Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: A cube with side length $ 1$ is sliced by a plane that passes through two diagonally opposite vertices $ A$ and $ C$ and the midpoints $ B$ and $ D$ of two opposite edges not containing $ A$ and $ C$, ac shown. What is the area of quadrilateral $ ABCD$?
<image1>
Choices:
A. $\frac{\sqrt{6}}{2}$
B. $\frac{5}{4}$
C. $\sqrt{2}$
D. $\frac{3}{2}$
E. $\sqrt{3}$
Answer: A
Problem: <image>Question: A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$?
<image1>
Choices:
A. $2\sqrt{5} - \sqrt{3}$
B. $3$
C. $\frac{3\sqrt{7} - \sqrt{3}}{2}$
D. $2\sqrt{3}$
E. $\frac{5 + 2\sqrt{3}}{2}$
Answer: C
Problem: <image>Question: A circle of radius $ 2$ is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
<image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{\pi}{6}$
C. $\frac{2}{\pi}$
D. $\frac{2}{3}$
E. $\frac{3}{\pi}$
Answer: A
Problem: <image>Question: Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD=3$, and $ DC=4$. What is the area of $ \triangle{ABC}$?
<image1>
Choices:
A. $4\sqrt{3}$
B. $7\sqrt{3}$
C. $21$
D. $14\sqrt{3}$
E. $42$
Answer: B
Problem: <image>Question: In quadrilateral $ ABCD$, $ AB = 5$, $ BC = 17$, $ CD = 5$, $ DA = 9$, and $ BD$ is an integer. What is $ BD$?
<image1>
Answer: 13
Problem: <image>Question: Four congruent rectangles are placed as shown. The area of the outer square is $ 4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
<image1>
Choices:
A. $3$
B. $\sqrt{10}$
C. $2 + \sqrt{2}$
D. $2\sqrt{3}$
E. $4$
Answer: A
Problem: <image>Question: The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n - 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$?
<image1>
Answer: 761
Problem: <image>Question: Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle, In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the four smaller circles to the area of the larger circle?
<image1>
Choices:
A. $3-2\sqrt{2}$
B. $2-\sqrt{2}$
C. $4(3-2\sqrt{2})$
D. $\frac{1}{2}(3-\sqrt{2})$
E. $2\sqrt{2}-2$
Answer: C
Problem: <image>Question: A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds?
<image1>
Choices:
A. $\frac{1}{8}$
B. $\frac{1}{6}$
C. $\frac{1}{5}$
D. $\frac{1}{4}$
E. $\frac{1}{3}$
Answer: C
Problem: <image>Question: Segment $ BD$ and $ AE$ intersect at $ C$, as shown, $ AB=BC=CD=CE$, and $ \angle A=\frac{5}{2}\angle B$. What is the degree measure of $ \angle D$?
<image1>
Answer: 52.5
Problem: <image>Question: As shown below, convex pentagon $ ABCDE$ has sides $ AB = 3$, $ BC = 4$, $ CD = 6$, $ DE = 3$, and $ EA = 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right along the $ x$-axis. Which side will touch the point $ x = 2009$ on the $ x$-axis?
<image1>
Choices:
A. $\overline{AB}$
B. $\overline{BC}$
C. $\overline{CD}$
D. $\overline{DE}$
E. $\overline{EA}$
Answer: C
Problem: <image>Question: Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $ (a,0)$ to $ (3,3)$, divides the entire region into two regions of equal area. What is $ a$?
<image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{3}{5}$
C. $\frac{2}{3}$
D. $\frac{3}{4}$
E. $\frac{4}{5}$
Answer: C
Problem: <image>Question: Triangle $ ABC$ has a right angle at $ B$, $ AB = 1$, and $ BC = 2$. The bisector of $ \angle BAC$ meets $ \overline{BC}$ at $ D$. What is $ BD$?
<image1>
Choices:
A. $\frac{\sqrt{3} - 1}{2}$
B. $\frac{\sqrt{5} - 1}{2}$
C. $\frac{\sqrt{5} + 1}{2}$
D. $\frac{\sqrt{6} + \sqrt{2}}{2}$
E. $2\sqrt{3} - 1$
Answer: B
Problem: <image>Question: A cubical cake with edge length $ 2$ inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where $ M$ is the midpoint of a top edge. The piece whose top is triangle $ B$ contains $ c$ cubic inches of cake and $ s$ square inches of icing. What is $ c+s$?
<image1>
Choices:
A. $\frac{24}{5}$
B. $\frac{32}{5}$
C. $8+\sqrt{5}$
D. $5+\frac{16\sqrt{5}}{5}$
E. $10+5\sqrt{5}$
Answer: B
Problem: <image>Question: The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $ 9$ trapezoids, let $ x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $ x$?
<image1>
Answer: 100
Problem: <image>Question: Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
<image1>
Choices:
A. $\frac{5}{4}$
B. $\frac{4}{3}$
C. $\frac{3}{2}$
D. $2$
E. $3$
Answer: B
Problem: <image>Question: Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$. What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?

<image1>
Choices:
A. $3 - \frac{\pi}{2}$
B. $\frac{\pi}{2}$
C. $2$
D. $\frac{3\pi}{4}$
E. $1+\frac{\pi}{2}$
Answer: C
Problem: <image>Question: The area of $\triangle EBD$ is one third of the area of $3-4-5$ $ \triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$?

<image1>
Choices:
A. $\frac{4}{3}$
B. $\sqrt{5}$
C. $\frac{9}{4}$
D. $\frac{4\sqrt{3}}{3}$
E. $\frac{5}{2}$
Answer: D
Problem: <image>Question: A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square?

<image1>
Choices:
A. $\frac{\sqrt{2} - 1}{2}$
B. $\frac{1}{4}$
C. $\frac{2 - \sqrt{2}}{2}$
D. $\frac{\sqrt{2}}{4}$
E. $2 - \sqrt{2}$
Answer: A
Problem: <image>Question: In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4:5$. What is the degree measure of angle $BCD$?

<image1>
Answer: 130
Problem: <image>Question: Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?

<image1>
Choices:
A. $\frac{1}{6}$
B. $\frac{1}{5}$
C. $\frac{2}{9}$
D. $\frac{1}{3}$
E. $\frac{\sqrt{2}}{4}$
Answer: B
Problem: <image>Question: The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve?

<image1>
Choices:
A. $2\pi+6$
B. $2\pi+4\sqrt{3}$
C. $3\pi+4$
D. $2\pi+3\sqrt{3}+2$
E. $\pi+6\sqrt{3}$
Answer: E
Problem: <image>Question: A circle of radius $5$ is inscribed in a rectangle as shown. The ratio of the the length of the rectangle to its width is $2\ :\ 1$. What is the area of the rectangle?

<image1>
Answer: 200
Problem: <image>Question: Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?

<image1>
Choices:
A. $10\pi+4\sqrt{3}$
B. $13\pi-\sqrt{3}$
C. $12\pi+\sqrt{3}$
D. $10\pi+9$
E. $13\pi$
Answer: A
Problem: <image>Question: A bug travels from $A$ to $B$ along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there?

<image1>
Answer: 2400
Problem: <image>Question: Square $ ABCD $ has side length $ 10 $. Point $ E $ is on $ \overline{BC} $, and the area of $ \bigtriangleup ABE $ is $ 40 $. What is $ BE $? 
<image1>
Answer: 8
Problem: <image>Question: In $\triangle ABC$, $AB=AC=28$ and $BC=20$.  Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively.  What is the perimeter of parallelogram $ADEF$?

<image1>
Answer: 56
Problem: <image>Question: In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$
<image1>
Answer: 13.5
Problem: <image>Question: The regular octagon $ABCDEFGH$ has its center at $J$.  Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal.  In how many ways can this be done? 
<image1>
Answer: 1152
Problem: <image>Question: A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region?

<image1>
Choices:
A. $27\sqrt{3}-9\pi$
B. $27\sqrt{3}-6\pi$
C. $54\sqrt{3}-18\pi$
D. $54\sqrt{3}-12\pi$
E. $108\sqrt{3}-9\pi$
Answer: C
Problem: <image>Question: Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?

<image1>
Choices:
A. $\frac{12+3\sqrt{3}}4$
B. $\frac{9}{2}$
C. $3+\sqrt{3}$
D. $\frac{6+3\sqrt{3}}2$
E. $6$
Answer: C
Problem: <image>Question: In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region?

<image1>
Choices:
A. $\frac{1}{12}$
B. $\frac{\sqrt{3}}{18}$
C. $\frac{\sqrt{2}}{12}$
D. $\frac{\sqrt{3}}{12}$
E. $\frac{1}{6}$
Answer: E
Problem: <image>Question: Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$?
<image1>
Choices:
A. $\frac{3\sqrt{33}}5$
B. $2\sqrt{3}$
C. $\frac{2\sqrt{33}}3$
D. $4$
E. $3\sqrt{2} $
Answer: A
Problem: <image>Question: A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $B:A$?

<image1>
Choices:
A. 1:2
B. 3:5
C. 2:3
D. 3:4
E. 4:5
Answer: C
Problem: <image>Question: Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window?

<image1>
Answer: 26
Problem: <image>Question: Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$?

<image1>
Choices:
A. $2\sqrt{3}$
B. $3\sqrt{3}$
C. $1+3\sqrt{2}$
D. $2+2\sqrt{3}$
E. $3+2\sqrt{3}$
Answer: B
Problem: <image>Question: In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?

<image1>
Choices:
A. $\frac{\sqrt{3}}{6}$
B. $\frac{\sqrt{6}}{8}$
C. $\frac{3\sqrt{3}}{16}$
D. $\frac{1}{3}$
E. $\frac{\sqrt{2}}{4}$
Answer: A
Problem: <image>Question: Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

<image1>
Choices:
A. $\frac{1+\sqrt{2}}4$
B. $\frac{\sqrt{5}-1}2$
C. $\frac{\sqrt{3}+1}4$
D. $\frac{2\sqrt{3}}5$
E. $\frac{\sqrt{5}}3$
Answer: B
Problem: <image>Question: A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?


<image1>
Choices:
A. $\frac{3}{2}$
B. $\frac{1+\sqrt{5}}{2}$
C. $\sqrt{3}$
D. $2$
E. $\frac{3+\sqrt{5}}{2}$
Answer: E
Problem: <image>Question: Ann made a 3-step staircase using 18 toothpicks as shown in the figure. How many toothpicks does she need to add to complete a 5-step staircase? 
<image1>
Answer: 22
Problem: <image>Question: The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?

<image1>
Choices:
A. $\text{2 o'clock}$
B. $\text{3 o'clock}$
C. $\text{4 o'clock}$
D. $\text{6 o'clock}$
E. $\text{8 o'clock}$
Answer: C
Problem: <image>Question: The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
<image1><image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\frac{3}{2}$ and center $(0,\frac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?

<image1>
Choices:
A. $\frac{4\pi}{5}$
B. $\frac{9\pi}{8}$
C. $\frac{4\pi}{3}$
D. $\frac{7\pi}{5}$
E. $\frac{3\pi}{2}$
Answer: B
Problem: <image>Question: The centers of the faces of the right rectangular prism shown below are joined to create an octahedron, What is the volume of the octahedron?

<image1>
Choices:
A. $\frac{75}{12}$
B. $10$
C. $12$
D. $10\sqrt{2}$
E. $15$
Answer: B
Problem: <image>Question: In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$?
<image1>
Choices:
A. $3$
B. $12-4\sqrt{5}$
C. $\frac{5+2\sqrt{5}}{3}$
D. $1+\sqrt{5}$
E. $\frac{11+11\sqrt{5}}{10}$
Answer: D
Problem: <image>Question: A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?
<image1>
Answer: 2
Problem: <image>Question: What is the area of the shaded region of the given $8 \times 5$ rectangle?

<image1>
Choices:
A. $4\frac{3}{5}$
B. $5$
C. $5\frac{1}{4}$
D. $6\frac{1}{2}$
E. $8$
Answer: D
Problem: <image>Question: 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 radius in inches of the scrap cookie?

<image1>
Choices:
A. $\sqrt{2}$
B. $1.5$
C. $\sqrt{\pi}$
D. $\sqrt{2\pi}$
E. $\pi$
Answer: A
Problem: <image>Question: Rectangle $ABCD$ has $AB=5$ and $BC=4$. Point $E$ lies on $\overline{AB}$ so that $EB=1$, point $G$ lies on $\overline{BC}$ so that $CG=1$. and point $F$ lies on $\overline{CD}$ so that $DF=2$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?


<image1>
Choices:
A. $\frac{\sqrt{13}}{16}$
B. $\frac{\sqrt{2}}{13}$
C. $\frac{9}{82}$
D. $\frac{10}{91}$
E. $\frac{1}{9}$
Answer: D
Problem: <image>Question: Tamara has three rows of two 6-feet by 2-feet flower beds in her garden. The beds are separated and also surrounded by 1-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
<image1>
Answer: 78
Problem: <image>Question: In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?

<image1>
Answer: 12
Problem: <image>Question: All of the triangles in the diagram below are similar to iscoceles triangle $ABC$, in which $AB=AC$. Each of the 7 smallest triangles has area 1, and $\triangle ABC$ has area 40. What is the area of trapezoid $DBCE$?

<image1>
Answer: 24
Problem: <image>Question: A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
<image1>
Choices:
A. $1+\frac{1}{2} \sqrt{2}$
B. $\sqrt{3}$
C. $\frac{7}{4}$
D. $\frac{15}{8}$
E. $2$
Answer: D
Problem: <image>Question: Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points  $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

<image1>
Answer: 69
Problem: <image>Question: Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted?

<image1>
Choices:
A. $\frac{25}{27}$
B. $\frac{26}{27}$
C. $\frac{73}{75}$
D. $\frac{145}{147}$
E. $\frac{74}{75}$
Answer: D
Problem: <image>Question: In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. The ratio $A:B$ is $1:18$. What is $N$?
<image1>
Answer: 19
Problem: <image>Question: Sara makes a staircase out of toothpicks as shown:<image1>
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?
Answer: 12
Problem: <image>Question: In the rectangular parallelpiped shown, $AB = 3, BC= 1,$ and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?

<image1>
Choices:
A. $1$
B. $\frac{4}{3}$
C. $\frac{3}{2}$
D. $\frac{5}{3}$
E. $2$
Answer: E
Problem: <image>Question: A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?
<image1>
Choices:
A. $2(w+h)^2$
B. $\frac{(w+h)^2}2$
C. $2w^2+4wh$
D. $2w^2$
E. $w^2h$
Answer: A
Problem: <image>Question: The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
<image1>
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$
Answer: 2
Problem: <image>Question: The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1 ?$

<image1>
Choices:
A. $4 \pi \sqrt{3}$
B. $7 \pi$
C. $\pi(3\sqrt{3} +2)$
D. $10 \pi (\sqrt{3} - 1)$
E. $\pi(\sqrt{3} + 6)$
Answer: A
Problem: <image>Question: The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

<image1>
Choices:
A. $4$
B. $12 - 4\sqrt{3}$
C. $3\sqrt{3}$
D. $4\sqrt{3}$
E. $16 - \sqrt{3}$
Answer: B
Problem: <image>Question: As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2$. Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
\[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\]where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$?

<image1>
Answer: 17
Problem: <image>Question: Triangle $AMC$ is isoceles with $AM = AC$. Medians $\overline{MV}$ and $\overline{CU}$ are perpendicular to each other, and $MV=CU=12$. What is the area of $\triangle AMC?$
<image1>
Answer: 96
Problem: <image>Question: As shown in the figure below a regular dodecahedron (the polyhedron consisting of 12 congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adjacent to the bottom face. How many ways are there to move from the top face to the bottom face via a sequence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?

<image1>
Answer: 810
Problem: <image>Question: A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
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Choices:
A. $3\pi \sqrt{5}$
B. $4\pi \sqrt{3}$
C. $3 \pi \sqrt{7}$
D. $6\pi \sqrt{3}$
E. $6\pi \sqrt{7}$
Answer: C
Problem: <image>Question: As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length $2$ so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region—inside the hexagon but outside all of the semicircles?

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Choices:
A. $6\sqrt{3}-3\pi$
B. $\frac{9\sqrt{3}}{2}-2\pi$
C. $\frac{3\sqrt{3}}{2}-\frac{\pi}{3}$
D. $3\sqrt{3}-\pi \$
E. $\frac{9\sqrt{3}}{2}-\pi$
Answer: D