2021 Malaysia IMONST 1

Part of Malaysia IMONST

Subcontests

(22)

1

last digit of number in cell (2021, 2021) 2021 Malaysia IMONST 1 Senior P20

2021\times 2021

table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate

(1, 1)

, contains the number

0

. For every other cell

C

, we consider a route from

(1, 1)

C

, where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell

C

. For example, the cell with coordinate

(2, 1)

1 = 1 + 0

, the cell with coordinate

(3, 1)

3 = 2 + 0 + 1

, and the cell with coordinate

(3, 2)

7 = 3 + 0 + 1 + 3

. What is the last digit of the number in the cell

(2021, 2021)

1

secret safe box that is locked by 6 lock 2021 Malaysia IMONST 1 Senior P19

A company has a secret safe box that is locked by six locks. Several copies of the keys are distributed among the directors of the company. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can unlock the safe. What is the maximum possible number of directors in the company?

1

|x - 2| - 4 =1/ |x - 3| 2021 Malaysia IMONST 1 Senior P18

How many real numbers

x

are solutions to the equation

|x - 2| - 4 =\frac{1}{|x - 3|}

1

6n+1 written in base 10 has all digits equal 2021 Malaysia IMONST 1 Senior P17

Determine the sum of all positive integers

n

that satisfy the following condition: when

6n + 1

is written in base

10

, all its digits are equal.

1

area of octagon, in a square 2021 Malaysia IMONST 1 Senior P16

ABCD

with side length

6

. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side

AB

C

D

. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?

1

2021 Malaysia IMONST 1 Juniors 20 problems 2.5 hours, integer >=0 answer only

IMONST = International Mathematical Olympiad National Selection Test Malaysia 2021 Round 1 Juniors
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. Adam draws

7

circles on a paper, with radii

1

2

3

4

5

6

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?
p2. 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

2 + 1 = 3

are all prime numbers). Among numbers from

2021

2041

, how many of them have this property?
p3. 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

24

62

legs. How many goldshes are there inside Clarissa's shop?
p4. A positive integer

n

is called special if

n

is divisible by

4

n+1

is divisible by

5

n + 2

is divisible by

6

. How many special integers smaller than

1000

are there?
p5. Suppose that this decade begins on

1

2020

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

1

2030

. How many Wednesdays are there in this decade?
Part B (2 points each)
p6. Given an isosceles triangle

ABC

AB = AC

. Let D be a point on

AB

CD

is the bisector of

\angle ACB

CB = CD

\angle ADC

, in degrees?
p7. 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.
p8. 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

?
p9. What is the smallest positive multiple of

24

that can be written using digits

4

5

only?
p10. 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?
Part C (3 points each)
p11. Dinesh has several squares and regular pentagons, all with side length

1

. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg
p12. If

x +\frac{1}{x} = 5

, what is the value of

x^3 +\frac{1}{x^3}

?
p13. There are

10

girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?
p14. The two diagonals of a rhombus have lengths with ratio

3 : 4

56

. What is the perimeter of the rhombus?
p15. How many integers

n

1 \le n \le 2021

) have the property that

8n + 1

is a perfect square?
Part D (4 points each)
p16. Given a segment of a circle, consisting of a straight edge and an arc. The length of the straight edge is

24

. The length between the midpoint of the straight edge and the midpoint of the arc is

6

. Find the radius of the circle.
p17. Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is

18

. How many passcodes satisfy these conditions?
p18. A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day?
p19. Find the sum of (decimal) digits of the number

(10^{2021} + 2021)^2

?
p20. Determine the number of integer solutions

(x, y, z)

0 \le x, y, z \le 100

, for the equation

(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.

1

2021 Malaysia IMONST 1 Primary 20 problems 2.5 hours, integer >=0 answer only

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

2021

^3

. Five of the cubes have side lengths

5

5

6

6

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

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

909

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

a, b, c

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

2

3

4

5

6

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

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

24

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

n + 2

is divisible by

6

. How many special integers smaller than

1000

are there?
p15. Suppose that this decade begins on

1

2020

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

1

2030

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

ABC

AB = AC

. Let D be a point on

AB

CD

is the bisector of

\angle ACB

CB = CD

\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

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.

1

several squares and regular pentagons 2021 Malaysia IMONST 1 Senior P1

Dinesh has several squares and regular pentagons, all with side length

1

. He wants to arrange the shapes alternately to form a closed loop (see diagram). How many pentagons would Dinesh need to do so? https://cdn.artofproblemsolving.com/attachments/8/9/6345d7150298fe26cfcfba554656804ed25a6d.jpg

1

n and n + 2021 are perfect squares 2021 Malaysia IMONST 1 Senior P15

Find the sum of all integers

n

with this property: both

n

n + 2021

are perfect squares.

1

p(3)=? in 5 degree polynomial with coeff. +-1 2021 Malaysia IMONST 1 Senior P14

Given a function

p(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f

. Each coefficient

a, b, c, d, e

f

is equal to either

1

-1

p(2) = 11

, what is the value of

p(3)

1

Jasmin has a mobile phone 2021 Malaysia IMONST 1 Senior P13

Jasmin has a mobile phone that runs on a battery. When the battery is dead, it takes

2

hours to recharge it fully, if she is not using the phone. If she uses the phone while recharging,

75\%

of the charge obtained is immediately consumed and the remaining is stored in the battery. One day, her battery died. Jasmin took

2

30

minutes to recharge the battery fully. For how many minutes did she use the phone while recharging?

1

diophantine xyz = 2(x + y + z) 2021 Malaysia IMONST 1 Senior P12

Determine the number of positive integer solutions

(x,y, z)

to the equation

xyz = 2(x + y + z)

1

angle wanted, 2 circles 2021 Malaysia IMONST 1 Senior P11

Given two points

A

B

and two circles,

\Gamma_1

A

and passing through

B

\Gamma_2

B

and passing through

A

AB

\Gamma_2

C

D

\Gamma_2

\angle CDB = 57^o

BD

\Gamma_1

E

\angle CAE

1

(x - y)^2 + (y + z)^2 = (x + y)^2 + (y- z)^2 2021 Malaysia IMONST 1 Senior P10

Determine the number of integer solutions

(x, y, z)

0 \le x, y, z \le 100

, for the equation

(x - y)^2 + (y + z)^2 = (x + y)^2 + (y - z)^2.

1

sum of digits (10^{2021} + 2021)^2 2021 Malaysia IMONST 1 Senior P9

Find the sum of (decimal) digits of the number

(10^{2021} + 2021)^2

1

how many leaves does the tree after 10 days 2021 Malaysia IMONST 1 Senior P8

A tree grows in the following manner. On the first day, one branch grows out of the ground. On the second day, a leaf grows on the branch and the branch tip splits up into two new branches. On each subsequent day, a new leaf grows on every existing branch, and each branch tip splits up into two new branches. How many leaves does the tree have at the end of the tenth day?

1

4-digit number with product of digits 18 2021 Malaysia IMONST 1 Senior P7

Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is

18

. How many passcodes satisfy these conditions?

1

computational geometry 2021 Malaysia IMONST 1 Senior P6

Given a segment of a circle, consisting of a straight edge and an arc. The length of the straight edge is

24

. The length between the midpoint of the straight edge and the midpoint of the arc is

6

. Find the radius of the circle.

1

8n + 1 is a perfect square 2021 Malaysia IMONST 1 Senior P5

How many integers

n

1 \le n \le 2021

) have the property that

8n + 1

is a perfect square?

1

perimeter of rhombus 2021 Malaysia IMONST 1 Senior P4

The two diagonals of a rhombus have lengths with ratio

3 : 4

56

. What is the perimeter of the rhombus?

1

10 girls in a class, with different heights 2021 Malaysia IMONST 1 Senior P3

10

girls in a class, all with different heights. They want to form a queue so that no girl stands directly between two girls shorter than her. How many ways are there to form the queue?

1

x^3 +1/x^3 =? if x +1/x=5 2021 Malaysia IMONST 1 Senior P2

x +\frac{1}{x} = 5

, what is the value of

x^3 +\frac{1}{x^3}