id
stringlengths 1
4
| question
stringlengths 31
709
| options
listlengths 6
6
| image_path
stringlengths 12
15
| image
imagewidth (px) 56
2.52k
| answer
stringclasses 5
values | solution
stringclasses 259
values | level
int64 1
5
| subject
stringclasses 16
values | Answer(Option)
stringclasses 5
values |
|---|---|---|---|---|---|---|---|---|---|
174
|
A parallelogram is divided into 4 triangles as shown in the figure. Of the following possibilities for the areas of the triangles at most one can be true. Which one is it?
<image1>
|
[
"$4,5,8,9$",
"$3,5,6,7$",
"$5,6,7,12$",
"$10,11,12,19$",
"$5,6,8,10$",
"I don't know"
] |
images/174.jpg
|
A
| null | 5
|
metric geometry - area
|
A
|
|
175
|
The figure shows graphs of functions $f$ and $g$ defined on real numbers. Each graph consists of two perpendicular halflines. Which equality is satisfied for every real number $x$?
<image1>
|
[
"$f(x)=-g(x)+2$",
"$f(x)=-g(x)-2$",
"$f(x)=-g(x+2)$",
"$f(x+2)=-g(x)$",
"$f(x+1)=-g(x-1)$",
"I don't know"
] |
images/175.jpg
|
C
| null | 4
|
analytic geometry
|
C
|
|
177
|
Let $A B C D$ be a convex quadrilateral with an area of 1 where $A B$ and $B D$ are the bases of two isosceles triangles $A D B$ and $B C D$ respectively (as shown). The product $A C \cdot B D$ is equal to:
<image1>
|
[
"$\\frac{\\sqrt{3}}{3}$",
"$\\frac{2 \\sqrt{3}}{3}$",
"$\\sqrt{3}$",
"$\\frac{4 \\sqrt{3}}{3}$",
"other answer",
"I don't know"
] |
images/177.jpg
|
D
| null | 5
|
metric geometry - length
|
D
|
|
179
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/179.jpg
|
A
| null | 5
|
combinatorial geometry
|
A
|
|
180
|
A $3 \times 3 \times 3$ cube weighs 810 grams. If we drill three holes through it as shown, each of which is a $1 \times 1 \times 3$ rectangular parallelepiped, the weight of the remaining solid is:
<image1>
|
[
"$540 \\mathrm{~g}$",
"$570 \\mathrm{~g}$",
"$600 \\mathrm{~g}$",
"$630 \\mathrm{~g}$",
"$660 \\mathrm{~g}$",
"I don't know"
] |
images/180.jpg
|
C
| null | 3
|
solid geometry
|
C
|
|
182
|
The diagram shows a rectangle $A B E F$ and a triangle $A B C$. We know that the angle $A C F$ equals angle $C B E$. If $F C=6$ and $C E=2$ then the area of $A B C$ is:
<image1>
|
[
"12",
"16",
"$8 \\sqrt{2}$",
"$8 \\sqrt{3}$",
"Another value",
"I don't know"
] |
images/182.jpg
|
D
| null | 5
|
metric geometry - area
|
D
|
|
183
|
In the quadrilateral $A B C D$ the diagonal $B D$ is the bisector of $\angle A B C$ and $A C=B C$. Given $\angle B D C=80^{\circ}$ and $\angle A C B=20^{\circ}, \angle B A D$ is equal to:
<image1>
|
[
"$90^{\\circ}$",
"$100^{\\circ}$",
"$110^{\\circ}$",
"$120^{\\circ}$",
"$135^{\\circ}$",
"I don't know"
] |
images/183.jpg
|
D
| null | 4
|
metric geometry - angle
|
D
|
|
186
|
Susan has two pendants made of the same material. They are equally thick and weigh the same. One of them has the shape of an annulus created from two concentric circles with the radii $6 \mathrm{~cm}$ and $4 \mathrm{~cm}$ (see the diagram). The second has the shape of a solid circle. What is the radius of the second pendant?
<image1>
|
[
"$4 \\mathrm{~cm}$",
"$2 \\sqrt{6} \\mathrm{~cm}$",
"$5 \\mathrm{~cm}$",
"$2 \\sqrt{5} \\mathrm{~cm}$",
"$\\sqrt{10} \\mathrm{~cm}$",
"I don't know"
] |
images/186.jpg
|
D
| null | 5
|
metric geometry - length
|
D
|
|
187
|
In the diagram, $A B$ has length $1 ; \angle A B C=\angle A C D=90^{\circ}$; $\angle C A B=\angle D A C=\theta$. What is the length of $A D$?
<image1>
|
[
"$\\cos \\beta+\\tg \\beta$",
"$\\frac{1}{\\cos (2 \\beta)}$",
"$\\cos ^{2} \\beta$",
"$\\cos (2 \\beta)$",
"$\\frac{1}{\\cos ^{2} \\beta}$",
"I don't know"
] |
images/187.jpg
|
E
| null | 5
|
metric geometry - length
|
E
|
|
188
|
The radius of the traffic sign is $20 \mathrm{~cm}$. Each of the dark pieces is a quarter of a circle. The area of all 4 quarters equals that of the light part of the sign. What is the radius of this circle in centimetres?
<image1>
|
[
"$10 \\sqrt{2}$",
"$4 \\sqrt{5}$",
"$\\frac{20}{3}$",
"12.5",
"10",
"I don't know"
] |
images/188.jpg
|
A
| null | 5
|
metric geometry - length
|
A
|
|
189
|
The ratio of the radii of the sector and the incircle in the picture is $3: 1$. Than the ratio of their areas is:
<image1>
|
[
"$3: 2$",
"$4: 3$",
"$\\sqrt{3}: 1$",
"$2: 1$",
"$9: 1$",
"I don't know"
] |
images/189.jpg
|
A
| null | 5
|
metric geometry - area
|
A
|
|
194
|
In how many ways can all the numbers $1,2,3,4,5,6$ be written in the squares of the figure (one in each square) so that there are no adjacent squares in which the difference of the numbers written is equal to 3? (Squares that share only a corner are not considered adjacent.)
<image1>
|
[
"$3 \\cdot 2^{5}$",
"$3^{6}$",
"$6^{3}$",
"$2 \\cdot 3^{5}$",
"$3 \\cdot 5^{2}$",
"I don't know"
] |
images/194.jpg
|
A
| null | 5
|
combinatorics
|
A
|
|
196
|
If each side of the regular hexagon has length $\sqrt{3}$ and $X A B C$ and $X P Q R$ are squares, what is the area of the shaded region?
<image1>
|
[
"$\\frac{5-\\sqrt{3}}{4}$",
"$\\frac{\\sqrt{3}+1}{2}$",
"$\\frac{\\sqrt{3}}{4}$",
"$\\frac{2-\\sqrt{3}}{4}$",
"$\\frac{2+\\sqrt{3}}{4}$",
"I don't know"
] |
images/196.jpg
|
A
| null | 5
|
metric geometry - area
|
A
|
|
197
|
The shaded area is equal to $\sqrt{3}$. What is the area of the triangle $A B C$?
<image1>
|
[
"$2 \\sqrt{3}$",
"2",
"5",
"6",
"$4 \\sqrt{3}$",
"I don't know"
] |
images/197.jpg
|
A
| null | 5
|
metric geometry - area
|
A
|
|
198
|
The billiard ball meets the board under $45^{\circ}$ as shown. Which pocket will it fall into?
<image1>
|
[
"$A$",
"$B$",
"$C$",
"$D$",
"Neither of the pockets",
"I don't know"
] |
images/198.jpg
|
C
| null | 5
|
transformation geometry
|
C
|
|
199
|
The segment $A E$ is divided into four equal parts and semicircles are drawn taking $A E, A D$ and $D E$ as diameters, creating two paths from $A$ to $E$ as shown. Determine the ratio of the length of the upper path to the length of the lower path.
<image1>
|
[
"$1: 2$",
"$2: 3$",
"$2: 1$",
"$3: 2$",
"$1: 1$",
"I don't know"
] |
images/199.jpg
|
E
| null | 5
|
metric geometry - length
|
E
|
|
201
|
Two semicircles are drawn as shown in the figure. The chord $C D$, of length 4 , is parallel to the diameter $A B$ of the greater semicircle and touches the smaller semicircle. Then the area of the shaded region equals
<image1>
|
[
"$\\pi$",
"$1.5 \\pi$",
"$2 \\pi$",
"$3 \\pi$",
"Not enough data",
"I don't know"
] |
images/201.jpg
|
C
| null | 5
|
metric geometry - area
|
C
|
|
202
|
Which is the graph of the function $y=\sqrt{|(1+x)(1-|x|)|}$?
<image1>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/202.jpg
|
D
| null | 4
|
analytic geometry
|
D
|
|
206
|
A river starts at point $A$. As it flows the river splits into two. The first branch takes $\frac{2}{3}$ of the water and the second takes the rest. Later the first branch splits into three, one taking $\frac{1}{8}$ of the branch's water, the second $\frac{5}{8}$ and the third one the rest. Further down this last branch meets again a branch of the river. The map below shows the situation. What part of the original water flows at point $B$?
<image1>
|
[
"$\\frac{1}{3}$",
"$\\frac{5}{4}$",
"$\\frac{2}{9}$",
"$\\frac{1}{2}$",
"$\\frac{1}{4}$",
"I don't know"
] |
images/206.jpg
|
D
| null | 5
|
graph theory
|
D
|
|
207
|
Given an isosceles triangle $A B C, C A=C B, A D=$ $=A C, D B=D C$ (see the fig.). Find the value of the angle $A C B$.
<image1>
|
[
"$98^{\\circ}$",
"$100^{\\circ}$",
"$104^{\\circ}$",
"$108^{\\circ}$",
"$110^{\\circ}$",
"I don't know"
] |
images/207.jpg
|
D
| null | 4
|
metric geometry - angle
|
D
|
|
209
|
Each of the cubes in the figure has the length of an edge equal to 1. What is the length of the segment $A B$?
<image1>
|
[
"$\\sqrt{17}$",
"7",
"$\\sqrt{13}$",
"$\\sqrt{7}$",
"$\\sqrt{14}$",
"I don't know"
] |
images/209.jpg
|
A
| null | 3
|
solid geometry
|
A
|
|
213
|
The square $A B C D$ has a side of length 1 and $M$ is the midpoint of $A B$. The area of the shaded region is
<image1>
|
[
"$\\frac{1}{24}$",
"$\\frac{1}{16}$",
"$\\frac{1}{8}$",
"$\\frac{1}{12}$",
"$\\frac{2}{13}$",
"I don't know"
] |
images/213.jpg
|
D
| null | 5
|
metric geometry - area
|
D
|
|
215
|
The suare in the diagram has side length 1. The radius of the small circle would
then be of the length
<image1>
|
[
"$\\sqrt{2}-1$",
"$\\frac{1}{4}$",
"$\\frac{\\sqrt{2}}{4}$",
"$1-\\frac{\\sqrt{2}}{2}$",
"$(\\sqrt{2}-1)^{2}$",
"I don't know"
] |
images/215.jpg
|
E
| null | 5
|
metric geometry - length
|
E
|
|
217
|
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>
|
[
"either A or B",
"only C",
"only D",
"either C or D",
"A, B, C or D",
"I don't know"
] |
images/217.jpg
|
D
| null | 5
|
graph theory
|
D
|
|
218
|
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>
|
[
"7",
"$2 \\sqrt{13}$",
"8",
"$4 \\sqrt{3}$",
"$2 \\cdot(\\sqrt{2}+\\sqrt{3})$",
"I don't know"
] |
images/218.jpg
|
B
| null | 5
|
metric geometry - length
|
B
|
|
220
|
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>
|
[
"$6+\\pi$",
"$3+2 \\pi$",
"$9+\\frac{\\pi}{3}$",
"$3 \\pi$",
"$9+\\pi$",
"I don't know"
] |
images/220.jpg
|
A
| null | 5
|
metric geometry - length
|
A
|
|
221
|
The adjacent diagram illustrates the graphs of the two functions f and g. How can we describe the relationship between f and g?
<image1>
|
[
"$g(x-2)=-f(x)$",
"$g(x)=f(x+2)$",
"$g(x)=-f(-x+2)$",
"$g(-x)=-f(-x-2)$",
"$g(2-x)=-f(x)$",
"I don't know"
] |
images/221.jpg
|
A
| null | 4
|
analytic geometry
|
A
|
|
222
|
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>
|
[
"I",
"II",
"III",
"IV",
"none of them",
"I don't know"
] |
images/222.jpg
|
D
| null | 1
|
descriptive geometry
|
D
|
|
223
|
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>
|
[
"16",
"51",
"54",
"55",
"110",
"I don't know"
] |
images/223.jpg
|
D
| null | 1
|
logic
|
D
|
|
224
|
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>
|
[
"$8+2 \\sqrt{2}$",
"$11-\\sqrt{2}$",
"10",
"$8+3 \\sqrt{2}$",
"$11+\\frac{\\sqrt{2}}{2}$",
"I don't know"
] |
images/224.jpg
|
A
| null | 5
|
metric geometry - length
|
A
|
|
226
|
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>
|
[
"$105^{\\circ}$",
"$108^{\\circ}$",
"$110^{\\circ}$",
"$120^{\\circ}$",
"$125^{\\circ}$",
"I don't know"
] |
images/226.jpg
|
D
| null | 4
|
metric geometry - angle
|
D
|
|
227
|
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>
|
[
"$4 \\cdot(3-2 \\sqrt{2}) \\cdot \\pi$",
"$\\sqrt{2} \\cdot \\pi$",
"$\\frac{\\sqrt{3}}{4} \\cdot \\pi$",
"$\\pi$",
"$\\frac{1}{4} \\cdot \\pi$",
"I don't know"
] |
images/227.jpg
|
A
| null | 5
|
metric geometry - area
|
A
|
|
228
|
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>
|
[
"$32 \\pi$",
"$63 \\pi$",
"$64 \\pi$",
"$32 \\pi^{2}$",
"It depends on the radius of the circles.",
"I don't know"
] |
images/228.jpg
|
C
| null | 5
|
metric geometry - area
|
C
|
|
229
|
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>
|
[
"$11 \\mathrm{~cm}^{2}$",
"$12 \\mathrm{~cm}^{2}$",
"$13 \\mathrm{~cm}^{2}$",
"$14 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"I don't know"
] |
images/229.jpg
|
A
| null | 5
|
combinatorial geometry
|
A
|
|
230
|
Which of the following graphs represents the solution set of $(x-|x|)^{2}+(y-|y|)^{2}=4$?
<image1>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/230.jpg
|
A
| null | 4
|
analytic geometry
|
A
|
|
231
|
A strip of paper is folded three times as shown. Determine $\beta$ if $\alpha=70^{\circ}$.
<image1>
|
[
"$140^{\\circ}$",
"$130^{\\circ}$",
"$120^{\\circ}$",
"$110^{\\circ}$",
"$100^{\\circ}$",
"I don't know"
] |
images/231.jpg
|
C
| null | 5
|
transformation geometry
|
C
|
|
233
|
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>
|
[
"$(2 \\cdot \\sqrt{3}): 1$",
"$(2+\\sqrt{3}): 1$",
"$(3+\\sqrt{2}): 1$",
"$(3 \\cdot \\sqrt{2}): 1$",
"2:1",
"I don't know"
] |
images/233.jpg
|
B
| null | 4
|
metric geometry - angle
|
B
|
|
235
|
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>
|
[
"Amy",
"Ben",
"Carol",
"David",
"It cannot be determined from the drawing.",
"I don't know"
] |
images/235.jpg
|
C
| null | 1
|
logic
|
C
|
|
236
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/236.jpg
|
C
| null | 1
|
descriptive geometry
|
C
|
|
238
|
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>
|
[
"9",
"10",
"12",
"13",
"None of these numbers is possible.",
"I don't know"
] |
images/238.jpg
|
E
| null | 1
|
logic
|
E
|
|
240
|
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>
|
[
"$\\frac{3 \\pi-\\alpha}{2}$",
"$\\pi-\\frac{\\alpha}{2}$",
"$\\frac{3 \\pi}{2}-\\alpha$",
"$\\frac{\\pi}{2}+\\alpha$",
"$\\pi+\\frac{\\alpha}{2}$",
"I don't know"
] |
images/240.jpg
|
B
| null | 4
|
metric geometry - angle
|
B
|
|
242
|
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>
|
[
"$a>0$",
"$b<0$",
"$a+b+c<0$",
"$b^{2}>4 a c$",
"$c<0$",
"I don't know"
] |
images/242.jpg
|
E
| null | 4
|
analytic geometry
|
E
|
|
244
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/244.jpg
|
A
| null | 5
|
algebra
|
A
|
|
247
|
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>
|
[
"$28 \\mathrm{~cm}^{2}$",
"$30 \\mathrm{~cm}^{2}$",
"$32 \\mathrm{~cm}^{2}$",
"$48 \\mathrm{~cm}^{2}$",
"$56 \\mathrm{~cm}^{2}$",
"I don't know"
] |
images/247.jpg
|
C
| null | 5
|
transformation geometry
|
C
|
|
248
|
How big is the angle $\alpha$ in the regular five-sided star shown?
<image1>
|
[
"$24^{\\circ}$",
"$30^{\\circ}$",
"$36^{\\circ}$",
"$45^{\\circ}$",
"$72^{\\circ}$",
"I don't know"
] |
images/248.jpg
|
C
| null | 4
|
metric geometry - angle
|
C
|
|
249
|
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>
|
[
"$114 \\mathrm{~m}^{2}$",
"$130 \\mathrm{~m}^{2}$",
"$144 \\mathrm{~m}^{2}$",
"$160 \\mathrm{~m}^{2}$",
"$186 \\mathrm{~m}^{2}$",
"I don't know"
] |
images/249.jpg
|
C
| null | 5
|
metric geometry - area
|
C
|
|
251
|
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>
|
[
"$3 \\sqrt{3}$",
"$2 \\sqrt{3}$",
"$4 \\sqrt{3}$",
"$2+\\sqrt{3}$",
"$12-3 \\sqrt{3}$",
"I don't know"
] |
images/251.jpg
|
C
| null | 5
|
metric geometry - length
|
C
|
|
252
|
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>
|
[
"70",
"71",
"142",
"143",
"It is impossible to obtain exactly 2012 points on the non-glued together sides.",
"I don't know"
] |
images/252.jpg
|
E
| null | 1
|
arithmetic
|
E
|
|
253
|
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>
|
[
"$\\mathrm{E}_{\\mathrm{x}}=\\mathrm{E}_{\\mathrm{y}}$ and $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)=\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}}=\\mathrm{E}_{\\mathrm{y}}$ but $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right) \\neq \\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}} \\neq \\mathrm{E}_{\\mathrm{y}}$ and $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)>\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}} \\neq \\mathrm{E}_{\\mathrm{y}}$ and $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)<\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"$\\mathrm{E}_{\\mathrm{x}} \\neq \\mathrm{E}_{\\mathrm{y}}$ but $\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{x}}\\right)=\\operatorname{Vol}\\left(\\mathrm{E}_{\\mathrm{y}}\\right)$",
"I don't know"
] |
images/253.jpg
|
C
| null | 3
|
solid geometry
|
C
|
|
254
|
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>
|
[
"$4 \\pi$",
"$\\frac{28}{3} \\pi$",
"$8 \\pi$",
"$\\frac{14}{3} \\pi$",
"$\\frac{21}{2} \\pi$",
"I don't know"
] |
images/254.jpg
|
B
| null | 5
|
transformation geometry
|
B
|
|
256
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/256.jpg
|
E
| null | 1
|
descriptive geometry
|
E
|
|
258
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/258.jpg
|
E
| null | 5
|
combinatorial geometry
|
E
|
|
259
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/259.jpg
|
A
| null | 4
|
analytic geometry
|
A
|
|
261
|
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>
|
[
"$86^{\\circ}$",
"$89^{\\circ}$",
"$90^{\\circ}$",
"$92^{\\circ}$",
"$94^{\\circ}$",
"I don't know"
] |
images/261.jpg
|
E
| null | 4
|
metric geometry - angle
|
E
|
|
262
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/262.jpg
|
A
| null | 3
|
solid geometry
|
A
|
|
264
|
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>
|
[
"It is 2-times as big.",
"It is 3-times as big.",
"It is $\\pi$-times as big.",
"It is 4-times as big.",
"It is 8-times as big.",
"I don't know"
] |
images/264.jpg
|
D
| null | 3
|
solid geometry
|
D
|
|
267
|
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>
|
[
"$2: 1$",
"$\\sqrt{3}: 1$",
"$3: 2$",
"$\\sqrt{2}: 1$",
"$5: 4$",
"I don't know"
] |
images/267.jpg
|
D
| null | 5
|
metric geometry - length
|
D
|
|
269
|
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>
|
[
"$\\frac{2}{5}$",
"$\\frac{1}{4}$",
"$\\frac{1}{\\sqrt{2}}$",
"$\\frac{1}{\\sqrt{5}}$",
"$\\frac{1}{2}$",
"I don't know"
] |
images/269.jpg
|
A
| null | 5
|
metric geometry - length
|
A
|
|
270
|
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>
|
[
"$720^{\\circ}$",
"$1080^{\\circ}$",
"$1200^{\\circ}$",
"$1440^{\\circ}$",
"$1800^{\\circ}$",
"I don't know"
] |
images/270.jpg
|
B
| null | 3
|
solid geometry
|
B
|
|
271
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/271.jpg
|
A
| null | 3
|
statistics
|
A
|
|
273
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/273.jpg
|
E
| null | 1
|
descriptive geometry
|
E
|
|
274
|
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>
|
[
"$X+Y<Z$",
"$\\sqrt{X}+\\sqrt{Y}=\\sqrt{Z}$",
"$X+Y=Z$",
"$X^{2}+Y^{2}=Z^{2}$",
"$X^{2}+Y^{2}=Z$",
"I don't know"
] |
images/274.jpg
|
C
| null | 5
|
metric geometry - area
|
C
|
|
276
|
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>
|
[
"-5",
"-16",
"-8",
"-3",
"This question has no solution.",
"I don't know"
] |
images/276.jpg
|
E
| null | 5
|
algebra
|
E
|
|
277
|
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>
|
[
"$\\sqrt{6}$",
"3",
"$\\frac{3 \\sqrt{3}}{2}$",
"$2 \\sqrt{2}$",
"6",
"I don't know"
] |
images/277.jpg
|
A
| null | 5
|
metric geometry - length
|
A
|
|
278
|
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>
|
[
"only 5",
"only 2",
"either 2 or 5",
"either 1, 2, 3 or 5",
"either 2,3 or 5",
"I don't know"
] |
images/278.jpg
|
A
| null | 3
|
solid geometry
|
A
|
|
279
|
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>
|
[
"$a$",
"$b$",
"$c$",
"$d$",
"none of them",
"I don't know"
] |
images/279.jpg
|
A
| null | 4
|
analytic geometry
|
A
|
|
281
|
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>
|
[
"$\\frac{7}{16}$",
"$\\frac{3}{16}$",
"$\\frac{7}{32}$",
"$\\frac{9}{32}$",
"$\\frac{1}{5}$",
"I don't know"
] |
images/281.jpg
|
C
| null | 5
|
metric geometry - area
|
C
|
|
282
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/282.jpg
|
B
| null | 5
|
metric geometry - length
|
B
|
|
283
|
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>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/283.jpg
|
A
| null | 4
|
analytic geometry
|
A
|
|
285
|
The rectangles $S_{1}$ and $S_{2}$ shown in the picture have the same area. Determine the ratio $x: y$.
<image1>
|
[
"$1: 1$",
"$3: 2$",
"$4: 3$",
"$7: 4$",
"$8: 5$",
"I don't know"
] |
images/285.jpg
|
E
| null | 4
|
analytic geometry
|
E
|
|
286
|
The diagram shows a circle with centre $O$ as well as a tangent that touches the circle in point $P$. The arc $A P$ has length 20, the arc $B P$ has length 16. What is the size of the angle $\angle A X P$?
<image1>
|
[
"$30^{\\circ}$",
"$24^{\\circ}$",
"$18^{\\circ}$",
"$15^{\\circ}$",
"$10^{\\circ}$",
"I don't know"
] |
images/286.jpg
|
E
| null | 4
|
metric geometry - angle
|
E
|
|
287
|
In this number pyramid each number in a higher cell is equal to the product of the two numbers in the cells immediately underneath that number. Which of the following numbers cannot appear in the topmost cell, if the cells on the bottom row hold natural numbers greater than 1 only?
<image1>
|
[
"56",
"84",
"90",
"105",
"220",
"I don't know"
] |
images/287.jpg
|
D
| null | 5
|
algebra
|
D
|
|
288
|
The square shown in the diagram has a perimeter of 4. The perimeter of the equilateral triangle is
<image1>
|
[
"4",
"$3+\\sqrt{3}$",
"3",
"$3+\\sqrt{2}$",
"$4+\\sqrt{3}$",
"I don't know"
] |
images/288.jpg
|
B
| null | 5
|
metric geometry - length
|
B
|
|
289
|
Each of the ten points in the diagram is labelled with one of the numbers 0,1 or 2. It is known that the sum of the numbers in the corner points of each white triangle is divisible by 3, while the sum of the numbers in the corner points of each black triangle is not divisible by 3. Three of the points are already labeled as shown in the diagram. With which numbers can the inner point be labeled?
<image1>
|
[
"only 0",
"only 1",
"only 2",
"only 0 and 1",
"either 0 or 1 or 2",
"I don't know"
] |
images/289.jpg
|
A
| null | 5
|
combinatorics
|
A
|
|
290
|
Bettina chooses five points $A, B, C, D$ and $E$ on a circle and draws the tangent to the circle at point $A$. She realizes that the five angles marked $x$ are all equally big. (Note that the diagram is not drawn to scale!) How big is the angle $\angle A B D$?
<image1>
|
[
"$66^{\\circ}$",
"$70.5^{\\circ}$",
"$72^{\\circ}$",
"$75^{\\circ}$",
"$77.5^{\\circ}$",
"I don't know"
] |
images/290.jpg
|
C
| null | 4
|
metric geometry - angle
|
C
|
|
292
|
A rectangular piece of paper $A B C D$ is $5 \mathrm{~cm}$ wide and $50 \mathrm{~cm}$ long. The paper is white on one side and grey on the other. Christina folds the strip as shown so that the vertex $B$ coincides with $M$ the midpoint of the edge $C D$. Then she folds it so that the vertex $D$ coincides with $N$ the midpoint of the edge $A B$. How big is the area of the visible white part in the diagram?
<image1>
<image2>
<image3>
|
[
"$50 \\mathrm{~cm}^{2}$",
"$60 \\mathrm{~cm}^{2}$",
"$62.5 \\mathrm{~cm}^{2}$",
"$100 \\mathrm{~cm}^{2}$",
"$125 \\mathrm{~cm}^{2}$",
"I don't know"
] |
images/292.jpg
|
B
| null | 5
|
transformation geometry
|
B
|
|
293
|
We consider a $5 \times 5$ square that is split up into 25 fields. Initially all fields are white. In each move it is allowed to change the colour of three fields that are adjacent in a horizontal or vertical line (i.e. white fields turn black and black ones turn white). What is the smallest number of moves needed to obtain the chessboard colouring shown in the diagram?
<image1>
|
[
"less than 10",
"10",
"12",
"more than 12",
"This colouring cannot be obtained.",
"I don't know"
] |
images/293.jpg
|
A
| null | 5
|
graph theory
|
A
|
|
296
|
Four of the following five pictures show pieces of the graph of the same quadratic function. Which piece does not belong?
<image1>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/296.jpg
|
C
| null | 4
|
analytic geometry
|
C
|
|
297
|
The diagram shows a circle with centre $O$ and the diameters $A B$ and $C X$. Let $O B=$ $B C$. Which fraction of the circle area is shaded?
<image1>
|
[
"$\\frac{2}{5}$",
"$\\frac{1}{3}$",
"$\\frac{2}{7}$",
"$\\frac{3}{8}$",
"$\\frac{4}{11}$",
"I don't know"
] |
images/297.jpg
|
B
| null | 5
|
metric geometry - area
|
B
|
|
298
|
A $4 \times 1 \times 1$ cuboid is made up of 2 white and 2 grey cubes as shown. Which of the following cuboids can be build entirely out of such $4 \times 1 \times 1$ cuboids?
<image1>
<image2>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/298.jpg
|
A
| null | 5
|
combinatorial geometry
|
A
|
|
299
|
Which quadrant contains no points of the graph of the linear function $f(x)=-3.5 x+7$?
<image1>
|
[
"I",
"II",
"III",
"IV",
"Every quadrant contains at least one point of the graph.",
"I don't know"
] |
images/299.jpg
|
C
| null | 4
|
analytic geometry
|
C
|
|
302
|
Julia has 2017 round discs available: 1009 black ones and 1008 white ones. Using them, she wants to lay the biggest square pattern (as shown) possible and starts by using a black disc in the left upper corner. Subsequently she lays the discs in such a way that the colours alternate in each row and column. How many discs are left over
when she has laid the biggest square possible?
<image1>
|
[
"none",
"40 of each colour",
"40 black and 41 white ones",
"41 of each colour",
"40 white and 41 black ones",
"I don't know"
] |
images/302.jpg
|
E
| null | 5
|
algebra
|
E
|
|
303
|
The diagram shows a regular hexagon with side length 1. The grey flower is outlined by circular arcs with radius 1 whose centre's lie in the vertices of the hexagon. How big is the area of the grey flower?
<image1>
|
[
"$\\frac{\\pi}{2}$",
"$\\frac{2 \\pi}{3}$",
"$2 \\sqrt{3}-\\pi$",
"$\\frac{\\pi}{2}+\\sqrt{3}$",
"$2 \\pi-3 \\sqrt{3}$",
"I don't know"
] |
images/303.jpg
|
E
| null | 5
|
metric geometry - area
|
E
|
|
304
|
We look at a regular tetrahedron with volume 1. Its four vertices are cut off by planes that go through the midpoints of the respective edges (see diagram). How big is the volume of the remaining solid?
<image1>
|
[
"$\\frac{4}{5}$",
"$\\frac{3}{4}$",
"$\\frac{2}{3}$",
"$\\frac{1}{2}$",
"$\\frac{1}{3}$",
"I don't know"
] |
images/304.jpg
|
D
| null | 3
|
solid geometry
|
D
|
|
306
|
In the diagram you can see the calendar page of a certain month. Unfortunately ink has run across parts of the page. Which day of the week does the 27th of that month fall on?
<image1>
|
[
"Monday",
"Wednesday",
"Thursday",
"Saturday",
"Sunday",
"I don't know"
] |
images/306.jpg
|
A
| null | 1
|
arithmetic
|
A
|
|
309
|
Four identical rhombuses (diamonds) and two squares are fitted together to form a regular octagon as shown. How big are the obtuse interior angles in the rhombuses?
<image1>
|
[
"$135^{\\circ}$",
"$140^{\\circ}$",
"$144^{\\circ}$",
"$145^{\\circ}$",
"$150^{\\circ}$",
"I don't know"
] |
images/309.jpg
|
A
| null | 3
|
solid geometry
|
A
|
|
310
|
The faces of the brick have the areas A, B and C as shown. How big is the volume of the brick?
<image1>
|
[
"$A B C$",
"$\\sqrt{A B C}$",
"$\\sqrt{A B+B C+C A}$",
"$\\sqrt[3]{A B C}$",
"$2(A+B+C)$",
"I don't know"
] |
images/310.jpg
|
B
| null | 3
|
solid geometry
|
B
|
|
311
|
Two dice with volumes $V$ and $W$ intersect each other as shown. $90 \%$ of the volume of the die with volume $V$ does not belong to both dice. $85 \%$ of the volume of the die with volume $W$ does not belong to both dice. What is the relationship between the volumes of the two dice?
<image1>
|
[
"$V=\\frac{2}{3} W$",
"$V=\\frac{3}{2} W$",
"$V=\\frac{85}{90} \\mathrm{~W}$",
"$V=\\frac{90}{85} W$",
"$V=W$",
"I don't know"
] |
images/311.jpg
|
B
| null | 3
|
solid geometry
|
B
|
|
312
|
The five vases shown are filled with water. The filling rate is constant. For which of the five vases does the graph shown describe the height of the water $h$ as a function of the time t?
<image1>
<image2>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/312.jpg
|
D
| null | 3
|
statistics
|
D
|
|
313
|
An octahedron is inscribed into a die with side length 1. The vertices of the octahedron are the midpoints of the faces of the die. How big is the volume of the octahedron?
<image1>
|
[
"$\\frac{1}{3}$",
"$\\frac{1}{4}$",
"$\\frac{1}{5}$",
"$\\frac{1}{6}$",
"$\\frac{1}{8}$",
"I don't know"
] |
images/313.jpg
|
D
| null | 3
|
solid geometry
|
D
|
|
315
|
A regular pentagon is cut out of a page of lined paper. Step by step this pentagon is then rotated $21^{\circ}$ counter clockwise about its midpoint. The result after step one is shown in the diagram. Which of the diagrams shows the situation when the pentagon fills the hole entirely again for the first time?
<image1>
<image2>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/315.jpg
|
B
| null | 5
|
transformation geometry
|
B
|
|
317
|
Two rectangles form the angles $40^{\circ}$ and $30^{\circ}$ respectively, with a straight line (see diagram). How big is angle $\alpha$?
<image1>
|
[
"$105^{\\circ}$",
"$120^{\\circ}$",
"$130^{\\circ}$",
"$135^{\\circ}$",
"another value",
"I don't know"
] |
images/317.jpg
|
E
| null | 4
|
metric geometry - angle
|
E
|
|
319
|
On an idealised rectangular billiard table with side lengths $3 \mathrm{~m}$ and $2 \mathrm{m}$ a ball (point-shaped) is pushed away from point $M$ on the long side $A B$. It is reflected exactly once on each of the other sides as shown. at which distance from the vertex $A$ will the ball hit this side again if $B M=1.2 \mathrm{~m}$ and $B N=$ $0.8 m$?
<image1>
|
[
"$2 \\mathrm{~m}$",
"$1.5 \\mathrm{~m}$",
"$1.2 \\mathrm{~m}$",
"$2.8 \\mathrm{~m}$",
"$1.8 \\mathrm{~m}$",
"I don't know"
] |
images/319.jpg
|
E
| null | 5
|
metric geometry - length
|
E
|
|
320
|
$A B C D E F$ is a regular hexagon, as shown in the diagram. $G$ is the midpoint of $A B. H$ and I are the intercepts of the line segments GD and GE respectively, with the line segment FC. How big is the ratio of the areas of the triangle GIF and the trapezium IHDE?
<image1>
|
[
"$\\frac{1}{2}$",
"$\\frac{1}{3}$",
"$\\frac{1}{4}$",
"$\\frac{\\sqrt{3}}{3}$",
"$\\frac{\\sqrt{3}}{4}$",
"I don't know"
] |
images/320.jpg
|
A
| null | 5
|
metric geometry - area
|
A
|
|
321
|
Archimedes has calculated 15!. The result is on the board. Unfortunately two of the digits, the second and the tenth, cannot be read. What are the two missing digits? (Remark: $15 !=15 \cdot 14 \cdot 13 \cdot \ldots \cdot 2 \cdot 1$ )
<image1>
|
[
"2 and 0",
"4 and 8",
"7 and 4",
"9 and 2",
"3 and 8",
"I don't know"
] |
images/321.jpg
|
E
| null | 1
|
arithmetic
|
E
|
|
322
|
The flag of Kangoraland is a rectangle which is split into three equal rectangles as shown. How big is the ratio of the side lengths of the white rectangle?
<image1>
|
[
"$1: 2$",
"$2: 3$",
"$2: 5$",
"$3: 7$",
"$4: 9$",
"I don't know"
] |
images/322.jpg
|
A
| null | 5
|
metric geometry - length
|
A
|
|
323
|
The numbers $1,2,3$ and 4 are inserted into different cells of the $2 \times 2$ table shown. Then the sums of the numbers in each row and column are determined. Two of these sums are 4 and 5. How big are the two remaining sums?
<image1>
|
[
"6 and 6",
"3 and 5",
"4 and 5",
"4 and 6",
"5 and 6",
"I don't know"
] |
images/323.jpg
|
E
| null | 5
|
algebra
|
E
|
|
324
|
A rectangle is coloured in five different ways as shown. In which picture is the grey area biggest?
<image1>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/324.jpg
|
E
| null | 5
|
metric geometry - area
|
E
|
|
325
|
Three triangles are connected to each other as shown. In which of the following pictures are the three triangles connected in the same way?
<image1>
<image2>
|
[
"A",
"B",
"C",
"D",
"E",
"I don't know"
] |
images/325.jpg
|
D
| null | 1
|
topology
|
D
|
|
326
|
Three four-digit numbers are written onto three separate pieces of paper as shown. The sum of the three numbers is 11126. Three of the digits in the picture are hidden. Which are the three hidden digits?
<image1>
|
[
"1,4 and 7",
"1,5 and 7",
"3,3 and 3",
"4,5 and 6",
"4,5 and 7",
"I don't know"
] |
images/326.jpg
|
B
| null | 5
|
algebra
|
B
|
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