Kangaroo 2013 Junior
(lukio 1st year)
page 1 / 9
Name ________________________ Group ______
Points: _______ Kangaroo leap: _____
Separate this answer sheet from the test.
Write your answer under each problem number.
For each wrong answer, 1/4 of the points of the problem will be deducted.
If you don’t want to answer a question, leave the space empty and no deduction
will be made.
PROBLEM
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ANSWER
PROBLEM
ANSWER
PROBLEM
ANSWER
Kangaroo 2013 Junior
(lukio 1st year)
page 2 / 9
3 points
1.
Mary drew figures on square paper sheets. How many of these figures have perimeter equal to
the perimeter of the paper sheet?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
2.
Mrs. Margareth bought four cobs of corn for everyone in her four-member family. In the shop she got
the discount the shop offered. How much did she pay?
(A) 0,80 €
(B) 1,20 €
(C) 2,80 €
(D) 3,20 €
(E) 3,40 €
3.
Kangoo loves travelling through tunnels by train. Yesterday he looked at his watch when he
entered the tunnel (12:30) and when he came out again (12:34). Which of the following is
certainly true? Kangoo was in the tunnel
(A) exactly four
minutes
(B) at most four
minutes
(C) at least four
minutes
(D) at least three
minutes
(E) more than
four minutes
Kangaroo 2013 Junior
(lukio 1st year)
page 3 / 9
4.
By drawing two circles, Mike obtained a figure, which consists of three regions (see picture): the
first region is inside the circle on the left, the second area is inside the circle on the right, and the
third region is inside both of the circles.
At most how many regions could he obtain by drawing two squares instead of two circles?
(A) 3
(B) 5
(C) 6
(D) 8
(E) 9
5.
Three of the numbers 2, 4, 16, 25, 50 and 125 have product 1000. What is their sum?
(A) 131
(B) 91
(C) 77
(D) 70
(E) 45
6.
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?
(A) 1/4
7.
Adding
it?
(A)
(B) 1/3
to
(C) 1/2
(D) 1
(E) 2
, Sarah has correctly obtained a number which is a power of 2. What number is
(B)
(C)
(D)
(E)
Kangaroo 2013 Junior
(lukio 1st year)
page 4 / 9
8.
A cube is painted on the outside with black and white squares as if it was built of four white and
four black smaller cubes. Which of the following is a correct building scheme for this cube?
(A)
(B)
(C)
(D)
(E)
9.
The number is the largest positive integer for which
is a 3-digit number, and
positive integer for which
is a 3-digit number. What is the value of
?
(A) 900
(B)
(C)
is the smallest
(D)
(E)
(D) √
(E) √
10.
Which of the numbers (A) - (E) is the largest?
(A)
√
(B) √
(C) √
√
Kangaroo 2013 Junior
(lukio 1st year)
page 5 / 9
4 points
11.
Triangle RZT is the image of the equilateral triangle AZC upon rotation around Z, whereby
. Determine the angle
.
(A) 20°
(B) 25°
(C) 30°
(D) 35°
(E) 40°
12.
The figure below shows a “zigzag” of six 1 cm x 1 cm squares. Its perimeter is 14 cm. What is the
perimeter of a “zigzag” made in the same way consisting of 2013 squares?
(A) 2022 cm
(B) 4028 cm
(C) 4032 cm
(D) 6038 cm
(E) 8050 cm
13.
The segment AB connects two opposite vertices of a regular hexagon. The segment CD, which is
perpendicular to AB, connects the midpoints of two opposite sides. Find the product of the lengths
AB and CD if the area of the hexagon is 60.
(A) 40
(B) 50
(C) 60
(D) 80
(E) 100
Kangaroo 2013 Junior
(lukio 1st year)
page 6 / 9
14.
A class of pupils had a test. If each boy had got 3 points more in the test, then the average
result in the class would had been 1,2 points higher than now. How many percent of the pupils of
the class are girls?
(A) 20 %
(B) 30 %
(C) 40 %
(D) 60 %
(E) Impossible to
determine
15.
Sides AB and CD of rectangle ABCD are parallel to the x-axis, and none of its vertexis lie on the
y-axis. The x-coordinate of A is smaller than the x-coordinate of B. The y-coordinate of A is smaller
than the y-coordinate of D. Which of the four points gives the smallest result for the fraction
(y-coordinate) : (x-coordinate)?
(A) A
(B) B
(C) C
(D) D
(E) it depends on
the situation
16.
Yesterday John and his son were celebrating their birthday. Today John multiplies correctly his age
by the age of his son and obtains the answer 2013. In what year was John born?
(A) 1952
(B) 1953
(C) 1981
(D) 1982
(E) impossible to
know
17.
Jack tried to draw two equilateral triangles attached to get a rhombus, but he did not measure all
the distances and angles correctly. Afterwards Jane measured the four angles correctly (see
picture). Which of the five segments of the figure is the longest?
(A) AD
(B) AC
(C) AB
(D) BC
(E) BD
Kangaroo 2013 Junior
(lukio 1st year)
page 7 / 9
18.
Five consecutive positive integers have the following property: three of them have the same
sum as the sum of the other two. How many such sets of integers exist?
(A) 0
(B) 1
(C) 2
(D) 3
(E) more than 3
19.
What is the number of all different paths leading from point A to point B in the given picture? It is
only allowed to move according to the arrows.
(A) 6
(B) 8
(C) 9
(D) 12
(E) 15
20.
A six-digit number is given. The sum of its digits is an even number, and the product of its digits is
an odd number. Which is the correct statement about the given number?
(A) Either two or four digits of the number are even.
(B) Such a number cannot exist.
(C) The amount of the odd digits of the number is odd.
(D) The number can consist of six digits different from each other.
(E) None of the above.
Kangaroo 2013 Junior
(lukio 1st year)
page 8 / 9
5 points
21.
How many decimal places are there in the decimal number
(A) 10
(B) 12
(C) 13
?
(D) 14
(E) 1024000
22.
What is the minimal number of chords on a circle, if the number of intersecting points among
them in the interior of the circle is exactly 50?
(A) 10
(B) 11
(C) 12
(D) 13
(E) 14
23.
100 students took part in the Math Olympiad, 50 students participated in the Physics Olympiad
and 48 students were involved in the Computer Sciences Olympiad. Each student was asked three
yes-no-questions: did you participate in 1) at least one 2) at least two 3) three competitions? The
answer “yes” was given 50 % less in question 2 than in question 1 and 2/3 less in question 3 than
in question 1. How many students participated in at least one of these competitions?
(A) 100
(B) 108
(C) 124
(D) 150
(E) 198
24.
From a list of three numbers we call "changesum" the procedure to make a new list by replacing
} "changesum" gives
each number by the sum of the other two. For example, from {
{
} and a new "changesum" leads to {
}. If we begin with the list {
}, how
many consecutive "changesums" will be required for the number 2013 to appear in the list?
(A) 8
(B) 9
(C) 10
(D) 2013 will appear several times.
(E) 2013 will never appear.
25.
Using all integers from 1 to 22, 11 fractions are made. Each integer is used exactly once. What is
the greatest number of these fractions that can have integer values?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11
Kangaroo 2013 Junior
(lukio 1st year)
page 9 / 9
26.
How many such triangles are there, whose vertices are chosen from the vertices of a regular
polygon with 13 sides and the centre of the circumscribed circle of the polygon is inside of the
triangle?
(A) 72
(B) 85
(C) 91
(D) 100
(E) Other number
27.
A spaceship left point A and flew straight at a constant speed of 50 km/h. Then each hour a
spaceship left point A and flew straight at a constant speed, and the next spaceship was always
1 km/h faster than the previous one. The last spaceship (at a speed of 100 km/h) left 50 hours
after the first one. What is the speed of the spaceship which was furthest from point A 100 hours
after the start of the first spaceship? (All the spaceships flew in slightly different directions, so that
they couldn’t crash.)
(A) 50 km/h
(B) 66 km/h
(C) 75 km/h
(D) 84 km/h
(E) 100 km/h
28.
The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are written in a circle in arbitrary order. If we add each
of these numbers with its neighbors (the numbers on both sides of it), we get 10 sums. What is the
maximum possible value of the smallest of these sums?
(A) 14
(B) 15
(C) 16
(D) 17
(E) 18
29.
100 trees (oaks and birches) grow along a road, all on the same side. The number of trees between
any two oaks does not equal 5. What is the greatest possible number of oaks among these 100
trees?
(A) 52
(B) 51
(C) 50
(D) 49
(E) 48
30.
Once upon a time there was a town, which was inhabited by only two kinds of people: knights who
always say the truth and liars who always lie. One day an inspector came to the town. He asked
every inhabitant one question about another person in the town – whether or not that person was
a liar. He never asked about the same person twice. Then he arrested all who were told to be liars
and left the town with the arrested. All the remaining knights whose answer caused an arrest got
very upset and left the town as well. The amount of the knights voluntarily leaving the town was
1/3 of the amount of the arrested knights. What part of all the people who disappeared from the
town in one way or another did the knights constitute?
(A) 4/7
(B) 2/3
(C) 3/5
(D) 4/9
(E) 5/11