 Trang chủ / Toán Kangaroo / Lớp 7-8

USA Math Kangaroo Grade 7-8 in 2009

1. 3 points questions

Among these numbers, which one is even?

2. At a party there were 4 boys and 4 girls. Boys danced only with girls and girls danced only with boys. Afterwards we asked all of them, how many dance partners they each had. The boys said: 3, 1, 2, 2. Three of girls said: 2, 2, 2.What number did the fourth girl say?

3. In the figure, the triangle consists of 9 identical equilateral triangles. The perimeter of the outer big triangle is 36 cm. What is the value of the perimeter of the shaded inner hexagon? 4. Harry is a postman. One day he has to deliver packages to Kangourou street delivering one package to each odd numbered house. The first house he visited was number 15 and the last one was number 53, while he visited all the houses in between with odd number in their address. In how many houses did Harry deliver a package? 5. The area of the big square is 1. What is the area of the black little square? 6. What is the remainder of the division of 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x 15 - 6 by 13?

7. In a garden there are cats and dogs. All the cats together have double number of legs than the noses of all dogs together. Then the number of cats is

8. In the triangle ΑΒΓ , the angle BA∆ is equal to ο32 . In addition ABΑ∆ Γ∆==. How many degrees is the angle ΒΑΓ? 9. Due to restrictions in weight, in an elevator it is only permitted to enter 12 adults maximum or 20 children maximum. It is understood the elevator can enter mixed adults and children. If 9 adults entered the elevator, what is the maximum number of children that can enter? (For practical reasons we assume that all adults have the same weight, all children have the same weight and 12 adults weigh as much as 20 children).

10. Which of the following links requires more than one piece of rope to construct? 11. 4 points questions

How many natural numbers from 1 to 30 inclusive, have the property that their square and cube have the same number of digits?

12. What is the smallest number of points in the figure one needs to remove so that no 3 of the remaining points are collinear? 13. Nick drew an acute and an obtuse triangle. The four of the angles of the two triangles were 120o ,80o ,55o and 10o . How many degrees is the smallest angle of the acute triangle?

14. What is the area of the shaded region, if the length of the outer square is 1? 15. In an island there are 3 inhabitants. Some of them always say the truth and the rest always say lies. One day, these 3 people stood in a queue. Every one of the last two in line said that the person in front of him is a liar. The first one on line said that the other two are liars. How many of the 3 people on this island are liars.

16. The product of four distinct natural numbers is 100. What is their sum?

17. n the equality of products Α x Β x Γ x ∆ x Ε = Α x Ζ x Η x Θ x Ι x Κ. Every letter represents a different number from the digits 0, 1, 2, 3, ..., 9. How many different values could the product Α x Γ Η x Κ have?

18. We want to colour the squares in the grid using colours A, B, C and D in such a way that neighbouring squares do not have the same colour (squares that share a vertex are considered neighbours). Some of the squares have been coloured as shown. What are the possibilities for the shaded square? 19. Andreas, Vasilis, Yiannis and Demetris have books in their bags. One of them has one book in his bag,another one has two, another has three and the last one has four books in his bag. Andreas, Vasilis and Demetris have together 6 books. Vasilis and Yiannis together have 6 books. Vasilis has in his bag less books than Andreas. Who is the one that has only one book in his bag;

20. The first three patterns are shown. Not including the square hole, how many unit squares are needed to build the 10th pattern in this sequence? 21. 5 points questions

Dino calculated the value of the expression 2009 - 2008 + 2007 - 2006 + ...  + 5 - 4 - 3 - 2 - 1 and Dina calculated the value of the expression 2008 - 2007 + 2006 - 2005 + ... + 4 - 3 + 2 - 1.
What is the sum of the values of both Dino and Dina?

22. How many four-digit numbers composed only of digits 1,2,3 exist, in which any two neighbouring digits differ by 1 ? (Repetition of digits is allowed).

23. In a straight road we mark the distances in Km from a tree. A sign shows 1/5 Km and another shows 1/3Km from the tree. What is the position of the sign that shows 1/4 Km from the tree? 24.

John and Demetri had the same cube. John painted the outer surface of his cube. Demetri made three cuts of his cube in order to make eight smaller cuboids and then he painted the outer surface of the eight smaller cuboids. How much more paint did Demetri used compared to John? 25. We place a square of dimensions 6 cm x 6 cm on top of a triangle. The shaded common region covers the 60% of the triangle. The same region covers the 2/3 of the square. What is the area of the triangle? 26. Costa wrote on a computer the products of the consecutive numbers 1 x 2, 1 x 2 x 3, 1 x 2 x 3 x 4, 1 x 2 x 3 x 4 x 5, ..., 1 x 2 x 3 x 4 x ... x 100.
Then he added all these numbers. What is the last digit of the number he found?

27. Tasia drew a strange windmill. He began drawing 5 lines passing through the same point and then she connected them with some smaller lines. In this way 5 triangles were established with a common vertex. What is the sum of the of the marked 10 angles of the 5 triangles?  28. Five friends, Anna, Viky, Yianna, Danae and Elli compared their height. We observe that
• Anna is the shortest of all
• Danae is taller than Viky but shorter than Elli
Which of the following is definitely wrong?

29. We write the natural number 1, 2, 3, 4, ..., consecutively in three columns of squares, as shown in the figure. In places where there is X, the square remains empty. The empty squares are in triples diagonal. What is the number in the 100th square of the middle column? 30. The product of three natural numbers is equal to 140. The second of the numbers is seven times the first one, and the third of the numbers is smaller than the second. What is the sum of the three natural numbers?