4. In ∆𝐴𝐵𝐶 , 𝐵𝑀 and 𝐶𝑁 intersect at 𝑃 .
Given the areas [𝐶𝑀𝑃] = 3, [𝐵𝑁𝑃] = 4
and [𝐵𝐶𝑃] = 5, find [𝐴𝐵𝐶].
QUESTIONS 1 TO 10 ARE WORTH
3 MARKS EACH
1. Evaluate
3
3
√8 + 3√21 + √8 − 3√21
(A) 1
(B)
(C)
(D)
(E)
3
√2
√2
2
None of the above
2. Suppose 𝑎 and 𝑏 are positive real
numbers such that (𝑎 + 𝑏)2 = 400 and
𝑎𝑏 = 50.
(A)
(B)
(C)
(D)
(E)
Find the value of 𝑎3 + 𝑏3 .
(A)
(B)
(C)
(D)
(E)
1250
2500
6400
8000
None of the above
5. In the figure below, a semicircle is
inscribed in an equilateral triangle,
which is inscribed in a larger
semicircle.
3. Following the direction of the arrows
in the figure below, how many ways
are there to get from 𝐴 to 𝐵?
(A)
(B)
(C)
(D)
(E)
Find the ratio of the radius of the
larger semicircle to that of the smaller
semicircle.
(A)
(B)
(C)
(D)
(E)
27
28
29
30
None of the above
SEAMO 2020 Paper E © SEAMO PTE LTD
18
25
47
48
None of the above
1
√3
2
√5
√6
None of the above
6. Evaluate
9. Evaluate
2020
cos 4 15° + sin4 15° + 2 sin2 15° cos 2 15°
.
cos 6 15° + sin6 15° + 3 sin2 15° cos 2 15°
(A)
1
2
(B)
2
3
√1 + √5
+
+
(A)
(B)
(C)
(D)
(E)
4
(C)
5
(D) √3
(E) None of the above
2020
√5 + √9
+
2020
√9 + √13
2020
√2021 + √2025
+⋯
.
2020
2025
2222
8080
None of the above
10. Find the minimum value of
7. Let 𝑝, 𝑞 and 𝑟 be the roots of the
equation 2𝑥 3 − 5𝑥 2 − 6𝑥 + 2 = 0.
𝑓(𝑥) =
Evaluate
for 0 < 𝑥 < 𝜋.
1 1 1
+ +
𝑝 𝑞 𝑟
(A)
(B)
(C)
(D)
(E)
(A)
(B)
(C)
(D)
(E)
1
2
3
6
None of the above
12
16
20
36
None of the above
QUESTIONS 11 TO 20 ARE WORTH
4 MARKS EACH
8. Suppose 𝑎, 𝑏 and 𝑐 are positive real
numbers such that
11. Let 𝑝(𝑥) = 𝑥 3 + 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 be a thirddegree polynomial with three real
roots 𝑥1 , 𝑥2 and 𝑥3.
𝑎𝑏𝑐(𝑎 + 𝑏 + 𝑐) = 100
Suppose 𝑐 + 𝑎 = 10 and 𝑏 = 5, find the
value of (𝑥12 − 1)(𝑥22 − 1)(𝑥32 − 1).
Find the minimum value of
(𝑎 + 𝑏)(𝑏 + 𝑐)
(A)
(B)
(C)
(D)
(E)
9𝑥 2 sin2 𝑥 + 4
,
𝑥 sin 𝑥
(A)
(B)
(C)
(D)
(E)
10
20
40
48
None of the above
2
64
72
80
81
None of the above
SEAMO 2020 Paper E © SEAMO PTE LTD
12. In the same 3-hour interval, two
people went to the gym at random
times.
14. Find the remainder when
1 1
1
2020! (1 + + + ⋯ +
)
2 3
2020
Given that they each spent 1.5 hours
working out there, what is the
probability that they met?
(A)
1
4
(B)
1
2
(C)
3
4
is divided by 2021.
(A)
(B)
(C)
(D)
(E)
1
2
2019
2020
None of the above
15. Amos, Bryan and Catherine play a
series of 1-on-1 chess games.
(D) 1
At any one time, 2 people play while 1
watches.
(E) None of the above
The winner continues playing against
the next competitor while the loser
switches out.
13. Consider the following array, in
which the integers 1 to 9 are placed
as shown below.
Amos played 7 games in total.
Bryan played 10 games in total.
Catherine played 13 games in total.
Who lost the second game?
(A) Amos
(B) Bryan
(C) Catherine
(D) Impossible to determine
(E) None of the above
The diagonal sum of integers is 15.
If we construct a similar array with
integers from 1 to 100, what is the
diagonal sum of integers?
(A)
(B)
(C)
(D)
(E)
5050
6250
6400
7280
None of the above
SEAMO 2020 Paper E © SEAMO PTE LTD
3
16. The figure below shows 5 connected
circles.
18. Point 𝑃 lies in equilateral triangle 𝐴𝐵𝐶.
Suppose ∠𝐴𝑃𝐵 = 120° and ∠𝐵𝑃𝐶 = 130°.
Given 4 distinct colours, how many
different ways are there to colour the
circles such that no two connected
circles are of the same colour?
(A)
(B)
(C)
(D)
(E)
If segments 𝐴𝑃, 𝐵𝑃 and 𝐶𝑃 were to
form a new triangle, what would be its
largest angle?
72
84
96
100
None of the above
(A)
(B)
(C)
(D)
(E)
17. Two circles 𝐶1 and 𝐶2 of different
sizes are internally tangential.
60°
70°
120°
130°
None of the above
19. Given that 𝑎 and 𝑏 are two real
numbers, where 𝑎𝑏 ≠ 1, satisfying
Given that 𝐴𝐵 = 18 and 𝐶𝐷 = 10 , find
the area of the shaded region.
3𝑎2 − 4𝑎 − 5 = 0 (1)
{ 2
5𝑏 + 4𝑏 − 3 = 0 (2)
Evaluate
(𝑎 + 1)(𝑏 + 1)
𝑏
A) 169 𝜋
B) 480 𝜋
C) 810 𝜋
D) 819 𝜋
E) None of the above
(A)
1
2
(B)
2
3
(C)
3
4
(D) 1
(E) None of the above
4
SEAMO 2020 Paper E © SEAMO PTE LTD
20. During a round-table meeting, the
CEO wanted to select a team of 3
out of 25 marketing specialists to
handle a new project.
23. When a conical bottle rests on its flat
base, the water surface in the bottle
is 8 𝑐𝑚 from the vertex.
When Leonard turns the conical bottle
upside down, the water surface is 2 𝑐𝑚
from the base.
If the probability that 2 of the
selected specialists were sitting
next to each other is
𝑚
𝑛
, where
Let the height of the bottle be
(𝑎 + 𝑏√𝑐) 𝑐𝑚 , where 𝑎, 𝑏 and 𝑐 are
𝑚 and 𝑛 are relatively prime, find
the value of (𝑚 + 𝑛).
(A)
(B)
(C)
(D)
(E)
integers and 𝑐 is not divisible by any
perfect square number other than 1.
49
53
55
57
None of the above
Find (𝑎 + 𝑏 + 𝑐).
QUESTIONS 21 TO 25 ARE WORTH
6 MARKS EACH
21. Suppose
112𝑛 + 223𝑛 + 334𝑛 + 445𝑛 + 556𝑛
is divisible by 10.
24. Suppose 𝑚 and 𝑛 are positive integers
satisfying 1! + 2! + 3! + ⋯ + 𝑛! = 𝑚2 .
Determine the maximum value of
(𝑚 + 𝑛).
How many possible two-digit positive
integer values of 𝑛 are there?
22. The composition of alloys A, B and C
are as shown in the table below.
Terry mixed the alloys to obtain a
new alloy T containing 15% of Lead.
If 𝑚 and 𝑛 are the maximum and
minimum percentages, respectively,
of Tin in the new alloy, find (𝑚 − 𝑛).
Lead
Tin
Zinc
Alloy A
30%
70%
0%
Alloy B
0%
80%
20%
Alloy C
10%
50%
40%
Alloy T
15%
25. What is the remainder when
2022
20202021
is divided by 7?
End of Paper
SEAMO 2020 Paper E © SEAMO PTE LTD
5
SEAMO 2020
Paper E – Answers
Multiple-Choice Questions
Questions 1 to 10 carry 3 marks each.
Q1
A
Q2
E
Q3
C
Q4
E
Q5
B
Q6
E
Q7
C
Q8
B
Q9
E
Q10
A
Questions 11 to 20 carry 4 marks each.
Q11
A
Q12
C
Q13
A
Q14
E
Q15
A
Q16
A
Q17
D
Q18
B
Q19
B
Q20
D
Free-Response Questions
Questions 21 to 25 carry 6 marks each.
© SEAMO 2020
21
22
23
24
25
68
20
87
6
4
Paper E
1