Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: We used metal rods to build this nice ensemble. We know there are 61 octagons. How many rods are there?
<image1>
Answer: 446
Problem: <image>Question: The suare in the diagram has side length 1. The radius of the small circle would
then be of the length
<image1>
Choices:
A. $\sqrt{2}-1$
B. $\frac{1}{4}$
C. $\frac{\sqrt{2}}{4}$
D. $1-\frac{\sqrt{2}}{2}$
E. $(\sqrt{2}-1)^{2}$
Answer: E
Problem: <image>Question: Each side of a triangle $A B C$ is being extended to the points $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}, \mathrm{T}$ and $\mathrm{U}$, so that $\mathrm{PA}=\mathrm{AB}=\mathrm{BS}, \mathrm{TC}=\mathrm{CA}$ $=\mathrm{AQ}$ and $\mathrm{UC}=\mathrm{CB}=\mathrm{BR}$. The area of $\mathrm{ABC}$ is 1. How big is the area of the hexagon PQRSTU?
<image1>
Answer: 13
Problem: <image>Question: In the diagram on the right we want to colour the fields with the colours A, B, C and D so that adjacent fields are always in different colours. (Even fields that share only one corner, count as adjacent.) Some fields have already been coloured in. In which colour can the grey field be coloured in?
<image1>
Choices:
A. either A or B
B. only C
C. only D
D. either C or D
E. A, B, C or D
Answer: D
Problem: <image>Question: A (very small) ball is kicked off from point A on a square billiard table with side length $2 \mathrm{~m}$. After moving along the shown path and touching the sides three times as indicated, the path ends in point $B$. How long is the path that the bal travels from A to B? (As indicated on the right: incident angle = emergent angle.)
<image1>
Choices:
A. 7
B. $2 \sqrt{13}$
C. 8
D. $4 \sqrt{3}$
E. $2 \cdot(\sqrt{2}+\sqrt{3})$
Answer: B
Problem: <image>Question: In the diagram to the right a $2 \times 2 \times 2$ cube is made up of four transparent $1 \times 1 \times 1$ cubes and four non-transparent black $1 \times 1 \times 1$ cubes. They are placed in a way so that the entire big cube is nontransparent; i.e. looking at it from the front to the back, the right to the left, the top to the bottom, at no point you can look through the cube. What is the minimum number of black $1 \times 1 \times 1$ cubes needed to make a $3 \times 3 \times 3$ cube non-transparent in the same way?
<image1>
Answer: 9
Problem: <image>Question: An equilateral triangle with side length 3 and a circle with radius 1 have the same centre. What is the perimeter of the figure that is created when the two are being put together?
<image1>
Choices:
A. $6+\pi$
B. $3+2 \pi$
C. $9+\frac{\pi}{3}$
D. $3 \pi$
E. $9+\pi$
Answer: A
Problem: <image>Question: The adjacent diagram illustrates the graphs of the two functions f and g. How can we describe the relationship between f and g?
<image1>
Choices:
A. $g(x-2)=-f(x)$
B. $g(x)=f(x+2)$
C. $g(x)=-f(-x+2)$
D. $g(-x)=-f(-x-2)$
E. $g(2-x)=-f(x)$
Answer: A
Problem: <image>Question: In the diagram on the right we see the birdô-eye view and front elevation of a solid that is defined by flat surfaces (i.e. view from obove and the front respectively). Bird' s-Eye View (view from above): <image1>. Front Elevation (view from the front): <image2>.
Which of the outlines I to IV can be the side elevation (i.e. view from the left) of the same object?
<image3>
Choices:
A. I
B. II
C. III
D. IV
E. none of them
Answer: D
Problem: <image>Question: The sum of the number in each line, column and diagonal in the Ămagic squareñon the right is always constant. Only two numbers are visible. Which number is missing in field $a$?
<image1>
Choices:
A. 16
B. 51
C. 54
D. 55
E. 110
Answer: D
Problem: <image>Question: In a rectangle JKLM the angle bisector in $\mathrm{J}$ intersects the diagonal KM in $\mathrm{N}$. The distance of $\mathrm{N}$ to $\mathrm{LM}$ is 1 and the distance of $\mathrm{N}$ to $\mathrm{KL}$ is 8. How long is LM?
<image1>
Choices:
A. $8+2 \sqrt{2}$
B. $11-\sqrt{2}$
C. 10
D. $8+3 \sqrt{2}$
E. $11+\frac{\sqrt{2}}{2}$
Answer: A
Problem: <image>Question: In the box are seven blockss. You want to rearrange the blocks so that another block can placed. What is the minimum number of blocks that have to be moved?
<image1>
Answer: 3
Problem: <image>Question: The triangle pictured is right-angled. $M$ is the midoint of the hypotenuse $\mathrm{AB}$ and $\angle \mathrm{BCA}=90^{\circ}$. How big is $\angle \mathrm{BMC}$?
<image1>
Choices:
A. $105^{\circ}$
B. $108^{\circ}$
C. $110^{\circ}$
D. $120^{\circ}$
E. $125^{\circ}$
Answer: D
Problem: <image>Question: In the figure the square has side length 2. The semi-circles pass through the midpoint of the square and have their centres on the corners of the square. The grey circles have their centres on the sides of the square and touch the semi-circles. How big is the total area of the grey parts?
<image1>
Choices:
A. $4 \cdot(3-2 \sqrt{2}) \cdot \pi$
B. $\sqrt{2} \cdot \pi$
C. $\frac{\sqrt{3}}{4} \cdot \pi$
D. $\pi$
E. $\frac{1}{4} \cdot \pi$
Answer: A
Problem: <image>Question: The chord $A B$ touches the smaller of the two concentric circles. The length $A B=$ 16. How big is the area of the grey part?
<image1>
Choices:
A. $32 \pi$
B. $63 \pi$
C. $64 \pi$
D. $32 \pi^{2}$
E. It depends on the radius of the circles.
Answer: C
Problem: <image>Question: The big equilateral triangle consists of 36 small equilateral triangles which each have an area of $1 \mathrm{~cm}^{2}$. Determine the area of $A B C$.
<image1>
Choices:
A. $11 \mathrm{~cm}^{2}$
B. $12 \mathrm{~cm}^{2}$
C. $13 \mathrm{~cm}^{2}$
D. $14 \mathrm{~cm}^{2}$
E. $15 \mathrm{~cm}^{2}$
Answer: A
Problem: <image>Question: Which of the following graphs represents the solution set of $(x-|x|)^{2}+(y-|y|)^{2}=4$?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: A strip of paper is folded three times as shown. Determine $\beta$ if $\alpha=70^{\circ}$.
<image1>
Choices:
A. $140^{\circ}$
B. $130^{\circ}$
C. $120^{\circ}$
D. $110^{\circ}$
E. $100^{\circ}$
Answer: C
Problem: <image>Question: A barcode as pictured is made up of alternate black and white stripes. The code always starts and ends with a black stripee. Each strip (black or white) has the width 1 or 2 and the total width of the barcode is 12. How many different barcodes if this kind are there if one reads from left to right?
<image1>
Answer: 116
Problem: <image>Question: The picture on the right shows a tile pattern. The side length of the bigger tiles is a and of the smaller ones b. The dotted lines (horizontal and tilted) include an angle of $30^{\circ}$. How big is the ratio a:b?
<image1>
Choices:
A. $(2 \cdot \sqrt{3}): 1$
B. $(2+\sqrt{3}): 1$
C. $(3+\sqrt{2}): 1$
D. $(3 \cdot \sqrt{2}): 1$
E. 2:1
Answer: B
Problem: <image>Question: In the picture on the right a number should be written next to each point. The sum of the numbers on the corners of each side of the hexagon should be equal. Two numbers have already been inserted. Which number should be in the place marked '$x$'?
<image1>
Answer: 1
Problem: <image>Question: Jan cannot draw very accurately but nevertheless he tried to produce a roadmap of his village. The relative position of the houses and the street crossings are all correct but three of the roads are actually straight and only Qurwik street is not. Who lives in Qurwik street?
<image1>
Choices:
A. Amy
B. Ben
C. Carol
D. David
E. It cannot be determined from the drawing.
Answer: C
Problem: <image>Question: A rectangular piece of paper is wrapped around a cylinder. Then an angled straight cut is made through the points $\mathrm{X}$ and $\mathrm{Y}$ of the cylinder as shown on the left. The lower part of the piece of paper is then unrolled. Which of the following pictures could show the result?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: Determine the area of the quadrilateral PQRS pictured on the right, where $\mathrm{PS}=\mathrm{RS}$, $\angle \mathrm{PSR}=\angle \mathrm{PQR}=90^{\circ}, \mathrm{ST}  \perp \mathrm{PQ}$, and $\mathrm{ST}=5$.
<image1>
Answer: 25
Problem: <image>Question: Michael wants to write whole numbers into the empty fields of the $3 \times 3$ table on the right so that the sum of the numbers in each $2 \times 2$ square equals 10. Four numbers have already been written down. Which of the following values could be the sum of the remaining five numbers?
<image1>
Choices:
A. 9
B. 10
C. 12
D. 13
E. None of these numbers is possible.
Answer: E
Problem: <image>Question: How many graphs of the functions $y=x^{2}, y=-x^{2}, y=+\sqrt{x}, y=-\sqrt{x}$, $y=+\sqrt{-x}, y=-\sqrt{-x}, y=+\sqrt{|x|}, y=-\sqrt{|x|}$ are included in the sketch on the right?
<image1>
Answer: 6
Problem: <image>Question: The rear window wiper of a car is made in a way so that the rod $r$ and the wiper blade $\mathrm{w}$ are equally long and are connected at an angle $\alpha$. The wiper rotates around the centre of rotation $\mathrm{O}$ and wipes over the area shown on the right. Calculate the angle $\beta$ between the right edge of the cleaned area and the tangent of the curved upper edge.
<image1>
Choices:
A. $\frac{3 \pi-\alpha}{2}$
B. $\pi-\frac{\alpha}{2}$
C. $\frac{3 \pi}{2}-\alpha$
D. $\frac{\pi}{2}+\alpha$
E. $\pi+\frac{\alpha}{2}$
Answer: B
Problem: <image>Question: We have three horizontal lines and three parallel, sloped lines. Both of the circles shown touch four of the lines. X, Y and Z are the areas of the grey regions. $\mathrm{D}$ is the area of the parallelogram PQRS. At least how many of the areas $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ and $\mathrm{D}$ does one have to know in order to be able to determine the area of the parallelogram $\mathrm{T}$ ?
<image1>
Answer: 1
Problem: <image>Question: In the (x,y)-plane the co-ordinate axes are positioned as usual. Point $A(1,-10)$ which is on the parabola $y=a x^{2}+b x+c$ was marked. Afterwards the co-ordinate axis and the majority of the parabola were deleted. Which of the following statements could be false?
<image1>
Choices:
A. $a>0$
B. $b<0$
C. $a+b+c<0$
D. $b^{2}>4 a c$
E. $c<0$
Answer: E
Problem: <image>Question: An archer tries his art on the target shown below on the right. With each of his three arrows he always hits the target. How many different scores could he total with three arrows?
<image1>
Answer: 19
Problem: <image>Question: A clock has three hands in different lengths (for seconds, minutes and hours). We don't know the length of each hand but we know that the clock shows the correct time. At 12:55:30 the hands are in the positions shown on the right. What does the clockface look like at 8:10:00?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: The water level in a port rises and falls on a certain day as shown in the diagram. How many hours on that day was the water level over $30 \mathrm{~cm}$?
<image1>
Answer: 13
Problem: <image>Question: In a list of five numbers the first number is 2 and the last one is 12. The product of the first three numbers is 30 , of the middle three 90 and of the last three 360. What is the middle number in that list?
<image1>
Answer: 5
Problem: <image>Question: A rectangular piece of paper $A B C D$ with the measurements $4 \mathrm{~cm} \times 16 \mathrm{~cm}$ is folded along the line $\mathrm{MN}$ so that point $C$ coincides with point $A$ as shown. How big is the area of the quadrilateral ANMD'?
<image1>
Choices:
A. $28 \mathrm{~cm}^{2}$
B. $30 \mathrm{~cm}^{2}$
C. $32 \mathrm{~cm}^{2}$
D. $48 \mathrm{~cm}^{2}$
E. $56 \mathrm{~cm}^{2}$
Answer: C
Problem: <image>Question: How big is the angle $\alpha$ in the regular five-sided star shown?
<image1>
Choices:
A. $24^{\circ}$
B. $30^{\circ}$
C. $36^{\circ}$
D. $45^{\circ}$
E. $72^{\circ}$
Answer: C
Problem: <image>Question: In the diagram we see a rose bed. White roses are growing in the squares that are equally big, red ones are in the big square and yellow ones in the right-angled triangle. The bed has width and height $16 \mathrm{~m}$. How big is the area of the bed?
<image1>
Choices:
A. $114 \mathrm{~m}^{2}$
B. $130 \mathrm{~m}^{2}$
C. $144 \mathrm{~m}^{2}$
D. $160 \mathrm{~m}^{2}$
E. $186 \mathrm{~m}^{2}$
Answer: C
Problem: <image>Question: A right-angled triangle with side lengths $a=8, b=15$ and $c=17$ is given. How big is the radius $r$ of the inscribed semicircle shown?
<image1>
Answer: 4.8
Problem: <image>Question: The clock shown has a rectangular clock face, the hands however move as usual in a constant circular pattern. How big is the distance $x$ of the digits 1 and 2 (in $\mathrm{cm}$ ), if the distance between the numbers 8 and 10 is given as $12 \mathrm{~cm}$?
<image1>
Choices:
A. $3 \sqrt{3}$
B. $2 \sqrt{3}$
C. $4 \sqrt{3}$
D. $2+\sqrt{3}$
E. $12-3 \sqrt{3}$
Answer: C
Problem: <image>Question: Renate wants to glue together a number of ordinary dice (whose number of points on opposite sides always adds up to 7) to form a "dicebar" as shown. Doing this she only wants to glue sides together with an equal number of points. She wants to make sure that the sum of all points on the non-glued sides equals 2012. How many dice does she have to glue together?
<image1>
Choices:
A. 70
B. 71
C. 142
D. 143
E. It is impossible to obtain exactly 2012 points on the non-glued together sides.
Answer: E
Problem: <image>Question: Let $a>b$. If the ellipse shown rotates about the $x$-axis an ellipsoid $E_{x}$ with volume $\operatorname{Vol}\left(E_{x}\right)$ is obtained. If it rotates about the $y$-axis an ellipsoid $E_{y}$ with volume $\operatorname{Vol}\left(E_{y}\right)$ is obtained. Which of the following statements is true?
<image1>
Choices:
A. $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
B. $\mathrm{E}_{\mathrm{x}}=\mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right) \neq \operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
C. $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)>\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
D. $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ and $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)<\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
E. $\mathrm{E}_{\mathrm{x}} \neq \mathrm{E}_{\mathrm{y}}$ but $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$
Answer: C
Problem: <image>Question: An equilateral triangle is being rolled around a unit square as shown. How long is the path that the point shown covers, if the point and the triangle are both back at the start for the first time?
<image1>
Choices:
A. $4 \pi$
B. $\frac{28}{3} \pi$
C. $8 \pi$
D. $\frac{14}{3} \pi$
E. $\frac{21}{2} \pi$
Answer: B
Problem: <image>Question: The regular eight-sided shape on the right has sides of length 10. A circle touches all inscribed diagonals of this eight-sided shape. What is the radius of this circle?
<image1>
Answer: 5
Problem: <image>Question: Inside the cube lattice pictured on the side one can see a solid, non-seethrough pyramid $A B C D S$ with square base $A B C D$, whose top $S$ is exactly in the middle of one edge of the cube. If you look at the pyramid from above, from below, from the front, from the back, from the right and from the left - which of the following views cannot be possible?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: Ralf has a number of equally big plastic plates each in the form of a regular five sided shape. He glues them together along the sides to form a complete ring (see picture). Out of how many of these plates is the ring made up?
<image1>
Answer: 10
Problem: <image>Question: A circular carpet is placed on a floor which is covered by equally big, square tiles. All tiles that have at least one point in common with the carpet are coloured in grey. Which of the following cannot be a result of this?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: Amongst the graphs shown below there is the graph of the function $f(x)=(a-x)(b-x)^{2}$ with $a<b$. Which is it?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Peter has drawn the graph of a function $f: R \rightarrow R$ which consists of two rays and a line segment as indicated on the right. How many solutions has the equation $f(f(f(x)))=0$ ?
<image1>
Answer: 4
Problem: <image>Question: In a triangle $A B C$ the points $M$ and $N$ are placed on side $A B$ so that $A N=A C$ and $B M=$ $B C$. Determine $\angle A C B$ if $\angle M C N=43^{\circ}$
<image1>
Choices:
A. $86^{\circ}$
B. $89^{\circ}$
C. $90^{\circ}$
D. $92^{\circ}$
E. $94^{\circ}$
Answer: E
Problem: <image>Question: The cube pictured on the side is intersected by a plane that passes through the three points adjacent to $A$, that is $D, E$ and $B$. In a similar way the cube is also intersected by those planes that go through the three points adjacent to each of the other seven vertices. These planes dissect the cube into several pieces. What does the piece that contains the centre of the cube look like?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: If one removes some $1 \times 1 \times 1$ cubes from a $5 \times 5 \times 5$ cube, you obtain the solid shown. It consists of several equally high pillars that are built upon a common base. How many little cubes have been removed?
<image1>
Answer: 64
Problem: <image>Question: The curved surfaces of two identical cylinders are cut open along the vertical dotted line, as shown and then stuck together to create the curved surface of one big cylinder. What can be said about the volume of the resulting cylinder compared to the volume of one of the small cylinders?
<image1>
Choices:
A. It is 2-times as big.
B. It is 3-times as big.
C. It is $\pi$-times as big.
D. It is 4-times as big.
E. It is 8-times as big.
Answer: D
Problem: <image>Question: The ratio of the radii of two concentric circles is $1: 3$. The line $A C$ a diameter of the biggest circle. A chord $B C$ of the big circle touches the small circle (see diagram). The line $A B$ has length 12. How big is the radius of the big circle?
<image1>
Answer: 18
Problem: <image>Question: The vertices of a die are numbered 1 to 8, so that the sum of the four numbers on the vertices of each face are the same. The numbers 1, 4 and 6 are already indicated in the picture. Which number is in position $x$?
<image1>
Answer: 2
Problem: <image>Question: PQRS is a rectangle. $T$ is the midpoint of $R S. Q T$ is normal to the diagonal $P R$. What is the ratio of the lengths $P Q: Q R$?
<image1>
Choices:
A. $2: 1$
B. $\sqrt{3}: 1$
C. $3: 2$
D. $\sqrt{2}: 1$
E. $5: 4$
Answer: D
Problem: <image>Question: The straight line $g$ runs through the vertex $A$ of the rectangle $A B C D$ shown. The perpendicular distance from $C$ to $g$ is 2 and from $D$ to $g$ is $6. A D$ is twice as long as $A B$. Determine the length of $A D$.
<image1>
Answer: 10
Problem: <image>Question: In the diagram on the right the following can be seen: a straight line, which is the common tangent of two touching circles with radius 1, and a square with one edge on the straight line and the other vertices one on each of the two circles. How big is the side length of the square?
<image1>
Choices:
A. $\frac{2}{5}$
B. $\frac{1}{4}$
C. $\frac{1}{\sqrt{2}}$
D. $\frac{1}{\sqrt{5}}$
E. $\frac{1}{2}$
Answer: A
Problem: <image>Question: In the diagram a closed polygon can be seen whose vertices are the midpoints of the edges of the die. The interior angles are as usual defined as the angle that two sides of the polygon describe in a common vertex. How big is the sum of all interior angles of the polygon?
<image1>
Choices:
A. $720^{\circ}$
B. $1080^{\circ}$
C. $1200^{\circ}$
D. $1440^{\circ}$
E. $1800^{\circ}$
Answer: B
Problem: <image>Question: Diana produces a bar chart which shows the number of four different types of trees which she has counted on a biology trip. Heinz believes that a pie chart would represent the ratio of the different types of trees in a better way. What would the pie chart look like?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: How many of the following shapes can be drawn using one continuous line (i.e. without lifting the pencil) and without going over a line twice?
<image1>
Answer: 3
Problem: <image>Question: A drinking glass is made in the shape of a truncated cone. The outside of the glass (without the upper or lower circle) should be covered with coloured paper. How do you need to cut the paper to completely cover the glass without an overlap?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The diameters of three semi-circles form the sides of a right-angled triangle. Their areas are $X \mathrm{~cm}^{2}, Y \mathrm{~cm}^{2}$ and $Z \mathrm{~cm}^{2}$ as pictured. Which of the following expressions is definitely correct?
<image1>
Choices:
A. $X+Y<Z$
B. $\sqrt{X}+\sqrt{Y}=\sqrt{Z}$
C. $X+Y=Z$
D. $X^{2}+Y^{2}=Z^{2}$
E. $X^{2}+Y^{2}=Z$
Answer: C
Problem: <image>Question: A square bit of paper is folded along the dashed lines in some order and direction. One of the corners of the resulting small square is cut off. The piece of paper is then unfolded. How many holes are on the inner area of the piece of paper?
<image1>
Answer: 1
Problem: <image>Question: Ella wants to write a number into each circle in the diagram on the right, in such a way that each number is equal to the sum, of its two direct neighbours. Which number does Ella need to write into the circle marked with "?".
<image1>
Choices:
A. -5
B. -16
C. -8
D. -3
E. This question has no solution.
Answer: E
Problem: <image>Question: The diagram shows three concentric circles and two perpendicular, common diameters of the three circles. The three grey sections are of equal area, the small circle has radius 1. What is the product of the radii of the three circles?
<image1>
Choices:
A. $\sqrt{6}$
B. 3
C. $\frac{3 \sqrt{3}}{2}$
D. $2 \sqrt{2}$
E. 6
Answer: A
Problem: <image>Question: On a standard die the sum of the numbers on opposite faces is always 7. Two identical standard dice are shown in the figure. How many dots could there be on the non-visible right-hand face (marked with "?")?
<image1>
Choices:
A. only 5
B. only 2
C. either 2 or 5
D. either 1, 2, 3 or 5
E. either 2,3 or 5
Answer: A
Problem: <image>Question: The curve in the diagram is defined by the equation
$$
\left(x^{2}+y^{2}-2 x\right)^{2}=2\left(x^{2}+y^{2}\right)
$$
Which of the lines $a, b, c, d$ is the $y$-axis?
<image1>
Choices:
A. $a$
B. $b$
C. $c$
D. $d$
E. none of them
Answer: A
Problem: <image>Question: The following table is the multiplication table of the numbers 1 to 10. What is the sum of all 100 products in the complete table?
<image1>
Answer: 3025
Problem: <image>Question: In the rectangle $A B C D$ pictured, $M_{1}$ is the midpoint of $D C, M_{2}$ the midpoint of $A M_{1}, M_{3}$ the midpoint of $B M_{2}$ and $M_{4}$ the midpoint of $C M_{3}$. Determine the ratio of the area of the quadrilateral $M_{1} M_{2} M_{3} M_{4}$ to the area of the rectangle $A B C D$.
<image1>
Choices:
A. $\frac{7}{16}$
B. $\frac{3}{16}$
C. $\frac{7}{32}$
D. $\frac{9}{32}$
E. $\frac{1}{5}$
Answer: C
Problem: <image>Question: Maria wants to build a bridge across a river. This river has the special feature that from each point along one shore the shortest possible bridge to the other shore has always got the same length. Which of the following diagrams is definitely not a sketch of this river?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: A scatter diagram on the $x y$-plane gives the picture of a kangaroo as shown on the right. Now the $x$- and the $y$-coordinate are swapped around for every point. What does the resulting picture look like?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Diana wants to write whole numbers into each circle in the diagram, so that for all eight small triangles the sum of the three numbers in the corners is always the same. What is the maximum amount of different numbers she can use?
<image1>
Answer: 3
Problem: <image>Question: The rectangles $S_{1}$ and $S_{2}$ shown in the picture have the same area. Determine the ratio $x: y$.
<image1>
Choices:
A. $1: 1$
B. $3: 2$
C. $4: 3$
D. $7: 4$
E. $8: 5$
Answer: E