Division

Mathematical Olympiad
P4 2015

Full Name:
Index Number:
Class:
School:

SASMO 2015 Primary 4 Contest

INSTRUCTIONS
1. Please DO NOT OPEN the contest booklet until the Proctor has given permission to
start.

2. TIME : 1 hour 30 minutes.

3. Attempt all 20 questions.

Questions 1 to 10 score 2 points each, no points are deducted for unanswered question
and 1 point is deducted for wrong answer.
Questions 11 to 20 score 4 points each. No points are deducted for unanswered or
wrong answers.

4. Shade your answers neatly using a pencil in the answer sheet.

5. PROCTORING : No one may help any student in any way during the contest.

6. No electronic devices capable of storing and displaying visual information is

allowed during the course of the exam.

7. Strictly No Calculators are allowed into the exam.

8. All students must fill and shade in their Name, Index number, Class and School in
the answer sheet and contest booklet.

9. MINIMUM TIME: Students must stay in the exam hall at least 1h 15 min.

10. Students must show detailed working and transfer answers to the answer sheet.

11. No exam papers and written notes can be taken out by any contestant.
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SASMO 2015, Primary 4 Contest
SASMO 2015 Primary 4 [10 MCQ + 10 non-MCQ = 20 Q]

Starting Score = 10 marks (to avoid negative marks); Max Possible Score = 70 marks

Section A (Correct answer = 2 marks; no answer = 0; incorrect answer = minus 1 mark)

1. The diagram shows a figure that contains 13 identical squares. The area of the
figure is 117 cm2. Find its perimeter.

(a) 54 cm
(b) 57 cm
(c) 60 cm
(d) 63 cm
(e) 66 cm

________________________________________________________________

2. 25 means 2 multiplied by itself 5 times, i.e. 25 = 2  2  2  2  2 = 32. What is

54 equal to?
(a) 9
(b) 20
(c) 125
(d) 625
(e) None of the above

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SASMO 2015, Primary 4 Contest
3. There are 4 types of cakes available in a cake shop: chocolate, banana,
strawberry and oreo. Pete wants to buy 2 cakes. How many different choices
does he have?
(a) 6
(b) 8
(c) 10
(d) 12
(e) 16

________________________________________________________________

4. What is the number obtained from subtracting the number just after 1 ten 4
ones from the number just before 1 ten 6 ones?
(a) 0
(b) 1
(c) 2
(d) 3
(e) None of the above

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SASMO 2015, Primary 4 Contest
5. Find the missing term in the following sequence: 1, 1, 2, 6, _____, 120.

(a) 12
(b) 24
(c) 26
(d) 30
(e) 60

_______________________________________________________________

6. In a basketball tournament, 8 teams play against each other twice. How many
games are there altogether?
(a) 16
(b) 28
(c) 36
(d) 56
(e) 72

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SASMO 2015, Primary 4 Contest
7. An operator  acts on two numbers to give the following outcomes:

3  2 = 15
5  3 = 28
6  1 = 57
9  4 = 513

What is 7  5 equal to?

(a) 112
(b) 122
(c) 211
(d) 212
(e) None of the above

________________________________________________________________
8. Bill wants to cut square cards of length 3 cm from a rectangular sheet 27 cm by
20 cm. What is the biggest number of cards that can be cut from the sheet?
(a) 15
(b) 54
(c) 60
(d) 63
(e) None of the above

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SASMO 2015, Primary 4 Contest
9. A bag contains some sweets that can be divided equally among 2, 3 or 4 children
with no remainder. What is the smallest possible number of sweets in the bag?
(a) 6
(b) 8
(c) 12
(d) 24
(e) None of the above

_____________________________________________________________________________

10. If the four-digit number 12N4 is divisible by 4, how many possible values are
there for N?
(a) 1
(b) 2
(c) 3
(d) 4
(e) 5

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SASMO 2015, Primary 4 Contest
Section B (Correct answer = 4 marks; incorrect or no answer = 0)

11. A whole number multiplied by itself will give a special type of numbers called
perfect squares. Examples of perfect squares are 9 (= 3  3) and 16 (= 4  4).
What is the smallest number that can be multiplied by 45 to give a perfect
square?

________________________________________________________________
12. Two numbers x and y are such that
 x is greater than or equal to 2, but less than or equal to 7
 y is greater than or equal to 3, but less than or equal to 5.

Find the greatest possible value of x  y.

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SASMO 2015, Primary 4 Contest
13. Find the day of the week that is 60 days from a Sunday.

________________________________________________________________
14. Charles and Denise are brother and sister. Charles has twice as many brothers as
he has sisters, but Denise has 5 times as many brothers as she has sisters. How
many boys and girls are there in their family?

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SASMO 2015, Primary 4 Contest
15. A circle and a triangle are drawn on a flat surface. What is the biggest number of
regions that can be formed on the surface?

______________________________________________________________________

16. What are the dimensions of a rectangle that is made from 91 one-centimetre
square tiles if all the sides of the rectangle are longer than 1 cm?

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SASMO 2015, Primary 4 Contest
17. In the following alphametic, all the different letters stand for different digits. Find
the four-digit product RATS.

S T A R
 9

R A T S

_________________________________________________________________________________

18. Elsa thinks of 5 different whole numbers. None of the numbers is 0, and none of
them is a multiple of each other. What is the least possible sum of these 5
numbers?

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SASMO 2015, Primary 4 Contest
19. The diagram shows a rectangle being divided into 3 smaller rectangles and a
square. If the perimeter of the unshaded rectangle is 24 cm and the area of the
square is 16 cm2, find the total area of the shaded rectangles.

_______________________________________________________________

20. Aileen drives 1000 km during a trip. She rotates the tyres (four tyres on the car
and one spare tyre) so that each tyre has been used for the same distance at the
end of the trip. How many kilometres are covered by each tyre?

End of Paper

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SASMO 2015, Primary 4 Contest

Solutions to SASMO 2015 Primary 4

Section A
1. The diagram shows a figure that contains 13 identical squares. The area of the
figure is 117 cm2. Find its perimeter.

(a) 54 cm
(b) 57 cm
(c) 60 cm [Ans]
(d) 63 cm
(e) 66 cm

Solution

Area of 13 identical squares = 117 cm2

Area of one square = 9 cm2
Length of square = 3 cm
Since the perimeter of the figure is made up of 20 sides of a square, then its
perimeter
= 20 × length of square
= 20 × 3
= 60 cm

2. 25 means 2 multiplied by itself 5 times, i.e. 25 = 2 × 2 × 2 × 2 × 2 = 32. What is

54 equal to?

(a) 9
(b) 20
(c) 125
(d) 625 [Ans]
(e) None of the above

Solution
54 = 5 × 5 × 5 × 5 = 625

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SASMO 2015, Primary 4 Contest
3. There are 4 types of cakes available in a cake shop: chocolate, banana,
strawberry and oreo. Pete wants to buy 2 cakes. How many different choices
does he have?

(a) 6
(b) 8
(c) 10 [Ans]
(d) 12
(e) 16

Solution

Method 1 (Systematic Listing)

If the 2 cakes are of the same type, then there are 4 choices.
If the 2 cakes are of different types, then there are 6 choices as shown below:

Chocolate   
Banana   
Strawberry   
Oreo   

 total no. of choices = 4 + 6 = 10

Method 2 (Rephrase the Problem)

If the 2 cakes are of the same type, then there are 4 choices.
If the 2 cakes are of different types, then choosing 2 different types of cakes
from 4 types is the same as the handshake problem of 4 people shaking
hands once with one another, i.e. there are 3 + 2 + 1 = 6 choices.
 total no. of choices = 4 + 6 = 10

4. What is the number obtained from subtracting the number just after 1 ten 4
ones from the number just before 1 ten 6 ones?

(a) 0 [Ans]
(b) 1
(c) 2
(d) 3
(e) None of the above

Solution

The number just after 1 ten 4 ones (i.e. 14) is 15.

The number just before 1 ten 6 ones (i.e. 16) is also 15.
 the final number is 15  15 = 0

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SASMO 2015, Primary 4 Contest
5. Find the missing term in the following sequence: 1, 1, 2, 6, _____, 120.

(a) 12
(b) 24 [Ans]
(c) 26
(d) 30
(e) 60

Solution

The pattern is as follows:

1, 1, 2, 6, _____, 120

1 2 3 4 5

 the missing term is 6  4 = 24.

6. In a basketball tournament, 8 teams play against each other twice. How many
games are there altogether?

(a) 16
(b) 28
(c) 36
(d) 56 [Ans]
(e) 72

Solution

Consider the 8 teams playing against each other once first.

The first team will play against 7 other teams, i.e. 7 games;
the second team will play against 6 other teams, i.e. 6 games;
the third team will play against 5 other teams, i.e. 5 games; etc.
Thus total no. of games = 7 + 6 + 5 + … + 3 + 2 + 1

1+7=8
2+6=8 3 pairs

3+5=8
4
 total no. of games = 8  3 + 4 = 28

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SASMO 2015, Primary 4 Contest
Since the 8 teams play against each other twice, then total no. of games
= 28  2
= 56

7. An operator  acts on two numbers to give the following outcomes:

3  2 = 15
5  3 = 28
6  1 = 57
9  4 = 513

What is 7  5 equal to?

(a) 112
(b) 122
(c) 211
(d) 212 [Ans]
(e) None of the above

Solution

a  b = (𝑎 − 𝑏)(𝑎 + 𝑏)
 7  5 = 212

8. Bill wants to cut square cards of length 3 cm from a rectangular sheet 27 cm by

20 cm. What is the biggest number of cards that can be cut from the sheet?

(a) 15
(b) 54 [Ans]
(c) 60
(d) 63
(e) None of the above

Solution

27 cm  3 cm = 9 cards along the length of the rectangular sheet.

2
20 cm  3 cm = 63, i.e. 6 cards along the breadth of the rectangular sheet.

 biggest number of cards that can be cut from the sheet = 9  6 = 54

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SASMO 2015, Primary 4 Contest

9. A bag contains some sweets that can be divided equally among 2, 3 or 4 children
with no remainder. What is the smallest possible number of sweets in the bag?

(a) 6
(b) 8
(c) 12 [Ans]
(d) 24
(e) None of the above

Solution

Method 1

The no. of sweets must be a multiple of 2, a multiple of 3 and a multiple of 4.

So the possible no. of sweets are 12, 24, 36, …
 smallest possible number of sweets = 12

Method 2

Smallest possible number of sweets = LCM (2, 3, 4) = 12

10. If the four-digit number 12N4 is divisible by 4, how many possible values are
there for N?

(a) 1
(b) 2
(c) 3
(d) 4
(e) 5 [Ans]

Solution

Using the divisibility test for 4, N4 is also divisible by 4.

So the possible values for N are 0, 2, 4, 6, and 8.
 no. of possible values for N = 5

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SASMO 2015, Primary 4 Contest
Section B
11. A whole number multiplied by itself will give a special type of numbers called
perfect squares. Examples of perfect squares are 9 (= 3  3) and 16 (= 4  4).
What is the smallest number that can be multiplied by 45 to give a perfect
square?

Solution

Since 45 = 3  3  5 = (3  5)  3, then the smallest number that can be

multiplied by 45 to give a perfect square is 5, so that (3  5)  (3  5) = 15  15
is a perfect square.

12. Two numbers x and y are such that

 x is greater than or equal to 2, but less than or equal to 7

 y is greater than or equal to 3, but less than or equal to 5.

Find the greatest possible value of x  y.

Solution

Greatest possible value of x  y

= greatest possible value of x  least possible value of y
=73

=4

13. Find the day of the week that is 60 days from a Sunday.

Solution

By counting, 7 days from a Sunday is Sunday.

So 56 days (= 8  7 day) from a Sunday is still Sunday.
 60 days from a Monday is Thursday.

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SASMO 2015, Primary 4 Contest
14. Charles and Denise are brother and sister. Charles has twice as many brothers as
he has sisters, but Denise has 5 times as many brothers as she has sisters. How
many boys and how many girls are there in their family?

Solution

Method 1 (Guess and Check)

Since Charles is a boy and he has twice as many brothers as he has sisters, then
the number of boys in the family is one more than twice the number of girls.

Denise has 5 times as many brothers as she has

sisters. True or false?

1 girl and 3 boys Denise has 3 brothers and 0 sister. Above statement is false.

2 girls and 5 boys Denise has 5 brothers and 1 sister. Above statement is true.

Using guess and check as shown in the above table, there are 5 boys and 2
girls in the family.

Method 2 (Model Method)

For Charles
Boys
Girls 1

For Denise

Boys
Girls 1

From the model for Charles, 1 unit for Girls = 1 + 1 = 2.

 there are 5 boys and 2 girls in the family.

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SASMO 2015, Primary 4 Contest
Method 3 (Algebra)
Let the number of girls in the family be x.
Since Charles is a boy and he has twice as many brothers as he has sisters, then
there are 2x + 1 boys in the family.
Since Denise is a girl, then he has 2x + 1 brothers and x  1 sisters.
Since she has 5 times as many brothers as she has sisters, then
2x + 1 = 5(x  1)
2x + 1 = 5x  5
3x = 6
x=2
 there are 5 boys and 2 girls in the family.

15. A circle and a triangle are drawn on a flat surface. What is the biggest number of
regions that can be formed on the surface?

Solution

To form the biggest number of regions, the triangle and the circle should
intersect as often as possible.
Since a line can cut a circle at most two times, then the following diagram shows
that the biggest number of regions that can be formed on the surface is 8.

2 4

7 8

1 5
6

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SASMO 2015, Primary 4 Contest
16. What are the dimensions of a rectangle that is made from 91 one-centimetre
square tiles if all the sides of the rectangle are longer than 1 cm?

Solution

Since 91 = 7 × 13, and 7 and 13 cannot be divided by any whole number greater
than 1, then the dimensions of the rectangle are 7 cm by 13 cm.

17. In the following alphametic, all the different letters stand for different digits. Find
the four-digit product RATS.

S T A R
 9
R A T S

Solution

Since the product of a 4-digit number and 9 is still a 4-digit number, then S = 1
and R = 9 in the thousands column.
In the hundreds column, if T ≥ 2, then T  9 will result in a carryover to the
thousands column, and then the product will be a 5-digit number.
Since S = 1 already, then T = 0.
In the ones column, R  9 = 9  9 = 81, i.e. there is a carryover of 8 to the tens
column.
In the tens column, A  9 + 8 = _0 implies A  9 = _2, so A = 8.
 RATS = 1089  9 = 9801.

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SASMO 2015, Primary 4 Contest
18. Elsa thinks of 5 different whole numbers. None of the numbers is 0, and none of
them is a multiple of each other. What is the least possible sum of these 5
numbers?

Solution

Since we want the least possible sum of these 5 numbers, we try to use as small
whole numbers as possible.
Since none of these numbers is 0, we start with 1. But any number is a multiple
of 1, contradicting that none of these numbers is a multiple of each other.
So the smallest of these numbers is 2. Then all the other multiples of 2 (e.g. 4,
6, 8, etc.) are not possible.
The second smallest number is 3. Then all the other multiples of 3 (e.g. 6, 9, 12,
etc.) are not possible.
Since 4, which is a multiple of 2, is no longer possible, then the third smallest
number is 5.
Similarly, 6 is not possible, so the fourth smallest number is 7.
Similarly, 8, 9 and 10 are not possible, so the fifth number is 11.
 least possible sum of these 5 numbers = 2 + 3 + 5 + 7 + 11 = 28

19. The diagram shows a rectangle being divided into 3 smaller rectangles and a
square. If the perimeter of the unshaded rectangle is 24 cm and the area of the
square is 16 cm2, find the total area of the shaded rectangles.

Solution
Put the two shaded rectangles to form a long rectangle as shown:

1
Length of long rectangle = × perimeter of unshaded rectangle
2
1
= × 24 cm
2

= 12 cm
Breadth of long rectangle = length of square = 4 cm
 total area of shaded rectangles = 12 cm × 4 cm = 48 cm2

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SASMO 2015, Primary 4 Contest
20. Aileen drives 1000 km during a trip. She rotates the tyres (four tyres on the car
and one spare tyre) so that each tyre has been used for the same distance at the
end of the trip. How many kilometres are covered by each tyre?

Solution

Since there are 4 tyres on the road at any one time, total distance covered by 4
tyres = 1000 km × 4 = 4000 km
Since 5 tyres share the total distance equally, then distance covered by each tyre
= 4000 km  5 = 800 km

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