SASMO 2021 Grade 11 and Grade 12

Question 1
Figure 1 is called a “stack map.” The numbers tell how many cubes are stacked in
each position. Figure 2 shows these cubes, and Figure 3 shows the view of the
stacked cubes as seen from the front.

4 3 1 3 4

1 2 2 2 1

Figure 1 Figure 2 Figure 3 Figure 4

A B C D E

Question 2
Find the value of the following expression.

3 3
√9 + √80 + √9 − √80

A. 1
B. 0
C. 3
D. 9
E. None of the above
Question 3
Given that log 3 𝑥 = 𝑎, find the value of log 3 (9𝑥) in terms of 𝑎.
A. 𝑎
B. 3𝑎
C. 9𝑎
1
D. 3 𝑎
E. None of the above

Question 4
Given 318 is equal to the 9-digit number 𝑎8742048𝑏, what is the value of 𝑎 × 𝑏?
A. 27
B. 21
C. 24
D. 12
E. None of the above
Question 5
Find the last two digits of 20212021 .
A. 61
B. 41
C. 21
D. 01
E. None of the above

Question 6
Find the next fraction in the sequence below.
3 5 7 1 11 13
, , , , , ,…
2 6 18 6 162 486
15
A. 21
1
B. 21
5
C. 486
17
D. 1458
E. None of the above
Question 7
If an integer is to be randomly selected from set A below and an integer is to be
randomly selected from set B, what is the probability that the product of the two
integers will be negative?
𝐴 = {−10, −8, −6, −4, −3}
𝐵 = {−4, −2, 0, 2, 4}
1
A. 2
B. 0
2
C. 5
1
D. 3
E. None of the above

Question 8
In the diagram, 𝐴𝐵𝐶 is a sector of a circle with
centre 𝐵 and a radius of 21 cm. The unshaded
circle touches the two radii, 𝐵𝐶 and 𝐵𝐴, and the
arc 𝐴𝐶 as shown. If ∠𝐴𝐵𝐶 = 60°, find the area
22
of the shaded region. (Use 𝜋 = )
7

A. 77 cm2
B. 231 cm2
C. 154 cm2
D. 1232 cm2
E. None of the above
Question 9
There are 4 boxes on a table. Each box contains only 1 item, and the 4 items in the
boxes are 1 red pencil, 1 green pencil, 1 blue pen and 1 black pen. The following are
the labels on the boxes:
• Box 1: This box does not contain the red pencil.
• Box 2: The blue pen is in either Box 1 or Box 3.
• Box 3: The black pen is Box 1.
• Box 4: This box contains the green pencil.
The boxes with the pens inside are labelled with a correct statement whereas the
boxes with the pencils inside are labelled with a wrong statement. Which box
contains the blue pen?
A. Box 4
B. Box 3
C. Box 2
D. Box 1
E. Impossible to determine

Question 10
What is the remainder when (𝑥 20 + 21) is divided by 𝑥 + 1?
A. 20
B. 21
C. 22
D. 2021
E. None of the above
Question 11
A standard dice is rolled twice and the rolled numbers are recorded as 𝑥 and 𝑦.
What is the probability that the 𝑥, 𝑦 and 5 are the 3 sides of an isosceles triangle?
11
A. 36
13
B. 36
5
C. 18
7
D. 18

E. None of the above

Question 12
Solve the equation below where 0 < 𝑥 < 𝜋.

√3 cos(3𝑥) = sin(3𝑥)
𝜋
A. 3
𝜋
B. 9
C. 𝜋
2𝜋
D. − 9
E. None of the above
Question 13
Derrick spent $200 in 5 days. Each day, he spent no more than in the previous day
and the amount he spent is an integer number of $. What is the smallest possible
amount of money (in $) he could have spent on the 1st, 3rd and 5th day in total?
A. 100
B. 50
C. 40
D. 20
E. None of the above

Question 14
1 1 𝜋
It is given that 2 cot(2𝑥) + sin 𝑥 cos 𝑥 = 2, where 0 < 𝑥 < 2 . Find the value of
1
(sin 𝑥 cos 𝑥 − 2 cot(2𝑥)).
1
A. 2
1
B. 4
1
C. 3
2
D. 3
E. None of the above
Question 15
Let 𝑓(𝑥) be a function such that
𝑥 + 2020
𝑓(𝑥) + 2𝑓 ( ) = 4040 − 𝑥
𝑥−1
for all real values 𝑥 ≠ 1. Find the value of 3𝑓(2022).
A. 12114
B. 6058
C. 6056
D. 4038
E. None of the above
Question 16
How many rectangles are there in the diagram such that the sum of the numbers
within the rectangle is a multiple of 5?

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

Question 17
The lengths of the sides of a right-angled triangle form 3 consecutive terms of a
𝑎+√𝑏
geometric sequence. The cosine of the largest of the angles in the triangle is
𝑐
where 𝑎 and 𝑐 are integers and 𝑏 is a prime number. Find the value of 𝑎 + 𝑏 − 𝑐.
Question 18
Tom uses the digits 0 to 9 exactly once to form the smallest possible 10-digit
number which is divisible by 495. Let 𝐴 be the 4-digit number formed by the first 4
digits of Tom’s number and 𝐵 be the 4-digit number formed by the last 4 digits of
Tom’s number. For example, if Tom’s number is 1234567890, then 𝐴 = 1234 and
𝐵 = 7890. Find the value of 𝐴 + 𝐵.

Question 19
Let 𝑎 be a positive integer such that the last 2 digits of 𝑎2021 is 21. Find the smallest
possible value of 𝑎.
Question 20
For any positive real number 𝑥, [𝑥] is the greatest integer which is less than or equal
to 𝑥. For example, [2.35] = 2. What is the value of the following sum?

[√1] + [√2] + [√3]+. . . +[√199] + [√200]

Question 21
What is the least number of distinct even numbers which must be randomly selected
from the list of 100 numbers in 21, 22, 23, …, 119, 120 to make sure that among
the selected numbers, there are two numbers with a sum of 100?
Question 22
Let 𝑎, 𝑏, 𝑐 and 𝑑 be real numbers such that |𝑎 − 𝑏| = 5, |𝑏 − 𝑐| = 4, and

|𝑐 − 𝑑| = 3. How many possible values of |𝑎 − 𝑑| are there?

Question 23

In the following cryptarithm, all the different letters stand for different digits.

S A S M O

– 2 0 2 1

A W A R D

If O = 6, find the value of the sum S+A+S+M+O.

Question 25
The roots of 𝑥 2 − 𝑎𝑥 + 𝑏 = 0 and 𝑥 2 − 𝑏𝑥 + 𝑎 = 0 are positive integers. How many
different values of (𝑎 − 𝑏) are there?