1.3*S); label(\"$7$\",(P+R)/2,NE); label(\"$6$\",(R+B)/2,NE); [/asy]", "answer": "$\\sqrt{117}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 225, @@ -1797,7 +2021,8 @@ "geo_code": "[asy] size(6cm); pair A = (0,10); label(\"$A$\", A, N); pair B = (0,0); label(\"$B$\", B, S); pair C = (10,0); label(\"$C$\", C, S); pair D = (10,10); label(\"$D$\", D, SW); pair EE = (15,11.8); label(\"$E$\", EE, N); pair F = (3,10); label(\"$F$\", F, N); filldraw(D--arc(D,2.5,270,380)--cycle,lightgray); dot(A^^B^^C^^D^^EE^^F); draw(A--B--C--D--cycle); draw(D--EE--F--cycle); label(\"$110^\\circ$\", (15,9), SW); [/asy]", "answer": "$\\frac{17}{11}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 226, @@ -1805,7 +2030,8 @@ "geo_code": "[asy] size(10cm); pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),D=(cos(pi/12),sin(pi/12)),E=rotate(150,D)*C,F=rotate(-30,E)*D,A=rotate(150,F)*E,B=rotate(-30,A)*F; draw(C--D--E--F--A--B--cycle,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(E,q); dot(F,q); label(\"$C$\",C,2*S); label(\"$D$\",D,2*S); label(\"$E$\",E,2*S); label(\"$F$\",F,2*dir(0)); label(\"$A$\",A,2*N); label(\"$B$\",B,2*W); [/asy]", "answer": "$12\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 227, @@ -1813,7 +2039,8 @@ "geo_code": "[asy] size(10cm); usepackage(\"mathptmx\"); import geometry; void perp(picture pic=currentpicture, pair O, pair M, pair B, real size=5, pen p=currentpen, filltype filltype = NoFill){ perpendicularmark(pic, M,unit(unit(O-M)+unit(B-M)),size,p,filltype); } pen p=black+linewidth(1),q=black+linewidth(5); pair C=(0,0),Y=(2,0),X=(3,0),A=(6,0),B=(2,sqrt(5.6)),D=(3,-sqrt(12.6)); draw(A--B--C--D--cycle,p); draw(A--C,p); draw(B--Y,p); draw(D--X,p); dot(A,q); dot(B,q); dot(C,q); dot(D,q); dot(X,q); dot(Y,q); label(\"2\",C--Y,S); label(\"1\",Y--X,S); label(\"3\",X--A,S); label(\"$A$\",A,2*E); label(\"$B$\",B,2*N); label(\"$C$\",C,2*W); label(\"$D$\",D,2*S); label(\"$Y$\",Y,2*sqrt(2)*NE); label(\"$X$\",X,2*N); perp(B,Y,C,8,p); perp(A,X,D,8,p); [/asy]", "answer": "$3\\sqrt{35}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 228, @@ -1821,7 +2048,8 @@ "geo_code": "[asy] usepackage(\"mathptmx\"); size(275); defaultpen(linewidth(0.8)); real r = 2, s = 2.5, theta = 14; pair G = (0,0), F = (r,0), C = (r,s), B = (0,s), M = (C+F)/2, I = M + s/2 * dir(-theta); pair N = (B+G)/2, J = N + s/2 * dir(180+theta); pair E = F + r * dir(- 45 - theta/2), D = I+E-F; pair H = J + r * dir(135 + theta/2), A = B+H-J; draw(A--B--C--I--D--E--F--G--J--H--cycle^^rightanglemark(F,I,C)^^rightanglemark(G,J,B)); draw(J--B--G^^C--F--I,linetype (\"4 4\")); dot(\"$A$\",A,N); dot(\"$B$\",B,1.2*N); dot(\"$C$\",C,N); dot(\"$D$\",D,dir(0)); dot(\"$E$\",E,S); dot(\"$F$\",F,1.5*dir(-100)); dot(\"$G$\",G,S); dot(\"$H$\",H,W); dot(\"$I$\",I,NE); dot(\"$J$\",J,1.5*S); [/asy]", "answer": "$192$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "volume" }, { "index": 229, @@ -1829,7 +2057,8 @@ "geo_code": "[asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label(\"$A$\",A,SW); label(\"$B$\", B, NW); label(\"$C$\",C,NE); label(\"$D$\",D,SE); label(\"$P$\",P,S); [/asy]", "answer": "$4$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 230, @@ -1837,7 +2066,8 @@ "geo_code": "[asy] import geometry; unitsize(3cm); draw(circle((0,0),1),linewidth(1.5)); for (int i = 0; i < 7; ++i) { for (int j = 0; j < i; ++j) { draw(dir(i * 360/7) -- dir(j * 360/7),linewidth(1.5)); } } for(int i = 0; i < 7; ++i) { dot(dir(i * 360/7),5+black); } [/asy]", "answer": "$147$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 231, @@ -1845,7 +2075,8 @@ "geo_code": "[asy] usepackage(\"mathptmx\"); size(9cm); draw((0,-.5)--(0,11),EndArrow(size=.15cm)); draw((1,0)--(1,11),mediumgray); draw((2,0)--(2,11),mediumgray); draw((3,0)--(3,11),mediumgray); draw((4,0)--(4,11),mediumgray); draw((5,0)--(5,11),mediumgray); draw((6,0)--(6,11),mediumgray); draw((7,0)--(7,11),mediumgray); draw((8,0)--(8,11),mediumgray); draw((9,0)--(9,11),mediumgray); draw((10,0)--(10,11),mediumgray); draw((11,0)--(11,11),mediumgray); draw((12,0)--(12,11),mediumgray); draw((13,0)--(13,11),mediumgray); draw((14,0)--(14,11),mediumgray); draw((15,0)--(15,11),mediumgray); draw((16,0)--(16,11),mediumgray); draw((-.5,0)--(17,0),EndArrow(size=.15cm)); draw((0,1)--(17,1),mediumgray); draw((0,2)--(17,2),mediumgray); draw((0,3)--(17,3),mediumgray); draw((0,4)--(17,4),mediumgray); draw((0,5)--(17,5),mediumgray); draw((0,6)--(17,6),mediumgray); draw((0,7)--(17,7),mediumgray); draw((0,8)--(17,8),mediumgray); draw((0,9)--(17,9),mediumgray); draw((0,10)--(17,10),mediumgray); draw((-.13,1)--(.13,1)); draw((-.13,2)--(.13,2)); draw((-.13,3)--(.13,3)); draw((-.13,4)--(.13,4)); draw((-.13,5)--(.13,5)); draw((-.13,6)--(.13,6)); draw((-.13,7)--(.13,7)); draw((-.13,8)--(.13,8)); draw((-.13,9)--(.13,9)); draw((-.13,10)--(.13,10)); draw((1,-.13)--(1,.13)); draw((2,-.13)--(2,.13)); draw((3,-.13)--(3,.13)); draw((4,-.13)--(4,.13)); draw((5,-.13)--(5,.13)); draw((6,-.13)--(6,.13)); draw((7,-.13)--(7,.13)); draw((8,-.13)--(8,.13)); draw((9,-.13)--(9,.13)); draw((10,-.13)--(10,.13)); draw((11,-.13)--(11,.13)); draw((12,-.13)--(12,.13)); draw((13,-.13)--(13,.13)); draw((14,-.13)--(14,.13)); draw((15,-.13)--(15,.13)); draw((16,-.13)--(16,.13)); label(scale(.7)*\"$1$\", (1,-.13), S); label(scale(.7)*\"$2$\", (2,-.13), S); label(scale(.7)*\"$3$\", (3,-.13), S); label(scale(.7)*\"$4$\", (4,-.13), S); label(scale(.7)*\"$5$\", (5,-.13), S); label(scale(.7)*\"$6$\", (6,-.13), S); label(scale(.7)*\"$7$\", (7,-.13), S); label(scale(.7)*\"$8$\", (8,-.13), S); label(scale(.7)*\"$9$\", (9,-.13), S); label(scale(.7)*\"$10$\", (10,-.13), S); label(scale(.7)*\"$11$\", (11,-.13), S); label(scale(.7)*\"$12$\", (12,-.13), S); label(scale(.7)*\"$13$\", (13,-.13), S); label(scale(.7)*\"$14$\", (14,-.13), S); label(scale(.7)*\"$15$\", (15,-.13), S); label(scale(.7)*\"$16$\", (16,-.13), S); label(scale(.7)*\"$1$\", (-.13,1), W); label(scale(.7)*\"$2$\", (-.13,2), W); label(scale(.7)*\"$3$\", (-.13,3), W); label(scale(.7)*\"$4$\", (-.13,4), W); label(scale(.7)*\"$5$\", (-.13,5), W); label(scale(.7)*\"$6$\", (-.13,6), W); label(scale(.7)*\"$7$\", (-.13,7), W); label(scale(.7)*\"$8$\", (-.13,8), W); label(scale(.7)*\"$9$\", (-.13,9), W); label(scale(.7)*\"$10$\", (-.13,10), W); dot((0,0),linewidth(4)); label(scale(.75)*\"$A$\", (0,0), NE); dot((3,1),linewidth(4)); label(scale(.75)*\"$B$\", (3,1), NE); dot((0,10),linewidth(4)); label(scale(.75)*\"$C$\", (0,10), NE); dot((2,9),linewidth(4)); label(scale(.75)*\"$D$\", (2,9), NE); draw((15,3)--(16,3)--(16,5)--(15,5)--cycle,linewidth(1.125)); dot((15,3),linewidth(4)); dot((16,3),linewidth(4)); dot((16,5),linewidth(4)); dot((15,5),linewidth(4)); [/asy]", "answer": "$1$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 232, @@ -1853,7 +2084,8 @@ "geo_code": "[asy] usepackage(\"mathptmx\"); size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6); draw((11,11)--(171,11),linetype(\"2 2\")+white+linewidth(1.2)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.85)*rotate(45)*\"Water 1\", (23,-13.5)); label(scale(.85)*rotate(45)*\"Water 2\", (43,-13.5)); label(scale(.85)*rotate(45)*\"Water 7\", (143,-13.5)); filldraw(circle((103,-13.5),.2)); filldraw(circle((98,-13.5),.2)); filldraw(circle((93,-13.5),.2)); filldraw(circle((88,-13.5),.2)); filldraw(circle((83,-13.5),.2)); label(scale(.85)*rotate(90)*\"Start\", (5.5,11)); label(scale(.85)*rotate(270)*\"Finish\", (176.5,11)); [/asy]", "answer": "$48$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 233, @@ -1861,7 +2093,8 @@ "geo_code": "[asy] import graph; // The Solid // To save processing time, do not use three (dimensions) // Project (roughly) to two size(15cm); pair Fr, Lf, Rt, Tp, Bt, Bk; Lf=(0,0); Rt=(12,1); Fr=(7,-1); Bk=(5,2); Tp=(6,6.7); Bt=(6,-5.2); draw(Lf--Fr--Rt); draw(Lf--Tp--Rt); draw(Lf--Bt--Rt); draw(Tp--Fr--Bt); draw(Lf--Bk--Rt,dashed); draw(Tp--Bk--Bt,dashed); label(rotate(-8.13010235)*slant(0.1)*\"$Q$\", (4.2,1.6)); label(rotate(21.8014095)*slant(-0.2)*\"$?$\", (8.5,2.05)); pair g = (-8,0); // Define Gap transform real a = 8; draw(g+(-a/2,1)--g+(a/2,1), Arrow()); // Make arrow // Time for the NET pair DA,DB,DC,CD,O; DA = (4*sqrt(3),0); DB = (2*sqrt(3),6); DC = (DA+DB)/3; CD = conj(DC); O=(0,0); transform trf=shift(3g+(0,3)); path NET = O--(-2*DA)--(-2DB)--(-DB)--(2DA-DB)--DB--O--DA--(DA-DB)--O--(-DB)--(-DA)--(-DA-DB)--(-DB); draw(trf*NET); label(\"$7$\",trf*DC); label(\"$Q$\",trf*DC+DA-DB); label(\"$5$\",trf*DC-DB); label(\"$3$\",trf*DC-DA-DB); label(\"$6$\",trf*CD); label(\"$4$\",trf*CD-DA); label(\"$2$\",trf*CD-DA-DB); label(\"$1$\",trf*CD-2DA); [/asy]", "answer": "$1$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 234, @@ -1869,7 +2102,8 @@ "geo_code": "[asy] filldraw(circle((0,0),8),gray); filldraw(circle((-1,0),7),white); filldraw(circle((-2,0),6),gray); filldraw(circle((-3,0),5),white); filldraw(circle((-4,0),4),gray); filldraw(circle((-5,0),3),white); filldraw(circle((-6,0),2),gray); filldraw(circle((-7,0),1),white); [/asy]", "answer": "$64$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 235, @@ -1877,7 +2111,8 @@ "geo_code": "[asy] size(200); defaultpen(linewidth(0.6pt)+fontsize(10pt)); real y = sqrt(3); pair A,B,C,D,E,F,G,H; A = (0,0); B = (0,y); C = (y,y); D = (y,0); E = ((y + 1)/2,y); F = (y, (y - 1)/2); G = ((y - 1)/2, 0); H = (0,(y + 1)/2); fill(H--B--E--cycle, gray); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); [/asy]", "answer": "$2-\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 236, @@ -1885,7 +2120,8 @@ "geo_code": "[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label(\"$C_4$\", D); label(\"$C_1$\", (-1.375, 0)); label(\"$C_2$\", (1.375,0)); label(\"$\\frac{1}{2}$\", (0, -.125)); label(\"$C_3$\", (-0.4, -0.4)); label(\"$1$\", (-.85, 0.70)); label(\"$1$\", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]", "answer": "$\\frac{3}{14}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 237, @@ -1893,7 +2129,8 @@ "geo_code": "[asy] size(150); filldraw((0,0)--(10,0)--(10,10)--(0,10)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(9,0)--(9,9)--(0,9)--cycle,white,linewidth(1)); filldraw((0,0)--(7,0)--(7,7)--(0,7)--cycle,gray(0.7),linewidth(1)); filldraw((0,0)--(4,0)--(4,4)--(0,4)--cycle,white,linewidth(1)); draw((11,0)--(11,4),linewidth(1)); draw((11,6)--(11,10),linewidth(1)); label(\"$10$\",(11,5),fontsize(14pt)); draw((10.75,0)--(11.25,0),linewidth(1)); draw((10.75,10)--(11.25,10),linewidth(1)); draw((0,11)--(3,11),linewidth(1)); draw((5,11)--(9,11),linewidth(1)); draw((0,11.25)--(0,10.75),linewidth(1)); draw((9,11.25)--(9,10.75),linewidth(1)); label(\"$9$\",(4,11),fontsize(14pt)); draw((-1,0)--(-1,1),linewidth(1)); draw((-1,3)--(-1,7),linewidth(1)); draw((-1.25,0)--(-0.75,0),linewidth(1)); draw((-1.25,7)--(-0.75,7),linewidth(1)); label(\"$7$\",(-1,2),fontsize(14pt)); draw((0,-1)--(1,-1),linewidth(1)); draw((3,-1)--(4,-1),linewidth(1)); draw((0,-1.25)--(0,-.75),linewidth(1)); draw((4,-1.25)--(4,-.75),linewidth(1)); label(\"$4$\",(2,-1),fontsize(14pt)); [/asy]", "answer": "$52$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 238, @@ -1901,7 +2138,8 @@ "geo_code": "[asy] draw((3,11)--(11,7)--(5,7)--(3,11)); dot((5,7)); label(\"$A(5,7)$\",(5,7),S); dot((11,7)); label(\"$B(11,7)$\",(11,7),S); dot((3,11)); label(\"$C(3,y)$\",(3,11),NW); [/asy]", "answer": "$4\\sqrt{5}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 239, @@ -1909,7 +2147,8 @@ "geo_code": "[asy] size(150); import graph; draw(circle((0,0),3)); real radius = 3; real angleStart = -54; // starting angle of the sector real angleEnd = 54; // ending angle of the sector label(\"$O$\",(0,0),W); pair O = (0, 0); filldraw(arc(O, radius, angleStart, angleEnd)--O--cycle, gray); filldraw(circle((0,0),2),gray); filldraw(circle((0,0),1),white); draw((1.763,2.427)--(0,0)--(1.763,-2.427)); label(\"$B$\",(1.763,2.427),NE); label(\"$C$\",(1.763,-2.427),SE); [/asy]", "answer": "$108$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 240, @@ -1917,7 +2156,8 @@ "geo_code": "[asy] unitsize(.3cm); filldraw((0,0)--(8,9)--(11,6)--(18,12)--(30,0)--cycle,gray(0.7),linewidth(1)); draw((-1,0)--(-1,9),linewidth(.75)); draw((-1.4,0)--(-.6,0),linewidth(.75)); draw((-1.4,9)--(-.6,9),linewidth(.75)); label(\"$8$\",(-1,4),W); label(\"$12$\",(31,6),E); draw((-1,9)--(8,9),dashed); draw((31,0)--(31,12),linewidth(.75)); draw((30.6,0)--(31.4,0),linewidth(.75)); draw((30.6,12)--(31.4,12),linewidth(.75)); draw((31,12)--(18,12),dashed); label(\"$45^{\\circ}$\",(.75,0),NE,fontsize(10pt)); label(\"$45^{\\circ}$\",(29.25,0),NW,fontsize(10pt)); draw((8,9)--(7.5,8.5)--(8,8)--(8.5,8.5)--cycle); draw((18,12)--(17.5,11.5)--(18,11)--(18.5,11.5)--cycle); draw((11,6)--(11,0),dashed); label(\"$h$\",(11,2.5),E); [/asy]", "answer": "$5$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 241, @@ -1925,7 +2165,8 @@ "geo_code": "[asy]\nsize(175);\ndefaultpen(linewidth(0.8));\npair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;\ndraw(A--B--C--cycle^^X--Y--Z--cycle);\nlabel(\"$A$\",A,N);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,E);\nlabel(\"$X$\",X,W);\nlabel(\"$Y$\",Y,S);\nlabel(\"$Z$\",Z,NE);[/asy]", "answer": "$122$", "category": "Local Relation Composition", - "source": "olympiads" + "source": "olympiads", + "problem_type": "area" }, { "index": 242, @@ -1933,7 +2174,8 @@ "geo_code": "[asy] size(250); pair A=(0,12), E=(0,8), B=origin, C=(24*sqrt(2),0), D=(6*sqrt(2),0), F=A+10*dir(A--C), G=intersectionpoint(E--F, A--D); draw(A--B--C--A--D^^E--F); pair point=G+1*dir(250); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(point--D)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$G$\", G, dir(point--G)); markscalefactor=0.1; draw(rightanglemark(A,B,C)); label(\"10\", A--F, dir(90)*dir(A--F)); label(\"27\", F--C, dir(90)*dir(F--C)); label(\"3\", (0,10), W); label(\"9\", (0,4), W); [/asy]", "answer": "$148$", "category": "Local Relation Composition", - "source": "AIME-83-24" + "source": "AIME-83-24", + "problem_type": "area" }, { "index": 243, @@ -1941,7 +2183,8 @@ "geo_code": "[asy]\nint n=9;\ndraw(polygon(n));\nfor (int i = 0; ic+r) a=c; else b=c;\n}\ndraw(circle(Q,c));[/asy]", "answer": "$169$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 269, @@ -2149,7 +2417,8 @@ "geo_code": "[asy]\ndefaultpen(linewidth(0.7)); size(120);\npair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }\npair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2);\ndraw(A--B--C--cycle);\nlabel(\" $A$ \",A,N); label(\" $B$ \",B,S); label(\" $C$ \",C,S);\nfilldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));\nfilldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));\nfilldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]", "answer": "$49$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 270, @@ -2157,7 +2426,8 @@ "geo_code": "[asy]\nsize(120);defaultpen(linewidth(0.7)+fontsize(10));\npair O=origin, A=dir(90), K=dir(180), M=1/2*dir(180), N=0.4*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);\ndraw(K--O--A--M--N--B--A^^circle((0,0),1));\nlabel(\" $A$ \", A, dir(O--A));\nlabel(\" $K$ \", K, dir(O--K));\nlabel(\" $B$ \", B, dir(O--B));\nlabel(\" $N$ \", N, E);\nlabel(\" $M$ \", M, S);\nlabel(\" $O$ \", O, SE);[/asy]", "answer": "$36$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "angle" }, { "index": 271, @@ -2165,7 +2435,8 @@ "geo_code": "[asy]\nsize(175);\ndefaultpen(linewidth(0.8));\npair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;\ndraw(A--B--C--cycle^^X--Y--Z--cycle);\nlabel(\" $A$ \",A,N);\nlabel(\" $B$ \",B,S);\nlabel(\" $C$ \",C,E);\nlabel(\" $X$ \",X,W);\nlabel(\" $Y$ \",Y,S);\nlabel(\" $Z$ \",Z,NE);[/asy]", "answer": "$183$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 272, @@ -2173,7 +2444,8 @@ "geo_code": "[asy] pair incenter(pair A=(0,0), pair B=(0,0), pair C=(0,0)){pair P,Q;P=rotate((angle(C-A)-angle(B-A))*90/pi,A)*B;Q=rotate((angle(A-B)-angle(C-B))*90/pi,B)*C;return extension(A,P,B,Q);}pair foot(pair P,pair A, pair B){real s;s=dot(P-A,unit(B-A));return (scale(s)*unit(B-A)+A);}defaultpen(fontsize(10)); size(7cm);pair A = (5.6,5), B = (0,0), C = (5,0), M = midpoint(B--C), I = incenter(A,B,C), P = extension(A, A+dir(I--A)*dir(-90), B,C), K = foot(M,A,P), F = extension(M, (M.x, M.x+1), A,P);draw(K--M--F--P--B--A--C);pair point = I;pair[] p={A,B,C,M,P,F,K};string s = \"A,B,C,M,P,F,K\";int size = p.length;real[] d; real[] mult; for(int i = 0; i=1;i=i-1)\n{\nif (floor(i/2)==i/2)\n{\nfilldraw(circle(origin,4*i),white);\n}\nelse\n{\nfilldraw(circle(origin,4*i),red);\n}\n}\n[/asy]", "answer": "$41$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 283, @@ -2261,7 +2543,8 @@ "geo_code": "[asy]\ndefaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);\ndraw(unitcircle,dg);\nfor(int i = 0; i < 12; ++i) {\n draw(dir(30*i+theta)--dir(30*(i+1)+theta), db);\n dot(dir(30*i+theta),Fill(rgb(0.8,0,0)));\n} dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr));\n[/asy]", "answer": "$1800$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 284, @@ -2269,7 +2552,8 @@ "geo_code": "[asy]\nsize(3cm);\nreal lw=0.4, dr=0.3;\nreal r1=14, r2=9;\npair A=(0,0), B=(r1-r2,0);\ndraw(A--B,dashed);\ndraw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw));\nfilldraw(circle(A,dr)); filldraw(circle(B,dr));\nlabel(\" $5$ \",(A+B)/2,dir(-90));\nlabel(\" $\\gamma$ \",A+r1*dir(135),dir(135)); label(\" $\\omega$ \",B+r2*dir(135),dir(135));\n[/asy]", "answer": "$23$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 285, @@ -2277,7 +2561,8 @@ "geo_code": "[asy]\nimport graph;\nsize(3cm);\npair A = (0,0);\npair temp = (1,0);\npair B = rotate(45,A)*temp;\npair C = rotate(90,B)*A;\npair D = rotate(270,C)*B;\npair E = rotate(270,D)*C;\npair F = rotate(90,E)*D;\npair G = rotate(270,F)*E;\npair H = rotate(270,G)*F;\npair I = rotate(90,H)*G;\npair J = rotate(270,I)*H;\npair K = rotate(270,J)*I;\npair L = rotate(90,K)*J;\ndraw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);\n[/asy]", "answer": "$500$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "area" }, { "index": 286, @@ -2285,7 +2570,8 @@ "geo_code": "[asy]\nsize(7cm);\npair A=(0,0), B=(64,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);\ndraw(A--B--C--D--cycle);\ndraw(A--C);\ndraw(B--D);\ndot(\" $A$ \", A, SW);\ndot(\" $B$ \", B, SE);\ndot(\" $C$ \", C, NE);\ndot(\" $D$ \", D, NW);\ndot(\" $E$ \", E, S);\ndot(\" $F$ \", F, S);\ndraw(circle((A + C)/2, abs(A - C)/2));\ndraw(circle((B + D)/2, abs(B - D)/2));\ndraw(circle((A + B)/2, abs(A - B)/2));\nlabel(\" $\\mathcal P$ \", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));\nlabel(\" $\\mathcal Q$ \", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));\nlabel(\" $\\mathcal R$ \", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));\n[/asy]", "answer": "$961$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "count" }, { "index": 287, @@ -2293,7 +2579,8 @@ "geo_code": "[asy]\n\nsize(120);\npair A = (0,3);\npair B = (0,0);\npair C = (3,0);\npair D = (0,1.5);\npair E = (0.35,0);\npair F = (1.1,1.9);\npair J = (0.17,0);\npair Y = (0.17,0.75);\n\npair Z = (1.6,0.4);\ndraw(A--B);\ndraw(B--C);\ndraw(C--A);\ndraw(D--F--Z--E--D);\ndraw(\" $O$ \", B, dir(180));\ndraw(\" $B$ \", A, dir(45));\ndraw(\" $A$ \", C, dir(45));\ndraw(\" $Q$ \", E, dir(45));\ndraw(\" $P$ \", D, dir(45));\ndraw(\" $R$ \", Z, dir(45));\ndraw(\" $S$ \", F, dir(45));\ndraw(\" $a$ \", Y, dir(210));\ndraw(\" $b$ \", J, dir(100));\n[/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 288, @@ -2301,7 +2588,8 @@ "geo_code": "[asy] draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((5,5)--(12,-2)--(5,-9)--(-2,-2)--cycle); label(\"A\", (0,0), W); label(\"B\", (10,0), E); label(\"C\", (10,10), NE); label(\"D\", (0,10), NW); label(\"G\", (5,5), N); label(\"F\", (12,-2), E); label(\"E\", (5,-9), S); label(\"H\", (-2,-2), W); dot((-2,-2)); dot((5,-9)); dot((12,-2)); dot((0,0)); dot((10,0)); dot((10,10)); dot((0,10)); dot((5,5)); [/asy]", "answer": "$\\frac{175}{4}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 289, @@ -2309,7 +2597,8 @@ "geo_code": "[asy] draw(Circle((0,0), 10)); draw((0,10)--(6,-8)--(-6,-8)--cycle); label(\"A\", (0,10), N); label(\"B\", (-6,-8), SW); label(\"C\", (6,-8), SE); dot((0,10)); dot((6,-8)); dot((-6,-8)); [/asy]", "answer": "$3$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 290, @@ -2317,7 +2606,8 @@ "geo_code": "[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy]", "answer": "$10$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 291, @@ -2325,7 +2615,8 @@ "geo_code": "[asy] pair A,B,C,D,P,Q,R; A = (0,4); B = (8,4); C = (8,0); D = (0,0); P = (2,2); Q = (4,2); R = (6,2); dot(A); dot(B); dot(C); dot(D); dot(P); dot(Q); dot(R); draw(A--B--C--D--cycle); draw(circle(P,2)); draw(circle(Q,2)); draw(circle(R,2)); label(\"$A$\",A,NW); label(\"$B$\",B,NE); label(\"$C$\",C,SE); label(\"$D$\",D,SW); label(\"$P$\",P,W); label(\"$Q$\",Q,W); label(\"$R$\",R,W); [/asy]", "answer": "$8$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 292, @@ -2333,7 +2624,8 @@ "geo_code": "[asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 4; ++b) { dot((a,b)); } } draw((0,0)--(3,2)--(4,3)--cycle); label(\"$A$\",(0,0),SW); label(\"$B$\",(3,2),SE); label(\"$C$\",(4,3),NE); [/asy]", "answer": "$\\frac{9}{2}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 293, @@ -2341,7 +2633,8 @@ "geo_code": "[asy] pair A,B,C,D; A = (0,0); B = (9,10); C = (10,0); D = (6.66,3); dot(A); dot(B); dot(C); dot(D); draw(A--B--C--cycle); draw(A--D--C); label(\"$A$\",A,SW); label(\"$B$\",B,N); label(\"$C$\",C,SE); label(\"$D$\",D,N); label(\"$60^\\circ $\",(9.4,8.8),SW); [/asy]", "answer": "$120$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 294, @@ -2349,7 +2642,8 @@ "geo_code": "[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((1,0)--(1,0.2)); draw((2,0)--(2,0.2)); draw((3,1)--(2.8,1)); draw((3,2)--(2.8,2)); draw((1,3)--(1,2.8)); draw((2,3)--(2,2.8)); draw((0,1)--(0.2,1)); draw((0,2)--(0.2,2)); draw((2,0)--(3,2)--(1,3)--(0,1)--cycle); [/asy]", "answer": "$5$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 295, @@ -2357,7 +2651,8 @@ "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = (0,2); pair B = origin; pair C = (2,0); pair D = (2,2); pair E = midpoint(A--D); pair F = foot(C,B,E); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label(\"$A$\",A,N);label(\"$B$\",B,S);label(\"$C$\",C,S);label(\"$D$\",D,N);label(\"$E$\",E,N);label(\"$F$\",F,NW); draw(A--B--C--D--cycle); draw(B--E); draw(C--F); draw(rightanglemark(B,F,C,4));[/asy]", "answer": "$\\frac{11}{20}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 296, @@ -2365,7 +2660,8 @@ "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (1.25,1); pair C = (2,0); pair D = midpoint(A--C); pair E = midpoint(A--B); pair G = intersectionpoint(E--C,B--D); dot(A);dot(B);dot(C);dot(D);dot(E);dot(G); label(\"$A$\",A,S);label(\"$B$\",B,N);label(\"$C$\",C,S);label(\"$D$\",D,S);label(\"$E$\",E,NW);label(\"$G$\",G,NE); draw(A--B--C--cycle); draw(B--D); draw(E--C); draw(rightanglemark(C,G,D,3));[/asy]", "answer": "$16$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 297, @@ -2373,7 +2669,8 @@ "geo_code": "[asy] defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33); dot(A);dot(B);dot(C);dot(O); label(\"$A$\",A,SW);label(\"$B$\",B,SE);label(\"$C$\",C,W);label(\"$O$\",O,NW); label(\"$1$\",midpoint(A--B),S);label(\"$60^\\circ$\",B,2W + N); draw((3,0)--A--(0,3)); draw(B--C); draw(Arc(O,2.33,163,288.5));[/asy]", "answer": "$\\frac{3}{2}+\\frac{\\sqrt{3}}{2}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 298, @@ -2381,7 +2678,8 @@ "geo_code": "[asy] size(6cm); defaultpen(linewidth(.8pt)+fontsize(10pt)); draw((-1,1)--(2,1)); draw((-1,0)--(1,0)); draw((-1,1)--(-1,0)); draw((0,-1)--(0,3)); draw((1,2)--(1,0)); draw((-1,1)--(1,1)); draw((0,2)--(1,2)); draw((0,3)--(1,2)); draw((0,-1)--(2,1)); draw((0,-1)--((0,-1) + sqrt(2)*dir(-15))); draw(((0,-1) + sqrt(2)*dir(-15))--(1,0)); label(\"$\\textbf{A}$\",foot((0,2),(0,3),(1,2)),SW); label(\"$\\textbf{B}$\",midpoint((0,1)--(1,2))); label(\"$\\textbf{C}$\",midpoint((-1,0)--(0,1))); label(\"$\\textbf{D}$\",midpoint((0,0)--(1,1))); label(\"$\\textbf{E}$\",midpoint((1,0)--(2,1)),NW); label(\"$\\textbf{F}$\",midpoint((0,-1)--(1,0)),NW); label(\"$\\textbf{G}$\",midpoint((0,-1)--(1,0)),2SE);[/asy]", "answer": "$frac{20}{3}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "volume" }, { "index": 299, @@ -2389,7 +2687,8 @@ "geo_code": "[asy] pointpen = black; pathpen = black; D(unitsquare); D((0,0)); D((1,0)); D((1,1)); D((0,1)); D(D((.5,.5))--D((1,.5))); D(D((.17,1))--(.5,.5)--D((.17,0))); MP(\"x\",(.58,1),N); [/asy]", "answer": "$\\frac{5}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 300, @@ -2397,7 +2696,8 @@ "geo_code": "[asy]pathpen = black+linewidth(0.7); D((0,0)--(7,0)--(7,7)--(0,7)--cycle); D((1,0)--(1,6)); D((0,6)--(6,6)); D((1,1)--(7,1)); D((6,7)--(6,1)); [/asy]", "answer": "$81$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 301, @@ -2405,7 +2705,8 @@ "geo_code": "[asy] unitsize(12); draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle); draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW)); draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW)); draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW)); label(rotate(30)*\"$40^\\circ$\",(2,-8.9),ENE); label(\"$100^\\circ$\",(21/3,-2/3),SE); label(\"$110^\\circ$\",(900/83,-317/83),NNW); label(\"$A$\",(0,0),NW); [/asy]", "answer": "$30$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 302, @@ -2413,7 +2714,8 @@ "geo_code": "[asy] pair A,B,C,D,M,N; A = (0,0); B = (0,3); C = (3,3); D = (3,0); M = (0,1); N = (1,0); draw(A--B--C--D--cycle); draw(M--C--N); label(\"$A$\",A,SW); label(\"$M$\",M,W); label(\"$B$\",B,NW); label(\"$C$\",C,NE); label(\"$D$\",D,SE); label(\"$N$\",N,S); [/asy]", "answer": "$frac{\\sqrt{13}}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 303, @@ -2421,7 +2723,8 @@ "geo_code": "[asy] draw((0,0)--(2,0)--(3,1)--(3,3)--(2,2)--(0,2)--cycle); draw((2,0)--(2,2)); draw((0,2)--(1,3)); draw((1,7/3)--(1,10/3)--(2,10/3)--(2,7/3)--cycle); draw((2,7/3)--(5/2,17/6)--(5/2,23/6)--(3/2,23/6)--(1,10/3)); draw((2,10/3)--(5/2,23/6)); draw((3,3)--(5/2,3));[/asy]", "answer": "$\\frac{1}{6}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 304, @@ -2429,7 +2732,8 @@ "geo_code": "[asy] draw((2,0)--(8,0)--(6,4)--cycle); draw((4,2)--(7,2)); draw((1,4)--(9,4),Arrows); label(\"$A$\",(2,0),SW); label(\"$B$\",(8,0),SE); label(\"$M$\",(4,2),W); label(\"$N$\",(7,2),E); label(\"$P$\",(6,4),N); [/asy]", "answer": "$3$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 305, @@ -2437,7 +2741,8 @@ "geo_code": "[asy] label(\"A\", (0,0), W); label(\"B\", (64,0), E); label(\"C\", (32, 32*sqrt(3)), N); draw(arc((0,0),64,0,60)); draw(arc((64,0),64,120,180)); draw((0,0)--(64,0)); draw(circle((32, 24), 24)); [/asy]", "answer": "$\\frac{27}{\\pi}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 306, @@ -2445,7 +2750,8 @@ "geo_code": "[asy] draw((0,8)--(0,0)--(4,0)--(4,8)--(0,8)--(3.5,8.5)--(3.5,8)); draw((2,-1)--(2,9),dashed); [/asy]", "answer": "$\\frac{6}{5}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 307, @@ -2453,7 +2759,8 @@ "geo_code": "[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy]", "answer": "$\\frac{5}{9}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 308, @@ -2461,7 +2768,8 @@ "geo_code": "[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label(\"5 in\", (2.5,0), S); label(\"3 in\", (5,1.5), E); label(\"3 in\", (3.5,3), N);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label(\"4 in\", (2,0), S); label(\"2 in\", (4,1), E);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label(\"3 in\", (1.5,0), S); label(\"2 in\", (2.5,1), W);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label(\"3 in\", (1.5,0), S); label(\"4 in\", (3,2), E);[/asy]", "answer": "$40$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 309, @@ -2469,7 +2777,8 @@ "geo_code": "[asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label(\"5 in\", (2.5,0), S); label(\"3 in\", (5,1.5), E); label(\"3 in\", (3.5,3), N);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label(\"4 in\", (2,0), S); label(\"2 in\", (4,1), E);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label(\"3 in\", (1.5,0), S); label(\"2 in\", (2.5,1), W);[/asy][asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label(\"3 in\", (1.5,0), S); label(\"4 in\", (3,2), E);[/asy]", "answer": "$24$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 310, @@ -2477,7 +2786,8 @@ "geo_code": "[asy] size(4inch,2inch); draw((0,0)--(31,0)--(16,8)--(6,8)--cycle); draw((11,8)--(11,0), linetype(\"8 4\")); draw((11,1)--(12,1)--(12,0)); label(\"$A$\", (0,0), SW); label(\"$D$\", (31,0), SE); label(\"$B$\", (6,8), NW); label(\"$C$\", (16,8), NE); label(\"10\", (3,5), W); label(\"8\", (11,4), E); label(\"17\", (22.5,5), E);[/asy]", "answer": "$10$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 311, @@ -2485,7 +2795,8 @@ "geo_code": "[asy]defaultpen(linewidth(.8pt)+fontsize(8pt)); pair B = (0,0); pair A = 2*dir(60); pair C = (2,0); pair D = (4,0); pair M = midpoint(A--C); label(\"$A$\",A,NW);label(\"$B$\",B,SW);label(\"$C$\",C, SE);label(\"$M$\",M,NE);label(\"$D$\",D,SE); draw(A--B--C--cycle); draw(C--D--M--cycle);[/asy]", "answer": "$\\frac {\\sqrt {3}}{8}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 312, @@ -2493,7 +2804,8 @@ "geo_code": "[asy] unitsize(3mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((8,4)--(8,0)--(10,0)--(10,-4)); label(\"$A$\",(0,4),NW); label(\"$B$\",(18,4),NE); label(\"$C$\",(18,-4),SE); label(\"$D$\",(0,-4),SW); label(\"$x$\",(4,4),S); label(\"$x$\",(14,-4),N); label(\"$18$\",(9,4),N); label(\"$18$\",(9,-4),S); label(\"$8$\",(0,0),W); label(\"$8$\",(18,0),E); dot((0,4)); dot((18,4)); dot((18,-4)); dot((0,-4));[/asy]", "answer": "$6$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 313, @@ -2501,7 +2813,8 @@ "geo_code": "[asy] size(150); defaultpen(linewidth(0.7)+fontsize(8)); draw(circle((2,4),4));draw(circle((14,9),9)); draw((0,-2)--(0,20));draw((-6,0)--(25,0)); draw((2,4)--(2,4)+4*expi(pi*4.5/11)); draw((14,9)--(14,9)+9*expi(pi*6/7)); label(\"4\",(2,4)+2*expi(pi*4.5/11),(-1,0)); label(\"9\",(14,9)+4.5*expi(pi*6/7),(1,1)); label(\"(2,4)\",(2,4),(0.5,-1.5));label(\"(14,9)\",(14,9),(1,-1)); draw((-4,120*-4/119+912/119)--(11,120*11/119+912/119)); dot((2,4)^^(14,9)); [/asy]", "answer": "$\\frac{1032}{119}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 314, @@ -2509,7 +2822,8 @@ "geo_code": "[asy] size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label(\"$\\mathcal{T}$\",(2.1,-1.6)); label(\"$\\mathcal{P}$\",(0,-1),NE); label(\"$\\mathcal{Q}$\",(4.2,-1),NW); label(\"$\\mathcal{R}$\",(0,-2.2),SE); label(\"$\\mathcal{S}$\",(4.2,-2.2),SW); [/asy]", "answer": "$89$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 315, @@ -2517,7 +2831,8 @@ "geo_code": "[asy] unitsize(13mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(1,3), B=(2,3), C=(2,2), D=(3,2), Ep=(3,1), F=(2,1), G=(2,0), H=(1,0), I=(1,1), J=(0,1), K=(0,2), L=(1,2); pair M=intersectionpoints(A--G,H--C)[0]; draw(A--B--C--D--Ep--F--G--H--I--J--K--L--cycle); draw(A--G); draw(H--C); dot(M); label(\"$A$\",A,NW); label(\"$B$\",B,NE); label(\"$C$\",C,NE); label(\"$D$\",D,NE); label(\"$E$\",Ep,SE); label(\"$F$\",F,SE); label(\"$G$\",G,SE); label(\"$H$\",H,SW); label(\"$I$\",I,SW); label(\"$J$\",J,SW); label(\"$K$\",K,NW); label(\"$L$\",L,NW); label(\"$M$\",M,W); [/asy]", "answer": "$110$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 316, @@ -2525,7 +2840,8 @@ "geo_code": "[asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy]", "answer": "$\\frac{4\\sqrt{3}-1}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 317, @@ -2533,7 +2849,8 @@ "geo_code": "[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21)); pair D=foot(B,A,C); pair[] ps={B,C,A,D}; draw(A--B--C--cycle); draw(B--D); draw(rightanglemark(B,D,C)); dot(ps); label(\"$A$\",A,NW); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$3$\",midpoint(A--D),NE); label(\"$4$\",midpoint(D--C),NE); [/asy]", "answer": "$7\\sqrt3$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 318, @@ -2541,7 +2858,8 @@ "geo_code": "[asy] unitsize(2cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), Ep=dir(35), D=dir(-35), B=dir(145); pair A=intersectionpoints(Circle(B,1),C--(-1*Ep))[0]; pair[] ds={A,B,C,D,Ep}; dot(ds); draw(A--Ep--D--B--cycle); label(\"$A$\",A,SW); label(\"$B$\",B,NW); label(\"$C$\",C,N); label(\"$E$\",Ep,E); label(\"$D$\",D,E); [/asy]", "answer": "$52.5$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 319, @@ -2549,7 +2867,8 @@ "geo_code": "[asy] unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6; path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0)); [/asy]", "answer": "$100$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 320, @@ -2557,7 +2876,8 @@ "geo_code": "[asy] unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D}; draw(D--A--B--C--D--B); dot(dotted); label(\"$D$\",D,NW); label(\"$C$\",C,W); label(\"$B$\",B,E); label(\"$A$\",A,NE); [/asy]", "answer": "$12$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 321, @@ -2565,7 +2885,8 @@ "geo_code": "[asy] unitsize(45); import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,0.15)--(1.85,0.15)--(1.85,0)--(2,0)--cycle); draw(circle((2,1),2.24)); draw(circle((2,1),1)); draw((0,0)--(4,0)); draw((0,0)--(2,1)); draw((2,1)--(2,0)); draw((2,1)--(4,0)); dot((0,0),ds); label(\"$A$\", (-0.19,-0.23),NE*lsf); dot((2,0),ds); label(\"$B$\", (1.97,-0.31),NE*lsf); dot((2,1),ds); label(\"$C$\", (1.96,1.09),NE*lsf); dot((4,0),ds); label(\"$D$\", (4.07,-0.24),NE*lsf); clip((-3.1,-7.72)--(-3.1,4.77)--(11.74,4.77)--(11.74,-7.72)--cycle); [/asy]", "answer": "$16 \\pi$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 322, @@ -2573,7 +2894,8 @@ "geo_code": "[asy] filldraw((0,0)--(25,0)--(25,15)--(0,15)--cycle,white,black); label(\"D\",(0,0),S); label(\"R\",(25,0),S); label(\"Q\",(25,15),N); label(\"A\",(0,15),N); filldraw((10,0)--(15,0)--(15,15)--(10,15)--cycle,mediumgrey,black); label(\"S\",(10,0),S); label(\"C\",(15,0),S); label(\"B\",(15,15),N); label(\"P\",(10,15),N);[/asy]", "answer": "$225$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 323, @@ -2581,7 +2903,8 @@ "geo_code": "[asy] pair A,B,C,D; A=(3,20); B=(35,20); C=(47,0); D=(0,0); draw(A--B--C--D--cycle); dot((0,0)); dot((3,20)); dot((35,20)); dot((47,0)); label(\"A\",A,N); label(\"B\",B,N); label(\"C\",C,S); label(\"D\",D,S); draw((19,20)--(19,0)); dot((19,20)); dot((19,0)); draw((19,3)--(22,3)--(22,0)); label(\"12\",(21,10),E); label(\"50\",(19,22),N); label(\"15\",(1,10),W); label(\"20\",(41,12),E);[/asy]", "answer": "$75$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 324, @@ -2589,7 +2912,8 @@ "geo_code": "[asy] filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,gray,black); filldraw(Circle((0,0),1), mediumgray,black); filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]", "answer": "$\\frac{\\pi-1}{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 325, @@ -2597,7 +2921,8 @@ "geo_code": "[asy] unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r=3; pair A=(-3cos(80),-3sin(80)); pair D=(3cos(80),3sin(80)), C=(-3cos(80),3sin(80)); pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r); draw(outer); draw(E--B); draw(E--A); draw(B--A); draw(E--D); draw(C--D); draw(B--C); pair[] ps={A,B,C,D,E,O}; dot(ps); label(\"$A$\",A,N); label(\"$B$\",B,NE); label(\"$C$\",C,S); label(\"$D$\",D,S); label(\"$E$\",E,NW); label(\"$$\",O,N); [/asy]", "answer": "$130$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 326, @@ -2605,7 +2930,8 @@ "geo_code": "[asy] pair A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R; A=(4,0); B=(7,0); C=(7,4); D=(8,4); E=(8,5); F=(10,5); G=(10,7); H=(7,7); I=(7,8); J=(5,8); K=(5,7); L=(4,7); M=(4,6); N=(0,6); O=(0,5); P=(2,5); Q=(2,3); R=(4,3); draw(A--B--C--D--E--F--G--H--I--J--K--L--M--N--O--P--Q--R--cycle); label(\"$X$\",(3.4,1.5)); label(\"6\",(7.6,1.5)); label(\"1\",(7.6,3.5)); label(\"1\",(8.4,4.6)); label(\"2\",(9.4,4.6)); label(\"2\",(10.4,6)); label(\"3\",(8.4,7.4)); label(\"1\",(7.5,7.8)); label(\"2\",(6,8.5)); label(\"1\",(4.7,7.8)); label(\"1\",(4.3,7.5)); label(\"1\",(3.5,6.5)); label(\"4\",(1.8,6.5)); label(\"1\",(-0.5,5.5)); label(\"2\",(0.8,4.5)); label(\"2\",(1.5,3.8)); label(\"2\",(2.8,2.6));[/asy]", "answer": "$5$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 327, @@ -2613,7 +2939,8 @@ "geo_code": "[asy] draw((0,2)--(2,2)--(2,0)--(0,0)--cycle); draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle); label(\"$a$\",(-0.1,0.15)); label(\"$b$\",(-0.1,1.15));[/asy]", "answer": "$4$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 328, @@ -2621,7 +2948,8 @@ "geo_code": "[asy] draw((0,0)--(0,10)--(20,10)--(20,0)--cycle); draw(circle((10,5),5)); [/asy]", "answer": "$50$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 329, @@ -2629,7 +2957,8 @@ "geo_code": "[asy] filldraw((0,0)--(2,0)--(1,sqrt(3))--cycle,gray,gray); filldraw(circle((1,sqrt(3)),1),gray); filldraw(circle((0,0),1),gray); filldraw(circle((2,0),1),grey);[/asy]", "answer": "$40\\pi+16\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 330, @@ -2637,7 +2966,8 @@ "geo_code": "[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]", "answer": "$\\frac{375}{4}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "volume" }, { "index": 331, @@ -2645,7 +2975,8 @@ "geo_code": "[asy] size(100);import graph; pair A,B,C; A=(0,8); B=(0,0); C=(18,0); draw((0,8)..(-4,4)..(0,0)--(0,8)); draw((0,0)..(9,-9)..(18,0)--(0,0)); real theta = aTan(8/18); draw(arc((9,4),10,-theta,180-theta)); draw((0,8)--(18,0)); dot(A); dot(B); dot(C); label(\"$A$\", A, NW); label(\"$B$\", B, SW); label(\"$C$\", C, SE); [/asy]", "answer": "$15$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 332, @@ -2653,7 +2984,8 @@ "geo_code": "[asy]\nunitsize (2 cm);\n\npair A, B, C, D, Bp, Cp, Dp, P;\n\nA = (0,0);\nB = (-1,0);\nC = (-1,1);\nD = (0,1);\nBp = rotate(-30)*(B);\nCp = rotate(-30)*(C);\nDp = rotate(-30)*(D);\nP = extension(C, D, Bp, Cp);\n\nfill(A--Bp--P--D--cycle, gray(0.8));\ndraw(A--B--C--D--cycle);\ndraw(A--Bp--Cp--Dp--cycle);\n\nlabel(\" $30^\\circ$ \", (-0.5,0.1), fontsize(10));\n[/asy]", "answer": "$12\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 333, @@ -2661,7 +2993,8 @@ "geo_code": "[asy]\n\npair A = (2,6);\npair B = (0,0);\npair C = (10,0);\npair D = (3.5,0) ;\npair E = (3.1,2);\ndraw(A--B);\ndraw(B--C);\ndraw(C--A);\ndraw (A--D);\ndot ((3.1,1.7));\nlabel (\"E\", E, dir(45));\nlabel (\"A\", A, dir(45));\nlabel (\"B\", B, dir(45));\nlabel (\"C\", C, dir(45));\nlabel (\"D\", D, dir(45));\ndraw(circle((1.8,1.3),1.3)); \ndraw(circle((4.9,1.7),1.75)); \n[/asy]", "answer": "$48$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 334, @@ -2669,7 +3002,8 @@ "geo_code": "[asy] unitsize(7); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E; A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10); draw(A--B--C--cycle); draw(D--E); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,S); label(\"$E$\",E,NE); draw(rightanglemark(B,D,E,30)); [/asy]", "answer": "$37\\frac{1}{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 335, @@ -2677,7 +3011,8 @@ "geo_code": "[asy] defaultpen(linewidth(.8pt)); unitsize(2.5cm); pair A = origin; pair B = (2,0); pair C = (0.5,0.75); pair D = midpoint(C--B); draw(A--B--C--cycle); draw(A--D); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,N); label(\"$D$\",D,NE);[/asy]", "answer": "$20$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 336, @@ -2685,7 +3020,8 @@ "geo_code": "[asy] size(6cm); pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7); draw(A--B--C--D--cycle); draw(F--C--E--B); label(\"$A$\", A, SW); label(\"$B$\", B, S); label(\"$C$\", C, N); label(\"$D$\", D, NW); label(\"$E$\", E, SE); label(\"$F$\", F, W); [/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 337, @@ -2693,7 +3029,8 @@ "geo_code": "[asy] pair A, B, C, D, P; A = (0, 0); B = (6.5, 0); C = (6.5, 4.5); D = (0, 4.5); P = (2.5, 1.5); draw(A--B--C--D--cycle); draw(A--P); draw(C--P); draw(D--P); draw(B--P, dashed); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$C$\", C, NE); label(\"$D$\", D, NW); label(\"$P$\", P, S); label(\"$6$\", midpoint(A--P), NW); label(\"$8$\", midpoint(D--P), NE); label(\"$10$\", midpoint(C--P), NW); [/asy]", "answer": "$6\\sqrt{2}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 338, @@ -2701,7 +3038,8 @@ "geo_code": "[asy] draw((0,0)--(10,20*sqrt(3)/2)--(20,0)--cycle,black+linewidth(.75)); draw((20,0)--(20,12)--(32,12)--(32,0)--cycle,black+linewidth(.75)); draw((32,0)--(37,10*sqrt(3)/2)--(42,0)--cycle,black+linewidth(.75)); MP(\"I\",(10,0),N);MP(\"II\",(26,0),N);MP(\"III\",(37,0),N); MP(\"A\",(0,0),S);MP(\"B\",(20,0),S);MP(\"C\",(32,0),S);MP(\"D\",(42,0),S); [/asy]", "answer": "$24$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 339, @@ -2709,7 +3047,8 @@ "geo_code": "[asy] draw(arc((0,-1),2,30,150),dashed+linewidth(.75)); draw((-1.7,0)--(0,0)--(1.7,0),dot); draw((0,0)--(0,.98),dot); MP(\"A\",(-1.7,0),W);MP(\"B\",(1.7,0),E);MP(\"M\",(0,0),S);MP(\"C\",(0,1),N); [/asy]", "answer": "$7$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 340, @@ -2717,7 +3056,8 @@ "geo_code": "[asy] size(2.5inch); pair A, B, C, E, F, G; A = (0,3); B = (-1,0); C = (4,0); E = (0,0); F = (1.14,2.14); G = intersectionpoint(B--F,A--E); draw(A--B--C--cycle); draw(A--E); draw(B--F); label(\"$A$\",A,N); label(\"$B$\",B,W); label(\"$C$\",C,dir(0)); label(\"$E$\",E,S); label(\"$F$\",F,NE); label(\"$G$\",G,SE); [/asy]", "answer": "$3$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 341, @@ -2725,7 +3065,8 @@ "geo_code": "[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label(\"D\",(0,1),NW);label(\"E\",(.4,1),N);label(\"C\",(1,1),NE); label(\"P\",(0,.6),W);label(\"M\",(.25,.55),E);label(\"Q\",(1,.2),E); label(\"A\",(0,0),SW);label(\"B\",(1,0),SE); [/asy]", "answer": "$\\frac{19}{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 342, @@ -2733,7 +3074,8 @@ "geo_code": "[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((.82,0)--(1,1)--(0,.76)--cycle); label(\"A\", (0,0), S); label(\"B\", (1,0), S); label(\"C\", (1,1), N); label(\"D\", (0,1), N); label(\"M\", (0,.76), W); label(\"N\", (.82,0), S);[/asy]", "answer": "$8\\sqrt{3}-12$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 343, @@ -2741,7 +3083,8 @@ "geo_code": "[asy] unitsize(45); pair O = (0,0); pair T = dir(90); pair T1 = dir(270); pair T2 = dir(25); pair P = (.61,1); pair Q = (1.61, -1); draw(unitcircle); dot(O); label(\"O\",O,W); label(\"T\",T,N); label(\"T'\",T1,S); label(\"T''\",T2,NE); label(\"P\",P,NE); label(\"Q\",Q,S); draw(O--T2); label(\"$r$\",midpoint(O--T2),NW); draw(T--P); label(\"8\",midpoint(T--P),N); draw(T1--Q); label(\"18\",midpoint(T1--Q),S); draw(P--Q);[/asy]", "answer": "$12$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 344, @@ -2749,7 +3092,8 @@ "geo_code": "[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label(\"A\", (0,0), S); label(\"B\", (5,0), S); label(\"C\", (6,3), NE); label(\"D\", (1,3), NW); label(\"P\", (10,0), E); label(\"Q\", (-5,0), W); label(\"M\", (.5,1.5), NW); label(\"N\", (5.65, 1.5), NE); label(\"O\", (3.4,1.75));[/asy]", "answer": "$9$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 345, @@ -2757,7 +3101,8 @@ "geo_code": "[asy] draw((-4,0)--(4,0)--(-1,4)--cycle); draw((-1, 4)--(0, 0.00001)); label(\"B\", (-4,0), S); label(\"C\", (4,0), S); label(\"A\", (-1, 4), N); label(\"M\", (0, 0.0001), S); [/asy]", "answer": "$\\sqrt{31}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 346, @@ -2765,7 +3110,8 @@ "geo_code": "[asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label(\"A\", (0,0), S); label(\"B\", (12,0), S); label(\"C\", (14,7.75), E); label(\"E\", (8.75,4.84), N); label(\"F\", (5,0), S); label(\"M\", (13,3.875), E); label(\"G\", (7,1)); [/asy]", "answer": "$\\frac{2}{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 347, @@ -2773,7 +3119,8 @@ "geo_code": "[asy] size(150); pair K=(0,0),B=(1,0),A=(-1,0),L=(0,0.5),M=(sqrt(2)/2,.25); draw(circle(K,1)^^A--B); draw(circle(L,0.5)^^circle(M,.25)); label(\"$A$\", A, W); label(\"$K$\", K, S); label(\"$B$\", B, E); label(\"$L$\", L); label(\"$M$\", M); [/asy]", "answer": "$4$", "category": "Primitive Recognition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "ratio" }, { "index": 348, @@ -2781,7 +3128,8 @@ "geo_code": "[asy] size(120); real t = 2/sqrt(3); real x = 1 + sqrt(3); pair A = t*dir(90), D = x*A; pair B = t*dir(210), E = x*B; pair C = t*dir(330), F = x*C; draw(D--E--F--cycle); draw(Circle(A, 1)); draw(Circle(B, 1)); draw(Circle(C, 1)); [/asy]", "answer": "$12+12\\sqrt{3}$", "category": "Local Relation Composition", - "source": "aops_forum" + "source": "aops_forum", + "problem_type": "length" }, { "index": 349, @@ -2789,7 +3137,8 @@ "geo_code": "[asy] size(100); real a=4, b=3; // import cse5; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP(\"A\",A)--MP(\"B\",B,N)--MP(\"C\",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP(\"R\",IP(CP((X+C)/2,C),B--C),NW)--MP(\"Q\",IP(CP((X+C)/2,C),A--C+(0.1,0)))); [/asy]", "answer": "$2.4$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 350, @@ -2797,7 +3146,8 @@ "geo_code": "[asy] draw((-2,1)--(2,1)--(2,-1)--(-2,-1)--cycle); draw((0,0)--(0,-1)--(-2,-1)--(-2,0)--cycle); label(\"$F$\",(0,0),E); label(\"$A$\",(-2,1),W); label(\"$B$\",(2,1),E); label(\"$C$\", (2,-1),E); label(\"$D$\",(-2,-1),WSW); label(\"$E$\",(-2,0),W); label(\"$G$\",(0,-1),S); [/asy]", "answer": "$30$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 351, @@ -2805,7 +3155,8 @@ "geo_code": "[asy] real s=sqrt(3)/2; draw(box((0,0),(1,1))); draw((1+s,0.5)--(1,1)); draw((1+s,0.5)--(1,0)); draw((0,1)--(1+s,0.5)); label(\"$A$\",(1,1),N); label(\"$B$\",(1,0),S); label(\"$C$\",(0,0),W); label(\"$D$\",(0,1),W); label(\"$E$\",(1+s,0.5),E); [/asy]", "answer": "$45$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 352, @@ -2813,7 +3164,8 @@ "geo_code": "[asy] size(150); import patterns; pair D=(0,0),C=(1,-1),B=(2.5,-0.2),A=(1,2),AA,BB,CC,DD,P,Q,aux; add(\"hatch\",hatch()); draw(rotate(100,D)*(A--B--C--D--cycle)); AA=rotate(100,D)*A; BB=rotate(100,D)*D; CC=rotate(100,D)*C; DD=rotate(100,D)*B; aux=midpoint(AA--BB); draw(BB--DD); P=midpoint(AA--aux); aux=midpoint(CC--DD); Q=midpoint(CC--aux); draw(AA--CC,dashed); dot(P); dot(Q); fill(DD--BB--CC--cycle,pattern(\"hatch\")); label(\"$A$\",AA,W); label(\"$B$\",BB,S); label(\"$C$\",CC,E); label(\"$D$\",DD,N); label(\"$P$\",P,S); label(\"$Q$\",Q,E); [/asy]", "answer": "$\\sqrt{2}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 353, @@ -2821,7 +3173,8 @@ "geo_code": "[asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP(\"$A$\",A)); D(MP(\"$B$\",B)); D(MP(\"$C$\",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP(\"$B'$\",BB[0],NW);MP(\"$C'$\",CC[1],NE); [/asy]", "answer": "$1+\\sqrt{3(r^2-1)}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 354, @@ -2829,7 +3182,8 @@ "geo_code": "[asy] size(200); import cse5; pathpen=black; anglefontpen=black; pointpen=black; anglepen=black; dotfactor=3; pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE; EE=(B+C)/2; D(MP(\"$A$\",A,W)--MP(\"$B$\",B,N)--MP(\"$C$\",C,E)--cycle); D(MP(\"$E$\",EE,N)--MP(\"$D$\",D,S)); D(D);D(EE); MA(\"80^\\circ\",8,D,EE,C,0.1); MA(\"20^\\circ\",8,EE,C,D,0.3,2,shift(1,3)*C); draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow()); MA(\"100^\\circ\",8,A,B,C,0.1,0); MA(\"60^\\circ\",8,C,A,B,0.1,0); [/asy]", "answer": "$\\frac{\\sqrt{3}}{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 355, @@ -2837,7 +3191,8 @@ "geo_code": "[asy] size(100);defaultpen(linewidth(0.7)+fontsize(10)); pair D=(0,0), C=D+dir(230), E=D+dir(310), F=E+dir(40), G=D+dir(40), A=D+dir(140), B=C+dir(140); draw(E--D--G--F--E--C--D--A--B--C); pair point=(0,0.5); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(-15)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$G$\", G, dir(point--G));[/asy]", "answer": "$150$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 356, @@ -2845,7 +3200,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(-1,0), B=(1,0), C=(0,1), D=(0,-1), Q=origin, P=(-0.5,0); draw(P--C--D^^A--B^^Circle(Q,1)); label(\"$A$\", A, W); label(\"$B$\", B, E); label(\"$C$\", C, N); label(\"$D$\", D, S); label(\"$P$\", P, S); label(\"$Q$\", Q, SE); label(\"$60^\\circ$\", P+0.0.5*dir(30), dir(30));[/asy]", "answer": "$\\frac{2\\sqrt{3}}{3}$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 357, @@ -2853,7 +3209,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); real r=degrees((12,5)), s=degrees((3,4)); pair D=origin, A=(13,0), C=D+12*dir(r), B=A+3*dir(180-(90-r+s)); draw(A--B--C--D--cycle); markscalefactor=0.05; draw(rightanglemark(A,B,C)); pair point=incenter(A,C,D); label(\"$A$\", A, dir(A)); label(\"$B$\", B, dir(B)); label(\"$C$\", C, dir(C)); label(\"$D$\", D, dir(D)); label(\"$6$\", A--B, dir(A--B)*dir(-90)); label(\"$8$\", B--C, dir(B--C)*dir(-90)); label(\"$24$\", C--D, dir(C--D)*dir(-90)); label(\"$26$\", D--A, dir(D--A)*dir(-90));[/asy]", "answer": "$144$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 358, @@ -2861,7 +3218,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(15,3), D=(5,1), A=7*dir(72)*dir(B--C), E=midpoint(A--C), F=intersectionpoint(A--D, B--E); draw(E--B--A--C--B^^A--D); label(\"$A$\", A, dir(D--A)); label(\"$B$\", B, dir(E--B)); label(\"$C$\", C, dir(0)); label(\"$D$\", D, SE); label(\"$E$\", E, N); label(\"$F$\", F, dir(80));[/asy]", "answer": "$5$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 359, @@ -2869,7 +3227,8 @@ "geo_code": "[asy] size(150); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(42), C=intersectionpoint(B--(30,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$M$\", M, dir(point--M)); label(\"$N$\", N, dir(30)); label(rotate(angle(dir(A--C)))*\"$38$\", A--C, dir(A--C)*dir(90)); label(rotate(angle(dir(A--B)))*\"$28$\", A--B, dir(A--B)*dir(90)); [/asy]", "answer": "$5$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 360, @@ -2877,7 +3236,8 @@ "geo_code": "[asy] defaultpen(linewidth(.8pt)); pair A = (0,11); pair B = (2,0); pair D = (4,0); pair E = (7,0); pair C = (13,0); label(\"$A$\",A,N); label(\"$B$\",B,SW); label(\"$C$\",C,SE); label(\"$D$\",D,S); label(\"$E$\",E,S); label(\"$4$\",midpoint(B--D),N); label(\"$6$\",midpoint(D--E),NW); label(\"$12$\",midpoint(E--C),NW); draw(A--B--C--cycle); draw(A--D); draw(A--E); [/asy]", "answer": "$4\\sqrt{10}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 361, @@ -2885,7 +3245,8 @@ "geo_code": "[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0)); draw(B--M--O--B--C--O--N--C^^N--A--M); label(\"$A$\", A, dir(90)); label(\"$B$\", B, dir(O--B)); label(\"$C$\", C, dir(O--C)); label(\"$M$\", M, dir(90)*dir(B--A)); label(\"$N$\", N, dir(90)*dir(A--C)); label(\"$O$\", O, dir(90));[/asy]", "answer": "$15$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 362, @@ -2893,7 +3254,8 @@ "geo_code": "[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10));real r=54.72; pair B=origin, C=dir(r), A=intersectionpoint(B--(9,0), C--C+4*dir(r-90)), M=midpoint(B--A), N=midpoint(A--C), P=intersectionpoint(B--N, C--M); draw(M--C--A--B--C^^B--N); pair point=P; markscalefactor=0.01; draw(rightanglemark(B,C,N)); draw(rightanglemark(C,P,B)); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$M$\", M, S); label(\"$N$\", N, dir(C--A)*dir(90)); label(\"$2$\", B--C, NW); [/asy]", "answer": "$\\sqrt{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 363, @@ -2901,7 +3263,8 @@ "geo_code": "[asy] size(250);defaultpen(linewidth(0.7)); real alpha=5.797939254, x=71.191836; int i; for(i=0; i<5; i=i+1) { real r=8*(sqrt(6)/2)^i; draw(Circle((x+r)*dir(alpha), r)); x=x+2r; } real x=71.191836+40+20*sqrt(6), r=18; pair A=tangent(origin, (x+r)*dir(alpha), r, 1), B=tangent(origin, (x+r)*dir(alpha), r, 2); pair A1=300*dir(origin--A), B1=300*dir(origin--B); draw(B1--origin--A1); pair X=(69,-5), X1=reflect(origin, (x+r)*dir(alpha))*X, Y=(200,-5), Y1=reflect(origin, (x+r)*dir(alpha))*Y, Z=(130,0), Z1=reflect(origin, (x+r)*dir(alpha))*Z; clip(X--Y--Y1--X1--cycle); label(\"$L_2$\", Z, S); label(\"$L_1$\", Z1, dir(2*alpha)*dir(90));[/asy]", "answer": "$4\\sqrt{6}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 364, @@ -2909,7 +3272,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,0), C=(8,7), F=7*dir(A--C), E=(10,0)+4*dir(B--C), D=4*dir(A--B); draw(A--B--C--A--E--F--D); pair point=incenter(A,B,C); label(\"$A$\", A, dir(point--A)); label(\"$B$\", B, dir(point--B)); label(\"$C$\", C, dir(point--C)); label(\"$D$\", D, dir(point--D)); label(\"$E$\", E, dir(point--E)); label(\"$F$\", F, dir(point--F)); label(\"$2$\", (2,0), S); label(\"$3$\", (7,0), S);[/asy]", "answer": "$12$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 365, @@ -2917,7 +3281,8 @@ "geo_code": "[asy] size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label(\"2\", (sqrt(3)/4, 1/4), NW); label(\"$60^\\circ$\", (1,0));[/asy]", "answer": "$\\frac{10}{3}\\pi+4$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 366, @@ -2925,7 +3290,8 @@ "geo_code": "[asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7));[/asy]", "answer": "$24$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 367, @@ -2933,7 +3299,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label(\"A\", A, NE); label(\"B\", B, NE); label(\"C\", C, dir(0)); label(\"D\", D, dir(180)); label(\"$L$\", (x/2,0), SW); label(\"$L^\\prime$\", C, SW); label(\"2\", D--F, NW); label(\"2\", F--E, SE); label(\"2\", E--B, SE); clip(W--X--Y--Z--cycle);[/asy]", "answer": "$12\\sqrt{2}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 368, @@ -2941,7 +3308,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(12,0), C=(9,5); draw(A--B--C--cycle); label(\"$A$\", A, SW); label(\"$B$\", B, SE); label(\"$C$\", C, N); label(\"$a$\", B--C, dir(B--C)*dir(-90)); label(\"$b$\", A--C, dir(C--A)*dir(-90)); label(\"$c$\", A--B, dir(A--B)*dir(-90));[/asy]", "answer": "$35$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 369, @@ -2949,7 +3317,8 @@ "geo_code": "[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13pt)); path table = origin--(1,0)--(1,6)--(6,6)--(6,0)--(7,0)--(7,7)--(0,7)--cycle; path block = origin--(3,0)--(3,1.5)--(0,1.5)--cycle; path rotblock = origin--(1.5,0)--(1.5,3)--(0,3)--cycle; draw(table^^shift((14,0))*table); filldraw(shift((7,0))*block^^shift((5.5,7))*rotblock^^shift((21,0))*rotblock^^shift((18,7))*block,gray); draw((7.25,1.75)--(8.5,3.5)--(8.5,8)--(7.25,9.75),Arrows(size=5)); draw((21.25,3.25)--(22,3.5)--(22,8)--(21.25,8.25),Arrows(size=5)); unfill((8,5)--(8,6.5)--(9,6.5)--(9,5)--cycle); unfill((21.5,5)--(21.5,6.5)--(23,6.5)--(23,5)--cycle); label(\"$r$\",(8.5,5.75)); label(\"$s$\",(22,5.75)); [/asy]", "answer": "$30$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 370, @@ -2957,7 +3326,8 @@ "geo_code": "[asy] draw((0,0)--(2,0)--(2.5,.87)--(1.5,2.6)--cycle, linewidth(1)); draw((2,0)--(3,0)--(2.5,.87)); label(\"6\", (0.75,1.3), NW); label(\"2\", (2.5, 0), S); label(\"2\", (2.75,.44), NE); label(\"A\", (1.5,2.6), N); label(\"B\", (3,0), S); label(\"C\", (0,0), W); label(\"D\", (2.5,.87), NE); label(\"E\", (2,0), S); [/asy]", "answer": "$16$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 371, @@ -2965,7 +3335,8 @@ "geo_code": "[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label(\"O\",(-2.5,0),W); label(\"A\",A,W); label(\"B\",B,S); label(\"C\",C,W); label(\"D\",D,E); label(\"0\",(-1.8,0),W); label(\"20\",(-1.7,.5),NW); label(\"160\",(1.6,.5),NE); label(\"180\",(1.7,0),E); [/asy]", "answer": "$140$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 372, @@ -2973,7 +3344,8 @@ "geo_code": "[asy] draw((0,0)--(1,0)--(1,4)--(0,4)--(0,0)--(0,1)--(-1,1)--(-1,2)); draw((-1,2)--(0,2)--(0,4)--(-1,4)--(-1,5)--(1,5)--(1,6)--(0,6)); draw((0,6)--(0,5)--(3,5)--(3,6)--(4,6)--(4,2)--(5,2)); draw((5,2)--(5,1)--(1,1)--(3,1)--(3,0)--(4,0)--(4,1)); draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)); draw((3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4)); [/asy]", "answer": "$64$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 373, @@ -2981,7 +3353,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,0), B=(16,0), C=(16,16), D=(0,16), E=(32,0), F=(48,0), G=(48,16), H=(32,16), I=(0,8), J=(10,8), K=(10,16), L=(32,6), M=(40,6), N=(40,16); draw(A--B--C--D--A^^E--F--G--H--E^^I--J--K^^L--M--N); label(\"S\", (18,8)); label(\"S\", (50,8)); label(\"Figure 1\", (A+B)/2, S); label(\"Figure 2\", (E+F)/2, S); label(\"10'\", (I+J)/2, S); label(\"8'\", (12,12)); label(\"8'\", (L+M)/2, S); label(\"10'\", (42,11)); label(\"table\", (5,12)); label(\"table\", (36,11)); [/asy]", "answer": "$9\\sqrt{2}$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 374, @@ -2989,7 +3362,8 @@ "geo_code": "[asy] defaultpen(linewidth(0.7)+fontsize(10)); pair H=origin, B=(1,-(1/sqrt(3))), C=(-1,-(1/sqrt(3))), A=(0,(2/sqrt(3))), E=(2,-(2/sqrt(3))), F=(-2,-(2/sqrt(3))), D=(0,(4/sqrt(3))); draw(A--B--C--A^^D--E--F--D); label(\"A'\", A, N); label(\"B'\", B, SE); label(\"C'\", C, SW); label(\"A\", D, E); label(\"B\", E, E); label(\"C\", F, W); [/asy]", "answer": "$4$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 375, @@ -2997,7 +3371,8 @@ "geo_code": "[asy] defaultpen(fontsize(10)+0.8); size(175); pair A,B,C,D,M,P,Q; C=origin; B=(8,0); D=IP(CR(C,6.5),CR(B,8)); A=(4,-3); P=midpoint(A--B); Q=midpoint(C--D); draw(B--C--D--B--A--C^^A--D); draw(D--P--C^^P--Q, gray+dashed+0.5); pen p=fontsize(12)+linewidth(3); dot(\"$A$\",A,down,p); dot(\"$B$\",B,right,p); dot(\"$C$\",C,left,p); dot(\"$D$\",D,up,p); dot(\"$M$\",P,dir(-45),p); dot(\"$N$\",Q,0.2*(Q-P),p); label(\"$27$\",B--D,2*dir(30),fontsize(10)); label(\"$7$\",A--C,2*dir(210),fontsize(10)); label(\"$18$\",A--D,1.5*dir(30),fontsize(10)); label(\"$36$\",(3,0),up,fontsize(10)); [/asy]", "answer": "$137$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 376, @@ -3005,7 +3380,8 @@ "geo_code": "[asy] draw((0,0)--(0,3)--(3,3)--(3,0)--cycle); draw((0,2)--(2,2)--(2,0)); draw((0,1)--(1,1)--(1,0)); draw((0,0)--(3,3)); fill((0,0)--(0,1)--(1,1)--cycle,grey); fill((1,0)--(1,1)--(2,2)--(2,0)--cycle,grey); fill((0,2)--(2,2)--(3,3)--(0,3)--cycle,grey); [/asy]", "answer": "$8$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 377, @@ -3013,7 +3389,8 @@ "geo_code": "[asy] draw((0,2)--(2,2)--(2,1)--(3,1)--(3,0)--(1,0)--(1,1)--(0,1)--cycle,linewidth(1)); draw((1,2)--(1,1)--(2,1)--(2,0),dashed); [/asy]", "answer": "$100$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 378, @@ -3021,7 +3398,8 @@ "geo_code": "[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1)); [/asy]", "answer": "$36$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 379, @@ -3029,7 +3407,8 @@ "geo_code": "[asy] draw((0,0)--(16,0)--(21,5*sqrt(3))--(5,5*sqrt(3))--cycle,dot); draw((5,5*sqrt(3))--(1,5*sqrt(3))--(16,0),dot); MP(\"A\",(0,0),S);MP(\"B\",(16,0),S);MP(\"C\",(21,5sqrt(3)),NE);MP(\"D\",(5,5sqrt(3)),N);MP(\"E\",(1,5sqrt(3)),N); MP(\"16\",(9,0),S);MP(\"10\",(18.5,5sqrt(3)/2),E);MP(\"4\",(3,5sqrt(3)),N); dot((4,4sqrt(3))); MP(\"F\",(4,4sqrt(3)),dir(210)); [/asy]", "answer": "$8$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 380, @@ -3037,7 +3416,8 @@ "geo_code": "[asy] unitsize(36); fill((0,0)--(2,0)--(1,sqrt(3))--cycle,gray); draw((0,0)--(2,0)--(1,sqrt(3))--cycle,linewidth(1)); fill((4,0)--(6,0)--(5,sqrt(3))--cycle,gray); fill((5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--cycle,white); draw((5,sqrt(3))--(4,0)--(5,0)--(9/2,sqrt(3)/2)--(11/2,sqrt(3)/2)--(5,0)--(6,0)--cycle,linewidth(1)); fill((8,0)--(10,0)--(9,sqrt(3))--cycle,gray); fill((9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--cycle,white); fill((17/2,0)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--cycle,white); fill((9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--cycle,white); fill((19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--cycle,white); draw((9,sqrt(3))--(35/4,3*sqrt(3)/4)--(37/4,3*sqrt(3)/4)--(9,sqrt(3)/2)--(35/4,3*sqrt(3)/4)--(33/4,sqrt(3)/4)--(35/4,sqrt(3)/4)--(17/2,0)--(33/4,sqrt(3)/4)--(8,0)--(9,0)--(17/2,sqrt(3)/2)--(19/2,sqrt(3)/2)--(9,0)--(19/2,0)--(37/4,sqrt(3)/4)--(39/4,sqrt(3)/4)--(19/2,0)--(10,0)--cycle,linewidth(1)); label(\"Change 1\",(3,3*sqrt(3)/4),N); label(\"$\\Longrightarrow $\",(3,5*sqrt(3)/8),S); label(\"Change 2\",(7,3*sqrt(3)/4),N); label(\"$\\Longrightarrow $\",(7,5*sqrt(3)/8),S); [/asy]", "answer": "$\\frac{81}{256}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 381, @@ -3045,7 +3425,8 @@ "geo_code": "[asy] draw((0,0)--(2,2)--(2,1)--(5,1)--(5,-1)--(2,-1)--(2,-2)--cycle,dot); MP(\"A\",(0,0),W);MP(\"B\",(2,2),N);MP(\"C\",(2,1),S);MP(\"D\",(5,1),NE);MP(\"E\",(5,-1),SE);MP(\"F\",(2,-1),NW);MP(\"G\",(2,-2),S); MP(\"5\",(2,1.5),E);MP(\"5\",(2,-1.5),E);MP(\"30\",(3.5,1),N);MP(\"30\",(3.5,-1),S);MP(\"10\",(5,0),E); [/asy]", "answer": "$400$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 382, @@ -3053,7 +3434,8 @@ "geo_code": "[asy] for (int a=0; a <= 3; ++a) { for (int b=0; b <= 3-a; ++b) { fill((a,b)--(a,b+1)--(a+1,b)--cycle,grey); } } for (int c=0; c <= 3; ++c) { draw((c,0)--(c,4-c),linewidth(1)); draw((0,c)--(4-c,c),linewidth(1)); draw((c+1,0)--(0,c+1),linewidth(1)); } label(\"$16$\",(2,0),S); label(\"$16$\",(0,2),W); [/asy]", "answer": "$80$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 383, @@ -3061,7 +3443,8 @@ "geo_code": "[asy] draw((-7,0)--(7,0),black+linewidth(.75)); draw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75)); draw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75)); [/asy]", "answer": "$4\\sqrt{3}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 384, @@ -3069,7 +3452,8 @@ "geo_code": "[asy] draw(circle((0,0),18),black+linewidth(.75)); draw(circle((0,0),6),black+linewidth(.75)); draw((-18,0)--(18,0)--(-14,8*sqrt(2))--cycle,black+linewidth(.75)); dot((-18,0));dot((18,0));dot((-14,8*sqrt(2))); MP(\"A\",(-18,0),W);MP(\"C\",(18,0),E);MP(\"B\",(-14,8*sqrt(2)),W); [/asy]", "answer": "$6\\sqrt{5}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "length" }, { "index": 385, @@ -3077,7 +3461,8 @@ "geo_code": "[asy] draw((1,0)--(2*cos(pi/8),2*sin(pi/8))--(cos(pi/4),sin(pi/4))--(2*cos(3*pi/8),2*sin(3*pi/8))--(cos(pi/2),sin(pi/2))--(2*cos(5*pi/8),2*sin(5*pi/8))--(cos(3*pi/4),sin(3*pi/4))--(2*cos(7*pi/8),2*sin(7*pi/8))--(-1,0),black+linewidth(.75)); MP(\"A_1\",(2*cos(5*pi/8),2*sin(5*pi/8)),N);MP(\"A_2\",(2*cos(3*pi/8),2*sin(3*pi/8)),N);MP(\"A_3\",(2*cos(1*pi/8),2*sin(1*pi/8)),N); MP(\"A_n\",(2*cos(7*pi/8),2*sin(7*pi/8)),N); MP(\"B_1\",(cos(4*pi/8),sin(4*pi/8)),S);MP(\"B_2\",(cos(2*pi/8),sin(2*pi/8)),S);MP(\"B_n\",(cos(6*pi/8),sin(6*pi/8)),S); [/asy]", "answer": "$18$", "category": "Global Abstract Integration", - "source": "HARP" + "source": "HARP", + "problem_type": "count" }, { "index": 386, @@ -3085,7 +3470,8 @@ "geo_code": "[asy] fill((0,0)--(20,0)--(20,5)--(0,5)--cycle,lightgray); fill((20,0)--(20+5*sqrt(2),5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)--cycle,lightgray); draw((0,0)--(20,0)--(20,5)--(0,5)--cycle); draw((0,5)--(5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5+5*sqrt(2))--(20,5)); draw((20+5*sqrt(2),5+5*sqrt(2))--(20+5*sqrt(2),5*sqrt(2))--(20,0)); draw((5*sqrt(2),5+5*sqrt(2))--(5*sqrt(2),5*sqrt(2))--(5,5),dashed); draw((5*sqrt(2),5*sqrt(2))--(15+5*sqrt(2),5*sqrt(2)),dashed); [/asy]", "answer": "$500$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 387, @@ -3093,7 +3479,8 @@ "geo_code": "[asy] pair A,B,C,D,EE,F; A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10); draw(A--C--D--EE--cycle); draw(B--D--F); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); label(\"$A$\",A,NW); label(\"$B$\",B,N); label(\"$C$\",C,NE); label(\"$D$\",D,SE); label(\"$E$\",EE,SW); label(\"$F$\",F,W); [/asy]", "answer": "$306$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 388, @@ -3101,7 +3488,8 @@ "geo_code": "[asy] draw((-5,0)--(5,0)--(2,14)--cycle,black+linewidth(.75)); draw((-2.25,5.5)--(4,14/3),black+linewidth(.75)); MP(\"A\",(-5,0),S);MP(\"C\",(5,0),S);MP(\"B\",(2,14),N);MP(\"E\",(4,14/3),E);MP(\"D\",(-2.25,5.5),W); MP(\"55^\\circ\",(-4.5,0),NE);MP(\"75^\\circ\",(5,0),NW); [/asy]", "answer": "$115$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 389, @@ -3109,7 +3497,8 @@ "geo_code": "[asy] draw((-1,0)--(1,0)--(1+sqrt(2),sqrt(2))--(0,sqrt(2)+sqrt(13-2*sqrt(2)))--(-1-sqrt(2),sqrt(2))--cycle,black+linewidth(.75)); MP(\"A\",(-1,0),SW);MP(\"B\",(1,0),SE);MP(\"C\",(1+sqrt(2),sqrt(2)),E);MP(\"D\",(0,sqrt(2)+sqrt(13-2*sqrt(2))),N);MP(\"E\",(-1-sqrt(2),sqrt(2)),W); dot((-1,0));dot((1,0));dot((1+sqrt(2),sqrt(2)));dot((-1-sqrt(2),sqrt(2)));dot((0,sqrt(2)+sqrt(13-2*sqrt(2)))); [/asy]", "answer": "$\\frac{3}{4}$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "ratio" }, { "index": 390, @@ -3117,7 +3506,8 @@ "geo_code": "[asy] draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75)); draw((0,-1)--(0,1), black+linewidth(.75)); draw((-1,0)--(1,0), black+linewidth(.75)); draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75)); draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75)); draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75)); draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75)); [/asy]", "answer": "$2\\sqrt{3}-2$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 391, @@ -3125,7 +3515,8 @@ "geo_code": "[asy] pair A,B,C,D,EE; A = origin; B = (2,0); C = (5,0); EE = (1.5,3); D = (1.75,1.5); draw(A--C--D); draw(B--EE--A); dot(A); dot(B); dot(C); dot(D); dot(EE); label(\"$A$\",A,SW); label(\"$B$\",B,S); label(\"$C$\",C,SE); label(\"$D$\",D,NE); label(\"$E$\",EE,N); [/asy]", "answer": "$130$", "category": "Primitive Recognition", - "source": "HARP" + "source": "HARP", + "problem_type": "angle" }, { "index": 392, @@ -3133,7 +3524,8 @@ "geo_code": "[asy] pair A,B,C,D; A = origin; B = (4,0); C = (4,4); D = (0,4); draw(A--B--C--D--cycle); draw(arc((2,1),(1,1),(3,1),CCW)--arc((3,2),(3,1),(3,3),CCW)--arc((2,3),(3,3),(1,3),CCW)--arc((1,2),(1,3),(1,1),CCW)); draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); dot(A); dot(B); dot(C); dot(D); dot((1,1)); dot((3,1)); dot((1,3)); dot((3,3)); label(\"$A$\",A,SW); label(\"$B$\",B,SE); label(\"$C$\",C,NE); label(\"$D$\",D,NW); [/asy]", "answer": "$256$", "category": "Local Relation Composition", - "source": "HARP" + "source": "HARP", + "problem_type": "area" }, { "index": 393, @@ -3141,7 +3533,8 @@ "geo_code": "[asy] import graph; unitsize(0.1cm); pair A = (0,0);pair B = (70,0);pair C = (70,16);pair D = (0,16);pair E = (3,16);pair F = (90,16);pair G = (90,33);pair H = (3,33); dot(A^^B^^C^^D^^E^^F^^G^^H); label(\"$A$\", A, S);label(\"$B$\", B, S);label(\"$C$\", C, N);label(\"$D$\", D, N);label(\"$E$\", E, S);label(\"$F$\", F, S);label(\"$G$\", G, N);label(\"$H$\", H, N); draw(E--D--A--B--C--E--H--G--F--C); [/asy]", "answer": "$104$", "category": "Local Relation Composition", - "source": "AIME-24" + "source": "AIME-24", + "problem_type": "length" }, { "index": 394, @@ -3149,7 +3542,8 @@ "geo_code": "[asy] size(10cm); usepackage(\"tikz\");label(\"\\begin{tikzpicture}[scale=.5]\\draw(0,0)grid(8,8);\\draw[line width=2,red](0,0)--(2,0)--(2,3)--(5,3)--(5,8)--(8,8);\\end{tikzpicture}\",origin); label(\"\\begin{tikzpicture}[scale=.5]\\draw(0,0)grid(8,8);\\draw[line width=2,red](0,0)--(0,3)--(3,3)--(3,5)--(8,5)--(8,8);\\end{tikzpicture}\",E); [/asy]", "answer": "$294$", "category": "Local Relation Composition", - "source": "AIME-24" + "source": "AIME-24", + "problem_type": "count" }, { "index": 395, @@ -3157,7 +3551,8 @@ "geo_code": "[asy] pair A = (2,1); pair B = (0,0); pair C = (3,0); dot(A^^B^^C); label(\"$A$\", A, N); label(\"$B$\", B, S); label(\"$C$\", C, S); draw(A--B--C--cycle); for(real i=0.62; i<2.7; i+=0.29){ draw(circle((i,0.145), 0.145)); } [/asy]", "answer": "$197$", "category": "Global Abstract Integration", - "source": "AIME-24" + "source": "AIME-24", + "problem_type": "count" }, { "index": 396, @@ -3165,7 +3560,8 @@ "geo_code": "[asy] unitsize(0.3 inch); draw(ellipse((0,0), 3, 1.75)); draw((-1.2,0.1)..(-0.8,-0.03)..(-0.4,-0.11)..(0,-0.15)..(0.4,-0.11)..(0.8,-0.03)..(1.2,0.1)); draw((-1,0.04)..(-0.5,0.12)..(0,0.16)..(0.5,0.12)..(1,0.04)); draw((0,2.4)--(0,-0.15)); draw((0,-0.15)--(0,-1.75), dashed); draw((0,-1.75)--(0,-2.25)); draw(ellipse((2,0), 1, 0.9)); draw((2.03,-0.02)--(2.9,-0.4)); [/asy]", "answer": "$127$", "category": "Global Abstract Integration", - "source": "AIME-24" + "source": "AIME-24", + "problem_type": "count" }, { "index": 397, @@ -3173,7 +3569,8 @@ "geo_code": "[asy] unitsize(0.6 inch); for(int i=0; i<360; i+=30) { dot(dir(i), 4+black); draw(dir(i)--dir(i+30)); } draw(dir(120)--dir(330)); filldraw(dir(210)--dir(240)--dir(30)--dir(60)--cycle, mediumgray, linewidth(1.5)); draw((0,0.366)--(0.366,0), linewidth(1.5)); [/asy]", "answer": "$315$", "category": "Global Abstract Integration", - "source": "AIME-24" + "source": "AIME-24", + "problem_type": "count" }, { "index": 398, @@ -3181,7 +3578,8 @@ "geo_code": "[asy]\nfor ( int i = 1; i <= 7; ++i )\n{\n\ndraw((i,0)--(i,6));\n}\n\nfor ( int i = 1; i <= 5; ++i )\n{\n\ndraw((0,i)--(8,i));\n}\ndraw((-0.5,0)--(8,0), linewidth(1));\ndraw((0,-0.5)--(0,6), linewidth(1));\nlabel(\"$O$\", (0,0), SW);\nlabel(scale(.85)*rotate(90)*\"distance\", (0, 3), W);\nlabel(scale(.85)*\"time\", (4, 0), S);\ndot((1.25, 4.5));\nlabel(scale(.85)*\"1\", (1.25, 4.8), N);\ndot((2.5, 2.2));\nlabel(scale(.85)*\"2\", (2.5, 2.2), S);\ndot((4.25,5.2));\nlabel(scale(.85)*\"3\", (4.25, 5.2), SE);\ndot((5.6, 2.8));\nlabel(scale(.85)*\"4\", (5.6, 2.8), N);\ndot((6.8, 1.4));\nlabel(scale(.85)*\"5\", (6.8, 1.4), E);\n[/asy]", "answer": "$1$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 399, @@ -3189,7 +3587,8 @@ "geo_code": "[asy]\nsize(120);\ndraw(shift(2.2,0)*yscale(0.3)*Circle((0,0), 1.2));\n\ndraw((1,0)--(1,-2));\ndraw((3.4,0)--(3.4,-2));\n\ndraw((1,-2)..(2.2,-2.36)..(3.4,-2));\n\nlabel(\"$h$\",midpoint((3.4,0)--(3.4,-2)),E);\n\ndraw (((2.2,0)--(3.4,0)));\n\nlabel(\"$r=3$\",midpoint((2.2,0)--(3.4,0)),N);\n\n[/asy]", "answer": "$5$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 400, @@ -3197,7 +3596,8 @@ "geo_code": "[asy]\npair D,E,F;\nF = (0,0);\nD = (7,7);\nE = (0,7);\ndraw(D--E--F--D);\ndraw(rightanglemark(D,E,F,15));\nlabel(\"$D$\",D,NE);\nlabel(\"$E$\",E,NW);\nlabel(\"$F$\",F,SW);\nlabel(\"$7$\",(E+F)/2,W);\n[/asy]", "answer": "$\\sqrt{51}$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 401, @@ -3205,7 +3605,8 @@ "geo_code": "[asy]\nunitsize(0.6 cm);\n\npair C, W, Z;\n\nZ = (2 + sqrt(2), -3 - 3*sqrt(2));\nC = (2,-3);\nW = rotate(45,C)*(Z);\n\ndraw(Z--C--W);\n\ndot(\"$c$\", C, N);\ndot(\"$w$\", W, SE);\ndot(\"$z$\", Z, S);\nlabel(\"$\\frac{\\pi}{4}$\", C + (0.6,-1));\n[/asy]", "answer": "$6 - 5i$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 402, @@ -3213,7 +3614,8 @@ "geo_code": "[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]", "answer": "$\\pi$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 403, @@ -3221,7 +3623,8 @@ "geo_code": "[asy]\ndraw((0,0)--(10,0));\ndraw((0,3)--(10,3));\ndraw((2,3)--(8,0));\ndraw((2,3)--(4,0));\nlabel(\"$A$\",(2,3),N);\nlabel(\"$B$\",(4,0),S);\nlabel(\"$C$\",(8,0),S);\nlabel(\"$124^{\\circ}$\",(2,3),SW);\nlabel(\"$x^{\\circ}$\",(4.5,3),S);\n[/asy]", "answer": "$28$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 404, @@ -3229,7 +3632,8 @@ "geo_code": "[asy]size(80); pair A = dir(120); pair B=dir(60); pair M=(A+B)/2; draw(dir(360)--B--A--dir(180)--dir(240)--dir(300)--cycle); label(\"16 cm\", M, N);[/asy]", "answer": "$12$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 405, @@ -3237,7 +3641,8 @@ "geo_code": "[asy]\nsize(150);\npair A , B, C, D; A = (0,0); B = (2, 4); C = (7,4); D = (7, -2);\ndraw( (0,0)--(2,4) -- (7,4) -- (7, -2)-- cycle);\nlabel(\"$A$\", A, SW);\nlabel(\"$B$\", B, NW);\nlabel(\"$C$\", C, NE);\nlabel(\"$D$\", D, SE);\npair E, F;\nE = (4.5-.2,1-.2); F = (5, 3);\ndraw(A--E--D); draw(A--F--D);\nlabel(\"$E$\", E, N); label(\"$F$\", F, NW);\ndot(A);dot(B);dot(C);dot(D);dot(E);dot(F);\nlabel(\"$x$\", (1, 1.5), S); label(\"$x$\", (2, 1), S+W); label(\"$x$\", (2, -1), N+N+N+W);\nlabel(\"$y$\", (5.5+.3, .5-.3), S); label(\"$y$\", (6.5+.3, 0)); label(\"$y$\", (5+.5, -1.5+.3));\nlabel(\"$110^{\\circ}$\",(2.5,3.5)); label(\"$100^{\\circ}$\",(6.5-.2,3.5));\n[/asy]", "answer": "$80$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 406, @@ -3245,7 +3650,8 @@ "geo_code": "[asy]\npair pA, pB, pC, pD, pE, pF, pO;\npO = (0, 0);\npA = pO + dir(-10);\npB = pO + dir(60);\npC = pO + dir(130);\npD = pO + dir(170);\npE = pO + dir(-160);\npF = pO + dir(-80);\ndraw(pA--pB--pC--pD--pE--pF--pA);\nlabel(\"$105^\\circ$\", pF, N * 2);\nlabel(\"$110^\\circ$\", pB, SW * 1.5);\nlabel(\"$\\alpha$\", pD, E);\ndraw(circle(pO, 1));\n[/asy]", "answer": "$145$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 407, @@ -3253,7 +3659,8 @@ "geo_code": "[asy]\nsize(180); defaultpen(linewidth(.7pt)+fontsize(10pt));\npair A, B, C, D, E, F;\nA=(0,6);\nB=(0,0);\nC=(9,0);\nD=(0,3);\nE=(4,0);\nF=(3,2);\ndraw(E--A--C--D);\ndraw((-1,0)--(10,0), EndArrow);\ndraw((0,-1)--(0,8), EndArrow);\nlabel(\"$A(0,6)$\", A, W);\nlabel(\"$B(0,0)$\", B, SW);\nlabel(\"$C(8,0)$\", C, S);\nlabel(\"$D$\", D, W);\nlabel(\"$E$\", E, S);\nlabel(\"$F$\", F, SW);\nlabel(\"$x$\", (10,0), dir(0));\nlabel(\"$y$\", (0,8), dir(90));\n[/asy]", "answer": "$8$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 408, @@ -3261,7 +3668,8 @@ "geo_code": "[asy]\nsize(50);\nfor (int i=0; i<3; ++i) {\nfor (int j=0; j<3; ++j) {\ndot((i,j));};}\n[/asy]", "answer": "$\\frac{2}{21}$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "ratio" }, { "index": 409, @@ -3269,7 +3677,8 @@ "geo_code": "[asy]\nunitsize(2 cm);\n\npair A, B, O, P;\n\nA = (0.4,0);\nB = (1.2,0);\nO = (0,0);\nP = (0,1);\n\ndraw((-0.5,0)--(2,0));\ndraw(O--P);\ndraw(P--A);\ndraw(P--B);\n\nlabel(\"$A$\", A, S);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, S);\nlabel(\"$P$\", P, N);\n[/asy]", "answer": "$30$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 410, @@ -3277,7 +3686,8 @@ "geo_code": "[asy]\nfor(int i=0; i <=7; ++i) {\ndraw(dir(360*i/7+90)--dir(360*(i+1)/7+90));\n}\npair A = dir(360*3/7+90);\npair F = dir(360*4/7+90);\npair C = A+dir(90)*(F-A);\npair D = C+F-A;\npair B = dir(360*2/7+90);\n\ndraw(A--C--D--F);\n\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,W);\nlabel(\"$C$\",C,SE);\nlabel(\"$D$\",F,S);\n\n[/asy]", "answer": "$\\frac{270}7$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 411, @@ -3285,7 +3695,8 @@ "geo_code": "[asy]\n\nunitsize(0.4 inch);\n\ndot((0,0),linewidth(9bp));\ndot((1,0),linewidth(9bp));\ndot((2,0),linewidth(9bp));\ndot((0,1),linewidth(9bp));\ndot((0,2),linewidth(9bp));\ndot((1,1),linewidth(9bp));\ndot((2,1),linewidth(9bp));\ndot((1,2),linewidth(9bp));\ndot((2,2),linewidth(9bp));\n\nfilldraw((2.95,-0.05)--(3.05,-0.05)--(3.05,0.05)--(2.95,0.05)--cycle,black);\nfilldraw((2.45,-0.05)--(2.55,-0.05)--(2.55,0.05)--(2.45,0.05)--cycle,black);\nfilldraw((3.45,-0.05)--(3.55,-0.05)--(3.55,0.05)--(3.45,0.05)--cycle,black);\n\nfilldraw((2.95,0.95)--(3.05,0.95)--(3.05,1.05)--(2.95,1.05)--cycle,black);\nfilldraw((2.45,0.95)--(2.55,0.95)--(2.55,1.05)--(2.45,1.05)--cycle,black);\nfilldraw((3.45,0.95)--(3.55,0.95)--(3.55,1.05)--(3.45,1.05)--cycle,black);\n\nfilldraw((2.95,1.95)--(3.05,1.95)--(3.05,2.05)--(2.95,2.05)--cycle,black);\nfilldraw((2.45,1.95)--(2.55,1.95)--(2.55,2.05)--(2.45,2.05)--cycle,black);\nfilldraw((3.45,1.95)--(3.55,1.95)--(3.55,2.05)--(3.45,2.05)--cycle,black);\n\ndot((4,0),linewidth(9bp));\ndot((5,0),linewidth(9bp));\ndot((4,1),linewidth(9bp));\ndot((5,1),linewidth(9bp));\ndot((4,2),linewidth(9bp));\ndot((5,2),linewidth(9bp));\n\n[/asy]", "answer": "$19$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 412, @@ -3293,7 +3704,8 @@ "geo_code": "[asy]\nsize(200);\npair A, B, C, P, Q, R, S;\nR=(0,0);\nQ=(-2,0);\nS=(2,0);\nP=(1,1.732);\nB=(-5.73,-1);\nC=(3.732,-1);\nA=(1.366,3.098);\ndraw(A--B--C--A);\ndraw(circle(P, 1));\ndraw(circle(Q, 1));\ndraw(circle(R, 1));\ndraw(circle(S, 1));\nlabel(\"A\", A, N);\nlabel(\"B\", B, SW);\nlabel(\"C\", C, SE);\ndot(P);\ndot(Q);\ndot(R);\ndot(S);\nlabel(\"P\", P, N);\nlabel(\"Q\", Q, SW);\nlabel(\"R\", R, SW);\nlabel(\"S\", S, SE);\n[/asy]", "answer": "$30$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 413, @@ -3301,7 +3713,8 @@ "geo_code": "[asy]\nsize(250);\npair A, B, C, D, O, H, W, X, Y, Z;\nO=(0,0);\nA=(1,1);\nD=(1.5,-.3);\nB=(-1.5,.3);\nC=(-1,-1);\nH=(0,2.5);\nW=(5/3)*(A+D);\nX=(5/3)*(A+B);\nY=(-1)*(W);\nZ=(-1)*(X);\ndraw(W--X--Y--Z--W);\ndraw(A--C);\ndraw(B--D);\ndraw(O--H, linewidth(1));\ndraw(A--H, dashed);\ndraw(B--H, dashed);\ndraw(C--H, dashed);\ndraw(D--H, dashed);\ndot(A);\ndot(B);\ndot(C);\ndot(D);\ndot(O);\ndot(H);\nlabel(\"A\", A, NE);\nlabel(\"B\", B, SW);\nlabel(\"C\", C, SE);\nlabel(\"D\", D, NE);\nlabel(\"O\", O, SE);\nlabel(\"H\", H, NW);\n[/asy]", "answer": "$160$", "category": "Global Abstract Integration", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 414, @@ -3309,7 +3722,8 @@ "geo_code": "[asy]\nsize(4cm);defaultpen(linewidth(0.75));\n\n// Filled portions\nfill((0, 4)--(0, 0)--(2, 0)--cycle, gray(0.75));\nfill((0, 4)--(3, 4)--(3, 0)--cycle, gray(0.75));\n\n// grid\nint j;\nfor (j = 0; j < 4; ++j) {draw((j, 0)--(j, 4));}\nfor (j = 0; j < 5; ++j) {draw((0, j)--(3, j));}\n\n//diagonals\ndraw((0, 4)--(3, 0)); draw((0, 4)--(2, 0));\n[/asy]", "answer": "$10$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 415, @@ -3317,7 +3731,8 @@ "geo_code": "[asy]\nunitsize(1cm);\ndraw(Circle((0,0),2),linewidth(0.7));\ndraw((1.7,1)--(-1.7,-1),linewidth(0.7));\ndraw((1.7,-1)--(-1.7,1),linewidth(0.7));\ndraw((0,2)--(0,-2));\nlabel(\"1\",(0.8,0.5),NW);\nlabel(\"2\",(0.8,-0.5),SW);\nlabel(\"6\",(-0.8,0.5),NE);\nlabel(\"9\",(-0.8,-0.5),SE);\nlabel(\"3\",(-0.7,0),W);\nlabel(\"7\",(0.7,0),E);\ndraw((-2.8,0)--(-2.1,0),Arrow);\nlabel(\"Pointer\",(-2.8,0),W);\nfill((3,0)--(3,1)--(4,1)--(4,0)--cycle,gray(0.7));\nfill((3,-2)--(3,-1)--(4,-1)--(4,-2)--cycle,gray(0.7));\nfill((4,1)--(4,2)--(5,2)--(5,1)--cycle,gray(0.7));\nfill((4,-1)--(4,0)--(5,0)--(5,-1)--cycle,gray(0.7));\nfill((5,0)--(5,1)--(6,1)--(6,0)--cycle,gray(0.7));\nfill((5,-2)--(5,-1)--(6,-1)--(6,-2)--cycle,gray(0.7));\ndraw((3,-2)--(3,2)--(6,2)--(6,-2)--cycle,linewidth(0.7));\ndraw((3,-1)--(6,-1),linewidth(0.7));\ndraw((3,0)--(6,0),linewidth(0.7));\ndraw((3,1)--(6,1),linewidth(0.7));\ndraw((4,-2)--(4,2),linewidth(0.7));\ndraw((5,-2)--(5,2),linewidth(0.7));\nlabel(\"1\",(3.5,-2),S);\nlabel(\"2\",(4.5,-2),S);\nlabel(\"3\",(5.5,-2),S);\nlabel(\"1\",(3,-1.5),W);\nlabel(\"2\",(3,-0.5),W);\nlabel(\"3\",(3,0.5),W);\nlabel(\"4\",(3,1.5),W);\n[/asy]", "answer": "$\\frac{1}{2}$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "ratio" }, { "index": 416, @@ -3325,7 +3740,8 @@ "geo_code": "[asy]\ndraw((-.3,-3)--(.1,1)--(-1,0)--(3,0)--cycle);\nlabel(\"$A$\",(.1,1),N);\nlabel(\"$B$\",(-1,0),W);\nlabel(\"$C$\",(0,0),NE);\nlabel(\"$D$\",(3,0),E);\nlabel(\"$E$\",(-.3,-3),S);\n[/asy]", "answer": "$54$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 417, @@ -3333,7 +3749,8 @@ "geo_code": "[asy]\ndraw((0,0)--(10,0),black+linewidth(1));\ndraw((0,0)--(10,0),MidArrow);\ndraw((10,0)--(20,0),black+linewidth(1));\ndraw((0,0)--(-7,10)--(7,10)--(10,0),black+linewidth(1));\ndraw((-5,10)--(7,10),MidArrow);\nlabel(\"$x^{\\circ}$\",(-6,10),SE);\nlabel(\"$2x^{\\circ}$\",(7,10),SW);\nlabel(\"$128^{\\circ}$\",(10,0),NE);\nlabel(\"$P$\",(-7,10),N);\nlabel(\"$T$\",(7,10),N);\nlabel(\"$R$\",(10,0),S);\nlabel(\"$Q$\",(0,0),S);\n[/asy]", "answer": "$116$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 418, @@ -3341,7 +3758,8 @@ "geo_code": "[asy]\nunitsize(5mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\n\npair A=(0,0), B=(3,0), C=(6,0), D=(9,0), Ep=(9,3), G=(6,3);\npair F0=bisectorpoint(B,2*Ep-B), H0=bisectorpoint(Ep,2*B-Ep);\npair H=extension(B,H0,A,G);\npair F=extension(Ep,F0,A,G);\n\ndraw(H--B--Ep--F--A--D--Ep--G--C);\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,S);\nlabel(\"$D$\",D,S);\nlabel(\"$E$\",Ep,E);\nlabel(\"$F$\",F,N);\nlabel(\"$G$\",G,NW);\nlabel(\"$H$\",H,NW);\n[/asy]", "answer": "$1\\frac{4}{5}$", "category": "Global Abstract Integration", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 419, @@ -3349,7 +3767,8 @@ "geo_code": "[asy]\nsize(150);\nimport graph;\npair J = (0,0), H = (6,0), O, N;\npath circ = Circle(J,3);\npair M = midpoint(J--H);\npath secCirc = Circle(M,3);\npair[] tangentPoints = intersectionpoints(circ,secCirc);\nO = tangentPoints[0]; N = tangentPoints[1];\ndraw(J--N--H--O--cycle);\ndraw(circ);\nlabel(\"$H$\",H,E);\nlabel(\"$J$\",J,W);\nlabel(\"$N$\",N,S);\nlabel(\"$O$\",O,NE);\n[/asy]", "answer": "$180$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 420, @@ -3357,7 +3776,8 @@ "geo_code": "[asy]\nimport graph;\nfilldraw(circle((0,0),7), lightgray, black+linewidth(1));\nfilldraw(circle((0,0),6), gray, black+linewidth(1));\nfilldraw(circle((0,0),4), white, black+linewidth(1));\ndot((0,0));\nlabel(\"$X$\",(2,0));\nlabel(\"$Y$\",(5,0));\nlabel(\"$Z$\",(6.5,0));\n[/asy]", "answer": "$7\\pi$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 421, @@ -3365,7 +3785,8 @@ "geo_code": "[asy]\ndraw((-42.4,30.8)--(-10,30.8)--(0,63.2)--(10,30.8)--(42.4,30.8)--(16.2,11.8)--(24.9,-18.1)--(0,0)--(-24.9,-18.1)--(-16.2,11.8)--cycle,linewidth(1));\ndraw((-10,30.8)--(10,30.8)--(16.2,11.8)--(0,0)--(-16.2,11.8)--cycle,linewidth(1));\nlabel(\"$A$\",(-42.4,30.8),W);\nlabel(\"$F$\",(-10,30.8),NW);\ndot((-10,30.8));\nlabel(\"$G$\",(10,30.8),NE);\ndot((10,30.8));\nlabel(\"$H$\",(16.2,11.8),E);\ndot((16.2,11.8));\nlabel(\"$I$\",(0,0),S);\ndot((0,0));\nlabel(\"$J$\",(-16.2,11.8),WSW);\ndot((-16.2,11.8));\n[/asy]", "answer": "$36$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 422, @@ -3373,7 +3794,8 @@ "geo_code": "[asy]\n/* AMC8 2002 #8, 9, 10 Problem */\nsize(3inch, 1.5inch);\nfor ( int y = 0; y <= 5; ++y )\n{\n\ndraw((0,y)--(18,y));\n}\ndraw((0,0)--(0,5));\ndraw((6,0)--(6,5));\ndraw((9,0)--(9,5));\ndraw((12,0)--(12,5));\ndraw((15,0)--(15,5));\ndraw((18,0)--(18,5));\n\nlabel(scale(0.8)*\"50s\", (7.5,4.5));\nlabel(scale(0.8)*\"4\", (7.5,3.5));\nlabel(scale(0.8)*\"8\", (7.5,2.5));\nlabel(scale(0.8)*\"6\", (7.5,1.5));\nlabel(scale(0.8)*\"3\", (7.5,0.5));\n\nlabel(scale(0.8)*\"60s\", (10.5,4.5));\nlabel(scale(0.8)*\"7\", (10.5,3.5));\nlabel(scale(0.8)*\"4\", (10.5,2.5));\nlabel(scale(0.8)*\"4\", (10.5,1.5));\nlabel(scale(0.8)*\"9\", (10.5,0.5));\n\nlabel(scale(0.8)*\"70s\", (13.5,4.5));\nlabel(scale(0.8)*\"12\", (13.5,3.5));\nlabel(scale(0.8)*\"12\", (13.5,2.5));\nlabel(scale(0.8)*\"6\", (13.5,1.5));\nlabel(scale(0.8)*\"13\", (13.5,0.5));\n\nlabel(scale(0.8)*\"80s\", (16.5,4.5));\nlabel(scale(0.8)*\"8\", (16.5,3.5));\nlabel(scale(0.8)*\"15\", (16.5,2.5));\nlabel(scale(0.8)*\"10\", (16.5,1.5));\nlabel(scale(0.8)*\"9\", (16.5,0.5));\n\nlabel(scale(0.8)*\"Country\", (3,4.5));\nlabel(scale(0.8)*\"Brazil\", (3,3.5));\nlabel(scale(0.8)*\"France\", (3,2.5));\nlabel(scale(0.8)*\"Peru\", (3,1.5));\nlabel(scale(0.8)*\"Spain\", (3,0.5));\n\nlabel(scale(0.9)*\"Juan's Stamp Collection\", (9,0), S);\nlabel(scale(0.9)*\"Number of Stamps by Decade\", (9,5), N);\n[/asy]", "answer": "$5.4$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 423, @@ -3381,7 +3803,8 @@ "geo_code": "[asy]\n\npair R,P,Q,SS;\n\nSS = (-2,0);\n\nP = (0,0);\n\nQ = (2,0);\n\nR = rotate(aSin(7/25))*(1.5,0);\n\ndot(\"$S$\",SS,S);\n\ndot(\"$Q$\",Q,S);\n\ndot(\"$R$\",R,N);\n\ndot(\"$P$\",P,S);\n\ndraw(Q--SS);\n\ndraw(P--R);\n\n[/asy]", "answer": "$-\\frac{24}{25}$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 424, @@ -3389,7 +3812,8 @@ "geo_code": "[asy]\nsize(4cm);\ndefaultpen(linewidth(0.75));\nreal adc = 100;\npair d = (0, 0); pair a = 2 * dir(100); pair c = (2, 0);\npath inner = arc(d, a/2, c/2, CW);\n\npath outer = arc(d, c, a, CCW);\nguide region1 = (a--a/2)..inner..(c/2--c)..outer..cycle;\nguide region2 = arc(d, a/2, c/2, CCW)..(c/2--d--a/2)..cycle;\nfill(region1, gray(0.75));\nfill(region2, gray(0.75));\ndraw(unitcircle); draw(scale(2) * unitcircle);\ndraw(a--d--c);\nlabel(\"$A$\", a, N); label(\"$C$\", c, E); label(\"$D$\", d, NE);\n[/asy]", "answer": "$120$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 425, @@ -3397,7 +3821,8 @@ "geo_code": "[asy]\nunitsize(0.4 cm);\n\npair A, B, C, D, M;\n\nA = (0,11);\nD = (0,1);\nB = (-11/2,0);\nC = (11/2,0);\nM = (B + C)/2;\n\ndraw(A--B--C--cycle);\ndraw(A--M);\ndraw(B--D--C);\n\nlabel(\"$A$\", A, N);\nlabel(\"$B$\", B, SW);\nlabel(\"$C$\", C, SE);\nlabel(\"$D$\", D, NW);\nlabel(\"$M$\", M, S);\n[/asy]", "answer": "$11 \\sqrt{5} + 11$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 426, @@ -3405,7 +3830,8 @@ "geo_code": "[asy]\nunitsize(0.4 cm);\n\npair A, B, L, R;\nint i, n;\n\nfor (i = -8; i <= 8; ++i) {\n draw((i,-8)--(i,8),gray(0.7));\n draw((-8,i)--(8,i),gray(0.7));\n}\n\ndraw((-8,0)--(8,0),Arrows(6));\ndraw((0,-8)--(0,8),Arrows(6));\n\nA = (-5,4);\nB = (-1,3);\nL = extension(A, B, (-8,0), (-8,1));\nR = extension(A, B, (8,0), (8,1));\n\ndraw(L--R, red);\n\nlabel(\"$x$\", (8,0), E);\nlabel(\"$y$\", (0,8), N);\n[/asy]", "answer": "$\\frac{7}{4}$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 427, @@ -3413,7 +3839,8 @@ "geo_code": "[asy]\ndefaultpen(linewidth(0.7));\ndraw((0,0)--(27,0)--(15,9)--(0,9)--cycle);\nlabel(\"5 cm\",(19,4.5),NE);\nlabel(\"5 cm\",(7.5,9),N);\nlabel(\"3 cm\",(0,4.5),W);\nlabel(\"9 cm\",(13.5,0),S);\ndraw(rightanglemark((0,9),(0,0),(27,0),35));\ndraw(rightanglemark((0,0),(0,9),(15,9),35));\n[/asy]", "answer": "$21$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 428, @@ -3421,7 +3848,8 @@ "geo_code": "[asy]\ndraw((0,0)--(10,0)--(8.2635,9.8481)--cycle,black+linewidth(1));\ndraw((10,0)--(20,0)--(8.2635,9.8481),black+linewidth(1));\ndraw((5,-0.5)--(5,0.5),black+linewidth(1));\ndraw((15,-0.5)--(15,0.5),black+linewidth(1));\ndraw((8.6318,4.8359)--(9.6317,5.0122),black+linewidth(1));\nlabel(\"$A$\",(8.2635,9.8481),N);\nlabel(\"$B$\",(0,0),SW);\nlabel(\"$C$\",(20,0),SE);\nlabel(\"$D$\",(10,0),S);\n[/asy]", "answer": "$90$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 429, @@ -3429,7 +3857,8 @@ "geo_code": "[asy]\nunitsize(1 cm);\n\nfilldraw((0,0)--(3,0)--(3,2)--(0,2)--cycle,lightgreen);\ndraw((0,0)--(3,0),linewidth(2*bp));\ndraw((0,2)--(3,2),linewidth(2*bp));\ndraw(arc((3,1),1,-90,90),linewidth(2*bp));\ndraw(arc((0,1),1,90,270),linewidth(2*bp));\n[/asy]", "answer": "$\\frac{20000}{\\pi}$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 430, @@ -3437,7 +3866,8 @@ "geo_code": "[asy]\npair R,S,T;\nT = (0,0);\nS = (2,0);\nR = (2,sqrt(21));\ndraw(R--S--T--R);\ndraw(rightanglemark(R,S,T,10));\nlabel(\"$T$\",T,SW);\nlabel(\"$S$\",S,SE);\nlabel(\"$R$\",R,NE);\nlabel(\"$5$\",(R+T)/2,NW);\n[/asy]", "answer": "$\\frac{\\sqrt{21}}{5}$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 431, @@ -3445,7 +3875,8 @@ "geo_code": "[asy]\nimport three;\n\nsize(125);\ncurrentprojection = perspective(6,3,1);\n\ntriple A, B, C, D, E, F, P;\n\nA = (1,0,0);\nB = (-1,0,0);\nC = (0,1,0);\nD = (0,-1,0);\nE = (0,0,1);\nF = (0,0,-1);\nP = (1.2,1.5,1);\n\ndraw(A--P,red);\ndraw(B--P,red);\ndraw(C--P,red);\ndraw(D--P,red);\ndraw(E--P,red);\ndraw(F--P,red);\n\ndraw(A--C);\ndraw(A--D);\ndraw(A--E);\ndraw(A--F);\ndraw(C--E--D--F--cycle);\ndraw(D--B--C,dashed);\ndraw(B--C,dashed);\ndraw(B--D,dashed);\ndraw(B--E,dashed);\ndraw(B--F,dashed);\n\nlabel(\"$P$\", P, NE);\n[/asy]", "answer": "$\\sqrt{66}$", "category": "Global Abstract Integration", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 432, @@ -3453,7 +3884,8 @@ "geo_code": "[asy]\nfill((6,0)--(12,6)--(12,0)--cycle,gray(.7));\ndraw((0,0)--(0,12)--(12,12)--(12,0)--cycle,linewidth(1));\ndraw((0,0)--(12,12),linewidth(1));\ndraw((3,3)--(6,0)--(12,6),linewidth(1));\ndraw((0,12)--(9,3)--(9,9),linewidth(1));\nlabel(\"$A$\",(0,12),W);\nlabel(\"$B$\",(12,12),E);\nlabel(\"$C$\",(12,0),E);\nlabel(\"$D$\",(0,0),W);\nlabel(\"e\",(6,3));\n\n[/asy]", "answer": "$2$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 433, @@ -3461,7 +3893,8 @@ "geo_code": "[asy]\ndefaultpen(linewidth(1));\ndraw((0,0)--(1,0)--(1,1)--(0,1)--cycle);\npair a = (1.25,0)+1.25*dir(60);\npair b = a+1.25*dir(-60);\ndraw((1.25,0)--a--b--cycle);\n\n[/asy]", "answer": "$6\\sqrt{2}$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 434, @@ -3469,7 +3902,8 @@ "geo_code": "[asy]\nsize(200);\ndraw((-250,100)--(250,100)--(250,-100)--(-250,-100)--cycle);\ndot((0,0));\nlabel(\"$O$\",(0,0),N);\nlabel(\"$A$\",(-250,100),NW); label(\"$B$\",(250,100),NE); label(\"$C$\",(250,-100),SE); label(\"$D$\",(-250,-100),SW);[/asy]", "answer": "$\\frac{1}{2}$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "ratio" }, { "index": 435, @@ -3477,7 +3911,8 @@ "geo_code": "[asy]\nunitsize(0.8 cm);\n\nreal upperparab (real x) {\n return (sqrt(8*x));\n}\n\nreal lowerparab (real x) {\n return (-sqrt(8*x));\n}\n\npair A, B, C, D;\n\nA = (-1,1);\nB = (2,4);\nC = (-1,-1);\nD = (2,-4);\n\ndraw(graph(upperparab,0,3));\ndraw(graph(lowerparab,0,3));\ndraw(Circle((0,0),sqrt(2)));\ndraw(interp(A,B,-0.2)--interp(A,B,1.2));\ndraw(interp(C,D,-0.2)--interp(C,D,1.2));\ndraw(A--C);\ndraw(B--D);\n\ndot(A);\ndot(B);\ndot(C);\ndot(D);\n[/asy]", "answer": "$15$", "category": "Local Relation Composition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 436, @@ -3485,7 +3920,8 @@ "geo_code": "[asy]\npair P,Q,R,SS,T;\nQ = (0,0);\nR = (1,0);\nP = (1.1,0.5);\nSS = 0.6*P;\nT = R + 0.6*(P-R);\ndraw(T--SS--P--R--Q--SS);\nlabel(\"$P$\",P,N);\nlabel(\"$S$\",SS,NW);\nlabel(\"$Q$\",Q,S);\nlabel(\"$R$\",R,S);\nlabel(\"$T$\",T,ENE);\n[/asy]", "answer": "$75$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "angle" }, { "index": 437, @@ -3493,7 +3929,8 @@ "geo_code": "[asy]\nfill((0,0)--(2,3)--(10,0)--cycle,gray);\ndraw((0,0)--(10,0)--(10,3)--(0,3)--cycle,linewidth(1));\ndraw((0,0)--(2,3)--(10,0),linewidth(1));\nlabel(\"10 cm\",(5,3),N);\nlabel(\"10 cm\",(5,0),S);\nlabel(\"3 cm\",(0,1.5),W);\nlabel(\"3 cm\",(10,1.5),E);\ndraw((0,2.5)--(.5,2.5)--(.5,3));\ndraw((10,2.5)--(9.5,2.5)--(9.5,3));\n[/asy]", "answer": "$15$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "area" }, { "index": 438, @@ -3501,7 +3938,8 @@ "geo_code": "[asy]size(125);\nfor(int i = 0; i<4; ++i)\n{\n\ndraw((0,i)--(3,i),linewidth(1));\n}\n\nfor(int j = 0; j<4; ++j)\n{\n\ndraw((j,0)--(j,3),linewidth(1));\n}\n\nlabel(\"$n-3$\",(.5,.5));\nlabel(\"3\",(.5,1.5));\nlabel(\"$n+1$\",(.5,2.5));\n\nlabel(\"$n+2$\",(1.5,.5));\nlabel(\"$2n-9$\",(1.5,1.5));\nlabel(\"$1$\",(1.5,2.5));\n\nlabel(\"$2$\",(2.5,.5));\nlabel(\"$n$\",(2.5,1.5));\nlabel(\"$n-1$\",(2.5,2.5));\n[/asy]", "answer": "$7$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "count" }, { "index": 439, @@ -3509,7 +3947,8 @@ "geo_code": "[asy]\npair A,B,C, D;\nA = (-3,3); B = (3,0); C = (0, -4); D = (0,0);\ndraw(D--B--C--cycle); draw(D--A--B--cycle);draw(D--A--C--cycle);\nlabel(\"$A$\", A, NW);label(\"$B$\", B, E); label(\"$C$\", C, S);label(\"$D$\", D, NE);\nlabel(\"3\", D--B, S); label(\"6\", A--B, NE); label(\"6\", A--C, SW); label(\"4\", D--C, NW+N);\nlabel(\"5\", A--D, SE+NE); label(\"5\", C--B, E);\n[/asy]", "answer": "$13$", "category": "Primitive Recognition", - "source": "MATH-500" + "source": "MATH-500", + "problem_type": "length" }, { "index": 440, @@ -3517,7 +3956,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n h = 9 \n l = 11\n r = np.sqrt(l**2 - h**2) \n \n \n plt.plot([0, -r], [0, -h], 'k', lw=3)\n plt.plot([0, r], [0, -h], 'k', lw=3)\n \n plt.plot([0, 0], [0, -h], 'k--', lw=2)\n \n \n theta = np.linspace(0, 2 * np.pi, 100)\n x = r * np.cos(theta)\n y = (r/2) * np.sin(theta) - h\n plt.plot(x, y, 'navy', lw=3)\n \n \n plt.plot([-1.5*r, -1.5*r], [0, -h], 'k', lw=2)\n plt.plot([-1.5*r-0.2, -1.5*r+0.2], [0, 0], 'k', lw=2)\n plt.plot([-1.5*r-0.2, -1.5*r+0.2], [-h, -h], 'k', lw=2)\n plt.text(-1.65*r, -h/2, '12 m', fontsize=14, va='center')\n \n \n plt.plot([0.5, 1.1*r], [0, -h], 'k', lw=3)\n plt.plot([0, 0.2*r-0.1], [0, 0.1*r], 'k', lw=3)\n plt.plot([1*r, 1.2*r], [-h-0.2, -h+0.2], 'k', lw=3)\n plt.text(r/2 + 0.6, -h/2, '13 m', fontsize=14)\n \n \n plt.plot([0, r], [-h, -h], 'k--', lw=2)\n plt.plot([0.1*r, 0.5, 0.1], [-h, -h+0.5, -h+0.5], 'teal', lw=4)\n plt.text(r/2, -h - 0.9, 'r', fontsize=14, ha='center')\n \n \n plt.plot([0], [-h], 'o', color='black')\n \n plt.axis('equal')\n plt.axis('off')\n plt.ylim(-15, 2)\n plt.xlim(-15, 15) \n[/python]", "answer": "$10$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "length" }, { "index": 441, @@ -3525,7 +3965,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n h = 9\n s = 10\n \n r = (s**2 - h**2)**0.5\n \n theta = np.linspace(0, 2*np.pi, 200)\n x_circ = r * np.cos(theta)\n y_circ = r/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(6,10))\n \n ax.plot(x_circ, y_circ, 'k')\n \n ax.plot([0, -r], [h, 0], 'k')\n ax.plot([0, r], [h, 0], 'k')\n \n ax.plot([0, 0], [0, h], 'k--')\n \n ax.plot([-r, r], [0, 0], 'b', linewidth=1)\n \n ax.plot([0.4, 0.4, 0], [0, 0.4, 0.4], 'b', linewidth=1)\n \n ax.text(-0.2, h/2, \"$12m$\", fontsize=14, color=\"k\")\n ax.text(r-2.5, h/2, \"$13m$\", fontsize=14, color=\"k\")\n ax.text(r/2-0.1, -0.7, \"$r$\", fontsize=14, color=\"b\")\n ax.text(0-0.4, -0.5, \"$O$\", fontsize=14, color=\"b\")\n \n ax.set_aspect('equal')\n ax.axis('off')\n plt.ylim(-r-1, h+1)\n plt.xlim(-r-2, r+2) \n[/python]", "answer": "$5$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "length" }, { "index": 442, @@ -3533,7 +3974,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig = plt.figure(figsize=(9,6))\n ax = fig.add_subplot(121, projection='3d')\n \n # Vertices as (x, y, z)\n A = [0, 0, 12]\n B = [0, 8, 12]\n C = [17, 8, 12]\n D = [17, 0, 12]\n E = [0, 0, 0]\n F = [0, 8, 0]\n G = [17, 8, 0]\n H = [17, 0, 0]\n \n edges = [\n (A,D), (D,C), (C,B), (B,A),\n (H,G), (E,H),\n (A,E), (D,H), (C,G)\n ]\n \n for e in edges:\n x, y, z = zip(*e)\n ax.plot(x, y, z, color='black')\n \n ax.plot([B[0],F[0]], [B[1],F[1]], [B[2],F[2]], ls='dashed', color='black')\n ax.plot([E[0],F[0]], [E[1],F[1]], [E[2],F[2]], ls='dashed', color='black')\n ax.plot([F[0],G[0]], [F[1],G[1]], [F[2],G[2]], ls='dashed', color='black')\n \n ax.plot([D[0],F[0]], [D[1],F[1]], [D[2],G[2]], ls='dotted', color='grey')\n ax.plot([H[0],F[0]], [H[1],F[1]], [H[2],G[2]], ls='dotted', color='grey')\n \n verts = {'A':A, 'B':B, 'C':C, 'D':D, 'E':E, 'F':F, 'G':G, 'H':H}\n for k, v in verts.items():\n ax.scatter(*v, color='navy')\n d = dict(A=(-0.5,-0.5,0.5), D=(0.5,-0.5,0.5), C=(0.5,0.5,0.5), B=(-0.5,0.5,0.5),\n H=(0.5,-0.5,-0.25), G=(0.7,0.7,-0.2), F=(-0.5,0.5,-0.2), E=(-0.8,-0.7,-0.2))\n o = d.get(k, (0,0,0))\n ax.text(v[0]+o[0], v[1]+o[1], v[2]+o[2], f'{k}', color='navy', fontsize=17, fontweight='bold')\n \n ax.text(1, 6, (A[2]+D[2])/2, r\"$11\\,\\mathit{cm}$\", fontsize=17, color='black', va='center', ha='right', rotation=90)\n ax.text(17, 6, 6, r\"$11\\,\\mathit{cm}$\", fontsize=17, color='black', va='center', ha='left', rotation=90)\n \n ax.text(7.5, 8.7, 0, r\"$15\\,\\mathit{cm}$\", fontsize=17, color='black', ha='center', va='bottom')\n \n ax.text((D[0]+F[0])/2, (D[1]+F[1])/2+0.5, (D[2]+F[2])/2+0.9, r\"$y\\,cm$\", fontsize=17, color='black')\n ax.text((E[0]+G[0])/2-1, (E[1]+G[1])/2-1, (E[2]+G[2])/2, r\"$z\\,cm$\", fontsize=17, color='black')\n \n ax.set_xlim(-3, 19)\n ax.set_ylim(-3, 12)\n ax.set_zlim(-3, 16)\n ax.set_xticks([])\n ax.set_yticks([])\n ax.set_zticks([])\n ax.set_box_aspect([2,1,1])\n ax.view_init(azim=-60, elev=22)\n plt.tight_layout()\n ax.axis('off') \n[/python]", "answer": "$21.61$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "length" }, { "index": 443, @@ -3541,7 +3983,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n import numpy as np\n \n A = np.array([0, 0, 0])\n B = np.array([20, 0, 0])\n C = np.array([20, 15, 0])\n D = np.array([20, 15, 15])\n E = np.array([0, 0, 15])\n F = np.array([20, 0, 15])\n G = np.array([0, 15, 0])\n H = np.array([0, 15, 15])\n \n fig = plt.figure(figsize=(8, 6))\n ax = fig.add_subplot(111, projection='3d')\n \n edges = [\n [A, B], [B, C], [C, G], [G, A], # bottom\n [E, F], [F, D], [D, H], [H, E], # top\n [A, E], [B, F], [C, D], [G, H] # vertical\n ]\n for edge in edges:\n ax.plot(*zip(*edge), c='k')\n \n face = [A, C, D]\n ax.add_collection3d(Poly3DCollection([[A, C, D]], facecolors='cyan', linewidths=1, edgecolors='cyan', alpha=0.4))\n \n ax.plot([A[0], C[0]], [A[1], C[1]], [A[2], C[2]], color='r', linestyle='--', linewidth=2)\n \n def annotate_point(point, label, offset=(0,0.5,0)):\n ax.text(point[0]+offset[0], point[1]+offset[1], point[2]+offset[2], label, fontsize=15, color='navy')\n \n annotate_point(A, 'A')\n annotate_point(B, 'B')\n annotate_point(C, 'C')\n annotate_point(D, 'D')\n \n ax.text(8.5, -1.5, 0, r'$17cm$', fontsize=13)\n ax.text(8.5, 6, 0, r'$zcm$', fontsize=13)\n ax.text(20.8, 6, 0, r'$12cm$', fontsize=13)\n ax.text(20.8, 15, 6, r'$12cm$', fontsize=13)\n \n ax.set_xlim([0, 20])\n ax.set_ylim([0, 15])\n ax.set_zlim([0, 15])\n ax.set_box_aspect([20, 15, 15])\n \n ax.axis('off') \n[/python]", "answer": "$124.85$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 444, @@ -3549,7 +3992,8 @@ "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n \n r = 10\n h = 12\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = r * np.cos(theta)\n y = r/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(6,6))\n ax.plot(x, y + h, 'k')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([r, r], [h, 0], 'k')\n ax.plot([-r, -r], [h, 0], 'k')\n \n ax.plot(x[(theta > np.pi)], y[(theta > np.pi)] + h, 'k--', alpha=0.5)\n \n ax.plot([0, r], [h, h], 'k')\n ax.text(r/2, h+0.5, '6 cm', fontsize=12, ha='center')\n \n ax.text(r+0.5, h/2, '8 cm', fontsize=12, va='center')\n \n ax.axis('equal')\n ax.axis('off') \n[/python]", "answer": "$904.78$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 445, @@ -3557,7 +4001,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(10, 10))\n \n center = [0, 0]\n radius = 1\n height = 5\n \n theta = np.linspace(0, 2 * np.pi, 200)\n x = radius * np.cos(theta)\n y = 0.5 * radius * np.sin(theta)\n \n ax.plot(x, y, 'k')\n \n ax.plot([0, -radius], [height, 0], 'k')\n ax.plot([0, radius], [height, 0], 'k')\n \n ax.plot([0, 0], [0, height], 'k')\n \n ax.plot([0, radius], [0, 0], 'k')\n \n ax.plot([0.2, 0.2, 0], [0, 0.2, 0.2], 'k')\n \n ax.text(0.1, height/2, r\"$6cm$\", fontsize=14, ha='left', va='center')\n ax.text(radius/2, -0.3, r\"$2cm$\", fontsize=14, ha='center')\n \n ax.set_xlim(-3, 3)\n ax.set_ylim(-1, 7)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$25.13$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 446, @@ -3565,7 +4010,8 @@ "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n \n radius = 3\n slant_height = 9\n height = np.sqrt(slant_height**2 - radius**2)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_circle = radius * np.cos(theta)\n y_circle = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(6, 6))\n \n ax.plot([0, -radius], [height, 0], 'k')\n ax.plot([0, radius], [height, 0], 'k')\n \n ax.plot(x_circle, y_circle, 'k')\n \n ax.plot([0, radius], [0, 0], 'k', lw=1)\n \n ax.text(radius * 0.5, height/2 + 0.1, '8 cm', fontsize=12, style='italic')\n ax.text(radius/4, -0.4, '2 cm', fontsize=12, style='italic')\n \n ax.plot(0, height, 'ko')\n \n ax.set_aspect('equal')\n ax.axis('off')\n plt.ylim(-2.5, height+1)\n plt.xlim(-3, 3) \n[/python]", "answer": "$32.45$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 447, @@ -3573,7 +4019,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n A = np.array([0, 0])\n B = np.array([4, 0]) \n \n mid_BC = (A + B) / 2\n height = 9\n C = mid_BC + np.array([0, height])\n \n plt.plot([A[0], B[0]], [A[1], B[1]], 'k')\n plt.plot([A[0], C[0]], [A[1], C[1]], 'k')\n plt.plot([B[0], C[0]], [B[1], C[1]], 'k')\n \n plt.plot([mid_BC[0], C[0]], [mid_BC[1], C[1]], 'k--')\n \n theta = np.linspace(np.pi, 2 * np.pi, 100)\n r = 2\n x_semicircle = mid_BC[0] + r * np.cos(theta)\n y_semicircle = mid_BC[1] + r * np.sin(theta)\n plt.plot(x_semicircle, y_semicircle, 'k')\n \n plt.text(1, -0.5, r'$3cm$', fontsize=14)\n plt.text(3.75, 4, r'$9cm$', fontsize=14)\n plt.plot([2, 2.5], [0.5, 0.5], 'k')\n plt.plot([2.5, 2.5], [0.5, 0], 'k')\n \n plt.axis('equal')\n plt.axis('off') \n[/python]", "answer": "$10.45$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 448, @@ -3581,7 +4028,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,6))\n \n ax.fill([0, 4.3, -4.3, 0], [0, -13.5, -13.5, 0], color='olive')\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_ellipse = 4.3 * np.cos(theta)\n y_ellipse = -13.5 + 1 * np.sin(theta)\n ax.plot(x_ellipse[50:], y_ellipse[50:], color='black')\n ax.plot(x_ellipse[:50], y_ellipse[:50], color='black', linestyle=\"--\")\n ax.fill(x_ellipse, y_ellipse, color=\"olive\")\n \n ax.plot([0, 0], [0, -13.5], 'k--', lw=1)\n ax.plot([0, 4.3], [-13.5, -13.5], 'k--', lw=1)\n \n offset = 0.3\n ax.plot([0, offset], [-13.5, -13.5], color='black')\n ax.plot([offset, offset], [-13.5, -13.5+offset], color='black')\n ax.plot([0, offset], [-13.5+offset, -13.5+offset], color='black')\n \n ax.text(-1.18, -6.25, r\"$12.5cm$\", fontsize=14, va='center', color='black', style='italic')\n ax.text(2.05, -13.1, r\"4.1cm\", fontsize=14, ha='center', color='black', style='italic')\n \n ax.set_aspect('equal')\n plt.axis('off') \n[/python]", "answer": "$220.0$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 449, @@ -3589,7 +4037,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,6))\n \n ax.fill([0, 4.1, -4.1, 0], [0, -12.5, -12.5, 0], color='olive')\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_ellipse = 4.1 * np.cos(theta)\n y_ellipse = -12.5 + 1 * np.sin(theta)\n ax.plot(x_ellipse[50:], y_ellipse[50:], color='black')\n ax.plot(x_ellipse[:50], y_ellipse[:50], color='black', linestyle=\"--\")\n ax.fill(x_ellipse, y_ellipse, color=\"olive\")\n \n \n ax.plot([0, 0], [0, -12.5], 'k--', lw=1)\n ax.plot([-4.1, 0], [-12.5, -12.5], 'k--', lw=1)\n \n offset = 0.3\n #ax.plot([0, offset], [-12.5, -12.5], color='black')\n ax.plot([-offset, -offset], [-12.5, -12.5+offset], color='black')\n ax.plot([-offset, 0], [-12.5+offset, -12.5+offset], color='black')\n \n ax.text(0.2, -6.25, r\"$6$\", fontsize=14, va='center', color='black', style='italic')\n ax.text(-2.05, -13.1, r\"$3$\", fontsize=14, ha='center', color='black', style='italic')\n \n ax.set_aspect('equal')\n plt.axis('off') \n[/python]", "answer": "$56.55$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 450, @@ -3597,7 +4046,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(8,9))\n \n circle = plt.Circle((0, 0), 3.5, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3.5 * np.cos(theta)\n y = 1.75 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3.5], [0, 0], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(1.75, 0.1, '3 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]", "answer": "$113.10$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 451, @@ -3605,7 +4055,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n y = 3 * np.cos(theta)\n x = 1.1 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 0], [3, -3], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(0.2, 0.1, '4 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]", "answer": "$33.51$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 452, @@ -3613,7 +4064,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(8,8))\n \n R = 9\n r = 4.5\n \n theta = np.linspace(0, np.pi, 500)\n x_outer = R * np.cos(theta)\n y_outer = R * np.sin(theta)\n \n theta_inner = np.linspace(0, np.pi, 200)\n x_inner = r * np.cos(theta_inner)\n y_inner = r * np.sin(theta_inner) + (R-r)\n \n ax.plot(x_outer, -y_outer, color='navy', lw=3, zorder=1)\n \n ellipse_x = R * np.cos(np.linspace(0, 2*np.pi, 200))\n ellipse_y = 0.33 * R * np.sin(np.linspace(0, 2*np.pi, 200))\n ax.plot(ellipse_x, ellipse_y, color='navy', lw=3)\n \n dash_theta = np.linspace(0, 2*np.pi, 400)\n dash_x = r * np.cos(dash_theta)\n dash_y = 0.33 * r * np.sin(dash_theta)\n ax.plot(dash_x[:200], dash_y[:200], 'k--', lw=2)\n ax.plot(dash_x[200:], dash_y[200:], 'k', lw=2)\n \n ax.plot(x_inner, y_inner-R+r, color='navy', lw=3, zorder=2)\n \n ax.annotate('', xy=(-r, 5), xytext=(r, 5), arrowprops=dict(arrowstyle='<->', lw=1.8))\n ax.text(0, 5.25, \"6 cm\", ha='center', va='bottom', fontsize=14)\n \n ax.annotate('', xy=(-R, -9.2), xytext=(R, -9.2), arrowprops=dict(arrowstyle='<->', lw=1.8))\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$508.94$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 453, @@ -3621,7 +4073,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3 * np.cos(theta)\n y = 1.5 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3], [0, 0], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(1, 0.1, '8.8 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]", "answer": "$2854.54$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 454, @@ -3629,7 +4082,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3 * np.cos(theta)\n y = 1.5 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3], [0, 0], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(1, 0.1, '22.36m', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]", "answer": "$46827.83$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 455, @@ -3637,7 +4091,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='#C6E6EB', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='#8ED7DE', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(0.3, -radius/2, '6', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$452.389$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 456, @@ -3645,7 +4100,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 6\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='#C6E6EB', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='#8ED7DE', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(-3.3, -radius/2, '$25.36cm$', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$34159.095$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 457, @@ -3653,7 +4109,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 20\n height = 70\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = radius * np.cos(theta)\n y = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(5,8))\n \n ax.plot(x, y + height, 'k')\n \n ax.plot(x[:50], y[:50], 'k--')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([radius, radius], [height, 0], 'k')\n ax.plot([-radius, -radius], [height, 0], 'k')\n \n ax.annotate(\n '', xy=(radius + 4, height), xytext=(radius + 4, 0),\n arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([radius+3, radius+5], [height, height], color=\"black\")\n ax.plot([radius+3, radius+5], [0, 0], color=\"black\")\n ax.text(radius + 6, height/2, f'{35}', va='center', fontsize=12)\n \n ax.annotate(\n '', xy=(0, height+0.5), xytext=(-radius-0.5, height+0.5),\n arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([0, 0], [height-0.2, height+1.2], color=\"black\")\n ax.plot([-radius, -radius], [height-0.2, height+1.2], color=\"black\")\n ax.text(-radius/2, height + 1, f'{10}', ha='center', fontsize=12)\n \n ax.set_xlim(-radius*1.5, radius*1.5)\n ax.set_ylim(-radius, height + radius)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$2827.43$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 458, @@ -3661,7 +4118,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Cylinder dimensions\n radius = 17\n length = 36\n \n fig, ax = plt.subplots(figsize=(8, 5))\n \n # Draw the side edges of the cylinder\n ax.plot([0, length], [radius, radius], 'k')\n ax.plot([0, length], [-radius, -radius], 'k')\n \n # Draw the front ellipse (solid)\n theta = np.linspace(0, 2 * np.pi, 100)\n x_front = length + radius/2 * np.cos(theta)\n y_front = radius * np.sin(theta)\n ax.plot(x_front, y_front, 'k')\n \n # Draw the back ellipse (dashed)\n x_back = 0 + radius/2 * np.cos(theta)\n y_back = radius * np.sin(theta)\n ax.plot(x_back, y_back, 'k--')\n \n # Draw the radius on the front face\n ax.plot([length, length], [0, radius], 'k')\n ax.plot(length, 0, 'bo', markersize=3) # Center point\n \n # Add dimension lines and labels\n # Length\n ax.annotate('', xy=(0, -radius-5), xytext=(length, -radius-5),\n arrowprops=dict(arrowstyle='<->', lw=1.5))\n ax.text(length/2, -radius-8, '49', ha='center', va='top', fontsize=12)\n \n # Radius\n ax.annotate('', xy=(length, 0), xytext=(length, radius),\n arrowprops=dict(arrowstyle='<->', lw=1.5))\n ax.text(length+3, radius/2, '21', va='center', fontsize=12)\n \n # Set aspect, limits and remove axes\n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-radius*0.7, length+radius*1.2)\n ax.set_ylim(-radius*1.3, radius*1.3)\n \n plt.show() \n[/python]", "answer": "$9236.28$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 459, @@ -3669,7 +4127,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Cylinder parameters\n radius = 80\n height = 82\n \n # Plot setup\n fig, ax = plt.subplots(figsize=(7,5))\n ax.set_xlim(-120, 160)\n ax.set_ylim(-70, 90)\n \n # Angles for ellipse\n theta = np.linspace(0, 2*np.pi, 100)\n \n # Top ellipse (at y = height/2)\n x_top = radius * np.cos(theta)\n y_top = 0.25 * radius * np.sin(theta) + height/2\n \n # Bottom ellipse (at y = -height/2)\n x_bot = radius * np.cos(theta)\n y_bot = 0.25 * radius * np.sin(theta) - height/2\n \n # Draw top and bottom ellipses\n ax.plot(x_top, y_top, 'k')\n ax.plot(x_bot, y_bot, 'k', linestyle='dashed')\n \n # Draw sides\n ax.plot([-radius, -radius], [-height/2, height/2], 'k')\n ax.plot([radius, radius], [-height/2, height/2], 'k')\n \n # Draw radius annotation\n ax.plot([0, radius], [height/2, height/2], 'b', marker='o')\n ax.text(radius/2, height/2+7, '98', ha='center', va='bottom', fontsize=12)\n \n # Draw height annotation with double arrow\n ax.annotate(\n '', xy=(radius+15, height/2), xytext=(radius+15, -height/2),\n arrowprops=dict(arrowstyle='<->', linewidth=2))\n ax.text(radius+22, 0, '80', va='center', fontsize=12)\n \n # Remove axes\n ax.axis('off')\n \n plt.show() \n[/python]", "answer": "$109603.88$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 460, @@ -3677,7 +4136,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 20\n height = 70\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = radius * np.cos(theta)\n y = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(5,8))\n \n ax.plot(x, y + height, 'k')\n \n ax.plot(x[:50], y[:50], 'k--')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([radius, radius], [height, 0], 'k')\n ax.plot([-radius, -radius], [height, 0], 'k')\n \n ax.annotate(\n '', xy=(radius + 4, height), xytext=(radius + 4, 0),\n arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([radius+3, radius+5], [height, height], color=\"black\")\n ax.plot([radius+3, radius+5], [0, 0], color=\"black\")\n ax.text(radius + 6, height/2, '$hmm$', va='center', fontsize=12)\n \n ax.annotate(\n '', xy=(0, height+0.5), xytext=(-radius-0.5, height+0.5),\n arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([0, 0], [height-0.2, height+1.2], color=\"black\")\n ax.plot([-radius, -radius], [height-0.2, height+1.2], color=\"black\")\n ax.text(-radius/2, height + 1, '$43mm$', ha='center', fontsize=12)\n \n ax.set_xlim(-radius*1.5, radius*1.5)\n ax.set_ylim(-radius, height + radius)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$58$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "length" }, { "index": 461, @@ -3685,7 +4145,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 20\n height = 70\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = radius * np.cos(theta)\n y = radius/2 * np.sin(theta)\n \n fig, ax = plt.subplots(figsize=(5,8))\n \n ax.plot(x, y + height, 'k')\n \n ax.plot(x[:50], y[:50], 'k--')\n ax.plot(x[50:], y[50:], 'k')\n \n ax.plot([radius, radius], [height, 0], 'k')\n ax.plot([-radius, -radius], [height, 0], 'k')\n \n ax.annotate(\n '', xy=(radius + 4, height), xytext=(radius + 4, 0),\n arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([radius+3, radius+5], [height, height], color=\"black\")\n ax.plot([radius+3, radius+5], [0, 0], color=\"black\")\n ax.text(radius + 6, height/2, '$hmm$', va='center', fontsize=12)\n \n ax.annotate(\n '', xy=(0, height+0.5), xytext=(-radius-0.5, height+0.5),\n arrowprops=dict(arrowstyle='-', lw=1.5)\n )\n ax.plot([0, 0], [height-0.2, height+1.2], color=\"black\")\n ax.plot([-radius, -radius], [height-0.2, height+1.2], color=\"black\")\n ax.text(-radius/2, height + 1, '$79mm$', ha='center', fontsize=12)\n \n ax.set_xlim(-radius*1.5, radius*1.5)\n ax.set_ylim(-radius, height + radius)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$30$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "length" }, { "index": 462, @@ -3693,7 +4154,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n radius = 7\n height = 12\n \n fig, ax = plt.subplots(figsize=(6,10))\n \n theta = np.linspace(0, 2*np.pi, 100)\n x_top = radius * np.cos(theta)\n y_top = radius/2 * np.sin(theta) + height\n \n x_bottom = radius * np.cos(theta)\n y_bottom = radius/2 * np.sin(theta)\n \n ax.plot([radius, radius], [height, 0], color='k')\n ax.plot([-radius, -radius], [height, 0], color='k')\n \n ax.plot(x_top, y_top, color='k')\n ax.plot(x_bottom[50:], y_bottom[50:], color='k')\n \n theta_hidden = np.linspace(np.pi, 2*np.pi, 100)\n x_hidden = radius * np.cos(theta_hidden)\n y_hidden = radius/2 * np.sin(theta_hidden) + height\n ax.plot(x_hidden, y_hidden, color='k', alpha=0.3, linestyle='dashed')\n \n ax.plot([0, radius], [height, height], color='k')\n ax.text(radius/2, height+0.5, '6 cm', fontsize=12, style='italic', ha='center')\n \n ax.text(radius+1.5, height/2, '10 cm', fontsize=12, style='italic', ha='center')\n \n ax.set_xlim(-10, 12)\n ax.set_ylim(-6, 16)\n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$603.19$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 463, @@ -3701,7 +4163,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#19bfa7')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n x = 3 * np.cos(theta)\n y = 1.5 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#19bfa7')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 3], [0, 0], linestyle='--', linewidth=3, color=\"blue\")\n \n ax.text(1, 0.1, '11cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]", "answer": "$1520.53$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 464, @@ -3709,7 +4172,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(5,5))\n \n circle = plt.Circle((0, 0), 3, fill=False, linewidth=3, color='#2c3e50')\n ax.add_patch(circle)\n \n theta = np.linspace(0, 2*np.pi, 100)\n y = 3 * np.cos(theta)\n x = 1.1 * np.sin(theta)\n ax.plot(x, y, linestyle=\"--\", color='#2c3e50')\n ax.plot(0, 0, 'o', color='#2c3e50', markersize=10)\n \n ax.plot([0, 0], [3, -3], color='#2c3e50', linestyle='-', linewidth=3)\n \n ax.text(0.2, 0.1, '9 cm', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off')\n ax.set_xlim(-3.5, 3.5)\n ax.set_ylim(-3.5, 3.5) \n[/python]", "answer": "$254.47$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 465, @@ -3717,7 +4181,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Set up the plot\n fig, ax = plt.subplots(figsize=(6,6))\n radius = 30\n \n # Draw the main circle (sphere)\n circle = plt.Circle((0, 0), radius, fill=False, color='black')\n ax.add_artist(circle)\n \n # Draw the dashed radius (equator perspective)\n theta = np.linspace(0, 2*np.pi, 200)\n x_ellipse = radius * np.cos(theta)\n y_ellipse = 0.4 * radius * np.sin(theta) # Slightly \"flatten\" to suggest equator\n \n ax.plot(x_ellipse, y_ellipse, 'k--')\n \n # Draw the radius line\n ax.plot([0, radius], [0, 0], 'k')\n \n # Add the radius text\n plt.text(radius/2-5, 1, '56.17 $mm$', fontsize=14, fontstyle='italic')\n \n # Formatting\n ax.set_aspect('equal')\n ax.set_xlim(-radius*1.2, radius*1.2)\n ax.set_ylim(-radius*1.2, radius*1.2)\n ax.axis('off')\n \n plt.show() \n[/python]", "answer": "$39647.77$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 466, @@ -3725,7 +4190,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='olive', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='olive', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(0.3, -radius/2, '6', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$113.097$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 467, @@ -3733,7 +4199,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='olive', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='olive', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(0.3, -radius/2, '8', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$603.186$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 468, @@ -3741,7 +4208,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n fig, ax = plt.subplots(figsize=(6,4))\n \n radius = 7\n \n theta = np.linspace(np.pi, 2*np.pi, 100)\n x_semi = radius * np.cos(theta)\n y_semi = radius * np.sin(theta)\n ax.fill(x_semi, -radius-y_semi, color='olive', zorder=1)\n ax.plot(x_semi, -radius-y_semi, color='#22313F', linewidth=3)\n \n \n \n a = radius\n b = radius/3\n t = np.linspace(0, 2*np.pi, 100)\n x_ellipse = a * np.cos(t)\n y_ellipse = -radius + b * np.sin(t)\n ax.plot(x_ellipse, y_ellipse, color='#22313F', linewidth=2, linestyle='--')\n ax.fill_between(x_ellipse, y_ellipse, -radius, color='olive', alpha=0.7, zorder=2)\n \n ax.plot([0, 0], [0, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.plot([0, a], [-radius, -radius], color='#22313F', linestyle='--', linewidth=2)\n \n ax.text(-2, -radius/2, '$74.74cm$', fontsize=16)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$52647.45$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 469, @@ -3749,7 +4217,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n import matplotlib.patches as patches\n \n \n fig, ax = plt.subplots(figsize=(8,4))\n \n width, height = 15, 20\n \n # \u68af\u5f62\u524d\u5e95\u4e3a14\uff0c\u4e0a\u5e9510\uff0c\u9ad83\uff0c\u5de6\u504f\u79fb2\uff0c\u53f3\u504f\u79fb2\n # \u524d\u4fa7\u68af\u5f62\n A = np.array([0, 0])\n B = np.array([width, 0])\n D = A + np.array([0, height])\n C = B + np.array([0, height])\n \n # \u68f1\u67f1\u539a\u5ea6\n dx, dy = -5, 6\n dxy = np.array([dx, dy])\n # \u540e\u9762\u56db\u4e2a\u70b9\n A1 = A + dxy\n B1 = B + dxy\n C1 = C + dxy\n D1 = D + dxy\n \n x0 = np.array([25, 0])\n A2 = A + x0\n B2 = B + x0\n C2 = C + x0\n D2 = D + x0\n \n A3, B3, C3, D3 = A1 + x0, B1 + x0, C1 + x0, D1 + x0\n \n for X, Y in [(A, B), (B, C), (C, D), (D, A)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(C1, D1), (D1, A1), (D, D1), (C, C1), (A, A1), (A2, A3)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(A2, B2), (B2, C2), (C2, D2), (D2, A2)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(C3, D3), (D3, A3), (D2, D3), (C2, C3)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n for X, Y in [(A1, B1), (B1, C1), (A3, B3), (B3, C3), (B, B1), (B2, B3)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k--')\n \n # Big cylinder base\n main1 = patches.Arc(((B1[0] + B[0])/2, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2), 3, 13, theta1=0, theta2=360, color='black', linewidth=2, linestyle='--')\n #ax.plot([0, 0], [0, 0], 'ko')\n ax.add_patch(main1)\n \n main2 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=270, theta2=90, color='black', linewidth=2)\n main3 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=90, theta2=270, color='black', linewidth=2, linestyle='--')\n #ax.plot([0, 0], [0, 0], 'ko')\n ax.add_patch(main2)\n ax.add_patch(main3)\n \n ax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5], 'k')\n ax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5], 'k')\n \n \n ax.plot([B[0], A2[0]], [-2, -2], 'k')\n ax.plot([B[0], B[0]], [-1, -3], 'k')\n ax.plot([A2[0], A2[0]], [-1, -3], 'k')\n ax.plot([(A2[0] + A3[0])/2, (A2[0] + A3[0])/2], [((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2+6.5], 'k')\n \n ax.text(B2[0] + 1, C[1]/2, r\"$20$\", fontsize=12, style='italic')\n ax.text((B2[0] + A2[0])/2, -2, r\"$25$\", fontsize=12, style='italic')\n \n ax.text((A2[0] + A3[0])/2-1.2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2+0.5, r\"$6$\", fontsize=10, style='italic')\n \n ax.set_aspect('equal')\n ax.axis('off')\n plt.xlim(-8, 60)\n plt.ylim(-5, 30)\n plt.tight_layout()\n plt.show() \n[/python]", "answer": "$8128.50$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 470, @@ -3757,7 +4226,8 @@ "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n \n radius = 4 \n height = 9 \n \n fig = plt.figure(figsize=(6, 12))\n ax = fig.add_subplot(111)\n \n theta = np.linspace(0, np.pi, 100)\n x_hemisphere = radius * np.cos(theta)\n y_hemisphere = -radius * np.sin(theta)\n \n \n x_cone = np.array([-radius, 0, radius])\n y_cone = np.array([0, height, 0])\n \n ax.plot(x_hemisphere, y_hemisphere, color='deepskyblue', linewidth=4)\n ax.fill_between(x_hemisphere, y_hemisphere, 0, color='deepskyblue', alpha=0.6)\n \n \n theta = np.linspace(0, 2*np.pi, 200)\n x_circle = radius * np.cos(theta)\n y_circle = radius/ 5 * np.sin(theta)\n ax.plot(x_circle, y_circle, color=\"deepskyblue\", linewidth=4)\n ax.fill_between(x_circle, y_circle, 0, color=\"gray\")\n ax.plot([-radius, 0], [0, height], color='gray')\n ax.plot([radius, 0], [0, height], color='gray')\n \n ax.plot([0, 0], [0, height], 'k--', lw=1)\n ax.text(0.1, height/1.7, r'$10cm$', fontsize=15)\n \n ax.plot([0, radius], [0, 0], 'black', linestyle='dashed')\n \n ax.plot([0.5, 0.5], [0, 0.5], color='black')\n ax.plot([0, 0.5], [0.5, 0.5], color='black')\n \n ax.axis('off')\n \n ax.set_aspect('equal')\n ax.set_xlim(-7, 7)\n ax.set_ylim(-6, 13) \n[/python]", "answer": "$235.87$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 471, @@ -3765,7 +4235,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n import numpy as np\n \n # Parameters\n diameter = 9 \n radius = diameter / 2\n height = 18 \n \n fig, ax = plt.subplots(figsize=(4, 6))\n \n theta = np.linspace(np.pi, 2 * np.pi, 100)\n x = radius * np.cos(theta)\n y = radius * 0.25 * np.sin(theta) - height \n \n ax.plot(x, y, color='teal', linewidth=2)\n \n top_y = 0\n top_ellipse = radius * np.cos(theta)\n top_ellipse_y = radius * 0.25 * np.sin(theta) + top_y\n \n ax.plot(top_ellipse, top_ellipse_y, color='teal', linewidth=2, alpha=1, solid_capstyle='round')\n \n theta = np.linspace(0, np.pi, 100)\n ax.plot(radius * np.cos(theta), \n radius * 0.25 * np.sin(theta) - height, \n '--', color='teal', alpha=0.5, linewidth=2) \n \n ax.plot(radius * np.cos(theta), \n radius * 0.25 * np.sin(theta), \n '--', color='teal', alpha=0.5, linewidth=2) \n \n ax.plot([radius, radius], [top_y, -height], color='teal', linewidth=2)\n ax.plot([-radius, -radius], [top_y, -height], color='teal', linewidth=2)\n \n theta2 = np.linspace(0, np.pi, 100)\n x2 = radius * np.cos(theta2)\n y2 = radius * np.sin(theta2)\n \n ax.plot(x2, y2, color='teal', linewidth=2)\n \n ax.annotate(\n '', xy=(0, -height-2), xytext=(radius, -height-2),\n arrowprops=dict(arrowstyle='<->', color='#43536D', linewidth=2)\n )\n ax.text(radius/2, -height-3, f\"10.5 mm\", ha='center', va='top', fontsize=14, color=\"#43536D\")\n \n ax.plot([radius+1.2, radius+1.2], [-height, 0], color='#43536D', linewidth=2)\n ax.plot([radius+0.8, radius+1.6], [0, 0], color='#43536D', linewidth=2)\n ax.plot([radius+0.8, radius+1.6], [-height, -height], color='#43536D', linewidth=2)\n \n ax.set_aspect('equal')\n ax.axis('off') \n[/python]", "answer": "$3222.80$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 472, @@ -3773,7 +4244,8 @@ "geo_code": "[python]\n import numpy as np\n import matplotlib.pyplot as plt\n from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n \n length = 9 \n width = 4\n height = 4\n \n C = np.array([0, width, height])\n B = np.array([length, width, height])\n A = np.array([length, 0, height])\n D = np.array([0, 0, height])\n F = np.array([length, width, 0])\n E = np.array([length, 0, 0])\n H = np.array([0, 0, 0])\n G = np.array([0, width, 0])\n \n verts = [A, B, C, D, E, F, G, H]\n \n fig = plt.figure(figsize=(8,6))\n ax = fig.add_subplot(111, projection='3d')\n \n for point, name in zip(verts, ['A','B','C','D','E','F','G','H']):\n ax.scatter(*point, s=40)\n ax.text(*(point + np.array([0.2,0.2,0.2])), name, fontsize=12, color='k')\n \n edges = [\n (C, B), (B, A), (A, D), (D, C),\n (E, F), (E, H), (G, H), (G, F),\n (C, G), (B, F), (A, E), (D, H) \n ]\n for edge in edges:\n ax.plot(*zip(*edge), color='k')\n \n ax.plot(*zip(*[D, F]), 'r--')\n ax.plot(*zip(*[H, F]), 'g--')\n ax.plot(*zip(*[E, F]), 'b--')\n \n arc_radius = 0.5\n angle = np.arctan(width/length)\n angle_values = np.linspace(0, angle, 30)\n arc_x = length - arc_radius * np.cos(angle_values)\n arc_y = width - arc_radius * np.sin(angle_values)\n arc_z = np.zeros_like(arc_x)\n ax.plot(arc_x, arc_y, arc_z, 'b')\n \n ax.text(length-0.3, width-1, 0, r'$\\theta$', color='b', fontsize=16)\n \n ax.set_xlim([-.5, length+1])\n ax.set_ylim([-.5, width+1])\n ax.set_zlim([-.5, height+1])\n \n ax.text(0, width, height/2, '3 cm', color='k', fontsize=12)\n ax.text(-0.5, width/3, height+0.3, '3 cm', color='k', fontsize=12)\n ax.text(length/2, 1.5, -0.5, 'z cm', color='k', fontsize=12)\n \n ax.set_box_aspect([2,1,1])\n ax.axis('off')\n plt.tight_layout() \n[/python]", "answer": "$11.83$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "angle" }, { "index": 473, @@ -3781,7 +4253,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n \n fig, ax = plt.subplots(figsize=(4, 3))\n \n \n O = [0, 0]\n A = [-5.1, 4.1]\n B = [5.1, 4.1]\n C = [5.1, -4.1]\n D = [-5.1, -4.1]\n E = [2, 6]\n F = [12.2, 6]\n G = [12.2, -2.2]\n H = [2, -2.2]\n \n ax.plot([A[0], B[0]], [A[1], B[1]], color='black')\n ax.plot([B[0], C[0]], [B[1], C[1]], color='black')\n ax.plot([C[0], D[0]], [C[1], D[1]], color='black')\n ax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n \n ax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n ax.plot([F[0], G[0]], [F[1], G[1]], color='black')\n ax.plot([G[0], C[0]], [G[1], C[1]], color='black')\n \n ax.plot([A[0], E[0]], [A[1], E[1]], color='black')\n ax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n \n for X, Y in [(H, E), (H, D), (H, G)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n \n ax.text(A[0]-1, A[1], r\"$D$\", fontsize=12, style='italic')\n ax.text(B[0]-0.5, B[1]+0.5, r\"$C$\", fontsize=12, style='italic')\n ax.text(C[0], C[1]-1, r\"$G$\", fontsize=12, style='italic')\n ax.text(D[0], D[1]-1, r\"$H$\", fontsize=12, style='italic')\n ax.text(E[0], E[1]+0.5, r\"$A$\", fontsize=12, style='italic')\n ax.text(F[0], F[1]+0.5, r\"$B$\", fontsize=12, style='italic')\n ax.text(G[0], G[1]-1, r\"$F$\", fontsize=12, style='italic')\n ax.text(H[0]-1, H[1], r\"$E$\", fontsize=12, style='italic')\n \n ax.axis('off')\n plt.tight_layout()\n plt.show() \n[/python]", "answer": "$54.74$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "angle" }, { "index": 474, @@ -3789,7 +4262,8 @@ "geo_code": "[python]\n import matplotlib.pyplot as plt\n from mpl_toolkits.mplot3d.art3d import Poly3DCollection\n import numpy as np\n \n A = np.array([0, 0, 0])\n B = np.array([1, 0, 0])\n C = np.array([1, 1, 0])\n D = np.array([0, 1, 0])\n E = np.array([1, 1, 1.5])\n F = np.array([0, 1, 1.5])\n \n fig = plt.figure()\n ax = fig.add_subplot(111, projection='3d')\n \n verts = [A, B, C, D, E, F]\n \n edges = [\n [A, B], [B, C],\n [E, F], [A, E],\n [A, F], [B, E], [C, E]\n ]\n for edge in edges:\n ax.plot(*zip(*edge), color='black', linestyle='-', linewidth=1)\n \n edges = [[A,D], [C, D], [F, D]]\n for edge in edges:\n ax.plot(*zip(*edge), color='black', linestyle='--', linewidth=1)\n \n ax.plot(*zip(A, E), color='red', linewidth=2)\n \n labels = ['A', 'B', 'D', 'C', 'E', 'F']\n label_indices = [0, 1, 3, 2, 4, 5]\n for label, idx in zip(labels, label_indices):\n ax.text(*(verts[idx] + 0.05), label, fontsize=12)\n \n ax.set_box_aspect([1,1,1.5])\n ax.view_init(20, -30)\n ax.axis('off') \n[/python]", "answer": "$7$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "length" }, { "index": 475, @@ -3797,7 +4271,8 @@ "geo_code": "[python]\n\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(4,4))\n\ntheta1 = 24\ntheta2 = 66\nr = 9\n\nangle = np.deg2rad(np.linspace(theta1, theta2, 200))\nx_arc = r * np.cos(angle)\ny_arc = r * np.sin(angle)\n\nax.plot(x_arc, y_arc, color=\"black\")\n\nax.plot([0, x_arc[0]], [0, y_arc[0]], color=\"black\")\nax.plot([0, x_arc[-1]], [0, y_arc[-1]], color=\"black\")\nax.plot([0, 0], [0, -0.3], color=\"black\")\nax.plot([x_arc[0], x_arc[0]], [y_arc[0], y_arc[0]-0.3], color=\"black\")\nax.plot([0, x_arc[0]], [-0.3, y_arc[0]-0.3], color=\"black\")\n\n\nax.text(1.2, 1.3, r\"$42^\\circ$\", fontsize=16)\n\narc_mid = 0\nx_label = (r+0.2)*np.cos(np.deg2rad(arc_mid))/2\ny_label = (r+0.2)*np.sin(np.deg2rad(arc_mid)) +1\nax.text(x_label, y_label-0.4, \"9\", fontsize=16, ha='center')\n\nax.set_aspect('equal')\nax.set_xlim(-1.5, r+1)\nax.set_ylim(-1.5, r+2)\nax.axis('off')\n\nplt.tight_layout()\nplt.show()[/python]", "answer": "$89.1$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 476, @@ -3805,7 +4280,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nmain = patches.Arc((0, 0), 18, 4, theta1=180, theta2=360, color='black', linewidth=2)\nax.add_patch(main)\n\ntop_main = patches.Arc((0, 3), 18, 4, theta1=-204, theta2=24, color='black', linewidth=2)\nax.add_patch(top_main)\n\nsmall = patches.Arc((0, 3), 8, 2, theta1=180, theta2=360, color='black', linewidth=2)\nax.add_patch(small)\n\nax.plot([-4, -4], [3, 7], color='black', linewidth=2)\nax.plot([4, 4], [3, 7], color='black', linewidth=2)\n\n\nax.plot([-9, -9], [0, 3], color='black', linewidth=2)\nax.plot([9, 9], [0, 3], color='black', linewidth=2)\n\ntop_small = patches.Ellipse((0, 7), 8, 2, fill=False, color='black', linewidth=2)\nax.add_patch(top_small)\n\nax.plot([0, 9], [0, 0], color='black')\nax.plot([0, 0], [0, 0], 'ko')\nax.text(4.5, 0.1, \"$\\\\bf{9\\\\ cm}$\", fontsize=14, style='italic')\n\nax.plot([0, 4], [7, 7], color='black')\nax.plot([0, 0], [7, 7], 'ko')\nax.text(2, 7.3, \"$\\\\bf{4\\\\ cm}$\", fontsize=14, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-12, 15)\nax.set_ylim(-4, 12)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$609.47$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 477, @@ -3813,7 +4289,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\nfrom matplotlib.patches import Arrow\n\nfig, ax = plt.subplots(figsize=(6, 6))\n\nellipse_width = 8.1\nellipse_height = 1\nrect_height = 2.1\nsmall_cylinder_width = 3.1\nsmall_ellipse_height = 0.8\nsmall_cylinder_height = 6\n\ntop_center = (0, small_cylinder_height/2 + rect_height)\ntop_ellipse = patches.Arc(\n top_center, ellipse_width, ellipse_height*2, \n angle=0, theta1=0, theta2=360, linewidth=1.5, color='black'\n)\nax.plot([0, ellipse_width / 2], [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height], color='black')\nax.text(ellipse_width / 4, small_cylinder_height/2 + rect_height + 0.1, \"$\\\\bf{8\\\\ cm}$\", fontsize=8)\nax.add_patch(top_ellipse)\n\nax.add_line(plt.Line2D(\n [ellipse_width/2, ellipse_width/2], \n [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height - rect_height],\n color='black'\n))\nax.add_line(plt.Line2D(\n [-ellipse_width/2, -ellipse_width/2], \n [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height - rect_height],\n color='black'\n))\nax.plot([ellipse_width/2 + 1, ellipse_width/2 + 1], [small_cylinder_height/2 + rect_height, -small_cylinder_height/2 - rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [small_cylinder_height/2, small_cylinder_height/2], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [small_cylinder_height/2 + rect_height, small_cylinder_height/2 + rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [-small_cylinder_height/2 - rect_height, -small_cylinder_height/2 - rect_height], color='black')\nax.plot([ellipse_width/2 + 1.1, ellipse_width/2 + 0.9], [-small_cylinder_height/2, -small_cylinder_height/2], color='black')\nax.text(ellipse_width/2 + 1.2, (small_cylinder_height + rect_height)/2, \"$\\\\bf{2\\\\ cm}$\", fontsize=8)\nax.text(ellipse_width/2 + 1.2, -(small_cylinder_height + rect_height)/2, \"$\\\\bf{2\\\\ cm}$\", fontsize=8)\n\narrow1 = Arrow(\n x=0, y=-rect_height/2,\n dx=small_cylinder_width/2, dy=0,\n width=0.2,\n color='black'\n)\narrow2 = Arrow(\n x=0, y=-rect_height/2,\n dx=-small_cylinder_width/2, dy=0,\n width=0.2,\n color='black'\n)\nax.add_patch(arrow1)\nax.add_patch(arrow2)\nbottom_center = (0, -small_cylinder_height/2 - rect_height)\nbottom_ellipse = patches.Arc(\n bottom_center, ellipse_width, ellipse_height*2, \n angle=0, theta1=180, theta2=360, linewidth=1.5, color='black'\n)\nax.add_patch(bottom_ellipse)\n\nax.add_line(plt.Line2D(\n [ellipse_width/2, ellipse_width/2], \n [-small_cylinder_height/2 - rect_height, -small_cylinder_height/2],\n color='black'\n))\nax.add_line(plt.Line2D(\n [-ellipse_width/2, -ellipse_width/2], \n [-small_cylinder_height/2 - rect_height, -small_cylinder_height/2],\n color='black'\n))\n\ntop_inner_ellipse = patches.Arc(\n (0, small_cylinder_height/2), ellipse_width, ellipse_height*2,\n angle=0, theta1=180, theta2=360, linewidth=1.2, color='black'\n)\nax.plot([0, ellipse_width / 2], [-small_cylinder_height/2 - rect_height - 1.3, -small_cylinder_height/2 - rect_height - 1.3], color='black')\nax.text(ellipse_width / 4, -small_cylinder_height/2 - rect_height - 1.7, \"$\\\\bf{8\\\\ cm}$\", fontsize=8)\nax.plot([0, 0], [-small_cylinder_height/2 - rect_height - 1.2, -small_cylinder_height/2 - rect_height - 1.4], color='black')\nax.plot([ellipse_width / 2, ellipse_width / 2], [-small_cylinder_height/2 - rect_height - 1.2, -small_cylinder_height/2 - rect_height - 1.4], color='black')\nax.add_patch(top_ellipse)\nax.add_patch(top_inner_ellipse)\n\nbottom_inner_ellipse = patches.Arc(\n (0, -small_cylinder_height/2), ellipse_width, ellipse_height*2, theta1=-211, theta2=31, linewidth=1.2, color='black'\n)\nax.add_patch(bottom_inner_ellipse)\n\nax.add_line(plt.Line2D(\n [small_cylinder_width/2, small_cylinder_width/2], \n [-small_cylinder_height/2, small_cylinder_height/2 - 1],\n color='black'\n))\nax.add_line(plt.Line2D(\n [-small_cylinder_width/2, -small_cylinder_width/2], \n [-small_cylinder_height/2, small_cylinder_height/2 - 1],\n color='black'\n))\n\nmid_top = (0, small_cylinder_height/2)\nmid_bottom = (0, -small_cylinder_height/2)\n\nmid_ellipse_bottom = patches.Arc(\n mid_bottom, small_cylinder_width, small_ellipse_height, theta1=180, theta2=360, color='black'\n)\nax.add_patch(mid_ellipse_bottom)\n\nax.set_xlim(-6, 6)\nax.set_ylim(-10, 7)\nax.set_aspect('equal')\nax.axis('off')\n\nplt.show()[/python]", "answer": "$874.93$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 478, @@ -3821,7 +4298,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nO = [0, 0]\nA = [-5, 4]\nB = [5, 4]\nC = [5, -4]\nD = [-5, -4]\nE = [2, 10]\nF = [12, 10]\nG = [12, 2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.text(B[0] + 0.5, 0, \"$\\\\bf{8\\\\ cm}$\", fontsize=11)\nax.plot([C[0], D[0]], [C[1], D[1]], color='black')\nax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\nax.plot([F[0], G[0]], [F[1], G[1]], color='black')\nax.plot([G[0], C[0]], [G[1], C[1]], color='black')\nax.text((C[0] + G[0]) / 2 + 0.5, (C[1] + G[1]) / 2 - 0.5, \"$\\\\bf{14\\\\ cm}$\", fontsize=11)\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n\n\ntop_small = patches.Circle((0, 0), 2, fill=False, color='black', linewidth=2)\nax.plot([0, 2], [0, 0], color='black')\nax.add_patch(top_small)\n\nax.text(0 - 1.3, -A[1] - 1, \"$\\\\bf{10\\\\ cm}$\", fontsize=11)\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]", "answer": "$944.07$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 479, @@ -3829,7 +4307,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nO = [0, 0]\nA = [-5.1, 4.1]\nB = [5.1, 4.1]\nC = [5.1, -4.1]\nD = [-5.1, -4.1]\nE = [2, 6]\nF = [12.2, 6]\nG = [12.2, -2.2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], D[0]], [C[1], D[1]], color='black')\nax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\nax.plot([F[0], G[0]], [F[1], G[1]], color='black')\nax.plot([G[0], C[0]], [G[1], C[1]], color='black')\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n\nax.plot([0, 0], [-3.8, -4.4], color='black')\n\nax.plot([0, 0], [3.8, 4.4], color='black')\nax.plot([0, 0], [-3.8, -4.4], color='black')\nax.plot([7.1, 7.1], [5.7, 6.3], color='black')\n\nax.plot([-4.8, -5.4], [0, 0], color='black')\nax.plot([11.9, 12.4], [2, 2], color='black')\nax.plot([4.8, 5.4], [0, 0], color='black')\nax.plot([-1.6, -1.3], [5.40, 4.80], color='black')\nax.plot([8.6, 8.8], [-2.8, -3.4], color='black')\nax.text(0 - 1.3, -A[1] - 1, \"$\\\\bf{6\\\\ cm}$\", fontsize=11)\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]", "answer": "$216$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 480, @@ -3837,7 +4316,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nO = [0, 0]\nA = [-5.1, 3.1]\nB = [5.1, 3.1]\nC = [5.1, -3.1]\nD = [-5.1, -3.1]\nE = [5, 10]\nF = [15.2, 10]\nG = [15.2, 3.8]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], D[0]], [C[1], D[1]], color='black')\nax.plot([D[0], A[0]], [D[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\nax.plot([F[0], G[0]], [F[1], G[1]], color='black')\nax.plot([G[0], C[0]], [G[1], C[1]], color='black')\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([B[0], F[0]], [B[1], F[1]], color='black')\n\nax.text(0 - 1.3, -A[1] - 1, \"$\\\\bf{6\\\\ m}$\", fontsize=11)\nax.text(A[0] - 4, 0, \"$\\\\bf{4\\\\ m}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]", "answer": "$288$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 481, @@ -3845,7 +4325,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nA = [0, 4.8]\nB = [-2.5, 0]\nC = [2.5, 0]\nE = [10, 10]\nF = [12.5, 5.2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], A[0]], [C[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.text(-1, -1, \"$\\\\bf{12\\\\ cm}$\", fontsize=11)\nax.text(1.7, 2.4, \"$\\\\bf{10\\\\ cm}$\", fontsize=11)\nax.plot([B[0] - 0.5, B[0] - 0.5], [A[1], 0], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [A[1], A[1]], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [0, 0], color='black')\n\nax.plot([0, 0.5, 0.5], [0.5, 0.5, 0], color='black')\n\n\nax.plot([0, 0], [A[1], 0], color='black', linestyle='--')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\nax.plot([B[0]/2 - 0.4, B[0]/2 + 0.4], [2.6, 2.3], color='black')\nax.plot([C[0]/2 - 0.4, C[0]/2 + 0.4], [2.3, 2.6], color='black')\n\nax.text(B[0] - 3, 2.4, \"$\\\\bf{8\\\\ cm}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]", "answer": "$768$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 482, @@ -3853,7 +4334,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,7))\n\nA = np.array([0, 0])\nB = np.array([10, 0])\nC = np.array([3.6, -4.8])\n\nA0 = C * 0.9\nB0 = A0 + 0.1* (B - C)\nC0 = C + 0.1* (B - C)\n\n\ndy = 12\nA1 = A + np.array([0, dy])\nB1 = B + np.array([0, dy])\nC1 = C + np.array([0, dy])\n\nA10 = A0 + np.array([0, dy])\nB10 = B0 + np.array([0, dy])\nC10 = C0 + np.array([0, dy])\n\nfor p1, p2 in [(B,C), (C,A),\n (A1,B1), (B1,C1), (C1,A1),\n (A,A1), (B,B1), (C,C1),\n (A0, B0), (B0, C0), (A10, B10), (B10, C10)]:\n ax.plot([p1[0], p2[0]],[p1[1], p2[1]],'k', lw=1.5)\n\nax.plot([A[0], B[0]], [A[1], B[1]], 'k--')\nax.text(1, -2.5, r'$6\\,cm$', fontsize=14, ha='center', va='top', style='italic')\nax.text(4, dy+0.45, r'$10\\,cm$', fontsize=14, ha='left', va='center', style='italic')\nax.text(9, -2.65, r'$8\\,cm$', fontsize=14, ha='right', va='center', style='italic')\n\nax.axis('equal')\nax.axis('off')\nplt.show()[/python]", "answer": "$528$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 483, @@ -3861,7 +4343,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(8,4))\n\nA = np.array([0, 0])\nB = np.array([14, 0])\nC = np.array([12, 3])\nD = np.array([2, 3])\n\nw = 5\nA1 = A + np.array([w, w])\nB1 = B + np.array([w, w])\nC1 = C + np.array([w, w])\nD1 = D + np.array([w, w])\n\nfront = np.array([A,B,C,D,A])\nax.plot(front[:,0], front[:,1], 'k', lw=3)\nback = np.array([B1,C1,D1])\nax.plot(back[:,0], back[:,1], 'k', lw=3)\nax.plot([B[0],B1[0]], [B[1],B1[1]], 'k', lw=3)\nax.plot([C[0],C1[0]], [C[1],C1[1]], 'k', lw=3)\nax.plot([D[0],D1[0]], [D[1],D1[1]], 'k', lw=3)\n\nax.plot([7,7],[0,3],'k--', lw=2)\nax.text(7.3,1.1,'3 cm',fontsize=13, va='center')\n\nax.text(7, 3.5, '10 cm', fontsize=13, ha='center', va='bottom')\nax.text(0, 2.2, '5 cm', fontsize=13, ha='right', va='center')\nax.text(18, 7, '5 cm', fontsize=13, ha='left', va='center')\n\nax.set_aspect('equal')\nax.axis('off')\nax.set_xlim(-2,20)\nax.set_ylim(-1,10)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$338$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 484, @@ -3869,7 +4352,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\n\ndx, dy = -3, 3\n\nA = np.array([0, 0])\nB = np.array([w1, 0])\nC = np.array([w1-w2, h1])\nD = np.array([0, h1])\n\nE = np.array([w1, h2])\nF = np.array([w1-w2, h2])\n\nA_ = A + np.array([dx, dy])\nB_ = B + np.array([dx, dy])\nC_ = C + np.array([dx, dy])\nD_ = D + np.array([dx, dy])\nF_ = F + np.array([dx, dy])\nE_ = E + np.array([dx, dy])\n\nax = plt.gca()\nfor p1, p2 in [(A,B),(B,E),(E,F),(F,C), (C, D), (D, A)]:\n ax.plot([p1[0], p2[0]], [p1[1], p2[1]], color='k')\n\nfor p1, p2 in [(A,A_),(C,C_),(D,D_), (E,E_), (F,F_), (A_, D_), (D_, C_), (F_, C_), (E_, F_)]:\n ax.plot([p1[0],p2[0]], [p1[1],p2[1]], color='k')\n\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='k')\n\nax.text(-4, h1/2+dy, '8cm', va='center', ha='right', fontsize=13, fontstyle='italic')\nax.text(w1/2, -1.4, '17cm', va='top', ha='center', fontsize=13, fontstyle='italic')\nax.text(w1 + 1, h2/2, '16cm', va='center', ha='left', fontsize=13, fontstyle='italic')\nax.text(w1-w2/2+dx, h2+dy+0.6, '4cm', va='bottom', ha='center', fontsize=13, fontstyle='italic')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-6, 28)\nplt.ylim(-4, 22)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$534$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 485, @@ -3877,7 +4361,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(7,7))\n\nheight = 10\ndepth = 3\nwidth = 13\n\ndx, dy = 5, 3.5\n\nA = np.array([0,0])\nB = np.array([width,0])\nC = np.array([width, depth])\nD = np.array([0,height])\n\nA1 = A + np.array([dx,dy])\nB1 = B + np.array([dx,dy])\nC1 = C + np.array([dx,dy])\nD1 = D + np.array([dx,dy])\n\n\nax.plot([A[0],B[0]],[A[1],B[1]],'k')\nax.plot([B[0],C[0]],[B[1],C[1]],'k')\nax.plot([C[0],D[0]],[C[1],D[1]],'k')\nax.plot([D[0],A[0]],[D[1],A[1]],'k')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]],'k--')\nax.plot([B1[0],C1[0]],[B1[1],C1[1]],'k')\nax.plot([B1[0],B[0]],[B1[1],B[1]],'k')\nax.plot([C1[0],D1[0]],[C1[1],D1[1]],'k')\nax.plot([D1[0],A1[0]],[D1[1],A1[1]],'k--')\nax.plot([D[0],D1[0]],[D[1],D1[1]],'k')\nax.plot([C[0],C1[0]], [C[1],C1[1]], 'k')\n\nax.plot([A[0], A1[0]], [A[1], A1[1]], 'k--')\n\nax.text(width/2-0.8, -1.1, r'$13\\,cm$', fontsize=15, style='italic')\nax.text(-1.0, height/2, r'$10\\,cm$', fontsize=15, style='italic', va='center', ha='right')\nax.text(width+dx+0.5, dy+depth/2, r'$3\\,cm$', fontsize=15, style='italic', va='center', ha='left')\nax.text((A1[0]+B1[0])/2+0.5, (C1[1]+D1[1])/2+1, r'$y\\,cm$', fontsize=15, style='italic', ha='center')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-4, width+dx+depth+2)\nplt.ylim(-3, max(height,dy+height)+depth+2)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$577.0$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 486, @@ -3885,7 +4370,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\n\nA = [0, 4.8]\nB = [-2.5, 0]\nC = [2.5, 0]\nE = [10, 10]\nF = [12.5, 5.2]\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], A[0]], [C[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.text(-1, -1, \"$\\\\bf{12\\\\ cm}$\", fontsize=11)\nax.text(1.7, 2.4, \"$\\\\bf{10\\\\ cm}$\", fontsize=11)\nax.plot([B[0] - 0.5, B[0] - 0.5], [A[1], 0], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [A[1], A[1]], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [0, 0], color='black')\n\nax.plot([0, 0.5, 0.5], [0.5, 0.5, 0], color='black')\n\n\nax.plot([0, 0], [A[1], 0], color='black', linestyle='--')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\nax.plot([B[0]/2 - 0.4, B[0]/2 + 0.4], [2.6, 2.3], color='black')\nax.plot([C[0]/2 - 0.4, C[0]/2 + 0.4], [2.3, 2.6], color='black')\n\nax.text(B[0] - 3, 2.4, \"$\\\\bf{x\\\\ cm}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]", "answer": "$608$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 487, @@ -3893,7 +4379,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(7,7))\n\nheight = 10\nwidth = 15\nwidth1 = 8\n\ndx, dy = 5, 3.5\n\nA = np.array([0,0])\nB = np.array([width,0])\nC = np.array([width1, height])\nD = np.array([0, height])\n\nA1 = A + np.array([dx,dy])\nB1 = B + np.array([dx,dy])\nC1 = C + np.array([dx,dy])\nD1 = D + np.array([dx,dy])\n\n\nax.plot([A[0],B[0]],[A[1],B[1]],'k')\nax.plot([B[0],C[0]],[B[1],C[1]],'k')\nax.plot([C[0],D[0]],[C[1],D[1]],'k')\nax.plot([D[0],A[0]],[D[1],A[1]],'k')\nax.plot([B1[0],C1[0]],[B1[1],C1[1]],'k')\nax.plot([B1[0],B[0]],[B1[1],B[1]],'k')\nax.plot([C1[0],D1[0]],[C1[1],D1[1]],'k')\nax.plot([D[0],D1[0]],[D[1],D1[1]],'k')\nax.plot([C[0],C1[0]], [C[1],C1[1]], 'k')\n\n\nax.text(width/2-0.8, -1.1, r'$12\\,cm$', fontsize=15, style='italic')\nax.text(-1.0, height/2, r'$5\\,cm$', fontsize=15, style='italic', va='center', ha='right')\nax.text((A[0]+C[0])/2+0.5, C[1]+0.5, r'$9\\,cm$', fontsize=15, style='italic', ha='center')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-4, width+dx+depth+2)\nplt.ylim(-3, max(height,dy+height)+depth+2)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$200.49$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 488, @@ -3901,7 +4388,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(8,6))\n\nbar_width = 16\nbar_depth = 3\nbar_height = 3\nvert_width = 4\nvert_height = 7\nvert_depth = 1\n\ndx, dy = 2, 1.2\n\nA = np.array([0, 0])\nB = np.array([bar_width, 0])\nC = np.array([bar_width, bar_height])\nD = np.array([0, bar_height])\n\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\nC1 = C + np.array([dx, dy])\nD1 = D + np.array([dx, dy])\n\nleft_margin = (bar_width - vert_width) / 2\nright_margin = left_margin + vert_width\n\nE = np.array([left_margin, 0])\nF = E + np.array([vert_width, 0])\nG = F + np.array([0, -vert_height])\nH = E + np.array([0, -vert_height])\n\nE1 = E + np.array([dx, dy])\nF1 = F + np.array([dx, dy])\nG1 = G + np.array([dx, dy])\nH1 = H + np.array([dx, dy])\n\nfor X, Y in [(A,B), (B,C), (C,D), (D,A), (B,B1), (C,C1), (D,D1), (B1,C1), (C1,D1)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(F,G),(G,H),(H,E),(G,G1)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nax.plot([F1[0],G1[0]], [F1[1]-1.3,G1[1]], 'k') \n\n\nax.text(bar_width/2, 2.5, r\"$12\\,cm$\", ha='center', va='top', fontsize=14, style='italic')\nax.text(-0.7, bar_height/2, r\"$1\\,cm$\", ha='right', va='center', fontsize=14, style='italic')\nax.text(bar_width+dx-1, bar_height/2-dy-0.5, r\"$1\\,cm$\", ha='left', va='center', fontsize=14, style='italic')\nax.text((left_margin+vert_width/2 + 4), -vert_height-0.5, r\"$1\\,cm$\", ha='center', va='bottom', fontsize=14, style='italic')\n\nax.text((left_margin+vert_width/2), -vert_height-1, r\"$4\\,cm$\", ha='center', va='bottom', fontsize=14, style='italic')\nax.text(left_margin-0.2, -vert_height/2, r\"$7\\,cm$\", ha='right', va='center', fontsize=14, style='italic')\n\nax.plot([bar_width/6, bar_width/6], [0.35, -0.35], 'k')\nax.plot([bar_width*5/6, bar_width*5/6], [0.35, -0.35], 'k')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-2, 20)\nplt.ylim(-15, 10)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$120$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 489, @@ -3909,7 +4397,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,4))\n\na = 10\nh = a * np.sqrt(3) / 2\n\nO = np.array([0, 0])\nE = np.array([0, 10])\nA = np.array([-a/2 - h/2, -h/2])\nB = np.array([a - h/2, -h/2])\n\ndx, dy = h/2, h\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\n\n\nfor X, Y in [(A, A1), (B,B1), (A,B), (A,E), (A1,E), (B,E), (B1,E)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.plot([E[0],0],[E[1],0], 'k--')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]], 'k--')\n\n\nax.text((B[0]+B1[0])/2+0.5, 0, r\"$10\\,cm$\", fontsize=12, style='italic')\nax.plot([(A[0]+B[0])/2, (A[0]+B[0])/2], [A[1]-0.3, A[1]+0.3], 'k')\nax.plot([(A1[0]+B1[0])/2, (A1[0]+B1[0])/2], [A1[1]-0.3, A1[1]+0.3], 'k')\nax.plot([(A[0]+A1[0])/2-0.25, (A[0]+A1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\nax.plot([(B[0]+B1[0])/2-0.25, (B[0]+B1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\n\nax.plot([0, 1, 1], [1, 1, 0], color='black')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-a, a)\nplt.ylim(-a, h+2)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$256.2$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 490, @@ -3917,7 +4406,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(4,3))\n\n\nellipse = patches.Ellipse((0, 0), 16, 6, fill=False, color='black', linewidth=2)\nax.add_patch(ellipse)\nax.plot([-8, 0, 8], [0, 14, 0], color='black')\nax.plot([0, 8], [0, 0], color='black')\n\nax.text(4.5, 7.5, r\"$10\\,cm$\", fontsize=12, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-12, 12)\nax.set_ylim(-4, 18)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$122.52$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 491, @@ -3925,7 +4415,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(5,5))\n\n\nwidth = 20\nheight = 4\nellipse1 = patches.Ellipse((0, 0), width, height, fill=False, color='black', linewidth=2)\nellipse2 = patches.Ellipse((0, 10), width/2, height/2, fill=False, color='black', linewidth=2)\nax.add_patch(ellipse1)\nax.add_patch(ellipse2)\nax.plot([-width/2, -width/4, width/4, width/2, -width/2], [0, 10, 10, 0, 0], color='black')\nax.plot([0, 8], [0, 0], color='black')\nax.plot([-width/4, 0, width/4], [10, 20, 10], 'k--', color='black')\nax.plot([1, width/2 + 1.5], [21, 0.5], color='black')\nax.plot([0.5, 1.5], [20.75, 21.5], color='black')\nax.plot([width/4 + 0.75, width/4 + 1.75], [10.5, 11.25], color='black')\nax.plot([width/2 + 1, width/2 + 2], [0, 0.5], color='black')\n\nax.plot([0, 0], [10, 10], 'ko')\nax.plot([0, 0], [0, 0], 'ko')\n\nax.text(width/3 + 2.5, 5, r\"$8\\,cm$\", fontsize=12, style='italic')\nax.text(width/4 - 0.5, 15, r\"$8\\,cm$\", fontsize=12, style='italic')\nax.text(-1, 7.5, r\"$2\\,cm$\", fontsize=11, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-width, width)\nax.set_ylim(-4, 30)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$91.11$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 492, @@ -3933,7 +4424,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,4))\n\na = 11\nh = a * np.sqrt(3) / 2\n\nO = np.array([0, 0])\nE = np.array([0, 20])\nA = np.array([-a/2 - h/2, -h/2])\nB = np.array([a - h/2, -h/2])\n\ndx, dy = h/2, h\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\n\n\nfor X, Y in [(B,B1), (A,B), (A,E), (B,E), (B1,E)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.plot([E[0],0],[E[1],0], 'k--')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]], 'k--')\nax.plot([A[0], A1[0]], [A[1], A1[1]], 'k--')\nax.plot([A1[0], E[0]], [A1[1], E[1]], 'k--')\n\n\nax.text((B[0]+B1[0])/2+1, 0, r\"$10\\,mm$\", fontsize=11, style='italic')\nax.text(-dx + 1, A[1]-1.5, r\"$10\\,mm$\", fontsize=11, style='italic')\nax.text(0-4, E[1]/2-2, r\"$8\\,mm$\", fontsize=11, style='italic')\n\nax.plot([0, 1, 1], [1, 1, 0], color='black')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-a, a)\nplt.ylim(-a, 20)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$288.6$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 493, @@ -3941,7 +4433,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(5,4))\n\n\nellipse = patches.Ellipse((0, 0), 7, 20, fill=False, color='black', linewidth=2)\nax.add_patch(ellipse)\nE = np.array([-25, 0])\nax.plot([0, E[0], 0], [10, 0, -10], color='black', linewidth=2)\nax.plot([0, 0], [0, -10], 'k--', linewidth=2)\n\nax.text(-2, 1, r\"$9.1\\,cm$\", fontsize=10, style='italic')\nax.text(E[0]/2-3, 6.5, r\"$13.7\\,cm$\", fontsize=10, style='italic')\nax.plot([0, 0], [0, 0], 'ko')\n\nax.set_aspect('equal')\nax.set_xlim(-28, 5)\nax.set_ylim(-16, 18)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$651.82$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 494, @@ -3949,7 +4442,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots(figsize=(4, 3))\n\nh = 8\n\nA = np.array([0, 4.8])\nB = np.array([-2.5, 0])\nC = np.array([2.5, 0])\nE = np.array([10, 10])\nF = np.array([12.5, 5.2])\nB1 = B + np.array([0, -h])\nC1 = C + np.array([0, -h])\nF1 = F + np.array([0, -h])\n\n\nfor X, Y in [(B,B1), (C,C1), (B1,C1), (C1,F1), (F,F1)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k', color='black')\n\n\nax.plot([A[0], B[0]], [A[1], B[1]], color='black')\nax.plot([B[0], C[0]], [B[1], C[1]], color='black')\nax.plot([C[0], A[0]], [C[1], A[1]], color='black')\n\nax.plot([E[0], F[0]], [E[1], F[1]], color='black')\n\n\nax.plot([A[0], E[0]], [A[1], E[1]], color='black')\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([B[0] - 0.5, B[0] - 0.5], [A[1], 0], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [A[1], A[1]], color='black')\nax.plot([B[0] - 0.7, B[0] - 0.3], [0, 0], color='black')\n\nax.plot([0, 0.5, 0.5], [0.5, 0.5, 0], color='black')\n\n\nax.plot([0, 0], [A[1], 0], color='black', linestyle='--')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\n\nax.plot([C[0], F[0]], [C[1], F[1]], color='black')\nax.plot([B[0]/2 - 0.4, B[0]/2 + 0.4], [2.6, 2.3], color='black')\nax.plot([C[0]/2 - 0.4, C[0]/2 + 0.4], [2.3, 2.6], color='black')\n\nax.text(B[0] - 3, 2.4, \"$\\\\bf{3\\\\ m}$\", fontsize=11)\nax.text(B[0] - 3, -h/2, \"$\\\\bf{6\\\\ m}$\", fontsize=11)\nax.text(-1, -h-1.5, \"$\\\\bf{5\\\\ m}$\", fontsize=11)\nax.text((C1[0] + F1[0])/2, (C1[1] + F1[1])/2-1, \"$\\\\bf{10\\\\ m}$\", fontsize=11)\n\n\nax.axis('off')\nplt.tight_layout()\nplt.show()[/python]", "answer": "$323.10$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 495, @@ -3957,7 +4451,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 9\nmain = patches.Arc(O, r, r, theta1=90, theta2=360, color='black', linewidth=2)\nax.add_patch(main)\n\ndx, dy = 12, 6\ndxy = np.array([dx, dy])\nO = np.array([0, 0])\nO1 = O + np.array([dx, dy])\nbackend1 = patches.Arc(O1, r, r, theta1=90, theta2=120, color='black', linewidth=2)\nbackend2 = patches.Arc(O1, r, r, theta1=120, theta2=300, color='black', linewidth=2, linestyle='--')\nbackend3 = patches.Arc(O1, r, r, theta1=300, theta2=360, color='black', linewidth=2)\nax.add_patch(backend1)\nax.add_patch(backend2)\nax.add_patch(backend3)\n\nA = [0, r/2]\nB = [r/2, 0]\nC = [-r/4, r*1.7/4]\nD = [r/4, -r*1.7/4]\n\nA1 = A + dxy\nB1 = B + dxy\nC1 = C + dxy\nD1 = D + dxy\n\nfor X, Y in [(O,O1), (A, O), (B, O), (A1, O1), (B1, O1), (A, A1), (B, B1), (C, C1), (D, D1)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.text(1, -1, \"$\\\\bf{5\\\\ cm}$\", fontsize=10, style='italic')\n\nax.set_aspect('equal')\nax.set_xlim(-12, 20)\nax.set_ylim(-6, 12)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$520.55$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 496, @@ -3965,7 +4460,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(8,4))\n\nwidth, height = 15, 20\n\nA = np.array([0, 0])\nB = np.array([width, 0])\nD = A + np.array([0, height])\nC = B + np.array([0, height])\n\ndx, dy = -5, 6\ndxy = np.array([dx, dy])\nA1 = A + dxy\nB1 = B + dxy\nC1 = C + dxy\nD1 = D + dxy\n\nx0 = np.array([25, 0])\nA2 = A + x0\nB2 = B + x0\nC2 = C + x0\nD2 = D + x0\n\nA3, B3, C3, D3 = A1 + x0, B1 + x0, C1 + x0, D1 + x0\n\nfor X, Y in [(A, B), (B, C), (C, D), (D, A)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(C1, D1), (D1, A1), (D, D1), (C, C1), (A, A1), (A2, A3)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(A2, B2), (B2, C2), (C2, D2), (D2, A2)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(C3, D3), (D3, A3), (D2, D3), (C2, C3)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\nfor X, Y in [(A1, B1), (B1, C1), (A3, B3), (B3, C3), (B, B1), (B2, B3)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k--')\n\nmain1 = patches.Arc(((B1[0] + B[0])/2, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2), 3, 13, theta1=0, theta2=360, color='black', linewidth=2, linestyle='--')\nax.add_patch(main1)\n\nmain2 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=270, theta2=90, color='black', linewidth=2)\nmain3 = patches.Arc(((A2[0] + A3[0])/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2), 3, 13, theta1=90, theta2=270, color='black', linewidth=2, linestyle='--')\nax.add_patch(main2)\nax.add_patch(main3)\n\nax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 + 6.5], 'k')\nax.plot([(B1[0] + B[0])/2, (A2[0] + A3[0])/2], [((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5, ((B1[1] + C1[1])/2 + (B[1] + C[1])/2)/2 - 6.5], 'k')\n\nax.plot([B[0], A2[0]], [-2, -2], 'k')\nax.plot([B[0], B[0]], [-1, -3], 'k')\nax.plot([A2[0], A2[0]], [-1, -3], 'k')\nax.plot([(A2[0] + A3[0])/2, (A2[0] + A3[0])/2], [((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2, ((A1[1] + D1[1])/2 + (A[1] + D[1])/2)/2+6.5], 'k')\n\nax.text(B2[0] + 1, C[1]/2, r\"$20\\,cm$\", fontsize=12, style='italic')\nax.text((B2[0] + A2[0])/2, -2, r\"$25\\,cm$\", fontsize=12, style='italic')\nax.text((B2[0] + B3[0])/2, (C3[1] + C2[1])/2, r\"$33\\,cm$\", fontsize=12, style='italic')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-8, 60)\nplt.ylim(-5, 30)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$8128.50$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "area" }, { "index": 497, @@ -3973,7 +4469,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfig, ax = plt.subplots(figsize=(5,4))\n\na = 10\nh = a * np.sqrt(3) / 2\n\nO = np.array([0, 0])\nE = np.array([0, 10])\nA = np.array([-a/2 - h/2, -h/2])\nB = np.array([a - h/2, -h/2])\n\ndx, dy = h/2, h\nA1 = A + np.array([dx, dy])\nB1 = B + np.array([dx, dy])\n\n\nfor X, Y in [(A, A1), (B,B1), (A,B), (A,E), (A1,E), (B,E), (B1,E)]:\n ax.plot([X[0],Y[0]], [X[1],Y[1]], 'k')\n\n\nax.plot([E[0],0],[E[1],0], 'k--')\nax.plot([A1[0],B1[0]],[A1[1],B1[1]], 'k--')\n\n\nax.text((B[0]+B1[0])/2+0.5, 0, r\"$10\\,cm$\", fontsize=12, style='italic')\nax.text(-3, 2, r\"$6\\,cm$\", fontsize=12, style='italic')\nax.plot([(A[0]+B[0])/2, (A[0]+B[0])/2], [A[1]-0.3, A[1]+0.3], 'k')\nax.plot([(A1[0]+B1[0])/2, (A1[0]+B1[0])/2], [A1[1]-0.3, A1[1]+0.3], 'k')\nax.plot([(A[0]+A1[0])/2-0.25, (A[0]+A1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\nax.plot([(B[0]+B1[0])/2-0.25, (B[0]+B1[0])/2+0.25], [0.20, -0.20], 'k', lw=2)\n\nax.plot([0, 1, 1], [1, 1, 0], color='black')\n\nax.set_aspect('equal')\nax.axis('off')\nplt.xlim(-a, a)\nplt.ylim(-a, h+2)\nplt.tight_layout()\nplt.show()[/python]", "answer": "$200$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 498, @@ -3981,7 +4478,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 6\nR = 10\nOA = [0, 0]\nOB = [10, 0]\nmainA1 = patches.Arc(OA, r, r, theta1=00, theta2=360, color='black', linewidth=2)\nmainA2 = patches.Arc(OA, r, 2, theta1=00, theta2=360, color='black', linewidth=2)\nax.add_patch(mainA1)\nax.add_patch(mainA2)\nax.plot([0, r/2], [0, 0], color='black')\n\nmainB1 = patches.Arc(OB, R, R, theta1=0, theta2=360, color='black', linewidth=2)\nmainB2 = patches.Arc(OB, r, R, theta1=0, theta2=360, color='black', linewidth=2)\nax.add_patch(mainB1)\nax.add_patch(mainB2)\nax.plot([OB[0], OB[0]], [0, R/2], color='black')\n\n\nax.text(-1, -0.1, r\"$3$\", fontsize=10, style='italic')\nax.text(0, r/2+1, r\"$A$\", fontsize=12, style='italic')\nax.text(OB[0], R/2+1, r\"$B$\", fontsize=12, style='italic')\n\n\nax.set_aspect('equal')\nax.set_xlim(-5, 20)\nax.set_ylim(-6, 6)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$36$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "volume" }, { "index": 499, @@ -3989,7 +4487,8 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 6\nR = 10\nOA = [0, 0]\nOB = [10, 0]\nmainA1 = patches.Arc(OA, r, r, theta1=00, theta2=360, color='black', linewidth=2)\nmainA2 = patches.Arc(OA, r, 2, theta1=00, theta2=360, color='black', linewidth=2)\nax.add_patch(mainA1)\nax.add_patch(mainA2)\nax.plot([0, r/2], [0, 0], color='black')\n\nmainB1 = patches.Arc(OB, R, R, theta1=0, theta2=360, color='black', linewidth=2)\nmainB2 = patches.Arc(OB, r, R, theta1=0, theta2=360, color='black', linewidth=2)\nax.add_patch(mainB1)\nax.add_patch(mainB2)\nax.plot([OB[0], OB[0]], [0, R/2], color='black')\n\n\nax.text(-1, -0.1, r\"$3$\", fontsize=10, style='italic')\nax.text(0, r/2+1, r\"$A$\", fontsize=12, style='italic')\nax.text(OB[0], R/2+1, r\"$B$\", fontsize=12, style='italic')\n\n\nax.set_aspect('equal')\nax.set_xlim(-5, 20)\nax.set_ylim(-6, 6)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$64$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "ratio" }, { "index": 500, @@ -3997,6 +4496,7 @@ "geo_code": "[python]\nimport matplotlib.pyplot as plt\nimport matplotlib.patches as patches\n\nfig, ax = plt.subplots(figsize=(6,6))\n\nr = 5\nR = 10\nH, h = 15, 10\nOA = [0, 0]\nOA1 = [0, H]\nOB = [15, 0]\nOB1 = [15, h]\nmainA1 = patches.Arc((0, 0), R, r, theta1=0, theta2=180, color='black', linewidth=2, linestyle='--')\nmainA2 = patches.Arc((0, 0), R, r, theta1=180, theta2=360, color='black', linewidth=2)\nmainA3 = patches.Arc((0, H), R, r, theta1=0, theta2=360, color='black', linewidth=2)\nmainB1 = patches.Arc((15, 0), r, 3, theta1=0, theta2=180, color='black', linewidth=2, linestyle=\"--\")\nmainB2 = patches.Arc((15, 0), r, 3, theta1=180, theta2=360, color='black', linewidth=2)\nmainB3 = patches.Arc((15, h), r, 3, theta1=0, theta2=360, color='black', linewidth=2)\nax.add_patch(mainA1)\nax.add_patch(mainA2)\nax.add_patch(mainA3)\nax.add_patch(mainB1)\nax.add_patch(mainB2)\nax.add_patch(mainB3)\nax.plot([-R/2, -R/2], [0, H], color='black')\nax.plot([R/2, R/2], [0, H], color='black')\n\nax.plot([15-r/2, 15-r/2], [0, h], color='black')\nax.plot([15+r/2, 15+r/2], [0, h], color='black')\n\n\nax.text(12, -4, r\"$Volume\\,= \\,90\\,cm^3$\", fontsize=12, style='italic')\nax.text(R/2+1, H/2, r\"$c\\,\\,cm$\", fontsize=12, style='italic')\nax.text(15+r/2+1, h/2, r\"$20\\,\\,cm$\", fontsize=12, style='italic')\n\n\nax.set_aspect('equal')\nax.set_xlim(-8, 20)\nax.set_ylim(-6, 20)\nax.axis('off')\n\nplt.show()[/python]", "answer": "$100$", "category": "Global Abstract Integration", - "source": "Mathverse" + "source": "Mathverse", + "problem_type": "ratio" } ] \ No newline at end of file