Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: A large square of side-length $10 \mathrm{~cm}$ contains a smaller square of side-length $4 \mathrm{~cm}$, as shown in the diagram. The corresponding sides of the two squares are parallel. What percentage of the area of the large square is shaded? <image1>
Choices:
A. $25 \%$
B. $30 \%$
C. $40 \%$
D. $42 \%$
E. $45 \%$
Answer: D
Problem: <image>Question: Mary had to run to catch the train, got off two stops later and then walked to school. Which of the following speed-time graphs would best represent her journey? <image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The diagram shows three squares of side-length $3 \mathrm{~cm}, 5 \mathrm{~cm}$ and $8 \mathrm{~cm}$. What is the area, in $\mathrm{cm}^{2}$, of the shaded trapezium? <image1>
Choices:
A. $13$
B. $\frac{55}{4}$
C. $\frac{61}{4}$
D. $\frac{65}{4}$
E. $\frac{69}{4}$
Answer: B
Problem: <image>Question: Points $M$ and $N$ are the midpoints of two sides of the rectangle, shown in the diagram. What fraction of the rectangle's area is shaded? <image1>
Choices:
A. $\frac{1}{6}$
B. $\frac{1}{5}$
C. $\frac{1}{4}$
D. $\frac{1}{3}$
E. $\frac{1}{2}$
Answer: C
Problem: <image>Question: The Pentagon $P Q R S T$ is divided into four triangles with equal perimeters. The triangle $P Q R$ is equilateral. $P T U, S U T$ and $R S U$ are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon $P Q R S T$ to the perimeter of the triangle $P Q R$? <image1>
Choices:
A. $2: 1$
B. $3: 2$
C. $4: 3$
D. $5: 3$
E. $5: 2$
Answer: D
Problem: <image>Question: On the table there is a tower made of blocks numbered from 1 to 90 , as shown on the left of the diagram. Yett takes blocks from the top of the tower, three at a time, to build a new tower, as shown on the right of the diagram. How many blocks will be between blocks 39 and 40 when he has finished building the new tower? <image1>
Answer: 4
Problem: <image>Question: A square of side-length $30 \mathrm{~cm}$ is divided into nine smaller identical squares. The large square contains three circles with radii $5 \mathrm{~cm}$ (bottom right), $4 \mathrm{~cm}$ (top left) and $3 \mathrm{~cm}$ (top right), as shown. What is the total area of the shaded part? <image1>
Choices:
A. $400 \mathrm{~cm}^{2}$
B. $500 \mathrm{~cm}^{2}$
C. $(400+50 \pi) \mathrm{cm}^{2}$
D. $(500-25 \pi) \mathrm{cm}^{2}$
E. $(500+25 \pi) \mathrm{cm}^{2}$
Answer: B
Problem: <image>Question: The figure shows two touching semicircles of radius 1 , with parallel diameters $P Q$ and $R S$. What is the square of the distance $P S$ ? <image1>
Choices:
A. $16
B. $8+4 \sqrt{3}$
C. $12$
D. $9$
E. $5+2 \sqrt{3}$
Answer: B
Problem: <image>Question: Seven different single-digit numbers are written in the circles of the diagram shown with one number in each circle. The product of the three numbers in each of the three lines of three numbers is the same. Which number is written in the circle containing the question mark? <image1>
Answer: 2
Problem: <image>Question: Lancelot has drawn a closed path on a cuboid and unfolded it into a net. Which of the nets shown could not be the net of Lancelot's cuboid? <image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: In how many different ways can the word BANANA be read from the following table by moving from one cell to another cell with which it shares an edge? Cells may be visited more than once. <image1>
Answer: 84
Problem: <image>Question: The diagram shows a map of a park. The park is divided into regions. The number inside each region gives its perimeter, in $\mathrm{km}$. What is the outer perimeter of the park? <image1>
Choices:
A. $22 \mathrm{~km}$
B. $26 \mathrm{~km}$
C. $28 \mathrm{~km}$
D. $32 \mathrm{~km}$
E. $34 \mathrm{~km}$
Answer: B
Problem: <image>Question: Vumos wants to write the integers 1 to 9 in the nine boxes shown so that the sum of the integers in any three adjacent boxes is a multiple of 3 . In how many ways can he do this? <image1>
Choices:
A. $6 \times 6 \times 6 \times 6$
B. $6 \times 6 \times 6$
C. $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
D. $6 \times 5 \times 4 \times 3 \times 2 \times 1$
E. $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Answer: A
Problem: <image>Question: The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown.
<image1>
What is the sum of the other three numbers he will write?
Answer: 11
Problem: <image>Question: The diagram shows two circles and a square with sides of length $10 \mathrm{~cm}$. One vertex of the square is at the centre of the large circle and two sides of the square are tangents to both circles. The small circle touches the large circle. The radius of the small circle is $(a-b \sqrt{2}) \mathrm{cm}$.
<image1>
What is the value of $a+b$ ?
Answer: 50
Problem: <image>Question: The diagram shows a triangle $A B C$ with area $12 \mathrm{~cm}^{2}$. The sides of the triangle are extended to points $P, Q, R, S, T$ and $U$ as shown so that $P A=A B=B S, Q A=A C=C T$ and $R B=B C=C U$.
<image1>
What is the area (in $\mathrm{cm}^{2}$ ) of hexagon $P Q R S T U$ ?
Answer: 156
Problem: <image>Question: A ball is propelled from corner $A$ of a square snooker table of side 2 metres. After bouncing off three cushions as shown, the ball goes into a pocket at $B$. The total distance travelled by the ball is $\sqrt{k}$ metres. What is the value of $k$ ?
<image1>
(Note that when the ball bounces off a cushion, the angle its path makes with the cushion as it approaches the point of impact is equal to the angle its path makes with the cushion as it moves away from the point of impact as shown in the diagram below.)
<image2>
Answer: 52
Problem: <image>Question: In rectangle $J K L M$, the bisector of angle $K J M$ cuts the diagonal $K M$ at point $N$ as shown. The distances between $N$ and sides $L M$ and $K L$ are $8 \mathrm{~cm}$ and $1 \mathrm{~cm}$ respectively. The length of $K L$ is $(a+\sqrt{b}) \mathrm{cm}$. What is the value of $a+b$ ?
<image1>
Answer: 16
Problem: <image>Question: In quadrilateral $A B C D, \angle A B C=\angle A D C=90^{\circ}, A D=D C$ and $A B+B C=20 \mathrm{~cm}$.
<image1>
What is the area in $\mathrm{cm}^{2}$ of quadrilateral $A B C D$ ?
Answer: 100
Problem: <image>Question: Using this picture we can observe that
$1+3+5+7=4 \times 4$.
What is the value of
$1+3+5+7+9+11+13+15+17+19+21$ ?
<image1>
Answer: 121
Problem: <image>Question: Both rows of the following grid have the same sum. What is the value of $*$ ?
<image1>
Answer: 950
Problem: <image>Question: <image1>
In the diagram, $P T$ and $P S$ are tangents to a circle with centre $O$. The point $Y$ lies on the circumference of the circle; and the point $Z$ is where the line $P Y$ meets the radius $O S$.
Also, $\angle S P Z=10^{\circ}$ and $\angle T O S=150^{\circ}$.
How many degrees are there in the sum of $\angle P T Y$ and $\angle P Y T$ ?
Answer: 160
Problem: <image>Question: The diagram shows two concentric circles. Chord $A B$ of the larger circle is tangential to the smaller circle.
The length of $A B$ is $32 \mathrm{~cm}$ and the area of the shaded region is $k \pi \mathrm{cm}^{2}$.
What is the value of $k$ ?
<image1>
Answer: 256
Problem: <image>Question: Delia is joining three vertices of a square to make four right-angled triangles.
She can create four triangles doing this, as shown.
<image1>
How many right-angled triangles can Delia make by joining three vertices of a regular polygon with 18 sides?
Answer: 144
Problem: <image>Question: The large equilateral triangle shown consists of 36 smaller equilateral triangles. Each of the smaller equilateral triangles has area $10 \mathrm{~cm}^{2}$.
The area of the shaded triangle is $K \mathrm{~cm}^{2}$. Find $K$.
<image1>
Answer: 110
Problem: <image>Question: A barcode of the type shown in the two examples is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either colour) has a width of 1 or 2 . The total width of the barcode is 12 . The barcodes are always read from left to right. How many distinct barcodes are possible?
<image1>
Answer: 116
Problem: <image>Question: An integer is to be written in each circle of the network shown. The integers must be written so that the sum of the numbers at the end of each line segment is the same. Two of the integers have already been written. What is the total of all the integers in the completed diagram?
<image1>
Answer: 132
Problem: <image>Question: The figure shows a shape consisting of a regular hexagon of side $18 \mathrm{~cm}$, six triangles and six squares. The outer perimeter of the shape is $P \mathrm{~cm}$. What is the value of $P$ ?
<image1>
Answer: 216
Problem: <image>Question: The figure shows a quadrilateral $A B C D$ in which $A D=D C$ and $\angle A D C=\angle A B C=90^{\circ}$. The point $E$ is the foot of the perpendicular from $D$ to $A B$. The length $D E$ is 25 . What is the area of quadrilateral $A B C D$ ?
<image1>
Answer: 625
Problem: <image>Question: Priti is learning a new language called Tedio. During her one hour lesson, which started at midday, she looks at the clock and notices that the hour hand and the minute hand make exactly the same angle with the vertical, as shown in the diagram. How many whole seconds remain until the end of the lesson?
<image1>
Answer: 276
Problem: <image>Question: Robin shoots three arrows at a target. He earns points for each shot as shown in the figure. However, if any of his arrows miss the target or if any two of his arrows hit adjacent regions of the target, he scores a total of zero. How many different scores can he obtain?
<image1>
Answer: 13
Problem: <image>Question: At each of the vertices of a cube sits a Bunchkin. Two Bunchkins are said to be adjacent if and only if they sit at either end of one of the cube's edges. Each Bunchkin is either a 'truther', who always tells the truth, or a 'liar', who always lies. All eight Bunchkins say 'I am adjacent to exactly two liars'. What is the maximum number of Bunchkins who are telling the truth?
<image1>
Answer: 4
Problem: <image>Question: The pattern shown in the diagram is constructed using semicircles. Each semicircle has a diameter that lies on the horizontal axis shown and has one of the black dots at either end. The distance between each pair of adjacent black dots is $1 \mathrm{~cm}$. The area, in $\mathrm{cm}^{2}$, of the pattern that is shaded in grey is $\frac{1}{8} k \pi$. What is the value of $k$ ?
<image1>
Answer: 121
Problem: <image>Question: Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.
<image1>
ACROSS
1. A square
3. The answer to this Kangaroo question
5. A square
DOWN
1. 4 down minus eleven
2. One less than a cube
4. The highest common factor of 1 down and 4 down is greater than one
Answer: 829
Problem: <image>Question: The diagram shows a semicircle with diameter $P Q$ inscribed in a rhombus $A B C D$. The rhombus is tangent to the arc of the semicircle in two places. Points $P$ and $Q$ lie on sides $B C$ and $C D$ of the rhombus respectively. The line of symmetry of the semicircle is coincident with the diagonal $A C$ of the rhombus. It is given that $\angle C B A=60^{\circ}$. The semicircle has radius 10 . The area of the rhombus can be written in the form $a \sqrt{b}$ where $a$ and $b$ are integers and $b$ is prime. What is the value of
<image1>
$a b+a+b ?$
Answer: 603
Problem: <image>Question: The line segments $P Q R S$ and $W X Y S$ intersect circle $C_{1}$ at points $P, Q, W$ and $X$.
<image1>
The line segments intersect circle $C_{2}$ at points $Q, R, X$ and $Y$. The lengths $Q R, R S$ and $X Y$ are 7, 9 and 18 respectively. The length $W X$ is six times the length $Y S$.
What is the sum of the lengths of $P S$ and $W S$ ?
Answer: 150
Problem: <image>Question: The diagram shows a 16 metre by 16 metre wall. Three grey squares are painted on the wall as shown.
<image1>
The two smaller grey squares are equal in size and each makes an angle of $45^{\circ}$ with the edge of the wall. The grey squares cover a total area of $B$ metres squared.
What is the value of $B$ ?
Answer: 128
Problem: <image>Question: Identical regular pentagons are arranged in a ring. The partially completed ring is shown in the diagram. Each of the regular pentagons has a perimeter of 65 . The regular polygon formed as the inner boundary of the ring has a perimeter of $P$. What is the value of $P$ ?
<image1>
Answer: 130
Problem: <image>Question: The function $J(x)$ is defined by:
$$
J(x)= \begin{cases}4+x & \text { for } x \leq-2 \\ -x & \text { for }-2<x \leq 0 \\ x & \text { for } x>0\end{cases}
$$
<image1>
How many distinct real solutions has the equation $J(J(J(x)))=0$ ?
Answer: 4
Problem: <image>Question: In the triangle $A B C$ the points $M$ and $N$ lie on the side $A B$ such that $A N=A C$ and $B M=B C$.
We know that $\angle M C N=43^{\circ}$.
Find the size in degrees of $\angle A C B$.
<image1>
Answer: 94
Problem: <image>Question: Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct. What is the answer to 3 ACROSS?
<image1>
\section*{ACROSS}
1. A composite factor of 1001
3. Not a palindrome
5. $p q^{3}$ where $p, q$ prime and $p \neq q$
\section*{DOWN}
1. One more than a prime, one less than a prime
2. A multiple of 9
4. $p^{3} q$ using the same $p, q$ as 5 ACROSS
Answer: 295
Problem: <image>Question: Two identical cylindrical sheets are cut open along the dotted lines and glued together to form one bigger cylindrical sheet, as shown. The smaller sheets each enclose a volume of 100. What volume is enclosed by the larger
<image1>
Answer: 400
Problem: <image>Question: Margot writes the numbers $1,2,3,4,5,6,7$ and 8 in the top row of a table, as shown. In the second row she plans to write the same set of numbers, in any order.
Each number in the third row is obtained by finding the sum of the two numbers above it.
<image1>
In how many different ways can Margot complete row 2 so that every entry in row 3 is even?
Answer: 576
Problem: <image>Question: A square fits snugly between the horizontal line and two touching circles of radius 1000, as shown. The line is tangent to the circles.
What is the side-length of the square?
<image1>
Answer: 400
Problem: <image>Question: Each square in this cross-number can be filled with a non-zero digit such that all of the conditions in the clues are fulfilled. The digits used are not necessarily distinct.
What is the answer to 3 ACROSS?
<image1>
\section*{ACROSS}
1. A multiple of 7
3. The answer to this Question
5. More than 10
\section*{DOWN}
1. A multiple of a square of an odd prime; neither a square nor a cube
2. The internal angle of a regular polygon; the exterior angle is between $10^{\circ}$ and $20^{\circ}$
4. A proper factor of $5 \mathrm{ACROSS}$ but not a proper factor of $1 \mathrm{DOWN}$
Answer: 961
Problem: <image>Question: How many of the figures shown can be drawn with one continuous line without drawing a segment twice?
<image1>
Answer: 3
Problem: <image>Question: Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
<image1>
On the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively.
We know that $c=b e, a+b=d$ and $e-d=a$.
Frankie picks up the card which has the largest integer on its reverse. What number is on the front of Frankie's card?
Answer: 103
Problem: <image>Question: In the figure shown there are three concentric circles and two perpendicular diameters. The three shaded regions have equal area. The radius of the small circle is 2 . The product of the three radii is $Y$.
What is the value of $Y^{2}$ ?
<image1>
Answer: 384
Problem: <image>Question: Five cards have the numbers $101,102,103,104$ and 105 on their fronts.
<image1>
On the reverse, each card has one of five different positive integers: $a, b, c, d$ and $e$ respectively. We know that $a+2=b-2=2 c=\frac{d}{2}=e^{2}$.
Gina picks up the card which has the largest integer on its reverse. What number is on the front of Gina's card?
Answer: 104
Problem: <image>Question: The perimeter of the square in the figure is 40 . The perimeter of the larger equilateral triangle in the figure is $a+b \sqrt{p}$, where $p$ is a prime number. What is the value of $7 a+5 b+3 p$ ?
<image1>
Answer: 269
Problem: <image>Question: A particular flag is in the shape of a rectangle divided into five smaller congruent rectangles as shown. When written in its lowest terms, the ratio of the side lengths of the smaller rectangle is $\lambda: 1$, where $\lambda<1$. What is the value of $360 \lambda$ ? <image1>
Answer: 120
Problem: <image>Question: Five cards have the numbers $101,102,103,104$ and 105 on their fronts. <image1>
On the reverse, each card has a statement printed as follows:
101: The statement on card 102 is false
102: Exactly two of these cards have true statements
103: Four of these cards have false statements
104: The statement on card 101 is false
105: The statements on cards 102 and 104 are both false
What is the total of the numbers shown on the front of the cards with TRUE statements?
Answer: 206
Problem: <image>Question: The smallest four two-digit primes are written in different squares of a $2 \times 2$ table.

The sums of the numbers in each row and column are calculated.

Two of these sums are 24 and 28.

The other two sums are $c$ and $d$, where $c<d$.

What is the value of $5 c+7 d$ ?

<image1>
Answer: 412
Problem: <image>Question: Stephen's calculator displays only one digit, as shown in the diagram. Unfortunately, the calculator is broken. Each time he switches it on, each of the seven bars will either illuminate (show up) or not, with probability 0.5 . The resultant display correctly shows one of the ten digits $0-9$ with probability $\frac{a}{b}$.

Given that $\frac{a}{b}$ is written in its lowest terms, what is the value of $9 a+2 b$ ?

<image1>
Answer: 173
Problem: <image>Question: Each cell in this cross-number can be filled with a non-zero digit so that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.
<image1>
\section*{ACROSS}
1. Four less than a factor of 105.
3. One more than a palindrome.
5. The square-root of the answer to this Kangaroo question.
\section*{DOWN}
1. Two less than a square.
2. Four hundred less than a cube.
4. Six less than the sum of the answers to two of the other clues.
What is the square of the answer to 5 ACROSS?
Answer: 841
Problem: <image>Question: A machine-shop cutting tool has the shape of a notched circle, as shown.  The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm.  The angle $ABC$ is a right angle.  Find the square of the distance (in centimeters) from $B$ to the center of the circle.
<image1>
Answer: 26
Problem: <image>Question: The solid shown has a square base of side length $s$.  The upper edge is parallel to the base and has length $2s$.  All other edges have length $s$.  Given that $s = 6 \sqrt{2}$, what is the volume of the solid?
<image1>
Answer: 288
Problem: <image>Question: In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart.  At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length.  Find the square of the length of $QP$.

<image1>
Answer: 130
Problem: <image>Question: The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$.  Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$.  Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$.  It follows that the sine of the minor arc $AB$ is a rational number.  If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$?
<image1>
Answer: 175
Problem: <image>Question: A point $P$ is chosen in the interior of $\triangle ABC$ so that when lines are drawn through $P$ parallel to the sides of $\triangle ABC$, the resulting smaller triangles, $t_1$, $t_2$, and $t_3$ in the figure, have areas 4, 9, and 49, respectively. Find the area of $\triangle ABC$.
<image1>
Answer: 144
Problem: <image>Question: A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices.  Find the value of $n$ if the the area of the small square is exactly 1/1985.
<image1>
Answer: 32
Problem: <image>Question: As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point.  The areas of four of these triangles are as indicated.  Find the area of triangle $ABC$.

<image1>
Answer: 315
Problem: <image>Question: Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides.  These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron.  What is the volume (in $\text{cm}^3$) of this polyhedron?

<image1>
Answer: 864
Problem: <image>Question: Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$.  Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm.

<image1>
Answer: 193
Problem: <image>Question: Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$.  Find $PC$.

<image1>
Answer: 33
Problem: <image>Question: Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below.  Find $AC + CB$ if area$(S_1) = 441$ and area$(S_2) = 440$.

<image1>
Answer: 462
Problem: <image>Question: One commercially available ten-button lock may be opened by depressing -- in any order -- the correct five buttons.  The sample shown below has $\{1, 2, 3, 6, 9\}$ as its combination.  Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations.  How many additional combinations would this allow?

<image1>
Answer: 770
Problem: <image>Question: It is possible to place positive integers into the vacant twenty-one squares of the $5 \times 5$ square shown below so that the numbers in each row and column form arithmetic sequences.  Find the number that must occupy the vacant square marked by the asterisk (*).

<image1>
Answer: 142
Problem: <image>Question: Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides.  Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure.  Find the product $abc$ if $a + b + c = 43$ and $d = 3$.

<image1>
Answer: 441
Problem: <image>Question: Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $B$ at a speed of $7$ meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?

<image1>
Answer: 160
Problem: <image>Question: Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.

<image1>
Answer: 137