4. Find the sum of all possible positive
integers 𝑛 such that the expression
below is an integer.
QUESTIONS 1 TO 10 ARE WORTH
3 MARKS EACH
1. Find the last two digits of 112020.
(A)
(B)
(C)
(D)
(E)
4𝑛 3 − 16𝑛2 + 29𝑛 + 60
2𝑛 − 3
01
41
71
91
None of the above
(A)
(B)
(C)
(D)
(E)
2. The quadratic equation
42
69
75
81
None of the above
𝑥 2 − 52𝑥 + 𝑘 = 0
5. Evaluate the sum
has roots that are prime numbers.
𝑆=
Find the maximum value of 𝑘.
(A)
(B)
(C)
(D)
(E)
520
576
640
667
None of the above
+⋯+
(A)
1
2
(B)
210
421
(C)
105
211
3. Let 𝑓(𝑥) = 𝑥 2 + 2020𝑥 + 20.
How many ordered pairs of positive
integers (𝑚, 𝑛) are there such that
𝑓(𝑚 + 𝑛) = 𝑓(𝑚) + 𝑓(𝑛)?
(A)
(B)
(C)
(D)
(E)
20
1 + 202 + 204
(D) 1
(E) None of the above
1
2
3
4
None of the above
SEAMO 2020 Paper F © SEAMO PTE LTD
1
2
3
+
+
1 + 12 + 14 1 + 22 + 24 1 + 32 + 34
1
6. 2 white, 3 black and 4 grey marbles
are shared equally among 9 students.
9. How many positive integers less than
2020 with the property that the sum of
its digits equals 9?
Find the number of ways the marbles
can be distributed so that Bran and
Sansa gets the same colour and Arya
gets a grey marble.
(A)
(B)
(C)
(D)
(E)
(A)
(B)
(C)
(D)
(E)
120
130
140
150
None of the above
50
100
102
202
None of the above
10. The sequence {𝑎𝑛 } is defined by
𝑎𝑛+2 =
7. Given that 𝑎 is a real number such
that 𝑎4 + 𝑎3 + 𝑎2 + 𝑎 + 1 = 0.
with 𝑎1 = 1 and 𝑎2 = 2.
Evaluate 𝑎2020 + 2𝑎2010 + 3𝑎2000.
Evaluate 𝑎2020.
(A)
(B)
(C)
(D)
(E)
(A)
(B)
(C)
(D)
(E)
2
4
6
8
None of the above
11. Find the smallest prime factor of
1
1
1
=𝑏+ =𝑐+
𝑏
𝑐
𝑎
1000 … 01
⏟
2020 𝑧𝑒𝑟𝑜𝑠
What is the largest possible value of
𝑎𝑏𝑐?
(A)
(A)
(B)
(C)
(D)
(E)
1
2
(B) 2
(C)
1
2
3
4
None of the above
QUESTIONS 11 TO 20 ARE WORTH
4 MARKS EACH
8. Given that 𝑎. 𝑏 and 𝑐 are three distinct
real numbers such that
𝑎+
1 + 𝑎𝑛+1
𝑎𝑛
3
5
7
11
None of the above
5
2
(D) 3
(E) None of the above
2
SEAMO 2020 Paper F © SEAMO PTE LTD
12. In the expansion of
15. Given that 𝑎 and 𝑏 are real numbers
satisfying
𝑓(𝑥) = (1 + 𝑎𝑥)4 (1 + 𝑏𝑥)5
2
2
{ 6 − 5𝑎 + 4𝑏 − 3𝑎 + 2𝑎𝑏 − 𝑏 = 0
𝑎−𝑏=1
where 𝑎 and 𝑏 are positive integers,
the coefficient of 𝑥 2 is 66.
Find the sum of all possible values of
30𝑎
Evaluate 𝑎 + 𝑏.
(A)
(B)
(C)
(D)
(E)
𝑏
2
3
4
5
None of the above
(A)
(B)
(C)
(D)
(E)
13. The equation 𝑥 3 − 𝑎𝑥 2 + 𝑏𝑥 − 2020 has
three positive integer roots.
.
−15
−10
15
30
None of the above
16. The figure below shows a 5 × 6
rectangular board with a missing 1 × 2
rectangle in the center.
Find the least possible value of 𝑎.
(A)
(B)
(C)
(D)
(E)
101
110
202
220
None of the above
How many squares are there in the
board?
14. Evaluate the sum
𝑆 = sin2 0° + sin2 2° + sin2 4° + ⋯
+ sin2 180°
(A)
(B)
(C)
(D)
(E)
(A)
(B)
(C)
(D)
(E)
80
81
88
90
None of the above
SEAMO 2020 Paper F © SEAMO PTE LTD
3
14
30
54
56
None of the above
17. In ∆𝐴𝐵𝐶,
20. 𝐴, 𝐵, 𝐶 and 𝐷 are four distinct points
lying on the circumference of a circle
such
that chords 𝐴𝐵 and 𝐶𝐷 are
perpendicular at point 𝐸.
(sin 𝐴 + sin 𝐵) ∶ (sin 𝐵 + sin 𝐶) ∶ (sin 𝐶 + sin 𝐴)
= 19 ∶ 20 ∶ 21
Given that 𝐸𝐴 = 4, 𝐸𝐵 = 24 and 𝐸𝐶 = 6 ,
find the radius of the circle.
Find the value of 99cos 𝐴.
(A)
(B)
(C)
(D)
(E)
39
41
51
60
None of the above
18. Find the least positive integer 𝑛 such
that
the
equation
10𝑛
⌊
𝑥
⌋ = 98 has
integer solution 𝑥.
⌊𝑘⌋ is the largest integer smaller than
or equal to 𝑘.
(A)
(B)
(C)
(D)
(E)
(A)
(B)
(C)
(D)
(E)
3
4
5
6
None of the above
√270
18
None of the above
QUESTIONS 21 TO 25 ARE WORTH
6 MARKS EACH
21. You need to tile a 10 × 1 hallway with
a supply of 1 × 1 red, 2 × 1 red tiles
and 2 × 1 blue tiles. Find the number
of ways you can tile the 10 × 1
hallway.
19. How many positive integers 𝑘 < 100
such
that 2(56𝑛 ) + 𝑘(23𝑛+2 ) − 1 is
divisible by 7 for any positive integer
𝑛?
(A)
(B)
(C)
(D)
(E)
√221
15
12
14
18
19
None of the above
22. 𝑥, 𝑦 and 𝑧 are real numbers such that
𝑥+𝑦+𝑧 = 7
{𝑥 + 𝑦 2 + 𝑧 2 = 19
𝑥 3 + 𝑦 3 + 𝑧 3 = 64
2
Evaluate 𝑥 4 + 𝑦 4 + 𝑧 4.
4
SEAMO 2020 Paper F © SEAMO PTE LTD
23. In ∆𝐴𝐵𝐶 shown below, 𝐴𝐷, 𝐵𝐸 and 𝐶𝐹
intersect at 𝑃 . Suppose 𝐴𝑃 = 𝑎,
𝐵𝑃 = 𝑏, 𝐶𝑃 = 𝑐 and 𝐷𝑃 = 𝐸𝑃 = 𝐹𝑃 = 𝑥.
Given that 𝑥 = 3 and 𝑎 + 𝑏 + 𝑐 = 20 ,
find 𝑎𝑏𝑐.
24. Positive integers 𝑎, 𝑏 and 𝑐
randomly selected from the
{1,2,3, … ,2020} with replacement.
are
set
Find the probability that 𝑎𝑏𝑐 + 𝑎𝑏 + 2𝑎
is divisible by 5.
25. 𝐴𝐵𝐶𝐷 is a convex quadrilateral such
that 𝐴𝐶 intersects 𝐵𝐷 at 𝐸 . 𝐻 is a
point lying in the segment 𝐷𝐸 such
that 𝐴𝐻 is perpendicular to 𝐷𝐸.
Suppose 𝐵𝐸 = 𝐸𝐷, 𝐶𝐸 = 9, 𝐸𝐻 = 12, 𝐴𝐻 =
32 and ∠𝐵𝐶𝐴 = 90° . Evaluate the
length of 𝐶𝐷.
End of Paper
SEAMO 2020 Paper F © SEAMO PTE LTD
5
SEAMO 2020
Paper F – Answers
Multiple-Choice Questions
Questions 1 to 10 carry 3 marks each.
Q1
A
Q2
D
Q3
D
Q4
B
Q5
B
Q6
C
Q7
C
Q8
E
Q9
C
Q10
A
Questions 11 to 20 carry 4 marks each.
Q11
D
Q12
B
Q13
B
Q14
E
Q15
A
Q16
D
Q17
C
Q18
B
Q19
B
Q20
A
Free-Response Questions
Questions 21 to 25 carry 6 marks each.
© SEAMO 2020
21
22
23
24
25
686
247
234
33
125
30
Paper F
1