Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: In this diagram, not drawn to scale, figures $\text{I}$ and $\text{III}$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $\text{II}$ is a square region with area $32$ sq. in. Let the length of segment $AD$ be decreased by $12\frac{1}{2} \%$ of itself, while the lengths of $AB$ and $CD$ remain unchanged. The percent decrease in the area of the square is:
<image1>
Choices:
A. $12\frac{1}{2}$
B. $25$
C. $50$
D. $75$
E. $87\frac{1}{2}$
Answer: D
Problem: <image>Question: In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$. Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ become arbitrarily close to:
<image1>
Choices:
A. $0$
B. $1$
C. $\sqrt{2}$
D. $\frac{1}{\sqrt{2}}+\frac{1}{2}$
E. $\frac{1}{\sqrt{2}}+1$
Answer: D
Problem: <image>Question: <image1>

A parabolic arch has a height of $16$ inches and a span of $40$ inches.  The height, in inches, of the arch at a point $5$ inches from the center of $M$ is:
Choices:
A. $1$
B. $15$
C. $15\frac{1}{3}$
D. $15\frac{1}{2}$
E. $15\frac{3}{4}$
Answer: B
Problem: <image>Question: Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$.  Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane.  The transform (or image) of the square is:
<image1>
<image2>
<image3>
<image4>
<image5>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: In the accompanying figure, segments $AB$ and $CD$ are parallel, the measure of angle $D$ is twice the measure of angle $B$, and the measures of segments $AB$ and $CD$ are $a$ and $b$ respectively.
<image1>
Then the measure of $AB$ is equal to
Choices:
A. $\frac{1}{2}a+2b$
B. $\frac{3}{2}b+\frac{3}{4}a$
C. $2a-b$
D. $4b-\frac{1}{2}a$
E. $a+b$
Answer: E
Problem: <image>Question: Points $A,B,Q,D,$ and $C$ lie on the circle shown and the measures of arcs $\widehat{BQ}$ and $\widehat{QD}$ are $42^\circ$ and $38^\circ$ respectively.
<image1>
The sum of the measures of angles $P$ and $Q$ is
Choices:
A. $80^\circ$
B. $62^\circ$
C. $40^\circ$
D. $46^\circ$
E. $\text{None of these}$
Answer: C
Problem: <image>Question: <image1>

Pascal's triangle is an array of positive integers(See figure), in which the first row is $1$, the second row is two $1$'s, each row begins and ends with $1$, and the $k^\text{th}$ number in any row when it is not $1$, is the sum of the $k^\text{th}$ and $(k-1)^\text{th}$ numbers in the immediately preceding row. The quotient of the number of numbers in the first $n$ rows which are not $1$'s and the number of $1$'s is
Choices:
A. $\frac{n^2-n}{2n-1}$
B. $\frac{n^2-n}{4n-2}$
C. $\frac{n^2-2n}{2n-1}$
D. $\frac{n^2-3n+2}{4n-2}$
E. $\text{None of these}$
Answer: D
Problem: <image>Question: <image1>

In triangle $ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoints of $BF$. The point $E$ divides side $BC$ in the ratio
Choices:
A. 1:4
B. 1:3
C. 2:5
D. 4:11
E. 3:8
Answer: B
Problem: <image>Question: <image1>

Quadrilateral $ABCD$ is inscribed in a circle with side $AD$, a diameter of length $4$. If sides $AB$ and $BC$ each have length $1$, then side $CD$ has length
Choices:
A. $\frac{7}{2}$
B. $\frac{5\sqrt{2}}{2}$
C. $\sqrt{11}$
D. $\sqrt{13}$
E. $2\sqrt{3}$
Answer: A
Problem: <image>Question: <image1>

Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE$, $AD$, and $BC$ at points $M$, $P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is
Choices:
A. 5:12
B. 5:13
C. 5:19
D. 1:4
E. 5:21
Answer: C
Problem: <image>Question: <image1>

If the sum of the measures in degrees of angles $A,~B,~C,~D,~E$ and $F$ in the figure above is $90n$, then $n$ is equal to
Answer: 4
Problem: <image>Question: <image1>

The radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is
Choices:
A. $\sqrt{2}$
B. $\sqrt{1.25}$
C. $1.25$
D. $\frac{5\sqrt{17}}{16}$
E. $\text{None of these}$
Answer: D
Problem: <image>Question: <image1>

In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to
Choices:
A. 3x+2
B. 3x+1
C. 2x+3
D. 2x+2
E. 2x+1
Answer: E
Problem: <image>Question: <image1>

A rectangular piece of paper $6$ inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease $L$ in terms of angle $\theta$ is
Choices:
A. $3\sec ^2\theta\csc\theta$
B. $6\sin\theta\sec\theta$
C. $3\sec\theta\csc\theta$
D. $6\sec\theta\csc ^2\theta$
E. $\text{None of these}$
Answer: A
Problem: <image>Question: <image1>

Chords $AB$ and $CD$ in the circle above intersect at $E$ and are perpendicular to each other. If segments $AE$, $EB$, and $ED$ have measures $2$, $3$, and $6$ respectively, then the length of the diameter of the circle is
Choices:
A. $4\sqrt{5}$
B. $\sqrt{65}$
C. $2\sqrt{17}$
D. $3\sqrt{7}$
E. $6\sqrt{2}$
Answer: B
Problem: <image>Question: <image1>

Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to
Choices:
A. $20\pi/3$
B. $32\pi/3$
C. $12\pi$
D. $40\pi/3$
E. $15\pi$
Answer: D
Problem: <image>Question: A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below.
<image1>

The inequalities
\[ |x| + |y| \leq \sqrt{2(x^2 + y^2)} \leq 2\mbox{Max}(|x|, |y|)\]
are represented geometrically* by the figure numbered
Choices:
A. $I$
B. $II$
C. $III$
D. $IV$
E. $\mbox{none of these}*An inequality of the form  f(x, y) \leq g(x, y), for all  x and  y is represented geometrically by a figure showing the containment \{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}for a typical real number  a$.
Answer: B
Problem: <image>Question: In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.
<image1>
Of the three equations
\[ \textbf{I.}\ d-s=1, \qquad \textbf{II.}\ ds=1, \qquad \textbf{III.}\ d^2-s^2=\sqrt{5} \]those which are necessarily true are
Choices:
A. $\textbf{I} \text{only}$
B. $\textbf{II} \text{only}$
C. $\textbf{III} \text{only}$
D. $\textbf{I} \text{and} \textbf{II} \text{only}$
E. $\textbf{I, II} \text{and} \textbf{III}$
Answer: E
Problem: <image>Question: In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is
<image1>
Choices:
A. $2\sqrt{3}-3$
B. $1-\frac{\sqrt{3}}{3}$
C. $\frac{\sqrt{3}}{4}$
D. $\frac{\sqrt{2}}{3}$
E. $4-2\sqrt{3}$
Answer: A
Problem: <image>Question: In the adjoining figure $ TP$ and $ T'Q$ are parallel tangents to a circle of radius $ r$, with $ T$ and $ T'$ the points of tangency. $ PT''Q$ is a third tangent with $ T''$ as point of tangency. If $ TP=4$ and $ T'Q=9$ then $ r$ is

<image1>
Choices:
A. $25/6$
B. $6$
C. $25/4 \$
D. $\text{a number other than }25/6, 6, 25/4 \$
E. $\text{not determinable from the given information}$
Answer: B
Problem: <image>Question: In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area of the triangle $QPO$ is equal to
<image1>
Choices:
A. $k$
B. $\frac{6k}{5}$
C. $\frac{9k}{8}$
D. $\frac{5k}{4}$
E. $2k$
Answer: C
Problem: <image>Question: In the adjoining figure triangle $ ABC$ is such that $ AB = 4$ and $ AC = 8$.  If $ M$ is the midpoint of $ BC$ and $ AM = 3$, what is the length of $ BC$?
<image1>
Choices:
A. $2\sqrt{26}$
B. $2\sqrt{31}$
C. $9$
D. $4+2\sqrt{13}$
E. $\text{not enough information given to solve the problem}$
Answer: B
Problem: <image>Question: In the adjoining figure $AB$ and $BC$ are adjacent sides of square $ABCD$; $M$ is the midpoint of $AB$; $N$ is the midpoint of $BC$; and $AN$ and $CM$ intersect at $O$. The ratio of the area of $AOCD$ to the area of $ABCD$ is
<image1>
Choices:
A. $\frac{5}{6}$
B. $\frac{3}{4}$
C. $\frac{2}{3}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{(\sqrt{3}-1)}{2}$
Answer: C
Problem: <image>Question: In triangle $ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC$, $AB=12$ and $AC=16$. Points $E$ and $F$ are taken on $AC$ and $AB$, respectively, and lines $EF$ and $AM$ intersect at $G$. If $AE=2AF$ then $\frac{EG}{GF}$ equals
<image1>
Choices:
A. $\frac{3}{2}$
B. $\frac{4}{3}$
C. $\frac{5}{4}$
D. $\frac{6}{5} \$
E. $\text{not enough information to solve the problem}$
Answer: A
Problem: <image>Question: <image1>

In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is
Choices:
A. $3+\sqrt{3}$
B. $15/\pi$
C. $9/2$
D. $2\sqrt{6}$
E. $\sqrt{22}$
Answer: E
Problem: <image>Question: <image1>

In the adjoining figure, circle $\mathit{K}$ has diameter $\mathit{AB}$; cirlce $\mathit{L}$ is tangent to circle $\mathit{K}$ and to $\mathit{AB}$ at the center of circle $\mathit{K}$; and circle $\mathit{M}$ tangent to circle $\mathit{K}$, to circle $\mathit{L}$ and $\mathit{AB}$. The ratio of the area of circle $\mathit{K}$ to the area of circle $\mathit{M}$ is
Choices:
A. $12$
B. $14$
C. $16$
D. $18$
E. $\text{not an integer}$
Answer: C
Problem: <image>Question: <image1>

In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is
Choices:
A. $\text{the average of the lengths of the internal and external common tangents}$
B. $\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}$
C. $\text{always equal to the length of an external common tangent}$
D. $\text{greater than the length of an external common tangent}$
E. $\text{the geometric mean of the lengths of the internal and external common tangents}$
Answer: C
Problem: <image>Question: <image1>

In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
Choices:
A. $30^\circ$
B. $40^\circ$
C. $50^\circ$
D. $65^\circ$
E. $\text{none of these}$
Answer: C
Problem: <image>Question: <image1>

In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$.
Choices:
A. $10^\circ$
B. $15^\circ$
C. $20^\circ$
D. $\left(\frac{45}{2}\right)^\circ$
E. $30^\circ$
Answer: B
Problem: <image>Question: <image1>

Each of the three circles in the adjoining figure is externally tangent to the other two, and each side of the triangle is tangent to two of the circles. If each circle has radius three, then the perimeter of the triangle is
Choices:
A. $36+9\sqrt{2}$
B. $36+6\sqrt{3}$
C. $36+9\sqrt{3}$
D. $18+18\sqrt{3}$
E. $45$
Answer: D
Problem: <image>Question: <image1>

If $a,b,$ and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then
Choices:
A. $d=a+b$
B. $d^2=a^2+b^2$
C. $d^2=a^2+ab+b^2$
D. $b=\frac{a+d}{2}$
E. $b^2=ad$
Answer: A
Problem: <image>Question: The following four statements, and only these are found on a card:
<image1>
(Assume each statement is either true or false.) Among them the number of false statements is exactly
Answer: 3
Problem: <image>Question: <image1>

Vertex $E$ of equilateral triangle $ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is
Choices:
A. $1$
B. $\frac{\sqrt{2}}{2}$
C. $\frac{\sqrt{3}}{2}$
D. $4-2\sqrt{3}$
E. $\frac{1}{2}+\frac{\sqrt{3}}{4}$
Answer: C
Problem: <image>Question: <image1>
In $\triangle ABC$, $AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with 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. The length of segment $QR$ is
Choices:
A. $4.75$
B. $4.8$
C. $5$
D. $4\sqrt{2}$
E. $3\sqrt{3}$
Answer: B
Problem: <image>Question: <image1>

If $\triangle A_1A_2A_3$ is equilateral and $A_{n+3}$ is the midpoint of line segment $A_nA_{n+1}$ for all positive integers $n$, then the measure of $\measuredangle A_{44}A_{45}A_{43}$ equals
Choices:
A. $30^\circ$
B. $45^\circ$
C. $60^\circ$
D. $90^\circ$
E. $120^\circ$
Answer: E
Problem: <image>Question: <image1>

If rectangle $ABCD$ has area $72$ square meters and $E$ and $G$ are the midpoints of sides $AD$ and $CD$, respectively, then the area of rectangle $DEFG$ in square meters is
Answer: 18
Problem: <image>Question: <image1>

In the adjoining figure, $ABCD$ is a square, $ABE$ is an equilateral triangle and point $E$ is outside square $ABCD$. What is the measure of $\measuredangle AED$ in degrees?
Answer: 15
Problem: <image>Question: <image1>

In the adjoining figure, $CD$ is the diameter of a semi-circle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semi-circle, and $B$ is the point of intersection (distinct from $E$ ) of line segment $AE$ with the semi-circle. If length $AB$ equals length $OD$, and the measure of $\measuredangle EOD$ is $45^\circ$, then the
measure of $\measuredangle BAO$ is
Choices:
A. $10^\circ$
B. $15^\circ$
C. $20^\circ$
D. $25^\circ$
E. $30^\circ$
Answer: B
Problem: <image>Question: <image1>

Points $A , B, C$, and $D$ are distinct and lie, in the given order, on a straight line. Line segments $AB, AC$, and $AD$ have lengths $x, y$, and $z$ , respectively. If line segments $AB$ and $CD$ may be rotated about points $B$ and $C$, respectively, so that points $A$ and $D$ coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?

$\textbf{I. }x<\frac{z}{2}\qquad\textbf{II. }y<x+\frac{z}{2}\qquad\textbf{III. }y<\frac{z}{2}$
Choices:
A. $\textbf{I. }\text{only}$
B. $\textbf{II. }\text{only}$
C. $\textbf{I. }\text{and }\textbf{II. }\text{only}$
D. $\textbf{II. }\text{and }\textbf{III. }\text{only}$
E. $\textbf{I. },\textbf{II. },\text{and }\textbf{III. }$
Answer: C
Problem: <image>Question: The edges of a regular tetrahedron with vertices $A ,~ B,~ C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$.
<image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{3}{4}$
C. $\frac{\sqrt{2}}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{\sqrt{3}}{3}$
Answer: C
Problem: <image>Question: Circles with centers $A ,~ B,$ and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$.
<image1>
If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, then length $B'C'$ equals
Choices:
A. $3r-2$
B. $r^2$
C. $r+\sqrt{3(r-1)}$
D. $1+\sqrt{3(r^2-1)}$
E. $\text{none of these}$
Answer: D
Problem: <image>Question: <image1>

In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals
Choices:
A. $\frac{1}{4}\cos 10^\circ$
B. $\frac{\sqrt{3}}{8}$
C. $\frac{1}{4}\cos 40^\circ$
D. $\frac{1}{4}\cos 50^\circ$
E. $\frac{1}{8}$
Answer: B
Problem: <image>Question: In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares.
<image1>
The measure of $\angle GDA$ is
Choices:
A. $90^\circ$
B. $105^\circ$
C. $120^\circ$
D. $135^\circ$
E. $150^\circ$
Answer: C
Problem: <image>Question: If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\measuredangle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is
<image1>
Choices:
A. $\frac{\sqrt{3}}{2}$
B. $\frac{\sqrt{3}}{3}$
C. $\frac{\sqrt{2}}{2}$
D. $\frac{1}{2}$
E. $\frac{2}{3}$
Answer: B
Problem: <image>Question: Sides $AB,BC,CD$ and $DA$ of convex polygon $ABCD$ have lengths 3,4,12, and 13, respectively, and $\measuredangle CBA$ is a right angle. The area of the quadrilateral is

<image1>
Answer: 36
Problem: <image>Question: In triangle $ABC$, $\measuredangle CBA=72^\circ$, $E$ is the midpoint of side $AC$, and $D$ is a point on side $BC$ such that $2BD=DC$; $AD$ and $BE$ intersect at $F$. The ratio of the area of triangle $BDF$ to the area of quadrilateral $FDCE$ is

<image1>
Choices:
A. $\frac{1}{5}$
B. $\frac{1}{4}$
C. $\frac{1}{3}$
D. $\frac{2}{5}$
E. $\text{none of these}$
Answer: A
Problem: <image>Question: In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$.  If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals

<image1>
Choices:
A. $2$
B. $\frac{5}{2}$
C. $\frac{5}{2} - \sin \theta$
D. $\frac{5}{2} - \frac{1}{2} \sin \theta$
E. $\frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$
Answer: B
Problem: <image>Question: <image1>

Equilateral $ \triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $ T$ and tangent to sides $ AB$ and $ AC$ at points $ P$ and $ Q$. If side $ BC$ has length $ 12$, then segment $ PQ$ has length
Choices:
A. $6$
B. $6\sqrt{3}$
C. $8$
D. $8\sqrt{3}$
E. $9$
Answer: C
Problem: <image>Question: <image1>

In triangle $ ABC$ in the adjoining figure, $ AD$ and $ AE$ trisect $ \angle BAC$. The lengths of $ BD$, $ DE$ and $ EC$ are $ 2$, $ 3$, and $ 6$, respectively. The length of the shortest side of $ \triangle ABC$ is
Choices:
A. $2\sqrt{10}$
B. $11$
C. $6\sqrt{6}$
D. $6$
E. $\text{not uniquely determined by the given information}$
Answer: A
Problem: <image>Question: In the adjoining figure triangle $ ABC$ is inscribed in a circle. Point $ D$ lies on $ \stackrel{\frown}{AC}$ with $ \stackrel{\frown}{DC} = 30^\circ$, and point $ G$ lies on $ \stackrel{\frown}{BA}$ with $ \stackrel{\frown}{BG}\, > \, \stackrel{\frown}{GA}$. Side $ AB$ and side $ AC$ each have length equal to the length of chord $ DG$, and $ \angle CAB = 30^\circ$. Chord $ DG$ intersects sides $ AC$ and $ AB$ at $ E$ and $ F$, respectively. The ratio of the area of $ \triangle AFE$ to the area of $ \triangle ABC$ is
<image1>
Choices:
A. $\frac{2 - \sqrt{3}}{3}$
B. $\frac{2\sqrt{3} - 3}{3}$
C. $7\sqrt{3} - 12$
D. $3\sqrt{3} - 5$
E. $\frac{9 - 5\sqrt{3}}{3}$
Answer: C
Problem: <image>Question: In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, then the perimeter of $\triangle AMN$ is

<image1>
Answer: 30
Problem: <image>Question: In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circle $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has length

<image1>
Choices:
A. $20$
B. $15\sqrt{2}$
C. $24$
D. $25$
E. $\text{none of these}$
Answer: C
Problem: <image>Question: In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ$. Find the cosine of $\angle BHD$.

<image1>
Choices:
A. $\frac{\sqrt{3}}{6}$
B. $\frac{\sqrt{2}}{6}$
C. $\frac{\sqrt{6}}{3}$
D. $\frac{\sqrt{6}}{4}$
E. $\frac{\sqrt{6}-\sqrt{2}}{4}$
Answer: D
Problem: <image>Question: In the adjoining figure, the triangle $ABC$ is a right triangle with $\angle BCA=90^\circ$. Median $CM$ is perpendicular to median $BN$, and side $BC=s$. The length of $BN$ is

<image1>
Choices:
A. $s\sqrt{2}$
B. $\frac{3}{2}s\sqrt{2}$
C. $2s\sqrt{2}$
D. $\frac{1}{2}s\sqrt{5}$
E. $\frac{1}{2}s\sqrt{6}$
Answer: E
Problem: <image>Question: In the adjoining figure, the circle meets the sides of an equilateral triangle at six points. If $AG=2$, $GF=13$, $FC=1$, and $HJ=7$, then $DE$ equals

<image1>
Choices:
A. $2\sqrt{22}$
B. $7\sqrt{3}$
C. $9$
D. $10$
E. $13$
Answer: A
Problem: <image>Question: The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$.

<image1>
Choices:
A. $\frac{11}{32}$
B. $\frac{1}{2}$
C. $\frac{4}{7}$
D. $\frac{21}{32}$
E. $\frac{3}{4}$
Answer: D
Problem: <image>Question: In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, and sides $BC$ and $ED$. Each side has length of 1.  Also, $\measuredangle FAB = \measuredangle BCD = 60^\circ$. The area of the figure is

<image1>
Choices:
A. $\frac{\sqrt{3}}{2}$
B. $1$
C. $\frac{3}{2}$
D. $\sqrt{3}$
E. $2$
Answer: D
Problem: <image>Question: In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$ ($L_1$ is the line that is above the circles and $L_2$ is the line that goes under the circles). If the radius of the largest circle is 18 and that of the smallest one is 8, then the radius of the middle circle is

<image1>
Answer: 12
Problem: <image>Question: Triangle $\triangle ABC$ in the figure has area $10$.  Points $D$, $E$ and $F$, all distinct from $A$, $B$ and $C$, are on sides $AB$, $BC$ and $CA$ respectively, and $AD = 2$, $DB = 3$.  If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is

<image1>
Choices:
A. $4$
B. $5$
C. $6$
D. $\frac{5}{3}\sqrt{10}$
E. $\text{not uniquely determined}$
Answer: C
Problem: <image>Question: Distinct points $A$ and $B$ are on a semicircle with diameter $MN$ and center $C$.  The point $P$ is on $CN$ and $\angle CAP = \angle CBP = 10^{\circ}$.  If $\stackrel{\frown}{MA} = 40^{\circ}$, then $\stackrel{\frown}{BN}$ equals

<image1>
Choices:
A. $10^{\circ}$
B. $15^{\circ}$
C. $20^{\circ}$
D. $25^{\circ}$
E. $30^{\circ}$
Answer: C
Problem: <image>Question: A rectangle intersects a circle as shown: $AB=4$, $BC=5$, and $DE=3$. Then $EF$ equals:

<image1>
Choices:
A. $6$
B. $7$
C. $\frac{20}{3}$
D. $8$
E. $9$
Answer: B
Problem: <image>Question: A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ And $HB$, with $HB = 16$. The area of $\triangle ABC$ is:
<image1>
Choices:
A. $120$
B. $144$
C. $150$
D. $216$
E. $144\sqrt{5}$
Answer: C