# Problems And Solutions Future Intelligence Student Olympiad (11-12 Grades)

## “Future Intelligence Students Olympiad”

During this global pandemic periods we witness, students’ shift to the online learning system. A Mathematics, English and Science competition for students of grades 1 – 12, with six levels. In addition, Art competition for students of grades 1 – 12, with two levels. The key competence tested by FISO online is logical combination, not just pure knowledge of formulas.Aiming to encourage students to study and critical thinking.  FISO Olympiad team decided to conduct an online Olympiad of FISO Mathematics, English, Science, Chemistry, Biology, Physics and Art Olympiad to boost the motivation of both teachers and students who have been bored, lost goal-oriented character and stimulus for study under this difficult global pandemic times, when most of the Olympiads and competitions are forced to be cancelled or postponed until unknown time. FISO Olympiad team also hopes this competition will help all types of schools to keep their students engaged and to observe their Mathematics, English, Science and Art education in an international arena. (sc : www.fisoolympiad.com)

Berikut ini Soal dan Solusi contoh soal lomba FISO level Grade 11 and 12

1. $$365m72$$ is a six-digit number where $$m$$ is a digit. If $$365m72$$ is divisible by $$9$$, find the sum of all possible values of $$m$$?

A) 1
B) 2
C) 3
D) 4

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2. Find $$x$$ in the following equation.

$$\frac{2x}{3+\frac{4}{3+\frac{4}{3+\frac{4}{…}}}}+\frac{3x}{2+\frac{3}{2+\frac{3}{2+\frac{3}{…}}}} = 3$$

A) 1
B) 2
C) 3
D) 4

$$3+\frac{4}{3+\frac{4}{3+\frac{4}{…}}}=𝑎 ⇒ 3 +\frac{4}{𝑎}= 𝑎 ⇒ 𝑎 = 4$$

$$2+\frac{3}{2+\frac{3}{2+\frac{3}{…}}}=b ⇒ 2 +\frac{3}{b}= b ⇒ b = 3$$

Diperoleh

$$\frac{2𝑥}{4}+\frac{3𝑥}{3}= 3 ⇒\frac{3}{2}𝑥 = 3 ⟹ 𝑥 = 2$$

3. An electrical cut 53-meter-long piece of wire into three pieces, such that the longest pieces is four times as long as the shortest piece and the middle-sized piece is three meters shorter than twice the length of the shortest piece. Find the length of longest piece?

A) 8
B) 13
C) 22
D) 32

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4. Two pipes can fill a pool in six hours. The larger pipe can fill the pool twice as fast as the smaller one. How long does it take the smaller pipe to fill the pool alone?

A) 3
B) 6
C) 9
D) 18

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5. Solve

$$\left(\frac{1-\frac{1}{a^2}}{\frac{1}{a}+1}\right)\left(\frac{a^2}{1-a}\right)=?$$

A) $$a + 1$$
B) $$– a$$
C) $$a$$
D) $$– a – 1$$

$$\left(\frac{1-\frac{1}{a^2}}{\frac{1}{a}+1}\right)\left(\frac{a^2}{1-a}\right)$$

$$=\left(\frac{\frac{a^2-1}{a^2}}{\frac{1+a}{a}}\right)\left(\frac{a^2}{1-a}\right)$$

$$=\left(\frac{a^2-1}{a^2}\right)(\frac{a}{a+1})\left(\frac{a^2}{1-a}\right)$$

$$=\frac{(a+1)(a-1)}{a^2}(\frac{a}{a+1})\left(\frac{a^2}{1-a}\right)=-a$$

6. 6. A café offers chocolate, lemon, sour cherry and vanilla flavors of ice cream. A costumer can choose one, two or three scoops but the flavors must all be different. How many different possible ice creams can a customer order?

A) 4
B) 6
C) 10
D) 14

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7. Simplify the expression

$$\frac{a^3 – b^3}{a^2b+ab^2+b^3}·\frac{2b^2+2ab}{a^2-b^2}=?$$

A) 1
B) 2
C) $$\frac{2}{ab}$$
D) $$\frac{2(a+b)}{ab}$$

$$\frac{a^3 – b^3}{a^2b+ab^2+b^3}·\frac{2b^2+2ab}{a^2-b^2}$$

$$=\frac{(a-b)(a^2+ab+b^2)}{b(a^2+ab+b^2}·\frac{2b(b+a}{(a-b)(a+b)}=2$$

8. Simplify

$$\frac{\log_x 8}{\log_y 3}·\frac{\log_y(\frac{1}{9})}{\log_x(\frac{1}{2})}=?$$

A) 6
B) 3
C) -4
D) $$\frac{\log_x 4}{\log_y 3}$$

$$\frac{\log_x 8}{\log_y 3}·\frac{\log_y 3^{-2}}{\log_x 2^{-1}}$$

$$=\frac{-2\log_x 8 × \log_y 3}{-1\log_y 3 × \log_x 2}$$

$$=2\log_2 8 × \log_3 3=2×3×1=6$$

9. In a polygon $$ABCDE, EK$$ and $$BK$$ are the bisectors of angles $$E$$ and $$B$$ respectively. Find the measure of angle $$EKB$$?

A) 125
B) 120
C) 110
D) 100

$$70° + 120° + 130° + 2(𝑎 + 𝑏) = 540°$$
$$320° + 2(𝑎 + 𝑏) = 540°$$
$$2(𝑎 + 𝑏) = 220°$$
$$𝑎 + 𝑏 = 110°$$
Dari segiempat $$EKAB$$
$$130° + 𝑎 + 𝑏 + ∠𝐸𝐾𝐵 = 360°$$
$$130° + 110° + ∠𝐸𝐾𝐵 = 360°$$
$$∠𝐸𝐾𝐵 = 360° − 240° = 120°$$

10. In the figure, $$ABCD$$ is a square. Find the measure of angle $$DCF$$?

A) 20
B) 30
C) 45
D) 55

11. O is the center of circle

A) 20
B) 40
C) 60
D) 75

jadi jumlah $$∠CEB + ∠CAB = 20°+20°=40°$$

12. In accordance with the relationship. Find the number which corresponds to the place indicated by the question mark?

A) 156
B) 438
C) 1032
D) 2811

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13. Find the number, which corresponds to the place indicated by question mark.

A) 14
B) 17
C) 20
D) 23

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14. Find the number, which corresponds to the place indicated by question mark.

A) 84
B) 58
C) 40
D) 34

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15. Simplify trigonometric expression.

$$\frac{(\sin x + \cos x)^2 – 1}{(\sin x – \cos x)^2 – 1}$$

A) – 1
B) 0
C) 1
D) sinx + cosx

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16. Simplify $$\frac{4·\cos 50°·\sin 50°·\cos 100°}{\sin 200°}$$

A) 1
B) $$\frac{\sqrt 3}{16} \cot 10°$$
C) $$\cot 10°$$
D) -1

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17. Solve

$$f(x)=5^{2x-1}$$
$$fοf^{-1}(x)+f^{-1}οf(x)=?$$

A) $$x$$
B) $$5^x$$
C) $$2x$$
D) $$1$$

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18. With the given information, find $$k$$?

$$x+4=y$$
$$y+5=z$$
$$z+x=2k$$
$$x+y+z=21$$

A) -6
B) $$\frac{43}{6}$$
C) $$7$$
D) $$\frac{51}{7}$$

$$𝑥 + 𝑦 + 𝑧 + 𝑥 + 9 = 𝑦 + 𝑧 + 2𝑘$$
$$2𝑥 + 9 = 2𝑘$$
$$𝑥 =\frac{2𝑘 − 9}{2}$$
Selanjutnya ubah $$y$$ dan $$z$$ dalam $$k$$
$$𝑧 = 𝑦 + 5 = 𝑥 + 4 + 5 = 𝑥 + 9 =\frac{2𝑘 − 9}{2}+ 9 =\frac{2𝑘 + 9}{2}$$
$$𝑦 = 𝑥 + 4 =\frac{2𝑘 − 9}{2}+ 4 =\frac{2𝑘 − 1}{2}$$
Subtitusi nilai $$x, y$$ dan $$z$$ ke $$𝑥 + 𝑦 + 𝑧 = 21$$

19. Find the quotient

$$\frac{x^2+7x+12}{x^2-9}÷\frac{x^2+5x+4}{x^2-3x}$$

A) 1
B) $$x$$
C) $$\frac{x}{x+1}$$
D) $$\frac{x}{x-1}$$

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20. Solve

$$\frac{(x^2-y^2)(\sqrt[3]{x}+\sqrt[3]{y})}{\sqrt[3]{x^5}+\sqrt[3]{x^3y^3}-\sqrt[3]{x^3y^2}-\sqrt[3]{y^5}}-(\sqrt[3]{xy}+\sqrt[3]{y^2})=?$$

A) 1
B) $$x^2$$
C) $$\sqrt[3]{x^2}$$
D) $$x^2y$$

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21. Simplify trigonometric expression.

$$\frac{1+\sin x}{\cos x}-\frac{\cos x}{1-\sin x}$$

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22. Find the quotient

$$\frac{x^2 + 49x – 56}{x^2-4x-5}÷\frac{7x^2-56x}{4x^3-20x^2}$$

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23. If $$x=\frac{\sqrt[4]{125}}{5}$$, then find

$$(x^4-7x^2+1)^{-2}·\left[(x^2+\frac{1}{x^2})^2-14·(x+\frac{1}{x})^2+77\right]$$

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24. Find sum of all results of $$x$$

$$2x^2 + (2\sqrt 3 – 9)x – (2\sqrt 3 – 12x + 6\sqrt 3)=0$$

$$2x^2 + (2\sqrt 3 – 9)x – (2\sqrt 3 – 12x + 6\sqrt 3)=0$$
$$⇒2𝑥^2 + (2\sqrt 3 − 9 + 12)𝑥 − (2\sqrt 3 + 6\sqrt 3) = 0$$
$$⇒2𝑥^2 + (2\sqrt 3 +3)𝑥 − (8\sqrt 3) = 0$$

Misalkan nilai $$x$$ yang memenuhi $$a$$ dan $$b$$, berdasarkan rumus vieta Maka nilai dari $$𝑎 + 𝑏 =\frac{−(2√3+3)}{2}$$

25. Find $$x+y=?$$

$$(x+y)^x = (x-y)^y$$
$$\log_2 x – \log_2 y = 1$$

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