# Problems and Solutions SEAMO PAPER E 2020

The Southeast Asia Mathematical Olympiad (SEAMO) is an international Math Olympiad competition that originated in Singapore and was founded by Mr. Terry Chew in 2016 in 8 Southeast Asian Countries. Since then, it is growing its popularity around the world. In 2019 it was recognized by 18 countries. In 2020 total number of participating countries increased to 22, including students from Indonesia, Brazil, China, Newzealand, and Taiwan students enrolled in SEAMO 2022-23.

Problem and Solution SEAMO 2020 paper E. Soal ini bersumber dari seamo-official.org

1. Evaluate

$$\sqrt[3]{8 + 3\sqrt 21} +\sqrt[3]{8 – 3\sqrt 21}$$

(A) $$1$$
(B) $$\sqrt[3]{2}$$
(C) $$\sqrt 2$$
(D) $$2$$
(E) None of the above

Belum tersedia

2. Suppose 𝑎 and 𝑏 are positive real numbers such that $$(𝑎 + 𝑏)^2 = 400$$ and $$𝑎𝑏 = 50$$. Find the value of $$𝑎^3 + 𝑏^3$$.
(A) 1250
(B) 2500
(C) 6400
(D) 8000
(E) None of the above

\begin{align} 𝑎^3+𝑏^3 &=(𝑎+𝑏)(𝑎^2+𝑏^2−𝑎𝑏)\\ &=(𝑎+𝑏)((𝑎+𝑏)^2−2𝑎𝑏−𝑎𝑏)\\ &=(20)(400−3𝑎𝑏)\\ &=(20)(400−150)\\ &=20(250)\\ &=5000 \end{align}

3. Following the direction of the arrows in the figure below, how many ways are there to get from 𝐴 to 𝐵?

(A) 27
(B) 28
(C) 29
(D) 30
(E) None of the above

Belum tersedia

4. In $$Δ𝐴𝐵𝐶 , 𝐵𝑀$$ and $$𝐶𝑁$$ intersect at $$𝑃$$ . Given the areas $$[𝐶𝑀𝑃] = 3, [𝐵𝑁𝑃] = 4$$ and $$[𝐵𝐶𝑃] = 5$$, find $$[𝐴𝐵𝐶]$$.

(A) 18
(B) 25
(C) 47
(D) 48
(E) None of the above

Belum tersedia

5. In the figure below, a semicircle is inscribed in an equilateral triangle, which is inscribed in a larger semicircle. Find the ratio of the radius of the larger semicircle to that of the smaller semicircle.

(A) $$\sqrt 3$$
(B) $$2$$
(C) $$\sqrt 5$$
(D) $$\sqrt 6$$
(E) None of the above

Belum tersedia

6. Evaluate

$$\frac{\cos^4 15° + \sin^4 15° + 2 \sin^2 15° \cos^2 15°}{\cos^6 15° + \sin^6 15° + 3 \sin^2 15° \cos^2 15°}$$.

(A) $$\frac{1}{2}$$
(B) $$\frac{2}{3}$$
(C) $$\frac{4}{5}$$
(D) $$3$$
(E) N one of the above

Belum tersedia

7. Let $$𝑝, 𝑞$$ and $$𝑟$$ be the roots of the equation $$2𝑥^3 − 5𝑥^2 − 6𝑥 + 2 = 0$$.
Evaluate

$$\frac{1}{𝑝}+\frac{1}{𝑞}+\frac{1}{𝑟}$$

(A) 1
(B) 2
(C) 3
(D) 6
(E) None of the above

Dalil vieta
$$𝑝+𝑞+𝑟=52$$
$$𝑝𝑞+𝑝𝑟+𝑞𝑟=−3$$
$$𝑝𝑞𝑟=−1$$
Selanjutnya
$$\frac{1}{𝑝}+\frac{1}{𝑞}+\frac{1}{𝑟}=\frac{𝑞𝑟+𝑝𝑟+𝑝𝑞}{𝑝𝑞𝑟}=\frac{−3}{−1}=3$$

8. Suppose $$𝑎, 𝑏$$ and $$𝑐$$ are positive real numbers such that

$$𝑎𝑏𝑐(𝑎 + 𝑏 + 𝑐) = 100$$

Find the minimum value of

$$(𝑎 + 𝑏)(𝑏 + 𝑐)$$

(A) 10
(B) 20
(C) 40
(D) 48
(E) None of the above

Belum tersedia

Pages ( 1 of 3 ): 1 23Next »