MathDB

IMONST = International Mathematical Olympiad National Selection Test Malaysia 2020 Round 1 Juniors
Time: 2.5 hours

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For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.

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No mark is deducted for a wrong answer.

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The maximum number of points is (1 + 2 + 3 + 4) x 5 = 50 points.
Part A (1 point each)
p1. The number

N

is the smallest positive integer with the sum of its digits equal to

2020

. What is the first (leftmost) digit of

N

?
p2. At a food stall, the price of

16

banana fritters is

k

RM , and the price of

k

banana fritters is

1

RM . What is the price of one banana fritter, in sen?
Note:

1

RM is equal to

100

sen.
p3. Given a trapezium

ABCD

with

AD \parallel

BC

, and

\angle A = \angle B = 90^o

. It is known that the area of the trapezium is 3 times the area of

\vartriangle ABD

. Find

\frac{area \,\, of \,\, \vartriangle ABC}{area \,\, of \,\, \vartriangle ABD}.

p4. Each

\vartriangle

symbol in the expression below can be substituted either with

+

-

\vartriangle 1 \vartriangle 2 \vartriangle 3 \vartriangle 4.

How many possible values are there for the resulting arithmetic expression?
Note: One possible value is

-2

, which equals

-1 - 2 - 3 + 4

.
p5. How many

3

-digit numbers have its sum of digits equal to

4

?
Part B (2 points each)
p6. Find the value of

+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

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 circle lies inside the square, such that the circle touches all sides of the square. Another square with area

B

lies inside the circle, such that all its vertices lie on the circle. Find the value of

A/B

.
p9. This sequence lists the perfect squares in increasing order:

0, 1, 4, 9, 16, ... ,a, 10^8, b, ...

Determine the value of

b - a

.
p10. Determine the last digit of

5^5 + 6^6 + 7^7 + 8^8 + 9^9

.
Part C (3 points each)
p11. Find the sum of all integers between

-\sqrt{1442}

and

\sqrt{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

months if I work alone". Wahib says, "I can paint this building in

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

month only. What is

k

?
p13. Given a rectangle

ABCD

with a point

P

inside it. It is known that

PA = 17

PB = 15

, and

PC = 6

. What is the length of

PD

?
p14. What is the smallest positive multiple of

225

that can be written using digits

0

and

1

only?
p15. Given positive integers

a, b

, and

c

with

a + b + c = 20

. Determine the number of possible integer values for

\frac{a + b}{c}

.
Part D (4 points each)
p16. If we divide

2020

by a prime

p

, the remainder is

6

. Determine the largest possible value of

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

hexagons. Each hexagons is bordered by

3

pentagons and

3

hexagons. We need

12

pentagons to make one football. How many hexagons are needed to make one football?
p18. Given a right-angled triangle with perimeter

18

. The sum of the squares of the three side lengths is

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

that fulfill the equation

2^n(6 - n) = 8n

Problem Statement