Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: A piece of string is lying on the table. It is partially covered by three coins as seen in the figure. Under each coin the string is equally likely to pass over itself like this: <image1>
or like this: <image2>. What is the probability that the string is knotted after its ends are pulled?
<image3>
Choices:
A. $\frac{1}{2}$
B. $\frac{1}{4}$
C. $\frac{1}{8}$
D. $\frac{3}{4}$
E. $\frac{3}{8}$
Answer: B
Problem: <image>Question: A naughty pup grabs the end of a roll of toilet paper and walks away at a constant speed. Which of the functions below best describes the thickness $y$ of the roll as a function of the unrolled part $x$?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The diagram shows three squares, $P Q R S, T U V R$ and $U W X Y$. They are placed together, edge to edge. Points $P, T$ and $X$ lie on the same straight line. The area of $P Q R S$ is 36 and the area of TUVR is 16. What is the area of triangle PXV?
<image1>
Answer: 16
Problem: <image>Question: The figure shows the graph of a function $f:[-5,5] \rightarrow R$. How many distinct solutions does the equation $f(f(x))=0$ have?
<image1>
Answer: 8
Problem: <image>Question: Five kangaroos named A, B, C, D and E have one child each, named a, b, c, d and e, not necessarily in that order. In the first group photo shown exactly 2 of the children are standing next to their mothers. In the second group photo exactly 3 of the children are standing next to their mothers. Whose child is a?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: The solid shown in the diagram has 12 regular pentagonal faces, the other faces being either equilateral triangles or squares. Each pentagonal face is surrounded by 5 square faces and each triangular face is surrounded by 3 square faces. John writes 1 on each triangular face, 5 on each pentagonal face and -1 on each square. What is the total of the numbers written on the solid?
<image1>
Answer: 50
Problem: <image>Question: On a circle 15 points are equally spaced. We can form triangles by joining any 3 of these. Congruent triangles, by rotation or reflection, are counted as only one triangle. How many different triangles can be drawn?
<image1>
Answer: 19
Problem: <image>Question: A triangle $A B C$ is divided into four parts by two straight lines, as shown. The areas of the smaller triangles are 1,3 and 3. What is the area of the original triangle?
<image1>
Answer: 12
Problem: <image>Question: Two plane mirrors $O P$ and $O Q$ are inclined at an acute angle (diagram is not to scale). A ray of light $X Y$ parallel to $O Q$ strikes mirror $O P$ at $Y$. The ray is reflected and hits mirror $O Q$, is reflected again and hits mirror $O P$ and is reflected for a third time and strikes mirror $O Q$ at right angles at $R$ as shown. If $O R=5 \mathrm{~cm}$, what is the distance $d$ of the ray $X Y$ from the mirror $O Q$?
<image1>
Choices:
A. $4 \mathrm{~cm}$
B. $4.5 \mathrm{~cm}$
C. $5 \mathrm{~cm}$
D. $5.5 \mathrm{~cm}$
E. $6 \mathrm{~cm}$
Answer: C
Problem: <image>Question: Martin's smartphone displays the diagram on the right. It shows how long he has worked with four different apps in the previous week. The apps are sorted from top to bottom according to the amount of time they have been used. This week he has spent only half the amount of time using two of the apps and the same amount of time as last week using the other two apps. Which of the following pictures cannot be the diagram for the current week?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: Four circles with radius 1 intersect each other as seen in the diagram. What is the perimeter of the grey area?
<image1>
Choices:
A. $\pi$
B. $\frac{3 \pi}{2}$
C. a number between $\frac{3 \pi}{2}$ and $2 \pi$
D. $2 \pi$
E. $\pi^{2}$
Answer: D
Problem: <image>Question: The points $A, B, C$ and $D$ are marked on a straight line in this order as shown in the diagram. We know that $A$ is $12 \mathrm{~cm}$ from $C$ and that $B$ is $18 \mathrm{~cm}$ from $D$. How far apart from each other are the midpoints of the line segments $A B$ and $C D$?
<image1>
Choices:
A. $6 \mathrm{~cm}$
B. $9 \mathrm{~cm}$
C. $12 \mathrm{~cm}$
D. $13 \mathrm{~cm}$
E. $15 \mathrm{~cm}$
Answer: E
Problem: <image>Question: Four straight lines that intersect in one single point form eight equal angles (see diagram). Which one of the black arcs has the same length as the circumference of the little (grey) circle?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: We check the water meter and see that all digits on the display are different. What is the minimum amount of water that has to be used before this happens again?
<image1>
Choices:
A. $0.006 \mathrm{~m}^{3}$
B. $0.034 \mathrm{~m}^{3}$
C. $0.086 \mathrm{~m}^{3}$
D. $0.137 \mathrm{~m}^{3}$
E. $1.048 \mathrm{~m}^{3}$
Answer: D
Problem: <image>Question: The square pictured, is split into two squares and two rectangles. The vertices of the shaded quadrilateral with area 3 are the midpoints of the sides of the smaller squares. What is the area of the non-shaded part of the big square?
<image1>
Answer: 21
Problem: <image>Question: The diagram shows a map with 16 towns which are connected via roads. The government is planning to build power plants in some towns. Each power plant can generate enough electricity for the town in which it stands as well as for its immediate neighbouring towns (i.e. towns that can be reached via a direct connecting road). What is the minimum number of power plants that have to be built?
<image1>
Answer: 4
Problem: <image>Question: A cuboid with surface area $X$ is cut up along six planes parallel to the sides (see diagram). What is the total surface area of all 27 thus created solids?
<image1>
Choices:
A. $X$
B. $2 X$
C. $3 X$
D. $4 X$
E. It depends on the position of the planes.
Answer: C
Problem: <image>Question: A circle with midpoint $(0 \mid 0)$ has a radius of 5. How many points are there on the circumference where both co-ordinates are integers?
<image1>
Answer: 12
Problem: <image>Question: Which two building blocks can be joined together so that the object shown is created?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: A rectangle is split into 11 smaller rectangles as shown. All 11 small rectangles are similar to the initial rectangle. The smallest rectangles are aligned like the original rectangle (see diagram). The lower sides of the smallest rectangles have length 1. How big is the perimeter of the big rectangle?
<image1>
Answer: 30
Problem: <image>Question: Two rectangles are inscribed into a triangle as shown in the diagram. The dimensions of the rectangles are $1 \times 5$ and $2 \times 3$ respectively. How big is the height of the triangle in $A$?
<image1>
Choices:
A. 3
B. $\frac{7}{2}$
C. $\frac{8}{3}$
D. $\frac{6}{5}$
E. another number
Answer: B
Problem: <image>Question: The numbers 1 to 10 were written into the ten circles in the pattern shown in the picture. The sum of the four numbers in the left and the right column is 24 each and the sum of the three numbers in the bottom row is 25. Which number is in the circle with the question mark?
<image1>
Choices:
A. 2
B. 4
C. 5
D. 6
E. another number
Answer: E
Problem: <image>Question: A square is placed in a co-ordinate system as shown. Each point $(x \mid y)$ of the square is deleted and replaced by the point $\left(\frac{1}{x} \mid \frac{1}{y}\right)$. Which diagram shows the resulting shape?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: Two circles intersect a rectangle AFMG as shown in the diagram. The line segments along the long side of the rectangle that are outside the circles have length $A B=8, C D=26, E F=22, G H=12$ and $J K=24$. How long is the length $x$ of the line segment $L M$?
<image1>
Answer: 16
Problem: <image>Question: A cylindrical tin is $15 \mathrm{~cm}$ high. The circumference of the base circle is $30 \mathrm{~cm}$. An ant walks from point $A$ at the base to point $B$ at the top. Its path is partly vertically upwards and partly along horizontal circular arcs. Its path is drawn in bold on the diagram (with a solid line on the front and a dashed line at the back). How long is the total distance covered by the ant? <image1>
Choices:
A. $45 \mathrm{~cm}$
B. $55 \mathrm{~cm}$
C. $60 \mathrm{~cm}$
D. $65 \mathrm{~cm}$
E. $75 \mathrm{~cm}$
Answer: E
Problem: <image>Question: Emma should colour in the three strips of the flag shown. She has four colours available. She can only use one colour for each strip and immediately adjacent strips are not to be of the same colour. How many different ways are there for her to colour in the flag? <image1>
Answer: 36
Problem: <image>Question: What is the value of the following sum? <image1>
Answer: 12
Problem: <image>Question: A square with area 84 is split into four squares. The upper left square is coloured in black. The lower right square is again split into four squares and so on. The process is repeated infinitely many times. How big is the area coloured in black? <image1>
Answer: 28
Problem: <image>Question: The numbers from 1 to 9 are to be distributed to the nine squares in the diagram according to the following rules: There is to be one number in each square. The sum of three adjacent numbers is always a multiple of 3 . The numbers 7 and 9 are already written in. How many ways are there to insert the remaining numbers? <image1>
Answer: 24
Problem: <image>Question: Two equilateral triangles of different sizes are placed on top of each other so that a hexagon is formed on the inside whose opposite sides are parallel. Four of the side lengths of the hexagon are stated in the diagram. How big is the perimeter of the hexagon? <image1>
Answer: 70
Problem: <image>Question: In a three-sided pyramid all side lengths are integers. Four of the side lengths can be seen in the diagram. What is the sum of the two remaining side lengths? <image1>
Answer: 11
Problem: <image>Question: Leon has drawn a closed path on the surface of a cuboid. Which net can represent his path?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: A pentagon is cut into smaller parts as shown in the diagram. The numbers in the triangles state the area of the according triangle. How big is the area $P$ of the grey quadrilateral? <image1>
Answer: 16
Problem: <image>Question: The diagram shows a spiral of consecutive numbers starting with 1. In which order will the numbers 625, 626 and 627 appear in the spiral? <image1>
Choices:
A. $625 \uparrow 626 \uparrow 627$
B. $625 \uparrow 626 \rightarrow$
C. $625 \rightarrow 626 \rightarrow 627$
D. $625 \rightarrow 626 \uparrow 627$
E. $625 \downarrow 626 \downarrow 627$
Answer: B
Problem: <image>Question: A game marker in the shape of a regular tetrahedron has one marked area. That side is placed on the triangle marked START. The marker is then moved within the diagram always to the next adjacent triangle by rolling it around an edge. On which triangle is the marker when it is on the marked side again for the first time? <image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: A part of a polynomial of degree five is illegible due to an ink stain. It is known that all zeros of the polynomial are integers. What is the highest power of $x-1$ that divides this polynomial? <image1>
Choices:
A. $(x-1)^{1}$
B. $(x-1)^{2}$
C. $(x-1)^{3}$
D. $(x-1)^{4}$
E. $(x-1)^{5}$
Answer: D
Problem: <image>Question: The big square shown is split into four small squares. The circle touches the right side of the square in its midpoint. How big is the side length of the big square? (Hint: The diagram is not drawn to scale.) <image1>
Choices:
A. $18 \mathrm{~cm}$
B. $20 \mathrm{~cm}$
C. $24 \mathrm{~cm}$
D. $28 \mathrm{~cm}$
E. $30 \mathrm{~cm}$
Answer: A
Problem: <image>Question: The numbers from 1 to 11 are written in the empty hexagons. The sums of the three numbers in three hexagons with a common bold point are always equal. Three of the eleven numbers are already written in (see diagram). Which number is written in the hexagon with the question mark? <image1>
Answer: 9
Problem: <image>Question: Two identical cylindrical glasses contain the same amount of water. The left glass is upright, while the right one rests against the other one at a slant. The water level in both glasses is at the same height. The water level in the leaning glass touches its bottom in exactly one point (see diagram). The bases of both glasses have an area of $3 \pi \mathrm{cm}^{2}$. How much water is in each glass? <image1>
Choices:
A. $9 \pi \mathrm{cm}^{3}$
B. $6 \pi \mathrm{cm}^{3}$
C. $3 \sqrt{3} \pi \mathrm{cm}^{3}$
D. $\frac{3 \pi}{4} \mathrm{~cm}^{3}$
E. It cannot be uniquely determined from this information.
Answer: A
Problem: <image>Question: There are 10 boxes in the first van. Every further van contains twice as many boxes as the previous one. How many boxes are there in the fifth van?
<image1>
Answer: 160
Problem: <image>Question: In the picture the distance $KM=10, LN=15, KN=22$. Find the distance $LM$.
<image1>
Answer: 3
Problem: <image>Question: Chris constructed the brick on the picture using red and blue cubes of the same size. The outside of the brick is completely red, but all cubes used inside are blue. How many blue cubes did Chris use?
<image1>
Answer: 12
Problem: <image>Question: This table shows the quantity of different types of flowers in the botanical garden. Ted was told by the gardener that there were 35 azaleas, 50 irises and 85 roses in the garden. What is the number of gerberas growing in the garden?
<image1>
Answer: 110
Problem: <image>Question: The construction in the picture is built of cubes of the same size and weighs 189 grams. How many grams does one cube weigh?
<image1>
Answer: 21
Problem: <image>Question: If the length of the side of a little square is 1, what is the area of the letter N?
<image1>
Answer: 18
Problem: <image>Question: I surrounded the wooden circle (see picture) using $a \mathrm{~cm}$ of thread. After that I surrounded by thread the wooden square –– $b \mathrm{~cm}$ of thread was enough for that. How much thread (in $\mathrm{cm}$ ) would be enough to surround the three wooden circles without moving them?
<image1>
Choices:
A. 3a
B. 2a+b
C. a+2b
D. 3b
E. a+b
Answer: E
Problem: <image>Question: The picture on the right has been drawn on paper and cut out to make a house. Which of the houses does it make?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: A bar-code is formed by 17 alternating black and white bars (the first and the last bars are black). The black bars are of two types: wide and narrow. The number of white bars is greater by 3 than the number of wide black bars. Then the number of narrow black bars is
<image1>
Answer: 4
Problem: <image>Question: In the picture below you can see a road from town $M$ to town $N$ (a solid line) and a detour (a dashed line) of segment $K L$, which is under repair. How many more kilometers does one have to travel from $M$ to $N$ using the detour?
<image1>
Answer: 6
Problem: <image>Question: Which numbers are written in the area that belongs to the rectangle and to the circle but doesn't belong to the triangle?
<image1>
Choices:
A. 5 and 11
B. 1 and 10
C. 13
D. 3 and 9
E. 6, 7, and 4
Answer: B
Problem: <image>Question: How many white squares must you paint grey so that the number of grey squares is exactly half that of the white squares?
<image1>
Answer: 3
Problem: <image>Question: Which of the rectangles $\mathbf{A}$ to $\mathbf{E}$ can be covered by the pattern on the right-hand side in such a way that the result is a totally black rectangle?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: In this picture there is what I saw on four different clocks at the same time. Only one of them had the right time. One was 20 minutes fast. Another 20 minutes slow. One had stopped some time ago.
<image1>
What was the right time?
Choices:
A. 4:45
B. 5:05
C. 5:25
D. 5:40
E. 12:00
Answer: B
Problem: <image>Question: A cube (on the right) is colored in three colors so that each face has exactly one color and the opposite face has the same color. Which of the following developments is the development of this cube?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: This figure is made of squares. What is the side of the biggest square?
<image1>
Answer: 64
Problem: <image>Question: There are five houses on Color Street: a blue, a red, a yellow, a pink, and a green one. The houses are numbered from 1 to 5 (see picture). The red house is the neighbor of the blue house only. The blue house stands between the green and red houses.
<image1>
Which color is the house with number 3?
Choices:
A. Blue
B. Red
C. Yellow
D. Pink
E. Green
Answer: E
Problem: <image>Question: A large cube consists of 125 small black and white cubes, such that any two adjacent faces of the small cubes have different colors, the corner cubes being black. How many small black cubes are used?
<image1>
Answer: 63
Problem: <image>Question: This is a multiplication table. Which two letters represent the same number?
<image1>
Choices:
A. $L$ and $M$
B. $P$ and $N$
C. $R$ and $S$
D. $K$ and $R$
E. $M$ and $T$
Answer: E
Problem: <image>Question: A butterfly sat down on a correctly solved exercise. What number is the butterfly covering?
<image1>
Answer: 500
Problem: <image>Question: At noon the minute hand of a clock is in the position shown on the left and after the quarter of an hour -- in the position shown on the right. Which position the minute hand will take after seventeen quarters from the noon?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: In the diagram every of the eight kangaroos can jump to any empty square. What is the least number of kangaroos that must jump so that each row and each column have exactly two kangaroos?
<image1>
Answer: 1
Problem: <image>Question: John has a chocolate tablet consisting of square pieces of $1 \mathrm{~cm} \times 1 \mathrm{~cm}$. He has eaten already some pieces in a corner (see the picture). How many pieces John still have?
<image1>
Answer: 60
Problem: <image>Question: A square piece of paper has been cut in three pieces. Two of them are in the picture on the right. What is the third one?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: A frame of a rectangular picture is made from planks of equal width. What is the width of these planks (in centimetres) if the outside perimeter of the frame is $8 \mathrm{~cm}$ more than the inside perimeter?
<image1>
Answer: 1
Problem: <image>Question: You can make only one rectangle with the perimeter consisting of 6 matches (see the picture). How many different rectangles with the perimeter consisting of 14 matches can you compose?
<image1>
Answer: 3
Problem: <image>Question: Two traffic signs mark the bridge in my village. These marks indicate the maximum width and the maximum possible weight. Which one of the following trucks is allowed to cross that bridge?
<image1>
Choices:
A. The one $315 \mathrm{~cm}$ wide and weighing $4307 \mathrm{~kg}$
B. The one $330 \mathrm{~cm}$ wide and weighing $4250 \mathrm{~kg}$
C. The one $325 \mathrm{~cm}$ wide and weighing $4400 \mathrm{~kg}$
D. The one $322 \mathrm{~cm}$ wide and weighing $4298 \mathrm{~kg}$
E. No one of these
Answer: D
Problem: <image>Question: The figure shows a rectangular garden with dimensions $16 \mathrm{~m}$ and $20 \mathrm{~m}$. The gardener has planted six identical flowerbeds (they are gray in the diagram). What is the perimeter (in metres) of each of the flowerbeds?
<image1>
Answer: 24
Problem: <image>Question: Five cards are lying on the table in the order 5, 1, 4, 3, 2. You must get the cards in the order 1, 2, 3, 4, 5. Per move, any two cards may be interchanged. How many moves do you need at least?
<image1>
Answer: 3
Problem: <image>Question: Which of the following cubes has been folded out of the plan on the right?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: Betty keeps drawing three different figures in the same order. Which figure should be the next?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: How many cubes have been taken from the block?
<image1>
Answer: 7
Problem: <image>Question: A kangaroo enters a building. He only passes through triangular rooms. Where does he leave the building?
<image1>
Choices:
A. a
B. b
C. c
D. d
E. e
Answer: E
Problem: <image>Question: From which rectangular can you cut the figure shown on the right side out?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: Numbers in the picture are ticket prices between neighbouring towns. Peter wants to go from $A$ to $B$ as cheaply as possible. What is the lowest price he has to pay?
<image1>
Answer: 90
Problem: <image>Question: Six numbers are written on the following cards, as shown.
<image1>
What is the smallest number you can form with the given cards?
Answer: 2309415687
Problem: <image>Question: Between two points four routes are drawn. Which route is the shortest?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: In the picture you can see a number flower. Mary pulled out all the leaves with numbers which give remainder 2 when divided by 6. What is the sum of the numbers on the leaves that Mary pulled out?
<image1>
Answer: 46
Problem: <image>Question: You can move or rotate each shape as you like, but you are not allowed to flip them over. What shape is not used in the puzzle?
<image1>
<image2>
Answer: C
Problem: <image>Question: John is building houses of cards. On the picture there are houses of one, two, and three layers that John built. How many cards does he need to build a 4-layer house?
<image1>
Answer: 26
Problem: <image>Question: The structure shown in the picture is glued together from 10 cubes. Roman painted the entire structure, including the bottom. How many faces of the cubes are painted?
<image1>
Answer: 36
Problem: <image>Question: In each of the nine cells of the square we will write down one of the digits 1 , 2 or 3. We will do this in such a way that in each horizontal row and vertical column each of the digits 1, 2 and 3 will be written. In the upper left cell we will start with 1. How many different squares can we then make?
<image1>
Answer: 4
Problem: <image>Question: A child's toy hangs from the ceiling and it is in balance at all places. The same shapes have the same weight. The weight of a circle is 30 grams. What is the weight of a square?
<image1>
Answer: 20
Problem: <image>Question: Zita walks from left to right and puts the numbers into her basket. Which of the following numbers can be in her basket?
<image1>
Choices:
A. 1, 2 and 4
B. 2, 3 and 4
C. 2, 3 and 5
D. 1, 5 and 6
E. 1, 2 and 5
Answer: C
Problem: <image>Question: In which figure can you find the largest number of small squares?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: What is the piece that fits completely to the given one to form a rectangle?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: Which number has to be put into the dark cloud to have all the given calculations right?
<image1>
Answer: 5
Problem: <image>Question: Daniela has got cubes with their edges $1 \mathrm{dm}$ long. She has put some of them into the aquarium of the shape of a cube with the edges $3 \mathrm{dm}$ long as you see in the picture. How much more cubes can she put into the aquarium?
<image1>
Answer: 17
Problem: <image>Question: A cube with the edge $3 \mathrm{~cm}$ long is painted grey and cut into smaller cubes each with an edge of $1 \mathrm{~cm}$ long. How many smaller cubes will have exactly 2 faces painted?
<image1>
Answer: 12
Problem: <image>Question: We count the number of white cells. How many white cells has the next square?
<image1>
8 white cells
<image2>
21 white cells
<image3>
40 white cells
Answer: 65