2017 Kangaroo Math Student Test

Q1. On the number wall shown the number on each tile is equal to th e sum of the numbers on the two tiles directly below it. Which number is on the tile marked with “?” ?

3 points

Q2. Many model railways use the H0-scale 1:87. For his railway Benj amin owns a 2 cm high model of his brother in H0-scale. How tall is his b rother in reality?

3 points

Q3. In the diagram we see 10 islands that are connected by 15 bridg es. What is the minimum number of bridges th at need to be closed off so tha t there is no connection from A to B anymore?

3 points

Q4. : “The number is less than 30.” 5: “The number is divisible b y 3.” 6: “One digit of the number is a 7.” How big is the sum of the digits of the nu mber, Lilli is thinki ng of?

3 points

Q6. The diagram shows a circle with centre ܱ and the diameters ܣܤ and ܥܺ . Let ܱܤ ൌ ܤܥ . Which fraction of the circle area is shaded? ଶ ଵ ଶ ଷ ସ

3 points

Q7. A 4  1  1 cuboid is made up of 2 white a nd 2 grey cubes as shown. Whic h of the following cuboids can be build entirely out of such 4  1  1 cuboids?

3 points

Q10. The graph of which of the followi ng functions has the most inte rsections with the graph of the function ݂ሺݔሻ ൌ ݔ ? ଶ ଷ ସ ସ

3 points

Q11. Three circles with centres ܣ , ܤ , ܥ touch each other in pairs from the outside (see diagram). Their radii are 3, 2 and 1. How big is the area of the triangle ܣܤܥ ?

4 points

Q13. Two cylinders ܣ and ܤ have the same volume. The radius of the base of ܤ is 10 % bigger than that of ܣ . How much is the height of ܣ greater than that of ܤ ?

4 points

Q14. Each face of the polyhedron shown is either a triangle or a squ are. Each square borders 4 triangles, and each triangle borders 3 squares. The polyhedron has 6 squares. How many triangles does it have?

4 points

Q15. The four faces of a regular tetr ahedron are labelled with the f our digits 2, 0, 1 and 7 (one digit on each face). For a game, four such tetrahedrons are used as fair dice. All four di ce are thrown simultaneously. Three of the four faces of each die can then be seen from above. What is the probability that we can form the number 2017 using exactly one of the three visible digits of each die? ଵ ଺ଷ ଼ଵ ଷ ଶଽ

4 points

Q16. The polynomial 5ݔ ൅ܽݔ ൅ܾݔ൅24 has whole number coefficients ܽ and ܾ . ଷ ଶ Which of the following numbers is definitely not a solution to the equation 5ݔ ൅ܽݔ ൅ܾݔ൅24ൌ0 ?

4 points

Q17. Julia has 2017 round discs availa ble: 1009 black ones and 1008 white ones. Using them, she wants to lay the biggest square pattern (as shown) po ssible and starts by using a black disc in the left upper corner. Subsequently she l ays the discs in such a way that the colours alternate in each row and column. How many dis cs are left over when she has laid the biggest square possible?

4 points

Q18. Two consecutive positive whole n umbers are written on a board. The sum of the digits of e ach number is divisible by 7. What is the minimum number of digits the smaller of the t wo numbers has to have?

4 points

Q19. In a convex quadrilateral ܣܤܥܦ the diagonals are perpendi cular to each other. The length of the edges are ܣܤ ൌ 2017 , ܤܥ ൌ 2018 and ܥܦ ൌ 2019 (diagram not to scale). How long is side ܣܦ ?

4 points

Q20. Lilli tries to be a well-behaved kangaroo but she is having jus t too much fun not to lie every now and then. Therefore every third statement of hers is a lie and the rest is true. Sometimes she starts with a lie and sometimes with one or two true statements. Lilli thin ks of a two-digit number and says to her friend:

4 points

Q21. How many positive whole numbers have the property that, if you delete the last digit you obtain a new number, which is exactly equal to 1/14 of the original number?

5 points

Q22. The diagram shows a regular hexagon with side length 1. The gre y flower is outlined by circular arcs with radius 1 whose centre’s lie in the vertices of the hexagon. How big is the area of the grey flower? గ ଶగ గ

5 points

Q24. We look at a regular tetrahedron with volume 1. Its four vertic es are cut off by planes that go through the midpoints of the respective edges (see diag ram). How big is the volume of the remaining solid? ସ ଷ ଶ ଵ ଵ

5 points

Q25. The sum of the three side lengths of a right-angled triangle eq uals 18. The sum of the squares of these three side lengths equals 128. How big i s the area of the triangle?

5 points

Q26. Anna has five boxes, as well as five black balls and five white balls. She is allowed to d ecide how she shares out the balls between the boxes as long as she puts at least one ba ll into each box. Beate randomly chooses one box and takes one ball without looking. Beate wins if she draws a white ball. Otherwise Anna wins. How should Anna distribute the balls in order to get the highes t probability of winning?

5 points

Q27. Nine whole numbers were written into the cells of a 3  3-table. The sum of these n ine numbers is 500. We know that the numbers in two adjacent cells (with a common sideline) differ by exactly 1. Which number is in the middle cell?

5 points

Q30. 17 people live on an island. Eac h person is either a liar (wh o always lies) or a nobleman (who always tells the truth). Over a thousand of them attend a banquet where they all sit together around one big round table. Everyone is saying, “Of my two neighbours, one is a liar and one is a noble man.” What is the maximum number of noblemen on the island?

5 points

2017 Student (Grade 11 - 12)

Q1. On the number wall shown the number on each tile is equal to th e sum of the numbers on the two tiles directly below it. Which number is on the tile marked with “?” ?

Q2. Many model railways use the H0-scale 1:87. For his railway Benj amin owns a 2 cm high model of his brother in H0-scale. How tall is his b rother in reality?

Q3. In the diagram we see 10 islands that are connected by 15 bridg es. What is the minimum number of bridges th at need to be closed off so tha t there is no connection from A to B anymore?

Q4. : “The number is less than 30.” 5: “The number is divisible b y 3.” 6: “One digit of the number is a 7.” How big is the sum of the digits of the nu mber, Lilli is thinki ng of?

Q5. Four of the following five pictures show pieces of the graph of the same quadratic function. Which piece does not belong?

Q6. The diagram shows a circle with centre ܱ and the diameters ܣܤ and ܥܺ . Let ܱܤ ൌ ܤܥ . Which fraction of the circle area is shaded? ଶ ଵ ଶ ଷ ସ

Q7. A 4  1  1 cuboid is made up of 2 white a nd 2 grey cubes as shown. Whic h of the following cuboids can be build entirely out of such 4  1  1 cuboids?

Q8. Which quadrant contains no points of the graph of the linear fu nction ݂ሺݔሻ ൌ െ3.5ݔ ൅7 ?

Q9. In each of the five boxes

Q10. The graph of which of the followi ng functions has the most inte rsections with the graph of the function ݂ሺݔሻ ൌ ݔ ? ଶ ଷ ସ ସ

Q11. Three circles with centres ܣ , ܤ , ܥ touch each other in pairs from the outside (see diagram). Their radii are 3, 2 and 1. How big is the area of the triangle ܣܤܥ ?

Q12. The positive number ݌ is smaller than 1, and the number ݍ is greater than 1. Which of the following numbers is biggest? ௣

Q13. Two cylinders ܣ and ܤ have the same volume. The radius of the base of ܤ is 10 % bigger than that of ܣ . How much is the height of ܣ greater than that of ܤ ?

Q14. Each face of the polyhedron shown is either a triangle or a squ are. Each square borders 4 triangles, and each triangle borders 3 squares. The polyhedron has 6 squares. How many triangles does it have?

Q16. The polynomial 5ݔ ൅ܽݔ ൅ܾݔ൅24 has whole number coefficients ܽ and ܾ . ଷ ଶ Which of the following numbers is definitely not a solution to the equation 5ݔ ൅ܽݔ ൅ܾݔ൅24ൌ0 ?

Q18. Two consecutive positive whole n umbers are written on a board. The sum of the digits of e ach number is divisible by 7. What is the minimum number of digits the smaller of the t wo numbers has to have?

Q19. In a convex quadrilateral ܣܤܥܦ the diagonals are perpendi cular to each other. The length of the edges are ܣܤ ൌ 2017 , ܤܥ ൌ 2018 and ܥܦ ൌ 2019 (diagram not to scale). How long is side ܣܦ ?

Q21. How many positive whole numbers have the property that, if you delete the last digit you obtain a new number, which is exactly equal to 1/14 of the original number?

Q22. The diagram shows a regular hexagon with side length 1. The gre y flower is outlined by circular arcs with radius 1 whose centre’s lie in the vertices of the hexagon. How big is the area of the grey flower? గ ଶగ గ

Q23. We look at the sequence 〈 ܽ ௡ 〉 with ܽ ଵ ൌ 2017 and ܽ ௡ାଵ ൌ . Then: ܽ ଽଽଽ ൌ ௔ ೙ ଶ଴ଵ଺ ଵ

Q24. We look at a regular tetrahedron with volume 1. Its four vertic es are cut off by planes that go through the midpoints of the respective edges (see diag ram). How big is the volume of the remaining solid? ସ ଷ ଶ ଵ ଵ

Q25. The sum of the three side lengths of a right-angled triangle eq uals 18. The sum of the squares of these three side lengths equals 128. How big i s the area of the triangle?

Q27. Nine whole numbers were written into the cells of a 3  3-table. The sum of these n ine numbers is 500. We know that the numbers in two adjacent cells (with a common sideline) differ by exactly 1. Which number is in the middle cell?

Q28. How big is ݔ ൅ݕ , if |ݔ|൅ݔ ൅ݕ ൌ 5 as well as ݔ൅| ݕ|െݕൌ1 0 holds true?

Q29. How many different three-digit numbers ܣܤܥ are there so that ሺܣ൅ܤሻ is a three-digit power of two?