index
int64 1
500
| problem
stringlengths 37
987
| geo_code
stringlengths 81
4.63k
| answer
stringlengths 3
32
| category
stringclasses 3
values | source
stringclasses 7
values | problem_type
stringclasses 6
values |
|---|---|---|---|---|---|---|
201
|
Seven cookies of radius $1$ inch are cut from a circle of cookie dough, as shown. Neighboring cookies are tangent, and all except the center cookie are tangent to the edge of the dough. The leftover scrap is reshaped to form another cookie of the same thickness. What is the diameter in inches of the scrap cookie?
|
[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]
|
$2\sqrt{2}$
|
Local Relation Composition
|
HARP
|
length
|
202
|
Let $ABCD$ be a rectangle with $AB = 5$ and $BC = 4$. On side $\overline{AB}$, let $E$ be the point such that $EB = 1$. On side $\overline{BC}$, let $G$ be the point such that $CG = 1$. On side $\overline{CD}$, let $F$ be the point such that $DF = 2$. The lines $\overline{AG}$ and $\overline{AC}$ meet $\overline{EF}$ at points $Q$ and $P$, respectively. Compute the value of $\dfrac{PQ}{EF}$.
|
[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]
|
$\frac{10}{91}$
|
Primitive Recognition
|
HARP
|
ratio
|
203
|
In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. Compute the value of $\dfrac{AF}{FD}$.
|
[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]
|
$2$
|
Local Relation Composition
|
HARP
|
ratio
|
204
|
In $\triangle ABC$, let $AB=7$, $BC=8$, and $CA=9$. The altitude from $A$ to $BC$ is $\overline{AH}$. Let points $D$ and $E$ be located on $\overline{AC}$ and $\overline{AB}$, respectively, such that $\overline{BD}$ and $\overline{CE}$ are the angle bisectors of $\angle ABC$ and $\angle BCA$, respectively. These angle bisectors intersect $\overline{AH}$ at points $Q$ and $P$. Find the length of segment $PQ$.
|
[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]
|
$\frac{8}{15}\sqrt{5}$
|
Primitive Recognition
|
HARP
|
length
|
205
|
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. How many times as large is the area of quadrilateral $ABCD$ as the area of $\triangle BCD$?
|
[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]
|
$4$
|
Primitive Recognition
|
HARP
|
ratio
|
206
|
Tamara’s garden contains six rectangular flower beds, each measuring \$6$ feet by \$2$ feet, arranged in three rows with two beds per row. There are \$1$-foot-wide paths both between each bed and around the entire arrangement. What is the combined area, in square feet, of all the walkways?
|
[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]
|
$78$
|
Local Relation Composition
|
HARP
|
area
|
207
|
Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the side length of $ABCD?$
|
[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]
|
$6\sqrt{3}$
|
Local Relation Composition
|
HARP
|
length
|
208
|
Let $ABCDEFGH$ be a cube where $C$ and $E$ are a pair of opposite vertices. Points $J$ and $I$ are the midpoints of $\overline{FB}$ and $\overline{HD}$, respectively. Consider the cross-section of the cube determined by points $E$, $J$, $C$, and $I$. If $R$ is the ratio of the area of quadrilateral $EJCI$ to the area of a face of the cube, what is the value of $R^2$?
|
[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]
|
$\frac{3}{2}$
|
Global Abstract Integration
|
HARP
|
ratio
|
209
|
In the figure below, $N$ congruent semicircles lie along the diameter of a larger semicircle, with their diameters exactly covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of all the small semicircles and $B$ be the area of the region inside the large semicircle but outside the small semicircles. If the ratio $A:B$ is \$1:18$ and if the area of each small semicircle is \$2$, what is the value of $A$?
|
[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]
|
$38$
|
Local Relation Composition
|
HARP
|
area
|
210
|
Consider a rectangular box in which $AB = 3$, $BC = 1$, and $CG = 2$. Let $M$ denote the midpoint of edge $\overline{FG}$. Find the volume of the pyramid whose base is the rectangle $BCHE$ and whose apex is the point $M$.
|
[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]
|
$2$
|
Global Abstract Integration
|
HARP
|
volume
|
211
|
Farmer Pythagoras owns a plot of land shaped like a right triangle, with legs measuring \$3$ and \$4$ units, respectively. At the vertex where these sides form the right angle, he reserves a small square region $S$, unplanted, making it appear from above like a right angle marker. The remainder of the field is planted. If the minimum distance from the square $S$ to the hypotenuse of the triangle is \$2$ units, what fraction of the triangular field has been planted?
|
[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]
|
$\frac{145}{147}$
|
Local Relation Composition
|
HARP
|
ratio
|
212
|
Start with a \$23 \times 27$ rectangle and remove four congruent right triangles with side lengths \$6$, \$8$, and \$10$ from its corners, creating an octagon $ABCDEFGH$ such that $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$, with $\overline{AH}$ positioned along one of the rectangle’s shorter sides. Let $J$ be the midpoint of $\overline{AH}$. Connect $J$ to vertices $B$, $C$, $D$, $E$, $F$, and $G$ to partition the octagon into \$7$ triangles. Determine the area of the convex polygon whose vertices are the centroids of these \$7$ triangles.
|
[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]
|
$184$
|
Local Relation Composition
|
HARP
|
area
|
213
|
Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, What is the perimeter, in feet, of rectangle $ABCD$?
|
[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]
|
$50$
|
Primitive Recognition
|
HARP
|
length
|
214
|
A square contains \$81$ equally spaced grid points, including those on its boundary. Let $P$ be the grid point at the exact center of the square. If point $Q$ is randomly selected from the remaining \$80$ grid points, what is the probability that the segment $PQ$ forms a line of symmetry of the square?
|
[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]
|
$\frac{2}{5}$
|
Primitive Recognition
|
HARP
|
ratio
|
215
|
In triangle $\triangle ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $\overline{BC}$ and line $\overline{AE}$. How many times as large is the area of $\triangle ABC$ compared to the area of $\triangle EBF$
|
[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]
|
$12$
|
Local Relation Composition
|
HARP
|
ratio
|
216
|
The figure below depicts line $\ell$ and a regular, infinite, repeating arrangement made up of squares and line segments. Excluding the identity transformation, how many of the following types of rigid motions will map the figure onto itself?
|
[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]
|
$2$
|
Local Relation Composition
|
HARP
|
count
|
217
|
Let $ABC$ be an isosceles right triangle with legs of length \$1$, and let $C$ be the right angle. On the hypotenuse $\overline{AC}$, construct triangle $ACD$ externally such that $ACD$ is a right triangle with the right angle at $C$ and the perimeters of triangles $ABC$ and $ACD$ are equal. What is the value of $\sin(2\angle BAD)$?
|
[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]
|
$\frac{7}{9}$
|
Local Relation Composition
|
HARP
|
count
|
218
|
Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ Find $AB$
|
[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]
|
$15$
|
Local Relation Composition
|
HARP
|
length
|
219
|
A large square is entirely covered with $n^2$ gray square tiles, each with side length $s$ inches. Each tile is separated from adjacent tiles and the boundary of the large square by a uniform border of width $d$ inches. The diagram below illustrates the arrangement for $n=3$. If $n=24$, so there are \$576$ gray tiles, and these tiles collectively occupy \$64\%$ of the total area of the entire large square, what is the value of the ratio $\frac{d}{s}$ for this configuration?
|
[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]
|
$\frac{6}{25}$
|
Local Relation Composition
|
HARP
|
ratio
|
220
|
Let $\triangle AMC$ be an isosceles triangle with $AM = AC$. The medians $\overline{MV}$ and $\overline{CU}$ are equal in length and are perpendicular to each other, where $MV=CU=12$. Find the area of triangle $AMC$.
|
[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]
|
$96$
|
Local Relation Composition
|
HARP
|
area
|
221
|
A square sheet of paper with vertices labeled $A$, $B$, $C$, and $D$ in order and with side length \$1$ is folded so that vertex $C$ lands on edge $\overline{AD}$ at the point $C'$. The fold causes edge $\overline{BC}$ to cross edge $\overline{AB}$ at point $E$. If the distance from $C'$ to $D$ is $\frac{1}{3}$, what is the perimeter of triangle $AEC'$?
|
[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]
|
$2$
|
Local Relation Composition
|
HARP
|
length
|
222
|
Let $ABCD$ be an isosceles trapezoid with $AB$ and $CD$ as its parallel sides, where $AB > CD$. Suppose a point is chosen inside the trapezoid, and segments are drawn from this point to each of the four vertices, subdividing $ABCD$ into four triangles. Starting with the triangle whose base is $\overline{CD}$ and proceeding clockwise, the areas of the triangles are \$2$, \$3$, \$4$, and \$5$, respectively. Find the value of the ratio $\dfrac{AB}{CD}$.
|
[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]
|
$2+\sqrt{2}$
|
Local Relation Composition
|
HARP
|
ratio
|
223
|
Let $ABC$ be an equilateral triangle with side length \$840$. Point $D$ is positioned on the same side of $BC$ as $A$ so that $\overline{BD}$ is perpendicular to $\overline{BC}$. Through $D$, draw a line $\ell$ parallel to $BC$; let $\ell$ intersect $AB$ at $E$ and $AC$ at $F$. Let $G$ be a point on $\ell$ such that $F$ lies strictly between $E$ and $G$, and triangle $AFG$ is isosceles. If the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is \$8:9$, find the length $AF$.
|
[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]
|
$336$
|
Local Relation Composition
|
HARP
|
length
|
224
|
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the side length of the square?
|
[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]
|
$\sqrt{117}$
|
Local Relation Composition
|
HARP
|
length
|
225
|
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What multiple of $\angle CDE$ is the measure of $\angle AFE$?
|
[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]
|
$\frac{17}{11}$
|
Local Relation Composition
|
HARP
|
ratio
|
226
|
In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\circ$. The enclosed area of the hexagon is $6\sqrt{3}$. What is the perimeter of the hexagon?
|
[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]
|
$12\sqrt{3}$
|
Local Relation Composition
|
HARP
|
length
|
227
|
Let $ABCD$ be an isosceles trapezoid with $AB = CD$ and $\overline{BC} \parallel \overline{AD}$. Points $X$ and $Y$ are located on diagonal $\overline{AC}$ with $X$ lying between $A$ and $Y$. It is given that $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. Determine the area of the trapezoid $ABCD$.
|
[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]
|
$3\sqrt{35}$
|
Local Relation Composition
|
HARP
|
area
|
228
|
The diagram below presents a polygon $ABCDEFGH$ made from rectangles and right triangles. If this shape is cut out and folded along the dashed lines, it forms a triangular prism. Given that $AH = EF = 8$ and $GH = 14$, find the volume of the resulting prism.
|
[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]
|
$192$
|
Global Abstract Integration
|
HARP
|
volume
|
229
|
In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$, $AP = 3$, and $PD = 2$. Find $BP$?
|
[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]
|
$4$
|
Local Relation Composition
|
HARP
|
length
|
230
|
A regular heptagon (regular \$7$-gon) is inscribed in a circle of radius \$1$. Consider all \$21$ distinct edges and diagonals of the heptagon. What is the sum of the fourth powers of their lengths?
|
[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]
|
$147$
|
Local Relation Composition
|
HARP
|
count
|
231
|
Consider a rectangle whose sides are parallel to the coordinate axes and whose opposite vertices are at $(15, 3)$ and $(16, 5)$. One line passes through points $A(0, 0)$ and $B(3, 1)$, and a second line passes through points $C(0, 10)$ and $D(2, 9)$. How many points lying on the rectangle also lie on at least one of these two lines?
|
[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]
|
$1$
|
Local Relation Composition
|
HARP
|
count
|
232
|
A bicycle race course has \$7$ water stations that are placed at equal intervals between the start and finish. Additionally, \$2$ repair stations are positioned at equal intervals along the same route. If the \$3$rd water station is \$2$ miles beyond the \$1$st repair station, what is the total length of the race, in miles?
|
[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]
|
$48$
|
Primitive Recognition
|
HARP
|
length
|
233
|
A regular octahedron consists of eight equilateral triangular faces, with four faces coming together at each vertex. Jun constructs a regular octahedron by folding a paper pattern whose faces are numbered. After assembly, which numbered face will be immediately to the right of vertex $Q$?
|
[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]
|
$1$
|
Global Abstract Integration
|
HARP
|
count
|
234
|
A set of concentric circles is drawn, starting with a circle of radius \(1\), and each next circle has its radius increased by \(1\) from the previous one. There are an even number of circles, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius \(2\) but outside the circle of radius \(1\), then the region inside the circle of radius \(4\) but outside the circle of radius \(3\), and so on. What is the least even number of circles needed so that the total shaded area is at least \(2023\pi\)?
|
[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]
|
$64$
|
Local Relation Composition
|
HARP
|
count
|
235
|
A square of area \(2\) is inscribed, with its sides not parallel, inside a larger square of area \(3\) such that their centers coincide. This arrangement creates four congruent right triangles at the corners of the larger square. In one of these triangles, what is the ratio of the shorter leg to the longer leg?
|
[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]
|
$2-\sqrt{3}$
|
Local Relation Composition
|
HARP
|
ratio
|
236
|
Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the diameter of $C_4$?
|
[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]
|
$\frac{3}{14}$
|
Local Relation Composition
|
HARP
|
length
|
237
|
Four squares with side lengths \(4\), \(7\), \(9\), and \(10\) are placed so that their left sides and bottom sides are all aligned, and their sizes increase from smallest to largest. The colors of the squares alternate: the smallest square is white, the next is gray, then white again, and the largest is gray. What is the total area of the gray region that remains visible (not covered by any other square), in square units?
|
[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]
|
$52$
|
Local Relation Composition
|
HARP
|
area
|
238
|
The coordinates of $\triangle ABC$ are $A(5,7)$, $B(11,7)$, and $C(3,y)$, with $y>7$. The area of $\triangle ABC$ is 12. Find the length of $BC$?
|
[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]
|
$4\sqrt{5}$
|
Primitive Recognition
|
HARP
|
length
|
239
|
Three concentric circles with center \(O\) have radii \(1\), \(2\), and \(3\). Points \(B\) and \(C\) both lie on the outermost circle. The entire annulus between the circles of radius \(1\) and \(2\) is shaded, and an additional sector (defined by central angle \(BOC\)) of the annulus between the circles of radius \(2\) and \(3\) is also shaded. If the total shaded area is equal to the unshaded area, what is the degree measure of \(\angle BOC\)?
|
[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]
|
$108$
|
Local Relation Composition
|
HARP
|
angle
|
240
|
Jean creates a stained glass artwork shaped like two adjacent mountains, as shown in the diagram below. The first peak has a height of $8$ feet and the second peak has a height of $12$ feet. Each mountain peak forms a right angle ($90^\circ$), and both of the straight sides of the peaks rise at an angle of $45^\circ$ from the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$
|
[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]
|
$5$
|
Local Relation Composition
|
HARP
|
length
|
241
|
Let \(\triangle ABC\) be a triangle with area \(240\). Points \(X\), \(Y\), and \(Z\) are located on sides \(AB\), \(BC\), and \(CA\) respectively, such that \(\frac{AX}{BX} = 3\), \(\frac{BY}{CY} = 4\), and \(\frac{CZ}{AZ} = 5\). What is the area of triangle \(XYZ\)?
|
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;
draw(A--B--C--cycle^^X--Y--Z--cycle);
label("$A$",A,N);
label("$B$",B,S);
label("$C$",C,E);
label("$X$",X,W);
label("$Y$",Y,S);
label("$Z$",Z,NE);[/asy]
|
$122$
|
Local Relation Composition
|
olympiads
|
area
|
242
|
In the figure below, \(\angle ABC\) is a right angle. Point \(D\) lies on segment \(\overline{BC}\), and segment \(\overline{AD}\) is the angle bisector of \(\angle CAB\). Points \(E\) and \(F\) are located on segments \(\overline{AB}\) and \(\overline{AC}\), respectively, such that \(AE = 3\), \(AF = 10\), \(EB = 9\), and \(FC = 27\). Let \(G\) denote the point where \(\overline{DF}\) intersects \(\overline{CE}\). Determine the integer nearest to the area of quadrilateral \(DCFG\).
|
[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]
|
$148$
|
Local Relation Composition
|
AIME-83-24
|
area
|
243
|
Consider a fixed regular $n$ -gon of unit side. When a second regular $n$ -gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure.
Let $10$ be the area of a regular $n$ -gon of unit side, and let $5$ be the area of a regular $n$ -gon of unit circumradius. Calculate the area enclosed by $\kappa$.
|
[asy]
int n=9;
draw(polygon(n));
for (int i = 0; i<n;++i) {
draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4));
draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2));
}
[/asy]
|
$50$
|
Global Abstract Integration
|
aops_forum
|
area
|
244
|
Let $n\ge 2$ be an integer. Thibaud the Tiger lays $n$ $2\times 2$ overlapping squares out on a table, such that the centers of the squares are equally spaced along the line $y=x$ from $(0,0)$ to $(1,1)$ (including the two endpoints). For example, for $n=4$ the resulting figure is shown below, and it covers a total area of $\frac{23}{3}$ .
Find the minimum $n$ such that the figure covers an area of at least $\sqrt{63}$ .
|
[asy]
fill((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--cycle, lightgrey);
fill((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--cycle, lightgrey);
fill((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--cycle, lightgrey);
fill((1,1)--(3,1)--(3,3)--(1,3)--cycle, lightgrey);
draw((0.33333333333333,2)--(2,2)--(2,0.333333333333), dashed+grey+linewidth(0.4));
draw((0.66666666666666,2.3333333333333)--(2.3333333333333,2.3333333333333)--(2.3333333333333,0.66666666666), dashed+grey+linewidth(0.4));
draw((1,2.666666666666)--(2.666666666666,2.666666666666)--(2.666666666666,1), dashed+grey+linewidth(0.4));
draw((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--(0,0),linewidth(0.4));
draw((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--(0.333333333333,.333333333333),linewidth(0.4));
draw((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--(0.6666666666666,0.6666666666666),linewidth(0.4));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle,linewidth(0.4));
[/asy]
|
$17$
|
Local Relation Composition
|
aops_forum
|
count
|
245
|
In the figure, AQPB and ASRC are squares, and AQS is an equilateral triangle. If QS = 12 and BC = x, what is the value of x?
|
[asy]
unitsize(16);
pair A,B,C,P,Q,R,T;
A=(3.4641016151377544, 2);
B=(0, 0);
C=(6.928203230275509, 0);
P=(-1.9999999999999991, 3.464101615137755);
Q=(1.4641016151377544, 5.464101615137754);
R=(8.928203230275509, 3.4641016151377544);
T=(5.464101615137754, 5.464101615137754);
dot(A);dot(B);dot(C);dot(P);
dot(Q);dot(R);dot(T);
label(" $A$ ", (3.4641016151377544, 2),E);
label(" $B$ ", (0, 0),S);
label(" $C$ ", (6.928203230275509, 0),S);
label(" $P$ ", (-1.9999999999999991, 3.464101615137755), W);
label(" $Q$ ", (1.4641016151377544, 5.464101615137754),N);
label(" $R$ ", (8.928203230275509, 3.4641016151377544),E);
label(" $S$ ", (5.464101615137754, 5.464101615137754),N);
draw(B--C--A--B);
draw(B--P--Q--A--B);
draw(A--C--R--T--A);
draw(Q--T--A--Q);
label(" $x$ ", (3.4641016151377544, 0), S);
label(" $4$ ", (Q+T)/2, N);[/asy]
|
$12\sqrt{3}$
|
Local Relation Composition
|
aops_forum
|
length
|
246
|
In an acute triangle $ABC$ let $K,L,$ and $M$ be the midpoints of sides $AB,BC,$ and $CA,$ respectively. From each of $K,L,$ and $M$ drop two perpendiculars to the other two sides of the triangle; e.g., drop perpendiculars from $K$ to sides $BC$ and $CA,$ etc. The resulting $6$ perpendiculars intersect at points $Q,S,$ and $T$ as in the figure to form a hexagon $KQLSMT$ inside triangle $ABC.$ The area of triangle $ABC$ is $n$ times of the area of hexagon $KQLSMT$. Find the value of $n$
|
[asy]import graph; size(10cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
real xmin = 11.888712276357234, xmax = 17.841346447833423, ymin = 10.61620970860601, ymax = 15.470685507068502;
pen zzttqq = rgb(0.6,0.2,0.); pen qqwuqq = rgb(0.,0.39215686274509803,0.);
pair A = (12.488234161849352,12.833838721895551), B = (16.50823416184936,15.093838721895553), C = (16.28823416184936,11.353838721895551), K = (14.498234161849355,13.963838721895552), L = (16.39823416184936,13.223838721895552), M = (14.388234161849356,12.093838721895551), D = (13.615830174638527,13.467760858438725), F = (15.75135711740064,11.562938202365055), G = (15.625830174638523,14.597760858438724), H = (16.435061748056253,13.849907687412797), T = (14.02296781802369,12.74356027153236), Q = (16.032967818023693,13.873560271532357), O = (16.325061748056253,11.979907687412794);
draw(A--B--C--cycle, zzttqq);
draw((13.426050287639166,13.361068683160477)--(13.532742462917415,13.171288796161116)--(13.722522349916774,13.277980971439364)--D--cycle, qqwuqq);
draw((14.054227993863618,12.223925334689998)--(14.133240861538676,12.426796211152979)--(13.930369985075695,12.505809078828037)--(13.851357117400637,12.302938202365056)--cycle, qqwuqq);
draw((16.337846386707046,12.19724654447628)--(16.12050752964356,12.210031183127075)--(16.107722890992765,11.992692326063588)--O--cycle, qqwuqq);
draw((15.830369985075697,11.765809078828037)--(15.627499108612716,11.844821946503092)--(15.54848624093766,11.641951070040111)--F--cycle, qqwuqq);
draw((15.436050287639164,14.491068683160476)--(15.542742462917412,14.301288796161115)--(15.73252234991677,14.407980971439365)--G--cycle, qqwuqq);
draw((16.217722890992764,13.86269232606359)--(16.20493825234197,13.645353469000101)--(16.42227710940546,13.63256883034931)--H--cycle, qqwuqq);
Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true);
yaxis(ymin, ymax, Ticks(laxis, Step = 1., Size = 2, NoZero),EndArrow(6), above = true);
draw(A--B, zzttqq);
draw(B--C, zzttqq);
draw(C--A, zzttqq);
draw(M--D);
draw(K--(13.851357117400637,12.302938202365056));
draw(F--L);
draw(L--G);
draw(K--H);
draw(M--O);
dot(A,dotstyle);
label(" $A$ ", (12.52502834296331,12.93568440300881), NE * labelscalefactor);
dot(B,dotstyle);
label(" $B$ ", (16.548187989892043,15.193580123223922), NE * labelscalefactor);
dot(C,dotstyle);
label(" $C$ ", (16.332661580235147,11.457789022504372), NE * labelscalefactor);
dot(K,linewidth(3.pt) + dotstyle);
label(" $K$ ", (14.536608166427676,14.02357961365791), NE * labelscalefactor);
dot(L,linewidth(3.pt) + dotstyle);
label(" $L$ ", (16.43529320388129,13.28463192340569), NE * labelscalefactor);
dot(M,linewidth(3.pt) + dotstyle);
label(" $M$ ", (14.433976542781535,12.155684063298134), NE * labelscalefactor);
dot(D,linewidth(3.pt) + dotstyle);
dot((13.851357117400637,12.302938202365056),linewidth(3.pt) + dotstyle);
dot(F,linewidth(3.pt) + dotstyle);
dot(G,linewidth(3.pt) + dotstyle);
dot(H,linewidth(3.pt) + dotstyle);
dot((15.922967818023695,12.003560271532354),linewidth(3.pt) + dotstyle);
label(" $S$ ", (15.96318773510904,12.063315602016607), NE * labelscalefactor);
dot(T,linewidth(3.pt) + dotstyle);
label(" $T$ ", (14.064502697655428,12.802263292268826), NE * labelscalefactor);
dot(Q,linewidth(3.pt) + dotstyle);
label(" $Q$ ", (16.076082521119794,13.931211152376383), NE * labelscalefactor);
dot(O,linewidth(3.pt) + dotstyle);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]
|
$2$
|
Local Relation Composition
|
aops_forum
|
ratio
|
247
|
In the figure below, each side of the rhombus has length 8 centimeters.
The circle lies entirely within the rhombus. The area of the circle is $n$ square centimeters, where $n$ is a positive integer. Compute the number of possible values of $n$ .
|
[asy]
import graph;
unitsize(2.0cm);
real w = sqrt(3);
draw((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- cycle);
filldraw(Circle((-0.5, 0), 0.8 / sqrt(pi)), gray);
label(" $60^\circ$ ", (w - 0.1, 0), W);
[/asy]
|
$37$
|
Local Relation Composition
|
aops_forum
|
count
|
248
|
The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically with equal distance 13. Find the area of the irregularly shaped ten sided figure shown.
|
[asy]
import graph; size(5cm);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
draw((-2,5)--(-3,4), linewidth(1.6));
draw((-3,4)--(-2,1), linewidth(1.6));
draw((-2,1)--(1,0), linewidth(1.6));
draw((1,0)--(2,1), linewidth(1.6));
draw((2,1)--(1,3), linewidth(1.6));
draw((1,3)--(1,4), linewidth(1.6));
draw((1,4)--(2,5), linewidth(1.6));
draw((2,5)--(0,5), linewidth(1.6));
draw((-2,5)--(-1,4), linewidth(1.6));
draw((-1,4)--(0,5), linewidth(1.6));
dot((-3,5),linewidth(6pt) + dotstyle);
dot((-2,5),linewidth(6pt) + dotstyle);
dot((-1,5),linewidth(6pt) + dotstyle);
dot((0,5),linewidth(6pt) + dotstyle);
dot((1,5),linewidth(6pt) + dotstyle);
dot((2,5),linewidth(6pt) + dotstyle);
dot((2,4),linewidth(6pt) + dotstyle);
dot((2,3),linewidth(6pt) + dotstyle);
dot((2,2),linewidth(6pt) + dotstyle);
dot((2,1),linewidth(6pt) + dotstyle);
dot((2,0),linewidth(6pt) + dotstyle);
dot((-3,4),linewidth(6pt) + dotstyle);
dot((-3,3),linewidth(6pt) + dotstyle);
dot((-3,2),linewidth(6pt) + dotstyle);
dot((-3,1),linewidth(6pt) + dotstyle);
dot((-3,0),linewidth(6pt) + dotstyle);
dot((-2,0),linewidth(6pt) + dotstyle);
dot((-2,1),linewidth(6pt) + dotstyle);
dot((-2,2),linewidth(6pt) + dotstyle);
dot((-2,3),linewidth(6pt) + dotstyle);
dot((-2,4),linewidth(6pt) + dotstyle);
dot((-1,4),linewidth(6pt) + dotstyle);
dot((0,4),linewidth(6pt) + dotstyle);
dot((1,4),linewidth(6pt) + dotstyle);
dot((1,3),linewidth(6pt) + dotstyle);
dot((0,3),linewidth(6pt) + dotstyle);
dot((-1,3),linewidth(6pt) + dotstyle);
dot((-1,2),linewidth(6pt) + dotstyle);
dot((-1,1),linewidth(6pt) + dotstyle);
dot((-1,0),linewidth(6pt) + dotstyle);
dot((0,0),linewidth(6pt) + dotstyle);
dot((1,0),linewidth(6pt) + dotstyle);
dot((1,1),linewidth(6pt) + dotstyle);
dot((1,2),linewidth(6pt) + dotstyle);
dot((0,2),linewidth(6pt) + dotstyle);
dot((0,1),linewidth(6pt) + dotstyle); [/asy]
|
$214.5$
|
Primitive Recognition
|
aops_forum
|
area
|
249
|
In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$ , $GF=13$ , $FC=1$ , and $HJ=7$, the length of $DE$ is ?
|
[asy]
size(200);
defaultpen(fontsize(10));
real r=sqrt(22);
pair 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);
dot(A^^B^^C^^D^^E^^F^^G^^H^^J);
draw(Circle(O, abs(O-D))^^A--B--C--cycle, linewidth(0.7));
label(" $A$ ", A, N);
label(" $B$ ", B, dir(210));
label(" $C$ ", C, dir(330));
label(" $D$ ", D, SW);
label(" $E$ ", E, SE);
label(" $F$ ", F, dir(170));
label(" $G$ ", G, dir(250));
label(" $H$ ", H, SE);
label(" $J$ ", J, dir(0));
label("2", A--G, dir(30));
label("13", F--G, dir(180+30));
label("1", F--C, dir(30));
label("7", H--J, dir(-30));[/asy]
|
$2\sqrt{22}$
|
Local Relation Composition
|
aops_forum
|
length
|
250
|
An equilateral triangle has sides of length $6$ cm. At each vertex, a circle with radius $3$ cm is drawn, as shown in the figure below. The total area of the unshaded regions of the three circles is $a\cdot \pi \text{cm}^2$ . Determine $a$ .
[center][/center]
|
[asy]
size(2.5cm);
draw(circle((0,2sqrt(3)/3),1));
draw(circle((1,-sqrt(3)/3),1));
draw(circle((-1,-sqrt(3)/3),1));
draw((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle);
fill(((0,2sqrt(3)/3) -- arc((0,2sqrt(3)/3), 1, 240, 300) -- cycle),mediumgray);
draw((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle);
fill(((1,-sqrt(3)/3) -- arc((1,-sqrt(3)/3), 1, 180, 120) -- cycle),mediumgray);
draw((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle);
fill(((-1,-sqrt(3)/3) -- arc((-1,-sqrt(3)/3), 1, 0, 60) -- cycle),mediumgray);
[/asy]
|
$22.5$
|
Local Relation Composition
|
aops_forum
|
count
|
251
|
Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $6$ and $8$ , respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$ , respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$ . If $O_1O_2=50$ , find the lenght of line $AB$
|
[asy]
import graph; size(12cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
real xmin = -12.81977592804657, xmax = 32.13023014338037, ymin = -14.185056097058798, ymax = 12.56855801985179;
draw(circle((-3.4277328104418046,-1.4524996726688195), 3), linewidth(1.2));
draw(circle((21.572267189558197,-1.4524996726688195), 4), linewidth(1.2));
draw((-2.5877328104418034,1.4275003273311748)--(20.452267189558192,-5.2924996726687885), linewidth(1.2));
dot((-3.4277328104418046,-1.4524996726688195),linewidth(3pt) + dotstyle);
label(" $O_1$ ", (-4.252707018231291,-1.545940604327141), N * labelscalefactor);
dot((21.572267189558197,-1.4524996726688195),linewidth(3pt) + dotstyle);
label(" $O_2$ ", (21.704189347819636,-1.250863978037686), NE * labelscalefactor);
dot((-2.5877328104418034,1.4275003273311748),linewidth(3pt) + dotstyle);
label(" $A$ ", (-2.3937351324858342,1.6999022848568643), NE * labelscalefactor);
dot((20.452267189558192,-5.2924996726687885),linewidth(3pt) + dotstyle);
label(" $B$ ", (20.671421155806545,-4.9885012443707835), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]
|
$48$
|
Primitive Recognition
|
aops_forum
|
length
|
252
|
Suppose circles $\mathit{W}_1$ and $\mathit{W}2$ , with centres $\mathit{O}_1$ and $\mathit{O}_2$ respectively, intersect at points $\mathit{M}$ and $\mathit{N}$ . Let the tangent on $\mathit{W}_2$ at point $\mathit{N}$ intersect $\mathit{W}_1$ for the second time at $\mathit{B}_1$ . Similarly, let the tangent on $\mathit{W}_1$ at point $\mathit{N}$ intersect $\mathit{W}_2$ for the second time at $\mathit{B}_2$ . Let $\mathit{A}_1$ be a point on $\mathit{W}_1$ which is on arc $\mathit{B}_1\mathit{N}$ not containing $\mathit{M}$ and suppose line $\mathit{A}_1\mathit{N}$ intersects $\mathit{W}_2$ at point $\mathit{A}_2$ . Denote the incentres of triangles $\mathit{B}_1\mathit{A}_1\mathit{N}$ and $\mathit{B}_2\mathit{A}_2\mathit{N}$ by $\mathit{I}_1$ and $\mathit{I}_2$ , respectively.If the degree of $\angle\mathit{I}_1\mathit{MI}_2$ is 37, find the degree of angle $\angle\mathit{O}_1\mathit{MO}_2 + \angle\mathit{I}_1\mathit{MI}_2$.
|
[asy]import graph; size(10.1cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
real xmin = -0.9748626324969808, xmax = 13.38440254515721, ymin = 0.5680051903627492, ymax = 10.99430986899034;
pair O_2 = (7.682929606970993,6.084708172218866), O_1 = (2.180000000000002,6.760000000000007), M = (4.560858774883258,8.585242858926296), B_2 = (10.07334553576748,9.291873850408265), A_2 = (11.49301008867042,4.866805580476367), B_1 = (2.113311869970955,9.759258690628950), A_1 = (0.2203184186713625,4.488514120712773);
draw(circle(O_2, 4.000000000000000));
draw(circle(O_1, 3.000000000000000));
draw((4.048892687647541,4.413249028538064)--B_2);
draw(B_2--A_2);
draw(A_2--(4.048892687647541,4.413249028538064));
draw((4.048892687647541,4.413249028538064)--B_1);
draw(B_1--A_1);
draw(A_1--(4.048892687647541,4.413249028538064));
dot(O_2,dotstyle);
label(" $O_2$ ", (7.788512439159622,6.243082420501817), NE * labelscalefactor);
dot(O_1,dotstyle);
label(" $O_1$ ", (2.298205165350667,6.929370829727937), NE * labelscalefactor);
dot(M,dotstyle);
label(" $M$ ", (4.383466101076183,8.935444641311980), NE * labelscalefactor);
dot((4.048892687647541,4.413249028538064),dotstyle);
label(" $N$ ", (3.855551940133015,3.761885864068922), NE * labelscalefactor);
dot(B_2,dotstyle);
label(" $B_2$ ", (10.19052187145104,9.463358802255147), NE * labelscalefactor);
dot(A_2,dotstyle);
label(" $A_2$ ", (11.80066006232771,4.659339937672310), NE * labelscalefactor);
dot(B_1,dotstyle);
label(" $B_1$ ", (1.981456668784765,10.09685579538695), NE * labelscalefactor);
dot(A_1,dotstyle);
label(" $A_1$ ", (0.08096568938935705,3.973051528446190), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]
|
$74$
|
Global Abstract Integration
|
aops_forum
|
angle
|
253
|
The diagram below shows nine points on a circle where $AB=BC=CD=DE=EF=FG=GH$ . Given that $\angle GHJ=117^\circ$ and $\overline{BH}$ is perpendicular to $\overline{EJ}$ , there are relatively prime positive integers $m$ and $n$ so that the degree measure of $\angle AJB$ is $\frac{m}{n}$ . Find $m imes n$ .
|
[asy]
size(175);
defaultpen(linewidth(0.6));
draw(unitcircle,linewidth(0.9));
string labels[] = {"A","B","C","D","E","F","G"};
int start=110,increment=20;
pair J=dir(210),x[],H=dir(start-7*increment);
for(int i=0;i<=6;i=i+1)
{
x[i]=dir(start-increment*i);
draw(J--x[i]--H);
dot(x[i]);
label(" $"+labels[i]+"$ ",x[i],dir(origin--x[i]));
}
draw(J--H);
dot(H^^J);
label(" $H$ ",H,dir(origin--H));
label(" $J$ ",J,dir(origin--J));
[/asy]
|
$54$
|
Local Relation Composition
|
aops_forum
|
count
|
254
|
In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find the degree of angle $\gamma.$
|
[asy]
unitsize(40);
import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0);
draw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){
return (0.6*cos(t)+0.64,0.6*sin(t)+2.24);
}
draw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){
return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);
}
draw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){
return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);
}
draw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq);
dot((0,0),ds); label(" $O$ ", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label(" $A$ ", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label(" $B$ ", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label(" $C$ ", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label(" $E$ ", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label(" $D$ ", (-0.92,0.7), NE*lsf); label(" $\alpha$ ", (0.48,1.38),NE*lsf); label(" $\beta$ ", (-0.02,0.94),NE*lsf); label(" $\gamma$ ", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle);
[/asy]
|
$50$
|
Primitive Recognition
|
aops_forum
|
angle
|
255
|
Square $ABCD$ is divided into four rectangles by $EF$ and $GH$ . $EF$ is parallel to $AB$ and $GH$ parallel to $BC$ . $\angle BAF = 18^\circ$ . $EF$ and $GH$ meet at point $P$ . The area of rectangle $PFCH$ is twice that of rectangle $AGPE$ . Given that the value of $\angle FAH$ in degrees is $3y$ , find the nearest integer to $y$ .
|
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
pair 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];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H);
label(" $A$ ",D2(A),NW);
label(" $B$ ",D2(B),SW);
label(" $C$ ",D2(C),SE);
label(" $D$ ",D2(D),NE);
label(" $E$ ",D2(E),plain.N);
label(" $F$ ",D2(F),S);
label(" $G$ ",D2(G),W);
label(" $H$ ",D2(H),plain.E);
label(" $P$ ",D2(P),SE);
[/asy]
|
$15$
|
Local Relation Composition
|
aops_forum
|
count
|
256
|
In the following diagram (not to scale), $A$ , $B$ , $C$ , $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$ . Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$ . Find the degree of $\angle OPQ$.
|
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15;
pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);
[/asy]
|
$30$
|
Local Relation Composition
|
aops_forum
|
angle
|
257
|
In the diagram below $ \angle CAB, \angle CBD$ , and $\angle CDE$ are all right angles with side lengths $AC = 3$ , $BC = 5$ , $BD = 12$ , and $DE = 84$ . The distance from point $E$ to the line $AB$ can be expressed as the ratio of two relatively prime positive integers, $m$ and $n$ . Find $m imes n$ .
|
[asy]
size(300);
defaultpen(linewidth(0.8));
draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8));
label(" $A$ ",origin,SW);
label(" $B$ ",(0,4),dir(160));
label(" $C$ ",(3,0),S);
label(" $D$ ",(6,8),dir(80));
label(" $E$ ",(30,-4),E);[/asy]
|
$346320$
|
Local Relation Composition
|
aops_forum
|
count
|
258
|
As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side. If $CE = 18$ , $AB > 19$ , compute the minimal area of $\triangle AEG$ .
|
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(real x, real y) {
pair P = (x,y);
dot(P,linewidth(3)); return P;
}
int big = 30, small = 14;
filldraw((0,big)--(big+small,0)--(big,small)--cycle, rgb(0.9,0.5,0.5));
draw(scale(big)*unitsquare); draw(shift(big,0)*scale(small)*unitsquare);
label(" $A$ ",D2(0,big),NW);
label(" $B$ ",D2(0,0),SW);
label(" $C$ ",D2(big,0),SW);
label(" $D$ ",D2(big,big),N);
label(" $E$ ",D2(big+small,0),SE);
label(" $F$ ",D2(big+small,small),NE);
label(" $G$ ",D2(big,small),NE);
[/asy]
|
$162$
|
Local Relation Composition
|
aops_forum
|
area
|
259
|
In the diagram, the circle has radius $\sqrt{\frac{31}{3}}$ and and centre $O.$ Points $A, B$ and $C$ are on the circle. If $\angle BOC=120^\circ$ and $AC = AB + 1,$ determine the length of $AB.$
|
[asy]
import graph; size(120); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttff = rgb(0,0.2,1); pen xdxdff = rgb(0.49,0.49,1); pen fftttt = rgb(1,0.2,0.2);
draw(circle((2.34,2.4),2.01),qqttff); draw((2.34,2.4)--(1.09,0.82),fftttt); draw((2.34,2.4)--(4.1,1.41),fftttt); draw((1.09,0.82)--(1.4,4.18),fftttt); draw((4.1,1.41)--(1.4,4.18),fftttt);
dot((2.34,2.4),ds); label(" $O$ ", (2.1,2.66),NE*lsf); dot((1.09,0.82),ds); label(" $B$ ", (0.86,0.46),NE*lsf); dot((4.1,1.41),ds); label(" $C$ ", (4.2,1.08),NE*lsf); dot((1.4,4.18),ds); label(" $A$ ", (1.22,4.48),NE*lsf); clip((-4.34,-10.94)--(-4.34,6.3)--(16.14,6.3)--(16.14,-10.94)--cycle);
[/asy]
|
$5$
|
Local Relation Composition
|
aops_forum
|
length
|
260
|
The diagram below shows the regular hexagon $BCEGHJ$ surrounded by the rectangle $ADFI$ . Let $\theta$ be the measure of the acute angle between the side $\overline{EG}$ of the hexagon and the diagonal of the rectangle $\overline{AF}$ . There are relatively prime positive integers $m$ and $n$ so that $\sin^2\theta = \frac{m}{n}$ . Find $n - m$ .
|
[asy]
import graph; size(3.2cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((-1,3)--(-1,2)--(-0.13,1.5)--(0.73,2)--(0.73,3)--(-0.13,3.5)--cycle);
draw((-1,3)--(-1,2));
draw((-1,2)--(-0.13,1.5));
draw((-0.13,1.5)--(0.73,2));
draw((0.73,2)--(0.73,3));
draw((0.73,3)--(-0.13,3.5));
draw((-0.13,3.5)--(-1,3));
draw((-1,3.5)--(0.73,3.5));
draw((0.73,3.5)--(0.73,1.5));
draw((-1,1.5)--(0.73,1.5));
draw((-1,3.5)--(-1,1.5));
label(" $ A $ ",(-1.4,3.9),SE*labelscalefactor);
label(" $ B $ ",(-1.4,3.28),SE*labelscalefactor);
label(" $ C $ ",(-1.4,2.29),SE*labelscalefactor);
label(" $ D $ ",(-1.4,1.45),SE*labelscalefactor);
label(" $ E $ ",(-0.3,1.4),SE*labelscalefactor);
label(" $ F $ ",(0.8,1.45),SE*labelscalefactor);
label(" $ G $ ",(0.8,2.24),SE*labelscalefactor);
label(" $ H $ ",(0.8,3.26),SE*labelscalefactor);
label(" $ I $ ",(0.8,3.9),SE*labelscalefactor);
label(" $ J $ ",(-0.25,3.9),SE*labelscalefactor); [/asy]
|
$1$
|
Local Relation Composition
|
aops_forum
|
count
|
261
|
The diagram below shows an isosceles triangle with base $42$ and height $56$ . Inscribed in the triangle is a square. Find the area of the shaded regione.
|
[asy]
size(170);
defaultpen(linewidth(0.8));
draw((0,0)--(1,1));
pair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-1,2)};
pair intsquare[]={extension(origin,invis[0],A,C),extension(origin,invis[1],B,C)};
path triangle=A--B--C--cycle,square=(intsquare[0]--intsquare[1]--(intsquare[1].x,0)--(intsquare[0].x,0)--cycle);
fill(triangle,gray);
unfill(square);
draw(triangle^^square);
[/asy]
|
$600$
|
Local Relation Composition
|
aops_forum
|
area
|
262
|
A square is inscribed in a circle of radius $8$ . A quarter circle is inscribed in the square, as shown in the diagram below. Given the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$ , what is $n$ ?
|
[asy]
size(5 cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw(circle((1,1),1.41));
draw(arc((0,0),2,0,90));[/asy]
|
$32$
|
Local Relation Composition
|
aops_forum
|
count
|
263
|
In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. How many times of the area of square $ABCD$ compared to the area of quadrilateral $A'B'C'D'$
|
[asy]
import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1);
draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt));
dot((0,4),ds); label(" $A$ ", (0.07,4.12), NE*lsf); dot((0,0),ds); label(" $D$ ", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label(" $C$ ", (4.14,-0.39), NE*lsf); dot((4,4),ds); label(" $B$ ", (4.08,4.12), NE*lsf); dot((2,4),ds); label(" $M$ ", (2.08,4.12), NE*lsf); dot((4,2),ds); label(" $N$ ", (4.2,1.98), NE*lsf); dot((2,0),ds); label(" $P$ ", (1.99,-0.49), NE*lsf); dot((0,2),ds); label(" $Q$ ", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label(" $A'$ ", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label(" $B'$ ", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label(" $C'$ ", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label(" $D'$ ", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle);
[/asy]
|
$5$
|
Local Relation Composition
|
aops_forum
|
ratio
|
264
|
In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively. Find $AC$ .
|
[asy]
import geometry;
import graph;
unitsize(1.5 cm);
pair A, B, C, D;
B = (0,0);
D = (3,0);
A = 2*dir(120);
C = extension(B,dir(30),A,D);
draw(A--B--D--cycle);
draw(B--C);
draw(arc(B,0.5,0,30));
label(" $A$ ", A, NW);
label(" $B$ ", B, SW);
label(" $C$ ", C, NE);
label(" $D$ ", D, SE);
label(" $30^\circ$ ", (0.8,0.2));
label(" $90^\circ$ ", (0.1,0.5));
perpendicular(B,NE,C-B);
[/asy]
|
$\sqrt[3]{2}$
|
Local Relation Composition
|
aops_forum
|
length
|
265
|
In the following diagram, a chord $AN$ intersects its diameter $MN$ at $B$. Given that $MB : BN = 2 : 3$ and $MN = 10$ . If $AN = x$ , find $x^2$ .
|
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
real r = sqrt(80)/5;
pair M=(-1,0), N=(1,0), A=intersectionpoints(arc((M+N)/2, 1, 0, 180),circle(N,r))[0], C=intersectionpoints(circle(A,1),circle(N,1))[0], B=intersectionpoints(circle(C,1),M--N)[0];
draw(arc((M+N)/2, 1, 0, 180)--cycle); draw(A--N); draw(arc(C,1,180,180+2*aSin(r/2)));
label(" $A$ ",D2(A),NW);
label(" $B$ ",D2(B),SW);
label(" $M$ ",D2(M),S);
label(" $N$ ",D2(N),SE);
[/asy]
|
$80$
|
Local Relation Composition
|
aops_forum
|
length
|
266
|
$DEB$ is a chord of a circle such that $DE=6$ and $EB=5$ . Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. Given $EC=2$ , find the radius of the circle.
|
[asy]
size(6cm);
pair O = (0,0), B = dir(110), D = dir(30), E = 0.4 * B + 0.6 * D, C = intersectionpoint(O--2*E, unitcircle);
draw(unitcircle);
draw(O--C);
draw(B--D);
dot(O);
dot(B);
dot(C);
dot(D);
dot(E);
label(" $B$ ", B, B);
label(" $C$ ", C, C);
label(" $D$ ", D, D);
label(" $E$ ", E, dir(280));
label(" $O$ ", O, dir(270));
[/asy]
|
$14$
|
Local Relation Composition
|
aops_forum
|
length
|
267
|
The figure below shows a $9\times7$ arrangement of $3\times3$ squares. As shown in the diagram, find the area of the shaded region in the grid.
|
[asy]
size(5cm);
defaultpen(linewidth(.6));
fill((0,1)--(1,1)--(1,0)--cycle^^(0,3)--(1,3)--(1,2)--cycle^^(1,2)--(2,2)--(2,1)--cycle^^(2,1)--(3,1)--(3,0)--cycle,rgb(.76,.76,.76));
fill((0,5)--(1,5)--(1,4)--cycle^^(1,4)--(2,4)--(2,3)--cycle^^(2,3)--(3,3)--(3,2)--cycle^^(3,2)--(4,2)--(4,1)--cycle^^(4,1)--(5,1)--(5,0)--cycle,rgb(.76,.76,.76));
fill((0,7)--(1,7)--(1,6)--cycle^^(1,6)--(2,6)--(2,5)--cycle^^(2,5)--(3,5)--(3,4)--cycle^^(3,4)--(4,4)--(4,3)--cycle^^(4,3)--(5,3)--(5,2)--cycle^^(5,2)--(6,2)--(6,1)--cycle^^(6,1)--(7,1)--(7,0)--cycle,rgb(.76,.76,.76));
fill((2,7)--(3,7)--(3,6)--cycle^^(3,6)--(4,6)--(4,5)--cycle^^(4,5)--(5,5)--(5,4)--cycle^^(5,4)--(6,4)--(6,3)--cycle^^(6,3)--(7,3)--(7,2)--cycle^^(7,2)--(8,2)--(8,1)--cycle^^(8,1)--(9,1)--(9,0)--cycle,rgb(.76,.76,.76));
fill((4,7)--(5,7)--(5,6)--cycle^^(5,6)--(6,6)--(6,5)--cycle^^(6,5)--(7,5)--(7,4)--cycle^^(7,4)--(8,4)--(8,3)--cycle^^(8,3)--(9,3)--(9,2)--cycle,rgb(.76,.76,.76));
fill((6,7)--(7,7)--(7,6)--cycle^^(7,6)--(8,6)--(8,5)--cycle^^(8,5)--(9,5)--(9,4)--cycle,rgb(.76,.76,.76));
fill((8,7)--(9,7)--(9,6)--cycle,rgb(.76,.76,.76));
draw((0,0)--(0,7)^^(1,0)--(1,7)^^(2,0)--(2,7)^^(3,0)--(3,7)^^(4,0)--(4,7)^^(5,0)--(5,7)^^(6,0)--(6,7)^^(7,0)--(7,7)^^(8,0)--(8,7)^^(9,0)--(9,7));
draw((0,0)--(9,0)^^(0,1)--(9,1)^^(0,2)--(9,2)^^(0,3)--(9,3)^^(0,4)--(9,4)^^(0,5)--(9,5)^^(0,6)--(9,6)^^(0,7)--(9,7));
draw((0,1)--(1,0)^^(0,3)--(3,0)^^(0,5)--(5,0)^^(0,7)--(7,0)^^(2,7)--(9,0)^^(4,7)--(9,2)^^(6,7)--(9,4)^^(8,7)--(9,6));
[/asy]
|
$144$
|
Global Abstract Integration
|
aops_forum
|
area
|
268
|
A semicircle with diameter length $16$ contains a circle radius $3$ tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent to the inside of the semicircle, the outside of the circle, and the diameter of the semicircle. The diameter of the second circle can be written as $\frac{n + k\sqrt{2}}{m}$ where $m$ , $n$ , and $k$ are positive integers and $m$ and $n$ have no factors in common. Find $m + n - k$ .
|
[asy]
size(200);
pair O=(0,0);
real R=10, r=4.7;
draw(arc(O,R,0,180)--cycle);
pair P=(sqrt((R-r)^2-r^2),r),Q;
draw(circle(P,r));
real a=0,b=r,c;
for(int k=0;k<20;++k)
{
c=(a+b)/2;
Q=(-sqrt((R-c)^2-c^2),c);
if(abs(P-Q)>c+r) a=c; else b=c;
}
draw(circle(Q,c));[/asy]
|
$169$
|
Local Relation Composition
|
aops_forum
|
count
|
269
|
Three lines are drawn parallel to each of the three sides of $\triangle ABC$ so that the three lines intersect in the interior of $ABC$ . The resulting three smaller triangles have areas $1$ , $4$ , and $16$ . Find the area of $\triangle ABC$ .
|
[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }
pair 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);
draw(A--B--C--cycle);
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);
filldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));
filldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));
filldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]
|
$49$
|
Local Relation Composition
|
aops_forum
|
area
|
270
|
In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$ , $BN$ is parallel to $OK$ , and $\angle AMN=\angle NMO$ . Determine the measure of $\angle ABN$ in degrees.
|
[asy]
size(120);defaultpen(linewidth(0.7)+fontsize(10));
pair 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);
draw(K--O--A--M--N--B--A^^circle((0,0),1));
label(" $A$ ", A, dir(O--A));
label(" $K$ ", K, dir(O--K));
label(" $B$ ", B, dir(O--B));
label(" $N$ ", N, E);
label(" $M$ ", M, S);
label(" $O$ ", O, SE);[/asy]
|
$36$
|
Local Relation Composition
|
aops_forum
|
angle
|
271
|
$\triangle ABC$ has area $360$ . Points $X, Y, Z$ lie on sides $AB$ , $BC$ , and $CA$ , respectively. Given that $\frac{AX}{BX} = 3$ , $\frac{BY}{CY} = 4$ , and $\frac{CZ}{AZ} = 5$ , find the area of $\triangle XYZ$ .
|
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;
draw(A--B--C--cycle^^X--Y--Z--cycle);
label(" $A$ ",A,N);
label(" $B$ ",B,S);
label(" $C$ ",C,E);
label(" $X$ ",X,W);
label(" $Y$ ",Y,S);
label(" $Z$ ",Z,NE);[/asy]
|
$183$
|
Local Relation Composition
|
aops_forum
|
area
|
272
|
In acute triangle $ABC$ , $AB > AC$ . Let $M$ be the midpoint of side $BC$ . The exterior angle bisector of $\widehat{BAC}$ meet ray $BC$ at $P$ . Point $K$ and $F$ lie on line $PA$ such that $MF \perp BC$ and $MK \perp PA$ . What is $\frac{BC^2}{PF\cdot AK}$ ?
|
[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<size; ++i) { d[i] = 0; mult[i] = 1;}string[] k= split(s,",");for(int i = 0;i<p.length;++i) {label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i]));}[/asy]
|
$4$
|
Local Relation Composition
|
aops_forum
|
ratio
|
273
|
Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. How many times of the area of the 10-sided polygon compared to the shaded area?
|
[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705);
filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8));
draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle);
pair O = (A_1+A_2+A_3+A_4+A_5)/5;
label(" $A_1$ ",A_1, 2dir(A_1-O));
label(" $A_2$ ",A_2, 2dir(A_2-O));
label(" $A_3$ ",A_3, 2dir(A_3-O));
label(" $A_4$ ",A_4, 2dir(A_4-O));
label(" $A_5$ ",A_5, 2dir(A_5-O));
label(" $B_1$ ",B_1, 2dir(B_1-O));
label(" $B_2$ ",B_2, 2dir(B_2-O));
label(" $B_3$ ",B_3, 2dir(B_3-O));
label(" $B_4$ ",B_4, 2dir(B_4-O));
label(" $B_5$ ",B_5, 2dir(B_5-O));
[/asy]
|
$2$
|
Local Relation Composition
|
aops_forum
|
ratio
|
274
|
Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$ . $AN$ and $MB$ intersect at $X$ . Given that $\angle AXB$ measures $127^{\circ}$ and the minor arc $AM$ measures $14^{\circ}$ , compute the measure of the angle at $P$ .
|
[asy]
size(200);
defaultpen(fontsize(10pt));
pair P=(60,10),C=(-20,10),K=(-20,-10);
path CC=circle((0,0),20), PC=P--C, PK=P--K;
pair A=intersectionpoints(CC,PC)[0],
B=intersectionpoints(CC,PC)[1],
M=intersectionpoints(CC,PK)[0],
N=intersectionpoints(CC,PK)[1],
X=intersectionpoint(A--N,B--M);
draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M);
label(" $A$ ",A,plain.NE);label(" $B$ ",B,plain.NW);label(" $M$ ",M,SE);
label(" $P$ ",P,E);label(" $N$ ",N,dir(250));label(" $X$ ",X,plain.N);[/asy]
|
$39$
|
Local Relation Composition
|
aops_forum
|
angle
|
275
|
Equilateral triangle $ T$ is inscribed in circle $ A$ , which has radius $ 10$ . Circle $ B$ with radius $ 3$ is internally tangent to circle $ A$ at one vertex of $ T$ . Circles $ C$ and $ D$ , both with radius $ 2$ , are internally tangent to circle $ A$ at the other two vertices of $ T$ . Circles $ B$ , $ C$ , and $ D$ are all externally tangent to circle $ E$ , which has radius $ \frac {m}{n}$ , where $ m$ and $ n$ are relatively prime positive integers. Find $ m - n$ .
|
[asy]unitsize(2.2mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90);
pair Ep=(0,4-27/5);
pair[] dotted={A,B,C,D,Ep};
draw(Circle(A,10));
draw(Circle(B,3));
draw(Circle(C,2));
draw(Circle(D,2));
draw(Circle(Ep,27/5));
dot(dotted);
label(" $E$ ",Ep,E);
label(" $A$ ",A,W);
label(" $B$ ",B,W);
label(" $C$ ",C,W);
label(" $D$ ",D,E);[/asy]
|
$22$
|
Local Relation Composition
|
aops_forum
|
count
|
276
|
Let $ABCD$ be a rectangle with $AB=10$ and $BC=66$ . Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$ . Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$ ?
|
[asy]
size(10cm);
draw((0,0)--(26,0)--(26,10)--(0,10)--cycle);
draw((1,0)--(25,10));
draw(circle((0,5),5));
draw(circle((26,5),5));
dot((1,0));
dot((25,10));
label(" $E$ ",(1,0),SE);
label(" $F$ ",(25,10),NW);
label(" $A$ ", (0,0), SW);
label(" $B$ ", (0,10), NW);
label(" $C$ ", (26,10), NE);
label(" $D$ ", (26,0), SE);
dot((0,0));
dot((0,10));
dot((26,0));
dot((26,10));
[/asy]
|
$66$
|
Local Relation Composition
|
aops_forum
|
length
|
277
|
Point $P$ is in the interior of $\triangle ABC$ . The side lengths of $ABC$ are $AB = 7$ , $BC = 8$ , $CA = 9$ . The three foots of perpendicular lines from $P$ to sides $BC$ , $CA$ , $AB$ are $D$ , $E$ , $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b}\sqrt{c}$ , where $\gcd(a,b) = 1$ and $c$ is square free, calculate $a+b+c$ .
|
[asy]
size(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black;
pair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B);
D(A--B--C--cycle); D(P--D); D(P--E); D(P--F);
D(MP("A",A,N)); D(MP("B",B)); D(MP("C",C)); D(MP("D",D)); D(MP("E",E,NE)); D(MP("F",F,NW)); D(MP("P",P,SE));
[/asy]
|
$34$
|
Local Relation Composition
|
aops_forum
|
count
|
278
|
A rectangular piece of paper $ABCD$ has sides of lengths $AB = 3$ , $BC = 6$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How large, in degrees, is $\angle AMB$ ?
|
[asy]
size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;
D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd);
[/asy]
|
$60$
|
Global Abstract Integration
|
aops_forum
|
angle
|
279
|
Two concentric circles have radii $1$ and $4$ . Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$ . The radius of the six congruent circles can be written $\frac{k+\sqrt m}{n}$ , where $k,m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k imes m imes n$ .
|
[asy]
size(150);
defaultpen(linewidth(0.8));
real r = (sqrt(133)-9)/2;
draw(circle(origin,1)^^circle(origin,4));
for(int i=0;i<=2;i=i+1)
{
filldraw(circle(dir(90 + i*120)*(4-r),r),gray);
}
for(int j=0;j<=2;j=j+1)
{
filldraw(circle(dir(30+j*120)*(1+r),r),darkgray);
}
[/asy]
|
$-2394$
|
Local Relation Composition
|
aops_forum
|
count
|
280
|
Inside the $7\times 10$ rectangle below, one point is chosen a distance $\sqrt2$ from the left side and a distance $\sqrt7$ from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.
|
[asy]
import graph;
size(4cm);
pair A = (0,0);
pair B = (9,0);
pair C = (9,7);
pair D = (0,7);
pair P = (1.5,3);
draw(A--B--C--D--cycle,linewidth(1.5));
filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
[/asy]
|
$35$
|
Primitive Recognition
|
aops_forum
|
area
|
281
|
The pattern in the figure below continues inward infinitely. The base of the biggest triangle is 1. All triangles are equilateral. The area of the shaded region can be written as $\frac{\sqrt{m}{n}}$ where $m$ and $n$ are relatively prime integers. Find the value of $m+n$.
|
[asy]
defaultpen(linewidth(0.8));
pen blu = rgb(0,112,191);
real r=sqrt(3);
fill((8,0)--(0,8r)--(-8,0)--cycle, blu);
fill(origin--(4,4r)--(-4,4r)--cycle, white);
fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);
fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]
|
$8$
|
Global Abstract Integration
|
aops_forum
|
count
|
282
|
The target below is made up of concentric circles with diameters $2$ , $4$ , $8$ , $12$ , and $16$ . The area of the dark region is $n\pi$ . Find $n$ .
|
[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red);
}
}
[/asy]
|
$41$
|
Primitive Recognition
|
aops_forum
|
count
|
283
|
Find 600 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.
|
[asy]
defaultpen(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);
draw(unitcircle,dg);
for(int i = 0; i < 12; ++i) {
draw(dir(30*i+theta)--dir(30*(i+1)+theta), db);
dot(dir(30*i+theta),Fill(rgb(0.8,0,0)));
} dot(dir(theta),Fill(dr)); dot((0,0),Fill(dr));
[/asy]
|
$1800$
|
Primitive Recognition
|
aops_forum
|
area
|
284
|
Circles $\omega$ and $\gamma$ are drawn such that $\omega$ is internally tangent to $\gamma$ , the distance between their centers are $12$ , and the area inside of $\gamma$ but outside of $\omega$ is $360\pi$ . What is the sum of the radii of the circles?
|
[asy]
size(3cm);
real lw=0.4, dr=0.3;
real r1=14, r2=9;
pair A=(0,0), B=(r1-r2,0);
draw(A--B,dashed);
draw(circle(A,r1),linewidth(lw)); draw(circle(B,r2),linewidth(lw));
filldraw(circle(A,dr)); filldraw(circle(B,dr));
label(" $5$ ",(A+B)/2,dir(-90));
label(" $\gamma$ ",A+r1*dir(135),dir(135)); label(" $\omega$ ",B+r2*dir(135),dir(135));
[/asy]
|
$23$
|
Local Relation Composition
|
aops_forum
|
length
|
285
|
The cross below is made up of five congruent squares. The perimeter of the cross is $120$ . Find its area.
|
[asy]
import graph;
size(3cm);
pair A = (0,0);
pair temp = (1,0);
pair B = rotate(45,A)*temp;
pair C = rotate(90,B)*A;
pair D = rotate(270,C)*B;
pair E = rotate(270,D)*C;
pair F = rotate(90,E)*D;
pair G = rotate(270,F)*E;
pair H = rotate(270,G)*F;
pair I = rotate(90,H)*G;
pair J = rotate(270,I)*H;
pair K = rotate(270,J)*I;
pair L = rotate(90,K)*J;
draw(A--B--C--D--E--F--G--H--I--J--K--L--cycle);
[/asy]
|
$500$
|
Primitive Recognition
|
aops_forum
|
area
|
286
|
As shown in figure, circle $\Omega$ has $\overline{AB}=65$ and $CD=16$. Moreover, let $AD, BC, AC,$ and $BD$ all have integer lengths. Two other circles, $\omega_1$ and $\omega_2$ , have $\overline{AC}$ and $\overline{BD}$ as their diameters, respectively. Let circle $\omega_1$ intersect $AB$ at a point $E \neq A$ and let circle $\omega_2$ intersect $AB$ at a point $F \neq B$ . Then $EF=\frac{m}{n}$ , for relatively prime integers $m$ and $n$ . Find $m+n$ .
|
[asy]
size(7cm);
pair A=(0,0), B=(64,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
dot(" $A$ ", A, SW);
dot(" $B$ ", B, SE);
dot(" $C$ ", C, NE);
dot(" $D$ ", D, NW);
dot(" $E$ ", E, S);
dot(" $F$ ", F, S);
draw(circle((A + C)/2, abs(A - C)/2));
draw(circle((B + D)/2, abs(B - D)/2));
draw(circle((A + B)/2, abs(A - B)/2));
label(" $\mathcal P$ ", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));
label(" $\mathcal Q$ ", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));
label(" $\mathcal R$ ", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));
[/asy]
|
$961$
|
Local Relation Composition
|
aops_forum
|
count
|
287
|
In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is $\frac{2}{5}$ that of triangle AOB. How many time of a compared to b?
|
[asy]
size(120);
pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.1,1.9);
pair J = (0.17,0);
pair Y = (0.17,0.75);
pair Z = (1.6,0.4);
draw(A--B);
draw(B--C);
draw(C--A);
draw(D--F--Z--E--D);
draw(" $O$ ", B, dir(180));
draw(" $B$ ", A, dir(45));
draw(" $A$ ", C, dir(45));
draw(" $Q$ ", E, dir(45));
draw(" $P$ ", D, dir(45));
draw(" $R$ ", Z, dir(45));
draw(" $S$ ", F, dir(45));
draw(" $a$ ", Y, dir(210));
draw(" $b$ ", J, dir(100));
[/asy]
|
$2$
|
Local Relation Composition
|
aops_forum
|
ratio
|
288
|
Squares $ABCD$ and $EFGH$ are congruent, $G$ is the center of square $ABCD$ and $AB=5$. The area of the region in the plane covered by these squares is
|
[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]
|
$\frac{175}{4}$
|
Local Relation Composition
|
HARP
|
area
|
289
|
Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $3n$ sides inscribed in this circle, then $n=$
|
[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]
|
$3$
|
Local Relation Composition
|
HARP
|
count
|
290
|
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $n=5$, what is the value of $m$?
|
[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]
|
$10$
|
Global Abstract Integration
|
HARP
|
count
|
291
|
Three congruent circles with centers $P$, $Q$, and $R$ are tangent to the sides of rectangle $ABCD$ as shown. The circle centered at $Q$ has diameter $2$ and passes through points $P$ and $R$. The area of the rectangle is
|
[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]
|
$8$
|
Local Relation Composition
|
HARP
|
area
|
292
|
The horizontal and vertical distances between adjacent points equal 3 unit. What is the area of triangle $ABC$?
|
[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]
|
$\frac{9}{2}$
|
Primitive Recognition
|
HARP
|
area
|
293
|
The measure of angle $ABC$ is $60^\circ$, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is
|
[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]
|
$120$
|
Primitive Recognition
|
HARP
|
angle
|
294
|
Assume the square have side length of 3. Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The area of the inscribed square is
|
[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]
|
$5$
|
Local Relation Composition
|
HARP
|
area
|
295
|
In the figure, $E$ is the midpoint of $\overline{AD}$, $ABCD$ is a $1 \times 1$ square, and $F$ is on $\overline{BE}$. The area of quadrilateral $CDEF$ is
|
[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]
|
$\frac{11}{20}$
|
Local Relation Composition
|
HARP
|
area
|
296
|
$BD$ and $CE$ are medians of triangle $ABC$, $CE=6$, and $BD=4$. The area of triangle $ABC$ is
|
[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]
|
$16$
|
Primitive Recognition
|
HARP
|
area
|
297
|
A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown. To the nearest hundredth, what is the radius of the circle?
|
[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]
|
$\frac{3}{2}+\frac{\sqrt{3}}{2}$
|
Primitive Recognition
|
HARP
|
length
|
298
|
In the figure, $B$, $C$, and $D$ are squares with sides of length $2$; polygons $A$, $E$, and $F$ are isosceles right triangles; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is
|
[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]
|
$frac{20}{3}$
|
Global Abstract Integration
|
HARP
|
volume
|
299
|
A square with sides of length $2$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid.
|
[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]
|
$\frac{5}{3}$
|
Local Relation Composition
|
HARP
|
length
|
300
|
A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimter of each of the congruent rectangles is $18$. What is the area of the large square?
|
[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]
|
$81$
|
Local Relation Composition
|
HARP
|
area
|
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