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

1. Find the last two digits of $$11^{2020}$$.
(A) 01
(B) 41
(C) 71
(D) 91
(E) None of the above

$$11^1 \;mod\; 100 ≡ 11$$
$$11^2 \;mod\; 100 ≡ 21$$
$$11^3 \;mod\; 100 ≡ 31$$
$$11^4 \;mod\; 100 ≡ 41$$
$$11^5 \;mod\; 100 ≡ 51$$

$$11^9 \;mod\; 100 ≡ 91$$
$$11^{10} \;mod\; 100 ≡ 01$$
Jadi
$$11^{2020} \;mod\; 100 ≡ (11^{10})^{202} \;mod\; 100 ≡ 1^{202} \;mod\; 100 ≡ 01$$

2. The quadratic equation

$$𝑥^2 − 52𝑥 + 𝑘 = 0$$

has roots that are prime numbers. Find the maximum value of $$𝑘$$.
(A) 520
(B) 576
(C) 640
(D) 667
(E) None of the above

Misalkan akar-akar persamaan adalah $$p$$ dan $$q$$, keduanya bilangan prima,dengan menggunakan dalil vieta

$$𝑝 + 𝑞 = 52$$
$$𝑝𝑞 = 𝑘$$

Nilai maksimum $$pq$$ dicapai ketika selisih kedua bilangan minimum, maka nilai $$p$$ dan $$q$$
yang memenuhi adalah $$23$$ dan $$29$$. Nilai maksimum $$pq$$ adalah $$667$$

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) 1
(B) 2
(C) 3
(D) 4
(E) None of the above

Belum tersedia

4. Find the sum of all possible positive integers $$𝑛$$ such that the expression below is an integer.

$$\frac{4𝑛^3 − 16𝑛^2 + 29𝑛 + 60}{2𝑛 − 3}$$

(A) 42
(B) 69
(C) 75
(D) 81
(E) None of the above

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5. Evaluate the sum

$$𝑆 =\frac{1}{1 + 1^2 + 1^4} +\frac{2}{1 + 2^2 + 2^4} +\frac{3}{1 + 3^2 + 3^4}+ ⋯ +\frac{20}{1 + 20^2 + 20^4}$$

(A) $$\frac{1}{2}$$
(B) $$\frac{210}{421}$$
(C) $$\frac{105}{211}$$
(D) $$1$$
(E) None of the above

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6. 2 white, 3 black and 4 grey marbles are shared equally among 9 students. 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) 120
(B) 130
(C) 140
(D) 150
(E) None of the above

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7. Given that 𝑎 is a real number such that $$𝑎^4 + 𝑎^3 + 𝑎^2 + 1 = 0$$.
Evaluate $$𝑎^{2020} + 2𝑎^{2010} + 3𝑎^{2000}$$.
(A) 2
(B) 4
(C) 6
(D) 8
(E) None of the above

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8. Given that 𝑎. 𝑏 and 𝑐 are three distinct real numbers such that

$$𝑎+\frac{1}{𝑏}=𝑏+\frac{1}{c}=c+\frac{1}{a}$$

What is the largest possible value of 𝑎𝑏𝑐?
(A) $$\frac{1}{2}$$
(B) $$2$$
(C) $$\frac{5}{2}$$
(D) $$3$$
(E) None of the above

Belum tersedia

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