Problems and Solutions SEAMO PAPER E 2020

International MATH CONTEST SEAMO

9. Evaluate

\(\frac{2020}{\sqrt 1 +\sqrt 5}+\frac{2020}{\sqrt 5 +\sqrt 9}+\frac{2020}{\sqrt 9 +\sqrt {13}}+…+\frac{2020}{\sqrt {2021} +\sqrt {2025}}\).

(A) 2020
(B) 2025
(C) 2222
(D) 8080
(E) None of the above


\(\frac{2020}{\sqrt 1 +\sqrt 5}+\frac{2020}{\sqrt 5 +\sqrt 9}+\frac{2020}{\sqrt 9 +\sqrt {13}}+…+\frac{2020}{\sqrt {2021} +\sqrt {2025}}\).

\(=2020(\frac{1}{\sqrt 5 +\sqrt 1}+\frac{1}{\sqrt 9 +\sqrt 5}+\frac{1}{\sqrt {13} +\sqrt 9}+…+\frac{1}{\sqrt {2025} +\sqrt {2021}})\)

\(=2020(\frac{\sqrt 5−\sqrt 1}{4}+\frac{\sqrt 9−\sqrt 5}{4}+\frac{\sqrt {13}−\sqrt 9}{4}+⋯+\frac{\sqrt {2025}−\sqrt {2021}}{4})\)

\(=2020(\frac{\sqrt {2025}−\sqrt 1}{4})\)

\(=2020(\frac{45}{4}−\frac{1}{4})\)

\(=505(44)=22220\)


10. Find the minimum value of

\(𝑓(𝑥) =\frac{9𝑥^2 \sin^2 𝑥 + 4}{𝑥 \sin 𝑥},\)

for 0 < 𝑥 < 𝜋.
(A) 12
(B) 16
(C) 20
(D) 36
(E) None of the above


Belum tersedia


11. Let \(𝑝(𝑥) = 𝑥^3 + 𝑎𝑥^2 + 𝑏𝑥 + 𝑐\) be a third degree polynomial with three real roots \(𝑥_1, 𝑥_2\) and \(𝑥_3\).
Suppose \(𝑐 + 𝑎 = 10\) and \(𝑏 = 5\), find the value of

\(({𝑥_1}^2−1)({𝑥_2}^2−1)({𝑥_3}^2 – 1)\).

(A) 64
(B) 72
(C) 80
(D) 81
(E) None of the above


Belum tersedia


12. In the same 3-hour interval, two people went to the gym at random
times. Given that they each spent 1.5 hours working out there, what is the probability that they met?
(A) \(\frac{1}{4}\)

(B) \(\frac{1}{2}\)

(C) \(\frac{3}{4}\)

(D) 1

(E) None of the above


Belum tersedia


13. Consider the following array, in which the integers 1 to 9 are placed
as shown below.

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) 5050
(B) 6250
(C) 6400
(D) 7280
(E) None of the above


Jumlah diagonalanya \(1 + 12 + 23 + 34 + … + 100 =\frac{101×10}{2}= 505\)


14. Find the remainder when

\(2020! (1 +\frac{1}{2}+\frac{1}{3}+ ⋯ +\frac{1}{2020})\)

is divided by 2021.
(A) 1
(B) 2
(C) 2019
(D) 2020
(E) None of the above


\(2020!(1+\frac{1}{2}+\frac{1}{3}+⋯+\frac{1}{2020})\) mod \(2021\)

\(=1×2×3×…×2020(1+\frac{1}{2}+\frac{1}{3}+⋯+\frac{1}{2020})\) mod \(2021\)

Karena \(2021 =43×47\), angka \(43\) dan \(47\) bagian dari \(2020!\), maka jelas bahwa \(2020!\) mod \(2021 =0\).

Jadi \((1+\frac{1}{2}+\frac{1}{3}+⋯+\frac{1}{2020})\) mod \(2021=0\)


15. Amos, Bryan and Catherine play a series of 1-on-1 chess games.
At any one time, 2 people play while 1 watches. The winner continues playing against the next competitor while the loser switches out. 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


Belum tersedia


16. The figure below shows 5 connected circles. 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) 72
(B) 84
(C) 96
(D) 100
(E) None of the above


Ada 2 kemungkinan pilihan

  • A dan C berwarna sama banyak cara pewarnaan adalah 4 ∙ 3 ∙ 2 ∙ 2 ∙ 1 = 48 cara
    – Pilihan warna untuk A ada 4 pilihan
    – Pilihan warna untuk C ada 1 pilihan karena warnanya harus sama dengan A
    – Pilihan warna untuk E ada 3 pilihan karna harus berbeda dengan A atau C
    – Pilihan warna untuk D ada 2 pilihan karna harus berbeda dengan A=C dan E
    – Pilihan warna untuk B ada 2 pilihan karna harus berbeda dengan A=C dan E
    banyak cara pewarnaan adalah 4 ∙ 3 ∙ 2 ∙ 2 ∙ 1 = 48 cara
  • A dan C berbeda warna 
    – Pilihan warna untuk A ada 4 pilihan
    – Pilihan warna untuk C ada 3 pilihan karena harus berbeda warna dengan A
    – Pilihan warna untuk E ada 2 pilihan karna harus berbeda dengan A dan C
    – Pilihan warna untuk D ada 1 pilihan karna harus berbeda dengan A, C dan E
    – Pilihan warna untuk B ada 1 pilihan karna harus berbeda dengan A, C dan E
    banyak cara pewarnaan adalah 4 ∙ 3 ∙ 2 ∙ 1 ∙ 1 = 24 cara

Jadi total cara pewarnaan adalah 48 + 24 = 72 cara


17. Two circles \(𝐶_1\) and \(𝐶_2\) of different sizes are internally tangential. Given that \(𝐴𝐵 = 18\) and \(𝐶𝐷 = 10\) , find the area of the shaded region.


A) 169 𝜋
B) 480 𝜋
C) 810 𝜋
D) 819 𝜋
E) None of the above


Belum tersedia


18. Point \(𝑃\) lies in equilateral triangle \(𝐴𝐵𝐶\). Suppose \(∠𝐴𝑃𝐵 = 120°\) and \(∠𝐵𝑃𝐶 = 130°\). If segments \(𝐴𝑃, 𝐵𝑃\) and \(𝐶𝑃\) were to form a new triangle, what would be its largest angle?


(A) 60°
(B) 70°
(C) 120°
(D) 130°
(E) None of the above


Belum tersedia


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