Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$?
<image1>
Answer: 150
Problem: <image>Question: The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon?
<image1>
Choices:
A. $\displaystyle \frac{1}{2}$
B. $\displaystyle \frac{\sqrt{3}}{3}$
C. $\displaystyle \frac{2}{3}$
D. $\displaystyle \frac{3}{4}$
E. $\displaystyle \frac{\sqrt{3}}{2}$
Answer: D
Problem: <image>Question: Triangles $ABC$ and $ABD$ are isosceles with $AB =AC = BD$, and $BD$ intersects $AC$ at $E$. If $BD$ is perpendicular to $AC$, then $\angle C + \angle D$ is
<image1>
Choices:
A. $115^\circ$
B. $120^\circ$
C. $130^\circ$
D. $135^\circ$
E. $\text{not uniquely determined}$
Answer: D
Problem: <image>Question: The adjacent sides of the decagon shown meet at right angles. What is its perimeter?
<image1>
Answer: 44
Problem: <image>Question: A rectangle with perimeter $ 176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
<image1>
Answer: 80
Problem: <image>Question: In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is
<image1>
Choices:
A. $2$
B. $3 - \frac{\sqrt{3}}{2}$
C. $\frac{11}{5}$
D. $\sqrt{5}$
E. $\frac{9}{4}$
Answer: C
Problem: <image>Question: Medians $ BD$ and $ CE$ of triangle $ ABC$ are perpendicular, $ BD = 8$, and $ CE = 12$. The area of triangle $ ABC$ is
<image1>
Answer: 64
Problem: <image>Question: A circle with center $ O$ is tangent to the coordinate axes and to the hypotenuse of the $ 30^\circ$-$ 60^\circ$-$ 90^\circ$ triangle $ ABC$ as shown, where $ AB = 1$. To the nearest hundredth, what is the radius of the circle?
<image1>
Answer: 2.37
Problem: <image>Question: In the figure, polygons $ A$, $ E$, and $ F$ are isosceles right triangles; $ B$, $ C$, and $ D$ are squares with sides of length $ 1$; and $ G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces.
<image1>
The volume of this polyhedron is
Choices:
A. 1/2
B. 2/3
C. 3/4
D. 5/6
E. 4/3
Answer: D
Problem: <image>Question: Triangle $ ABC$ and point $ P$ in the same plane are given. Point $ P$ is equidistant from $ A$ and $ B$, angle $ APB$ is twice angle $ ACB$, and $ \overline{AC}$ intersects $ \overline{BP}$ at point $ D$.
<image1>
If $ PB = 3$ and $ PD = 2$, then $ AD\cdot CD =$
Answer: 5
Problem: <image>Question: <image1>
Each of the sides of the five congruent rectangles is labeled with an integer, as shown above. These five rectangles are placed, without rotating or reflecting, in positions $I$ through $V$ so that the labels on coincident sides are equal.
<image2>
Which of the rectangles is in position $I$?
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: A square $ ABCD$ with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points $ E,F,G$ where $ E$ is the midpoint of $ BC$, $ F,G$ are on $ AB$ and $ CD$, respectively, and they're positioned that $ AF < FB, DG < GC$ and $ F$ is the directly opposite of $ G$. If $ FB = x$, the length of the longer parallel side of each trapezoid, find the value of $ x$.
<image1>
Choices:
A. $\frac{3}{5}$
B. $\frac{2}{3}$
C. $\frac{3}{4}$
D. $\frac{5}{6}$
E. $\frac{7}{8}$
Answer: D
Problem: <image>Question: A large square is divided into a small square surrounded by four congruent rectangles as shown. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?
<image1>
Answer: 49
Problem: <image>Question: The figure shown is the union of a circle and two semicircles of diameters of $ a$ and $ b$, all of whose centers are collinear.
<image1>
The ratio of the area of the shaded region to that of the unshaded region is
Choices:
A. $\sqrt{\frac{a}{b}}$
B. $\frac{a}{b}$
C. $\frac{a^2}{b^2}$
D. $\frac{a + b}{2b}$
E. $\frac{a^2 + 2ab}{b^2 + 2ab}$
Answer: B
Problem: <image>Question: A $ 9\times9\times9$ cube is composed of twenty-seven $ 3\times3\times3$ cubes. The big cube is 'tunneled' as follows: First, the six $ 3\times3\times3$ cubes which make up the center of each face as well as the center of $ 3\times3\times3$ cube are removed. Second, each of the twenty remaining $ 3\times3\times3$ cubes is diminished in the same way. That is, the central facial unit cubes as well as each center cube are removed.
<image1>
The surface area of the final figure is
Answer: 1056
Problem: <image>Question: Consider all triangles $ ABC$ satisfying the following conditions: $ AB = AC$, $ D$ is a point on $ \overline{AC}$ for which $ \overline{BD} \perp \overline{AC}$, $ AD$ and $ CD$ are integers, and $ BD^2 = 57$. 
<image1>
Among all such triangles, the smallest possible value of $ AC$ is
Answer: 11
Problem: <image>Question: A circle centered at $ O$ has radius $ 1$ and contains the point $ A$. Segment $ AB$ is tangent to the circle at $ A$ and $ \angle{AOB} = \theta$. If point $ C$ lies on $ \overline{OA}$ and $ \overline{BC}$ bisects $ \angle{ABO}$, then $ OC =$

<image1>
Choices:
A. $\sec^2\theta - \tan\theta$
B. $\frac{1}{2}$
C. $\frac{\cos^2\theta}{1 + \sin\theta}$
D. $\frac{1}{1 + \sin\theta}$
E. $\frac{\sin\theta}{\cos^2\theta}$
Answer: D
Problem: <image>Question: The graph below shows a portion of the curve defined by the quartic polynomial $ P(x) = x^4 + ax^3 + bx^2 + cx + d$. Which of the following is the smallest?
<image1>
Choices:
A. $P( - 1)$
B. $\text{The product of the zeros of }P$
C. $\text{The product of the non - real zeros of }P$
D. $\text{The sum of the coefficients of }P$
E. $\text{The sum of the real zeros of }P$
Answer: C
Problem: <image>Question: If circular arcs $ AC$ and $ BC$ have centers at $ B$ and $ A$, respectively, then there exists a circle tangent to both $ \stackrel{\frown}{AC}$ and $ \stackrel{\frown}{BC}$, and to $ \overline{AB}$. If the length of $ \stackrel{\frown}{BC}$ is $ 12$, then the circumference of the circle is
<image1>
Answer: 27
Problem: <image>Question: Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
<image1>
Answer: 1680
Problem: <image>Question: A point $ P$ is selected at random from the interior of the pentagon with vertices $ A = (0,2)$, $B = (4,0)$, $C = (2 \pi + 1, 0)$, $D = (2 \pi + 1,4)$, and $ E = (0,4)$. What is the probability that $ \angle APB$ is obtuse?
<image1>
Choices:
A. $\frac{1}{5}$
B. $\frac{1}{4}$
C. $\frac{5}{16}$
D. $\frac{3}{8}$
E. $\frac{1}{2}$
Answer: C
Problem: <image>Question: A circle centered at $ A$ with a radius of 1 and a circle centered at $ B$ with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is
<image1>
Choices:
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{5}{12}$
D. $\frac{4}{9}$
E. $\frac{1}{2}$
Answer: D
Problem: <image>Question: In rectangle $ ABCD$, points $ F$ and $ G$ lie on $ \overline{AB}$ so that $ AF = FG = GB$ and $ E$ is the midpoint of $ \overline{DC}$. Also, $ \overline{AC}$ intersects $ \overline{EF}$ at $ H$ and $ \overline{EG}$ at $ J$. The area of the rectangle $ ABCD$ is $ 70$. Find the area of triangle $ EHJ$.
<image1>
Choices:
A. $\frac{5}{2}$
B. $\frac{35}{12}$
C. $3$
D. $\frac{7}{2}$
E. $\frac{35}{8}$
Answer: C
Problem: <image>Question: In $ \triangle ABC$, $ \angle ABC = 45^\circ$. Point $ D$ is on $ \overline{BC}$ so that $ 2 \cdot BD = CD$ and $ \angle DAB = 15^\circ$. Find $ \angle ACB$.
<image1>
Choices:
A. $54^\circ$
B. $60^\circ$
C. $72^\circ$
D. $75^\circ$
E. $90^\circ$
Answer: D
Problem: <image>Question: The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) = 6$ have?
<image1>
Answer: 6
Problem: <image>Question: Triangle $ ABC$ is a right triangle with $ \angle ACB$ as its right angle, $ m\angle ABC = 60^\circ$, and $ AB = 10$. Let $ P$ be randomly chosen inside $ \triangle ABC$, and extend $ \overline{BP}$ to meet $ \overline{AC}$ at $ D$. What is the probability that $ BD > 5\sqrt{2}$?

<image1>
Choices:
A. $\frac{2 - \sqrt{2}}{2}$
B. $\frac{1}{3}$
C. $\frac{3 - \sqrt{3}}{3}$
D. $\frac{1}{2}$
E. $\frac{5 - \sqrt{5}}{5}$
Answer: C
Problem: <image>Question: The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
<image1>
<image2>
<image3>
<image4>
<image5>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: Points $ K$, $ L$, $ M$, and $ N$ lie in the plane of the square $ ABCD$ so that $ AKB$, $ BLC$, $ CMD$, and $ DNA$ are equilateral triangles. If $ ABCD$ has an area of $ 16$, find the area of $ KLMN$.

<image1>
Choices:
A. $32$
B. $16 + 16\sqrt{3}$
C. $48$
D. $32 + 16\sqrt{3}$
E. $64$
Answer: D
Problem: <image>Question: Square $ ABCD$ has sides of length $ 4$, and $ M$ is the midpoint of $ \overline{CD}$. A circle with radius $ 2$ and center $ M$ intersects a circle with raidus $ 4$ and center $ A$ at points $ P$ and $ D$. What is the distance from $ P$ to $ \overline{AD}$?
<image1>
Choices:
A. $3$
B. $\frac{16}{5}$
C. $\frac{13}{4}$
D. $2\sqrt{3}$
E. $\frac{7}{2}$
Answer: B
Problem: <image>Question: Several figures can be made by attaching two equilateral triangles to the regular pentagon $ ABCDE$ in two of the five positions shown. How many non-congruent figures can be constructed in this way?
<image1>
Answer: 2
Problem: <image>Question: Part of the graph of $ f(x) = x^3 + bx^2 + cx + d$ is shown. What is $ b$?
<image1>
Choices:
A. $-\!4$
B. $-\!2$
C. $0$
D. $2$
E. $4$
Answer: B
Problem: <image>Question: Let $ ABCD$ be a rhombus with $ AC=16$ and $ BD=30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$?
<image1>
Answer: 7
Problem: <image>Question: The graph of the line $ y = mx + b$ is shown. Which of the following is true?
<image1>
Choices:
A. mb < - 1
B. - 1 < mb < 0
C. mb = 0
D. 0 < mb < 1
E. mb > 1
Answer: B
Problem: <image>Question: In the overlapping triangles $ \triangle{ABC}$ and $ \triangle{ABE}$ sharing common side $ AB$, $ \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: Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{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: Circles $ A$, $ B$ and $ C$ are externally tangent to each other and internally tangent to circle $ D$. Circles $ B$ and $ C$ are congruent. Circle $ A$ has radius $ 1$ and passes through the center of $ D$. What is the radius of circle $ B$?
<image1>
Choices:
A. $\frac{2}{3}$
B. $\frac{\sqrt{3}}{2}$
C. $\frac{7}{8}$
D. $\frac{8}{9}$
E. $\frac{1 + \sqrt{3}}{3}$
Answer: D
Problem: <image>Question: In $ \triangle ABC$ , $ AB = 13$, $ AC = 5$, and $ BC = 12$. Points $ M$ and $ N$ lie on $ \overline{AC}$ and $ \overline{BC}$, respectively, with $ CM = CN = 4$. Points $ J$ and $ K$ are on $ \overline{AB}$ so that $ \overline{MJ}$ and $ \overline{NK}$ are perpendicular to $ \overline{AB}$. What is the area of pentagon $ CMJKN$?
<image1>
Choices:
A. $15$
B. $\frac{81}{5}$
C. $\frac{205}{12}$
D. $\frac{240}{13}$
E. $20$
Answer: D
Problem: <image>Question: In $ \triangle ABC$, $ AB = BC$, and $ BD$ is an altitude. Point $ E$ is on the extension of $ \overline{AC}$ such that $ BE = 10$. The values of $ \tan CBE$, $ \tan DBE$, and $ \tan ABE$ form a geometric progression, and the values of $ \cot DBE$, $ \cot CBE$, $ \cot DBC$ form an arithmetic progression. What is the area of $ \triangle ABC$?
<image1>
Choices:
A. $16$
B. $\frac{50}{3}$
C. $10\sqrt{3}$
D. $8\sqrt{5}$
E. $18$
Answer: B
Problem: <image>Question: Three circles of radius $ s$ are drawn in the first quadrant of the $ xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $ x$-axis, and the third is tangent to the first circle and the $ y$-axis. A circle of radius $ r > s$ is tangent to both axes and to the second and third circles. What is $ r/s$?
<image1>
Answer: 9
Problem: <image>Question: A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $ W$?
<image1>
Choices:
A. $\frac{1}{12}$
B. $\frac{1}{9}$
C. $\frac{1}{8}$
D. $\frac{1}{6}$
E. $\frac{1}{4}$
Answer: A
Problem: <image>Question: The vertices of a $ 3 - 4 - 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?
<image1>
Choices:
A. $12\pi$
B. $\frac{25\pi}{2}$
C. $13\pi$
D. $\frac{27\pi}{2}$
E. $14\pi$
Answer: E
Problem: <image>Question: Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF = \sqrt{9 + 5\sqrt{2}}$. What is $ r/s$?
<image1>
Choices:
A. $\frac{1}{2}$
B. $\frac{5}{9}$
C. $\frac{3}{5}$
D. $\frac{5}{3}$
E. $\frac{9}{5}$
Answer: B
Problem: <image>Question: Circles with centers $ (2,4)$ and $ (14,9)$ have radii 4 and 9, respectively. The equation of a common external tangent to the circles can be written in the form $ y = mx + b$ with $ m > 0$. What is $ b$?

<image1>
Choices:
A. $\frac{908}{199}$
B. $\frac{909}{119}$
C. $\frac{130}{17}$
D. $\frac{911}{119}$
E. $\frac{912}{119}$
Answer: E
Problem: <image>Question: Isosceles $ \triangle ABC$ has a right angle at $ C$.  Point $ P$ is inside $ \triangle ABC$, such that $ PA = 11, PB = 7,$ and $ PC = 6$.  Legs $ \overline{AC}$ and $ \overline{BC}$ have length $ s = \sqrt{a + b\sqrt{2}}$, where $ a$ and $ b$ are positive integers.  What is $ a + b$?

<image1>
Answer: 127
Problem: <image>Question: The point $ O$ is the center of the circle circumscribed about $ \triangle ABC$, with $ \angle BOC = 120^\circ$ and $ \angle AOB = 140^\circ$, as shown. What is the degree measure of $ \angle ABC$?
<image1>
Answer: 50
Problem: <image>Question: A ship sails $ 10$ miles in a straight line from $ A$ to $ B$, turns through an angle between $ 45^{\circ}$ and $ 60^{\circ}$, and then sails another $ 20$ miles to $ C$. Let $ AC$ be measured in miles. Which of the following intervals contains $ AC^2$?
<image1>
Choices:
A. [400,500]
B. [500,600]
C. [600,700]
D. [700,800]
E. [800,900]
Answer: D
Problem: <image>Question: The set $ G$ is defined by the points $ (x,y)$ with integer coordinates, $ 3\le|x|\le7$, $ 3\le|y|\le7$. How many squares of side at least $ 6$ have their four vertices in $ G$?
<image1>
Answer: 225
Problem: <image>Question: Rectangle $ ABCD$, pictured below, shares $50\%$ of its area with square $ EFGH$. Square $ EFGH$ shares $20\%$ of its area with rectangle $ ABCD$. What is $ \frac{AB}{AD}$?
<image1>
Answer: 10
Problem: <image>Question: A big $ L$ is formed as shown. What is its area?
<image1>
Answer: 22
Problem: <image>Question: There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?

<image1>
Choices:
A. $(C, A, E, D, B)$
B. $(C, A, D, E, B)$
C. $(C, D, E, A, B) \ [1ex]$
D. $(C, E, A, D, B)$
E. $(C, E, D, A, B)$
Answer: E
Problem: <image>Question: A unit cube has vertices $P_1, P_2, P_3, P_4, P_1', P_2', P_3'$, and $P_4'$. Vertices $P_2, P_3$, and $P_4$ are adjacent to $P_1$, and for $1\leq i\leq 4$, vertices $P_i$ and $P_i'$ are opposite to each other. A regular octahedron has one vertex in each of the segments $P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'$, and $P_1'P_4'$. What is the octahedron's side length?
<image1>
Choices:
A. $\frac{3\sqrt{2}}{4}$
B. $\frac{7\sqrt{6}}{16}$
C. $\frac{\sqrt{5}}{2}$
D. $\frac{2\sqrt{3}}{3}$
E. $\frac{\sqrt{6}}{2}$
Answer: A
Problem: <image>Question: Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?

<image1>
Choices:
A. $29\sqrt{3}$
B. $\frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}$
C. $20\sqrt{3}+16$
D. $20\sqrt{2}+13\sqrt{3}$
E. $21\sqrt{6}$
Answer: A
Problem: <image>Question: Let $S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}$, and $(x,y) \neq (0,0) \}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)= \tan (\angle CBA)$. What is
\[ \prod_{t \in T} f(t) \] ?
<image1>
Choices:
A. $1$
B. $\frac{625}{144}$
C. $\frac{125}{24}$
D. $6$
E. $\frac{625}{24}$
Answer: B
Problem: <image>Question: Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\overline{AC}$ and points $D$ and $F$ are on $\overline{AB}$ such that both $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBCG$ all have the same perimeter. What is $DE+FG$?
<image1>
Choices:
A. $1$
B. $\frac{3}{2}$
C. $\frac{21}{13}$
D. $\frac{13}{8}$
E. $\frac{5}{3}$
Answer: C
Problem: <image>Question: Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$.  How many different routes are there from $A$ to $B$ that use each road exactly once?  (Such a route will necessarily visit cities more than once.)

<image1>
Answer: 16
Problem: <image>Question: A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$.  The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres.  What is $h$?

<image1>
Choices:
A. $2+2\sqrt{7}$
B. $3+2\sqrt{5}$
C. $4+2\sqrt{7}$
D. $4\sqrt{5}$
E. $4\sqrt{7}$
Answer: A
Problem: <image>Question: Convex quadrilateral $ABCD$ has $AB = 3, BC = 4, CD = 13, AD = 12,$ and $\angle ABC = 90^\circ,$ as shown. What is the area of the quadrilateral?

<image1>
Answer: 36
Problem: <image>Question: In the figure, $ABCD$ is a square of side length 1. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?
<image1>
Choices:
A. $\frac{1}{2}(\sqrt{6}-2)$
B. $\frac{1}{4}$
C. $2-\sqrt{3}$
D. $\frac{\sqrt{3}}{6}$
E. $1-\frac{\sqrt{2}}{2}$
Answer: C
Problem: <image>Question: A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows.  Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent.  For $k\geq 1$, the circles in $\textstyle\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tangency with the $x$-axis.  For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair.  Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way.  Let $S=\textstyle\bigcup_{j=0}^6 L_j$, and for every circle $C$ denote by $r(C)$ its radius.  What is \[\sum_{C\in S}\frac{1}{\sqrt{r(C)}}?\]

<image1>
Choices:
A. $\frac{286}{35}$
B. $\frac{583}{70}$
C. $\frac{715}{73}$
D. $\frac{143}{14}$
E. $\frac{1573}{146}$
Answer: D
Problem: <image>Question: The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is $\frac{a-\sqrt{2}}{b}$, where $a$ and $b$ are positive integers. What is $a+b$ ?

<image1>
Answer: 11
Problem: <image>Question: In $\triangle ABC$, $AB = 6$, $BC = 7$, and $CA = 8$. Point $D$ lies on $\overline{BC}$, and $\overline{AD}$ bisects $\angle BAC$. Point $E$ lies on $\overline{AC}$, and $\overline{BE}$ bisects $\angle ABC$. The bisectors intersect at $F$. What is the ratio $AF$ : $FD$?

<image1>
Choices:
A. 3:2
B. 5:3
C. 2:1
D. 7:3
E. 5:2
Answer: C
Problem: <image>Question: In $\triangle ABC$ shown in the figure, $AB=7$, $BC=8$, $CA=9$, and $\overline{AH}$ is an altitude. Points $D$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively, so that $\overline{BD}$ and $\overline{CE}$ are angle bisectors, intersecting $\overline{AH}$ at $Q$ and $P$, respectively. What is $PQ$?

<image1>
Choices:
A. $1$
B. $\frac{5}{8}\sqrt{3}$
C. $\frac{4}{5}\sqrt{2}$
D. $\frac{8}{15}\sqrt{5}$
E. $\frac{6}{5}$
Answer: D
Problem: <image>Question: In the figure below, semicircles with centers at $A$ and $B$ and with radii $2$ and $1$, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $\overline{JK}$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?

<image1>
Choices:
A. $\frac{3}{4}$
B. $\frac{6}{7}$
C. $\frac{1}{2}\sqrt{3}$
D. $\frac{5}{8}\sqrt{2}$
E. $\frac{11}{12}$
Answer: B
Problem: <image>Question: Square $ABCD$ has side length $30$. Point $P$ lies inside the square so that $AP = 12$ and $BP = 26$. The centroids of $\triangle{ABP}$, $\triangle{BCP}$, $\triangle{CDP}$, and $\triangle{DAP}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral?

<image1>
Choices:
A. $100\sqrt{2}$
B. $100\sqrt{3}$
C. $200$
D. $200\sqrt{2}$
E. $200\sqrt{3}$
Answer: C
Problem: <image>Question: Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two.  Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$.  See the figure below.  The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$.  What is $a+b$?

<image1>
Answer: 552
Problem: <image>Question: The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L,$ without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.)  How many different routes can Paula take?

<image1>
Answer: 4
Problem: <image>Question: Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\sin(2\angle BAD)$?
<image1>
Choices:
A. $\frac{1}{3}$
B. $\frac{\sqrt{2}}{2}$
C. $\frac{3}{4}$
D. $\frac{7}{9}$
E. $\frac{\sqrt{3}}{2}$
Answer: D