Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: Let $ABCD$ be a parallelogram with $\angle BAD < 90^{\circ}$. A circle tangent to sides $\overline{DA}$, $\overline{AB}$, and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ$, as shown. Suppose that $AP = 3$, $PQ = 9$, and $QC = 16$. Then the area of $ABCD$ can be expressed in the form $m\sqrtn$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

<image1>
Answer: 150
Problem: <image>Question: Let $\ell_A$ and $\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \allowbreak A_3, \allowbreak \ldots, \allowbreak A_m$ lie on $\ell_A$, and distinct points $B_1, B_2, B_3, \ldots, B_n$ lie on $\ell_B$. Additionally, when segments $\overline{A_iB_j}$ are drawn for all $i=1,2,3,\ldots, m$ and $j=1,\allowbreak 2,\allowbreak 3, \ldots, \allowbreak n$, no point strictly between $\ell_A$ and $\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$.
<image1>
Answer: 244
Problem: <image>Question: Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon.
<image1>
Answer: 140
Problem: <image>Question: Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\sqrt{21}$ and $\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\fracmn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
<image1>
Answer: 125
Problem: <image>Question: The following analog clock has two hands that can move independently of each other.
<image1>
Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move.

Let $N$ be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when $N$ is divided by 1000.
Answer: 608
Problem: <image>Question: Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside this region. The probability that the midpoint of $\overline{AB}$ also lies inside this L-shaped region can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
<image1>
Answer: 35
Problem: <image>Question: Let $N$ be the number of ways to place the integers $1$ through $12$ in the $12$ cells of a $2\times 6$ grid so that for any two cells sharing a side, the difference between the numbers in those cells is not divisible by $3$.  One way to do this is shown below.  Find the number of positive integer divisors of $N$.

<image1>
Answer: 144
Problem: <image>Question: A cube-shaped container has vertices $A$, $B$, $C$, and $D$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of the faces of the cube.  Vertex $A$ of the cube is set on a horizontal plane $\mathcal P$ so that the plane of the rectangle $ABCD$ is perpendicular to $\mathcal P$, vertex $B$ is $2$ meters above $\mathcal P$, vertex $C$ is $8$ meters above $\mathcal P$, and vertex $D$ is $10$ meters above $\mathcal P$.  The cube contains water whose surface is $7$ meters above $\mathcal P$.  The volume of the water is $\frac{m}{n}$ cubic meters, where $m$ and $n$ are relatively prime positive integers.  Find $m+n$.
<image1>
Answer: 751
Problem: <image>Question: Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?

$\mathrm{a.}\ \text{the length of the segment} MN$

$\mathrm{b.}\ \text{the perimeter of }\triangle PAB$

$\mathrm{c.}\ \text{ the area of }\triangle PAB$

$\mathrm{d.}\ \text{ the area of trapezoid} ABNM$


<image1>
Answer: 1
Problem: <image>Question: In rectangle $ ABCD$, $ AD = 1$, $ P$ is on $ \overline{AB}$, and $ \overline{DB}$ and $ \overline{DP}$ trisect $ \angle ADC$. What is the perimeter of $ \triangle BDP$?
<image1>
Choices:
A. $3 + \frac{\sqrt{3}}{3}$
B. $2 + \frac{4\sqrt{3}}{3}$
C. $2 + 2\sqrt{2}$
D. $\frac{3 + 3\sqrt{5}}{2}$
E. $2 + \frac{5\sqrt{3}}{3}$
Answer: B
Problem: <image>Question: Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$?
<image1>
Answer: 20201
Problem: <image>Question: There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?

<image1>
Choices:
A. $0$
B. $1$
C. $5!\cdot4!\cdot3!\cdot2!\cdot1!$
D. $\frac{15!}{5!\cdot4!\cdot3!\cdot2!\cdot1!}$
E. $15!$
Answer: B
Problem: <image>Question: The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.

<image1>
Choices:
A. $\frac{4\sqrt{5}}{3}$
B. $\frac{5\sqrt{5}}{3}$
C. $\frac{12\sqrt{5}}{7}$
D. $2\sqrt{5}$
E. $\frac{5\sqrt{65}}{9}$
Answer: B
Problem: <image>Question: How many of the twelve pentominoes pictured below have at least one line of symmetry?
<image1>
Answer: 6
Problem: <image>Question: Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains $ 8$ unit squares. The second ring contains $ 16$ unit squares.
<image1>
If we continue this process, the number of unit squares in the $ 100^\text{th}$ ring is
Choices:
A. $396$
B. $404$
C. $800$
D. $10,\!000$
E. $10,\!404$
Answer: C
Problem: <image>Question: Which of the cones listed below can be formed from a $ 252^\circ$ sector of a circle of radius $ 10$ by aligning the two straight sides?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: The plane is tiled by congruent squares and congruent pentagons as indicated.
<image1>
The percent of the plane that is enclosed by the pentagons is closest to
Choices:
A. 50
B. 52
C. 54
D. 56
E. 58
Answer: D
Problem: <image>Question: In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by $ v$, $ w$, $ x$, $ y$, and $ z$. Find $ y + z$.
<image1>
Answer: 46
Problem: <image>Question: Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.
<image1>
Choices:
A. $\pi$
B. $1.5\pi$
C. $2\pi$
D. $3\pi$
E. $3.5\pi$
Answer: C
Problem: <image>Question: Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $ B$ be the total area of the blue triangles, $ W$ the total area of the white squares, and $ R$ the area of the red square. Which of the following is correct?

<image1>
Choices:
A. B = W
B. W = R
C. B = R
D. 3B = 2R
E. 2R = W
Answer: A
Problem: <image>Question: In trapezoid $ ABCD$ with bases $ AB$ and $ CD$, we have $ AB=52$, $ BC=12$, $ CD=39$, and $ DA=5$. The area of $ ABCD$ is

<image1>
Answer: 210
Problem: <image>Question: Circles of radius $ 2$ and $ 3$ are externally tangent and are circumscribed by a third circle, as shown in the figure. Find the area of the shaded region.

<image1>
Choices:
A. $3\pi$
B. $4\pi$
C. $6\pi$
D. $9\pi$
E. $12\pi$
Answer: E
Problem: <image>Question: The polygon enclosed by the solid lines in the figure consists of $ 4$ congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

<image1>
Answer: 6
Problem: <image>Question: A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.
<image1>
Choices:
A. $\frac{1}{6}\pi - \frac{\sqrt{3}}{4}$
B. $\frac{\sqrt{3}}{4} - \frac{1}{12}\pi$
C. $\frac{\sqrt{3}}{4} - \frac{1}{24}\pi$
D. $\frac{\sqrt{3}}{4} + \frac{1}{24}\pi$
E. $\frac{\sqrt{3}}{4} + \frac{1}{12}\pi$
Answer: C
Problem: <image>Question: In rectangle $ ABCD$, we have $ AB=8$, $ BC=9$, $ H$ is on $ \overline{BC}$ with $ BH=6$, $ E$ is on $ \overline{AD}$ with $ DE=4$, line $ EC$ intersects line $ AH$ at $ G$, and $ F$ is on line $ AD$ with $ \overline{GF}\perp\overline{AF}$. Find the length $ GF$.
<image1>
Answer: 20
Problem: <image>Question: A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have $ 3$ rows of small congruent equilateral triangles, with $ 5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $ 2003$ small equilateral triangles?
<image1>
Choices:
A. $1,\!004,\!004$
B. $1,\!005,\!006$
C. $1,\!507,\!509$
D. $3,\!015,\!018$
E. $6,\!021,\!018$
Answer: C
Problem: <image>Question: Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $ \$$1 each, begonias $ \$$1.50 each, cannas $ \$$2 each, dahlias $ \$$2.50 each, and Easter lilies $ \$$3 each. What is the least possible cost, in dollars, for her garden?
<image1>
Answer: 108
Problem: <image>Question: Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $ 4 : 3$. The horizontal length of a “$ 27$-inch” television screen is closest, in inches, to which of the following?
<image1>
Choices:
A. 20
B. 20.5
C. 21
D. 21.5
E. 22
Answer: D
Problem: <image>Question: Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles?
<image1>
Choices:
A. $\pi-\sqrt{3}$
B. $\pi-\sqrt{2}$
C. $\frac{\pi+\sqrt{2}}{2}$
D. $\frac{\pi+\sqrt{3}}{2}$
E. $\frac{7}{6}\pi-\frac{\sqrt{3}}{2}$
Answer: E
Problem: <image>Question: In rectangle $ ABCD$, $ AB=5$ and $ BC=3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF=1$ and $ GC=2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$.
<image1>
Choices:
A. $10$
B. $\frac{21}{2}$
C. $12$
D. $\frac{25}{2}$
E. $15$
Answer: D
Problem: <image>Question: A regular octagon $ ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ ABEF$?
<image1>
Choices:
A. $1-\frac{\sqrt{2}}{2}$
B. $\frac{\sqrt{2}}{4}$
C. $\sqrt{2}-1$
D. $\frac{1}{2}$
E. $\frac{1+\sqrt{2}}{4}$
Answer: D
Problem: <image>Question: A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?
<image1>
Choices:
A. $\frac{1}{21}$
B. $\frac{1}{14}$
C. $\frac{2}{21}$
D. $\frac{1}{7}$
E. $\frac{2}{7}$
Answer: C
Problem: <image>Question: In the figure, $ \angle EAB$ and $ \angle ABC$ are right angles. $ AB = 4, BC = 6, AE = 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle ADE$ and $ \triangle BDC$?
<image1>
Answer: 4
Problem: <image>Question: The $ 5\times 5$ grid shown contains a collection of squares with sizes from $ 1\times 1$ to $ 5\times 5$. How many of these squares contain the black center square?
<image1>
Answer: 19
Problem: <image>Question: A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?

<image1>
Choices:
A. $120$
B. $180$
C. $240$
D. $360$
E. $480$
Answer: C
Problem: <image>Question: Points $E$ and $F$ are located on square $ABCD$ so that $\Delta BEF$ is equilateral. What is the ratio of the area of $\Delta DEF$ to that of $\Delta ABE$?

<image1>
Choices:
A. $\frac{4}{3}$
B. $\frac{3}{2}$
C. $\sqrt{3}$
D. $2$
E. $1+\sqrt{3}$
Answer: D
Problem: <image>Question: Two distinct lines pass through the center of three concentric circles of radii $3$, $2$, and $1$. The area of the shaded region in the diagram is $8/13$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)

<image1>
Choices:
A. $\frac{\pi}8$
B. $\frac{\pi}7$
C. $\frac{\pi}6$
D. $\frac{\pi}5$
E. $\frac{\pi}4$
Answer: B
Problem: <image>Question: Square $ABCD$ has side length 2. A semicircle with diameter $AB$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $AD$ at $E$. What is the length of $CE$?

<image1>
Choices:
A. $\frac{2+\sqrt{5}}2$
B. $\sqrt{5}$
C. $\sqrt{6}$
D. $\frac{5}{2}$
E. $5-\sqrt{5}$
Answer: D
Problem: <image>Question: An annulus is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a = XZ$, $ d = YZ$, and $ e = XY$. What is the area of the annulus?
<image1>
Choices:
A. $\pi a^2$
B. $\pi b^2$
C. $\pi c^2$
D. $\pi d^2$
E. $\pi e^2$
Answer: A
Problem: <image>Question: Three circles of radius $ 1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle?

<image1>
Choices:
A. $\frac{2 + \sqrt{6}}{3}$
B. $2$
C. $\frac{2 + 3\sqrt{2}}{3}$
D. $\frac{3 + 2\sqrt{3}}{3}$
E. $\frac{3 + \sqrt{3}}{2}$
Answer: D
Problem: <image>Question: In right triangle $ \triangle ACE$, we have $ AC = 12$, $ CE = 16$, and $ EA = 20$. Points $ B$, $ D$, and $ F$ are located on $ \overline{AC}$, $ \overline{CE}$, and $ \overline{EA}$, respectively, so that $ AB = 3$, $ CD = 4$, and $ EF = 5$. What is the ratio of the area of $ \triangle DBF$ to that of $ \triangle ACE$?
<image1>
Choices:
A. $\frac{1}{4}$
B. $\frac{9}{25}$
C. $\frac{3}{8}$
D. $\frac{11}{25}$
E. $\frac{7}{16}$
Answer: E
Problem: <image>Question: In $ \triangle ABC$ points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$, respectively. If $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT = 3$ and $ BT/ET = 4$, what is $ CD/BD$?
<image1>
Choices:
A. $\frac{1}{8}$
B. $\frac{2}{9}$
C. $\frac{3}{10}$
D. $\frac{4}{11}$
E. $\frac{5}{12}$
Answer: D
Problem: <image>Question: A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles?
<image1>
Choices:
A. $\frac{5}{3}\pi - 3\sqrt{2}$
B. $\frac{5}{3}\pi - 2\sqrt{3}$
C. $\frac{8}{3}\pi - 3\sqrt{3}$
D. $\frac{8}{3}\pi - 3\sqrt{2}$
E. $\frac{8}{3}\pi - 2\sqrt{3}$
Answer: B
Problem: <image>Question: Square $ EFGH$ is inside the square $ ABCD$ so that each side of $ EFGH$ can be extended to pass through a vertex of $ ABCD$. Square $ ABCD$ has side length $ \sqrt{50}$ and $ BE = 1$. What is the area of the inner square $ EFGH$?
<image1>
Answer: 36
Problem: <image>Question: The figure shown is called a trefoil and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $ 2$?
<image1>
Choices:
A. $\frac{1}{3}\pi+\frac{\sqrt{3}}{2}$
B. $\frac{2}{3}\pi$
C. $\frac{2}{3}\pi+\frac{\sqrt{3}}{4}$
D. $\frac{2}{3}\pi+\frac{\sqrt{3}}{3}$
E. $\frac{2}{3}\pi+\frac{\sqrt{3}}{2}$
Answer: B
Problem: <image>Question: In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order.  The sums of the numbers at the ends of the line segments $\overline{AB}$,$\overline{BC}$,$\overline{CD}$,$\overline{DE}$, and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order.  What is the middle term of the arithmetic sequence?

<image1>
Answer: 12
Problem: <image>Question: Three one-inch squares are palced with their bases on a line. The center square is lifted out and rotated $ 45^\circ$, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $ B$ from the line on which the bases of the original squares were placed?
<image1>
Choices:
A. $1$
B. $\sqrt{2}$
C. $\frac{3}{2}$
D. $\sqrt{2} + \frac{1}{2}$
E. $2$
Answer: D
Problem: <image>Question: Let $ \overline{AB}$ be a diameter of a circle and $ C$ be a point on $ \overline{AB}$ with $ 2 \cdot AC = BC$. Let $ D$ and $ E$ be points on the circle such that $ \overline{DC} \perp \overline{AB}$ and $ \overline{DE}$ is a second diameter. What is the ratio of the area of $ \triangle DCE$ to the area of $ \triangle ABD$?
<image1>
Choices:
A. $\frac{1}{6}$
B. $\frac{1}{4}$
C. $\frac{1}{3}$
D. $\frac{1}{2}$
E. $\frac{2}{3}$
Answer: C
Problem: <image>Question: An $ 8$-foot by $ 10$-foot floor is tiled with square tiles of size $ 1$ foot by $ 1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $ 1/2$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
<image1>
Choices:
A. $80-20\pi$
B. $60-10\pi$
C. $80-10\pi$
D. $60+10\pi$
E. $80+10\pi$
Answer: A
Problem: <image>Question: Equilateral $ \triangle ABC$ has side length $ 2$, $ M$ is the midpoint of $ \overline{AC}$, and $ C$ is the midpoint of $ \overline{BD}$. What is the area of $ \triangle CDM$?
<image1>
Choices:
A. $\frac{\sqrt{2}}{2}$
B. $\frac{3}{4}$
C. $\frac{\sqrt{3}}{2}$
D. $1$
E. $\sqrt{2}$
Answer: C
Problem: <image>Question: The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
<image1>
Answer: 6
Problem: <image>Question: Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet?

<image1>
Choices:
A. $\text{ I, by }8\pi$
B. $\text{ I, by }6\pi$
C. $\text{ II, by }4\pi$
D. $\text{II, by }8\pi$
E. $\text{ II, by }10\pi$
Answer: C
Problem: <image>Question: A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?
<image1>
Answer: 173
Problem: <image>Question: A circle of radius 1 is tangent to a circle of radius 2.  The sides of $ \triangle ABC$ are tangent to the circles as shown, and the sides $ \overline{AB}$ and $ \overline{AC}$ are congruent.  What is the area of $ \triangle ABC$?

<image1>
Choices:
A. $\frac{35}2$
B. $15\sqrt{2}$
C. $\frac{64}3$
D. $16\sqrt{2}$
E. $24$
Answer: D
Problem: <image>Question: In rectangle $ ADEH$, points $ B$ and $ C$ trisect $ \overline{AD}$, and points $ G$ and $ F$ trisect $ \overline{HE}$.  In addition, $ AH = AC = 2$.  What is the area of quadrilateral $ WXYZ$ shown in the figure?

<image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{\sqrt{2}}2$
C. $\frac{\sqrt{3}}2$
D. $\frac{2\sqrt{2}}3$
E. $\frac{2\sqrt{3}}3$
Answer: A
Problem: <image>Question: Circles with centers $ A$ and $ B$ have radii 3 and 8, respectively. A common internal tangent intersects the circles at $ C$ and $ D$, respectively. Lines $ AB$ and $ CD$ intersect at $ E$, and $ AE = 5$. What is $ CD$?
<image1>
Choices:
A. $13$
B. $\frac{44}{3}$
C. $\sqrt{221}$
D. $\sqrt{255}$
E. $\frac{55}{3}$
Answer: B
Problem: <image>Question: A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $ 2/\pi $, as shown. What is the perimeter of this region?

<image1>
Choices:
A. $\frac{4}\pi$
B. $2$
C. $\frac{8}\pi$
D. $4$
E. $\frac{16}{\pi}$
Answer: D
Problem: <image>Question: A square of area $40$ is inscribed in a semicircle as shown. What is the area of the semicircle?

<image1>
Choices:
A. $20\pi$
B. $25\pi$
C. $30\pi$
D. $40\pi$
E. $50\pi$
Answer: B
Problem: <image>Question: Rhombus $ ABCD$ is similar to rhombus $ BFDE$.  The area of rhombus $ ABCD$ is 24, and $ \angle BAD = 60^\circ$.  What is the area of rhombus $ BFDE$?

<image1>
Choices:
A. $6$
B. $4\sqrt{3}$
C. $8$
D. $9$
E. $6\sqrt{3}$
Answer: C
Problem: <image>Question: A circle of radius 2 is centered at $ O$. Square $ OABC$ has side length 1. Sides $ \overline{AB}$ and $ \overline{CB}$ are extended past $ b$ to meet the circle at $ D$ and $ E$, respectively. What is the area of the shaded region in the figure, which is bounded by $ \overline{BD}$, $ \overline{BE}$, and the minor arc connecting $ D$ and $ E$?

<image1>
Choices:
A. $\frac{\pi}3 + 1 - \sqrt{3}$
B. $\frac{\pi}2\left( 2 - \sqrt{3}\right)$
C. $\pi\left(2 - \sqrt{3}\right)$
D. $\frac{\pi}{6} + \frac{\sqrt{3} - 1}{2} \  \indent$
E. $\frac{\pi}{3} - 1 + \sqrt{3}$
Answer: A
Problem: <image>Question: A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?


<image1>
Choices:
A. $15$
B. $17$
C. $\frac{35}{2}$
D. $18$
E. $\frac{55}{3}$
Answer: D
Problem: <image>Question: Circles with centers $ O$ and $ P$ have radii 2 and 4, respectively, and are externally tangent.  Points $ A$ and $ B$ are on the circle centered at $ O$, and points $ C$ and $ D$ are on the circle centered at $ P$, such that $ \overline{AD}$ and $ \overline{BC}$ are common external tangents to the circles.  What is the area of hexagon $ AOBCPD$?
<image1>
Choices:
A. $18\sqrt{3}$
B. $24\sqrt{2}$
C. $36$
D. $24\sqrt{3}$
E. $32\sqrt{2}$
Answer: B
Problem: <image>Question: Older television screens have an aspect ratio of $ 4: 3$. That is, the ratio of the width to the height is $ 4: 3$. The aspect ratio of many movies is not $ 4: 3$, so they are sometimes shown on a television screen by 'letterboxing' - darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of $ 2: 1$ and is shown on an older television screen with a $ 27$-inch diagonal. What is the height, in inches, of each darkened strip?
<image1>
Answer: 2.7