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Contests
National and Regional Contests
Malaysia Contests
Malaysia National Olympiad
2018 Malaysia National Olympiad
2018 Malaysia National Olympiad
Part of
Malaysia National Olympiad
Subcontests
(9)
A1
3
Hide problems
OMK 2018 Sulong, Section A Problem 1
A cuboid has an integer volume. Three of the faces have different areas, namely
7
,
27
7, 27
7
,
27
, and
L
L
L
. What is the smallest possible integer value for
L
L
L
?
min side of equilateral, creating hexagon 2018 Malaysia OMK Intermediate A1
Hassan has a piece of paper in the shape of a hexagon. The interior angles are all
12
0
o
120^o
12
0
o
, and the side lengths are
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
5
5
5
,
6
6
6
, although not in that order. Initially, the paper is in the shape of an equilateral triangle, then Hassan has cut off three smaller equilateral triangle shapes, one at each corner of the paper. What is the minimum possible side length of the original triangle?
area of ABCD with integer sidelengths wanted 2018 Malaysia OMK Juniors A1
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is neither a kite nor a rectangle. It is known that its sidelengths are integers,
A
B
=
6
AB = 6
A
B
=
6
,
B
C
=
7
BC = 7
BC
=
7
, and
∠
B
=
∠
D
=
9
0
o
\angle B = \angle D = 90^o
∠
B
=
∠
D
=
9
0
o
. Find the area of
A
B
C
D
ABCD
A
BC
D
.
A2
3
Hide problems
OMK 2018 Sulong, Section A Problem 2
Let
a
a
a
and
b
b
b
be prime numbers such that
a
+
b
=
10000
a+b = 10000
a
+
b
=
10000
. Find the sum of the smallest possible value of
a
a
a
and the largest possible value of
a
a
a
.
integer has 2018 digits, divisible by 7 2018 Malaysia OMK Intermediate A2
An integer has
2018
2018
2018
digits and is divisible by
7
7
7
. The first digit is
d
d
d
, while all the other digits are
2
2
2
. What is the value of
d
d
d
?
max sum, of 10 integers with product 1024 , 2018 Malaysia OMK Juniors A2
The product of
10
10
10
integers is
1024
1024
1024
. What is the greatest possible sum of these
10
10
10
integers?
A3
3
Hide problems
OMK 2018 Sulong, Section A Problem 3
Given a regular polygon with
n
n
n
sides. It is known that there are
1200
1200
1200
ways to choose three of the vertices of the polygon such that they form the vertices of a right triangle. What is the value of
n
n
n
?
4 out of 15 points, vertices of equilateral 2018 Malaysia OMK Intermediate A3
On each side of a triangle,
5
5
5
points are chosen (other than the vertices of the triangle) and these
15
15
15
points are colored red. How many ways are there to choose four red points such that they form the vertices of a quadrilateral?
buying apples, mangos and papayas 2018 Malaysia OMK Juniors A3
Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs
3
3
3
RM ,each mango costs
4
4
4
RM , and each papaya costs
5
5
5
RM . He bought at least one of each fruit, and paid exactly
50
50
50
RM. What is the maximum number of fruits that he could have bought?
A4
3
Hide problems
OMK 2018 Sulong, Section A Problem 4
Given a circle with diameter
A
B
AB
A
B
. Points
C
C
C
and
D
D
D
are selected on the circumference of the circle such that the chord
C
D
CD
C
D
intersects
A
B
AB
A
B
inside the circle, at point
P
P
P
. The ratio of the arc length \overarc {AC} to the arc length \overarc {BD} is
4
:
1
4 : 1
4
:
1
, while the ratio of the arc length \overarc{AD} to the arc length \overarc {BC} is
3
:
2
3 : 2
3
:
2
. Find
∠
A
P
C
\angle{APC}
∠
A
PC
, in degrees.
sum of a areas, parts of regular octagon 2018 Malaysia OMK Intermediate A4
Given a regular octagon
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
with side length
3
3
3
. By drawing the four diagonals
A
F
AF
A
F
,
B
E
BE
BE
,
C
H
CH
C
H
, and
D
G
DG
D
G
, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.
max distance among collinear points 2018 Malaysia OMK Juniors A4
Given points
A
,
B
,
C
,
D
,
E
A, B, C, D, E
A
,
B
,
C
,
D
,
E
, and
F
F
F
on a line (not necessarily in that order) with
A
B
=
2
AB = 2
A
B
=
2
,
B
C
=
6
BC = 6
BC
=
6
,
C
D
=
8
CD = 8
C
D
=
8
,
D
E
=
10
DE = 10
D
E
=
10
,
E
F
=
20
EF = 20
EF
=
20
, and
F
A
=
22
FA = 22
F
A
=
22
. Find the distance between the two furthest points on the line.
A5
3
Hide problems
OMK 2018 Sulong, Section A Problem 5
Determine the value of
(
101
×
99
)
(101 \times 99)
(
101
×
99
)
-
(
102
×
98
)
(102 \times 98)
(
102
×
98
)
+
(
103
×
97
)
(103 \times 97)
(
103
×
97
)
−
(
104
×
96
)
(104 \times 96)
(
104
×
96
)
+ ... ... +
(
149
×
51
)
(149 \times 51)
(
149
×
51
)
−
(
150
×
50
)
(150 \times 50)
(
150
×
50
)
.
no of coloring some faces of a cube 2018 Malaysia OMK Intermediate A5
Daud want to paint some faces of a cube with green paint. At least one face must be painted. How many ways are there for him to paint the cube?Note: Two colorings are considered the same if one can be obtained from the other by rotation.
n^2 - [\sqrt{n} ]= 2018 2018 Malaysia OMK Juniors A5
Find the positive integer
n
n
n
that satisfi es the equation
n
2
−
⌊
n
⌋
=
2018
n^2 - \lfloor \sqrt{n} \rfloor = 2018
n
2
−
⌊
n
⌋
=
2018
A6
3
Hide problems
n^4 + 2n^3 + 2n^2 + 2n + 1 is prime 2018 Malaysia OMK Intermediate A6
How many integers
n
n
n
are there such that
n
4
+
2
n
3
+
2
n
2
+
2
n
+
1
n^4 + 2n^3 + 2n^2 + 2n + 1
n
4
+
2
n
3
+
2
n
2
+
2
n
+
1
is a prime number?
OMK 2018 Sulong, Section A Problem 6
Determine the smallest prime
p
p
p
such that
2018
!
2018!
2018
!
is divisible by
p
3
p^{3}
p
3
, but not divisible by
p
4
p^{4}
p
4
.
no of products of primes less than 100 2018 Malaysia OMK Juniors A6
A semiprime is a positive integer that is a product of two prime numbers. For example,
9
9
9
and
10
10
10
are semiprimes. How many semiprimes less than
100
100
100
are there?
B3
3
Hide problems
Removing a factorial to form a square
There are
200
200
200
numbers on a blackboard:
1
!
,
2
!
,
3
!
,
4
!
,
.
.
.
.
.
.
,
199
!
,
200
!
1! , 2! , 3! , 4! , ... ... , 199! , 200!
1
!
,
2
!
,
3
!
,
4
!
,
......
,
199
!
,
200
!
. Julia erases one of the numbers. When Julia multiplies the remaining
199
199
199
numbers, the product is a perfect square. Which number was erased?
OMK 2018 Muda, Section B Problem 3
Let
n
n
n
be an integer greater than
1
1
1
, such that
3
n
+
1
3n + 1
3
n
+
1
is a perfect square. Prove that
n
+
1
n + 1
n
+
1
can be expressed as a sum of three perfect squares.
2018 ones in a row 2018 Malaysia OMK Juniors B3
Given
2018
2018
2018
ones in a row:
1
1
1
1
.
.
.
1
1
1
1
⏟
2018
o
n
e
s
\underbrace{1\,\,\,1\,\,\,1\,\,\,1 \,\,\, ... \,\,\,1 \,\,\,1 \,\,\,1 \,\,\,1}_{2018 \,\,\, ones}
2018
o
n
es
1
1
1
1
...
1
1
1
1
in which plus symbols
(
+
)
(+)
(
+
)
are allowed to be inserted in between the ones. What is the maximum number of plus symbols
(
+
)
(+)
(
+
)
that need to be inserted so that the resulting sum is 8102?
B2
3
Hide problems
OMK 2015 Sulong, Section B Problem 2
A subset of
{
1
,
2
,
3
,
.
.
.
.
.
.
,
2015
}
\{1, 2, 3, ... ... , 2015\}
{
1
,
2
,
3
,
......
,
2015
}
is called good if the following condition is fulfilled: for any element
x
x
x
of the subset, the sum of all the other elements in the subset has the same last digit as
x
x
x
. For example,
{
10
,
20
,
30
}
\{10, 20, 30\}
{
10
,
20
,
30
}
is a good subset since
10
10
10
has the same last digit as
20
+
30
=
50
20 + 30 = 50
20
+
30
=
50
,
20
20
20
has the same last digit as
10
+
30
=
40
10 + 30 = 40
10
+
30
=
40
, and
30
30
30
has the same last digit as
10
+
20
=
30
10 + 20 = 30
10
+
20
=
30
. (a) Find an example of a good subset with 400 elements. (b) Prove that there is no good subset with 405 elements.
OMK 2018 Muda, Section B Problem 2
Let
a
a
a
and
b
b
b
be positive integers such that (i) both
a
a
a
and
b
b
b
have at least two digits; (ii)
a
+
b
a + b
a
+
b
is divisible by
10
10
10
; (iii)
a
a
a
can be changed into
b
b
b
by changing its last digit. Prove that the hundreds digit of the product
a
b
ab
ab
is even.
OMK 2018 Bongsu, Section B Problem 2
Prove that the number
9
(
a
1
+
a
2
)
(
a
2
+
a
3
)
(
a
3
+
a
4
)
.
.
.
(
a
98
+
a
99
)
(
a
99
+
a
1
)
9^{(a_1 + a_2)(a_2 + a_3)(a_3 + a_4)...(a_{98} + a_{99})(a_{99} + a_1)}
9
(
a
1
+
a
2
)
(
a
2
+
a
3
)
(
a
3
+
a
4
)
...
(
a
98
+
a
99
)
(
a
99
+
a
1
)
−
1
1
1
is divisible by
10
10
10
, for any choice of positive integers
a
1
,
a
2
,
a
3
,
.
.
.
,
a
99
a_1, a_2, a_3, . . . , a_{99}
a
1
,
a
2
,
a
3
,
...
,
a
99
.
B1
3
Hide problems
OMK 2018 Sulong, Section B Problem 1
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
D
D
D
be the reflection of point
B
B
B
with respect to the line
A
C
AC
A
C
. Let
E
E
E
be the reflection of point
C
C
C
with respect to the line
A
B
AB
A
B
. Let
Γ
1
\Gamma_1
Γ
1
be the circle that passes through
A
,
B
A, B
A
,
B
, and
D
D
D
. Let
Γ
2
\Gamma_2
Γ
2
be the circle that passes through
A
,
C
A, C
A
,
C
, and
E
E
E
. Let
P
P
P
be the intersection of
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
, other than
A
A
A
. Let
Γ
\Gamma
Γ
be the circle that passes through
A
,
B
A, B
A
,
B
, and
C
C
C
. Show that the center of
Γ
\Gamma
Γ
lies on line
A
P
AP
A
P
.
square by n sticks of lengths 1,2, 3,..., n 2018 Malaysia OMK Intermediate B1
Let
n
n
n
be an integer. Dayang are given
n
n
n
sticks of lengths
1
,
2
,
3
,
.
.
.
,
n
1,2, 3,..., n
1
,
2
,
3
,
...
,
n
. She may connect the sticks at their ends to form longer sticks, but cannot cut them. She wants to use all these sticks to form a square. For example, for
n
=
8
n = 8
n
=
8
, she can make a square of side length
9
9
9
using these connected sticks:
1
+
8
1 + 8
1
+
8
,
2
+
7
2 + 7
2
+
7
,
3
+
6
3 + 6
3
+
6
, and
4
+
5
4 + 5
4
+
5
. How many values of
n
n
n
, with
1
≤
n
≤
2018
1 \le n \le 2018
1
≤
n
≤
2018
, that allow her to do this?
OMK 2018 Bongsu, Section B Problem 1
Given two triangles with the same perimeter. Both triangles have integer side lengths. The first triangle is an equilateral triangle. The second triangle has a side with length 1 and a side with length
d
d
d
. Prove that when
d
d
d
is divided by 3, the remainder is 1.