EUROPEAN ‘KANGAROO’ MATHEMATICAL CHALLENGE ‘GREY’ - 2012 - Thursday 15th March 2012 Organised by the United Kingdom Mathematics Trust and the Association Kangourou Sans Frontières

1. A watch is placed face up on a table so that its minute hand points north-east. How many minutes pass before the minute hand points north-west for the first time?
20
15
45
30
40

2. The Slovenian hydra has five heads. Every time a head is chopped off, five new heads grow. Six heads are chopped off one by one. How many heads will the hydra finally have?
29
30
25
33
35

3. Each of the nine paths in a park is 100 m long. Ann wants to go from X to Y without going along any path more than once. What is the length of the longest route she can choose?

500 m
700 m
900 m
600 m
800 m

4. The diagram (which is drawn to scale) shows two triangles. In how many ways can you choose two vertices, one in each triangle, so that the straight line through the two vertices does not cross either triangle?

3
more than 4
2
1
4

5. Werner folds a sheet of paper as shown in the diagram and makes two straight cuts with a pair of scissors. He then opens up the paper again. Which of the following shapes cannot be the result?

6. In each of the following expressions, the number 8 is to be replaced by a fixed positive number other than 8. In which expression do you get the same result, whatever positive number 8 is replaced by?

7. Kanga forms two four-digit numbers using each of the digits 1, 2, 3, 4, 5, 6, 7 and 8 exactly once. Kanga wants the sum of the two numbers to be as small as possible. What is the value of this smallest possible sum?
4734
6912
3333
3825
2468

8. Mrs Gardner has beds for peas and strawberries in her rectangular garden. This year, by moving the boundary between them, she changed her rectangular pea bed to a square by lengthening one of its sides by 3 metres. As a result of this change, the area of the strawberry bed reduced by 15 m². What was the area of the pea bed before the change?

18 m²
10 m²
9 m²
5 m²
15 m²

9. Barbara wants to complete the diagram below by inserting three numbers, one into each empty cell. She wants the sum of the first three numbers to be 100, the sum of the middle three numbers to be 200 and the sum of the last three numbers to be 300. What number should Barbara insert into the middle cell of the diagram?

100
70
50
75
60

11. Four cards each have a number written on one side and a phrase written on the other. The four phrases are ‘divisible by 7’, ‘prime’, ‘odd’ and ‘greater than 100’ and the four numbers are 2, 5, 7 and 12. On each card, the number does not have the property given on the other side. What number is written on the same card as the phrase ‘greater than 100’?
7
5
12
2
impossible to determine

12. Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle with sides 6 cm as shown. The sum of the perimeters of the three small triangles is equal to the perimeter of the remaining hexagon. What is the side-length of one of the small triangles?

2 cm
1 cm
1.2 cm
1.5 cm
1.25 cm

13. A piece of cheese was cut into a large number of pieces. During the course of the day, a number of mice came and stole some pieces, watched by the lazy cat Ginger. Ginger noticed that each mouse stole a different number of pieces, that each mouse stole fewer than 10 pieces and that no mouse stole exactly twice as many pieces as any other mouse. What is the largest number of mice that Ginger could have seen stealing cheese?
5
7
8
4
6

14. At the airport there is a moving walkway 500 metres long, which moves with a speed of 4 km/hour. Andrew and Bill step onto the walkway at the same time. Andrew walks with a speed of 6 km/hour on the walkway while Bill stands still. When Andrew comes to the end of the walkway, how far is he ahead of Bill?
160 m
300 m
200 m
100 m
250 m

15. A cube is being rolled on a plane so it turns around its edges. Its bottom face passes through the positions 1, 2, 3, 4, 5, 6 and 7 in that order, as shown. Which of these two positions were occupied by the same face of the cube?

1 and 7
2 and 7
2 and 6
1 and 6
1 and 5

16. Rick has five cubes. When he arranges them from smallest to largest, the difference between the heights of two neighbouring cubes is always 2 cm. The largest cube is as high as a tower built of the two smallest cubes. How high is a tower built of all five cubes?
50 cm
6 cm
44 cm
14 cm
22 cm

17. In the diagram, WXYZ is a square, M is the midpoint of WZ and MN is perpendicular to WY. What is the ratio of the area of the shaded triangle MNY to the area of the square?

3:16
1:5
1:6
7:40
7:36

18. The tango is danced by couples, each consisting of one man and one woman. At a dance evening, fewer than 50 people were present. At one moment, 3/4 of the men were dancing with 4/3 of the women. How many people were dancing at that moment?
24
20
32
30
40

19. David wants to arrange the twelve numbers from 1 to 12 in a circle so that any two neighbouring numbers differ by either 2 or 3. Which of the following pairs of numbers have to be neighbours?
6 and 8
4 and 6
3 and 5
5 and 8
7 and 9

20. Some three-digit integers have the following property: if you remove the first digit of the number, you get a perfect square; if instead you remove the last digit of the number, you also get a perfect square. What is the sum of all the three-digit integers with this curious
property?
1993
2016
1013
1465
1177

21. A book contains 30 stories, each starting on a new page. The lengths of the stories are 1, 2, 3, ..., 30 pages in some order. The first story starts on the first page. What is the largest number of stories that can start on an odd-numbered page?
18
21
15
20
23

22. An equilateral triangle starts in a given position and is moved to new positions by a sequence of steps. At each step it is rotated clockwise about its centre; at the first step by 3°, at the second step by a further 9°; at the third by a further 27° and, in general, at the n-th step by a further (3ⁿ)°. How many different positions, including the initial position, will the triangle occupy?

5
360
6
4
3

23. A long thin ribbon is folded in half lengthways, then in half again and then in half again. Finally, the folded ribbon is cut through at right angles to its length forming several strands. The lengths of two of the strands are 4 cm and 9 cm. Which of the following could not have been the length of the original ribbon?
68 cm
all answers are possible
88 cm
72 cm
52 cm

24. A large triangle is divided into four smaller triangles and three quadrilaterals by three straight line segments. The sum of the perimeters of the three quadrilaterals is 25 cm. The sum of the perimeters of the four triangles is 20 cm. The perimeter of the original triangle is 19 cm. What is the sum of the lengths of the three straight line segments?

16 cm
12 cm
15 cm
11 cm
13 cm

25. Each cell of the 3 x 3 grid shown has placed in it a positive number so that: in each row and each column, the product of the three numbers is equal to 1; and in each 2 x 2 square, the product of the four numbers is equal to 2. What number should be placed in the central cell?

4
1/4
1/8
16
8