Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: In the obtuse triangle $ABC$, $AM = MB, MD \perp BC, EC \perp BC$. If the area of $\triangle ABC$ is 24, then the area of $\triangle BED$ is

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Choices:
A. $9$
B. $12$
C. $15$
D. $18$
E. $\text{not uniquely determined}$
Answer: B
Problem: <image>Question: In an arcade game, the "monster" is the shaded sector of a circle of radius $ 1$ cm, as shown in the figure.  The missing piece (the mouth) has central angle $ 60^{\circ}$.  What is the perimeter of the monster in cm?

<image1>
Choices:
A. $\pi + 2$
B. $2\pi$
C. $\frac{5}{3} \pi$
D. $\frac{5}{6} \pi + 2$
E. $\frac{5}{3} \pi + 2$
Answer: E
Problem: <image>Question: In right $ \triangle ABC$ with legs $ 5$ and $ 12$, arcs of circles are drawn, one with center $ A$ and radius $ 12$, the other with center $ B$ and radius $ 5$.  They intersect the hypotenuse at $ M$ and $ N$.  Then, $ MN$ has length:

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Choices:
A. $2$
B. $\frac{13}{5}$
C. $3$
D. $4$
E. $\frac{24}{5}$
Answer: D
Problem: <image>Question: The odd positive integers $1,3,5,7,\cdots,$ are arranged into in five columns continuing with the pattern shown on the right.  Counting from the left, the column in which $ 1985$ appears in is the

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Choices:
A. $\text{ first}$
B. $\text{ second}$
C. $\text{ third}$
D. $\text{ fourth}$
E. $\text{ fifth}$
Answer: B
Problem: <image>Question: Pegs are put in a board $ 1$ unit apart both horizontally and vertically.  A reubber band is stretched over $ 4$ pegs as shown in the figure, forming a quadrilateral.  Its area in square units is

<image1>
Answer: 6
Problem: <image>Question: Diagonal $ DB$ of rectangle $ ABCD$ is divided into $ 3$ segments of length $ 1$ by parallel lines $ L$ and $ L'$ that pass through $ A$ and $ C$ and are perpendicular to $ DB$.  The area of $ ABCD$, rounded to the nearest tenth, is

<image1>
Answer: 4.2
Problem: <image>Question: In a circle with center $ O$, $ AD$ is a diameter, $ ABC$ is a chord, $ BO = 5$, and $ \angle ABO = \stackrel{\frown}{CD} = 60^{\circ}$.  Then the length of $ BC$ is:

<image1>
Choices:
A. $3$
B. $3 + \sqrt{3}$
C. $5 - \frac{\sqrt{3}}{2}$
D. $5$
E. $\text{none of the above}$
Answer: D
Problem: <image>Question: In $ \triangle ABC$, we have $ \angle C = 3 \angle A$, $ a = 27$, and $ c = 48$.  What is $ b$?

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Choices:
A. $33$
B. $35$
C. $37$
D. $39$
E. $\text{not uniquely determined}$
Answer: B
Problem: <image>Question: $\triangle ABC$ is a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ is the bisector of $\angle ABC$, then $\angle BDC =$
<image1>
Choices:
A. $40^\circ$
B. $45^\circ$
C. $50^\circ$
D. $55^\circ$
E. $60^\circ$
Answer: D
Problem: <image>Question: Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1.  Length $r$ is found to be $32$ inches.  After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches.  How high is the table?

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Choices:
A. $28\text{ inches}$
B. $29\text{ inches}$
C. $30\text{ inches}$
D. $31\text{ inches}$
E. $32\text{ inches}$
Answer: C
Problem: <image>Question: In $\triangle ABC$, $AB = 13$, $BC = 14$ and $CA = 15$.  Also, $M$ is the midpoint of side $AB$ and $H$ is the foot of the altitude from $A$ to $BC$.  The length of $HM$ is

<image1>
Answer: 6.5
Problem: <image>Question: In $\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$ and side $BC$ is extended, as shown in the figure, to a point $P$ so that $\triangle PAB$ is similar to $\triangle PCA$.  The length of $PC$ is

<image1>
Answer: 9
Problem: <image>Question: In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$.
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A necessary and sufficient condition for the equality of the two shaded areas, given $0 < \theta < \frac{\pi}{2}$, is
Choices:
A. $\tan \theta = \theta$
B. $\tan \theta = 2\theta$
C. $\tan \theta = 4\theta$
D. $\tan 2\theta = \theta  \$
E. $\tan \frac{\theta}{2} = \theta$
Answer: B
Problem: <image>Question: In the adjoining figure, $AB$ is a diameter of the circle, $CD$ is a chord parallel to $AB$, and $AC$ intersects $BD$ at $E$, with $\angle AED = \alpha$.  The ratio of the area of $\triangle CDE$ to that of $\triangle ABE$ is

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Choices:
A. $\cos \alpha$
B. $\sin \alpha$
C. $\cos^2\alpha$
D. $\sin^2\alpha$
E. $1 - \sin \alpha$
Answer: C
Problem: <image>Question: $ABCDE$ is a regular pentagon.  $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively.  Let $O$ be the center of the pentagon.  If $OP = 1$, then $AO + AQ + AR$ equals

<image1>
Choices:
A. $3$
B. $1 + \sqrt{5}$
C. $4$
D. $2 + \sqrt{5}$
E. $5$
Answer: C
Problem: <image>Question: A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. The perimeter of the remaining quadrilateral is

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Choices:
A. $6$
B. $6\frac{1}{2}$
C. $7$
D. $7\frac{1}{2}$
E. $8$
Answer: E
Problem: <image>Question: In the $\triangle ABC$ shown, $D$ is some interior point, and $x, y, z, w$ are the measures of angles in degrees. Solve for $x$ in terms of $y, z$ and $w$.
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Choices:
A. $w-y-z$
B. $w-2y-2z$
C. $180-w-y-z \$
D. $2w-y-z$
E. $180-w+y+z$
Answer: A
Problem: <image>Question: In the figure the sum of the distances $AD$ and $BD$ is
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Choices:
A. $\text{between 10 and 11}$
B. $12$
C. $\text{between 15 and 16}$
D. $\text{between 16 and 17}$
E. $17$
Answer: C
Problem: <image>Question: $ABCD$ is a square and $M$ and $N$ are the midpoints of $BC$ and $CD$ respectively. Then $\sin \theta=$
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Choices:
A. $\frac{\sqrt{5}}{5}$
B. $\frac{3}{5}$
C. $\frac{\sqrt{10}}{5}$
D. $\frac{4}{5}$
E. $\text{none of these}$
Answer: B
Problem: <image>Question: There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \text{cm}^2$. What is the area (in $\text{cm}^2$) of the square inscribed in the same $\triangle ABC$ as shown in Figure 2 below?
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Answer: 392
Problem: <image>Question: 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 $\frac{AD}{AB}$ is
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Choices:
A. $\frac{1}{\sqrt{2}}$
B. $\frac{2}{2+\sqrt{2}}$
C. $\frac{1}{\sqrt{3}}$
D. $\frac{1}{\sqrt[3]{6}}$
E. $\frac{1}{\sqrt[4]{12}}$
Answer: E
Problem: <image>Question: Four rectangular paper strips of length $10$ and width $1$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?
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Answer: 36
Problem: <image>Question: An $8'\text{ X }10'$ table sits in the corner of a square room, as in Figure 1 below.  The owners desire to move the table to the position shown in Figure 2.  The side of the room is $S$ feet.  What is the smallest integer value of $S$ for which the table can be moved as desired without tilting it or taking it apart?

<image1>
Answer: 13
Problem: <image>Question: On each horizontal line in the figure below, the five large dots indicate the populations of cities $A$, $B$, $C$, $D$ and $E$ in the year indicated.  Which city had the greatest percentage increase in population from 1970 to 1980?

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Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: $ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as in the figure.  The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$.  The ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ is
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Choices:
A. $\frac{1}{36}$
B. $\frac{1}{6}$
C. $\frac{1}{4}$
D. $\frac{\sqrt{3}}{4}$
E. $\frac{9+8\sqrt{3}}{36}$
Answer: C
Problem: <image>Question: In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$.  These four pieces can then be reassembled into a rectangle as shown in the second figure.  The ratio of height to base, $XY$ / $YZ$, in this rectangle is

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Choices:
A. $4$
B. $1+2\sqrt{3}$
C. $2\sqrt{5}$
D. $\frac{8+4\sqrt{3}}{3}$
E. $5$
Answer: E
Problem: <image>Question: In the figure, $AB \perp BC$, $BC \perp CD$, and $BC$ is tangent to the circle with center $O$ and diameter $AD$.  In which one of the following cases is the area of $ABCD$ an integer?

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Choices:
A. AB=3, CD=1
B. AB=5, CD=2
C. AB=7, CD=3
D. AB=9, CD=4
E. AB=11, CD=5
Answer: D
Problem: <image>Question: A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is

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Answer: 81
Problem: <image>Question: In the figure, $ABCD$ is an isosceles trapezoid with side lengths $AD = BC = 5, AB = 4,$ and $DC = 10$. The point $C$ is on $\overline{DF}$ and $B$ is the midpoint of hypotenuse $\overline{DE}$ in the right triangle $DEF$. Then $CF =$
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Answer: 4.0
Problem: <image>Question: Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is

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Answer: 430
Problem: <image>Question: In $\triangle ABC, \angle A = 100^\circ, \angle B = 50^\circ, \angle C = 30^\circ, \overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC =$

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Choices:
A. $15^\circ$
B. $22.5^\circ$
C. $30^\circ$
D. $40^\circ$
E. $45^\circ$
Answer: C
Problem: <image>Question: Two strips of width 1 overlap at an angle of $\alpha$ as shown. The area of the overlap (shown shaded) is

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Choices:
A. $\sin \alpha$
B. $\frac{1}{\sin \alpha}$
C. $\frac{1}{1 - \cos \alpha}$
D. $\frac{1}{\sin^2 \alpha}$
E. $\frac{1}{(1 - \cos \alpha)^2}$
Answer: B
Problem: <image>Question: In $\triangle ABC$, $AB = 5, BC = 7, AC = 9$ and $D$ is on $\overline{AC}$ with $BD = 5$. Find the ratio of $AD: DC$.

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Choices:
A. 4:3
B. 7:5
C. 11:6
D. 13:5
E. 19:8
Answer: E
Problem: <image>Question: A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up $36\%$ of the area of the flag, what percent of the area of the flag is blue?
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Answer: 2
Problem: <image>Question: A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly $1989$ minutes?
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Choices:
A. (35,44)
B. (36,45)
C. (37,45)
D. (44,35)
E. (45,36)
Answer: D
Problem: <image>Question: Let $ABCD$ be a parallelogram with $\angle ABC=120^\circ$, $AB=16$ and $BC=10$.  Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. 
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If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then $FD$ is closest to
Answer: 2
Problem: <image>Question: An acute isosceles triangle, $ABC$ is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC=\angle ACB=2\angle D$ and $x$ is the radian measure of $\angle A$, then $x=$

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Choices:
A. $\frac{3}{7}\pi$
B. $\frac{4}{9}\pi$
C. $\frac{5}{11}\pi$
D. $\frac{6}{13}\pi$
E. $\frac{7}{15}\pi$
Answer: A
Problem: <image>Question: $ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$

<image1>
Answer: 4.2
Problem: <image>Question: Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person (not the original number the person picked). The number picked by the person who announced the average $6$ was
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Choices:
A. $1$
B. $5$
C. $6$
D. $10$
E. $\text{not uniquely determined from the given information}$
Answer: A
Problem: <image>Question: In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$.  The area of the polygon is closest to

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Choices:
A. 288
B. 291
C. 294
D. 297
E. 300
Answer: E
Problem: <image>Question: Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$.  Triangle $ABD$ has a right angle at $A$ and $AD = 12$.  Points $C$ and $D$ are on opposite sides of $\overline{AB}$.  The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$.  If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then $m + n = $
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Answer: 128
Problem: <image>Question: Two circles are externally tangent.  Lines $\overline{PAB}$ and $\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle.  If $PA = AB = 4$, then the area of the smaller circle is
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Choices:
A. $1.44\pi$
B. $2\pi$
C. $2.56\pi$
D. $\sqrt{8}\pi$
E. $4\pi$
Answer: B
Problem: <image>Question: If $ABCD$ is a $2\ X\ 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, then the area of quadrilateral $BEIH$ is

<image1>
Choices:
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{7}{15}$
D. $\frac{8}{15}$
E. $\frac{3}{5}$
Answer: C
Problem: <image>Question: Equilateral triangle $ABC$ has been creased and folded so that vertex $A$ now rests at $A'$ on $\overline{BC}$ as shown.  If $BA' = 1$ and $A'C = 2$ then the length of crease $\overline{PQ}$ is
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Choices:
A. $\frac{8}{5}$
B. $\frac{7}{20}\sqrt{21}$
C. $\frac{1+\sqrt{5}}{2}$
D. $\frac{13}{8}$
E. $\sqrt{3}$
Answer: B
Problem: <image>Question: A square floor is tiled with congruent square tiles.  The tiles on the two diagonals of the floor are black.  The rest of the tiles are white.  If there are 101 black tiles, then the total number of tiles is

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Answer: 2601
Problem: <image>Question: Five equilateral triangles, each with side $2\sqrt{3}$, are arranged so they are all on the same side of a line containing one side of each.  Along this line, the midpoint of the base of one triangle is a vertex of the next.  The area of the region of the plane that is covered by the union of the five triangular regions is

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Choices:
A. $10$
B. $12$
C. $15$
D. $10\sqrt{3}$
E. $12\sqrt{3}$
Answer: E
Problem: <image>Question: The ratio of the radii of two concentric circles is $1:3$.  If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is

<image1>
Answer: 18
Problem: <image>Question: Part of an "$n$-pointed regular star" is shown.  It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_{1}$, $A_{2}$, $\ldots$, $A_{n}$ are congruent and angles $B_{1}$, $B_{2}$, $\ldots$, $B_{n}$ are congruent.  If the acute angle at $A_{1}$ is $10^{\circ}$ less than the acute angle at $B_{1}$, then $n = $
<image1>
Answer: 36
Problem: <image>Question: Semicircle $\stackrel{\frown}{AB}$ has center $C$ and radius $1$.  Point $D$ is on $\stackrel{\frown}{AB}$ and $\overline{CD} \perp \overline{AB}$.  Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\stackrel{\frown}{AE}$ and $\stackrel{\frown}{BF}$ have $B$ and $A$ as their respective centers.  Circular arc $\stackrel{\frown}{EF}$ has center $D$.  The area of the shaded "smile", $AEFBDA$, is

<image1>
Choices:
A. $(2 - \sqrt{2})\pi$
B. $2\pi - \pi\sqrt{2} - 1$
C. $\left(1 - \frac{\sqrt{2}}{2}\right)\pi$
D. $\frac{5\pi}{2} - \pi\sqrt{2} - 1$
E. $(3 - 2\sqrt{2})\pi$
Answer: B
Problem: <image>Question: In $\triangle ABC$, $\angle A=55^{\circ}$, $\angle C=75^{\circ}$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle BED=$
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Choices:
A. $50^{\circ}$
B. $55^{\circ}$
C. $60^{\circ}$
D. $65^{\circ}$
E. $70^{\circ}$
Answer: D
Problem: <image>Question: The convex pentagon $ABCDE$ has $\angle A=\angle B=120^{\circ}$, $EA=AB=BC=2$ and $CD=DE=4$. What is the area of $ABCDE$?
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Choices:
A. $10$
B. $7\sqrt{3}$
C. $15$
D. $9\sqrt{3}$
E. $12\sqrt{5}$
Answer: B
Problem: <image>Question: Amy painted a dart board 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 $\frac{q}{t}=$
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Choices:
A. $2\sqrt{3}-2$
B. $\frac{3}{2}$
C. $\frac{\sqrt{5}+1}{2}$
D. $\sqrt{3}$
E. $2$
Answer: A
Problem: <image>Question: Twenty cubical blocks are arranged as shown. First, $10$ are arranged in a triangular pattern; then a layer of $6$, arranged in a triangular pattern, is centered on the $10$; then a layer of $3$, arranged in a triangular pattern, is centered on the $6$; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered $1$ through $10$ in some order. Each block in layers $2, 3$ and $4$ is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block.
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Answer: 114
Problem: <image>Question: Points $A, B, C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\overline{AD}$. If $BX=CX$ and $3 \angle BAC=\angle BXC=36^{\circ}$, then $AX=$
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Choices:
A. $\cos 6^{\circ}\cos 12^{\circ} \sec 18^{\circ}$
B. $\cos 6^{\circ}\sin 12^{\circ} \csc 18^{\circ}$
C. $\cos 6^{\circ}\sin 12^{\circ} \sec 18^{\circ} \$
D. $\sin 6^{\circ}\sin 12^{\circ} \csc 18^{\circ}$
E. $\sin 6^{\circ} \sin 12^{\circ} \sec 18^{\circ}$
Answer: B
Problem: <image>Question: Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are
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Choices:
A. $\text{exactly 2 such triangles} \$
B. $\text{exactly 3 such triangles} \$
C. $\text{exactly 7 such triangles} \$
D. $\text{exactly 15 such triangles} \$
E. $\text{more than 15 such triangles}$
Answer: E
Problem: <image>Question: The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled?
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Answer: 12
Problem: <image>Question: 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 area of the fourth rectangle?
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Answer: 15
Problem: <image>Question: Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is
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Answer: 175
Problem: <image>Question: In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is $56$. The area of the region bounded by the polygon is
<image1>
Answer: 100
Problem: <image>Question: In triangle $ABC$, $AB=AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP=PC=CB$, then $\angle A =$
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Choices:
A. $30^{\circ}$
B. $36^{\circ}$
C. $48^{\circ}$
D. $60^{\circ}$
E. $72^{\circ}$
Answer: B
Problem: <image>Question: Triangle $ABC$ is inscribed in a circle, and $\angle B = \angle C = 4\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$
<image1>
Answer: 9
Problem: <image>Question: 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 $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
<image1>
Choices:
A. $\frac{2}{7}$
B. $\frac{1}{3}$
C. $\frac{2}{3}$
D. $\frac{3}{4}$
E. $\frac{7}{9}$
Answer: E
Problem: <image>Question: A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
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Answer: 5
Problem: <image>Question: 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 $AB/BC=$
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Choices:
A. $\frac{1}{2}\csc{\frac{1}{4}}$
B. $2\cos{\frac{1}{2}}$
C. $4\sin{\frac{1}{2}}$
D. $\csc{\frac{1}{2}}$
E. $2\sec{\frac{1}{2}}$
Answer: A
Problem: <image>Question: The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$?

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Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: In $\triangle ABC$, $\angle C = 90^\circ, AC = 6$ and $BC = 8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED = 90^\circ$. If $DE = 4$, then $BD =$

<image1>
Choices:
A. $5$
B. $\frac{16}{3}$
C. $\frac{20}{3}$
D. $\frac{15}{2}$
E. $8$
Answer: C
Problem: <image>Question: Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is

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Answer: 16
Problem: <image>Question: Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point

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Answer: 4
Problem: <image>Question: Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. The ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$ is
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Choices:
A. $\frac{1}{6}$
B. $\frac{1}{4}$
C. $\frac{1}{3}$
D. $\frac{2}{5}$
E. $\frac{1}{2}$
Answer: C
Problem: <image>Question: In the figure, $\overline{AB}$ and $\overline{CD}$ are diameters of the circle with center $O$, $\overline{AB} \perp \overline{CD}$, and chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 6$ and $EF = 2$, then the area of the circle is
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Choices:
A. $23\pi$
B. $\frac{47}{2}\pi$
C. $24\pi$
D. $\frac{49}{2}\pi$
E. $25\pi$
Answer: C
Problem: <image>Question: Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is

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Answer: 184
Problem: <image>Question: A large cube is formed by stacking $27$ unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is
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Answer: 19
Problem: <image>Question: Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n = 5, m = 7$.]; What is the ratio of the area of triangle $A$ to the area of triangle $B$?

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Choices:
A. 1
B. m/n
C. n/m
D. 2m/n
E. 2n/m
Answer: B