Quiz

Kangaroo 2020 – Cadet (grades 8 and 9)

30 problems from the international Kangaroo mathematics competition (19 March 2020), Cadet category, for grades 8 and 9. The solutions are shown after checking.

Choose one of the five answers for each problem. Problems 1–10 are worth 3 points, 11–20 4 points and 21–30 5 points. A wrong answer loses a quarter of the problem’s points and no answer scores 0. You start with 30 points; the maximum is 150. In the competition the time is 75 minutes. When you finish, click Check answers.

1. How many of the numbers 2, 20, 202, 2020 are prime? (3 points)

2. Which of the marked angles in the regular polygons is the largest? (3 points)

3. Mihail solves 6 olympiad problems a day and Lazar 4. In how many days does Lazar solve as many problems as Mihail does in 4 days? (3 points)

4. Which fraction has the largest value? (3 points)

5. The big square is divided into smaller squares, and a diagonal is drawn in one of them (picture). What fraction of the big square is grey? (3 points) Figure for problem 5

6. Four teams play a football tournament, each meeting every other once. A win gives 3 points, a loss 0 and a draw 1 to each. Which total can NO team have at the end? (3 points)

7. The figure is made of 36 small congruent equilateral triangles. What is the smallest number of such triangles that must be added to get a hexagon? (3 points) Figure for problem 7

8. Koce wants to multiply three different numbers chosen from −5, −3, −1, 2, 4 and 6. What is the smallest product he can get? (3 points)

9. If Kosta goes to school by bus and walks home, the trips take 3 hours. If he takes the bus both ways, they take 1 hour. How long do they take if he walks both ways? (3 points)

10. A number is written in each cell of a 3 × 3 square, but ink has covered them. The sums of every row and of two columns are known (picture). What is the sum of the third column? (3 points) Figure for problem 10

11. The shortest road from Atown to Cetown goes through Betown. Both signposts stand on the road from Atown to Cetown. What distance was written on the broken sign? (4 points) Figure for problem 11

12. Ana wants to walk 5 km a day on average in March. On the evening of 16 March she worked out that she had walked 95 km so far. How far must she walk per day on average for the rest of the month? (4 points)

13. Which picture shows the object seen from above? (4 points) Figure for problem 13

14. Every pupil in a class swims or dances. Three fifths swim and three fifths dance. Five pupils do both. How many pupils are in the class? (4 points)

15. Sasha’s garden has the shape in the picture; all sides are parallel or perpendicular. Some lengths are given. What is the perimeter? (4 points) Figure for problem 15

16. Andrej bought 27 identical small cubes, each with two neighbouring faces painted red. He used all of them to build a big cube. At most how many faces of the big cube can be completely red? (4 points)

17. A big square is made of four identical rectangles and a small square. The big square has area 49 cm² and the diagonal AB of one rectangle is 5 cm. What is the area of the small square? (4 points) Figure for problem 17

18. Viktor’s salary is 20% of his boss’s salary. By what percentage is the boss’s salary larger than Viktor’s? (4 points)

19. Ivona built a “town” from identical wooden cubes. The right picture is the view from above and the left one the view from one of the sides, we don’t know which. What is the largest number of cubes she could have used? (4 points) Figure for problem 19

20. Aleksandra has a paper strip with the numbers 1, 2, 3, 4 and 5 in five squares (picture). She folds it so that the squares lie exactly on top of each other in five layers. Which order (top to bottom) is NOT possible? (4 points) Figure for problem 20

21. Twelve coloured cubes stand in a row: 3 blue, 2 yellow, 3 red and 4 green. One end is yellow and the other red. The red cubes are next to each other, and so are the green ones. The tenth cube from the left is blue. What colour is the sixth cube from the left? (5 points)

22. Zoran folded a square sheet of paper by folding two neighbouring sides onto the diagonal (left picture) and got a quadrilateral (right picture). What is its largest angle? (5 points) Figure for problem 22

23. How many four-digit numbers A are there such that half of A is divisible by 2, a third of A by 3 and a fifth of A by 5? (5 points)

24. In a dance final each of the three judges gives the five finalists 0, 1, 2, 3 or 4 points, and no judge gives two people the same mark. Adam knows the totals and some marks (table). How many points did Adam get from judge III?

	Adam	Berta	Klara	David	Emil
I	2	0
II		2	0
III
Total	7	5	3	4	11

(5 points)

25. Sanja wrote a positive whole number on each side of a square, and at each corner the product of the numbers on the two sides meeting there. The corner numbers add up to 15. What is the sum of the numbers on the sides? (5 points)

26. Sofia has 52 identical isosceles right triangles. How many squares of different sizes can she make using some of them? (5 points)

27. Leo built a pyramid of metal balls: the base is 4 × 4 (picture), then 3 × 3, 2 × 2 and one ball on top. There is a dot of glue wherever two balls touch. How many dots are there? (5 points) Figure for problem 27

28. Four children are in the four corners of a 10 m × 25 m pool. Their coach is somewhere on one of the sides. When he called, three children got out and walked to him along the edges by the shortest routes, 50 m in total. How far must the coach walk to the fourth child? (5 points)

29. Ane, Boris and Kire start a race together and run at constant speeds. When Ane finished, Boris still had 15 m to go and Kire 35 m. When Boris finished, Kire had 22 m left. How long was the track? (5 points)

30. The statements in the table are about a four-digit number. What is its units digit?

4	1	3	2	two digits are right but in the wrong places
9	8	2	6	one digit is right and in the right place
5	0	7	9	two digits are right: one in the right place, one in the wrong place
2	7	4	1	one digit is right but in the wrong place
7	6	4	2	no digit is right

(5 points)