Problems And Solutions SEAMO PAPER D 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 D. Soal ini bersumber dari seamo-official.org

1. Which of the following numbers is the largest?
(A) $$1$$
(B) $$\sqrt 2$$
(C) $$\sqrt [3]3$$
(D) $$\sqrt [4]4$$
(E) $$\sqrt [6]6$$

Belum tersedia

2. If $$|𝑥 − 10| + 𝑥 − 10 = 0$$ , what is the range of $$𝑥$$?
(A) $$𝑥 < 10$$
(B) $$𝑥 > 10$$
(C) $$𝑥 ≤ 10$$
(D) $$𝑥 ≥ 10$$
(E) None of the above

Belum tersedia

3. How many triangles are there in the figure below?

(A) 12
(B) 16
(C) 18
(D) 20
(E) None of the above

Belum tersedia

4. Using the digits 1,2,3 and 4, without repeat, one can form distinct 3-digit
numbers. What is the average of all possible numbers formed?
(A) 200
(B) 225.5
(C) 275
(D) 277.5
(E) None of the above

Banyak bilangan $$3$$ digit yang dapat dibentuk dari angka-angka $$1, 2, 3$$ dan $$4$$ adalah $$4×3×2=24$$ bilangan.

Jumlah semua bilangan $$3$$ angka yang terbentuk adalah
$$6×100(1+2+3+4)+6×10(1+2+3+4)+6×1(1+2+3+4)=6000+600+60=6660$$ .

Rata-ratanya adalah $$\frac{6660}{24}=277,5$$

5. Find the sum of the coefficients in the expansion of

$$(𝑥^2 + 2𝑥 + 3)^3(3𝑥^2 + 2𝑥 + 1)$$

(A) 1140
(B) 1200
(C) 1254
(D) 1296
(E) None of the above

Misalkan $$𝑃(𝑥)=(𝑥^2+ 2𝑥 + 3)^3(3𝑥^2 + 2𝑥 + 1)$$
$$𝑃(1)$$ hasilnya merupakan jumlah semua koefisien dari $$𝑃(𝑥)$$

$$𝑃(1)=(1+2+3)^3(3+2+1)=6^4=1296$$
Jadi jumlah koefisien dari $$𝑃(𝑥)$$ adalah $$1296$$

6. How many fractions are there from

$$\frac{1}{99}, \frac{2}{99}, \frac{3}{99}, … ,\frac{98}{99}$$

that cannot be reduced to a simpler
form?
(A) 45
(B) 60
(C) 75
(D) 90
(E) None of the above

Banyaknya sama dengan mencari nilai euler dari $$99$$ yaitu
$$𝜑(99)=99(1−\frac{1}{3})(1−\frac{1}{11})=99(\frac{2}{3})(\frac{10}{11})=60$$

7. What is the area of the square 𝐴𝐵𝐶𝐷 below?

(A) 64
(B) 72
(C) 84
(D) 100
(E) None of the above

$$𝐵𝐷 =\sqrt{𝐷𝐸^2+𝐵𝐸^2}=\sqrt{(5+4+3)^2+(2+2)^2}=\sqrt{144+16}=\sqrt 160$$

$$[𝐴𝐵𝐶𝐷]=\frac{𝐵𝐷.𝐴𝐶}{2}=\frac{\sqrt 160.\sqrt 160}{2}=\frac{160}{2}=80$$

8. A teacher wrote three numbers on the whiteboard: 4875, 4563 and 𝑁 , where 𝑁 is a positive integer. He then asked the class to compute the lowest common multiple of the three numbers. One student misread 4875 as 4275 . However, the student still gets the correct answer. What is the least possible value of 𝑁?
(A) 1755
(B) 2375
(C) 2535
(D) 3705
(E) None of the above

$$4875 = 3 × 5^3 × 13$$
$$4563 = 3^3 × 13^2$$
$$𝐾𝑃𝐾(4875,4563) = 3^3 × 5^3 ×13^2$$

$$4563 = 3^3 × 13^2$$
$$4275 = 3^2 × 5^2 × 19$$
$$𝐾𝑃𝐾(4275,4563)= 3^3 × 5^2 × 13^2 × 19$$

Terdapat nilai $$N$$ sehingga $$𝐾𝑃𝐾(4875,4563, 𝑁) = 𝐾𝑃𝐾(4275,4563, 𝑁) = 3^3 × 5^3 × 13^2 × 19$$, nilai terkecil $$N$$ yang memenuhi adalah  $$19 × 5^3 = 2375$$

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