Känguru der Mathematik 2016

Level Student (from grade 11)

Österreich – 17.03.2016
- 3 Point Questions -
1. 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?
(A) 10 (B) 11 (C) 12 (D) 13 (E) 14
1 1 1
2. The sum + + gives
10 100 1000
3 111 111 3 3
(A) (B) (C) (D) (E)
111 1110 1000 1000 1110

3. 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 this river?

(A) (B) (C) (D) (E)

4. How many whole numbers are bigger than 2015 × 2017 but smaller than2016 × 2016?
(A) 0 (B) 1 (C) 2015 (D) 2016 (E) 2017
5. A scatter diagram on the xy-plane gives the picture of a kangaroo as shown on the right.
Now the x- and the y-coordinate are swapped around for every point. What does the resulting
picture look like?

(A) (B)

(C) (D) (E)

6. What is the minimum number of planes necessary to border a certain region in a three-dimensional space?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
7. 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?
(A) 1 (B) 2 (C) 3 (D) 5 (E) 8
8. The rectangles 𝑆1 and 𝑆2 shown in the picture have the same area. Determine the
ratio 𝑥: 𝑦.
(A) 1:1 (B) 3:2 (C) 4:3 (D) 7:4 (E) 8:5
2 2
9. If 𝑥 − 4𝑥 + 2 = 0 then 𝑥 + equals
𝑥

(A) -4 (B) -2 (C) 0 (D) 2 (E) 4

10. The diagram shows a circle with centre O as well as a tangent that touches the circle in
point P. The arc AP has length 20, the arc BP has length 16. What is the size of the
angle ∠𝐴𝑋𝑃 ?
(A) 30° (B) 24° (C) 18° (D) 15° (E) 10°

- 4 Point Questions -
11. 𝑎, 𝑏, 𝑐, 𝑑 are positive whole numbers for which 𝑎 + 2 = 𝑏 − 2 = 𝑐 × 2 = 𝑑 ÷ 2 holds
true.
Which of the four numbers 𝑎, 𝑏, 𝑐 and 𝑑 is biggest?
(A) 𝑎 (B) 𝑏 (C) 𝑐 (D) 𝑑 (E) It is not uniquely defined.
12. 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?
(A) 56 (B) 84 (C) 90 (D) 105 (E) 220
13. Which value does 𝑥4 take if 𝑥1 = 2 and 𝑥𝑛+1 = 𝑥𝑛 𝑥𝑛 for 𝑛 ≥ 1?
3 4 11 16 768
(A) 22 (B) 22 (C) 22 (D) 22 (E) 22

14. In rectangle 𝐴𝐵𝐶𝐷 the side 𝐵𝐶 is exactly half as long as the diagonal ̅̅̅̅
𝐴𝐶 . Let 𝑋 be the point on ̅̅̅̅ ̅̅̅̅| = |𝑋𝐶
𝐶𝐷 for which |𝐴𝑋 ̅̅̅̅ |
holds true. How big is the angle ∠𝐶𝐴𝑋 ?
(A) 12.5° (B) 15° (C) 27.5° (D) 42.5° (E) another angle
15. 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.)
(A) 2 (B) 4 (C) 6 (D) 8 (E) 0
16. The square shown in the diagram has a perimeter of 4. The perimeter of the equilateral triangle is
(A) 4 (B) 𝟑 + √𝟑 (C) 3 (D) 𝟑 + √𝟐 (E) 𝟒 + √𝟑

17. 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 tell
you “I am sitting between two liars!”. How many liars are sitting around the bonfire?
(A) 3 (B) 4 (C) 5 (D) 6 (E) More information is necessary to make a decision.
18. Three 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?
(A) 1500 (B) 1503 (C) 1512 (D) 1521 (E) 1575
19. Each of the ten points in the diagram is labelled with 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 already labeled as shown in the diagram. With which numbers can the inner
point be labeled?
(A) only 0 (B) only 1 (C) only 2 (D) only 0 and 1 (E) either 0 or 1 or 2

20. Bettina chooses five points 𝐴, 𝐵, 𝐶, 𝐷 and 𝐸 on a circle and draws the tangent to the circle
at point 𝐴. She realizes that the five angles marked 𝑥 are all equally big. (Note that the
diagram is not drawn to scale!)
How big is the angle ∠𝐴𝐵𝐷 ?
(A) 66° (B) 70.5° (C) 72° (D) 75° (E) 77.5°
 - 5 Point Questions -
21. How many different real solutions does the following equation have?
2 +𝑥−30
(𝑥 2 − 4𝑥 + 5)𝑥 =1 ?
(A) 1 (B) 2 (C) 3 (D) 4 (E) infinitely many

22. 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
(A) 4: 𝜋 (B) 3√2: 𝜋 (C) 16: 9 (D) 𝜋: 3 (E) 4: 3

23. How many quadratic functions 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 (with 𝑎 ≠ 0) have graphs that go through at least 3 of
the marked points?
(A) 6 (B) 15 (C) 19 (D) 22 (E) 27

24. In the right-angled triangle 𝐴𝐵𝐶 (with the right angle in 𝐴) the angle bisectors of the acute angles intersect at point P. The
distance of 𝑃 to the hypotenuse is √8. What is the distance of 𝑃 to 𝐴?

(A) 8 (B) 3 (C) √10 (D) √12 (E) 4

25. The equations 𝑥 2 + 𝑎𝑥 + 𝑏 = 0 and 𝑥 2 + 𝑏𝑥 + 𝑎 = 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 𝑎 ≠ 𝑏.
𝑎 + 𝑏 equals
(A) 0 (B) -2 (C) 4 (D) -4 (E) The sum cannot be uniquely determined.

26. 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 and 14. What is the volume of the sixth
pyramid?
(A) 1 (B) 4 (C) 6 (D) 9 (E) 12

27. A rectangular piece of paper 𝐴𝐵𝐶𝐷 is 5 cm wide and 50 cm long. The paper is
white on one side and grey on the other. Christina folds the strip as shown so that the
vertex 𝐵 coincides with 𝑀 the midpoint of the edge 𝐶𝐷. Then she folds it so that the
vertex 𝐷 coincides with 𝑁 the midpoint of the edge 𝐴𝐵. How big is the area of the visible
white part in the diagram?
(A) 50 cm² (B) 60 cm² (C) 62.5 cm² (D) 100 cm² (E) 125 cm²

28. Anna chooses a positive whole number 𝑛 and writes down the sum of all positive whole numbers
from 1 to 𝑛. A prime number 𝑝 divides this sum but none of the summands. Which of the following
numbers is a possible value of 𝑛 + 𝑝?
(A) 217 (B) 221 (C) 229 (D) 245 (E) 269

29. We 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?
(A) less than 10 (B) 10 (C) 12 (D) more than 12 (E) This
colouring cannot be obtained.

30. The positive whole number 𝑁 has exactly six different (positive) factors including 1 and 𝑁. The product of five of these factors
is 648. Which of these numbers is the sixth factor of 𝑁?
(A) 4 (B) 8 (C) 9 (D) 12 (E) 24