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International Mathematical Olympiad National Selection Test Malaysia 2021 Round 1 Primary 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. Faris has six cubes on his table. The cubes have a total volume of

2021

^3

. Five of the cubes have side lengths

5

cm,

5

cm,

6

cm,

6

cm, and

11

cm. What is the side length of the sixth cube (in cm)?
p2. What is the sum of the first

200

even positive integers?
p3. Anushri writes down five positive integers on a paper. The numbers are all different, and are all smaller than

10

. If we add any two of the numbers on the paper, then the result is never

10

. What is the number that Anushri writes down for certain?
p4. If the time now is

10.00

AM, what is the time

1,000

hours from now? Note: Enter the answer in a

12

-hour system, without minutes and AM/PM. For example, if the answer is

9.00

PM, just enter

9

.
p5. Aminah owns a car worth

10,000

RM. She sells it to Neesha at a

10\%

profit. Neesha sells the car back to Aminah at a

10\%

loss. How much money did Aminah make from the two transactions, in RM?
Part B (2 points each)
p6. Alvin takes 250 small cubes of side length

1

cm and glues them together to make a cuboid of size

5

\times 5

\times 10

cm. He paints all the faces of the large cuboid with the color green. How many of the small cubes are painted by Alvin?
p7. Cikgu Emma and Cikgu Tan select one integer each (the integers do not have to be positive). The product of the two integers they selected is

2021

. How many possible integers could have been selected by Cikgu Emma?
p8. A three-digit number is called superb if the first digit is equal to the sum of the other two digits. For example,

431

and

909

are superb numbers. How many superb numbers are there?
p9. Given positive integers

a, b, c

, and

d

that satisfy the equation

4a = 5b =6c = 7d

. What is the smallest possible value of

b

?
p10. Find the smallest positive integer n such that the digit sum of n is divisible by

5

, and the digit sum of

n + 1

is also divisible by

5

.
Note: The digit sum of

1440

1 + 4 + 4 + 0 = 9

.
Part C (3 points each)
p11. Adam draws

7

circles on a paper, with radii

1

cm,

2

cm,

3

cm,

4

cm,

5

cm,

6

cm, and

7

cm. The circles do not intersect each other. He colors some circles completely red, and the rest of the circles completely blue. What is the minimum possible difference (in cm

^2

) between the total area of the red circles and the total area of the blue circles?
p12. The number

2021

has a special property that the sum of any two neighboring digits in the number is a prime number (

2 + 0 = 2

0 + 2 = 2

, and

2 + 1 = 3

are all prime numbers). Among numbers from

2021

2041

, how many of them have this property?
p13. Clarissa opens a pet shop that sells three types of pets: goldshes, hamsters, and parrots. The pets inside the shop together have a total of

14

wings,

24

heads, and

62

legs. How many goldshes are there inside Clarissa's shop?
p14. A positive integer

n

is called special if

n

is divisible by

4

n+1

is divisible by

5

, and

n + 2

is divisible by

6

. How many special integers smaller than

1000

are there?
p15. Suppose that this decade begins on

1

January

2020

(which is a Wednesday) and the next decade begins on

1

January

2030

. How many Wednesdays are there in this decade?
Part D (4 points each)
p16. Given an isosceles triangle

ABC

with

AB = AC

. Let D be a point on

AB

such that

CD

is the bisector of

\angle ACB

. If

CB = CD

, what is

\angle ADC

, in degrees?
p17. Determine the number of isosceles triangles with the following properties: all the sides have integer lengths (in cm), and the longest side has length

21

cm.
p18. Haz marks

k

points on the circumference of a circle. He connects every point to every other point with straight lines. If there are

210

lines formed, what is

k

?
p19. What is the smallest positive multiple of

24

that can be written using digits

4

and

5

only?
p20. In a mathematical competition, there are

2021

participants. Gold, silver, and bronze medals are awarded to the winners as follows: (i) the number of silver medals is at least twice the number of gold medals, (ii) the number of bronze medals is at least twice the number of silver medals, (iii) the number of all medals is not more than

40\%

of the number of participants. The competition director wants to maximize the number of gold medals to be awarded based on the given conditions. In this case, what is the maximum number of bronze medals that can be awarded?
PS. Problems 11-20 were also used in [url=https://artofproblemsolving.com/community/c4h2676837p23203256]Juniors as 1-10.

Problem Statement