– 3 Point Examples – 1. Alice draws lines between the beetles. She starts with the beetle with the fewest points. Then she continues drawing to the beetle with one more point. Which figure is formed?
(A) (B) (C) (D) (E)
2. The same amount of kangaroos should be in both parks. How many kangaroos have to be moved from the left park to the right park for that to happen?
(A) 4 (B) 5 (C) 6 (D) 8 (E) 9
3. Which beetle has to fly away so that the remaining beetles have 20 dots altogether?
(A) Beetle with 4 points (B) Beetle with 7 points (C) Beetle with 5 points (D) Beetle with 6 points (E) no beetle
4. Peter has drawn this pattern:
He draws exactly the same pattern once more. Which point is on his drawing?
(A) A (B) B (C) C (D) D (E) E
5. Theodor has built this tower made up of discs. He looks at the tower from above. How many discs does he see?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
– 4 Point Examples – 6. This diagram shows two see-through sheets. You place the sheets on top of each other. Which pattern do you get?
(A) (B) (C) (D) (E)
7. 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?
(A) (B) (C)
8. Lisa needs exactly 3 pieces to complete her jigsaw. Which of the 4 pieces is left over?
(D) (E)
(A) A (B) B (C) C (D) D
(E) C or D
9. 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?
(A) (B) (C) 10. How many of the hands pictured show a right hand?
(D) (E)
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
– 5 Point Examples – 11. The number of spots on the fly agarics (toadstools) shows how many dwarfs fit under it. We can see one side of the fungi. The other side has the same amount of spots. When it rains 36 dwarfs are trying to hide under the fungi. How many dwarfs get wet?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
12. You are forming two-digit numbers using the digits 2, 0, 1 or 8. They have to be bigger than 10 and smaller than 25. Every number is made up of two different digits. How many different numbers to you get? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 13. Alice has 3 white, 2 black and 2 grey pieces of paper. First she cuts every piece of paper that is not black into two pieces. Then she halves every piece of paper that is not white. How many pieces of paper does she obtain in total? (A) 14 (B) 16 (C) 17 (D) 18 (E) 20 14. Susi makes this pattern using ice-lolly sticks. Each stick is 5 cm long and 1 cm wide. How long is Susi’s pattern?
(A) 20 cm (B) 21 cm (C) 22 cm (D) 23 cm (E) 25 cm 15. The road from Anna’s to Mary’s house is 16 km long. The road from Mary’s to John’s house is 20 km long. The road from the crossing to Mary’s house is 9 km long. How long is the road from Anna’s to John’s house?
(A) 7 km
(B) 9 km
(C) 11 km
(D) 16 km
(E) 18 km
Känguru der Mathematik 2018 Level Ecolier (Grade 3 and 4)
Austria – 15. 3. 2018
- 3 Point Examples -
1. As seen in the diagram, 3 darts are flying towards 9 fixed balloons. If a balloon is hit by a dart, it bursts and the dart continues in the same direction it had beforehand. How many balloons are hit by the darts?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
2. Susanne is 6 years old. Her sister Lisa is 2 years younger. Brother Max is 2 years older than Susanne. How old are the 3 siblings altogether? (A) 15 (B) 16 (C) 17 (D) 18 (E) 19
3. The diagram shows a wooden block with 5 screws. 4 of which are equally long, one screw is shorter.
Which is the shorter screw? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
4. Leonie has one stamp for each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using them, she stamps the date of the kangaroo-competition.
How many of the stamps does Leonie use to do that? (A) 5 (B) 6 (C) 7 (D) 9 (E) 10
5. On the right you can see a picture of ladybird Sophie. Sophie turns.
Which of the pictures below is not Sophie? (A) (B) (C) (D)
6. Lucy folds a piece of paper exactly half way and then cuts out a figure: Then she unfolds the paper again. Which of the five pictures can she see? (A) (B) (C) (D)
7. Mike sets the table for 8 people: The fork has to lie to the left and the knife to the right of the plate.
For how many people is the cutlery set correctly? (A) 5 (B) 4 (C) 6 (D) 2 (E) 3
(E)
(E)
8. Using these tiles Robert makes different patterns. How many of the patterns shown below can he make?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
- 4 Point Examples - 9. Diana shoots 3 darts, three times at a target board with two fields. The first time she scores 12 points, the second time 15. The number of points depends on which field she has hit.
How many points does she score the third time? (A) 18 (B) 19 (C) 20 (D) 21 (E) 22 10.
12 Points 15 Points ?
Albert places these 5 figures , , , , on a 5x5-grid. Each figure is only allowed to appear once in every column and in every row.
Which figure does Albert have to place on the field with the question mark?
(A) (B) 11. Tom wants to completely cover his paper boat using the shapes 12. The two colours of this picture are swapped. Then the picture is turned.
Which of the pictures below is obtained?
and .
What is the smallest number of shapes he needs for that? (A) 5 (B) 6 (C) 7 (D) 8
(C)
(D)
(E)
(E) 9
(A) (B) (C) (D) (E) 13. Felix the rabbit has 20 carrots. Every day he eats 2 of them. He has eaten the 12th carrot on a Wednesday. On which day of the week did he start eating the carrots? (A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) Friday 14. A rose bush has 8 flowers on which butterflies and dragonflies are sitting. On every flower there is at most one insect sitting on it. More than half of the flowers are occupied.
15.
The number of butterflies is twice as big as the number of dragonflies. How many butterflies are sitting on the rose blossoms? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
(A) 17 km (B) 23 km (C) 26 km (D) 33 km (E) 35 km 16. Tobias glues 10 cubes together so that the following object is formed: He paints all of it, even the bottom.
How many cubes then have exactly 4 faces coloured in?
(A) 6 (B) 7 (C) 8 (D) 9 (E) 10 The map shows the roundtrip that Captain Bluebear covers during his journey. Three distances are given on the map.
He sails from island to island and starts at the island Berg. In total he covers a distance of 100 km. The
distances between the islands Wüste and Wald is equal to the distance between the islands Berg and Blume via Vulkan.
How big is the distance between Berg and Wald?
- 5 Point Examples - 17. The big rectangle consists of various squares of different sizes. Each of the three smallest squares has area 1. How big is the area of the big rectangle?
(A) 65 (B) 71 (C) 77 (D) 87 (E) 98 18. In order to slay a dragon, Mathias has to cut off all of its heads. As soon as he has cut off 3 heads, a new one grows back immediately. After Mathias has cut off 13 heads the dragon is dead. How many heads did the dragon have initially?
Start
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12 19. The rooms in Kanga’s house are numbered. Eva enters the house through the main entrance. Eva has to walk through the rooms in such a way that each room that she enters has a number higher than the previous one. Through which door does Eva leave the house? (A) A (B) B (C) C (D) D (E) E
20. The symbols It is known that
stand for one of the digits 1, 2, 3, 4 or 5.
Which symbol stands for the digit 3?
(D)
(E)
(A) (B) (C) 21. A belt can be joined together in 5 different ways.
How many cm is the belt longer if it is only closed in the first hole instead of in all 5 holes?
(A) 4 cm
(B) 8 cm
(C) 10 cm
(D) 16 cm (E) 20 cm
22. A decorated glass tile is mirrored several times along the boldly printed edge. The first mirror image is shown.
spiegeln
What does the tile on the far right look like after the third reflection?
(A) (B) (C) (D) (E) 23. Lea should write the numbers 1 to 7 in the fields of the given figure. There is only one number allowed in every field.
Two consecutive numbers are not allowed to be in adjacent fields. Two fields are adjacent if they have one edge or one corner in common.
Which numbers can she write into the field with the question mark?
(A) all 7 numbers (B) only odd numbers (C) only even numbers (D) the number 4 (E) the numbers 1 or 7 24. Each of the four balls weighs either 10 or 20 or 30 or 40 grams. Which ball weighs 30 grams?
(A) A
(B) B
(C) C
(D) D
(E) It can be A or B.
Känguru der Mathematik 2018 Group Benjamin (Grade 5 and 6)
Austria – 15. 3. 2018
- 3 Points Examples - 1. As seen in the diagram, three darts are thrown at nine fixed
balloons. If a balloon is hit it will burst and the dart continues in the same direction it had beforehand. How many balloons will not be hit by a dart? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
2. Peter places three
building blocks on a table, as shown.
What does he see when he is looking at them from above?
(A) (B) (C)
(D) (E)
3. If you hit the target board, you score points.
The number of points depends on which one of the three areas you hit. Diana throws two darts, three times at the target board. On the first attempt she scores 14 points and on the second 16 points. How many points does she score on the third attempt?
(A) 17 (B) 18 (C) 19 (D) 20 (E) 22 14 Points 16 Points ???
4. A garden is split into equally sized square-shaped lots. A fast and a slow snail crawl in different directions along the outside edge of the garden. Both start at the corner S. The slow snail crawls 1 m in one hour and the fast one crawls 2 m in one hour. In which position will the two snails meet for the first time? (A) A (B) B (C) C (D) D (E) E
5. A star consist of a square and four triangles. All
sides of the triangles are equally long. The perimeter of the square is 36 cm. What is the perimeter of the star?
(A) 144 cm (B) 120 cm (C) 104 cm (D) 90 cm (E) 72 cm
6. A big spot of ink covers most of a calendar page of a certain month. Which day of the week does the 25th day of that month fall on?
(A) Monday (B) Wednesday (C) Thursday (D) Saturday (E) Sunday
7. How many times do you have to roll an ordinary die in order to be certain that at least one number is rolled twice? (A) 5 (B) 6 (C) 7 (D) 12 (E) 18
8. A figure is made up of three squares. The side length of the smallest square is 6 cm. How long is the side length of the biggest square? (A) 8 cm (B) 10 cm (C) 12 cm (D) 14 cm (E) 16 cm
- 4 Point Examples - 9. Alice subtracts one two-digit number from another two-digit number. Afterwards she paints over two digits in the calculation. How big is the sum of the two painted digits? (A) 8 (B) 9 (C) 12 (D) 13 (E) 15
10. In the diagram the circles represent light bulbs which are connected to some other light bulbs. Initially all light bulbs are switched off. If you touch a light bulb then that light bulb and all directly adjacent light bulbs switch themselves on. What is the minimum number of light bulbs you have to touch in order to switch on all the light bulbs? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
11. Four equally big squares are partially coloured in black.
In which of the four squares is the total area of the black parts biggest?
(A) A D
(B) B (C) C (D)
(E) The total area of the black parts is always equally big.
12. The four smudges hide four of the numbers 1, 2, 3, 4, 5. The calculations along the two arrows are correct.
Which number hides behind the smudge with the star? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
13. A lion hides in one of three rooms. On the door to room number 1 a note reads: „The lion is not here“. On the door to room number 2 a note reads: „The lion is here“. On the door to room number 3 a note reads: „2 + 3 = 5“. Exactly one of the three notes is true. In which room is the lion? (A) Room 1 (B) Room 2 (C) Room 3 (D) It can be in any room. (E) It is either in room 1 or room 2.
14. The two girls Eva and Olga and the three boys Adam, Isaac and Urban play together with a ball. If a girl has the ball she throws it either to the second girl or to a boy. Every boy only throws the ball to another boy, however not to the one where the ball has just come from. The first throw is made by Eva to Adam. Who makes the 5th throw? (A) Adam (B) Eva (C) Isaac (D) Olga (E) Urban
15. The faces of a die are either white, grey or black. Opposite faces are always a different colour. Which of the following nets does not belong to such a die?
(A)
(B)
(C)
(D)
(E)
16. From a list with the numbers 1, 2, 3, 4, 5, 6, 7, Monika chooses 3 different numbers whose sum is 8. From the same list Daniel chooses 3 different numbers whose sum is 7. How many of the numbers were chosen by both Monika and Daniel? (A) none (B) 1 (C) 2 (D) 3 (E) It cannot be determined.
- 5 Point Examples - 17. Emily wants to write a number into every free small triangle. The sum of the numbers in two triangles with a common side should always be the same. Two numbers are already given. How big is the sum of all numbers in the figure? (A) 18 (B) 20 (C) 21 (D) 22 (E) it cannot be calculated
18. Instead of digits Hannes uses the letters A, B, C and D in a calculation. Different letters stand for different digits. Which digit does the letter B stand for? (A) 0 (B) 2 (C) 4 (D) 5 (E) 6
19. Four ladybirds each sit on a different cell of a 4 x 4 grid. One is asleep and does not move. On a
whistle the other three each move to an adjacent free cell.
They can crawl up, down, to the right or to the left but are not allowed on any
Initial
account to move back to the cell that
position
they have just come from.
Where could the ladybirds be after the fourth whistle?
After the first whistle After the second whistle After the third
whistle
(A)
(B) (C) (D) (E)
20. The five balls weigh 30 g, 50 g, 50 g, 50 g and 80 g. Which of the balls weighs 30 g?
(A) A (B) B (C) C (D) D (E) E
21. Three different digits A, B and C are chosen. Then the biggest possible six-digit number is built where the digit A appears 3 times, the digit B 2 times and the digit C 1 time.
Which representation is definitely not possible for this number? (A) AAABBC (B) CAAABB (C) BBAAAC (D) AAABCB (E) AAACBB
22. The sum of Kathi’s age and the age of her mother is 36. The sum of the age of her mother and the age of her grandmother is 81. How old was Kathi’s grandmother when Kathi was born? (A) 28 (B) 38 (C) 45 (D) 53 (E) 56
23. Nick wants to split the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10 into some groups so that the sum of the numbers in each group is equally big. What is the biggest number of groups he can build this way? (A) 2 (B) 3 (C) 4 (D) 6 (E) another number
24. The figure shown on the right consists of one square part and eight rectangular parts. Each part is 8 cm wide. Peter assembles all parts to form one long, 8 cm wide rectangle. How long is this rectangle? (A) 150 cm (B) 168 cm (C) 196 cm (D) 200 cm (E) 232 cm
Känguru der Mathematik 2018
Level Kadett (Grade 7 and 8)
Austria – 15. 3. 2018 - 3 Point Examples -
1. Which result is obtained by the calculation (20+18)∶(20−18)? (A) 18 (B) 19 (C) 20 (D) 34 (E) 36
2. If the letters of the Word MAMA are written underneath each other then the word has a vertical axis of symmetry. For which of these words does that also hold true? (A) ADAM (B) BAUM (C) BOOT (D) LOGO (E) TOTO
3. A triangle ABC has side lengths 6 cm, 10 cm and 11 cm. An equilateral triangle XYZ has the same perimeter as the triangle ABC. What are the side lengths of the triangle XYZ? (A) 6 cm (B) 9 cm (C) 10 cm (D) 11 cm (E) 27 cm
4. Which number has to replace the in the calculation so that it is true?
2 18 14 = 6 7 (A) 8
(B) 9 (C) 10 (D) 12 (E) 15
5. The fence on the right has many holes. One morning the fence falls over and lies on the floor. Which of the following pictures shows the fallen down fence?
(A)
(B) (C) (D) (E)
6. Bernd produces steps for a staircase which are 15 cm high and 15 cm deep (see diagram). The staircase should reach from the ground floor to the first floor which is 3 m higher. How many steps does Bernd have to produce?
(A) 8 (B) 10 (C) 15 (D) 20 (E) 25
7. In a game of luck, A ball rolls downwards towards hammered nails and is diverted either to the right or the left by a nail immediately below it. One possible path is shown in the diagram. How many different ways are there for the ball to reach the second compartment from the left? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
8. A large rectangle is made up of 9 equally big rectangles. The longer side of each small rectangle is 10 cm long. What is the perimeter of the large rectangle? (A) 40 cm (B) 48 cm (C) 76 cm (D) 81 cm (E) 90 cm
9. Two circles are inscribed into an 11 cm long and 7 cm wide rectangle so that they each touch three sides of the rectangle. How big is the distance between the centres of the two circles? (A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm (E) 5 cm
10. The square ABCD has side length 3 cm. The points M and N, which lie on the sides AD and AB respectively, are joined to the corner C. That way the square is split up into three parts with equal area. How long is the line segment DM? (A) 0.5 cm (B) 1 cm (C) 1.5 cm (D) 2 cm (E) 2.5 cm
- 4 Point Examples - 11. Martina multiplies two, two-digit numbers and then paints over some of the digits. How big is the sum of the three digits that Martina has painted over? (A) 5 (B) 6 (C) 9 (D) 12 (E) 14
12. A rectangle is split up into 40 equally big squares. The rectangle consists of more than one row of squares. Andreas colours in all squares of the middle row. How many squares did he not colour in? (A) 20 (B) 30 (C) 32 (D) 35 (E) 39
13. Philipp wants to know how much his book weighs correct to half a gram. However, his scale only shows correct to 10 g and therefore he weighs several identical books all together. What is the minimum number of identical books he has to put on the scale in order to reach his aim? (A) 5 (B) 10 (C) 15 (D) 20 (E) 50
14. A lion hides in one of three rooms. On the door to room number 1 a note reads: „The lion is here“. On the door to room number 2 a note reads: „The lion is not here“. On the door to room number 3 a note reads: „2 + 3 = 2 x 3“. Exactly one of the three notes is true. Which room is the lion in? (A) Room 1 (B) Room 2 (C) Room 3 (D) It can be in any room. (E) It is either in room 1 or room 2.
15. Valentin draws a zig-zag line inside
a rectangle as shown in the diagram. For that he uses the angles 10°, 14°, 33° and 26°. How big is angle ? (A) 11° (B) 12° (C) 16° (D) 17° (E) 33°
16. Alice writes down three prime numbers that are all less than 100. She only uses the digits 1, 2, 3, 4 and 5, in fact she uses each digit exactly once. Which of the following prime numbers did she definitely write down? (A) 2 (B) 5 (C) 31 (D) 41 (E) 53
17. A hotel in the carribean correctly advertises using the slogan: „350 days of sun in the year!” How many days does Mr. Happy have to spend in the hotel in a year with 365 days to be guaranteed to have two consecutive days of sunshine to enjoy? (A) 17 (B) 21 (C) 31 (D) 32 (E) 35
18. The diagram shows a rectangle and a straight line x, which is parallel to one of the sides of the rectangle. There are two points A and B on x inside the rectangle. The sum of the areas of the two triangles shaded in grey is 10 cm². How big is the area of the rectangle? (A) 18 cm2 (B) 20 cm2 (C) 22 cm2 (D) 24 cm2 (E) It depends on the position of the points A and B.
19. Jakob writes one of the natural numbers 1 to 9 into each cell of the 3x3-table. Then he works out the sum of the numbers in each row and in each column. Five of his results are 12, 13, 15, 16 and 17. What is the sixth sum? (A) 17 (B) 16 (C) 15 (D) 14 (E) 13
20. 11 points are marked left to right on a straight line and their distances recorded. The sum of the distances from the first point to every other point is 2018. The sum of all distances from the second point to every other point, including the first point, is 2000. What is the distance between the first and the second point? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
- 5 Point Examples - 21. At an election for student representatives there are three candidates. 130 students have voted. The candidate that has the most votes wins. Currently Samuel has 24, Kevin 29 and Alfred 37 votes. How many of the currently not yet counted votes does Alfred need to get in order to definitely win the election? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17
22. The diagram shows the net of a box consisting only of rectangles. How big is the volume of the box?
(A) 43 cm3 (B) 70 cm3 (C) 80 cm3 (D) 100 cm3
(E) 1820 cm3
23. Rita wants to write a number into every square of the diagram shown. Every number should be equal to the sum of the two
numbers from the adjacent squares. Squares
are adjacent if they share one edge. Two numbers are already given. Which number is she going to write into the square marked with x?
(A) 10 (B) 7 (C) 13 (D) −13 (E) −3
24. Simon runs along the edge round a 50 m long rectangular swimming pool, while at the same
time Jan swims lengths in the pool. Simon runs three times as fast as Jan swims. While Jan swims 6 lengths, Simon manages 5 rounds around the pool. How wide is the swimming pool? (A) 25 m (B) 40 m (C) 50 m (D) 80 m (E) 180 m
25. Lisas aviation club designs a flag with a flying „dove“ on a 4x6-grid. The area of the
„dove“ is 192 cm2. The perimeter of the „dove“ is made up of straight lines and circular arcs. What measurements does the flag have? (A) 6 cm x 4 cm (B) 12 cm x 8 cm (C) 20 cm x 12 cm (D) 24 cm x 16 cm (E) 30 cm x 20 cm
26. The points N, M and L lie on the sides of an equilateral triangle ABC so that NM BC, ML AB and LN AC holds true. The area of the triangle ABC is 36 cm2. What is the area of the triangle LMN? (A) 9 cm² (B) 12 cm² (C) 15 cm² (D) 16 cm² (E) 18 cm²
27. Anna, Bettina and Claudia go shopping. Bettina spends 85% less than Claudia. Anna spends 60% more than Claudia. Together they spend 55 €. How much money does Anna spend? (A) 3 € (B) 20 € (C) 25 € (D) 26 € (E) 32 €
28. Viola practices long-jumping. On average she has jumped 3.80 m so far. On the next jump she reaches 3.99 m and thus the mean increases to 3.81 m. How far does she have to jump on her next attempt in order to increase her mean to 3.82 m? (A) 3.97 m (B) 4.00 m (C) 4.01 m (D) 4.03 m (E) 4.04 m
29. In the isosceles triangle ABC (with base AC) the points K and L are added on the sides AB and BC respectively so that AK = KL = LB and KB = AC. How big is the angle ABC? (A) 30° (B) 35° (C) 36° (D) 40° (E) 44°
30. In a game of dominoes the tiles always have to be placed so that the touching halves of
two adjacent domino tiles show the same number of dots. Paul has six domino tiles in front of him (see diagram).
In several steps Paul tries to arrange them in a correct order. In each step he is either allowed to swap any two domino tiles or he is allowed to turn one domino tile 180° around. What is the minimum number of steps he needs in order to arrange the domino tiles correctly? (A) 1 (B) 2 (C) 3 (D) 4 (E) This is impossible.
Känguru der Mathematik 2018 Level Junior (Grade 9 and 10)
Austria – 15.3.2018
- 3 Point Examples -
1. Every child in my family has at least two brothers and at least one sister. What is the minimum number of children in my family? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 2. The rings shown are partially interlinked. How long is the longest chain built this way which also contains the thick light ring? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7
3. In a triangle one side has length 5 and another side has length 2. The length of the third side is an odd whole number. Determine the length of the third side. (A) 3 (B) 4 (C) 5 (D) 6 (E) 7
4. The distance between the top of the cat that is sitting on the table to the top of the cat that is sleeping on the floor is 150 cm. The distance from the top of the cat that is sleeping on the table to the top of the cat that is sitting on the floor is 110 cm. How high is the table?
(A) 110 cm (B) 120 cm (C) 130 cm (D) 140 cm (E) 150 cm
5. The sum of 5 consecutive whole numbers is 102018. What is the middle number of those numbers? (A) 102013 (B) 52017 (C) 102017 (D) 22018 (E) 2∙ 102017
6. In the three regular hexagons shown, X, Y and Z describe in this order the areas of the grey shaded parts. Which of the following statements is true?
(A) 𝑋=𝑌=𝑍 (B) 𝑌=𝑍≠ 𝑋 (C) 𝑍=𝑋≠ 𝑌 (D) 𝑋=𝑌≠ 𝑍 (E) Each of the areas has a different value. 7. Maria wants to divide 42 apples, 60 peaches and 90 cherries fairly amongst her friends. In order to do so she
divides the entire fruit into baskets, each with the same amount of apples, peaches and cherries, to then give each of her friends one such basket with fruit. At most, how many baskets of fruit can she fill this way? (A) 3 (B) 6 (C) 10 (D) 14 (E) 42 8. In the (correct) calculation shown, some of the digits were replaced by the letters P, Q, R and S. What is the value of P + Q + R + S? (A) 14 (B) 15 (C) 16 (D) 17 (E) 24 9. How big is the sum of 25 % of 2018 and 2018 % of 25? (A) 1009 (B) 2016 (C) 2018 (D) 3027
(E) 5045
10. In the diagram shown, you should follow the arrows to get from A to B. How many different ways are there that fulfill this condition? (A) 20 (B) 16 (C) 12 (D) 9 (E) 6
- 4 Point Examples - 11. The entrances of two student halls lie in a plain street 250 m apart from each other. There are 100 students in the first one and 150 students in the second one. Where should a bus stop be built if the total sum of the distances that each student of both halls has to cover to get to the bus stop should be a minimum? (A) directly in front of the first hall (B) 100 m away from the entrance of the first hall (C) 100 m away from the entrance of the second hall (D) directly in front of the second hall (E) in any place between the two hall entrances
12. 105 numbers are written in a row: 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,...Where each number n is written exactly n-times. How many of those numbers are divisible by 3? (A) 4 (B) 12 (C) 21 (D) 30 (E) 45
13. Eight congruent semi-circles are drawn inside a square with side length 4. How big is the area of the white part? (A) 2𝜋 (B) 8 (C) 6+𝜋 (D) 3𝜋−2 (E) 3𝜋
14. On one particular day there are a total of 40 trains from one of the towns M, N, O, P and Q to exactly one other of those towns. There are 10 trains either from or to M. There are 10 trains either from or to N. There are 10 trains either from or to O. There are 10 trains either from or to P. How many trains are there either from or to Q? (A) 0 (B) 10 (C) 20 (D) 30 (E) 40
15. At a humanistic university you can study languages, history and philosophy. Some of the students there study exactly one language. (Nobody studies several languages at the same time.) Amongst those, 35 % study English. Amongst all students of the university 13 % study a language other than English. Which percentage of the students studies a language? (A) 13 % (B) 20 % (C) 22 % (D) 48 % (E) 65 %
16. Peter wants to buy a book but has no money. He can only buy this book with his father's and his two brother's help. His father gives him half as much money as his brothers give him jointly. His older brother gives him a third of the sum that the two others give him. The youngest brother gives him 10 €. How expensive is the book? (A) 24 € (B) 26 € (C) 28 € (D) 30 € (E) 32 €
17. How many three-digit numbers are there with the property that the two-digit number obtained by deleting the middle number is exactly a ninth of the original number? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
18. How often does the summand 2018² appear under the root, if the following statement is correct?
√20182 + 20182 + … + 20182 = 201810
(A) 5 (B) 8 (C) 18 (D) 20188 (E) 201818
1
19. How many digits has the final result of the calculation ∙ 102018 ∙ (102018−1)?
9
(A) 2017 (B) 2018 (C) 4035 (D) 4036 (E) 4037
20. In a regular 2018-sided shape the vertices are numbered 1 to 2018 in order. Two diagonals of the polygon are drawn in, where one of them connects the vertices 18 and 1018 and the other one the vertices 1018 and 2000. How many vertices do the three resulting polygons have? (A) 38, 983, 1001 (B) 37, 983, 1001 (C) 38, 982, 1001 (D) 37, 982, 1000 (E) 37, 983, 1002
- 5 Point Examples - 21. Some whole numbers are written on a board, amongst them the number 2018. The sum of all these number is 2018. The product of all these number is also 2018. Which of the following numbers could be the amount of numbers on the board? (A) 2016 (B) 2017 (C) 2018 (D) 2019 (E) 2020
22. Given are four positive numbers. Take three of them, work out their mean and then add the fourth number. This can be done in four different ways. The results obtained this way are 17, 21, 23 and 29. Which number is the biggest of the four numbers? (A) 12 (B) 15 (C) 21 (D) 24 (E) 29
̅0̅̅̅̅̅23. The points 𝐴0, 𝐴1, 𝐴2, ... all lie on a straight line. It is true that ̅𝐴 𝐴1=1 and 𝐴𝑛 is the midpoint of every line
segment 𝐴𝑛+1𝐴𝑛+2 , for every non-negative index n. How long is the line segment 𝐴0𝐴11? (A) 171 (B) 341 (C) 512 (D) 587 (E) 683
24. Two concentric circles with radii 1 and 9 form an annulus. n circles without overlap are drawn inside this annulus, where every circle touches both circles of the annulus. (The diagram shows an example for n=1 and the other radii as given.) What is the biggest possible value of n? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
25. A number is to be written into every vertex of the 18-sided shape so that it is equal to
the sum of the two numbers from the adjacent vertices. Two of these numbers are given. Which number is written in vertex A? (A) 2018 (B) 20 (C) 18 (D) 38 (E) 38
26. Diana draws a rectangle made up of twelve squares onto a piece of squared paper. Some of the squares are coloured in black. She writes the number of adjacent black squares into every white square. The
diagram shows an example of such a rectangle. Now she does the same with a rectangle made up of 2018 squares. What is the biggest number that she can obtain as the sum of all numbers in the white squares? (A) 1262 (B) 2016 (C) 2018 (D) 3025 (E) 3027
27. Seven little dice were removed from a 3 x 3 x 3 die, as can be seen in the diagram. The remaining (completely symmetrical) figure is cut along a plane through the centre and perpendicular to one of the four space diagonals. What does the cross-section look like?
(A) (B) (C) (D) (E)
28. Every number of the set {1, 2, 3, 4, 5, 6} is written into exactly one cell of a 2 x 3 table. In how many ways can this be done so that the sum of the numbers in every column and every row is divisible by 3? (A) 36 (B) 42 (C) 45 (D) 48 (E) another number
29. Ed forms a big die using several identical small white dice and colours some of the faces of the big die, red. His sister Nicole drops the die and it again breaks into the original small dice. 45 of which do not have a red face. How many faces of the big die did Ed colour in red? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
30. Two chords 𝐴𝐵 and 𝐴𝐶 are drawn into a circle with diameter AD. ∠ 𝐵𝐴𝐶=60°, ̅̅̅̅= 24 cm, E lies on AC so that ̅̅̅̅= 3 cm, and BE is perpendicular to AC. How 𝐴𝐵𝐸𝐶long is the chord BD?
(A) √3 cm (B) 2 cm (C) 3 cm (D) 2√3 cm (E) 3√2 cm
Känguru der Mathematik 2018
Level Student (Grade 11 onwards)
Austria - 15. 3. 2018 - 3 Points Examples -
1. 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?
(A) Monday (B) Wednesday (C) Thursday (D) Saturday (E) Sunday
2. Which of the following expressions has the biggest value?
(A) 2 0 1 + 8 (B) 2 0 1 8 (C) 2 0 + 1 8 (D) 2 (0 + 1 + 8) (E) 2 0 + 1 + 8
3. The diagram shows the floor plan of Renate's house. Renate enters her house from the terrace (Terrasse) and walks through every door of the house exactly once. Which room does she end up in? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
4. Thor has seven stones and a hammer. With his hammer he hits a stone and it
breaks into five small stones. He does that a few times. Which of these numbers could be the number of stones he ends up with? (A) 17 (B) 20 (C) 21 (D) 23 (E) 25
5. The diagram shows an object made up of 12 dice glued-together. The object is
dipped into some colour so that the entire outside is coloured in this new colour. How many of the small dice will have exactly four faces coloured in? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12
6. The following two statements are true: Some aliens are green and all others are purple. Green aliens live on Mars only. Which one of the following logical conclusions can be made? (A) All aliens live on Mars. (B) There are only green aliens on Mars. (C) Some purple aliens live on Venus (D) All purple aliens live on Venus. (E) There are no green aliens on Venus.
7. 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? (A) 135° (B) 140° (C) 144° (D) 145° (E) 150°
8. There are 65 balls in a box, 8 of which are white, the rest are black. Up to 5 balls can be taken out of the box in one draw. It is not allowed to put any balls back into the box. What is the minimum number of draws which have to be made to be certain that at least one white ball is drawn from the box? (A) 11 (B) 12 (C) 13 (D) 14 (E) 15
9. The faces of the brick have the areas A, B and C as shown. How big is the volume of the brick?
(A) 𝐴𝐵𝐶
(B) √𝐴𝐵𝐶 (C) √𝐴𝐵+𝐵𝐶+𝐶𝐴 (D) √𝐴𝐵𝐶
3
(E) 2(𝐴+𝐵+𝐶)
10. How many ways are there to write the number 1001 as the sum of two prime numbers? (A) no way (B) one way (C) two ways (D) three ways (E) more than three ways
- 4 Point Examples- 11. 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?
(A) 𝑉= 𝑊
3
2
(B) 𝑉= 𝑊
2
3
(C) 𝑉=
8590
𝑊 (D) 𝑉=
9085
𝑊 (E) 𝑉=𝑊
12. 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?
(A)
(B) (C) (D) (E)
13. |√17−5|+|√17+5|= (A) 10
(B) 2√17 (C) √34−10 (D) 10−√34 (E) 0
14. 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?
(A)
13
(B)
4
1
(C)
5
1
(D)
6
1
(E)
8
1
15. The vertices of a triangle have the co-ordinates A(p|q), B(r|s) and C(t|u) as shown. The midpoints of the sides of the triangle are the points M(2|1), N(2|1) and P(3|2). Determine the value of the expression 𝑝+𝑞+𝑟+𝑠+𝑡+𝑢
(A) 2
(B)
2
5
(C) 3 (D) 5 (E) another value
16. Before the football game, Real Madrid vs. Manchester United, the following five predictions were made:
i) The game will not end in a draw.
ii) Real Madrid will score at least one goal. iv) Real Madrid will win. iii) Real Madrid will not lose. v) Exactly three goals will be scored.
It turns out that exactly three of these predictions then come true. How many goals did Real Madrid score?(A) 0 (B) 1 (C) 2 (D) 3 (E) This cannot be determined for certain.
17. A regular pentagon is cut out of a page of lined paper. Step by step this pentagon is then rotated 21° 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?
(A)
(B) (C) (D) (E)
18. Which of the following numbers is not a factor of 182017+182018? (A) 8 (B) 18 (C) 28 (D) 38 (E) 48
19. Three of the cards shown will be dealt to Nadia, the rest
to Riny. Nadia multiplies the three values of her cards and Riny multiplies the two values of his cards. It turns out that the sum of those two products is a prime number. Determine
the sum of the values of Nadia’s cards. (A) 12 (B) 13 (C) 15 (D) 17 (E) 18
20. Two rectangles form the angles 40° and 30° respectively, with a straight line (see diagram). How big is angle ? (A) 105° (B) 120° (C) 130° (D) 135° (E) another value
- 5 Point Examples -
21. The faces of the prism shown, are made up of two triangles and three squares. The six
vertices are labelled using the numbers 1 to 6. The sum of the four numbers around each square is always the same. The numbers 1 and 5 are given in the diagram. Which number is written at vertex X? (A) 2 (B) 3 (C) 4 (D) 6 (E) This situation is impossible.
22. 𝑚 and 𝑛 are the solutions of the equation 𝑥2−𝑥−2018=0. What is the value of the expression 𝑛2+𝑚? (A) 2016 (B) 2017 (C) 2018 (D) 2019 (E) 2020
23. Fours brothers with the harmonious names A, B, C and D are all of different heights. They make the following claims:
A: I am neither the tallest nor the smallest. B: I am not the smallest. C: I am the tallest. D: I am the smallest. Exactly one of them lies. Who is the tallest brother?
(A) A (B) B (C) C (D) D (E) Not enough information is given to be able to make a definite decision.
24. A function 𝑓 fulfills the property 𝑓(𝑥+𝑦)=𝑓(𝑥)∙𝑓(𝑦) for all whole numbers 𝑥 and 𝑦. Furthermore 𝑓(1)=1/2. Determine the value of the expression 𝑓(0)+𝑓(1)+𝑓(2)+𝑓(3). (A) 1/8 (B) 3/2 (C) 5/2 (D) 15/8 (E) 6
25. A quadratic function of the form 𝑓(𝑥)=𝑥2+𝑝𝑥+𝑞 intersects the x-axis and the y-axis in three different
points. The circle through these three points intersects the graph of the function f in a fourth point. What are the co-ordinates of this fourth point of intersection?
(A) (0|−𝑞)
(B) (𝑝|𝑞) (C) (–𝑝|𝑞) (D) (–|
𝑞𝑞2
𝑝𝑝2
) (E) (1|𝑝+𝑞+1)
26. On an idealised rectangular billiard table with side lengths 3 m and 2 m a ball (point-shaped) is pushed away from point M on the long side AB. 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 𝐵𝑀=1,2 𝑚 and 𝐵𝑁=0,8 𝑚 ?
(A) 2 𝑚
(B) 1,5 𝑚 (C) 1,2 𝑚 (D) 2,8 𝑚 (E) 1,8 𝑚
27. How many real solutions does the equation ||4𝑥−3|−2|=1 have? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
28. ABCDEF is a regular hexagon, as shown in the diagram. G is the midpoint of AB. 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?
(A)
12
(B)
3
1
(C)
4
1
(D)
√33
(E)
√34
29. In a class there are 40% more girls than boys. The probability that a student representative team of two students
1
randomly selected from this class is made up of exactly one girl and one boy is exactly . How many children are
2
there in this class? (A) 20 (B) 24 (C) 36 (D) 38 (E) This situation is not possible.
30. 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! = 151413…21) (A) 2 and 0 (B) 4 and 8 (C) 7 and 4 (D) 9 and 2 (E) 3 and 8
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