Quiz

Kangaroo 2020 – Student (secondary years 3 and 4)

30 problems from the international Kangaroo mathematics competition (19 March 2020), Student category, for secondary years 3 and 4. 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. What is the sum of the last two digits of the product \(1\cdot2\cdot3\cdot4\cdot5\cdot4\cdot3\cdot2\cdot1\)? (3 points)

2. Every day an ant walks in a straight line from A to B, 5 m apart. One day two obstacles, each 1 m high, are put in its way (picture). The ant keeps to the same line, climbing straight up and down the obstacles. How far does it walk now? (3 points) Figure for problem 2

3. Rene marked the points a and b on the number line as accurately as he could. Which of the points p, q, r, s, t best represents the product ab? (3 points) Figure for problem 3

4. The pie chart shows how the pupils of a school get there. About twice as many cycle as take public transport, and about the same number come by car as walk. The rest ride a scooter. What percentage ride a scooter? (3 points) Figure for problem 4

5. The five three-digit numbers in the picture add up to 2664. What is A + B + C + D + E? (3 points) Figure for problem 5

6. What is the value of \(\dfrac{1010^2+2020^2+3030^2}{2020}\)? (3 points)

7. a, b and c are integers with \(1\le a\le b\le c\) and \(abc=1\,000\,000\). What is the largest possible value of b? (3 points)

8. If D dogs weigh K kilograms and E elephants weigh as much as M dogs, how much does one elephant weigh? (3 points)

9. There are two dice, each with two red, two blue and two white faces. Both are thrown. What is the probability that they show the same colour? (3 points)

10. Which of these numbers is not divisible by 3 for any integer n? (3 points)

11. A blue and a red rectangle overlap; the picture shows four cases. B is the area of the blue part not shared with the red one, and R the area of the red part not shared with the blue one. What is true about B − R? (4 points) Figure for problem 11

A. B − R is largest in case 1

B. B − R is largest in case 2

C. B − R is largest in case 3

D. B − R is largest in case 4

E. B − R is the same in all cases

12. Five coins show tails. Each step turns over exactly three of them. What is the smallest number of steps to make them all show heads? (4 points)

13. Four identical boxes are glued into the solid in the picture. If painting the outside of one box takes 1 litre of paint, how many litres does the outside of the solid take? (4 points) Figure for problem 13

14. a, b and c are integers. Which value can \((a-b)^2+(b-c)^2+(c-a)^2\) certainly not take? (4 points)

15. The first two digits of a 100-digit number are 2 and 9. How many digits does its square have? (4 points) Figure for problem 15

16. Fifteen numbers are written round a wheel; only the 10 at the top is visible. Any 7 consecutive numbers (e.g. the grey ones) always have the same sum. How many of 75, 216, 365 and 2020 can be the sum of all 15 numbers? (4 points) Figure for problem 16

17. A square leans on two other squares as in the picture; the numbers in the small squares are their areas. What is the area of the big square? (4 points) Figure for problem 17

18. The sequence \(f_n\) has \(f_1=1\), \(f_2=3\) and \(f_{n+2}=f_n+f_{n+1}\) for \(n\ge1\). How many of the first 2020 terms are even? (4 points)

19. A circle touches two sides of a rectangle and passes through one of its corners (picture). One point of contact is 5 and 4 from the two neighbouring corners. What is the area of the rectangle? (4 points) Figure for problem 19

20. Three cuboids are joined into a bigger cuboid (picture). One of their dimensions is 6 and some faces have areas 14, 21, 16 and 30. What is the area of the face marked “?”? (4 points) Figure for problem 20

21. The picture shows part of the parabola \(y=ax^2+bx+c\). Which of these numbers is positive? (5 points) Figure for problem 21

22. A little kangaroo drew a line on a square grid and coloured the three triangles in the picture. What is the ratio of their areas? (5 points) Figure for problem 22

23. One side of a rectangular garden was increased by 20% and the other by 50%, and the garden became a square (picture). If the marked region between the diagonals of the rectangle and the square has area 30 m², what was the area of the rectangular garden? (5 points) Figure for problem 23

24. A large number N is divisible by all the numbers from 2 to 11 except two of them. Which two can the exceptions be? (5 points)

25. In the morning an ice-cream shop had 16 flavours. Ana wants a cone with two different flavours. By evening some flavours had run out, and Bela wants a cone with three different flavours from those left. Ana and Bela had the same number of choices. How many flavours had run out? (5 points)

26. Toni has 71 marbles in a box. He may take out exactly 30 or put back exactly 18, as often as he likes. What is the smallest number of marbles that can be left in the box? (5 points)

27. Vlado folded two sides of a square sheet with side 1 onto the diagonal (picture) and got a quadrilateral. What is its area? (5 points) Figure for problem 27

28. An iceberg is a cube. Exactly 90% of its volume is under water. Three edges of the cube are partly visible above the water, the visible parts being 24 m, 25 m and 27 m. How long is an edge of the cube? (5 points)

29. Different primes \(p_1,\dots,p_n\) are written in the bottom row (picture). The product of two neighbouring numbers goes in the box above them. At the top is \(K=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_n^{\alpha_n}\). If \(\alpha_2=8\), how many numbers in the table are divisible by \(p_4\)? (5 points) Figure for problem 29

30. Aleksandar and Blaže want to find out which of the figures (picture) is Kiril’s favourite. Kiril told Aleksandar its colour and Blaže its shape. Aleksandar: “I don’t know which it is, and I know Blaže doesn’t know either.” Blaže: “At first I didn’t know, but now I do.” Aleksandar: “Now I know too.” Which is Kiril’s favourite figure? (5 points) Figure for problem 30