y
23.
x
How many quadratic functions in
Lietuvos Respublikos svietimo ir mokslo ministerija
Keng
uros organizavimo komitetas
have a graph passing through at
least 3 of the marked points?
A)
B)
6
C)
15
D)
18
E)
22
27
O
24.
P . If
distance from P to A?
√
A) 8 B) 3 C) 10
intersect at point
25.
ABC
In a right-angled triangle
(right angle at
the distance from
D)
√
P
A)
the bisectors of the acute angles
√
to the hypotenuse is
8,
what is the
Three three-digits numbers are formed from digits from 1 to 9 (each digit is used
exactly once). Which of the following numbers couldn't be equal to the sum of the
three numbers?
A)
26.
B)
1500
C)
1503
D)
1512
1521
E)
B)
4
C)
D)
6
E)
9
A)
2.
3.
28.
Ann chose a positive integer
to
n.
29.
217
B)
n
and wrote down the sum of all positive integers from 1
p divides
n + p?
C)
221
4.
5.
1
100
E)
13
1
1000 is
111
C) 1000
14
+
111
1110
3
1000
D)
E)
3
1110
B)
C)
12
D)
15
E)
18
How many integers are greater than
B) 1
C) 2015
D) 2016
20
2015 · 2017
E) 2017
but less than
2016 · 2016?
y
xy -plane as shown.
swapped: (x, y) 7→ (y, x) .
A set of points forms a picture of a kangaroo in the
x
and
y
coordinates are
What is the result?
245
E)
y
269
y
y
y
x
y
Consider a 5×5 square divided into 25 cells. Initially all its cells
are white. In each move it is allowed to change the color of any
color (i.e. white cells become black and black ones become white).
What is the smallest possible number of moves needed to obtain the
6.
Less than 10
B)
The positive integer
N
10
C)
12
D)
E)
A)
It is impossible to do
1 and N .
648. Which one of the following is the sixth divisor of N ?
D) 12 E) 24
has exactly six distinct (positive) divisors including
x
C)
x
D)
7.
1
B)
00000
...
. What is the result of
C)
000000
D)
0000000
000 + 0000
E)
in his notation?
00000000
Diana wants to write nine numbers into the circles on the diagram so
The product of ve of these is
that, for the eight small triangles whose vertices are joined by segments
A) 4
the sums of the numbers in their vertices are identical. What is the
B) 8
C) 9
c 2016 Keng
uros konkurso organizavimo komitetas
x
E)
Little Lucas invented his own way to write down negative numbers before he learned
1, 0, 00, 000, 0000,
More than 12
x
B)
the usual way with the negative sign in front. Counting backwards, he'd write:
chessboard coloring shown in the gure?
A)
x
A)
three consecutive cells in a row or in a column to the opposite
30.
B)
+
For each point the
the sum, but not any of the summands. Which of the
D)
229
Snowboarding
10
A) 0
A prime number
following could be
A)
D)
1
10
D)
12
On a test consisting of 30 questions, Ruth had 50% more right answers than she had
A)
the hockey player`s left hand. Which sport did Eva do?
Ice hockey
3
111
C)
11
questions?
speed skater sat opposite Ben. Eva and Filip sat next to each other. A woman sat at
C)
B)
wrong answers. How many correct answers did Ruth have, assuming she answered all
snowboarder - had dinner at a round table. The skier sat at Andrea's left hand. The
Skiing
10
The sum of
A)
12
Four sportsmen and sportswomen - a speed skater, a skier, a hockey player and a
A) Speed skating
B)
E) Impossible to determine
The sum of the ages of Tom and John is 23, the sum of the ages of John and Alex is 24
and the sum of the ages of Tom and Alex is 25. What is the age of the oldest one?
11 and 14. What is the volume of the sixth pyramid?
1
11--12 grades
Questions for 3 points
A cube is dissected into 6 pyramids by connecting a given point in the interior of the
A)
Student
Time allowed: 75 minutes
Calculators are not permitted
1.
1575
cube with each vertex of the cube. The volumes of ve of these pyramids are 2, 5, 10,
27.
KANGAROO 2016
x
E) 4
12
VU Matematikos ir informatikos fakultetas
VU Matematikos ir informatikos institutas
largest number of dierent numbers she can use?
A)
1
B)
2
C)
3
D)
5
E)
8
. . .,
3, 2,
8.
The rectangles
S1
S2
and
x
Determine the ratio
y.
A) 1 B) 32
C) 49
9.
If
x2 − 4x + 2 = 0,
A) −4
10. a, b, c, d
C)
x+
13
5
B) b
a, b, c
C) c
4
17.
S1
y
9
4
2
E)
S2
x
9
18.
4
The lengths of arc
a + 2 = b − 2 = c · 2 = d : 2.
A) 30◦
B) 24◦
E)
Impossible to determine
19.
AP
and arc
BP
E)
0
B) −2
C)
D) −4
4
a+b
equals
If the perimeter of the square in the gure equals 4 then the perimeter
B) 3 +
√
C) 3
3
D) 3 +
√
E) 4 +
2
√
3
Each of ten points in the gure is marked with either 0
O
∠AXP equals
C) 18◦ D) 15◦
20
E) 10◦
divisible by 3. Three of the points are marked as shown in
B
B)
56
What is
A) 22
x4 ,
C)
84
if
x1 = 2
4
3
B) 22
D)
90
and
C) 22
E)
105
xn+1 = xxnn
11
16
D) 22
2
point?
16
A) Only 0 B)
E) Either 0 or 1
X
20.
C)
Only 1
D)
Only 2
2
Only 0 and 1
or 2
Betina draws ve points
A, B, C, D
α
ve angles marked with
∠ABD ?
A) 66◦
B) 70.5◦
E
and
as well as the tangent to the circle at
220
for
x
0
the gure. What numbers can be used to mark the central
In this pyramid of numbers each upper eld is the product of the
A)
A,
B
on a circle
α
are equal. How large is the
angle
◦
C) 72
D) 75
◦
E) 77.5
A
◦
n > 1?
C
?
such that all
α
α
α
α
D
768
E) 22
E
In rectangle
Let
M
ABCD
be a point on
A) 12.5◦
B) 15◦
BC is half
CD such that AM = M C . What
C) 27.5◦
D) 42.5◦
E) Some
the length of the side
the length of the diagonal
is the size of angle
AC .
∠CAM ?
other angle
Questions for 5 points
21.
How many dierent solutions are there to the equation
Diana cut up a rectangle of area 2016 into 56 equal squares. The lengths of the sides
x2 − 4x + 5
of the rectangle are integers. For how many dierent rectangles is it possible for her to
do this?
A)
16.
Then
Impossible to determine
sum of numbers in the vertices of any black triangle is not
A
in the
contain natural numbers bigger than 1?
15.
a 6= b.
A)
vertices of any white triangle is divisible by 3, while the
cannot appear in the top eld, if the three bottom elds only
14.
both have real roots. It is known
or 1 or 2. It is known that the sum of numbers in the
two elds directly underneath. Which of the following numbers
13.
x2 + bx + a = 0
of the roots of the second one, and
A) 4
Which is the largest
P
12.
and
d?
and
D) d
gure are 20 and 16, respectively. Then the
value of the angle
x2 + ax + b = 0
of the equilateral triangle equals
Questions for 4 points
11.
The equations
that the sum of squares of the roots of the rst equation is equal to the sum of squares
2
x equals
D)
0
E)
are positive integers satisfying
of the four numbers
A) a
D)
then
B) −2
in the picture have the same area.
2
B)
4
C)
6
D)
8
E)
A)
0
On the Island of Knights and Knaves every citizen is either a Knight (who always speaks
22.
A
1
B)
2
C)
quadrilateral
3
D)
contains
4
E)
an
inscribed
the truth) or a Knave (who always lies). During your travels on the island you meet
quadrilateral to that of the circle is
7 people sitting around a bonre. They all tell you I'm sitting between two Knaves!
to that of the circle is
How many Knaves are there?
A)
3
B)
4
C)
5
D)
6
E)
Impossible to determine
A) 4 : π
√
B) 3 2 : π
C) 16 : 9
x2 +x−30
= 1?
Innitely many
4 : 3.
circle.
The
ratio
of
the
perimeter
of
the
Then the ratio of the area of the quadrilateral
D) π : 3
E) 4 : 3