2016 Student Test | Test and Chat

Back to KangarooMath

2016 Student (Grade 11 - 12)

Questions: 30 | Answered: 0

Q1. The sum of the ages of Tom and Johann is 23. The sum of the ages of Johann and Alex is 24 and the sum of the ages of Alex and Tom is 25 . How old is the oldest of them ?

3 points

ABCDE

Question 1

Q2. 푥 2 + 푥 − 30 ( 푥 − 4 푥 + 5 ) = 1 ?

3 points

ABCDE

Question 2

Q3. Maria wants to build a bridge across a river. This river has the special feature that from each point along one shore the shortest possible bridge to the other shore has always got the same length . Which of the following diagrams is definitely not a sketch of th is river ?

3 points

ABCDE

Question 3

Q4. How many whole numbers are bigger than 2015 × 2017 but smaller than 2016 × 2016 ?

3 points

ABCDE

Question 4

Q5. A scatter diagram on the xy - plane gives the picture of a k angaroo as shown on the right . Now the x - a nd the y - coordi nate are swapped a round for every point . What does the resulting picture look like ?

3 points

ABCDE

Question 5

Q6. What is the minimum number of planes necessary to border a certain region in a three - dimensional space ?

3 points

ABCDE

Question 6

Q7. Diana wants to write whole numbers into each circle in the diagram, so that for all eight small triangles the sum of the three numbers in the corners is always the same. What is the maximum amount of different numbers she can use ?

3 points

ABCDE

Question 7

Q8. The rectangles 푆 1 a nd 푆 2 shown in the picture have the same area . Determine the ratio 푥 : 푦 .

3 points

ABCDE

Question 8

Q9. If 푥 2 − 4 푥 + 2 = 0 then 푥 + equals 푥

3 points

ABCDE

Question 9

Q10. The diagram shows a circle with centre O as well as a tangent that touches the circle in point P . The arc AP ha s length 20, the arc BP has length 16. What is the size of the angle ∠ 퐴푋푃 ?

3 points

ABCDE

Question 10

Q11. 푎 , 푏 , 푐 , 푑 are positive whole numbers for which 푎 + 2 = 푏 − 2 = 푐 × 2 = 푑 ÷ 2 holds true . Which of the four numbers 푎 , 푏 , 푐 a nd 푑 is biggest ?

4 points

ABCDE

Question 11

Q12. In this number pyramid each number in a higher cell is equal to the product of the two numbers in the cells immediately underneath that number. Which of the following numbers cannot appear in the topmost cell , if the cells on the bottom row hold natural numbers greater than 1 only ?

4 points

ABCDE

Question 12

Q13. W hich value does 푥 4 take if 푥 1 = 2 a nd 푥 푛 + 1 = 푥 푛 푥 푛 for 푛 ≥ 1 ?

4 points

ABCDE

Question 13

Q14. In rectangle 퐴퐵퐶퐷 the side 퐵퐶 is exactly half as long as the diagonal 퐴퐶 ̅ ̅ ̅ ̅ . Let 푋 be the point on ̅ 퐶퐷 ̅ ̅ ̅ for which | 퐴푋 ̅ ̅ ̅ ̅ | = | 푋퐶 ̅ ̅ ̅ ̅ | holds true . How big is the angle ∠ 퐶퐴푋 ?

4 points

ABCDE

Question 14

Q15. Diana cuts a rectangle of area 2016 into 56 identical squares . The side lengths of the rectangle and the squares are all whole numbers . For how many different rectangles can she do this ? ( Two rectangles are said to be different if they are not congruent .)

4 points

ABCDE

Question 15

Q16. T he square shown in the diagram has a perimeter of 4. T he perimeter of the equilateral triangle is

4 points

ABCDE

Question 16

Q17. On the island of knights and liars everybody is either a knight (who only tells the truth) or a liar (who always lies). On your journey on the island you meet 7 people who are sitting in a circle around a bonfire. They all tel l you “ I am sitting between two liars! ”. How many liars are sitting around the bonfire ?

4 points

ABCDE

Question 17

Q18. Thr ee three - digit numbers are built using the digits 1 to 9 so that each of the nine digits is used exactly once. Which of the following numbers cannot be the sum of the three numbers ?

4 points

ABCDE

Question 18

Q19. Each of the ten points in the diagram is labelled wit h one of the numbers 0, 1 or 2. It is known that the sum of the numbers in the corner points of each white triangle is divisible by 3, while the sum of the numbers in the corner points of each black triangle is not divisible by 3. Three of the points are a lready labeled as shown in the diagram. W ith which numbers can the inner point be labeled ?

4 points

ABCDE

Question 19

Q20. Bettina chooses five points 퐴 , 퐵 , 퐶 , 퐷 a nd 퐸 on a circle and draws the tangent to the circle at point 퐴 . She realizes that the five angles marked 푥 are all equally big . ( Note that the diagram is not draw n to scale !) How big is the angle ∠ 퐴퐵퐷 ?

4 points

ABCDE

Question 20

Q21. How many different real solutions does the following equation have ?

5 points

ABCDE

Question 21

Q22. A quadrilateral has an inner circle (i.e. all four sides of the quadrilateral are tangents to the circle ). The ratio of the perimeter of the quadrilateral to the circumference of the circle is 4:3. The ratio of the area of the quadrilateral to that of the circle is therefore

5 points

ABCDE

Question 22

Q23. How many quadratic functions 푦 = 푎푥 2 + 푏푥 + 푐 ( with 푎 ≠ 0 ) have graphs that go through at least 3 of the mark ed points ?

5 points

ABCDE

Question 23

Q24. I n the right - angled triangle 퐴퐵퐶 ( with the right angle in 퐴 ) the angle bisectors of the acute angles intersect at point P . The distance of 푃 to the hypotenuse is √ 8 . What is the distance of 푃 to 퐴 ?

5 points

ABCDE

Question 24

Q25. The equations 푥 2 + 푎푥 + 푏 = 0 a nd 푥 2 + 푏푥 + 푎 = 0 both have real solutions . It is known that the sum of the squares of the solutions of the first equation is equal to the sum of the squares of the solutions of the second equation and that 푎 ≠ 푏 . 푎 + 푏 equals

5 points

ABCDE

Question 25

Q26. In a solid cube P is a point on the inside . We cut the cube into 6 ( sloping ) pyramids. Each pyramid has one face of the cube as its base and point P as its top . The volumes of five of these pyramids are 2, 5, 10, 11 a nd 14. What is the vo lume of the sixth pyramid ?

5 points

ABCDE

Question 26

Q27. A rectangular piece of paper 퐴퐵퐶퐷 is 5 cm wide and 50 cm l o ng. The paper is white on one side and grey on the other . Christina f olds the strip as shown so that the vertex 퐵 coincides with 푀 the midpoint of the edge 퐶퐷 . Then she folds it so that the vertex 퐷 coincides with 푁 the midpoint of the edge 퐴퐵 . How big is the area of the visible white part in the diagram ?

5 points

ABCDE

Question 27

Q28. Anna chooses a positive whole number 푛 a nd writes down the sum of all positive whole numbers from 1 to 푛 . A prime number 푝 divides this sum but none of the summands . Which of the following numbers is a possible value of 푛 + 푝 ?

5 points

ABCDE

Question 28

Q29. W e consider a 5 × 5 square that is split up into 25 fields . Initially all fields are white. In each move it is allowed to change the colour of three fields that are adjacent in a horizontal or vertical line ( i.e. white fields turn black and black ones turn white ). What is the smallest number of moves needed to obtain the chessboard colouring shown in the diagram ?

5 points

ABCDE

Question 29

Q30. The positive whole number 푁 ha s exactly six different (positive) factors including 1 a nd 푁 . The product of five of these factors is 648 . Which of these numbers is the sixth factor of 푁 ?

5 points

ABCDE

Question 30

2016 Student Test | Test and Chat