KÄNGURU DER MATHEMATIK 2023

16. 3. 2023

Level: Kadett, Grade: Schulstufe 7 + 8

Name:
School:
Class:

Time: 75 min.
30 starting points
each correct answer to questions 1. – 10.: 3 points
each correct answer to questions 11. – 20.: 4 points
each correct answer to questions 21. – 30.: 5 points
each questions left unanswered: 0 points
each incorrect answer: minus ¼ of the points for the question

Please write the letter (A, B, C, D, E) of the correct

answer in the square under the question number
(1 bis 30). Write clearly and carefully!

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Information über den Känguruwettbewerb: www.kaenguru.at

Wenn du mehr in dieser Richtung machen möchtest,
gibt es die Österreichische Mathematikolympiade.
Infos unter: www.oemo.at
 Känguru der Mathematik 2023
Level Kadett (Schulstufe 7 and 8)
Austria – 16. 3. 2023

- 3 Point Examples -
01. The diagram shows a grid made of vertical and horizontal lines.
Which part was cut from the grid?
(A) (B) (C) (D) (E)
02. Which of the following shapes cannot be cut into two trapeziums with one single straight line?

(A) (triangle) (B) (rectangle) (C) (trapezium)

(D) (regular hexagon) (E) (square)

03. A dark disc with two holes is placed on top of a dial of a watch as shown. The dark
disc is now rotated so that the number 8 can be seen through one of the holes.
Which of the numbers could one see through the other hole now?
(A) 4 and 12 (B) 1 and 5 (C) 1 and 4 (D) 7 and 11 (E) 5 and 12
04. John throws 150 coins onto a table. 60 of them show „head“, the others show „tail“. He wants the same amount
of coins to show „head“ as „tail“. How many coins that show „head“ does he have to turn over?
(A) 10 (B) 15 (C) 20 (D) 25 (E) 30
05. Kristina has a piece of see-through foil on which some points and lines are drawn. She folds
the foil along the dotted line.
What can she see now?

(A) (B) (C) (D) (E)

06. A grid should be cut along the black lines into several identical shapes. No piece is to be left
over.
Into which of the following shapes is it not possible to cut this grid in this way?

(A) (B) (C) (D) (E)

07. The diagram shows the starting position, the direction and the distance covered within
5 seconds by four bumper cars.
Which two cars will first crash into each other?
(A) A and B (B) A and C (C) A and D (D) B and C (E) C and D
08. Werner wants to label each side and each corner point of the rhombus shown with exactly
one number. He wants the number on each side to be equal to the sum of the numbers on
the corner points of that sides.
Which number is he going to write in the place of the question mark?
(A) 11 (B) 12 (C) 13 (D) 14 (E) 15
09. Anna has five circular discs that are all of different sizes. She wants to build a tower using
three discs where a smaller disc always has to lie on top of a bigger disc.
How many ways are there for Anna to build the tower?
(A) 5 (B) 6 (C) 8 (D) 10 (E) 15
10. Evita wants to write the numbers from 1 to 8 with one number in each field. The sum of the
numbers in each row should be equal. The sum of the numbers in the four columns should also
be the same. She has already written in the numbers 3, 4 and 8 (see diagram).
Which number does she have to write in the dark field?
(A) 1 (B) 2 (C) 5 (D) 6 (E) 7
 - 4 Point Examples-
11. Dorli writes down three consecutive natural numbers in increasing order.
She replaces the digits using symbols and gets: □♦♦, ♡∆∆, ♡∆□.
What would be the next bigger number in this notation?
(A) ♡♡♦ (B) □♡□ (C) ♡∆♦ (D) ♡♦□ (E) ♡∆♡
12. The diagram shows 5 equally big semicircles and the length of 5
distances.
How big is the radius of one semicircle?
(A) 12 (B) 16 (C) 18 (D) 22 (E) 36
13. Some edges of a cube are coloured in red so that each sides of the cube has
at least one red edge.
What is the minimum number of red edges that the cube has?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
14. The digits 0 to 9 can be formed using matchsticks (see diagram).
How many different positive whole numbers can be formed
this way with exactly 6 matchsticks?
(A) 2 (B) 4 (C) 6 (D) 8 (E) 9
15. The side lengths of a square are 1 cm long.
How many points on a plane surface are there that are exactly 1 cm away from two corner points of the square?
(A) 4 (B) 5 (C) 6 (D) 10 (E) 12
16. Some kangaroos and three beavers are standing in a circle. No beaver stands directly next to another beaver.
There are exactly three kangaroos that are standing next to another kangaroo.
What is the biggest possible number of kangaroos in the circle?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8
17. Tom, John and Lily have each shot 6 arrows on a disc with three sections
(see diagram). The number of points of a hit depends on the section that
has been hit. Tom has 46 points and John has 34 points. How many
points did Lily get?
(A) 37 (B) 38 (C) 39 (D) 40 (E) 41
18. Max and two of his friends are standing in a line. The number of people in the line is a multiple of 3. He realises
that there are the same number of people in front of him as there are behind him. His two friends are both
behind him: one is in position 19, the other one in position 28 of the line. In which position of the line is Max?
(A) 14. (B) 15. (C) 16. (D) 17. (E) 18.
19. Two rays starting in S form a right angle. More rays starting in S are drawn within the right
angle so that each angle 10°, 20°, 30°, 40°, 50°, 60°, 70° and 80° is enclosed by two rays.
What is the minimum number of rays that have to be drawn?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
20. The sum of 2023 consecutive integers is 2023.
What is the sum of the digits of the biggest of those numbers?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

- 5 Point Examples -
21. The diagram shows a grey rectangle that lies within a bigger rectangle which sides it touches.
Two corner points of the grey rectangle are the midpoints of the shorter sides of the bigger
rectangle. The grey rectangle is made up of three squares that each have an area of 25 cm2.
How big is the area of the bigger rectangle in cm2?
(A) 125 (B) 136 (C) 149 (D) 150 (E) 172
22. Snow White organises a chess tournament for the seven dwarfs lasting several days. Every dwarf has to play
every other dwarf exactly once.
On Monday Grumpy plays 1 game, Sneezy plays 2, Sleepy 3, Bashful 4, Happy 5 and Doc 6 games.
How many games does Dopey, the 7th dwarf, play on Monday?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
23. The shown triangle 𝐴𝐵𝐶 is isosceles with ∡𝐴𝐵𝐶 = 40°.
The two angles indicated ∡𝐸𝐴𝐵 and ∡𝐷𝐶𝐴 are equally big.
How big is the angle ∡𝐶𝐹𝐸?
(A) 55° (B) 60° (C) 65° (D) 70° (E) 75°

24. An ant walks along the sides of an equilateral triangle (see diagram). Its velocity is
5 cm/min along the first side, 15 cm/min along the second and 20 cm/min along the
third.
With which average velocity in cm/min does the ant walk once around the entire
triangle?
80 180 40
(A) 10 (B) 11 (C) 19
(D) 15 (E) 3

25. Elisabeth wants to write the numbers 1 to 9 in the fields of the diagram shown so that the product of the
numbers of two fields next to each other is no greater than 15.
Two fields are called „next to each other“ if they share a common edge.

How many ways are there for Elisabeth to label the fields?
(A) 8 (B) 12 (C) 16 (D) 24 (E) 32
26. Several mice live in three houses. Last night every mouse left their house and
moved directly to one of the other two houses. The diagram shows how many
mice were in each house yesterday and today.
How many mice used the path that is indicated with an arrow?
(A) 9 (B) 11 (C) 12 (D) 16 (E) 19
27. Bart wrote the number 1015 as a sum of numbers that are made up of only the digit 7. In total he
uses the digit 7, 10 times. Now he wants to the write the number 2023 as a sum of numbers that
are made up of only the digit 7. He uses the digit 7, 19 times in total.
How often does he have to use the number 77?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6
28. A regular hexagon is split into four quadrilaterals and a smaller regular hexagon.
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑟𝑘 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑠 4
The ratio 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑚𝑎𝑙𝑙 ℎ𝑒𝑥𝑎𝑔𝑜𝑛 = 3.

𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑚𝑎𝑙𝑙 ℎ𝑒𝑥𝑎𝑔𝑜𝑛

How big is the ratio 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑖𝑔 ℎ𝑒𝑥𝑎𝑔𝑜𝑛
?
3 1 2 3 3
(A) 11 (B) 3
(C) 3 (D) 4 (E) 5
29. Jakob wrote six consecutive numbers on six little pieces of white paper, one number per piece of
paper. He stuck those six pieces of paper on the front and back of three coins. Then he threw the
coins three times. After the first throw the numbers 6, 7, 8 were on top (see diagram) which Jakob
then coloured in red.
After the second throw the sum of the numbers on top was 23 and after the third throw the sum
was 17.
How big is the sum of the numbers on the three white pieces of paper?
(A) 18 (B) 19 (C) 23 (D) 24 (E) 30
30. A rugby team scored 24, 17 and 25 points in their 7th , 8th and 9th game of the previous season. The average
number of points per game was higher after 9 games than after their first 6 games. Their average after 10 games
was more than 22 points. What is the minimum number of points they have scored in their 10th game?
(A) 22 (B) 23 (C) 24 (D) 25 (E) 26
 KÄNGURU DER MATHEMATIK 2022
17. 3. 2022

Level: Kadett, Grades 7 - 8

Name:
School:
Class:

Time: 75 min.
30 starting points
each correct answer to questions 1. – 10.: 3 points
each correct answer to questions 11. – 20.: 4 points
each correct answer to questions 21. – 30.: 5 points
each questions left unanswered: 0 points
each incorrect answer: minus ¼ of the points for the question

Please write the letter (A, B, C, D, E) of the correct

answer in the square under the question number
(1 bis 30). Write clearly and carefully!

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

Information über den Känguruwettbewerb: www.kaenguru.at

Wenn du mehr in dieser Richtung machen möchtest,
gibt es die Österreichische Mathematikolympiade.
Infos unter: www.oemo.at
 Känguru der Mathematik 2022
Level Kadett (Schulstufe 7 and 8)
Austria – 17. 3. 2022

- 3 Point Examples -

1. What is (20+22) ÷ (20−22) = ?

(A) -42 (B) -21 (C) -2 (D) 22 (E) 42
2. Meike paddles around five buoys with her boat (see diagram).
Which of the buoys does she paddle around in a clockwise direction?
(A) 2, 3 and 4 (B) 1, 2 and 3 (C) 1, 3 and 5 (D) 2, 4 and 5 (E) 2, 3 and 5

3. Beate arranges the five cards so that the smallest nine-digit number is
created. Which card is furthest on the right?

(A) (B) (C) (D) (E)

4. The numbers 3, 4, 5, 6, 7 are written inside the five circles of the shape. The product of the
numbers in the four outer circles is 360. Which number is in the inner circle?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

5. Anna, Beatrice and Clara altogether are 15 years old. Anna and Beatrice together are 11 years old.
Beatrice and Clara together are 12 years old. How old is the oldest of the three?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

6. Kengu likes to jump on the number line. He starts at 0, then always

starts with two big jumps and then three small jumps (see diagram). He
keeps repeating this in the same way, over and over again.
On which of the following numbers will he land in the course of his jumps?
(A) 82 (B) 83 (C) 84 (D) 85 (E) 86

7. Otto attaches the number plate to his car the wrong way round, i.e. upside down. Luckily it doesn’t matter
because the number plate looks exactly the same this way.
Which of the following number plates could be the one from Otto?
(A) 04 NSN 40 (B) 60 SOS 09 (C) 80 BNB 08 (D) 06 HNH 60 (E) 08 NBN 80

8. Sonja builds the cube shown, out of equally sized bricks. The shortest side of one
brick is 4 cm long. What dimensions in cm does one brick have?
(A) 4  6  12 (B) 4  6  16 (C) 4  8  12 (D) 4  8  16 (E) 4  12  16

9. The black-white caterpillar shown, rolls up to go to sleep.

Which diagram could show the rolled-up caterpillar?

(A) (B) (C) (D) (E)

10.Gerhard writes down the sum of the squares of two numbers. Unfortunately, some ink has run out (see diagram)
and therefore we cannot read all the digits.
What is the last digit of the first number?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

 - 4 Point Examples -

11. There are five gaps in the following calculation. Adriana wants to
write a „+“ into four of the gaps and a „−“ into one of the gaps so that the equation is correct. Where does she
have to insert the „−“?
(A) between 6 and 9 (B) between 9 and 12 (C) between 12 and 15
(D) between 15 and 18 (E) between 18 and 21
12. There are 5 trees and 3 paths in a park as shown on the map.
Another tree is planted so that there is an equal number of trees
on both sides of each path.
In which section of the park will the new tree be planted?
(A) A (B) B (C) C (D) D (E) E

13. The distance between two shelves in Monika’s kitchen is 36 cm. She knows that a stack of 8 identical
glasses is 42 cm high and a stack of 2 such glasses is 18 cm high.
How many glasses has the biggest stack that will fit between two shelves?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

14. On an ordinary die the numbers on opposite sides always add up to 7. Four such
dice are glued together as shown. All numbers that can still be seen on the outside
of the solid are added together. What is the minimum of that total?
(A) 52 (B) 54 (C) 56 (D) 58 (E) 60

15. How many integers between 100 and 300 have only odd digits?
(A) 25 (B) 50 (C) 75 (D) 100 (E) 150

16. Gardener Toni plants tulips and sunflowers in a square flowerbed with
side length 12 m, as shown in the diagram. How big is the entire area where
sunflowers are planted?
(A) 36 m² (B) 40 m² (C) 44 m² (D) 46 m² (E) 48 m²

17. There are two clocks in my office. One of which is one minute fast every hour and the other one is two minutes
behind every hour. Yesterday I have set them both on the correct time but when I checked today, one clock said
11:00 and the other 12:00.
At what time did I set the time yesterday?
(A) 23:00 (B) 19:40 (C) 15:40 (D) 14:00 (E) 11:20

18. Werner has written some numbers on a piece of paper whose sum is 22. Ria has then subtracted each number
from 7 and has also written down the results.
The sum of Ria’s numbers is 34. How many numbers has Werner written down?
(A) 7 (B) 8 (C) 9 (D) 10 (E) 11

19. The big rectangle ABCD is made up of 7 congruent smaller rectangles (see diagram).
𝐴𝐵
What is the ratio 𝐵𝐶 ?

1 4 8 12 7
(A) (B) (C) (D) (E)
2 3 5 7 3

20. Two identical bricks can be placed side by side in three different ways
as shown in the diagrams. The surface areas of the resulting cuboids
are 72, 96 and 102 cm².
What is the surface area (in cm²) of one brick?
(A) 36 (B) 48 (C) 52 (D) 54 (E) 60
 - 5 Point Examples -
21. Jenny writes numbers into a 3  3 table so that the sums of the four numbers in each 2  2 area
of the table are the same. The numbers in three of the cells in the corner can already be seen in
the diagram. Which number does she write into the cell in the fourth corner?
(A) 0 (B) 1 (C) 4 (D) 5 (E) 6

22. A shape is made up of a triangle and a circle that partially overlap.

The grey area is 45 % of the entire area of the shape.
The white part of the triangle is 40 % of the total area of the shape.
What percent of the area of the circle is the white part, outside the triangle?
(A) 20 % (B) 25 % (C) 30 % (D) 35 % (E) 50 %
23. The numbers 1 to 8 are written into the circles shown so that there is one number
in each circle. Along each of the five straight arrows the three numbers in the
circles are multiplied. Their product is written next to the tip of the arrow. How big
is the sum of the numbers in the three circles on the lowest row of the diagram?
(A) 11 (B) 12 (C) 15 (D) 17 (E) 19
24. By bike it takes Marc 20 minutes to go from home to school and back. He rides the
entire distance with a constant speed. By foot it takes him 60 minutes for the same distance. He also walks with a
constant speed.
Yesterday Marc took his bike to go to Eva’s house which is on the way to school. He left the bike there and
continued on foot to school. On the way home he first walked to Eva’s house and then cycled the rest of the way
back home. He needed 52 minutes for the entire journey (from home to school and back home). Which part of
his journey did he cover by bike?
1 1 1 1 1
(A) (B) (C) (D) (E)
6 5 4 3 2

25. The four villages A, B, C and D are situated (not necessarily in this order) along a straight road. The villages A and
C are 75 km away from each other, B and D 45 km away from each other and B and C 20 km away from each
other. Which of the following distances cannot be the distance from A to D?
(A) 10 km (B) 50 km (C) 80 km (D) 100 km (E) 140 km

26. A painter wants to mix 2 litres of blue paint with 3 litres of yellow paint to obtain 5 litres of green paint. He
accidentally uses 3 litres of blue paint and 2 litres of yellow paint and thus produces the wrong shade of green.
What is the minimum amount of this green paint he has to throw away so that he can use the rest to add blue or
yellow paint in order to get exactly 5 litres of the correct shade of green?
5 3 2 3 5
(A) litre (B) litre (C) litre (D) litre (E) litre
3 2 3 5 9
27. What is the minimum number of cells of a 5  5 grid that have to be coloured in so
that every possible 1  4 rectangle and every 4  1 rectangle respectively in the grid
has at least one cell coloured in?
(A) 5 (B) 6 (C) 7 (D) 8 (E) 9
28. Mowgli asks a bear and a panther which day of the week it is. The bear always lies on Monday, Tuesday and
Wednesday. The panther always lies on Thursday, Friday and Saturday. On all other days they both always speak
the truth. The bear says: „Yesterday was one of my lying days.“ The panther says: „Yesterday was also one of my
lying days.“ On which day of the week did this conversation take place?
(A) Thursday (B) Friday (C) Saturday (D) Sunday (E) Monday

29. Some points are marked on a straight line. Renate marks another point between every pair of adjacent points.
She repeats this process three more times.
Now 225 points are marked on the straight line. How many points were there to begin with?
(A) 10 (B) 12 (C) 15 (D) 16 (E) 25

30. In total there are 2022 kangaroos and some koalas living within seven parks. As many kangaroos live in each park
as there are koalas in all other parks together. How many koalas in total live in the seven parks?
(A) 288 (B) 337 (C) 576 (D) 674 (E) 2022
 KSF 2021 – Cadet (C)

3 points problems

1. Which of the following symbols for signs of the Zodiac has an axis of symmetry?

(A) Saggitarius (B) Scorpio (C) Leo (D) Cancer (E) Capricorn

2. The figure shows three concentric circles with four lines passing through their com-
mon centre. What percentage of the figure is shaded?

(A) 30% (B) 35% (C) 40% (D) 45% (E) 50%

20  21
3. What is the value of ?
2+ 0 +2+1

(A) 42 (B) 64 (C) 80 (D) 84 (E) 105

4. How many four-digit numbers have the property that their digits, from left to right, are consecutive and in
ascending order?

(A) 5 (B) 6 (C) 7 (D) 8 (E) 9

5. When the 5 pieces are fitted together correctly, the result is a rectangle
with a calculation written on it. What is the answer to this calculation?

(A) −100 (B) −8 (C) −1 (D) 199 (E) 208

6. Each of the five vases shown has the same height and each has a volume of 1 litre. Half a litre of water is
poured into each vase. In which vase would the level of the water be the highest?

(A) (B) (C) (D) (E)

7. A student correctly added the two two-digit numbers on the left of

the board and got the answer 137. What answer will he get if he adds
the two four-digit numbers on the right of the board?

(A) 13 737 (B) 13 837 (C) 14 747 (D) 23 737 (E) 137 137

CANGURU DE MATEMÁTICA BRASIL 2021 – CADET – COPYRIGHT 1

8. A cube 3  3  3 is made from 1  1  1 white, grey and black cu-
bes, as shown in the first diagram. The other two diagrams show
the white part and the black part of the cube. Which of the fol-
lowing diagrams shows the grey part?

(A) (B) (C) (D) (E)

9. A bike lock has four wheels numbered with the digits 0 to 9 in order. Each of the
four wheels is rotated by 180° from the code shown in the first diagram to get the
correct code. What is the correct code for the bike lock?

(A) (B) (C) (D) (E)

10. Byron is 5cm taller than Aaron, but 10cm shorter than Caron. Darren is 10cm taller than Caron, but 5cm
shorter than Erin. Which of the following statements is true?

(A) Aaron and Erin are equal heights (B) Aaron is 10cm taller than Erin (C) Aaron is 10cm shorter than Erin
(D) Aaron is 30cm taller than Erin (E) Aaron is 30cm shorter than Erin

4 points problems

11. A rectangular chocolate bar is made of equal squares. Neil breaks off two complete strips of squares and
eats the 12 squares he obtains. Later, Jack breaks off one complete strip of squares from the same bar and
eats the 9 squares he obtains. How many squares of chocolate are left in the bar?

(A) 72 (B) 63 (C) 54 (D) 45 (E) 36

12. A jar one fifth filled with water weighs 560 g. The same jar four fifths filled with water weighs 740 g.
What is the weight of the empty jar?

(A) 60 g (B) 112 g (C) 180 g (D) 300 g (E) 500 g

13. The area of the large square is 16 cm2 and the area of each small square is 1 cm2.What
is the total area of the central flower in cm2?

7 11
(A) 3 (B) (C) 4 (D) (E) 6
2 2

CANGURU DE MATEMÁTICA BRASIL 2021 – CADET – COPYRIGHT 2

14. Costa is building a new fence in his garden. He uses 25 planks of wood, each of which are 30 cm long.
He arranges these planks so that there is the same slight overlap between any two adjacent planks.

The total length of Costa's new fence is 6.9 metres. What is the length in centimetres of the overlap between
any pair of adjacent planks?

(A) 2,4 (B) 2,5 (C) 3 (D) 4,8 (E) 5

15. Five identical right-angled triangles can be arranged so that their larger acute angles
touch to form the star shown in the diagram. It is also possible to form a different star
by arranging more of these triangles so that their smaller acute angles touch. How many
triangles are needed to form the second star?

(A) 10 (B) 12 (C) 18 (D) 20 (E) 24

16. Five squares are positioned as shown. The small square indicated has area 1.
What is the value of ℎ?

(A) 3 m (B) 3.5 m (C) 4 m (D) 4.2 m (E) 4.5 m

17. There are 20 questions in a quiz. Each correct answer scores 7 points, each wrong answer scores −4 points,
and each question left blank scores 0 points. Eric took the quiz and scored 100 points. How many questions
did he leave blank?

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

18. A rectangular strip of paper of dimensions 4cm 13cm is folded as shown in the
diagram. 2 rectangles are formed with areas 𝑃 and 𝑄 where P = 2Q . What is the
value of x?

(A) 5 cm (B) 5,5 cm (C) 6 cm (D) 6,5 cm (E) 4 2 cm

19. A box of fruit contains twice as many apples as pears. Christy and Lily divided them up so that Christy had
twice as many pieces of fruit as Lily. Which one of the following statements is always true?

(A) Christy took at least one pear. (B) Christy took twice as many apples as pears.
(C) Christy took twice as many apples as Lily. (D) Christy took as many apples as Lily got pears.
(E) Christy took as many pears as Lily got apples.

CANGURU DE MATEMÁTICA BRASIL 2021 – CADET – COPYRIGHT 3

20. Three villages are connected by paths as shown. From Downend to Uphill, the
detour via Middleton is 1km longer than the direct path. From Downend to Middle-
ton, the detour via Uphill is 5km longer than the direct path. From Uphill to Middle-
ton, the detour via Downend is 7km longer than the direct path. How long is the
shortest of the three direct paths between the villages?

(A) 1 km (B) 2 km (C) 3 km (D) 4 km (E) 5 km

5 points problems

21. 5 friends talk about their collected astro-pins planets: , moons: or stars: . Xenia says: "I have
an even number of pins", Zach: "Half of my pins are planets, Sue: "I don't have any moons", Paul: "I have more
moons than stars" and Yvonne: "I have more stars than planets". Below are the collections of the 5 friends.
Which set of pins belongs to Yvonne?

(A) (B) (C)

(D) (E)

22. In a particular fraction the numerator and denominator are both positive. The numerator of this fraction
is increased by 40%. By what percentage should its denominator be decreased so that the new fraction is
double the original fraction?

(A) 10% (B) 20% (C) 30% (D) 40% (E) 50%

23. A triangular pyramid is built with 20 cannon balls, as shown. Each cannon ball is labelled
with one of A, B, C, D or E. There are 4 cannon balls with each type of label. The picture shows
the labels on the cannon balls on 3 of the faces of the pyramid. What is the label on the hidden
cannon ball in the middle of the fourth face?

(A) A (B) B (C) C (D) D (E) E

24. The 6-digit number 2ABCDE is multiplied by 3 and the result is the 6-digit number ABCDE2. What is the
sum of the digits of this number?

(A) 24 (B) 27 (C) 30 (D) 33 (E) 36

CANGURU DE MATEMÁTICA BRASIL 2021 – CADET – COPYRIGHT 4

25. A box contains only green, red, blue and yellow counters. There is always at least one green counter
amongst any 27 counters chosen from the box; always at least one red counter amongst any 25 counters
chosen; always at least one blue amongst any 22 counters chosen and always at least one yellow amongst any
17 counters chosen. What is the largest number of counters that could be in the box?

(A) 27 (B) 29 (C) 51 (D) 87 (E) 91

26. 2021 coloured kangaroos are arranged in a row and are numbered from 1 to 2021. Each kangaroo is col-
oured either red, grey or blue. Amongst any three consecutive kangaroos, there are always kangaroos of all
three colours. Bruce guesses the colours of five kangaroos. These are his guesses: Kangaroo 2 is grey; Kanga-
roo 20 is blue; Kangaroo 202 is red; Kangaroo 1002 is blue; Kangaroo 2021 is grey. Only one of his guesses is
wrong. What is the number of the kangaroo whose colour he guessed incorrectly?

(A) 2 (B) 20 (C) 202 (D) 1002 (E) 2021

27. A 3  4  5 cuboid consists of 60 identical small cubes. A termite eats its way along
the diagonal from 𝑃 to 𝑄.This diagonal does not intersect the edges of any small cube
inside the cuboid. How many of the small cubes does it pass through on its journey?
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12

28. In a town there are 21 knights who always tell the truth and 2000 knaves who always lie. A wizard divided
2020 of these 2021 people into 1010 pairs. Every person in a pair described the other person as either a knight
or a knave. As a result, 2000 people were called knights and 20 people were called knaves. How many pairs of
two knaves were there?

(A) 980 (B) 985 (C) 990 (D) 995 (E) 1000

29. In a tournament each of the 6 teams plays one match against every other
team. In each round of matches, 3 take place simultaneously. A TV station has
already decided which match it will broadcast for each round, as shown in the
diagram. In which round will team D play against team F?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

30. The diagram shows a quadrilateral divided into 4 smaller quadrilaterals with a com-
mon vertex K. The other labelled points divide the sides of the large quadrilateral into
three equal parts. The numbers indicate the areas of the corresponding small quadrilat-
erals. What is the area of the shaded quadrilateral?

(A) 4 (B) 5 (C) 6 (D) 6.5 (E) 7

CANGURU DE MATEMÁTICA BRASIL 2021 – CADET – COPYRIGHT 5

 Canguru de Matemática Brasil – Level C – 2020 – Second Application

3 points

1. What result of the following additions is not a prime number?

(A) 2 + 11 (B) 4 + 7 (C) 6 + 11 (D) 3 + 4 (E) 5 + 7

1
2. With the numbers 1, 2, 3 and 4, we can write several fractions whose value is less than 1, for example, .
3
How many different values, beyond the example, can be obtained?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 8

3. Miguel decided to solve three math problems a day. Eight days later, Daniel started solving five problems a
day, until the two of them tied in the amount of problems solved. How many problems each one solved until
that day?

(A) 12 (B) 20 (C) 60 (D) 80 (E) 120

4. One square was divided into four equal squares, containing other equal colored squares
and equal colored triangles, as shown in the picture. What fraction of the original square does
the colored part represent?
1 1 4 5 3
(A) (B) (C) (D) (E)
3 2 9 8 4

5. Three soccer teams compete in a championship. Each team plays exactly once with each of the other teams.
In each match, the victorious team wins 4 points, the loser loses 1 point, and in case of a tie, each team wins
2 points. Once the championship is over, what will be the largest possible sum of the points obtained by the
three teams?

(A) 8 (B) 9 (C) 10 (D) 11 (E) 12

6. The figure of side 1 is formed by six equal triangles, made with

12 sticks. How many matchsticks are needed to complete the fig-
ure of side 2, partially represented?

(A) 18 (B) 24 (C) 32 (D) 36 (E) 48

7. Carlos wants to square the sum of three chosen numbers from the list 5, 3, 1,0,2,7. What is the small-
est result he can get?
(A) 0 (B) 1 (C) 4 (D) 9 (E) 16

KSF BRASIL – LEVEL C – 2020 – SECOND APPLICATION – ALL RIGHTS RESERVED 1

 8. When Julia goes from home to school, she can walk half-way and half-way she can go by bus. If she only
walks, she will spend 45 minutes more. How much less time does it take her to go to school if she uses only
the bus?

(A) 25 minutos (B) 45 minutos (C) 1 hora (D) 1 hora e meia (E) 2 horas

9. Juca wrote a whole number greater than zero in each of the boxes on the 3 × 3
board on the right, so that the sums of the numbers in each row and in each column
are equal. The only thing Juca remembers is that there are no three numbers re-
peated. What number is written in the box of the center?

(A) 1 (B) 2 (C) 4 (D) 5 (E) 6

10. In the figure, formed by a square and an equilateral triangle, the letters indicate the
measurements of the angles. Which of the following equality is true?

(A) a = d (B) b + c = d (C) a + c = d + e (D) a + b = d + e (E) e + d = a

4 points

11. As soon as he left his city towards Caecá, Charles saw the sign on
the left. When he came back from Caecá, he saw the sign on the
right. At that point, how far was it to get to his city?

(A) 12 km (B) 21 km (C) 29 km (D) 41 km (E) 52 km

12. Ana planned to walk an average of 5 km per day in March. In the first 10 days she walked an average of
4,4 km per day and in the following 6 days she walked an average of 3,5 km per day. What is the average daily
distance she should walk on the remaining days in order to fulfill her plan?

(A) 5,4 km (B) 5,8 km (C) 6 km (D) 6,6 km (E) 7 km

13. Which of the pictures below shows what you will see if you look from above the piece
represented on the right?

(A) (B) (C) (D) (E)

KSF BRASIL – LEVEL C – 2020 – SECOND APPLICATION – ALL RIGHTS RESERVED 2

 14. In a class, students only swim or only dance or do both. Three eighths of the students in the class swim.
There are exactly five students who do both, that is, they swim and dance. At least how many students are in
class?

(A) 16 (B) 24 (C) 32 (D) 40 (E) 48

15. The garden of Sonia's house is shaped like a 12-meter square and is divided into
three lawns of the same area. The central lawn is shaped like a parallelogram, whose
smaller diagonal is parallel to two sides of the square, as shown in the picture. What is
the length of this diagonal, in meters?

(A) 7,2 (B) 7,6 (C) 8,0 (D) 8,4 (E) 8,8

16. Andrew bought 27 little cubes of the same color, each with three adjacent faces painted red
and the other three of another color. He wants to use all these little cubes to build a bigger cube.
What is the largest number of completely red faces that he can get for this cube?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

17. A square is formed by four identical rectangles and a central square, as in the figure.
The area of the square is 81 cm2 and the square formed by the diagonals of these
rectangles has an area equal to 64 cm2 . What is the area of the central square?

(A) 25 cm2 (B) 27 cm2 (C) 36 cm2 (D) 47 cm2 (E) 49 cm2

18. A store announced a 30% discount on a sale. However, one day before this promotion, the store increased
the prices of all its products by 20%. What was the real discount that this store gave on the day of the sale?
(A) 10% (B) 12% (C) 15% (D) 16% (E) 20%

19. Irene made a "city" using identical wooden cubes. We have, beside, a view
from above and a side view of this "city". We do not know which side of the
"city" is being shown. What is the smallest amount of cubes Irene may have
used to make its assembly?

(A) 10 (B) 12 (C) 13 (D) 14 (E) 15

20. Amelia has a paper strip with five equal cells containing different drawings,
according to the figure. She folds the strip in such a way that the cells overlap in
five layers. Which of the sequences of layers, from top to bottom, is not possible to obtain?

(A) (B) (C) (D) (E)

KSF BRASIL – LEVEL C – 2020 – SECOND APPLICATION – ALL RIGHTS RESERVED 3

 5 points

21. In each of the four corners of a swimming pool, 10 m wide by 25 meters long, there is a child. The swimming
instructor is sitting almost in the middle of one of the edges of the pool. When he calls the children, they all
choose the longest path along the edges to reach the instructor. What was the sum of the distances covered
by the four children?

(A) 90 m (B) 120 m (C) 140 m (D) 160 m (E) 210 m

22. Twelve colored cubes are lined side by side. There are three blue cubes, two yellow cubes, three red cubes,
and four green cubes, but not in that order. There is a red cube at one end and a yellow one at the other. The
red cubes are all together, and the green cubes are all together. The tenth cube from the left is blue. How
many ways can the cubes be queued?

(A) 2 (B) 3 (C) 6 (D) 9 (E) 12

23. Sofia has 52 isosceles triangles of 1 cm2 area. She wants to make a square using some of these triangles.
What is the area of the largest square she can make?

(A) 32 cm2 (B) 36 cm2 (C) 42 cm2 (D) 50 cm2 (E) 52 cm2

24. Let N be the smallest positive number such that half of N is divisible by 2, one-third of N is divisible by 3,
one-quarter of N is divisible by 4, one-fifth of N is divisible by 5, one-sixth of N is divisible by 6, one-eighth of
N is divisible by 8, and one-ninth of N is divisible by 9. The square root of N is a number of how many digits?

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

25. Jonas was traveling with his car and saw on the car display the following information: speed 90 km/h,
distance travelled 116,0 km and time 21h00min. Jonas continued driving at the same speed and that same
night he realized that the four-digit sequence showing the distance traveled was the same four-digit sequence
showing the time. At what time did this happen?

(A) 21h30min (B) 21h50min (C) 22h00min (D) 22h10min (E) 22h30min

26. Lady Josephine bought a pack of beans. The beans come mixed with impurities such as pebbles and sand,
and the label reads that these impurities correspond to 8% of the contents of the package. Lady Josephine
removes part of these impurities, which are reduced to 4% of the content of the package. What fraction of the
total amount of impurities was removed from the package?
1 25 7 5 25
(A) (B) (C) (D) (E)
2 48 12 8 36

27. Zilda took a square sheet of paper of side 1 and made two folds taking two
consecutive sides of the sheet to a diagonal of it, as shown in the picture,
obtaining a quadrilateral (highlighted outline). What is the area of this
quadrilateral?

7 3 2
(A) (B) 2 2 (C) (D) 2 1 (E)
10 5 2

KSF BRASIL – LEVEL C – 2020 – SECOND APPLICATION – ALL RIGHTS RESERVED 4

 28. Cleuza assembled the 2 × 2 × 2 block formed by equal balls beside, using one drop of glue
at each contact point between two balls, in a total of 12 drops. She then glued a few more
spheres until she completed a 4 × 3 × 2 block. How many extra drops of glue did she get to
use?

(A) 12 (B) 24 (C) 34 (D) 36 (E) 44

29. Sonia writes three consecutive whole numbers, one on each side of a triangle. Then she writes on each
vertex of the triangle the sum of the numbers written on the sides that touch this vertex and multiplies these
three numbers, obtaining the product 504. What is the product of the three numbers written on the sides of
the triangle?

(A) 24 (B) 60 (C) 120 (D) 210 (E) 336

30. The statements below give the clues to identifying a four-digit N number.
A digit is right, but it's in the wrong place.
Two digits are right, but they are in the wrong places.
None of the digits are right.
One digit is correct and in the right place.
Two digits are right, one is in the right place and the other is in the wrong place.
What is the digit of the hundreds of the number N?

(A) 0 (B) 1 (C) 3 (D) 5 (E) 9

KSF BRASIL – LEVEL C – 2020 – SECOND APPLICATION – ALL RIGHTS RESERVED 5