Lietuvos Respublikos svietimo ir mokslo ministerija VU Matematikos ir informatikos fakultetas
24. We call changesum the procedure to make from a list of three numbers the new list
Kenguros organizavimo komitetas Leidykla TEV VU Matematikos ir informatikos institutas
by replacing each number by the sum of the other two. For example, from {3, 4, 6}
changesum gives {10, 9, 7} and a new changesum leads to {16, 17, 19}. If we begin
with the list {1, 2, 3}, how many consecutive changesums will be required to get the
number 2013 in the list?
A) 8 B) 9 C) 10 D) 2013 E) 2013 will never appear
KANGAROO 2013
25. The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are written around the circle in arbitrary
order. By adding to each number it's two neighbours, we obtain 10 sums. What is the
maximum possible value of the smallest of these sums?
A) 14 B) 15 C) 16 D) 17 E) 18
Time allowed: 75 min Junior
Calculators are not permitted 9--10 grades
26. On 22 cards positive integers from 1 to 22 are written. With these cards 11 fractions Questions for 3 points
have been made. What is the greatest number of these fractions that can have integer
1. The number 200013 − 2013 is not divisible by:
values?
A) 2 B) 3 C) 5 D) 7 E) 11
A) 7 B) 8 C) 9 D) 10 E) 11
2. Mary shades various shapes on square sheets of paper, as shown below.
27. How many triangles are there, whose vertices are chosen from the vertices of a given
regular polygon with 13 sides, and such that the centre of the circumcircle of the polygon
is inside of the triangle?
How many of these gures have perimeter equal to the perimeter of the sheet of paper?
A) 72 B) 85 C) 91 D) 100 E) Other value
A) 2 B) 3 C) 4 D) 5 E) 6
28. A car left point A and drove along the straight road at a speed of 50 km/h. Then every 3. Mrs. Margareth bought 4 cobs of corn for everyone in her 4-member family. In the shop
hour a car left point A, and each next car was 1 km/h faster than the previous one. The she got the discount the shop oered: Corn sale! 1 cob for 20 cents! Every sixth cob is
last car (at a speed of 100 km/h) left 50 hours after the rst one. What is the speed of for free!. How much did she pay?
the car which was in front of all the column 100 hours later after the start of the rst A) 0,80 EUR B) 1,20 EUR C) 2,80 EUR D) 3,20 EUR E) 80 EUR
car?
4. Three of the numbers 2, 4, 16, 25, 50, 125 have product 1000. What is their sum?
A) 50 km/h B) 66 km/h C) 75 km/h D) 84 km/h E) 100 km/h
A) 70 B) 77 C) 131 D) 143 E) None of the previous
29. 100 trees (oaks and birches) grow along a road. The number of trees between any two
5. Six points are marked on a square grid with cell of size 1. What is
the smallest area of a triangle with vertices at marked points?
oaks does not equal 5. What greatest number of oaks can be among these 100 trees?
1 1 1
A) 48 B) 50 C) 52 D) 60 E) The situation is not possible
A) 4 B) 3 C) 2 D) 1 E) 2
6. Adding 4 15
to 8 10
, Mihai has obtained a number which is a power
30. Yurko was walking down the street when he saw a tractor that was pulling a long pipe. of 2. The number equals:
To measure its length, Yurko walked along the pipe against the movement of the tractor A) 210 B) 215 C) 220 D) 230 E) 231
and counted 20 steps. Then he walked along the pipe with the movement of the tractor 7. On the outside a cube is painted with black and white squares as if it was built
and counted 140 steps. Yurko's step equals 1 m. His and tractor's speed were constant. of four white and four black smaller cubes. Which of the following is a correct
What is the length of the pipe? building scheme for this cube?
A) 30 m B) 35 m C) 40 m D) 48 m E) 80 m
c 2013 Kenguros konkurso organizavimo komitetas A) B) C) D) E)
8. The number n is the largest positive integer for which 4n is a 3-digit number, and m 15. The sides of rectangle ABCD are parallel to the
is the smallest positive integer for which 4m is a 3-digit number. What is the value of coordinate-axes (see pic.). We calculate for each of these
4n − 4m? points the number y -coordinate ÷ x-coordinate. Which of
A) 900 B) 899 C) 896 D) 225 E) 224 the four points gives the smallest number?
A) A B) B C) C D) D
E) It is impossible to determine
9. Consider a three-quarter circle with center O and an orientation arrow
as indicated in the picture on the right. What is the position of the 16. Today John and his son are celebrating their birthday. John multiplied correctly his age
oriented three-quarter circle when it is rst rotated counterclockwise by by the age of his son and obtained the answer 2013. In which year was John born?
90◦ around O and then re ected at the x-axis? A) 1981 B) 1982 C) 1953 D) 1952 E) More information is needed
17. John wanted to draw two equilateral triangles attached to get a C
rhombus. But he did not hit correctly all the distances and, once he
60◦
had done, Jane measured the four angles and saw that they were not
equal (see pic.). Which of the ve segments of the gure is the longest?
61◦
A) AD B) AC C) AB D) BC E) BD A ◦ B
A) B) C) D) E) 60
18. Five consecutive positive integers have the following property: three
of them have the same sum as the sum of other two. How many such 59◦
10. Which of the following numbers is the largest?
√ √ √ √ √ √ sets of integers exist?
A) 20 · 13 B) 20 · 13 C) 20 · 13 D) 201 · 3 E) 2013 A) 0 B) 1 C) 2 D) 3 E) More than 3 D
Questions for 4 points 19. What is the number of all dierent paths going from the
B C point A to the point B at the given graph?
11. Triangle COD is the image of the equilateral triangle A) 6 B) 8 C) 9 D) 12 E) 15
AOB upon rotation around O, whereby β = ∠BOC =
70◦ . Determine the angle α = ∠BAC . 20. A six-digit positive integer is given. The sum of its digits is an even number, the product
β
A) 20◦ B) 25◦ C) 30◦ D) 35◦ E) 40◦ α of its digits is an odd number. Which statement about this number might be correct?
O A) Either two or four digits of the number are even B) Such a number cannot exist
A D
C) The amount of the odd digits of the number is odd
12. The gure below shows zigzag of six unit squares. Its perimeter D) The number has six dierent digits E) None of the above
is 14. What is the perimeter of a zigzag made in the same way
Questions for 5 points
consisting of 2013 squares?
A) 2022 B) 4028 C) 4032 D) 6038 E) 8050 21. How many decimal places are there in the decimal number
1
1024000 ?
A) 10 B) 12 C) 13 D) 14 E) 1024000
D A
13. The segment AB connects two opposite vertices of a regular hexagon. 22. How many positive integers are multiples of 2013 and have exactly 2013 positive divisors
(including 1 and the number itself )?
The segment CD connects the midpoints of two opposite sides. Find
the product of the lengths of AB and CD if the area of the hexagon
A) 0 B) 1 C) 3 D) 6 E) Another number
is 60.
23. The picture shows a polygon divided into ve isosceles triangles
A) 40 B) 50 C) 60 D) 80 E) 100 B C with top angles 24◦ , 48◦ , 72◦ , 96◦ and 120◦ − the rst multiples
of the smallest top angle. All top angles have an integer number
14. A class of students had a test. If each boy had got 3 points more for the test, then of degrees. We want to make a similar picture with as many non-
the average result of the class would had been 1,2 points higher than now. How many overlapping triangles as possible. How many degrees is the smallest
percent of the students of the class are girls? top angle in that case?
A) 20% B) 30% C) 40% D) 60% E) It is impossible to determine A) 1 B) 2 C) 3 D) 6 E) 8
