Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH$. Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. See the figure below. Triangle $AEH$, quadrilateral $BFIE$, quadrilateral $DHJG$, and pentagon $FCGJI$ each has area $1$. What is $FI^2$?
<image1>
Choices:
A. $\frac{7}{3}$
B. $8-4\sqrt{2}$
C. $1+\sqrt{2}$
D. $\frac{7}{4}\sqrt{2}$
E. $2\sqrt{2}$
Answer: B
Problem: <image>Question: As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$
<image1>
Answer: 170
Problem: <image>Question: A farmer's rectangular field is partitioned into  $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want to grow soybeans and potatoes in any two sections that share a border. Given these restrictions, in how many ways can the farmer choose crops to plant in each of the four sections of the field?
<image1>
Answer: 84
Problem: <image>Question: What is the area of the shaded figure shown below?

<image1>
Answer: 6
Problem: <image>Question: A square with side length $3$ is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length $2$ has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?

<image1>
Choices:
A. $19\frac{1}{4}$
B. $20\frac{1}{4}$
C. $21\frac{3}{4}$
D. $22\frac{1}{2}$
E. $23\frac{3}{4}$
Answer: B
Problem: <image>Question: In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square?

<image1>
Answer: 117
Problem: <image>Question: Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$

<image1>
Answer: 147
Problem: <image>Question: A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

<image1>
Answer: 67
Problem: <image>Question: Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3 \text{ cm}$ and $6 \text{ cm}$. Into each cone is dropped a spherical marble of radius $1 \text{ cm}$, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone?
<image1>
Choices:
A. 1:1
B. 47:43
C. 2:1
D. 40:13
E. 4:1
Answer: E
Problem: <image>Question: Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eight row and eight column) and places the other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest and the least number that appear in the second row from the top?

<image1>
Answer: 367
Problem: <image>Question: The figure below is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m}+\sqrt{n},$ where $m$ and $n$ are positive integers. What is $m+n?$

<image1>
Answer: 23
Problem: <image>Question: A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{BC}$ intersects edge $\overline{AB}$ at point $E$. Suppose that $C'D=\frac{1}{3}$. What is the perimeter of $\triangle AEC'$?

<image1>
Choices:
A. $2$
B. $1+\frac{2}{3}\sqrt{3}$
C. $\frac{13}{6}$
D. $1+\frac{3}{4}\sqrt{3}$
E. $\frac{7}{3}$
Answer: A
Problem: <image>Question: A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square, The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$, where $a$ and $b$ are positive integers. What is $a+b$?

<image1>
Answer: 68
Problem: <image>Question: Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2)$.
<image1>

Arjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth?
Choices:
A. (6,1,1)
B. (6,2,1)
C. (6,2,2)
D. (6,3,1)
E. (6,3,2)
Answer: B
Problem: <image>Question: A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
<image1>
Answer: 540
Problem: <image>Question: Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card?

<image1>
Choices:
A. $14$
B. $10\sqrt{2}$
C. $16$
D. $12\sqrt{2}$
E. $18$
Answer: E
Problem: <image>Question: A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?

<image1>
Choices:
A. $6$
B. $7$
C. $5+2\sqrt{2}$
D. $8$
E. $9$
Answer: B
Problem: <image>Question: Suppose that 13 cards numbered $1, 2, 3, \dots, 13$ are arranged in a row. The task is to pick them up in numerically increasing order, working repeatedly from left to right. In the example below, cards 1, 2, 3 are picked up on the first pass, 4 and 5 on the second pass, 6 on the third pass, 7, 8, 9, 10 on the fourth pass, and 11, 12, 13 on the fifth pass. For how many of the $13!$ possible orderings of the cards will the $13$ cards be picked up in exactly two passes?

<image1>
Answer: 8178
Problem: <image>Question: Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale).
<image1>

The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?
Answer: 337
Problem: <image>Question: In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ such that $BP\perp AD$, $AP = 3$, and $PD = 2$.  What is the area of $ABCD$?

<image1>
Choices:
A. $3\sqrt{5}$
B. $10$
C. $6\sqrt{5}$
D. $20$
E. $25$
Answer: D
Problem: <image>Question: The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

<image1>
Choices:
A. $15\frac{1}{8}$
B. $15\frac{3}{8}$
C. $15\frac{1}{2}$
D. $15\frac{5}{8}$
E. $15\frac{7}{8}$
Answer: D
Problem: <image>Question: Each square in a $5 \times 5$ grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring squares share either a side or a corner. The grid is transformed by the following rules:

Any filled square with two or three filled neighbors remains filled.
Any empty square with exactly three filled neighbors becomes a filled square.
All other squares remain empty or become empty.

A sample transformation is shown in the figure below.
<image1>

Suppose the $5 \times 5$ grid has a border of empty squares surrounding a $3 \times 3$ subgrid. How many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single transformation? (Rotations and reflections of the same configuration are considered different.)
<image2>
Answer: 22
Problem: <image>Question: A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
<image1>
Choices:
A. $\frac{1}{5}$
B. $\frac{1}{4}$
C. $2-\sqrt{3}$
D. $\sqrt{3}-\sqrt{2}$
E. $\sqrt{2}-1$
Answer: C
Problem: <image>Question: Each square in a $3\times 3$ grid of squares is colored red, white, blue, or green so that every $2\times 2$ square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
<image1>
Answer: 72
Problem: <image>Question: Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?

<image1>
Choices:
A. $\frac{1}{14}$
B. $\frac{1}{12}$
C. $\frac{1}{10}$
D. $\frac{3}{28}$
E. $\frac{1}{9}$
Answer: D
Problem: <image>Question: Six regular hexagonal blocks of side length 1 unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What is the area of the region inside the frame not occupied by the blocks?
<image1>
Choices:
A. $\frac{13 \sqrt{3}}{3}$
B. $\frac{216 \sqrt{3}}{49}$
C. $\frac{9 \sqrt{3}}{2}$
D. $\frac{14 \sqrt{3}}{3}$
E. $\frac{243 \sqrt{3}}{49}$
Answer: C
Problem: <image>Question: Square $ABCD$ is rotated $20^\circ$ clockwise about its center to obtain square $EFGH$, as shown below.  What is the degree measure of $\angle EAB$?
<image1>
Choices:
A. $20^\circ$
B. $30^\circ$
C. $32^\circ$
D. $35^\circ$
E. $45^\circ$
Answer: B
Problem: <image>Question: In the figure, it is given that angle $ C = 90^{\circ}, \overline{AD} = \overline{DB}, DE \perp AB, \overline{AB} = 20$, and $ \overline{AC} = 12$. The area of quadrilateral $ ADEC$ is:
<image1>
Choices:
A. $75$
B. $58\frac{1}{2}$
C. $48$
D. $37\frac{1}{2}$
E. $\text{none of these}$
Answer: B
Problem: <image>Question: In the figure, $ \overline{CD}, \overline{AE}$ and $ \overline{BF}$ are one-third of their respective sides. It follows that $ \overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} = 3: 3: 1$, and similarly for lines $ BE$ and $ CF$. Then the area of triangle $ N_1N_2N_3$ is:
<image1>
Choices:
A. $\frac{1}{10} \triangle ABC$
B. $\frac{1}{9} \triangle ABC$
C. $\frac{1}{7} \triangle ABC$
D. $\frac{1}{6} \triangle ABC$
E. $\text{none of these}$
Answer: C
Problem: <image>Question: In the right triangle shown the sum of the distances $ BM$ and $ MA$ is equal to the sum of the distances $ BC$ and $ CA$.  If $ MB = x$, $ CB = h$, and $ CA = d$, then $ x$ equals:
<image1>
Choices:
A. $\frac{hd}{2h + d}$
B. $d - h$
C. $\frac{1}{2}d$
D. $h + d - \sqrt{2d}$
E. $\sqrt{h^2 + d^2} - h$
Answer: A
Problem: <image>Question: Given triangle $ PQR$ with $ \overline{RS}$ bisecting $ \angle R$, $ PQ$ extended to $ D$ and $ \angle n$ a right angle, then:
<image1>
Choices:
A. $\angle m = \frac{1}{2}(\angle p - \angle q)$
B. $\angle m = \frac{1}{2}(\angle p + \angle q)$
C. $\angle d = \frac{1}{2} (\angle q + \angle p)$
D. $\angle d = \frac{1}{2}\angle m$
E. $\text{none of these is correct}$
Answer: B
Problem: <image>Question: In the diagram, if points $ A$, $ B$ and $ C$ are points of tangency, then $ x$ equals:
<image1>
Choices:
A. $\frac{3}{16}"$
B. $\frac{1}{8}"$
C. $\frac{1}{32}"$
D. $\frac{3}{32}"$
E. $\frac{1}{16}"$
Answer: E
Problem: <image>Question: In the figure, $ PA$ is tangent to semicircle $ SAR$; $ PB$ is tangent to semicircle $ RBT$; $ SRT$ is a straight line; the arcs are indicated in the figure.  Angle $ APB$ is measured by:
<image1>
Choices:
A. $\frac{1}{2}(a - b)$
B. $\frac{1}{2}(a + b)$
C. $(c - a) - (d - b)$
D. $a - b$
E. $a + b$
Answer: E
Problem: <image>Question: In circle $ O$ chord $ AB$ is produced so that $ BC$ equals a radius of the circle.  $ CO$ is drawn and extended to $ D$. $ AO$ is drawn.  Which of the following expresses the relationship between $ x$ and $ y$?

<image1>
Choices:
A. $x=3y \\$
B. $x=2y \\$
C. $x=60^\circ \\$
D. $\text{there is no special relationship between }x\text{ and }y \\$
E. $x=2y \text{ or }x=3y\text{, depending upon the length of }AB$
Answer: A
Problem: <image>Question: In the figure $ \overline{AB} = \overline{AC}$, angle $ BAD = 30^{\circ}$, and $ \overline{AE} = \overline{AD}$.
<image1>Then angle $ CDE$ equals:
Choices:
A. $7\frac{1}{2}^{\circ}$
B. $10^{\circ}$
C. $12\frac{1}{2}^{\circ}$
D. $15^{\circ}$
E. $20^{\circ}$
Answer: D
Problem: <image>Question: Circle $ O$ has diameters $ AB$ and $ CD$ perpendicular to each other. $ AM$ is any chord intersecting $ CD$ at $ P$. Then $ AP\cdot AM$ is equal to:
<image1>
Choices:
A. $AO\cdot OB$
B. $AO\cdot AB$
C. $CP\cdot CD$
D. $CP\cdot PD$
E. $CO\cdot OP$
Answer: B
Problem: <image>Question: In right triangle $ ABC$, $ BC = 5$, $ AC = 12$, and $ AM = x$; $ \overline{MN} \perp \overline{AC}$, $ \overline{NP} \perp \overline{BC}$; $ N$ is on $ AB$. If $ y = MN + NP$, one-half the perimeter of rectangle $ MCPN$, then:
<image1>
Choices:
A. $y = \frac{1}{2}(5 + 12)$
B. $y = \frac{5x}{12} + \frac{12}{5}$
C. $y = \frac{144 - 7x}{12}$
D. $y = 12    \,\,$
E. $y = \frac{5x}{12} + 6$
Answer: C
Problem: <image>Question: In triangle $ ABC$, $ AC = CD$ and $ \angle CAB - \angle ABC = 30^\circ$. Then $ \angle BAD$ is:
<image1>
Choices:
A. $30^\circ$
B. $20^\circ$
C. $22\frac{1}{2}^\circ$
D. $10^\circ$
E. $15^\circ$
Answer: E
Problem: <image>Question: In circle $ O$, the midpoint of radius $ OX$ is $ Q$; at $ Q$, $ \overline{AB} \perp \overline{XY}$. The semi-circle with $ \overline{AB}$ as diameter intersects $ \overline{XY}$ in $ M$. Line $ \overline{AM}$ intersects circle $ O$ in $ C$, and line $ \overline{BM}$ intersects circle $ O$ in $ D$. Line $ \overline{AD}$ is drawn. Then, if the radius of circle $ O$ is $ r$, $ AD$ is:
<image1>
Choices:
A. $r\sqrt{2}$
B. $r$
C. $\text{not a side of an inscribed regular polygon}$
D. $\frac{r\sqrt{3}}{2}$
E. $r\sqrt{3}$
Answer: A
Problem: <image>Question: Let $ ABC$ be an equilateral triangle inscribed in circle $ O$. $ M$ is a point on arc $ BC$. Lines $ \overline{AM}$, $ \overline{BM}$, and $ \overline{CM}$ are drawn. Then $ AM$ is:
<image1>
Choices:
A. $\text{equal to }{BM + CM}$
B. $\text{less than }{BM + CM}$
C. $\text{greater than }{BM + CM}$
D. $\text{equal, less than, or greater than }{BM + CM}\text{, depending upon the position of }{ {M} }$
E. $\text{none of these}$
Answer: A
Problem: <image>Question: The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:
<image1>
Choices:
A. 4: 3
B. 3: 2
C. 4: 1
D. 3: 1
E. 6: 1
Answer: C
Problem: <image>Question: In the accompanying figure $ \overline{CE}$ and $ \overline{DE}$ are equal chords of a circle with center $ O$. Arc $ AB$ is a quarter-circle. Then the ratio of the area of triangle $ CED$ to the area of triangle $ AOB$ is:
<image1>
Choices:
A. $\sqrt{2} : 1$
B. $\sqrt{3} : 1$
C. $4 : 1$
D. $3 : 1$
E. $2 : 1$
Answer: E
Problem: <image>Question: In a general triangle $ ADE$ (as shown) lines $ \overline{EB}$ and $ \overline{EC}$ are drawn. Which of the following angle relations is true?
<image1>
Choices:
A. $x + z = a + b$
B. $y + z = a + b$
C. $m + x = w + n  \\$
D. $x + z + n = w + c + m$
E. $x + y + n = a + b + m$
Answer: E
Problem: <image>Question: $ ABCD$ is a rectangle (see the accompanying diagram) with $ P$ any point on $ \overline{AB}$. $ \overline{PS} \perp \overline{BD}$ and $ \overline{PR} \perp \overline{AC}$. $ \overline{AF} \perp \overline{BD}$ and $ \overline{PQ} \perp \overline{AF}$. Then $ PR + PS$ is equal to:
<image1>
Choices:
A. PQ
B. AE
C. PT + AT
D. AF
E. EF
Answer: D
Problem: <image>Question: In this diagram a scheme is indicated for associating all the points of segment $ \overline{AB}$ with those of segment $ \overline{A'B'}$, and reciprocally. To described this association scheme analytically, let $ x$ be the distance from a point $ P$ on $ \overline{AB}$ to $ D$ and let $ y$ be the distance from the associated point $ P'$ of $ \overline{A'B'}$ to $ D'$. Then for any pair of associated points, if $ x = a,\, x + y$ equals:
<image1>
Choices:
A. $13a$
B. $17a - 51$
C. $17 - 3a$
D. $\frac{17 - 3a}{4}$
E. $12a - 34$
Answer: C
Problem: <image>Question: In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:
<image1>
Choices:
A. $AP^2 = PB \times AB$
B. $AP \times DO = PB \times AD$
C. $AB^2 = AD \times DE$
D. $AB \times AD = OB \times AO$
E. $\text{none of these}$
Answer: A
Problem: <image>Question: In this diagram $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which is inscribed equilateral triangle $DEF$. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. Then:
<image1>
Choices:
A. $b=\frac{a+c}{2}$
B. $b=\frac{a-c}{2}$
C. $a=\frac{b-c}{2}$
D. $a=\frac{b+c}{2}$
E. $\text{none of these}$
Answer: D
Problem: <image>Question: Chord $EF$ is the perpendicular bisector of chord $BC$, intersecting it in $M$. Between $B$ and $M$ point $U$ is taken, and $EU$ extended meets the circle in $A$. Then, for any selection of $U$, as described, triangle $EUM$ is similar to triangle:

<image1>
Choices:
A. EFA
B. EFC
C. ABM
D. ABU
E. FMC
Answer: A
Problem: <image>Question: Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of triangle $CEF$ is $200$ square inches. Then the number of inches in $BE$ is:

<image1>
Answer: 12
Problem: <image>Question: Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals:


<image1>
Answer: 16
Problem: <image>Question: In triangle $ABC$ lines $CE$ and $AD$ are drawn so that

$\frac{CD}{DB}=\frac{3}{1}$ and $\frac{AE}{EB}=\frac{3}{2}$. Let $r=\frac{CP}{PE}$

where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals:

<image1>
Choices:
A. $3$
B. $\frac{3}{2}$
C. $4$
D. $5$
E. $\frac{5}{2}$
Answer: D
Problem: <image>Question: In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\frac{1}{2}$ inches. The length of $RS$, in inches, is:

<image1>
Choices:
A. $\text{undetermined}$
B. $4$
C. $5\frac{1}{2}$
D. $6$
E. $6\frac{1}{4}$
Answer: E
Problem: <image>Question: $P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches.
<image1>
Then $PB$, in inches, equals:
Choices:
A. $2\sqrt{3}$
B. $3\sqrt{2}$
C. $3\sqrt{3}$
D. $4\sqrt{2}$
E. $2$
Answer: B
Problem: <image>Question: In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$. The circle is made to roll along the side $AB$, remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$, respectively. Let $n$ be the number of degrees in arc $MTN$.
<image1>
Then $n$, for all permissible positions of the circle:
Choices:
A. $\text{varies from }30^{\circ}\text{ to }90^{\circ}$
B. $\text{varies from }30^{\circ}\text{ to }60^{\circ}$
C. $\text{varies from }60^{\circ}\text{ to }90^{\circ}$
D. $\text{remains constant at }30^{\circ}$
E. $\text{remains constant at }60^{\circ}$
Answer: E
Problem: <image>Question: The magnitudes of the sides of triangle $ABC$ are $a$, $b$, and $c$, as shown, with $c\le b\le a$. Through interior point $P$ and the vertices $A$, $B$, $C$, lines are drawn meeting the opposite sides in $A'$, $B'$, $C'$, respectively.
<image1>
Let $s=AA'+BB'+CC'$. Then, for all positions of point $P$, $s$ is less than:
Choices:
A. 2a+b
B. 2a+c
C. 2b+c
D. a+2b
E. a+b+c
Answer: A
Problem: <image>Question: An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots,k,\cdots,n$, $n\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$)
<image1>

Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then $S$ equals:
Choices:
A. 180
B. 360
C. 180(n+2)
D. 180(n-2)
E. 180(n-4)
Answer: E
Problem: <image>Question: Triangle $ABC$ is inscribed in a circle with center $O'$.  A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$.
<image1>
 Then, we must have:
Choices:
A. $CD=BD=O'D$
B. $AO=CO=OD$
C. $CD=CO=BD  \\$
D. $CD=OD=BD$
E. $O'B=O'C=OD $
Answer: D
Problem: <image>Question: <image1>
In this figure $AB$ is a diameter of a circle, centered at $O$, with radius $a$.  A  chord $AD$ is drawn and extended to meet the tangent to the circle at $B$ in point $C$.  Point $E$ is taken on $AC$ so that $AE=DC$.  Denoting the distances of $E$ from the tangent through $A$ and from the diameter $AB$ by $x$ and $y$, respectively, we can deduce the relation:
Choices:
A. $y^2=\frac{x^3}{2a-x}$
B. $y^2=\frac{x^3}{2a+x}$
C. $y^4=\frac{x^2}{2-x} \\$
D. $x^2=\frac{y^2}{2a-x}$
E. $x^2=\frac{y^2}{2a+x}$
Answer: A
Problem: <image>Question: <image1>

In this diagram semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, so that they are mutually tangent.  If $\overline{CD} \bot \overline{AB}$, then the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius is:
Choices:
A. $1:2$
B. $1:3$
C. $\sqrt{3}:7$
D. $1:4$
E. $\sqrt{2}:6$
Answer: D