Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: Rita numbered the circles of the figure from 1 to 8 , so that the sum of the three numbers on each of the four sides of the square equals 13 . What is the sum of the four numbers written on the colored circles?
<image1>
Answer: 16
Problem: <image>Question: 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?
<image1>
Choices:
A. Person A
B. Person B
C. Person C
D. Person D
E. Person E
Answer: C
Problem: <image>Question: 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 $B$ ?
<image1>
Answer: 16
Problem: <image>Question: A kangaroo laid out 3 sticks like this to make a shape. It is not allowed to break or to bend the sticks. Which shape could the kangaroo make?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The picture shows 2 mushrooms. What is the difference between their heights?
<image1>
Answer: 5
Problem: <image>Question: Which of the paths shown in the pictures is the longest?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Four identical pieces of paper are placed as shown. Michael wants to punch a hole that goes through all four pieces. At which point should Michael punch the hole?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: Ella puts on this t-shirt and stands in front of a mirror. Which of these images does she see in the mirror?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: 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?
<image1>
Answer: 6
Problem: <image>Question: In the Kangaroo constellation, all stars have a number greater than 3 and their sum is 20 . Which is the Kangaroo constellation?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: Edmund cut a ribbon as shown in the picture. How many pieces of the ribbon did he finish with?
<image1>
Answer: 12
Problem: <image>Question: Rose the cat walks along the wall. She starts at point $B$ and follows the direction of the arrows shown in the picture. The cat walks a total of 20 metres. Where does she end up?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: Julia has two pots with flowers, as shown. She keeps the flowers exactly where they are. She buys more flowers and puts them in the pots. After that, each pot has the same number of each type of flower. What is the smallest number of flowers she needs to buy?
<image1>
Answer: 8
Problem: <image>Question: Tom encodes words using the board shown. For example, the word PIZZA has the code $A 2 A 4 C 1 C 1 B 2$. What word did Tom encode as B3B2C4D2?
<image1>
Choices:
A. MAZE
B. MASK
C. MILK
D. MATE
E. MATH
Answer: E
Problem: <image>Question: Which figure can be made from the 2 pieces shown on the right?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: The picture shows the five houses of five friends and their school. The school is the largest building in the picture. To go to school, Doris and Ali walk past Leo's house. Eva walks past Chole's house. Which is Eva's house?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: Mara built the square by using 4 of the following 5 shapes. Which shape was not used?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: Every time the witch has 3 apples she turns them in to 1 banana. Every time she has 3 bananas she turns them in to 1 apple. What will she finish with if she starts with 4 apples and 5 bananas?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: The picture beside shows two cogs, each with a black tooth. Where will the black teeth be after the small cog has made one full turn?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: Each participant in a cooking contest baked one tray of cookies like the one shown beside. What is the smallest number of trays of cookies needed to make the following plate?
<image1>
Answer: 3
Problem: <image>Question: Stan has five toys: a ball, a set of blocks, a game, a puzzle and a car. He puts each toy on a different shelf of the bookcase. The ball is higher than the blocks and lower than the car. The game is directly above the ball. On which shelf can the puzzle not be placed?
<image1>
Answer: 3
Problem: <image>Question: In which box are the most triangles?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: A sandwich and a juice cost 12 Euros together. A sandwich and two juices cost 14 Euros together. How many Euros does one juice cost?
<image1>
Answer: 2
Problem: <image>Question: Anna cuts the picture of a mushroom in two halves.
<image1>
She then arranges the two pieces together to form a new picture. What could this new picture look like?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: In the four squares of a row there always have to be exactly two coins. In the four squares below each other there also always have to be exactly two coins.
<image1>
On which square does one more coin have to be placed?
Choices:
A. square $A$
B. square $B$
C. square $C$
D. square $D$
E. square $E$
Answer: D
Problem: <image>Question: A monkey has torn off a piece of Captain Jack's map.
<image1>
What does the piece the monkey has torn off look like?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: These five animals are made up from different shapes. There is one shape which is only used on one animal. On which animal is this shape used?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: There is an animal asleep in each of the five baskets. The koala and the fox sleep in baskets with the same pattern and the same shape. The kangaroo and the rabbit sleep in baskets with the same pattern.
<image1>
In which basket does the mouse sleep?
Choices:
A. Basket 1
B. Basket 2
C. Basket 3
D. Basket 4
E. Basket 5
Answer: E
Problem: <image>Question: The picture shows one object made up of 5 identical building blocks.
<image1>
How many building blocks touch exactly 3 others?
Answer: 2
Problem: <image>Question: The kangaroo wants to visit the koala. On its way it is not allowed to jump through a square with water. Each arrow shows one jump on to a neighbouring field.
<image1>
Which path is the kangaroo allowed to take?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: Carl writes down a five-digit number.
He then places a shape on each of the five digits (see picture).
He places different shapes on different digits.
He places the same shape on the same digits.
Which number did Carl hide?
<image1>
Answer: 34426
Problem: <image>Question: Katrin forms a path around each square. For that she uses stones like this
<image1>
How many such stones does she need for a path around the square with side length 5?
Answer: 12
Problem: <image>Question: Below you see five pieces of lawn. Which one has the smallest area of grass?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: The numbers in the five circles around each house add up to 20 . Some numbers are missing.
<image1>
Which number does the question mark stand for?
Answer: 9
Problem: <image>Question: Dino walks from the entrance to the exit. He is only allowed to go through each room once. The rooms have numbers (see diagram). Dino adds up all the numbers of the rooms he walks through.
<image1>
What is the biggest result he can get this way?
Answer: 34
Problem: <image>Question: Out of how many circles is the beaver made of?
<image1>
Answer: 8
Problem: <image>Question: The picture shows 5 cubes from the front. What do they look like from above?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: Each bowl has 4 balls. Add up the numbers on the balls. In which bowl is the result biggest?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Mr Beaver re-arranges the parts to build a kangaroo.
<image1>
Which part is missing?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Sara says: "My boat has more than one circle. It also has 2 triangles more than squares." Which boat belongs to Sara?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The bee on the right has a few pieces missing. Each piece costs points (Punkte).
<image1>
How many points does Maya need to complete the bee?
Answer: 13
Problem: <image>Question: Susi folds a piece of paper in the middle. She stamps 2 holes.
<image1>
What does the piece of paper look like when she unfolds it again?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: Hansi sticks 12 cubes together to make this figure. He always puts one drop of glue between two cubes. How many drops of glue does he need?
<image1>
Answer: 11
Problem: <image>Question: The two markers with a question mark have the same number.
<image1>
Which number do you have to put instead of the question mark so that the calculation is correct?
Answer: 3
Problem: <image>Question: Max wants to complete the jigsaw shown. He has different pieces.
<image1>
Which pieces does he have to use?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Elvis has 6 triangles with this pattern
<image1>
Which picture can he make with them?
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: 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?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: Maria has a total of 19 apples in 3 bags. She takes the same amount of apples from each bag. Then there are 3, 4 and 6 apples in the bags.
<image1>
How many apples did Maria take from each bag?
Answer: 2
Problem: <image>Question: 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?
<image1>
Choices:
A. 154
B. 100
C. 90
D. 88
E. 60
Answer: A
Problem: <image>Question: Let $A B C$ be a triangle with area 30. Let $D$ be any point in its interior and let $e, f$ and $g$ denote the distances from $D$ to the sides of the triangle. What is the value of the expression $5 e+12 f+13 g$?
<image1>
Answer: 60
Problem: <image>Question: The diagram shows two squares: one has a side with a length of 2 and the other (abut on the first square) has a side with a length of 1. What is the area of the shaded zone?
<image1>
Answer: 1
Problem: <image>Question: 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?
<image1>
Answer: 248
Problem: <image>Question: 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
<image1>
Answer: 75
Problem: <image>Question: In a square of side $s$, where $s$ is an odd integer, the squares of side 1 on the diagonals are colored (like in the picture, where the square is of side 7). How many white squares are there?
<image1>
Choices:
A. $s^{2}+1-2 s$
B. $s^{2}+4-4 s$
C. $2 s^{2}+1-4 s$
D. $s^{2}-1-2 s$
E. $s^{2}-2 s$
Answer: A
Problem: <image>Question: On the circumference of radius $r$ three points $X, Y$ and $A$ are marked such that $X Y=r, X Y \perp A Y$ (see the figure). How many degrees has the angle $X A Y$?
<image1>
Answer: 30
Problem: <image>Question: The set of all pairs $(x, y)$ which satisfy conditions $x y \leqslant 0$ and $x^{2}+y^{2}=4$ is on the graph:
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: C
Problem: <image>Question: In the figure the two equilateral triangles $A B C$ and $E C D$ have sides of length 2 and 1 respectively. The area of the quadrilateral $A B C E$ is:
<image1>
Choices:
A. $\frac{5 \sqrt{3}}{3}$
B. $\frac{4+5 \sqrt{3}}{5}$
C. 3
D. $\frac{6+\sqrt{3}}{4}$
E. $\frac{3 \sqrt{3}}{2}$
Answer: E
Problem: <image>Question: A circle $K$ is inscribed in a quarter circle with radius 6 as shown in the figure. What is the radius of circle $K$?
<image1>
Choices:
A. $\frac{6-\sqrt{2}}{2}$
B. $\frac{3 \sqrt{2}}{2}$
C. 2.5
D. 3
E. $6(\sqrt{2}-1)$
Answer: E
Problem: <image>Question: 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>
Choices:
A. $4,5,8,9$
B. $3,5,6,7$
C. $5,6,7,12$
D. $10,11,12,19$
E. $5,6,8,10$
Answer: A
Problem: <image>Question: 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>
Choices:
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. $f(x+1)=-g(x-1)$
Answer: C
Problem: <image>Question: How many triangles can be drawn with vertices in the 18 points shown in the figure?
<image1>
Answer: 711
Problem: <image>Question: 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>
Choices:
A. $\frac{\sqrt{3}}{3}$
B. $\frac{2 \sqrt{3}}{3}$
C. $\sqrt{3}$
D. $\frac{4 \sqrt{3}}{3}$
E. other answer
Answer: D
Problem: <image>Question: Five cards are lying on the table in the order 1,3,5,4,2. You must get the cards in the order $1,2,3,4,5$. Per move, any two cards may be interchanged. How many moves do you need at least?
<image1>
Answer: 2
Problem: <image>Question: A square piece of paper has been cut in three pieces. Two of them are in the picture on the right. What is the third one?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: 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>
Choices:
A. $540 \mathrm{~g}$
B. $570 \mathrm{~g}$
C. $600 \mathrm{~g}$
D. $630 \mathrm{~g}$
E. $660 \mathrm{~g}$
Answer: C
Problem: <image>Question: 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:
<image1>
Answer: 8
Problem: <image>Question: 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>
Choices:
A. 12
B. 16
C. $8 \sqrt{2}$
D. $8 \sqrt{3}$
E. Another value
Answer: D
Problem: <image>Question: 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>
Choices:
A. $90^{\circ}$
B. $100^{\circ}$
C. $110^{\circ}$
D. $120^{\circ}$
E. $135^{\circ}$
Answer: D
Problem: <image>Question: We consider the perimeter and the area of the region corresponding to the grey squares. How many more squares can we colour grey for the grey area to increase without increasing its perimeter?
<image1>
Answer: 16
Problem: <image>Question: There are four cards on the table as in the picture. Every card has a letter on one side and a number on the other side. Peter said: "For every card on the table it is true that if there is a vowel on one side, there is an even number on the other side." What is the smallest number of cards Alice must turn in order to check whether Peter said the truth?
<image1>
Answer: 2
Problem: <image>Question: 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>
Choices:
A. $4 \mathrm{~cm}$
B. $2 \sqrt{6} \mathrm{~cm}$
C. $5 \mathrm{~cm}$
D. $2 \sqrt{5} \mathrm{~cm}$
E. $\sqrt{10} \mathrm{~cm}$
Answer: D
Problem: <image>Question: 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>
Choices:
A. $\cos \beta+\tg \beta$
B. $\frac{1}{\cos (2 \beta)}$
C. $\cos ^{2} \beta$
D. $\cos (2 \beta)$
E. $\frac{1}{\cos ^{2} \beta}$
Answer: E
Problem: <image>Question: 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>
Choices:
A. $10 \sqrt{2}$
B. $4 \sqrt{5}$
C. $\frac{20}{3}$
D. 12.5
E. 10
Answer: A
Problem: <image>Question: 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>
Choices:
A. $3: 2$
B. $4: 3$
C. $\sqrt{3}: 1$
D. $2: 1$
E. $9: 1$
Answer: A
Problem: <image>Question: 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?
<image1>
Answer: 4
Problem: <image>Question: In a church there is a rose window as in the figure, where the letters R, G and B represent glass of red colour, green colour and blue colour, respectively. Knowing that $400 \mathrm{~cm}^{2}$ of green glass have been used, how many $\mathrm{cm}^{2}$ of blue glass are necessary?
<image1>
Answer: 400
Problem: <image>Question: The lengths of the sides of triangle $X Y Z$ are $X Z=\sqrt{55}$, $X Y=8, Y Z=9$. Find the length of the diagonal $X A$ of the rectangular parallelepiped in the figure.
<image1>
Answer: 10
Problem: <image>Question: 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 "?".
<image1>
Answer: 25
Problem: <image>Question: 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>
Choices:
A. $3 \cdot 2^{5}$
B. $3^{6}$
C. $6^{3}$
D. $2 \cdot 3^{5}$
E. $3 \cdot 5^{2}$
Answer: A
Problem: <image>Question: 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?
<image1>
Answer: 3
Problem: <image>Question: 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>
Choices:
A. $\frac{5-\sqrt{3}}{4}$
B. $\frac{\sqrt{3}+1}{2}$
C. $\frac{\sqrt{3}}{4}$
D. $\frac{2-\sqrt{3}}{4}$
E. $\frac{2+\sqrt{3}}{4}$
Answer: A
Problem: <image>Question: The shaded area is equal to $\sqrt{3}$. What is the area of the triangle $A B C$?
<image1>
Choices:
A. $2 \sqrt{3}$
B. 2
C. 5
D. 6
E. $4 \sqrt{3}$
Answer: A
Problem: <image>Question: The billiard ball meets the board under $45^{\circ}$ as shown. Which pocket will it fall into?
<image1>
Choices:
A. $A$
B. $B$
C. $C$
D. $D$
E. Neither of the pockets
Answer: C
Problem: <image>Question: 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>
Choices:
A. $1: 2$
B. $2: 3$
C. $2: 1$
D. $3: 2$
E. $1: 1$
Answer: E
Problem: <image>Question: A mathematically skilled spider spins a cobweb and some of the strings have lengths as shown in the picture. If $x$ is an integer, determine the value of $x$.
<image1>
Answer: 13
Problem: <image>Question: 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>
Choices:
A. $\pi$
B. $1.5 \pi$
C. $2 \pi$
D. $3 \pi$
E. Not enough data
Answer: C
Problem: <image>Question: Which is the graph of the function $y=\sqrt{|(1+x)(1-|x|)|}$?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: D
Problem: <image>Question: 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 $b$?
<image1>
Answer: -2
Problem: <image>Question: The digits of the sequence $123451234512345 \ldots$ fill the cells on a sheet of paper in a spiral-like manner beginning with the marked cell (see the figure). Which digit is written in the cell being 100 cells above the marked one?
<image1>
Answer: 1
Problem: <image>Question: Numbers 3,4 and two other unknown numbers are written in the cells of the $2 \times 2$ table. It is known that the sums of numbers in the rows are equal to 5 and 10, and the sum of numbers in one of the columns is equal to 9. The larger number of the two unknown ones is
<image1>
Answer: 6
Problem: <image>Question: 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>
Choices:
A. $\frac{1}{3}$
B. $\frac{5}{4}$
C. $\frac{2}{9}$
D. $\frac{1}{2}$
E. $\frac{1}{4}$
Answer: D
Problem: <image>Question: 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>
Choices:
A. $98^{\circ}$
B. $100^{\circ}$
C. $104^{\circ}$
D. $108^{\circ}$
E. $110^{\circ}$
Answer: D
Problem: <image>Question: The figure shows a circle with the diameter $A B$ and point $D$ on it. Find $d$.
<image1>
Answer: 4
Problem: <image>Question: 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>
Choices:
A. $\sqrt{17}$
B. 7
C. $\sqrt{13}$
D. $\sqrt{7}$
E. $\sqrt{14}$
Answer: A
Problem: <image>Question: We take three points from the grid so that they were collinear. How many possibilities do we have?
<image1>
Answer: 20
Problem: <image>Question: In the figure each asterisk stands for one digit. The sum of the digits of the product is equal to
<image1>
Answer: 16
Problem: <image>Question: A circle is inscribed in the triangle $A B C$ (see the figure), $A C=5, A B=6, B C=3$. The segment $E D$ is tangent to the circle. The perimeter of the triangle $A D E$ is
<image1>
Answer: 8
Problem: <image>Question: 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>
Choices:
A. $\frac{1}{24}$
B. $\frac{1}{16}$
C. $\frac{1}{8}$
D. $\frac{1}{12}$
E. $\frac{2}{13}$
Answer: D