Problems And Solutions SEAMO PAPER F 2020

9. How many positive integers less than 2020 with the property that the sum of its digits equals 9?
(A) 50
(B) 100
(C) 102
(D) 202
(E) None of the above

10. The sequence \({𝑎_𝑛}\) is defined by

\(𝑎_{𝑛+2} = \frac{1 + 𝑎_{𝑛+1}}{𝑎_𝑛}\)

with \(𝑎_1 = 1\) and \(𝑎_2 = 2\).
Evaluate \(𝑎_{2020}\).
(A) 1
(B) 2
(C) 3
(D) 4
(E) None of the above

11. Find the smallest prime factor of

\(\underbrace{1000 … 01}_{{2020 zeroes}}\)

(A) 3
(B) 5
(C) 7
(D) 11
(E) None of the above

12. In the expansion of

\(𝑓(𝑥) = (1 + 𝑎𝑥)^4(1 + 𝑏𝑥)^5\)

where 𝑎 and 𝑏 are positive integers, the coefficient of \(𝑥^2\) is \(66\).
Evaluate \(𝑎 + 𝑏\).
(A) 2
(B) 3
(C) 4
(D) 5
(E) None of the above

13. The equation \(𝑥^3 − 𝑎𝑥^2 + 𝑏𝑥 − 2020\) has three positive integer roots.
Find the least possible value of \(𝑎\).
(A) 101
(B) 110
(C) 202
(D) 220
(E) None of the above

14. Evaluate the sum

\(𝑆 = \sin^2 0° + \sin^2 2° + \sin^2 4° + ⋯ + \sin^2 180°\)

(A) 80
(B) 81
(C) 88
(D) 90
(E) None of the above

15. Given that 𝑎 and 𝑏 are real numbers satisfying

\(6 − 5𝑎 + 4𝑏 − 3𝑎^2 + 2𝑎𝑏 − 𝑏^2 = 0\)
\(𝑎 − 𝑏 = 1\)

Find the sum of all possible values of \(\frac{30𝑎}{𝑏}\)
.
(A) −15
(B) −10
(C) 15
(D) 30
(E) None of the above

16. The figure below shows a \(5 × 6\) rectangular board with a missing \(1 × 2\) rectangle in the center.

How many squares are there in the board?
(A) 14
(B) 30
(C) 54
(D) 56
(E) None of the above

17. In \(Δ𝐴𝐵𝐶\),

\((sin 𝐴 + sin 𝐵) ∶ (sin 𝐵 + sin 𝐶) ∶ (sin 𝐶 + sin 𝐴) = 19 ∶ 20 ∶ 21\)

Find the value of \(99cos 𝐴\).
(A) 39
(B) 41
(C) 51
(D) 60
(E) None of the above