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<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many different digits can you find in this picture? | 6 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Which number do you have to write in the last daisy? | 61 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Misty the cat has five kittens: two of them are striped, one spotty, the rest of them are absolutely white. In which picture can we see the kittens of Misty, knowing that the ears of one of them are of different colour? | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many bricks are missing in the wall? | 6 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The sums of the all the three numbers on each side of the triangle are equal. Two numbers happened to be stained with ink. How much is the sum of these two numbers? | 2 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Four people can be seated at a square table. How many people at most could be seated if we pushed four tables of this kind together in one row? | 10 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Mike has built a construction, shown in the upper picture, from equal cubes. Lily has taken several cubes out of it, thus Mike's construction became such as we see in the lower picture. How many cubes has Lily taken? | 7 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many points are there in the three unseen sides of dice? | 11 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many plums (see the picture) weigh as much as an apple? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Tom bought a chocolate heart (see the picture) to Mary on her birthday.
How many grams did the chocolate weigh, if each square weighs 10 grams? | 140 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The ladybird would like to sit on his flower. The flower has five petals and the stem has three leaves. On which flower should the ladybird sit? | B | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Theresa moves a pencil along the line. She starts at the arrow shown. In which order will she go past the shapes?
(A) $\Delta, \square, \bullet$
(B) $\Delta, \bullet, \square$
(C) $\bullet, \Delta, \square$
(D) $\square, \Delta, \bullet$
(E) $\square, \bullet, \Delta$ | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
There are more grey squares than white. How many more? | 9 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
A big square is made from 25 small squares put together. A few of the small squares have been lost. How many have been lost? | 10 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The kangaroo is inside how many circles? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Seven sticks lay on top of each other. Stick 2 lays right at the bottom. Stick 6 lays right on top. Which stick lays exactly in the middle? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Each of the digits 2, 3, 4 and 5 will be placed in a square. Then there will be two numbers, which will be added together. What is the biggest number that they could make? | 95 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Ingrid has 4 red, 3 blue, 2 green and 1 yellow cube. She uses them to build the following object:
Cubes with the same colour don't touch each other. Which colour is the cube with the question mark?
(A) red
(B) blue
(C) green
(D) Yellow
(E) This cannot be worked out for certain. | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many triangles can you find in the picture? | 5 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many dots do all ladybirds have together? | 19 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Every one of these six building blocks consists of 5 little cubes. The little cubes are either white or grey. Cubes of equal colour don't touch each other. How many little white cubes are there in total? | 12 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Which point in the labyrinth can we get to, starting at point $O$? | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Max has 10 dice. Which one of the following solids can he build with them? | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Gerda walks along the road and writes down the letters she can see on her right hand side. Which word is formed while Gerda walks from point 1 to point 2?
(A) KNAO
(B) KNGO
(C) KNR
(D) AGRO
(E) KAO | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Into how many pieces will the string be cut? | 9 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many blocks are missing in this igloo? | 10 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Four of the numbers 1,3,4,5 and 7 are written into the boxes so that the calculation is correct.
Which number was not used? | 4 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Each one of the four keys locks exactly one padlock. Every letter on a padlock stands for exactly one digit. Same letters mean same digits.
Which letters must be written on the fourth padlock?
(A) GDA
(B) ADG
(C) GAD
(D) GAG
(E) DAD | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Peter has drawn this pattern:
He draws exactly the same pattern once more.
Which point is on his drawing? | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
In order to get to his bone, the dog has to follow the black line. In total he turns 3-times to the right and 2-times to the left.
Which path does he take? | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Charles cuts a rope into 3 equally long pieces. Then he makes one knot in one of the pieces, 2 in the next and in the third piece 3 knots. Then he lays the three pieces down in a random order. Which picture does he see? | B | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Jörg is sorting his socks. Two socks with the same number are one pair.
How many pairs can he find? | 5 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Five equally big square pieces of card are placed on a table on top of each other. The picture on the side is created this way. The cards are collected up from top to bottom. In which order are they collected?
(A) 5-4-3-2-1
(B) 5-2-3-4-1
(C) 5-4-2-3-1
(D) 5-3-2-1-4
(E) 5-2-3-1-4 | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The floor of a room is covered with equally big rectangular tiles (see picture). How long is the room?
(A) $6 \mathrm{~m}$
(B) $8 \mathrm{~m}$
(C) $10 \mathrm{~m}$
(D) $11 \mathrm{~m}$
(E) $12 \mathrm{~m}$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The picture shows a mouse and a piece of cheese. The mouse is only allowed to move to the neighbouring fields in the direction of the arrows. How many paths are there from the mouse to the cheese? | 6 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Maria made a block using white cubes and colored cubes in equal amounts. How many of the white cubes cannot be seen in the picture? | 2 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Ana draws some shapes on a sheet. Her drawing has fewer squares than triangles. What could be her drawing? | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
A village of 12 houses has four straight streets and four circular streets. The map shows 11 houses. In each straight street there are three houses and in each circular street there are also three houses. Where should the 12th house be placed on this map?
(A) On A
(B) On B
(C) On C
(D) On D
(E) On E | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Two equal trains, each with 31 numbered wagons, travel in opposite directions. When the wagon number 7 of a train is side by side with the wagon number 12 of the other train, which wagon is side by side with the wagon number 11 ? | 8 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Six different numbers, chosen from integers 1 to 9 , are written on the faces of a cube, one number per face. The sum of the numbers on each pair of opposite faces is always the same. Which of the following numbers could have been written on the opposite side with the number 8 ? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
In the figure, an arrow pointing from one person to another means that the first person is shorter than the second. For example, person $B$ is shorter than person $A$. Which person is the tallest?
(A) Person A
(B) Person B
(C) Person C
(D) Person D
(E) Person E | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Maia the bee can only walk on colorful houses. How many ways can you color exactly three white houses with the same color so that Maia can walk from $A$ to $B$ ? | 16 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The picture shows 2 mushrooms. What is the difference between their heights? | 5 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
These children are standing in a line. Some are facing forwards and others are facing backwards. How many children are holding another child's hand with their right hand? | 6 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
In the Kangaroo constellation, all stars have a number greater than 3 and their sum is 20 . Which is the Kangaroo constellation? | B | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Edmund cut a ribbon as shown in the picture. How many pieces of the ribbon did he finish with? | 12 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The numbers in the five circles around each house add up to 20 . Some numbers are missing.
Which number does the question mark stand for? | 9 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The two markers with a question mark have the same number.
Which number do you have to put instead of the question mark so that the calculation is correct? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Each of the children Ali, Lea, Josef, Vittorio and Sophie get a birthday cake. The number on top of the cake shows how old the child is. Lea is two years older than Josef, but one year younger than Ali. Vittorio is the youngest. Which cake belongs to Sophie? | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
According to the rule given in the left picture below, we construct a numerical triangle with an integer number greater than 1 in each cell. Which of the numbers given in the answers cannot appear in the shaded cell?
(A) 154
(B) 100
(C) 90
(D) 88
(E) 60 | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
We first draw an equilateral triangle, then draw the circumcircle of this triangle, then circumscribe a square to this circle. After drawing another circumcircle, we circumscribe a regular pentagon to this circle, and so on. We repeat this construction with new circles and new regular polygons (each with one side more than the preceding one) until we draw a 16 -sided regular polygon. How many disjoint regions are there inside the last polygon? | 248 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
In the picture $A B C D$ is a rectangle with $A B=16, B C=12$. Let $E$ be such a point that $A C \perp C E, C E=15$. If $F$ is the point of intersection of segments $A E$ and $C D$, then the area of the triangle $A C F$ is equal to | 75 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
A circle $K$ is inscribed in a quarter circle with radius 6 as shown in the figure. What is the radius of circle $K$?
(A) $\frac{6-\sqrt{2}}{2}$
(B) $\frac{3 \sqrt{2}}{2}$
(C) 2.5
(D) 3
(E) $6(\sqrt{2}-1)$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 $x$?
(A) $f(x)=-g(x)+2$
(B) $f(x)=-g(x)-2$
(C) $f(x)=-g(x+2)$
(D) $f(x+2)=-g(x)$
(E) $f(x+1)=-g(x-1)$ | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many triangles can be drawn with vertices in the 18 points shown in the figure? | 711 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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:
(A) $540 \mathrm{~g}$
(B) $570 \mathrm{~g}$
(C) $600 \mathrm{~g}$
(D) $630 \mathrm{~g}$
(E) $660 \mathrm{~g}$ | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
We are given three semi-circles as shown. $A B E F$ is a rectangle and the radius of each of the semi-circles is $2 \mathrm{~cm}$. $E$ and $F$ are the centers of the bottom semi-circles. The area of the shaded region (in $\mathrm{cm}^{2}$) is: | 8 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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:
(A) $90^{\circ}$
(B) $100^{\circ}$
(C) $110^{\circ}$
(D) $120^{\circ}$
(E) $135^{\circ}$ | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 $A D$?
(A) $\cos \beta+\tg \beta$
(B) $\frac{1}{\cos (2 \beta)}$
(C) $\cos ^{2} \beta$
(D) $\cos (2 \beta)$
(E) $\frac{1}{\cos ^{2} \beta}$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The cells of a $4 \times 4$ table are coloured black and white as shown in the left figure. One move allows us to exchange any two cells positioned in the same row or in the same column. What is the least number of moves necessary to obtain in the right figure? | 4 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Points $M$ and $N$ are given on the sides $A B$ and $B C$ of a rectangle $A B C D$. Then the rectangle is divided into several parts as shown in the picture. The areas of 3 parts are also given in the picture. Find the area of the quadrilateral marked with "?". | 25 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
A die is in the position shown in the picture. It can be rolled along the path of 12 squares as shown. How many times must the die go around the path in order for it to return to its initial position with all faces in the initial positions? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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
(A) $\pi$
(B) $1.5 \pi$
(C) $2 \pi$
(D) $3 \pi$
(E) Not enough data | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
We see in the diagram at the right a piece of the graphic of the function
$$
f(x)=a x^{3}+b x^{2}+c x+d.
$$
What is the value of $b$? | -2 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 $B$?
(A) $\frac{1}{3}$
(B) $\frac{5}{4}$
(C) $\frac{2}{9}$
(D) $\frac{1}{2}$
(E) $\frac{1}{4}$ | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
We take three points from the grid so that they were collinear. How many possibilities do we have? | 20 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
In the figure each asterisk stands for one digit. The sum of the digits of the product is equal to | 16 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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
(A) $\frac{1}{24}$
(B) $\frac{1}{16}$
(C) $\frac{1}{8}$
(D) $\frac{1}{12}$
(E) $\frac{2}{13}$ | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
We used metal rods to build this nice ensemble. We know there are 61 octagons. How many rods are there? | 446 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The suare in the diagram has side length 1. The radius of the small circle would
then be of the length
(A) $\sqrt{2}-1$
(B) $\frac{1}{4}$
(C) $\frac{\sqrt{2}}{4}$
(D) $1-\frac{\sqrt{2}}{2}$
(E) $(\sqrt{2}-1)^{2}$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 13 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) either A or B
(B) only C
(C) only D
(D) either C or D
(E) A, B, C or D | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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.)
(A) 7
(B) $2 \sqrt{13}$
(C) 8
(D) $4 \sqrt{3}$
(E) $2 \cdot(\sqrt{2}+\sqrt{3})$ | B | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 9 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 $a$?
(A) 16
(B) 51
(C) 54
(D) 55
(E) 110 | D | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) $8+2 \sqrt{2}$
(B) $11-\sqrt{2}$
(C) 10
(D) $8+3 \sqrt{2}$
(E) $11+\frac{\sqrt{2}}{2}$ | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
Which of the following graphs represents the solution set of $(x-|x|)^{2}+(y-|y|)^{2}=4$? | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
A strip of paper is folded three times as shown. Determine $\beta$ if $\alpha=70^{\circ}$.
(A) $140^{\circ}$
(B) $130^{\circ}$
(C) $120^{\circ}$
(D) $110^{\circ}$
(E) $100^{\circ}$ | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(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 | B | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 '$x$'? | 1 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) $a>0$
(B) $b<0$
(C) $a+b+c<0$
(D) $b^{2}>4 a c$
(E) $c<0$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 5 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 4.8 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(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 $\operatorname{Vol}\left(\mathrm{E}_{\mathrm{x}}\right)=\operatorname{Vol}\left(\mathrm{E}_{\mathrm{y}}\right)$ | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) $4 \pi$
(B) $\frac{28}{3} \pi$
(C) $8 \pi$
(D) $\frac{14}{3} \pi$
(E) $\frac{21}{2} \pi$ | B | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 5 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 $\angle M C N=43^{\circ}$
(A) $86^{\circ}$
(B) $89^{\circ}$
(C) $90^{\circ}$
(D) $92^{\circ}$
(E) $94^{\circ}$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 64 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) $\frac{2}{5}$
(B) $\frac{1}{4}$
(C) $\frac{1}{\sqrt{2}}$
(D) $\frac{1}{\sqrt{5}}$
(E) $\frac{1}{2}$ | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) $720^{\circ}$
(B) $1080^{\circ}$
(C) $1200^{\circ}$
(D) $1440^{\circ}$
(E) $1800^{\circ}$ | B | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) $\sqrt{6}$
(B) 3
(C) $\frac{3 \sqrt{3}}{2}$
(D) $2 \sqrt{2}$
(E) 6 | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) $a$
(B) $b$
(C) $c$
(D) $d$
(E) none of them | A | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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? | 3 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
The rectangles $S_{1}$ and $S_{2}$ shown in the picture have the same area. Determine the ratio $x: y$.
(A) $1: 1$
(B) $3: 2$
(C) $4: 3$
(D) $7: 4$
(E) $8: 5$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 $\angle A X P$?
(A) $30^{\circ}$
(B) $24^{\circ}$
(C) $18^{\circ}$
(D) $15^{\circ}$
(E) $10^{\circ}$ | E | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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 $\angle A B D$?
(A) $66^{\circ}$
(B) $70.5^{\circ}$
(C) $72^{\circ}$
(D) $75^{\circ}$
(E) $77.5^{\circ}$ | C | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
How many quadratic functions $y=a x^{2}+b x+c$ (with $a \neq 0$ ) have graphs that go through at least 3 of the marked points? | 22 | |
<image>Please solve the problem step by step and put your answer in one "\boxed{}". If it is a multiple choice question, only one letter is allowed in the "\boxed{}".
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?
(A) less than 10
(B) 10
(C) 12
(D) more than 12
(E) This colouring cannot be obtained. | A |
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