LiAuto-DSR/GeoGramBench huggingface dataset:
Let $ABC$ be an isosceles triangle with side lengths $AB = AC = 10\sqrt{2}$ and $BC = 10\sqrt{3}$. Semicircles $P$, $Q$, and $R$ are constructed with diameters $AB$, $AC$, and $BC$, respectively, so that the plane of each semicircle is perpendicular to the plane of $ABC$ and all semicircles are positioned on the same side of the plane $ABC$, as illustrated. There is a plane located above triangle $ABC$ that is tangent to each of the semicircles $P$, $Q$, and $R$ at points $D$, $E$, and $F$, respectively. Find the area of triangle $DEF$.



[asy]
size(200);
import three; defaultpen(linewidth(0.7)+fontsize(10)); currentprojection = orthographic(0,4,2.5); 
// 1.15 x-scale distortion factor 
triple A = (0,0,0), B = (75^.5/1.15,-125^.5,0), C = (-75^.5/1.15,-125^.5,0), D = (A+B)/2 + (0,0,abs((B-A)/2)), E = (A+C)/2 + (0,0,abs((C-A)/2)), F = (C+B)/2 + (0,0,abs((B-C)/2)); 
draw(D--E--F--cycle); draw(B--A--C); 
real r = 1.38; draw(B--(r*B+C)/(1+r)^^(B+r*C)/(1+r)--C,linetype("4 4")); draw((B+r*C)/(1+r)--(r*B+C)/(1+r));
draw(arc((A+B)/2,A,D)); draw(arc((A+B)/2,D,B)); draw(arc((A+C)/2,E,A)); draw(arc((A+C)/2,E,C)); draw(arc((C+B)/2,F,B)); draw(arc((C+B)/2,F,C)); 
label(" $A$ ",A,S); label(" $B$ ",B,W); label(" $C$ ",C,plain.E);
label(" $D$ ",D,SW); label(" $E$ ",E,SE); label(" $F$ ",F,N);[/asy]

Answer: $24$
In the figure below, there are two congruent circles with centers $A$ and $B$, which are tangent to each other. A third circle is tangent to the circle centered at $A$ at point $C$, and passes through point $B$. Points $C$, $A$, and $B$ lie on a straight line. The line segment $\overline{CDEFG}$ passes through the circles at the labeled points. Given that $DE = 6$ and $FG = 9$, determine the length $AG$.



[asy]
unitsize(5);
pair A = (-9 sqrt(3), 0);
pair B = (9 sqrt(3), 0);
pair C = (-18 sqrt(3), 0);
pair D = (-4 sqrt(3) / 3, 10 sqrt(6) / 3);
pair E = (2 sqrt(3), 4 sqrt(6));
pair F = (7 sqrt(3), 5 sqrt(6));
pair G = (12 sqrt(3), 6 sqrt(6));
real r = 9sqrt(3);
draw(circle(A, r));
draw(circle(B, r));
draw(circle((B + C) / 2, 3r / 2));
draw(C -- D);
draw(" $6$ ", E -- D);
draw(E -- F);
draw(" $9$ ", F -- G);
dot(A);
dot(B);
label(" $A$ ", A, plain.E);
label(" $B$ ", B, plain.E);
label(" $C$ ", C, W);
label(" $D$ ", D, dir(160));
label(" $E$ ", E, S);
label(" $F$ ", F, SSW);
label(" $G$ ", G, N);
[/asy]

Answer: $9\sqrt{19}$
A large kite is wrapped in gold foil, which is cut from a rectangular sheet that exactly covers the entire grid. How many square inches of foil are trimmed from the four corners as waste?



[asy]
for (int a = 0; a < 7; ++a)
{
for (int b = 0; b < 8; ++b)
{
dot((a,b));
}
}
draw((3,0)--(0,5)--(3,7)--(6,5)--cycle);[/asy]

Answer: $189$
A rubber band is stretched around two cylindrical pipes, one with a radius of \$3$ inches and the other with a radius of \$9$ inches, as illustrated. The length of the rubber band can be expressed as $a\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is square-free. Find the value of $a+b+c$.



[asy]
size(4cm);
draw(circle((0,0),3));
draw(circle((12,0),9));
draw(3*dir(120)--(12,0)+9*dir(120));
draw(3*dir(240)--(12,0)+9*dir(240));
[/asy]

Answer: $29$
A trapezoid has parallel sides of lengths \$3$ and \$9$, and non-parallel sides of lengths \$4$ and \$6$. A line parallel to the bases divides the original trapezoid into two smaller trapezoids with equal perimeters. In what integer ratio does this line divide each of the non-parallel sides?

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

pair A = origin;
pair B = (2.25,0);
pair C = (2,1);
pair D = (1,1);
pair E = waypoint(A--D,0.25);
pair F = waypoint(B--C,0.25);

draw(A--B--C--D--cycle);
draw(E--F);

label("6",midpoint(A--D),NW);
label("3",midpoint(C--D),N);
label("4",midpoint(C--B),NE);
label("9",midpoint(A--B),S);[/asy]

Answer: $4$
A semicircle has a diameter of length \$16$, and inside it there is a circle of radius \$3$ that is tangent to both the diameter and the semicircle itself. Another, larger circle is also placed inside the semicircle such that it is tangent to the semicircle, tangent to the diameter, and tangent externally to the smaller circle. If the diameter of the larger circle can be written in the form $\frac{n + k\sqrt{2}}{m}$, where $m$, $n$, and $k$ are positive integers with $\gcd(m, n) = 1$, 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]

Answer: $361$
In triangle $CAT$, suppose that $\angle ACT = \angle ATC$ and $\angle CAT = 36^\circ$. If $\overline{TR}$ is the angle bisector of $\angle ATC$, what is the measure of angle $\angle CRT$?

[asy]
pair A,C,T,R;
C = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = (3/2 - sqrt(5)/2,1.175570);
draw(C--A--T--cycle);
draw(T--R);
label(" $A$ ",A,N);
label(" $T$ ",T,SE);
label(" $C$ ",C,SW);
label(" $R$ ",R,NW);
[/asy]

Answer: $72$
In triangle $ABC$, suppose that $AB = AC$. If there exists a point $P$ located strictly between $A$ and $B$ such that $AP = PC = CB$, what is the value of $\angle A$?

[asy]
draw((0,0)--(8,0)--(4,12)--cycle);
draw((8,0)--(1.6,4.8));
label("A", (4,12), N);
label("B", (0,0), W);
label("C", (8,0), E);
label("P", (1.6,4.8), NW);
dot((0,0));
dot((4,12));
dot((8,0));
dot((1.6,4.8));
[/asy]

Answer: $36$
In $\triangle ABC$, the medians $\overline{AD}$ and $\overline{CE}$ meet at point $P$. If $PE = 1.5$, $PD = 2$, and $DE = 2.5$, what is the area of quadrilateral $AEDC$?

[asy]
unitsize(75);
pathpen = black; pointpen=black;
pair A = MP("A", D((0,0)), dir(200));
pair B = MP("B", D((2,0)), dir(-20));
pair C = MP("C", D((1/2,1)), dir(100));
pair D = MP("D", D(midpoint(B--C)), dir(30));
pair E = MP("E", D(midpoint(A--B)), dir(-90));
pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013);
draw(A--B--C--cycle);
draw(A--D--E--C);
[/asy]

Answer: $13.5$
Let $G$ be the centroid of triangle $ABC$, and let $D$ be the midpoint of side $BC$. Suppose triangle $BDG$ is equilateral with each side having length \$1$. Find the length of $BC$ in triangle $ABC$.

[asy]
size(200);
defaultpen(fontsize(10));
real r=100.8933946;
pair A=sqrt(7)*dir(r), B=origin, C=(3,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);
draw(A--B--C--A--D^^B--E^^C--F);
pair point=G;
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(20));
label("1", B--G, dir(150));
label("1", D--G, dir(30));
label("1", B--D, dir(270));[/asy]

Answer: $2$
The area of trapezoid $ABCD$ is \$164\,\text{cm}^2$. The height is \$8\,\text{cm}$, with $AB = 10\,\text{cm}$ and $CD = 17\,\text{cm}$. Determine the length of $BC$ in centimeters.

[asy]
size(4inch,2inch);
draw((0,0)--(31,0)--(16,8)--(6,8)--cycle);
draw((11,8)--(11,0), linetype("8 4"));
draw((11,1)--(12,1)--(12,0));
label(" $A$ ", (0,0), SW);
label(" $D$ ", (31,0), SE);
label(" $B$ ", (6,8), NW);
label(" $C$ ", (16,8), NE);
label("10", (3,5), W);
label("8", (11,4), E);
label("17", (22.5,5), E);[/asy]

Answer: $10$
In $\triangle ABC$, suppose that $AX = XY = YB = BC$, and $m\angle ABC = 120^\circ$. Find the measure of angle $BAC$.

[asy]
pair A, B, C, X, Y;
A = origin;
X = dir(30);
Y = X + dir(0);
B = Y + dir(60);
C = B + dir(330);
draw(A--B--C--cycle);
draw(X--Y--B);
label(" $A$ ",A,W);
label(" $B$ ",B,N);
label(" $C$ ",C,E);
label(" $X$ ",X,NW);
label(" $Y$ ",Y,SE);
[/asy]

Answer: $15$
In triangle $ABC$, the cevians $AD$, $BE$, and $CF$ meet at a common point $S$. Given that $AS : DS = 3 : 2$ and $BS : ES = 4 : 3$. By what factor is $CS$ greater than $FS$?

[asy]
unitsize (1 cm);

pair A, B, C, D, E, F, S;

A = (0,0);
B = (5,0);
C = (1,4);
S = (14*A + 15*B + 6*C)/35;
D = extension(A,S,B,C);
E = extension(B,S,C,A);
F = extension(C,S,A,B);

draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(C--F);

dot(" $A$ ", A, SW);
dot(" $B$ ", B, SE);
dot(" $C$ ", C, N);
dot(" $D$ ", D, NE);
dot(" $E$ ", E, W);
dot(" $F$ ", F, dir(270));
dot(" $S$ ", S, NE);
[/asy]

Answer: $2.889$
In the diagram below, $ABDE$ is a square and certain angles and segment lengths are given. Find the length of $\overline{CF}$. 




[asy]
markscalefactor=0.15;
size(8cm);
pair A = (0,0);
pair B = (17,0);
pair E = (0,17);
pair D = (17,17);
pair F = (-120/17,225/17);
pair C = (17+120/17, 64/17);
draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle);  
label(" $A$ ", A, S);
label(" $B$ ", B, S);
label(" $C$ ", C, dir(0));
label(" $D$ ", D, N);
label(" $E$ ", E, N);
label(" $F$ ", F, W);
label(" $8$ ", (F+E)/2, NW);
label(" $15$ ", (F+A)/2, SW);
label(" $8$ ", (C+B)/2, SE);
label(" $15$ ", (D+C)/2, NE);
draw(rightanglemark(E,F,A));
draw(rightanglemark(D,C,B));
[/asy]

Answer: $23\sqrt{2}$
Let $G$ denote the point of intersection of the medians of triangle $ABC$ (the centroid), and let $D$ be the midpoint of side $BC$. If triangle $BDG$ is equilateral with all sides equal to \$1$, find the length $AB$ of triangle $ABC$.



[asy]
size(200);
defaultpen(fontsize(10));
real r=100.8933946;
pair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);
draw(A--B--C--A--D^^B--E^^C--F);
pair point=G;
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(20));
label("1", B--G, dir(150));
label("1", D--G, dir(30));
label("1", B--D, dir(270));[/asy]

Answer: $\sqrt{7}$
Let $G$ be the centroid of triangle $ABC$, and let $D$ be the midpoint of side $BC$. Suppose that triangle $BDG$ is equilateral with each side of length \$1$. Find the length $CA$ in triangle $ABC$.



[asy]
size(200);
defaultpen(fontsize(10));
real r=100.8933946;
pair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);
draw(A--B--C--A--D^^B--E^^C--F);
pair point=G;
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(20));
label("1", B--G, dir(150));
label("1", D--G, dir(30));
label("1", B--D, dir(270));[/asy]

Answer: $\sqrt{13}$
Let $A$, $B$, and $C$ be the centers of three circles, each with radius $r$ where \$1 < r < 2$. The distance between every pair of centers is \$2$. Define $B'$ as the intersection point of circles centered at $A$ and $C$ that lies outside the circle centered at $B$, and let $C'$ be the intersection point of circles centered at $A$ and $B$ that is outside the circle centered at $C$. Find the value of the length $B'C'$.

[asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP("$A$",A)); D(MP("$B$",B)); D(MP("$C$",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP("$B'$",BB[0],NW);MP("$C'$",CC[1],NE); [/asy]

Answer: $1+\sqrt{3(r^2-1)}$
In $\triangle ABC$, let $E$ denote the midpoint of $BC$ and let $D$ be a point on $AC$. Suppose $AC = 1$ and the angle measures are $\angle BAC = 60^\circ$, $\angle ABC = 100^\circ$, $\angle ACB = 20^\circ$, and $\angle DEC = 80^\circ$. Then, the sum of the area of $\triangle ABC$ and twice the area of $\triangle CDE$ is equal to

[asy] size(200); import cse5; pathpen=black; anglefontpen=black; pointpen=black; anglepen=black; dotfactor=3; pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE; EE=(B+C)/2; D(MP("$A$",A,W)--MP("$B$",B,N)--MP("$C$",C,E)--cycle); D(MP("$E$",EE,N)--MP("$D$",D,S)); D(D);D(EE); MA("80^\circ",8,D,EE,C,0.1); MA("20^\circ",8,EE,C,D,0.3,2,shift(1,3)*C); draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow()); MA("100^\circ",8,A,B,C,0.1,0); MA("60^\circ",8,C,A,B,0.1,0); [/asy]

Answer: $\frac{\sqrt{3}}{8}$
In the triangle $ABC$ shown in the adjacent diagram, the rays $AD$ and $AE$ divide $\angle BAC$ into three equal parts. The segment lengths $BD$, $DE$, and $EC$ are \$2$, \$3$, and \$6$, respectively. What is the length of the shortest side of $\triangle ABC$? 

[asy] defaultpen(linewidth(.8pt)); pair A = (0,11); pair B = (2,0); pair D = (4,0); pair E = (7,0); pair C = (13,0); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,S); label("$E$",E,S); label("$2$",midpoint(B--D),N); label("$3$",midpoint(D--E),NW); label("$6$",midpoint(E--C),NW); draw(A--B--C--cycle); draw(A--D); draw(A--E); [/asy]

Answer: $2\sqrt{10}$
The number of square units contained in polygon $ABCDEF$ is

[asy] draw((1,9)--(6,9)--(6,0)--(2,0)--(2,4)--(1,4)--cycle); label("A",(1,9),NW); label("B",(6,9),NE); label("C",(6,0),SE); label("D",(2,0),SW); label("E",(2,4),NE); label("F",(1,4),SW); label("6",(3,9),N); label("9",(6,4.5),E); label("4",(4,0),S); label("5",(1,6.5),W); [/asy]

Answer: $46$
The diagonal $DB$ of the rectangle $ABCD$ is divided into three segments of length \$1$ each by two lines, $L$ and $L'$. These lines are perpendicular to $DB$ and pass through the vertices $A$ and $C$, respectively. Find the area of $ABCD$, rounded to one decimal place.

[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B)^^rightanglemark(N1,E,B)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135); label("A", A, NE); label("B", B, NE); label("C", C, dir(0)); label("D", D, dir(180)); label("$L$", (x/2,0), SW); label("$L^\prime$", C, SW); label("1", D--F, NW); label("1", F--E, SE); label("1", E--B, SE); clip(W--X--Y--Z--cycle);[/asy]

Answer: $4.2$
What is the perimeter of the polygon ?

[asy] size(4cm); draw((0,0)--(0,6)--(7,6)--(7,3)--(2.7,3)--(2.7,0)--cycle); label("$6$",(0,3),W); label("$8$",(4,6),N); draw((0.5,0)--(0.5,0.5)--(0,0.5)); draw((0.5,6)--(0.5,5.5)--(0,5.5)); draw((6.5,6)--(6.5,5.5)--(7,5.5)); draw((6.5,3)--(6.5,3.5)--(7,3.5)); draw((2.2,0)--(2.2,0.5)--(2.7,0.5)); draw((2.7,2.5)--(3.2,2.5)--(3.2,3)) [/asy]

Answer: $28$
The area of the rectangular region is 

[asy] size(4cm);draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(.5 mm)); label("2",(8,2),E); label("4",(4,0),S); [/asy]

Answer: $8$
What is area of the  shaded rectangular region?

[asy] draw((0,0)--(18,0)--(18,12)--(0,12)--cycle); draw((0,6)--(18,6)); for(int a=6; a<12; ++a) { draw((1.5a,0)--(1.5a,6)); } fill((15,0)--(18,0)--(18,6)--(15,6)--cycle,black); label("0",(0,0),W); label("9",(9,0),S); label("18",(18,0),S); label("6",(0,6),W); label("12",(0,12),W); [/asy]

Answer: $18$
$\text{ABCD}$ is a rectangle, $\text{D}$ is the center of the circle, and $\text{B}$ is on the circle. If $\text{AD}=4$ and $\text{CD}=3$, then the area of the shaded region is

[asy] pair A,B,C,D; A=(0,4); B=(3,4); C=(3,0); D=origin; draw(circle(D,5)); fill((0,5)..(1.5,4.7697)..B--A--cycle,black); fill(B..(4,3)..(5,0)--C--cycle,black); draw((0,5)--D--(5,0)); label("A",A,NW); label("B",B,NE); label("C",C,S); label("D",D,SW); [/asy]

Answer: $\frac{25}{4}\pi - 12$
In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$ The ratio $\frac{AB}{AD}$ is

[asy] size((220)); draw((0,0)--(20,0)--(7,6)--cycle); draw((6,6)--(10,-1)); label("A", (0,0), W); label("B", (20,0), E); label("C", (7,6), NE); label("D", (9.5,-1), W); label("E", (5.9, 6.1), SW); label("$45^{\circ}$", (2.5,.5)); label("$60^{\circ}$", (7.8,.5)); label("$30^{\circ}$", (16.5,.5)); [/asy]

Answer: $\sqrt[4]{12}$
If $\angle \text{CBD}$ is a right angle, then this protractor indicates that the measure of $\angle \text{ABC}$ is approximately

[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label("O",(-2.5,0),W); label("A",A,W); label("B",B,S); label("C",C,W); label("D",D,E); label("0",(-1.8,0),W); label("20",(-1.7,.5),NW); label("160",(1.6,.5),NE); label("180",(1.7,0),E); [/asy]

Answer: $50$
The shaded region formed by the two intersecting perpendicular rectangles, in square units, is

[asy] fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); label("2",(0,.9),W); label("3",(7.3,4.5),N); draw((0,-3.3)--(0,-5.3),linewidth(1)); draw((0,-4.3)--(3.7,-4.3),linewidth(1)); label("10",(4.7,-3.7),S); draw((5.7,-4.3)--(10,-4.3),linewidth(1)); draw((10,-3.3)--(10,-5.3),linewidth(1)); draw((11,4.5)--(13,4.5),linewidth(1)); draw((12,4.5)--(12,2),linewidth(1)); label("10",(11.3,1),E); draw((12,0)--(12,-3.5),linewidth(1)); draw((11,-3.5)--(13,-3.5),linewidth(1)); [/asy]

Answer: $44$
What is the ratio of the shaded part to the unshaded part in parallelogram $\text{ABCD}$?

[asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S); label("$10$",(9,8),N); label("$6$",(7,0),S); label("$8$",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy]

Answer: $4$
All six sides of a rectangular solid were rectangles. A one-foot cube was cut out of the rectangular solid as shown. The total number of square feet in the surface of the new solid is how many more or less than that of the original solid?

[asy] unitsize(24); draw((0,0)--(1,0)--(1,3)--(0,3)--cycle); draw((1,0)--(1+9*sqrt(3)/2,9/2)--(1+9*sqrt(3)/2,15/2)--(1+5*sqrt(3)/2,11/2)--(1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),4)--(1+2*sqrt(3),5)--(1,3)); draw((0,3)--(2*sqrt(3),5)--(1+2*sqrt(3),5)); draw((1+9*sqrt(3)/2,15/2)--(9*sqrt(3)/2,15/2)--(5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,5)); draw((1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),9/2)); draw((1+5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,11/2)); label("$1'$",(.5,0),S); label("$3'$",(1,1.5),E); label("$9'$",(1+9*sqrt(3)/4,9/4),S); label("$1'$",(1+9*sqrt(3)/4,17/4),S); label("$1'$",(1+5*sqrt(3)/2,5),E);label("$1'$",(1/2+5*sqrt(3)/2,11/2),S); [/asy]

Answer: $0$
Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is

[asy] fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray); fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white); fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white); fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white); fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white); draw(circle((3,3),3)); draw(circle((3,-3),3)); draw(circle((-3,-3),3)); draw(circle((-3,3),3)); draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle); [/asy]

Answer: $36-9\pi$
Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD}\perp\overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\widehat{EF}$ has center $D$. The area of the shaded "smile" $AEFBDA$, is

[asy] fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill((0,0)--arc((0,0),1,180,360)--cycle,white); draw((1,0)--arc((1,0),2,180,225)--(1,0),black+linewidth(1)); draw((-1,0)--arc((-1,0),2,315,360)--(-1,0),black+linewidth(1)); draw((0,0)--arc((0,0),1,180,360)--(0,0),black+linewidth(1)); draw(arc((0,-1),2-sqrt(2),225,315),black+linewidth(1)); draw((0,0)--(0,-1),black+linewidth(1)); MP("C",(0,0),N);MP("A",(-1,0),N);MP("B",(1,0),N); MP("D",(0,-.8),NW);MP("E",(1-sqrt(2),-sqrt(2)),SW);MP("F",(-1+sqrt(2),-sqrt(2)),SE); [/asy]

Answer: $2\pi-\pi \sqrt{2}-1$
The convex pentagon $ABCDE$ has $\angle{A}=\angle{B}=120^\circ,EA=AB=BC=4$ and $CD=DE=8$. What is the area of ABCDE?

[asy] draw((-1,0)--(1,0)--(1+sqrt(2),sqrt(2))--(0,sqrt(2)+sqrt(13-2*sqrt(2)))--(-1-sqrt(2),sqrt(2))--cycle,black+linewidth(.75)); MP("A",(-1,0),SW);MP("B",(1,0),SE);MP("C",(1+sqrt(2),sqrt(2)),E);MP("D",(0,sqrt(2)+sqrt(13-2*sqrt(2))),N);MP("E",(-1-sqrt(2),sqrt(2)),W); dot((-1,0));dot((1,0));dot((1+sqrt(2),sqrt(2)));dot((-1-sqrt(2),sqrt(2)));dot((0,sqrt(2)+sqrt(13-2*sqrt(2)))); [/asy]

Answer: $28\sqrt{3}$
Amy painted a dartboard over a square clock face using the "hour positions" as boundaries.[See figure.] If $t$ is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then $\frac{t}{q}=$

[asy] draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75)); draw((0,-1)--(0,1), black+linewidth(.75)); draw((-1,0)--(1,0), black+linewidth(.75)); draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75)); draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75)); draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75)); draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75)); [/asy]

Answer: $\frac{\sqrt{3}+1}{4}$
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the total area of the four rectangle?

[asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label("6", (1.5,6)); label("?", (1.5,2.5)); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy]

Answer: $70$
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. Suppose there is a straight line that divides the area of the region into two equal parts. What is the tangent of the angle formed between this line and the y-axis?

[asy] Label l; l.p=fontsize(6); xaxis("$x$",0,6,Ticks(l,1.0,0.5),EndArrow); yaxis("$y$",0,4,Ticks(l,1.0,0.5),EndArrow); draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy]

Answer: $\frac{9}{7}$
Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $BC/AB=$

[asy] draw(Circle((0,0), 13)); draw((-13,0)--(12,5)--(12,-5)--cycle); dot((-13,0)); dot((12,5)); dot((12,-5)); label("A", (-13,0), W); label("B", (12,5), NE); label("C", (12,-5), SE); [/asy]

Answer: $\2\sin\left(\frac{1}{4}\right)$
Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\overline{AM}$ with $AD=10$ and $\angle BDC=3\angle BAC.$ Then the perimeter of $\triangle ABC$ is $

[asy] import graph; size(5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.55,xmax=7.95,ymin=-4.41,ymax=5.3; draw((1,3)--(0,0)); draw((0,0)--(2,0)); draw((2,0)--(1,3)); draw((1,3)--(1,0)); draw((1,0.7)--(0,0)); draw((1,0.7)--(2,0)); label("$11$",(1,1.63),W); dot((1,3),ds); label("$A$",(1,3),N); dot((0,0),ds); label("$B$",(0,0),SW); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);[/asy]

Answer: $11+\sqrt{605}$
Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates (2,2). For what value of the x-coordinate of point $T$ will the area of triangle $PQT$ be exactly twice the area of square $OPQR$?

[asy] pair O,P,Q,R,T; O = (0,0); P = (2,0); Q = (2,2); R = (0,2); T = (-4,0); draw((-5,0)--(3,0)); draw((0,-1)--(0,3)); draw(P--Q--R); draw((-0.2,-0.8)--(0,-1)--(0.2,-0.8)); draw((-0.2,2.8)--(0,3)--(0.2,2.8)); draw((-4.8,-0.2)--(-5,0)--(-4.8,0.2)); draw((2.8,-0.2)--(3,0)--(2.8,0.2)); draw(Q--T); label("$O$",O,SW); label("$P$",P,S); label("$Q$",Q,NE); label("$R$",R,W); label("$T$",T,S); [/asy]

Answer: $-6$