# Asian Science And Math Olympiad (ASMO) 2018 For Grade 11

Asian Science and Maths Olympiad (ASMO) is a competition platform designed to challenge and evaluate student’s knowledge in Mathematics and Science at their grade level. The questions in the Olympiad will stretch their knowledge and understanding of the concepts. Our syllabus fits nicely into the syllabus that concentrates on non-routine problem-solution to prepare the students for the competition. With the expansion of STEM education worldwide, ASMO certainly answers the need of it. Students will be well prepared with the skills to meet the science and technology challenges.

In Malaysia, ASMO is officially endorsed by Ministry of Education and all participants will obtain curriculum marks. In 2018 alone, Asian Science and Mathematics Olympiad has received 70,000 entries from across the ASEAN countries. We are targeting for the number to increase at 80,000 for 2019.

We are also proud to present that ASMO International is a new effort by ASMO Malaysia which started in 2017 in Pattaya, Thailand. When it was initially launched, the competition was setup via collaboration with ASMOPSS and ASMO Thai was the host for the competition. In 2018, Malaysia has become the host for the competition and it was participated by 10 Asian countries.

The idea of opening up a new competition platform which is ASMO International is to expand the level of competition and to provide more opportunities for primary and secondary school students to experience international engagement. (sc : http://asmo2u.com/about-us)

Berikut ini problems and solution ASMO 2018 grade 11

1.If matriks $$A=\begin{pmatrix} -3 & -5\\ 2 & 4\\ \end{pmatrix}$$, matriks $$A^2 – A = k\begin{pmatrix} 1 & 0\\ 0 & 1\\ \end{pmatrix}$$, what is the value of $$k$$

$$A^2=\begin{pmatrix} -3 & -5\\ 2 & 4\\ \end{pmatrix}$$$$\begin{pmatrix} -3 & -5\\ 2 & 4\\ \end{pmatrix}$$$$=\begin{pmatrix} 9-10 & 15-20\\ -6+8 & -10+16\\ \end{pmatrix}$$$$=\begin{pmatrix} -1 & -5\\ 2 & 6\\ \end{pmatrix}$$

$$A^2-A=\begin{pmatrix} -1 & -5\\ 2 & 6\\ \end{pmatrix}$$$$-\begin{pmatrix} -3 & -5\\ 2 & 4\\ \end{pmatrix}$$$$=\begin{pmatrix} 2 & 0\\ 0 & 2\\ \end{pmatrix}$$$$=2\begin{pmatrix} 1 & 0\\ 0 & 1\\ \end{pmatrix}$$

Jadi nilai $$k$$ yang memenuhi adalah 2

2. A series $$a_1, a_2, a_3, a_4, …,$$ and has characteristic $$a_n · a_{n-2}=(a_{n-1})^2,$$, if $$a_1 + a_3 =10, a_4 + a_6=\frac{5}{4}$$, find the value of $$a_1 + a_2 + a_3 + a_4 + a_5$$

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3. If prime number $$x$$ and $$y$$ meet the requirement of $$3x+5y =21$$, then $$x^3+x^2y+xy^2+y^3 =?$$

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4. Vector $$\vec{p}, \vec{q}$$ and $$\vec{r}$$ has a relationship $$\vec{p}+\vec{q}+\vec{r}=0$$. If lenght of vector $$|\vec{p}|=1, |\vec{q}|=2$$ and $$|\vec{r}|=\sqrt 3$$, what is the angle between vector $$\vec{p}$$ and $$\vec{q}$$?

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5. A company purchase several ream of A4 papers at different month of a year with different unit cost and different amount of money, as shown in following:

$$\begin{matrix} Month & Cost\; per\; ream & Amount\; Spent\\ January & RM10 & RM60\\ May & RM11 & RM55\\ September & RM 9 & RM81\\ \end{matrix}$$

What is the average cost for every ream of A4 paper purchased by this company?

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6. $$\log_{10}(10^7+10^9+10^{10})-\log_{10}3-\log_{10}367=$$

$$\log_{10}(10^7 + 10^9 + 10^{10}) − \log_{10} 3 − \log_{10}{367}$$
$$= \log_{10}(10^7(1 + 100 + 1000) − \log_{10} 3 − \log_{10}{367}$$
$$= \log_{10}\left(\frac{10^7(1 + 100 + 1000)}{3 × 367}\right)$$
$$= \log_{10}\left(\frac{(107(1101)}{1101}\right)$$
$$= \log_{10} 10^7 = 7$$

7. Calculate the value of $$\frac{d}{dx}\int_{1}^{x} \log_{10}xdx$$ when $$x=100$$

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8. Given that $$\tan α$$ and $$\tan β$$ are the roots for quadratic equation $$x^2+5x+3=0,$$ calculate the value of $$\tan(α+β)$$.

Berdasarkan rumus dalil vieta

$$\tan 𝛼 + \tan 𝛽 = −5$$
$$\tan 𝛼 \tan 𝛽 = 3$$

$$\tan(𝛼 + 𝛽) =\frac{\tan 𝛼+\tan 𝛽}{1−\tan 𝛼 \tan 𝛽}=\frac{−5}{1−3}=\frac{−5}{−2}=\frac{5}{2}$$

9. $$\int_{0}^{100} \sqrt{x} dx + \int_{0}^{10} x^2 dx =$$

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10. For trigonometric relation
$$\sin(α+β)\sin(α-β)=K(\sinα + \sinβ)(\sinα-sinβ)$$, what is the value $$K$$

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11. If $$α$$ and $$β$$ are the solutions for equation $$x^2 +x -3=0$$, what is the value of $$α^3 -4β^2?$$

$$𝛼 + 𝛽 = −1$$ dan $$𝛼𝛽 = −3$$
$$𝛼^2 + 𝛼 − 3 = 0 ⇒ 𝛼^3 + 𝛼^2 − 3𝛼 = 0 ⇒ 𝛼^3 = −𝛼^2 + 3𝛼$$
$$𝛽^2 + 𝛽 − 3 = 0 ⇒ 𝛽^2 = −𝛽 + 3$$
Selanjutnya
\begin{align} 𝛼^3 − 4𝛽^2 &= −𝛼^2 + 3𝛼 − 4(−𝛽 + 3)\\ &= −(3 − 𝛼) + 3𝛼 + 4𝛽 − 12\\ &= −3 + 𝛼 + 3𝛼 + 4𝛽 − 12\\ &= −3 + 4(𝛼 + 𝛽) − 12\\ &= −3 + 4(−1) − 12\\ &= −3 − 4 − 12\\ &= −19\\ \end{align}

12. $$(x^2 +\frac{1}{2}x+\frac{1}{2})^{10}=a_0 + a_1x+a_2x^2+a_3x^3+…+a_{20}x^{20}$$
Calculate of $$a_0 + a_1 +2a_2 + 3a_3 + … +20a_{20}$$

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13. $$\frac{(1×2+2×2^2+3×2^3+…+2018×2^{2018})+(2+2^2+2^3+…+2^{2018})}{2^{2018}}=?$$

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14. $$\int_{-10}^{10}\sqrt{100-x^2}dx + \int_{0}^{20}(10-\sqrt{100-(x-10)^2)}dx=…$$

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15. If 3 integers are chosen from integer series 1, 2, 3, … ,48,49, and 50, how many possible combination are there if the sum of 3 integers is multiple of 3?

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16. Given that $$p=\log_{10}a, q=\log_{10}b$$ and $$r=\log_{10}c$$, what is the value of $$a^{q-r}b^{r-p}c^{p-q}$$

$$𝑝 = \log_{10} 𝑎 ⇒ 10^𝑝 = 𝑎$$
$$𝑞 = \log_{10} 𝑏 ⇒ 10^𝑞 = 𝑏$$
$$𝑟 = \log_{10} 𝑐 ⇒ 10^𝑟 = 𝑐$$
Yang dicari
$$𝑎^{𝑞−𝑟}𝑏^{𝑟−𝑝}𝑐^{𝑝−𝑞}$$
$$=\frac{𝑎^𝑞}{𝑎^𝑟} ·\frac{𝑏^𝑟}{𝑏^𝑝} ·\frac{𝑐^𝑝}{𝑐^𝑞}$$
$$=\frac{10^{𝑝𝑞}}{10^{𝑝𝑟}} ·\frac{10^{𝑞𝑟}}{10^{𝑞𝑝}} ·\frac{10^{𝑟𝑝}}{10^{𝑟𝑞}} = 1$$

17. Two dice are thrown, what is the probability that the sum of two numbers appear is multiple of 4 OR multiple of 6?

Jumlah kedua dadu kelipatan $$4 \{(1, 3), (3, 1), (2, 2), (2, 6), (6, 2), (3, 5), (5, 3), (4, 4), (6, 6)\}$$ ada $$9$$.
Jumlah kedua dadau kelipatan $$6\{(3, 3), (2, 4), (4, 2), (1, 5), (5, 1)\}$$ ada $$5$$, untuk $$(6, 6)$$ sudah terhitung di kelipatan 4.
Jadi peluangnya adalah $$\frac{14}{36}=\frac{7}{18}$$

18. Function $$f(x)$$ is defined as $$f(x)=x^3+ax^2+bx+c$$, where $$a, b$$ and $$c$$ are constant. If $$f(1)=1$$ and $$f(2)=2$$, what is the value of $$f(8)- f(-5)$$.

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19. If function $$f(x)=(2\sqrt x – 1)^3$$,  evaluate the value of

$$\lim_{x→1}\frac{xf(1)-f(x)}{x-1}$$

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20. How many positive integer solution set(s) of $$x$$ and $$y$$ are there for the following equation?

$$\begin{cases} x^{x+y}&=y^{12}\\ y^{x+y}&=x^3\\ \end{cases}$$

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21. Evaluate the value of $$\cos^4 20 + \cos^4 40 + \cos^4 60 + \cos^4 80$$

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22. The tangent equation of function $$f(x)=ax^3 + bx^2 + cx + d$$ when $$x = 1$$ is $$y=x-5$$. At the same time, $$f(x)$$ fulfill $$\lim_{x→2}\frac{f(x)}{x-2}=9$$. What is the value of $$f(-10)$$?

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23. Given that the common tangent line for curve $$y =x^2$$ and $$x=y^2$$ is $$x+y=-\frac{1}{4}$$ . What is the area enclosed by the two given curves and the common tangent line?

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24. As shown in the following diagram is a right angle triangle $$ΔABC$$, with side $$AC$$ equally divided by point $$D$$ and point $$E$$ into $$3$$ equal segment. If $$∠CAD = x, ∠DAE = y,$$ and $$∠EAB = z$$ , what is the value of $$\frac{\sin y}{\sin x·\sin z}?$$

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25. As shown in the following diagram is a cube $$ABCD-EFGH$$ with length of edge equal to $$2cm$$. If the angle between surface $$ΔAGC$$ and surface $$ΔBDH$$ is $$α$$, what is the value $$\cos α$$?

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