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National and Regional Contests
Malaysia Contests
Malaysia IMONST
2020 Malaysia IMONST 1
2020 Malaysia IMONST 1
Part of
Malaysia IMONST
Subcontests
(22)
Primary
1
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2020 Malaysia IMONST 1 Primary 20 problems 2.5 hours, integer >=0 answer only
International Mathematical Olympiad National Selection Test Malaysia 2020 Round 1 Primary Time: 2.5 hours
∙
\bullet
∙
For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
∙
\bullet
∙
No mark is deducted for a wrong answer.
∙
\bullet
∙
The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points. Part A (1 point each)p1. Annie asks his brother four questions, "What is
20
20
20
plus
20
20
20
? What is
20
20
20
minus
20
20
20
? What is
20
20
20
times
20
20
20
? What is
20
20
20
divided by
20
20
20
?". His brother adds the answers to these four questions, and then takes the (positive) square root of the result. What number does he get?p2. A broken watch moves slower than a regular watch. In every
7
7
7
hours, the broken watch lags behind a regular watch by
10
10
10
minutes. In one week, how many hours does the broken watch lags behind a regular watch?p3. Given a square
A
B
C
D
ABCD
A
BC
D
. A point
P
P
P
is chosen outside the square so that triangle
B
C
P
BCP
BCP
is equilateral. Find
∠
A
P
C
\angle APC
∠
A
PC
, in degrees.p4. Hussein throws 4 dice simultaneously, and then adds the number of dots facing up on all
4
4
4
dice. How many possible sums can Hussein get?Note: For example, he can get sum
14
14
14
, by throwing
4
4
4
,
6
6
6
,
3
3
3
, and
1
1
1
. Assume these are regular dice, with
1
1
1
to
6
6
6
dots on the faces.p5. Mrs. Sheila says, "I have
5
5
5
children. They were born one by one every
3
3
3
years. The age of my oldest child is
7
7
7
times the age of my youngest child." What is the age of her third child? Part B (2 points each) p6. The number
N
N
N
is the smallest positive integer with the sum of its digits equal to
2020
2020
2020
. What is the first (leftmost) digit of
N
N
N
? p7. At a food stall, the price of
16
16
16
banana fritters is
k
k
k
RM , and the price of
k
k
k
banana fritters is
1
1
1
RM . What is the price of one banana fritter, in sen?Note:
1
1
1
RM is equal to
100
100
100
sen. p8. Given a trapezium
A
B
C
D
ABCD
A
BC
D
with
A
D
∥
AD \parallel
A
D
∥
to
B
C
BC
BC
, and
∠
A
=
∠
B
=
9
0
o
\angle A = \angle B = 90^o
∠
A
=
∠
B
=
9
0
o
. It is known that the area of the trapezium is 3 times the area of
△
A
B
D
\vartriangle ABD
△
A
B
D
. Find
a
r
e
a
o
f
△
A
B
C
a
r
e
a
o
f
△
A
B
D
.
\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.
a
re
a
o
f
△
A
B
D
a
re
a
o
f
△
A
BC
.
p9. Each
△
\vartriangle
△
symbol in the expression below can be substituted either with
+
+
+
or
−
-
−
:
△
1
△
2
△
3
△
4.
\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.
△
1
△
2
△
3
△
4.
How many possible values are there for the resulting arithmetic expression?Note: One possible value is
−
2
-2
−
2
, which equals
−
1
−
2
−
3
+
4
-1 - 2 - 3 + 4
−
1
−
2
−
3
+
4
. p10. How many
3
3
3
-digit numbers have its sum of digits equal to
4
4
4
? Part C (3 points each) p11. Find the value of
+
1
+
2
+
3
−
4
−
5
−
6
+
7
+
8
+
9
−
10
−
11
−
12
+
.
.
.
−
2020
+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020
+
1
+
2
+
3
−
4
−
5
−
6
+
7
+
8
+
9
−
10
−
11
−
12
+
...
−
2020
where the sign alternates between
+
+
+
and
−
-
−
after every three numbers. p12. If Natalie cuts a round pizza with
4
4
4
straight cuts, what is the maximum number of pieces that she can get?Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting. p13. Given a square with area
A
A
A
. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area
B
B
B
lies inside the circle, such that all its vertices lie on the circle. Find the value of
A
/
B
A/B
A
/
B
. p14. This sequence lists the perfect squares in increasing order:
0
,
1
,
4
,
9
,
16
,
.
.
.
,
a
,
1
0
8
,
b
,
.
.
.
0, 1, 4, 9, 16, ... ,a, 10^8, b, ...
0
,
1
,
4
,
9
,
16
,
...
,
a
,
1
0
8
,
b
,
...
Determine the value of
b
−
a
b - a
b
−
a
. p15. Determine the last digit of
5
5
+
6
6
+
7
7
+
8
8
+
9
9
5^5 + 6^6 + 7^7 + 8^8 + 9^9
5
5
+
6
6
+
7
7
+
8
8
+
9
9
Part D (4 points each) p16. Find the sum of all integers between
−
1442
-\sqrt{1442}
−
1442
and
2020
\sqrt{2020}
2020
. p17. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, "I can paint this building in
3
3
3
months if I work alone". Wahib says, "I can paint this building in
2
2
2
months if I work alone". Wahub says, "I can paint this building in
k
k
k
months if I work alone". If they work together, they can finish painting the building in
1
1
1
month only. What is
k
k
k
? p18. Given a rectangle
A
B
C
D
ABCD
A
BC
D
with a point
P
P
P
inside it. It is known that
P
A
=
17
PA = 17
P
A
=
17
,
P
B
=
15
PB = 15
PB
=
15
, and
P
C
=
6
PC = 6
PC
=
6
. What is the length of
P
D
PD
P
D
? p19. What is the smallest positive multiple of
225
225
225
that can be written using digits
0
0
0
and
1
1
1
only? p20. Given positive integers
a
,
b
a, b
a
,
b
, and
c
c
c
with
a
+
b
+
c
=
20
a + b + c = 20
a
+
b
+
c
=
20
. Determine the number of possible integer values for
a
+
b
c
\frac{a + b}{c}
c
a
+
b
. PS. Problems 6-20 were also used in [url=https://artofproblemsolving.com/community/c4h2675966p23194287]Juniors as 1-15. Problems 11-20 were also used in Seniors 1-10.
Juniors
1
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2020 Malaysia IMONST 1 Juniors 20 problems 2.5 hours, integer >=0 answer only
IMONST = International Mathematical Olympiad National Selection Test Malaysia 2020 Round 1 JuniorsTime: 2.5 hours
∙
\bullet
∙
For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
∙
\bullet
∙
No mark is deducted for a wrong answer.
∙
\bullet
∙
The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points. Part A (1 point each) p1. The number
N
N
N
is the smallest positive integer with the sum of its digits equal to
2020
2020
2020
. What is the first (leftmost) digit of
N
N
N
? p2. At a food stall, the price of
16
16
16
banana fritters is
k
k
k
RM , and the price of
k
k
k
banana fritters is
1
1
1
RM . What is the price of one banana fritter, in sen?Note:
1
1
1
RM is equal to
100
100
100
sen. p3. Given a trapezium
A
B
C
D
ABCD
A
BC
D
with
A
D
∥
AD \parallel
A
D
∥
to
B
C
BC
BC
, and
∠
A
=
∠
B
=
9
0
o
\angle A = \angle B = 90^o
∠
A
=
∠
B
=
9
0
o
. It is known that the area of the trapezium is 3 times the area of
△
A
B
D
\vartriangle ABD
△
A
B
D
. Find
a
r
e
a
o
f
△
A
B
C
a
r
e
a
o
f
△
A
B
D
.
\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.
a
re
a
o
f
△
A
B
D
a
re
a
o
f
△
A
BC
.
p4. Each
△
\vartriangle
△
symbol in the expression below can be substituted either with
+
+
+
or
−
-
−
:
△
1
△
2
△
3
△
4.
\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.
△
1
△
2
△
3
△
4.
How many possible values are there for the resulting arithmetic expression?Note: One possible value is
−
2
-2
−
2
, which equals
−
1
−
2
−
3
+
4
-1 - 2 - 3 + 4
−
1
−
2
−
3
+
4
. p5. How many
3
3
3
-digit numbers have its sum of digits equal to
4
4
4
? Part B (2 points each) p6. Find the value of
+
1
+
2
+
3
−
4
−
5
−
6
+
7
+
8
+
9
−
10
−
11
−
12
+
.
.
.
−
2020
+1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 +... - 2020
+
1
+
2
+
3
−
4
−
5
−
6
+
7
+
8
+
9
−
10
−
11
−
12
+
...
−
2020
where the sign alternates between
+
+
+
and
−
-
−
after every three numbers. p7. If Natalie cuts a round pizza with
4
4
4
straight cuts, what is the maximum number of pieces that she can get?Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting. p8. Given a square with area
A
A
A
. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area
B
B
B
lies inside the circle, such that all its vertices lie on the circle. Find the value of
A
/
B
A/B
A
/
B
. p9. This sequence lists the perfect squares in increasing order:
0
,
1
,
4
,
9
,
16
,
.
.
.
,
a
,
1
0
8
,
b
,
.
.
.
0, 1, 4, 9, 16, ... ,a, 10^8, b, ...
0
,
1
,
4
,
9
,
16
,
...
,
a
,
1
0
8
,
b
,
...
Determine the value of
b
−
a
b - a
b
−
a
. p10. Determine the last digit of
5
5
+
6
6
+
7
7
+
8
8
+
9
9
5^5 + 6^6 + 7^7 + 8^8 + 9^9
5
5
+
6
6
+
7
7
+
8
8
+
9
9
. Part C (3 points each) p11. Find the sum of all integers between
−
1442
-\sqrt{1442}
−
1442
and
2020
\sqrt{2020}
2020
. p12. Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, "I can paint this building in
3
3
3
months if I work alone". Wahib says, "I can paint this building in
2
2
2
months if I work alone". Wahub says, "I can paint this building in
k
k
k
months if I work alone". If they work together, they can finish painting the building in
1
1
1
month only. What is
k
k
k
? p13. Given a rectangle
A
B
C
D
ABCD
A
BC
D
with a point
P
P
P
inside it. It is known that
P
A
=
17
PA = 17
P
A
=
17
,
P
B
=
15
PB = 15
PB
=
15
, and
P
C
=
6
PC = 6
PC
=
6
. What is the length of
P
D
PD
P
D
? p14. What is the smallest positive multiple of
225
225
225
that can be written using digits
0
0
0
and
1
1
1
only? p15. Given positive integers
a
,
b
a, b
a
,
b
, and
c
c
c
with
a
+
b
+
c
=
20
a + b + c = 20
a
+
b
+
c
=
20
. Determine the number of possible integer values for
a
+
b
c
\frac{a + b}{c}
c
a
+
b
. Part D (4 points each) p16. If we divide
2020
2020
2020
by a prime
p
p
p
, the remainder is
6
6
6
. Determine the largest possible value of
p
p
p
. p17. A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by
5
5
5
hexagons. Each hexagons is bordered by
3
3
3
pentagons and
3
3
3
hexagons. We need
12
12
12
pentagons to make one football. How many hexagons are needed to make one football? p18. Given a right-angled triangle with perimeter
18
18
18
. The sum of the squares of the three side lengths is
128
128
128
. What is the area of the triangle? p19. A perfect square ends with the same two digits. How many possible values of this digit are there? p20. Find the sum of all integers
n
n
n
that fulfill the equation
2
n
(
6
−
n
)
=
8
n
2^n(6 - n) = 8n
2
n
(
6
−
n
)
=
8
n
.
20
1
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Malaysia IMONST 1 Senior P20
Geetha wants to cut a cube of size
4
×
4
×
4
4 \times 4\times 4
4
×
4
×
4
into
64
64
64
unit cubes (of size
1
×
1
×
1
1\times 1\times 1
1
×
1
×
1
). Every cut must be straight, and parallel to a face of the big cube. What is the minimum number of cuts that Geetha needs? Note: After every cut, she can rearrange the pieces before cutting again. At every cut, she can cut more than one pieces as long as the pieces are on a straight line.
18
1
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Malaysia IMONST 1 Senior P18
In a triangle, the ratio of the interior angles is
1
:
5
:
6
1 : 5 : 6
1
:
5
:
6
, and the longest side has length
12
12
12
. What is the length of the altitude (height) of the triangle that is perpendicular to the longest side?
19
1
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3-element subsets
A set
S
S
S
has
7
7
7
elements. Several
3
3
3
-elements subsets of
S
S
S
are listed, such that any
2
2
2
listed subsets have exactly
1
1
1
common element. What is the maximum number of subsets that can be listed?
17
1
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Number of positive divisors
Given a positive integer
n
n
n
. The number
2
n
2n
2
n
has
28
28
28
positive factors, while the number
3
n
3n
3
n
has
30
30
30
positive factors. Find the number of positive divisors of
6
n
6n
6
n
.
16
1
Hide problems
Brute Force Number Theory
Find the number of positive integer solutions
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
to the equation
(
a
2
+
b
2
)
(
c
2
−
d
2
)
=
2020.
(a^2+b^2)(c^2-d^2)=2020.
(
a
2
+
b
2
)
(
c
2
−
d
2
)
=
2020.
Note: The solutions
(
10
,
1
,
6
,
4
)
(10,1,6,4)
(
10
,
1
,
6
,
4
)
and
(
1
,
10
,
6
,
4
)
(1,10,6,4)
(
1
,
10
,
6
,
4
)
are considered different.
15
1
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Malaysia IMONST 1 Senior P15
Find the sum of all integers
n
n
n
that fulfill the equation
2
n
(
6
−
n
)
=
8
n
.
2^n(6-n)=8n.
2
n
(
6
−
n
)
=
8
n
.
14
1
Hide problems
Malaysia IMONST 1 Senior P14
A perfect square ends with the same two digits. How many possible values of this digit are there?
13
1
Hide problems
Malaysia IMONST 1 Senior P13
Given a right-angled triangle with perimeter
18
18
18
. The sum of the squares of the three side lengths is
128
128
128
. What is the area of the triangle?
12
1
Hide problems
Malaysia IMONST 1 Senior P12
A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by 5 hexagons. Each hexagons is bordered by
3
3
3
pentagons and
3
3
3
hexagons. We need
12
12
12
pentagons to make one football. How many hexagons are needed to make one football?
11
1
Hide problems
Malaysia IMONST 1 Senior P11
If we divide
2020
2020
2020
by a prime
p
p
p
, the remainder is
6
6
6
. Determine the largest possible value of
p
p
p
.
10
1
Hide problems
Malaysia IMONST 1 Senior P10
Given positive integers
a
,
b
,
a, b,
a
,
b
,
and
c
c
c
with
a
+
b
+
c
=
20
a + b + c = 20
a
+
b
+
c
=
20
. Determine the number of possible integer values for
a
+
b
c
.
\frac{a + b}{c}.
c
a
+
b
.
9
1
Hide problems
Malaysia IMONST 1 Senior P9
What is the smallest positive multiple of
225
225
225
that can be written using digits
0
0
0
and
1
1
1
only?
8
1
Hide problems
Malaysia IMONST 1 Senior P8
Given a rectangle
A
B
C
D
ABCD
A
BC
D
with a point
P
P
P
inside it. It is known that
P
A
=
17
,
P
B
=
15
,
PA = 17, PB = 15,
P
A
=
17
,
PB
=
15
,
and
P
C
=
6.
PC = 6.
PC
=
6.
What is the length of
P
D
PD
P
D
?
7
1
Hide problems
Malaysia IMONST 1 Senior P7
Three brothers own a painting company called Tiga Abdul Enterprise. They are hired to paint a building. Wahab says, “I can paint this building in
3
3
3
months if I work alone”. Wahib says, “I can paint this building in
2
2
2
months if I work alone”. Wahub says, “I can paint this building in k months if I work alone”. If they work together, they can finish painting the building in
1
1
1
month only. What is
k
k
k
?
6
1
Hide problems
Malaysia IMONST 1 Senior P6
Find the sum of all integers between
−
1442
-\sqrt {1442}
−
1442
and
2020
\sqrt{2020}
2020
.
5
1
Hide problems
Malaysia IMONST 1 Senior P5
Determine the last digit of
5
5
+
6
6
+
7
7
+
8
8
+
9
9
5^5+6^6+7^7+8^8+9^9
5
5
+
6
6
+
7
7
+
8
8
+
9
9
.
4
1
Hide problems
Malaysia IMONST 1 Senior P4
This sequence lists the perfect squares in increasing order:
0
,
1
,
4
,
9
,
16
,
⋯
,
a
,
1
0
8
,
b
,
⋯
0,1,4,9,16,\cdots ,a,10^8,b,\cdots
0
,
1
,
4
,
9
,
16
,
⋯
,
a
,
1
0
8
,
b
,
⋯
Determine the value of
b
−
a
b-a
b
−
a
.
3
1
Hide problems
Malaysia IMONST 1 Senior P3
Given a square with area
A
A
A
. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area
B
B
B
lies inside the circle, such that all its vertices lie on the circle. Find the value of
A
B
.
\frac{A}{B}.
B
A
.
2
1
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Malaysia IMONST 1 Senior P2
If Natalie cuts a round pizza with
4
4
4
straight cuts, what is the maximum number of pieces that she can get? Note: Assume that all the cuts are vertical (perpendicular to the surface of the pizza). She cannot move the pizza pieces until she finishes cutting.
1
1
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Malaysia IMONST 1 Senior P1
Find the value of
+
1
+
2
+
3
−
4
−
5
−
6
+
7
+
8
+
9
−
10
−
11
−
12
+
⋯
−
2020
,
+1+2+3-4-5-6+7+8+9-10-11-12+\cdots -2020,
+
1
+
2
+
3
−
4
−
5
−
6
+
7
+
8
+
9
−
10
−
11
−
12
+
⋯
−
2020
,
where the sign alternates between
+
+
+
and
−
-
−
after every three numbers.