LiAuto-DSR/GeoGramBench huggingface dataset:
In $	riangle ABC$, $AB = 10$, $AC = 8$, and $BC = 6$. Circle $P$ is the circle with the smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$, of circle $P$ with sides $AC$ and $BC$, respectively. What is the length of segment $QR$?

[asy] size(100); real a=4, b=3; // import cse5; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0)))); [/asy]

Answer: $4.8$
In the figure below, let $ABCDE$ be a regular pentagon such that $AG = 1$. Find the value of $FG + JH + CD$.

[asy]
import cse5;pathpen=black;pointpen=black;
size(2inch);
pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162);
D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5));
D(A--C--E--B--D--cycle);
pair F=IP(A--D,B--E), G=IP(B--E,C--A), H=IP(C--A,B--D), I=IP(D--B,E--C), J=IP(C--E,D--A);
D(MP("F",F,dir(126))--MP("I",I,dir(270))--MP("G",G,dir(54))--MP("J",J,dir(198))--MP("H",H,dir(342))--cycle);
[/asy]

Answer: $1 + \sqrt{5}$
The diagram below shows four regular hexagons, each with side length \$1$ meter, attached to the sides of a square. This shape is drawn onto a thin sheet of metal and cut out. The hexagons are then folded upward along the sides of the square so that $A_1$ meets $A_2$, $B_1$ meets $B_2$, $C_1$ meets $C_2$, and $D_1$ meets $D_2$. If water is poured into the resulting dish, it fills to the height of the corner where $A_1$ and $A_2$ meet. There exist relatively prime positive integers $m$ and $n$ such that the number of cubic meters of water the dish will hold is $\sqrt{\frac{m}{n}}$. Find $m+n$.



[asy]
import graph; size(7cm); 
real labelscalefactor = 0.5; 
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); 
pen dotstyle = black; 
real xmin = -4.3, xmax = 14.52, ymin = -8.3, ymax = 6.3; 
draw((0,1)--(0,0)--(1,0)--(1,1)--cycle); 
draw((1,1)--(1,0)--(1.87,-0.5)--(2.73,0)--(2.73,1)--(1.87,1.5)--cycle); 
draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); 
draw((0,0)--(1,0)--(1.5,-0.87)--(1,-1.73)--(0,-1.73)--(-0.5,-0.87)--cycle); 
draw((0,1)--(0,0)--(-0.87,-0.5)--(-1.73,0)--(-1.73,1)--(-0.87,1.5)--cycle); 

draw((0,1)--(0,0)); 
draw((0,0)--(1,0)); 
draw((1,0)--(1,1)); 
draw((1,1)--(0,1)); 
draw((1,1)--(1,0)); 
draw((1,0)--(1.87,-0.5)); 
draw((1.87,-0.5)--(2.73,0)); 
draw((2.73,0)--(2.73,1)); 
draw((2.73,1)--(1.87,1.5)); 
draw((1.87,1.5)--(1,1)); 
draw((0,1)--(1,1)); 
draw((1,1)--(1.5,1.87)); 
draw((1.5,1.87)--(1,2.73)); 
draw((1,2.73)--(0,2.73)); 
draw((0,2.73)--(-0.5,1.87)); 
draw((-0.5,1.87)--(0,1)); 

dot((1.87,-0.5),dotstyle); 
label(" $C_1$ ", (1.72,-0.1), NE * labelscalefactor); 
dot((1.87,1.5),dotstyle); 
label(" $B_2$ ", (1.76,1.04), NE * labelscalefactor); 
dot((1.5,1.87),dotstyle); 
label(" $B_1$ ", (0.96,1.8), NE * labelscalefactor); 
dot((-0.5,1.87),dotstyle); 
label(" $A_2$ ", (-0.26,1.78), NE * labelscalefactor); 
dot((-0.87,1.5),dotstyle); 
label(" $A_1$ ", (-0.96,1.08), NE * labelscalefactor); 
dot((-0.87,-0.5),dotstyle); 
label(" $D_2$ ", (-1.02,-0.18), NE * labelscalefactor); 
dot((-0.5,-0.87),dotstyle); 
label(" $D_1$ ", (-0.22,-0.96), NE * labelscalefactor); 
dot((1.5,-0.87),dotstyle); 
label(" $C_2$ ", (0.9,-0.94), NE * labelscalefactor); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 
[/asy]

Answer: $67$
In the figure below, $E$ is the midpoint of arc $ABEC$ and segment $ED$ is perpendicular to chord $BC$ at $D$. If the length of chord $AB$ is \$5$ and the length of segment $BD$ is \$3$, find the length of $DC$.



[asy]
unitsize(1 cm);
pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C);
draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8));
label(" $A$ ",A,.5A); label(" $B$ ",B,.5B); label(" $C$ ",C,.5C); label(" $E$ ",E,.5E); label(" $D$ ",D,dir(-60));
[/asy]

Answer: $8$
What is the radius of the smallest circle that can contain the symmetric figure formed by the \$3$ unit squares shown above?



[asy]
draw((0,0)--(0,1)--(2,1)--(2,0)--cycle^^(.5,1)--(.5,2)--(1.5,2)--(1.5,1)--(.5,2)^^(.5,1)--(1.5,2)^^(1,2)--(1,0));
[/asy]

Answer: $\frac{5\sqrt{17}}{16} $
In the figure below, $\triangle ABC$, $\triangle DEF$, and $\triangle GHI$ are overlapping equilateral triangles. Points $C$ and $F$ lie on $\overline{BD}$, $F$ and $I$ lie on $\overline{EG}$, and $C$ and $I$ lie on $\overline{AH}$. Suppose $AB = 2FC$, $DE = 3FC$, and $GH = 4FC$. Given that the area of $\triangle FCI$ is \$3$, find the area of hexagon $ABGHDE$.



[asy]
size(5cm);
pen dps = fontsize(10);
defaultpen(dps);
pair A,B,C,D,E,F,G,H,I;
G=origin;
H=(4,0);
I=(2,2*sqrt(3));
F=(1.5,3*sqrt(3)/2);
C=F+(1,0);
B=F-(1,0);
D=C+(2,0);
A=F+(0,sqrt(3));
E=C+(0.5,3*sqrt(3)/2);
draw(A--H--G--E--D--B--cycle);
label(" $A$ ",A,N*.5);
label(" $B$ ",B,S*.5);
label(" $C$ ",C,SW*.5);
label(" $D$ ",D,S*.5);
label(" $E$ ",E,N*.5);
label(" $F$ ",F,SE*.5);
label(" $G$ ",G,S*.5);
label(" $H$ ",H,S*.5);
label(" $I$ ",I,N*2);
[/asy]

Answer: $114$
If the sum of the measures in degrees of angles $A$, $B$, $C$, $D$, $E$, and $F$ in the diagram above is \$90n$, what is the value of $n$?



[asy]
draw((3,-13)--(21.5,-5)--(19,-18)--(9,-18)--(10,-6)--(23,-14.5)--cycle);
label("A",(3,-13),W);label("C",(21.5,-5),N);label("E",(19,-18),E);label("F",(9,-18),W);label("B",(10,-6),N);label("D",(23,-14.5),E);
[/asy]

Answer: $4$
In the figure, circle $K$ has diameter $AB$. Circle $L$ is tangent to circle $K$ and to $AB$ at the center of circle $K$. Circle $M$ is tangent to circle $K$, to circle $L$, and to $AB$. What is the ratio of the area of circle $K$ to the area of circle $M$?



[asy]
size(150);
pair A=(0,0),B=(1,0),C=(0,1),D=(-1,0),E=(0,.5),F=(sqrt(2)/2,.25);
draw(circle(A,1)^^D--B);
draw(circle(E,.5)^^circle( F ,.25));
label(" $A$ ", D, W);
label(" $K$ ", A, S);
label(" $B$ ", B, dir(0));
label(" $L$ ", E, N);
label(" $M$ ",shift(-.05,.05)*F);
[/asy]

Answer: $16$
Pentagon $ABCDE$ consists of a square $ACDE$ and an equilateral triangle $ABC$ sharing the side $\overline{AC}$. A circle centered at $C$ has area \$24$. The intersection of the circle and the pentagon is exactly half the area of the pentagon. Find the area of the pentagon.



[asy]
import graph; size(4.26cm); 
real labelscalefactor = 0.5; 
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); 
pen dotstyle = black; 
real xmin = -1.52, xmax = 2.74, ymin = -2.18, ymax = 6.72;
draw((0,1)--(2,1)--(2,3)--(0,3)--cycle); 
draw((0,3)--(2,3)--(1,4.73)--cycle); 
draw((0,1)--(2,1)); 
draw((2,1)--(2,3)); 
draw((2,3)--(0,3)); 
draw((0,3)--(0,1)); 
draw((0,3)--(2,3)); 
draw((2,3)--(1,4.73)); 
draw((1,4.73)--(0,3)); 
draw(circle((0,3), 1.44)); 
label(" $C$ ",(-0.4,3.14),SE*labelscalefactor); 
label(" $A$ ",(2.1,3.1),SE*labelscalefactor); 
label(" $B$ ",(0.86,5.18),SE*labelscalefactor); 
label(" $D$ ",(-0.28,0.88),SE*labelscalefactor); 
label(" $E$ ",(2.1,0.8),SE*labelscalefactor); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Answer: $20$
In the figure, $AB$ and $BC$ are adjacent sides of square $ABCD$. Let $M$ be the midpoint of $AB$, and $N$ the midpoint of $BC$. The lines $AN$ and $CM$ intersect at $O$. What is the ratio of the area of $AOCD$ to the area of $ABCD$?

[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)--(2,1)--(2,2)--(1,0));
label("A", (0,0), S);
label("B", (2,0), S);
label("C", (2,2), N);
label("D", (0,2), N);
label("M", (1,0), S);
label("N", (2,1), E);
label("O", (1.2, .8));
[/asy]

Answer: $\frac{2}{3}$
Calculate  $\frac{area(CDF)}{area(CEF)}$  in the figure.



[asy]
import graph; size(5.75cm); 
real labelscalefactor = 0.5; 
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); 
pen dotstyle = black; 
real xmin = -2, xmax = 21, ymin = -2, ymax = 16;  
draw((0,0)--(20,0)); 
draw((13.48,14.62)--(7,0)); 
draw((0,0)--(15.93,9.12)); 
draw((13.48,14.62)--(20,0)); 
draw((13.48,14.62)--(0,0)); 
label("6",(15.16,12.72),SE*labelscalefactor); 
label("10",(18.56,5.1),SE*labelscalefactor); 
label("7",(3.26,-0.6),SE*labelscalefactor); 
label("13",(13.18,-0.71),SE*labelscalefactor); 
label("20",(5.07,8.33),SE*labelscalefactor); 
dot((0,0),dotstyle); 
label(" $B$ ", (-1.23,-1.48), NE * labelscalefactor); 
dot((20,0),dotstyle); 
label(" $C$ ", (19.71,-1.59), NE * labelscalefactor); 
dot((7,0),dotstyle); 
label(" $D$ ", (6.77,-1.64), NE * labelscalefactor); 
dot((13.48,14.62),dotstyle); 
label(" $A$ ", (12.36,14.91), NE * labelscalefactor); 
dot((15.93,9.12),dotstyle); 
label(" $E$ ", (16.42,9.21), NE * labelscalefactor); 
dot((9.38,5.37),dotstyle); 
label(" $F$ ", (9.68,4.5), NE * labelscalefactor); 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 
[/asy]

Answer: $\frac{14}{15}$
In the figure, a circle intersects the sides of an equilateral triangle at six points. Given that $AG = 2$, $GF = 13$, $FC = 1$, and $HJ = 7$, find the length of $DE$.



[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]

Answer: $2\sqrt{22}$
In the diagram below, points $A$, $B$, and $C$ are each \$6$ units apart. Define a point $X$ to be \emph{reachable} if there exists a path (not necessarily straight) from $A$ to $X$ with length at most \$8$ that does not pass through the interior of segment $\overline{BC}$. (Both $X$ and the path must lie in the plane determined by $A$, $B$, and $C$.) Let $R$ denote the set of all reachable points. What is the area of $R$?



[asy]
  unitsize(40);
  pair A = dir(90);
  pair B = dir(210);
  pair C = dir(330);
  dot(A); 
  dot(B);
  dot(C);
  draw(B -- C);
  label(" $A$ ", A, N);
  label(" $B$ ", B, W);
  label(" $C$ ", C, E);
[/asy]

Answer: $56\pi+9\sqrt{3}$
A net is shown below consisting of \$3$ squares, \$4$ equilateral triangles, and \$1$ regular hexagon, each with side length \$1$. If this net is folded to create a polyhedron, what is the volume of the resulting solid?

Net:





[asy]
pair A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P;
A = origin;
B = (0,3);
C = 3*dir(150);
D = (0,1);
E = (0,2);
F = C+2*dir(30);
G = C+dir(30);
H = 2*dir(150);
I = dir(150);
J = (1,1);
K = J+dir(30);
L = (1,2);
M = F+dir(120);
N = G+dir(120);
O = H+dir(240);
P = I+dir(240);
draw(A--B--C--cycle);
draw(D--E--F--G--H--I--cycle);
draw(D--E--L--J--cycle);
draw(F--G--N--M--cycle);
draw(H--I--P--O--cycle);
draw(J--K--L--cycle);
[/asy]

Answer: $\frac{5}{3\sqrt{2}}$
In the diagram, $BC$ is the diameter of a circle with $BC = \sqrt{257}$, $BD = 1$, and $DA = 12$. Determine the length of the altitude from $A$ to $BC$.



[asy]
import cse5;
size(200);
pair O=(2, 0), B=(0, 0), C=(4, 0), A=(1, 3), D, E;
D=MP("D",D(IP(D(CP(O,B)),D(MP("A",D(A),N)--MP("B",D(B),W)))),NW);
E=MP("E",D(IP(CP(O,B),D(MP("C",D(C),NE)--A),1)),NE);
D(B--C);
[/asy]

Answer: $7.18$
In the figure to the right, every point on circle $O'$ lies outside of circle $O$. Let $P$ and $Q$ be the intersection points of an internal common tangent and the two external common tangents. If the length of each external common tangent is \$6$, what is the length of $PQ$?



[asy]
size(150);
dotfactor=4;
draw(circle((0,0),4));
draw(circle((10,-6),3));
pair O,A,P,Q;
O = (0,0);
A = (10,-6);
P = (-.55, -4.12);
Q = (10.7, -2.86);
dot(" $O$ ", O, NE);
dot(" $O'$ ", A, SW);
dot(" $P$ ", P, SW);
dot(" $Q$ ", Q, NE);
draw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle);
draw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle);
draw(P--Q--cycle);
[/asy]

Answer: $6$
In the figure below, each vertex of a unit square is connected by a line segment to the midpoint of the opposite side. Determine the area of the quadrilateral $ABCD$ formed by the intersection of these segments.



[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((0,0)--(1,0.5));
draw((1,0)--(0.5,1));
draw((1,1)--(0,0.5));
draw((0,1)--(0.5,0));
label(" $A$ ",(0.21,0.6),N);
label(" $B$ ",(0.4,0.21),W);
label(" $C$ ",(0.8,0.4),S);
label(" $D$ ",(0.6,0.8),E);
[/asy]

Answer: $\frac{1}{5}$
The diagram below shows two parallel lines, $\ell$ and $m$, which are \$12$ units apart. A circle is tangent to line $\ell$ at point $A$, and a congruent circle is tangent to line $m$ at point $B$. The two circles are also tangent to each other. If the distance between $A$ and $B$ is \$13$, what is the radius of each circle?

[asy]unitsize(7);
  draw((-7, 0) -- (12, 0));
  draw((-7, 12) -- (12, 12));
  real r = 169 / 48;
  draw(circle((0, r), r));
  draw(circle((5, 12 - r), r));
  pair A = (0, 0);
  pair B = (5, 12);
  dot(A);
  dot(B);
  label(" $A$ ", A, plain.S);
  label(" $B$ ", B, plain.N);
  label(" $\ell$ ", (12, 0), plain.E);
  label(" $m$ ", (12, 12), plain.E);[/asy]

Answer: $\frac{169}{48}$
The figure below shows two congruent right triangles, each with angles \$40^\circ$, \$50^\circ$, and \$90^\circ$. What is the measure of the obtuse angle $\alpha$ created by the intersection of their hypotenuses?



[asy]
import olympiad;
	size(80);
defaultpen(linewidth(0.8));
draw((0,0)--(3,0)--(0,4.25)--(0,0)^^(0,3)--(4.25,0)--(3,0)^^rightanglemark((0,3),(0,0),(3,0),10));
pair P = intersectionpoint((3,0)--(0,4.25),(0,3)--(4.25,0));
draw(anglemark((4.25,0),P,(0,4.25),10));
label(" $\alpha$ ",P,2 * NE);
[/asy]

Answer: $170$
In rectangle $ADEH$, points $B$ and $C$ divide $\overline{AD}$ into three equal segments, and points $G$ and $F$ divide $\overline{HE}$ into three equal parts. Additionally, $AH = AC = 2$. What is the area of quadrilateral $WXYZ$ shown in the diagram?

[asy]defaultpen(linewidth(0.7));pointpen=black; pathpen=black;
size(7cm);
pair A,B,C,D,E,F,G,H,W,X,Y,Z;
A=(0,2); B=(1,2); C=(2,2); D=(3,2);
H=(0,0); G=(1,0); F=(2,0); E=(3,0);
D('A',A, N); D('B',B,N); D('C',C,N); D('D',D,N); D('E',E,NE); D('F',F,NE); D('G',G,NW); D('H',H,NW);
D(A--F); D(B--E); D(D--G); D(C--H);
Z=IP(A--F, C--H); Y=IP(A--F, D--G); X=IP(B--E,D--G); W=IP(B--E,C--H);
D('W',W,N); D('X',X,plain.E); D('Y',Y,S); D('Z',Z,plain.W);
D(A--D--E--H--cycle);[/asy]

Answer: $\frac{1}{2}$
In the figure, BC  is a diameter of the circle, where  $BC=\sqrt{901}, BD=1$ , and  $DA=16$ . If  $EC=x$ , what is the value of x?




[asy]size(2inch);
pair O,A,B,C,D,E;
B=(0,0);
O=(2,0);
C=(4,0);
D=(.333,1.333);
A=(.75,2.67);
E=(1.8,2);
draw(Arc(O,2,0,360));
draw(B--C--A--B);
label(" $A$ ",A,N);
label(" $B$ ",B,W);
label(" $C$ ",C,E);
label(" $D$ ",D,W);
label(" $E$ ",E,N);
label("Figure not drawn to scale",(2,-2.5),S);
[/asy]

Answer: $26$
Region $ABCDEFGHIJ$ is composed of 13 congruent squares and is inscribed within rectangle $PQRS$ such that $A$ lies on $\overline{PQ}$, $B$ on $\overline{QR}$, $E$ on $\overline{RS}$, and $H$ on $\overline{SP}$, as shown in the figure to the right. If $PQ = 28$ and $QR = 26$, find the area of region $ABCDEFGHIJ$.



[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(12)); pair P=(0,0), Q=(0,28), R=(26,28), S=(26,0), B=(3,28);
draw(P--Q--R--S--cycle);
picture p = new picture;
draw(p, (0,0)--(3,0)^^(0,-1)--(3,-1)^^(0,-2)--(5,-2)^^(0,-3)--(5,-3)^^(2,-4)--(3,-4)^^(2,-5)--(3,-5));
draw(p, (0,0)--(0,-3)^^(1,0)--(1,-3)^^(2,0)--(2,-5)^^(3,0)--(3,-5)^^(4,-2)--(4,-3)^^(5,-2)--(5,-3));
transform t = shift(B) * rotate(-aSin(1/26^.5)) * scale(26^.5);
add(t*p);
label(" $P$ ",P,SW); label(" $Q$ ",Q,NW); label(" $R$ ",R,NE); label(" $S$ ",S,SE); label(" $A$ ",t*(0,-3),W); label(" $B$ ",B,N); label(" $C$ ",t*(3,0),plain.ENE); label(" $D$ ",t*(3,-2),NE); label(" $E$ ",t*(5,-2),plain.E); label(" $F$ ",t*(5,-3),plain.SW); label(" $G$ ",t*(3,-3),(0.81,-1.3)); label(" $H$ ",t*(3,-5),plain.S); label(" $I$ ",t*(2,-5),NW); label(" $J$ ",t*(2,-3),SW);[/asy]

Answer: $338$
In the figure, $\overline{CE}$ and $\overline{DE}$ are equal chords of a circle with center $O$. The arc $AB$ represents a quarter of the circle. By what factor is the area of triangle $CED$ greater than the area of triangle $AOB$?

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

pair O = origin;
pair C = (-1,0);
pair D = (1,0);
pair E = (0,1);
pair A = dir(-135);
pair B = dir(-60);

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

label(" $A$ ",A,SW);
label(" $C$ ",C,W);
label(" $E$ ",E,N);
label(" $D$ ",D,NE);
label(" $B$ ",B,SE);
label(" $O$ ",O,N);[/asy]

Answer: $2$
In the given diagram, suppose that points $A$, $B$, and $C$ are points of tangency. Find 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: $\frac{1}{16}$
Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$, where $EF$ is parallel to $AB$ and $GH$ is parallel to $BC$. Given that $BF = \frac{1}{4}$ and point $P$ is the intersection of $EF$ and $GH$, suppose $BF + DH = FH$. Find the nearest integer to the measure, in degrees, of angle $\angle FAH$.



[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]

Answer: $45$
In the figure below, square $ABDE$ is shown along with certain angles and lengths. The length $\overline{CF}$ can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are integers, and $b$ is not divisible by the square of any integer greater than \$1$. Find the value of $a + b$.




[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: $25$
In the figure below, $\angle CAB$, $\angle CBD$, and $\angle CDE$ are all right angles, and the given side lengths are $AC = 3$, $BC = 5$, $BD = 12$, and $DE = 84$. The distance from point $E$ to line $AB$ can be written as a reduced fraction $\frac{m}{n}$, where $m$ and $n$ are positive integers with no common factors. Find $m+n$.



[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]

Answer: $5393$
Let $O$ be the centroid of an equilateral triangle $ABC$ with area $\frac{1}{\pi}$. As illustrated in the diagram below, a circle centered at $O$ intersects the triangle at points $D$, $E$, $F$, $G$, $H$, and $I$, which trisect the sides of the triangle. Find the total area of the six shaded regions.






[asy]
unitsize(90);
pair A = dir(0);
pair B = dir(120);
pair C = dir(240);
draw(A -- B -- C -- cycle);
pair D = (2*A + B)/3;
pair E = (A + 2*B)/3;
pair F = (2*B + C)/3;
pair G = (B + 2*C)/3;
pair H = (2*C + A)/3;
pair I = (C + 2*A)/3;
draw(E -- F);
draw(G -- H);
draw(I -- D);
draw(D -- G);
draw(E -- H);
draw(F -- I);
pair O = (0, 0);
real r = 1/sqrt(3);
draw(circle(O, r));
fill(O -- D -- E -- cycle, gray);
fill(O -- F -- G -- cycle, gray);
fill(O -- H -- I -- cycle, gray);
fill(arc(O, r, -30, 30) -- cycle, gray);
fill(arc(0, r, 90, 150) -- cycle, gray);
fill(arc(0, r, 210, 270) -- cycle, gray);
label(" $A$ ", A, A);
label(" $B$ ", B, B);
label(" $C$ ", C, C);
label(" $D$ ", D, unit(D));
label(" $E$ ", E, unit(E));
label(" $F$ ", F, unit(F));
label(" $G$ ", G, unit(G));
label(" $H$ ", H, unit(H));
label(" $I$ ", I, unit(I));
label(" $O$ ", O, C);
[/asy]

Answer: $\frac{2\sqrt{3}}{27}$
In the diagram, $ABCD$ and $EFGH$ are similar rectangles. Given that $DK:KC = 3:2$, by what factor is the area of rectangle $ABCD$ greater than the area of rectangle $EFGH$?



[asy]draw((75,0)--(0,0)--(0,50)--(75,50)--(75,0)--(55,0)--(55,20)--(100,20)--(100,0)--cycle);
draw((55,5)--(60,5)--(60,0));
draw((75,5)--(80,5)--(80,0));
label("A",(0,50),NW);
label("B",(0,0),SW);
label("C",(75,0),SE);
label("D",(75,50),NE);
label("E",(55,20),NW);
label("F",(55,0),SW);
label("G",(100,0),SE);
label("H",(100,20),NE);
label("K",(75,20),NE);[/asy]

Answer: $6.25$
In the figure below, square $ABCD$ with side length \$23$ is divided into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are given in the diagram. Determine the maximum possible area of the central rectangle.

[asy]
size(250);
 defaultpen (linewidth (0.7) + fontsize (10));
		 draw ((0,0)--(23,0)--(23,23)--(0,23)--cycle);
		 label(" $A$ ", (0,23), NW);
		 label(" $B$ ", (23, 23), NE);
		 label(" $C$ ", (23,0), SE);
		 label(" $D$ ", (0,0), SW);
		 draw((0,6)--(23,6));
		 draw((0,19)--(23,19));
		 draw((5,0)--(5,23));
		 draw((12,0)--(12,23));
		 label("13", (17/2, 21));
		 label("111",(35/2,25/2));
		 label("37",(17/2,3));
		 label("123",(2.5,12.5));[/asy]

Answer: $180$
In the diagram below, the regular hexagon $BCEGHJ$ is inscribed within rectangle $ADFI$. Let $\theta$ denote the acute angle between side $\overline{EG}$ of the hexagon and the diagonal $\overline{AF}$ of the rectangle. Suppose that $\sin^2 \theta = \frac{m}{n}$, where $m$ and $n$ are positive integers with no common factors. Find the value of $m+n$.



[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]

Answer: $55$