Lietuvos Respublikos švietimo ir mokslo ministerija VU Matematikos ir informatikos fakultetas

23. The picture shows the same cube from two different views.
Kengūros organizavimo komitetas Leidykla TEV VU Matematikos ir informatikos institutas
It is built from 27 small cubes, some of them are black and
some are white. What is the largest number of black cubes
there could be in this cube?
A) 5 B) 7 C) 8 D) 9 E) 10 KANGAROO 2014
24. On an island, frogs are always either green or blue. The number of blue frogs increased
by 60% while the number of green frogs decreased by 60%. It turns out that the new
ratio of blue frogs to green frogs is the same as the previous ratio in the opposite order
(green frogs to blue frogs). By what percentage did the overall number of frogs change?
A) 0% B) 20% C) 30% D) 40% E) 50% Time allowed: 75 minutes Junior
Calculators are not permitted 9--10 grades
25. Tom wrote down several distinct positive integers, not exceeding 100. Their product was
Questions for 3 points
not divisible by 18. At most how many numbers could he have written?
A) 5 B) 17 C) 68 D) 69 E) 90 1. Each year the day of the Kangaroo competition is the third Thursday of March. What
is the earliest possible date (day of March) of the competition?
26. Any three vertices of a cube form a triangle. What is the number of all such triangles A) 14 B) 15 C) 20 D) 21 E) 22
whose vertices are not all in the same face of the cube?
A) 16 B) 24 C) 32 D) 40 E) 48
2. The MSC Fabiola holds a record as being the largest container ship to enter San
√
n
√
n
Francisco Bay. It carries 12500 containers which if placed end to end would stretch
27. The equalities k = 2014 + m = 1024 + 1 are given for positive integers k, m, n with about 75 km. Roughly, what is the length of one container?
2 < n < 10. What is the sum of the digits of m? A) 6 m B) 16 m C) 60 m D) 160 m E) 600 m
A) 4 B) 8 C) 12 D) 16 E) 20

28. Consider the set of all the 7-digit numbers that can be obtained using, for each number, 3. If a, b and c denote the lengths of the lines in the picture, then which of the following
all the digits 1, 2, 3,..., 7. The numbers of the set are listed in increasing order. What is is correct?
the last number of the first half of this list?
A) 1234567 B) 3765421 C) 4123567 D) 4352617 E) 4376521 a b c
E
29. Let ABC be a triangle such that AB = 6, AC = 8 and A
BC = 10 and M be the midpoint of BC. AM DE is a
A) a < b < c B) a < c < b C) b < a < c D) b < c < a E) c < b < a
square, and M D intersects AC at point F (see picture).
Find the area of quadrilateral AF DE.
124 125 126 127 128 2 4
A) 8 B) 8 C) 8 D) 8 E) 8
D 4. Which number is in the middle of and ?
F 3 5
B M C 11 7 3 6 5
A) B) C) D) E)
30. There are 2014 persons in a row. Each of them is either a liar (who always lies) or a 15 8 4 15 8
knight (who always tells the truth). Each person says ’There are more liars to my left
than knights to my right’. How many liars are there in the row? 5. This year, in the year number 2014 the last digit is bigger than the sum of the other
A) 0 B) 1 C) 1007 D) 1008 E) 2014 three digits. How many years ago did this last occur?
A) 1 B) 3 C) 5 D) 7 E) 11

6. The length of the edges of the big regular hexagon is two times
the length of the edges of the small regular hexagon. The small
hexagon has an area of 4. What is the area of the big hexagon?
c 2014 Kengūros konkurso organizavimo komitetas A) 16 B) 14 C) 12 D) 10 E) 8
 7. What is the negation of the statement ”Everybody solved more than 20 problems”? 15. A new kind of crocodile has been discovered in Africa. The length of his tail is a third
A) Nobody solved more than 20 problems B) Somebody solved less than 21 problems of his entire length. His head is 93 cm long and its length is a quarter of the crocodile‘s
C) Everybody solved less than 21 problems D) Somebody solved exactly 20 problems length without his tail. How long is this crocodile in cm?
E) Somebody solved more than 20 problems A) 558 B) 496 C) 490 D) 372 E) 186
18

35 14
16. In the picture there is a special dice. Numbers on the opposite faces
8. In a coordinate system Tom drew a square. The coordinates of the two opposite vertices always make the same sum. The numbers that we cannot see in the
on the x-axis are (−1; 0) and (5; 0). Which of the following are the coordinates of another picture are all prime numbers. What is the digit sum of the number
vertex of this square? opposite to 14?
A) (2; 0) B) (2; 3) C) (2; −6) D) (3; 5) E) (3; −1) A) 9 B) 8 C) 7 D) 6 E) 5

9. In a certain village, the ratio between adult men and adult women is 2 : 3 and the ratio 17. In the picture, P B is tangent to the circle with center O and P C bisects the angle AP B.
between adult women and children is 8 : 1. What is the ratio between adults (men and Calculate the angle BCP .
B
women) and children?
A) 5 : 1 B) 10 : 3 C) 13 : 1 D) 12 : 1 E) 40 : 3 C

10. The big wheel of this bicycle has perimeter 4.2 metres. The
small wheel has perimeter 0.9 metres. At a certain moment, A O P
the valve of both wheels are at their lowest point. The bicycle
rolls to the left. After how many metres will both valves first
be at their lowest point together again?
A) 30◦ B) 45◦ C) 60◦ D) 75◦ E) It depends on the position of point P
A) 4.2 B) 6.3 C) 12.6 D) 25.2 E) 37.8
18. A chess player played 40 matches and scored 25 points (a win counts as one point, a
Questions for 4 points draw counts as half a point, and a loss counts as zero points). How many more matches
did he win than lose?
11. A grandmother, her daughter and her granddaughter can this year say that the sum of A) 5 B) 7 C) 10 D) 12 E) 15
their ages is 100. In which year was the granddaughter born if each age is a power of 2?
A) 1998 B) 2006 C) 2010 D) 2012 E) 2013 19. Triplets Jane, Danielle and Hannah wanted to buy identical hats. However, Jane lacked
a third of their price, Danielle a quarter and Hanna a fifth. When the hats became 9,40
12. Paul put some rectangular paintings on the wall. For each picture he EUR cheaper, the sisters joined their savings and each of them bought a hat. Not a cent
put one nail into the wall 2.5 m above the floor and attached a 2 m was left. What was the price of a hat before the price reduction?
long string at the two upper corners. Which of the following pictures is A) 12 EUR B) 16 EUR C) 28 EUR D) 36 EUR E) 112 EUR
closest to the floor (format: width in cm × height in cm)? 1 25
A) 60 × 40 B) 120 × 50 C) 120 × 90 D) 160 × 60 E) 160 × 100 20. Let p, q, r be positive integers and p + q+ r1
= 19 . Which of the following is equal to
p + q + r?
A) 6 B) 8 C) 10 D) 13 E) 14
13. Six girls share a flat with two bathrooms which they use every morning beginning at 7:00
o’clock. They use the bathroom one at a time, and sit down to eat breakfast together Questions for 5 points
as soon as the last girl has finished. They spend 9, 11, 13, 18, 22 and 23 minutes in the
bathroom respectively. Being well organized, what is the earliest they can have breakfast 21. In the equation, N × U × (M + B + E + R) = 33, each letter stands for a different digit.
together? How many different ways are there to choose the values of the letters?
A) 7:48 B) 7:49 C) 7:50 D) 7:51 E) 8:03 A) 12 B) 24 C) 30 D) 48 E) 60

22. On the picture shown Kaan wants to add some line segments so that each
14. In the following figure there is a regular octagon. The shaded area of the seven points has the same number of connections to other points.
measures√3 cm2 . Find the area
√ of the octagon in cm .
2
What is the least number of line segments Kaan must draw?
A) 8 + 4 2 B) 9 C) 8 2 D) 12 E) 14 A) 4 B) 5 C) 6 D) 9 E) 10