LiAuto-DSR/GeoGramBench huggingface dataset:
Let $\triangle ABC$ have 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 $\frac{CZ}{AZ} = 5$. 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]

Answer: $122$
In circle $O$, chord $AB$ is extended beyond $B$ to a point $C$ such that $BC$ equals the radius of the circle. $CO$ is drawn and extended to $D$, and $AO$ is also drawn. If $y = 1$, find the value of $x$.

[asy]size(200);defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, D=dir(195), A=dir(150), B=dir(30), C=B+1*dir(0);
draw(O--A--C--D);
dot(A^^B^^C^^D^^O);
pair point=O;
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(" $O$ ", O, dir(285));
label(" $x$ ", O+0.1*dir(172.5), dir(172.5));
label(" $y$ ", C+0.4*dir(187.5), dir(187.5));
draw(Circle(O,1));
[/asy]

Answer: $3$
In $\triangle ABC$, suppose $\angle C = 3\angle A$, $a = 27$, and $c = 48$. What is the value of $b$?

[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, B=(14,0), C=(10,6);
draw(A--B--C--cycle);
label(" $A$ ", A, SW);
label(" $B$ ", B, SE);
label(" $C$ ", C, N);
label(" $a$ ", B--C, dir(B--C)*dir(-90));
label(" $b$ ", A--C, dir(C--A)*dir(-90));
label(" $c$ ", A--B, dir(A--B)*dir(-90));
[/asy]

Answer: $35$
Square $ABCD$ has side length \$4$, and $M$ is the midpoint of $\overline{CD}$. A circle with radius \$2$ and center $M$ intersects a circle with radius \$4$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$?

[asy]unitsize(8mm);
defaultpen(linewidth(.8pt));
dotfactor=4;

draw(Circle((2,0),2));
draw(Circle((0,4),4));
clip(scale(4)*unitsquare);
draw(scale(4)*unitsquare);
filldraw(Circle((2,0),0.07));
label(" $A$ ",(0,4),NW);
label(" $B$ ",(4,4),NE);
label(" $C$ ",(4,0),SE);
label(" $D$ ",(0,0),SW);
label(" $M$ ",(2,0),S);
label(" $P$ ",(3.6,1.4),N);[/asy]

Answer: $\frac{16}{5}$
A circle with center $O$ and radius \$1$ contains point $A$. Segment $AB$ is tangent to the circle at $A$, and $\angle AOB = 30^\circ$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, find the length $OC$.

[asy]import olympiad;
unitsize(2cm);
defaultpen(fontsize(8pt)+linewidth(.8pt));
labelmargin=0.2;
dotfactor=3;

pair O=(0,0);
pair A=(1,0);
pair B=(1,1.5);
pair D=bisectorpoint(A,B,O);
pair C=extension(B,D,O,A);

draw(Circle(O,1));
draw(O--A--B--cycle);
draw(B--C);

label(" $O$ ",O,SW);
dot(O);
label(" $\theta$ ",(0.1,0.05),ENE);

dot(C);
label(" $C$ ",C,S);

dot(A);
label(" $A$ ",A,E);

dot(B);
label(" $B$ ",B,E);[/asy]

Answer: $\frac{2}{3}$
In right triangle $\triangle ABC$ with legs measuring \$5$ and \$12$, two arcs are drawn: one with center $A$ and radius \$12$, and one with center $B$ and radius \$5$. These arcs intersect the hypotenuse at points $M$ and $N$. Find the length of $MN$.

[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A);
real r=degrees(B);
draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270));
pair point=incenter(A,B,C);
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $M$ ", M, dir(point--M));
label(" $N$ ", N, dir(point--N));

label(" $12$ ", (6,0), S);
label(" $5$ ", (12,3.5), E);[/asy]

Answer: $4$
On square $ABCD$, points $E, F, G,$ and $H$ lie on sides $\overline{AB},\,\overline{BC},\,\overline{CD},$ and $\overline{DA}$, respectively, such that $\overline{EG} \perp \overline{FH}$ and $EG = FH = 34$. The segments $\overline{EG}$ and $\overline{FH}$ intersect at point $P$, and the areas of the quadrilaterals $AEPH$, $BFPE$, $CGPF$, and $DHPG$ are in the ratio \$269:275:405:411$. Find the area of square $ABCD$.

[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(10.6));
pair A = (0,sqrt(850));
pair B = (0,0);
pair C = (sqrt(850),0);
pair D = (sqrt(850),sqrt(850));
draw(A--B--C--D--cycle);
dotfactor = 3;
dot(" $A$ ",A,dir(135));
dot(" $B$ ",B,dir(215));
dot(" $C$ ",C,dir(305));
dot(" $D$ ",D,dir(45));
pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850));
pair F = ((2sqrt(850)+sqrt(306)+7)/6,0);
dot(" $H$ ",H,dir(90));
dot(" $F$ ",F,dir(270));
draw(H--F);
pair E = (0,(sqrt(850)-6)/2);
pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2);
dot(" $E$ ",E,dir(180));
dot(" $G$ ",G,dir(0));
draw(E--G);
pair P = extension(H,F,E,G);
dot(" $P$ ",P,dir(60));
label(" $w$ ", (H+E)/2,fontsize(15));
label(" $x$ ", (E+F)/2,fontsize(15));
label(" $y$ ", (G+F)/2,fontsize(15));
label(" $z$ ", (H+G)/2,fontsize(15));
label(" $w:x:y:z=269:275:405:411$ ",(sqrt(850)/2,-4.5),fontsize(11));
[/asy]

Answer: $850$
Let  $XYZ$  be an acute-angled triangle. Let  $s$  be the side-length of the square which has two adjacent vertices on side  $YZ$ , one vertex on side  $XY$  and one vertex on side  $XZ$ . Let  $h$  be the distance from  $X$  to the side  $YZ$  and let  $b$  be the distance from  $Y$  to  $Z$ .



If the vertices have coordinates  $X = (2, 4)$ ,  $Y = (0, 0)$  and  $Z = (4, 0)$ , find  $s$ .

[asy]
draw((4, 8)--(0, 0)--(14, 0)--cycle);
draw((2.54,0)--(2.54+5.1,0)--(2.54+5.1,5.1)--(2.54,5.1)--cycle);
label(" $X$ ",(4,8),N);
label(" $Y$ ",(0,0),W);
label(" $Z$ ",(14,0),E);
[/asy]

Answer: $2$
Let  $XYZ$  be an acute-angled triangle. Let  $s$  be the side-length of the square which has two adjacent vertices on side  $YZ$ , one vertex on side  $XY$  and one vertex on side  $XZ$ . Let  $h$  be the distance from  $X$  to the side  $YZ$  and let  $b$  be the distance from  $Y$  to  $Z$ .



Given the height  $h = 3$  and  $s = 2$ , find the base  $b$ .

[asy]
pair S, D;
D = 1.27;
S = 2.55;
draw((2, 4)--(0, 0)--(7, 0)--cycle);
draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);
label(" $X$ ",(2,4),N);
label(" $Y$ ",(0,0),W);
label(" $Z$ ",(7,0),E);
[/asy]

Answer: $6$
Let $XYZ$ be an acute-angled triangle. Let $s$ denote the side length of a square with two adjacent vertices on side $YZ$, one vertex on side $XY$, and one vertex on side $XZ$. Let $h$ be the distance from $X$ to side $YZ$, and let $b$ be the distance from $Y$ to $Z$.

If the area of the square is \$2017$, determine the minimum possible area of triangle $XYZ$.

[asy]
pair S, D;
D = 1.27;
S = 2.55;
draw((2, 4)--(0, 0)--(7, 0)--cycle);
draw((1.27,0)--(1.27+2.55,0)--(1.27+2.55,2.55)--(1.27,2.55)--cycle);
label(" $X$ ",(2,4),N);
label(" $Y$ ",(0,0),W);
label(" $Z$ ",(7,0),E);
[/asy]

Answer: $4034$
Two circles $A$ and $B$, each with radius \$1$, are externally tangent to one another. Four circles $P$, $Q$, $R$, and $S$, each with the same radius $r$, are arranged such that $P$ is externally tangent to $A$, $B$, $Q$, and $S$; $Q$ is externally tangent to $P$, $B$, and $R$; $R$ is externally tangent to $A$, $B$, $Q$, and $S$; and $S$ is externally tangent to $P$, $A$, and $R$. Find the value of $r$.

[asy]
unitsize(0.3 cm);

pair A, B, P, Q, R, S;
real r = (3 + sqrt(17))/2;

A = (-1,0);
B = (1,0);
P = intersectionpoint(arc(A,r + 1,0,180), arc(B,r + 1,0,180));
R = -P;
Q = (r + 2,0);
S = (-r - 2,0);

draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(P,r));
draw(Circle(Q,r));
draw(Circle(R,r));
draw(Circle(S,r));

label(" $A$ ", A);
label(" $B$ ", B);
label(" $P$ ", P);
label(" $Q$ ", Q);
label(" $R$ ", R);
label(" $S$ ", S);
[/asy]

Answer: $2$
In a circle with center $O$, $AD$ is a diameter and $ABC$ is a chord. If $BO = 5$ and $\angle ABO = \overset{\frown}{CD} = 60^\circ$, find the length of $BC$.

[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C);
draw(C--A--D^^B--O^^Circle(O,1));
pair point=O;
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(" $O$ ", O, dir(305));

label(" $5$ ", B--O, dir(O--B)*dir(90));
label(" $60^\circ$ ", dir(185), dir(185));
label(" $60^\circ$ ", B+0.05*dir(-25), dir(-25));[/asy]

Answer: $5$
A rectangle $HOMF$ has sides $HO = 11$ and $OM = 5$. Let $\triangle ABC$ be a triangle with $H$ as the orthocenter, $O$ as the circumcenter, $M$ as the midpoint of $BC$, and $F$ as the foot of the altitude from $A$. Find the length of $BC$.

[asy]
unitsize(0.3 cm);
pair F, H, M, O;
F = (0,0);
H = (0,5);
O = (11,5);
M = (11,0);
draw(H--O--M--F--cycle);
label(" $F$ ", F, SW);
label(" $H$ ", H, NW);
label(" $M$ ", M, SE);
label(" $O$ ", O, NE);
[/asy]

Answer: $28$
A circle with radius \$1$ is surrounded by \$4$ circles of radius $r$, as shown. What is the value of $r$?

[asy]defaultpen(linewidth(.9pt));

real r = 2 + sqrt(2);
pair A = dir(45)*(r + sqrt(2));
pair B = dir(135)*(r + sqrt(2));
pair C = dir(-135)*(r + sqrt(2));
pair D = dir(-45)*(r + sqrt(2));

draw(Circle(origin,sqrt(2)));
draw(Circle(A,r));draw(Circle(B,r));draw(Circle(C,r));draw(Circle(D,r));

draw(A--(dir(45)*r + A));
draw(B--(dir(45)*r + B));
draw(C--(dir(45)*r + C));
draw(D--(dir(45)*r + D));
draw(origin--(dir(25)*sqrt(2)));

label(" $r$ ",midpoint(A--(dir(45)*r + A)), SE);
label(" $r$ ",midpoint(B--(dir(45)*r + B)), SE);
label(" $r$ ",midpoint(C--(dir(45)*r + C)), SE);
label(" $r$ ",midpoint(D--(dir(45)*r + D)), SE);
label(" $1$ ",origin,W);[/asy]

Answer: $1 + \sqrt{2}$
Older television screens have an aspect ratio of \$4:3$, meaning the width is to the height as \$4$ is to \$3$. Many movies have a different aspect ratio, so when shown on such a screen, black bars of equal height (called 'letterboxing') appear at the top and bottom. Suppose a movie with an aspect ratio of \$2:1$ is displayed on a television screen with a \$27$-inch diagonal and \$4:3$ aspect ratio. What is the height, in inches, of each black bar?

[asy]unitsize(1mm);
defaultpen(linewidth(.8pt));
filldraw((0,0)--(8,0)--(8,1)--(0,1)--cycle,grey,black);
filldraw((0,5)--(8,5)--(8,6)--(0,6)--cycle,grey,black);
draw((0,1)--(0,5));
draw((8,1)--(8,5));[/asy]

Answer: $2.7$
In right triangle $ABC$, $BC = 5$, $AC = 12$, and $AM = 5$. Let $\overline{MN} \perp \overline{AC}$ and $\overline{NP} \perp \overline{BC}$, with $N$ lying on $AB$. If $y = MN + NP$, which is one-half the perimeter of rectangle $MCPN$, find the value of $y$.

[asy]defaultpen(linewidth(.8pt));
unitsize(2cm);
pair A = origin;
pair M = (1,0);
pair C = (2,0);
pair P = (2,0.5);
pair B = (2,1);
pair Q = (1,0.5);

draw(A--B--C--cycle);
draw(M--Q--P);

label(" $A$ ",A,SW);
label(" $M$ ",M,S);
label(" $C$ ",C,SE);
label(" $P$ ",P,E);
label(" $B$ ",B,NE);
label(" $N$ ",Q,NW);[/asy]

Answer: $\frac{109}{12}$
A rectangle intersects a circle as shown such that $AB = 4$, $BC = 5$, and $DE = 3$. Find the length of $EF$.

[asy]size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F);
draw(D--G--H--A--cycle);
draw(Circle(O, abs(O-C)));
label(" $A$ ", A, NW);
label(" $B$ ", B, NW);
label(" $C$ ", C, NE);
label(" $D$ ", D, SW);
label(" $E$ ", E, SE);
label(" $F$ ", F, SW);

label("4", (2,0.85), N);
label("3", D--E, S);
label("5", (6.5,0.85), N);
[/asy]

Answer: $7$
Two concentric circles have radii \$1$ and \$4$. Between them, six congruent circles are arranged in a ring so that each is tangent to its two neighboring circles. Of these six, three (shown lightly shaded) are tangent on the inside to the circle of radius \$4$, and the other three (shown darkly shaded) are tangent on the outside to the circle of radius \$1$. The radius of each of the six congruent circles can be expressed as $\frac{k + \sqrt{m}}{n}$, where $k$, $m$, and $n$ are integers with $k$ and $n$ relatively prime. Compute the value of $k + m + n$.

[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]

Answer: $126$
If $\angle A = 20^\circ$ and $\angle AFG = \angle AGF$, what is the value of $\angle B + \angle D$?

[asy]
pair A,B,C,D,EE,F,G;
A = (0,0);
B = (9,4);
C = (21,0);
D = (13,-12);
EE = (4,-16);
F = (13/2,-6);
G = (8,0);

draw(A--C--EE--B--D--cycle);

label(" $A$ ",A,W);
label(" $B$ ",B,N);
label(" $C$ ",C,E);
label(" $D$ ",D,SE);
label(" $E$ ",EE,SW);
label(" $F$ ",F,WSW);
label(" $G$ ",G,NW);
[/asy]

Answer: $80$
The pattern in the figure below continues inward infinitely. The base of the largest triangle is \$1$, and all triangles are equilateral. Find the total shaded area.

[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]

Answer: $\frac{\sqrt{3}}{5}$
Points $K$, $L$, $M$, and $N$ are located in the plane of square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are all equilateral triangles. If the area of $ABCD$ is \$16$, find the area of quadrilateral $KLMN$.

[asy]unitsize(2cm);
defaultpen(fontsize(8)+linewidth(0.8));
pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);
pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0);
draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A);
label(" $A$ ",A,SE);
label(" $B$ ",B,SW);
label(" $C$ ",C,NW);
label(" $D$ ",D,NE);
label(" $K$ ",K,NNW);
label(" $L$ ",L,E);
label(" $M$ ",M,S);
label(" $N$ ",N,W);[/asy]

Answer: $32 + 16\sqrt{3}$
The area of the dark gray triangle depicted below is  $35$ , and a segment is divided into lengths  $14$  and  $10$  as shown below. What is the area of the light gray triangle?

[asy] 
size(150);
filldraw((0,0)--(0,12)--(24,-60/7)--cycle, lightgray);
filldraw((14,0)--(14,5)--(0,12)--cycle, gray);
draw((0,0)--(24,0)--(0,12)--cycle);
draw((0,0)--(24,0)--(24,-60/7)--cycle);
draw((0,12)--(24,-60/7));
draw((14,5)--(14,0));
dot((0,0));
dot((0,12));
dot((14,5));
dot((24,0));
dot((14,0));
dot((24,-60/7));
label(" $14$ ", (7,0), S);
label(" $10$ ", (19,0), S);
draw((0,2/3)--(2/3,2/3)--(2/3,0));
draw((14,2/3)--(14+2/3,2/3)--(14+2/3,0));
draw((24-2/3,0)--(24-2/3,-2/3)--(24,-2/3));
[/asy]

Answer: $144$
The centers of the faces of the right rectangular prism shown below are connected to form an octahedron. What is the volume of this octahedron?

[asy]
import three; size(2inch);
currentprojection=orthographic(4,2,2);
draw((0,0,0)--(0,0,3),dashed);
draw((0,0,0)--(0,4,0),dashed);
draw((0,0,0)--(5,0,0),dashed);
draw((5,4,3)--(5,0,3)--(5,0,0)--(5,4,0)--(0,4,0)--(0,4,3)--(0,0,3)--(5,0,3));
draw((0,4,3)--(5,4,3)--(5,4,0));
label("3",(5,0,3)--(5,0,0),W);
label("4",(5,0,0)--(5,4,0),S);
label("5",(5,4,0)--(0,4,0),SE);
[/asy]

Answer: $10$
Six cubes, each with edge length \$1$ inch, are joined together as shown. Find the total surface area, in square inches, including the top, bottom, and sides.

[asy]
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
draw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));
draw((1,0)--(1.5,0.5)--(1.5,1.5));
draw((0.5,1.5)--(1,2)--(1.5,2));
draw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));
draw((1.5,3.5)--(2.5,3.5));
draw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));
draw((3,4)--(3,3)--(2.5,2.5));
draw((3,3)--(4,3)--(4,2)--(3.5,1.5));
draw((4,3)--(3.5,2.5));
draw((2.5,.5)--(3,1)--(3,1.5));[/asy]

Answer: $26$
Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is a right angle. Points $C$ and $D$ are located on radii $OA$ and $OB$, respectively, such that $AC = T - 3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.

[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
draw((0,3)--(4,0));
label("O",(0,0),SW);
label("C",(0,3),W);
label("A",(0,10),N);
label("D",(4,0),S);
label("B",(10,0),E);
[/asy]

Answer: $44$
In an isosceles right triangle $AOB$, points $P$, $Q$, and $S$ are chosen on sides $OB$, $OA$, and $AB$, respectively, so that a square $PQRS$ is constructed as shown. If $OP = a$ and $OQ = b$, and the area of $PQRS$ is $\dfrac{2}{5}$ the area of triangle $AOB$, determine the ratio $a : b$.

[asy]

pair A = (0,3);
pair B = (0,0);
pair C = (3,0);
pair D = (0,1.5);
pair E = (0.35,0);
pair F = (1.2,1.8);
pair J = (0.17,0);
pair Y = (0.17,0.75);

pair Z = (1.6,0.2);
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]

Answer: $2$
In circle $O$, let $Q$ be the midpoint of radius $OX$. At $Q$, $\overline{AB} \perp \overline{XY}$. The semicircle with diameter $\overline{AB}$ intersects $\overline{XY}$ at $M$. Line $\overline{AM}$ meets circle $O$ at $C$, and line $\overline{BM}$ meets circle $O$ at $D$. Line $\overline{AD}$ is drawn. If the radius of circle $O$ is \$1$, what is the length of $AD$?

[asy]defaultpen(linewidth(.8pt));
unitsize(2.5cm);

real m = 0;
real b = 0;

pair O = origin;
pair X = (-1,0);
pair Y = (1,0);
pair Q = midpoint(O--X);
pair A = (Q.x, -1*sqrt(3)/2);
pair B = (Q.x, -1*A.y);
pair M = (Q.x + sqrt(3)/2,0);

m = (B.y - M.y)/(B.x - M.x);
b = (B.y - m*B.x);

pair D = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));

m = (A.y - M.y)/(A.x - M.x);
b = (A.y - m*A.x);

pair C = intersectionpoint(Circle(O,1),M--(1.5,1.5*m + b));

draw(Circle(O,1));
draw(Arc(Q,sqrt(3)/2,-90,90));
draw(A--B);
draw(X--Y);
draw(B--D);
draw(A--C);
draw(A--D);
dot(O);dot(M);

label(" $B$ ",B,NW);
label(" $C$ ",C,NE);
label(" $Y$ ",Y,E);
label(" $D$ ",D,SE);
label(" $A$ ",A,SW);
label(" $X$ ",X,W);
label(" $Q$ ",Q,SW);
label(" $O$ ",O,SW);
label(" $M$ ",M,NE+2N);[/asy]

Answer: $\sqrt{2}$
In $\triangle ABC$, points $D$ and $E$ are located on $\overline{BC}$ and $\overline{AC}$, respectively. Suppose $\overline{AD}$ and $\overline{BE}$ intersect at $T$ such that $AT/DT = 3$ and $BT/ET = 4$. Find the value of $CD/BD$.

[asy]unitsize(2cm);
defaultpen(linewidth(.8pt));

pair A = (0,0);
pair C = (2,0);
pair B = dir(57.5)*2;
pair E = waypoint(C--A,0.25);
pair D = waypoint(C--B,0.25);
pair T = intersectionpoint(D--A,E--B);

label(" $B$ ",B,NW);label(" $A$ ",A,SW);label(" $C$ ",C,SE);label(" $D$ ",D,NE);label(" $E$ ",E,S);label(" $T$ ",T,2*W+N);

draw(A--B--C--cycle);
draw(A--D);
draw(B--E);[/asy]

Answer: $\frac{4}{11}$
Two squares, each with side length \$12$, are positioned so that one lies exactly on top of the other. One square is then rotated about a vertex by \$30^\circ$ with respect to the other. Determine the area of the region common to both squares.

[asy]
unitsize (2 cm);

pair A, B, C, D, Bp, Cp, Dp, P;

A = (0,0);
B = (-1,0);
C = (-1,1);
D = (0,1);
Bp = rotate(-30)*(B);
Cp = rotate(-30)*(C);
Dp = rotate(-30)*(D);
P = extension(C, D, Bp, Cp);

fill(A--Bp--P--D--cycle, gray(0.8));
draw(A--B--C--D--cycle);
draw(A--Bp--Cp--Dp--cycle);

label(" $30^\circ$ ", (-0.5,0.1), fontsize(10));
[/asy]

Answer: $48\sqrt{3}$
In the figure, $\angle C = 90^\circ$, $AD = DB$, $DE \perp AB$, $AB = 20$, and $AC = 12$. What is the area of quadrilateral $ADEC$?

[asy] unitsize(7); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E; A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10); draw(A--B--C--cycle); draw(D--E); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE); draw(rightanglemark(B,D,E,30)); [/asy]

Answer: $58\frac{1}{2}$
Given the diagram, where $A$, $B$, and $C$ are the points of tangency, determine the value of $x$.

[asy] unitsize(5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); pair O=(0,3/8); draw((-2/3,9/16)--(2/3,9/16)); draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); draw(Circle(O,3/16)); draw((-2/3,0)--(2/3,0)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$\frac{3}{8}$",O); draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); label("$\frac{1}{2}$",(.5,.25)); draw((.5,.33)--(.5,.5),EndArrow(3)); draw((.5,.17)--(.5,0),EndArrow(3)); label("$x$",midpoint((.5,.5)--(.5,9/16))); draw((.5,5/8)--(.5,9/16),EndArrow(3)); label("$60^{\circ}$",(0.01,0.12)); dot(A); dot(B); dot(C);[/asy]

Answer: $\frac{1}{16}$
Point $F$ is taken on side $AD$ of square $ABCD$. At $C$, a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is \$256$ square inches and the area of $\triangle CEF$ is \$200$ square inches. Find the number of inches in $BE$.

[asy] size(6cm); pair A = (0, 0), B = (1, 0), C = (1, 1), D = (0, 1), E = (1.3, 0), F = (0, 0.7); draw(A--B--C--D--cycle); draw(F--C--E--B); label("$A$", A, SW); label("$B$", B, S); label("$C$", C, N); label("$D$", D, NW); label("$E$", E, SE); label("$F$", F, W); [/asy]

Answer: $12$
Inside square $ABCD$ (see figure) with side length \$12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ that is \$5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and meets $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$, respectively. Find $\dfrac{PM}{MQ}$.

[asy] draw(unitsquare);draw((0,0)--(.4,1)^^(0,.6)--(1,.2)); label("D",(0,1),NW);label("E",(.4,1),N);label("C",(1,1),NE); label("P",(0,.6),W);label("M",(.25,.55),E);label("Q",(1,.2),E); label("A",(0,0),SW);label("B",(1,0),SE); [/asy]

Answer: $\frac{5}{19}$
In parallelogram $ABCD$ shown in the diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ intersect at $O$. If the area of parallelogram $ABCD$ is \$1$, then the area of triangle $QPO$ is equal to

[asy] size((400)); draw((0,0)--(5,0)--(6,3)--(1,3)--cycle); draw((6,3)--(-5,0)--(10,0)--(1,3)); label("A", (0,0), S); label("B", (5,0), S); label("C", (6,3), NE); label("D", (1,3), NW); label("P", (10,0), E); label("Q", (-5,0), W); label("M", (.5,1.5), NW); label("N", (5.65, 1.5), NE); label("O", (3.4,1.75));[/asy]

Answer: $\frac{9}{8}$
In $\triangle ABC$ shown in the figure, $M$ is the midpoint of side $BC$, $AB = 12$, and $AC = 16$. Points $E$ and $F$ are located on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE = 2AF$, then $\dfrac{EG}{GF}$ equals

[asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label("A", (0,0), S); label("B", (12,0), S); label("C", (14,7.75), E); label("E", (8.75,4.84), N); label("F", (5,0), S); label("M", (13,3.875), E); label("G", (7,1)); [/asy]

Answer: $\frac{3}{2}$
In the figure shown, $\measuredangle E = 40^\circ$ and arcs $AB$, $BC$, and $CD$ all have equal length. Find the measure of $\measuredangle ACD$.

[asy] size(120); path c = Circle((0, 0), 1); pair A = dir(20), B = dir(130), C = dir(240), D = dir(330); draw(c); pair F = 3(A-B) + B; pair G = 3(D-C) + C; pair E = intersectionpoints(B--F, C--G)[0]; draw(B--E--C--A); label("$A$", A, NE); label("$B$", B, NW); label("$C$", C, SW); label("$D$", D, SE); label("$E$", E, E); [/asy]

Answer: $15$
Each of the three circles in the figure shown is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius $3$, what is the perimeter of the triangle?

[asy] size(120); real t = 2/sqrt(3); real x = 1 + sqrt(3); pair A = t*dir(90), D = x*A; pair B = t*dir(210), E = x*B; pair C = t*dir(330), F = x*C; draw(D--E--F--cycle); draw(Circle(A, 1)); draw(Circle(B, 1)); draw(Circle(C, 1)); [/asy]

Answer: $18+18\sqrt{3}$