# Problems And Solutions SEAMO PAPER F 2016

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 2016 paper E. Soal ini bersumber dari seamo-official.org

1. $$S_n$$ is the sum of a number sequence of common difference $$a_n$$. Given $$a_3 = 4$$ and $$S_3 = 9$$, the value of $$S_5 − a_5$$ is _____.

(A) 12
(B) 14
(C) 19
(D) 23
(E) 32

2. In triangle $$ABC, ∠ABC = 30°$$ and $$AB = 10$$$$AC = AD$$ can be $$5, 7, 9$$ or $$11$$. How many scalene triangles are there?

(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

3. Suppose $$y ∈ N$$ and $$x + y ≤ 5$$. How many possible groups of $$(x, y)$$ are there?

(A) 4
(B) 6
(C) 8
(D) 10
(E) 12

4. Evaluate

$$\frac{1}{(x+1)(x+3)}+\frac{1}{(x+3)(x+5)}+\frac{1}{(x+5)(x+7)}+\frac{1}{(x+7)(x+9)}$$

(A) $$\frac{2}{(x+1)(x+9)}$$
(B) $$\frac{3}{(x+1)(x+9)}$$
(C) $$\frac{4}{(x+1)(x+9)}$$
(D) $$\frac{5}{(x+1)(x+9)}$$
(E) None of the above

5. For functions $$f(x)$$ and $$g(x)$$.

$$fg(x)>gf(x)$$ is satisfied when the value of $$x$$ is _____.

(A)−1
(B) 0
(C) 1
(D) 2
(E) 3

6. Evaluate

$$\frac{(2016^2 – 2015)×2017}{2016^2 – 2016·2015+2015^2}$$

(A) 2014
(B) 2015
(C) 2016
(D) 2017
(E) None of the above

7. In the triangle given below, $$AD ⊥ AB , BC = \sqrt{3}.BD$$ and $$AD = 1$$. Find $$AB. AD$$.

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

8. Let $$(2x-3)^6=a_0+a_1(x-1)+a_2(x-1)^2+…+a_6(x-1)^6$$, find the value of $$a_4$$

(A) 220
(B) 235
(C) 240
(D) 255
(E) 270

9. If a 6-digit number, $$m8949n$$, is divisible by 11, find its largest possible value.

(A) 189493
(B) 189497
(C) 389499
(D) 989494
(E) None of the above

10. For $$f(x_0) = x_0$$, where $$x_0∈R$$ , then $$x_0$$ is called an “Excellent Point” on $$f(x)$$. Find the range of values of $$a$$, such that there exists no “Excellent Point” for $$f(x)= x^2 + 2ax + 1$$.

(A) $$(−1, 1)$$
(B) $$(− \frac{3}{2}, \frac{1}{2}$$
(C) $$(−\frac{1}{2}, \frac{3}{2})$$
(D) $$(−1, ∞)$$
(E) $$(−\frac{5}{2}, 3)$$

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