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 2019 grade 10

1. The sum of 18 consecutive positive integers is a perfect square. Determine the smallest possible value of this sum.

$$𝑥 + (𝑥 + 1) + (𝑥 + 2) + (𝑥 + 3) + ⋯ + (𝑥 + 17) = 𝑛^2$$
$$18𝑥 + (1 + 2 + 3 + ⋯ + 17) = 𝑛^2$$
$$18𝑥 +\frac{18(17)}{2}= 𝑛^2$$
$$18𝑥 + 9(17) = 𝑛^2$$
$$9(2𝑥 + 17) = 𝑛^2$$

$$9$$ merupakan bilangan kuadrat maka $$(2𝑥 + 17)$$ juga harus merupakan bilangan kuadrat. Nilai $$x$$ yang memenuhi adalah $$4$$. Jadi bilangan terkecil pada penjumlahan di atas adalah $$4$$

2. Simplify $$\sqrt{2\left(\sqrt{1+\left(\frac{x^4-1}{2x^2})\right)^2}\right)}$$ ,where $$x$$ is any positive real number.

$$\sqrt{2\left(\sqrt{1+\left(\frac{x^4-1}{2x^2})\right)^2}\right)}$$

$$=\sqrt{2\left(\sqrt{1+\frac{x^8-2x^4+1}{4x^4}}\right)}$$

$$=\sqrt{2\left(\sqrt{\frac{4x^4}{4x^4}+\frac{x^8-2x^4+1}{4x^4}}\right)}$$

$$=\sqrt{2\left(\sqrt{\frac{x^8+2x^4+1}{4x^4}}\right)}$$

$$=\sqrt{2\left(\sqrt{\left(\frac{x^4+1}{2x^2}\right)^2}\right)}$$

$$=\sqrt{2\left(\frac{x^4+1}{2x^2}\right)}$$

$$=\frac{1}{x}\sqrt{x^4+1}$$

3. Determine the value of the positive integer x if

$$\frac{1}{\sqrt 4 +\sqrt 5}+\frac{1}{\sqrt 5 +\sqrt 6}+\frac{1}{\sqrt 6 +\sqrt 7}+…+\frac{1}{\sqrt x +\sqrt {x+1}}=10$$

$$\frac{1}{\sqrt 4 + \sqrt 5} =\frac{1}{\sqrt 4 + \sqrt 5}×\frac{\sqrt 5 – \sqrt 4}{\sqrt 5 – \sqrt 4}=\frac{\sqrt 5 – \sqrt 4}{5-4}={\sqrt 5 – \sqrt 4}$$

Dengan cara yang sama penjumlahan

$$\frac{1}{\sqrt 4 +\sqrt 5}+\frac{1}{\sqrt 5 +\sqrt 6}+\frac{1}{\sqrt 6 +\sqrt 7}+…+\frac{1}{\sqrt x +\sqrt {x+1}}=10$$

$${\sqrt 5 – \sqrt 4}+{\sqrt 6 – \sqrt 5}+{\sqrt 7 – \sqrt 6}+…+{\sqrt {x+1} – \sqrt x}=10$$
$$⇒-\sqrt 4 + \sqrt{x+1}=10$$
$$⇒-2+\sqrt{x+1}=10$$

$$⇒\sqrt{x+1}=12$$
$$⇒x+1=144$$
$$⇒x=143$$

4. Let $$ABC$$ be a triangle with $$∠A=90°$$  and $$AB = AC$$. Let $$D$$ and $$E$$ be points on the segment $$BC$$ such that $$BD : DE : EC = 3 : 5 : 4$$. Determine the angle $$∠DAE$$.

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5. What is the units digit in the answer to the sum $$3^{2025} + 9^{677}$$ ?

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6. Determine the greatest integer $$x$$ such that $$\sqrt{2007^2 – 20070 +31} ≥ x$$.

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7. Let $$x, y$$ and $$z$$ be three real numbers such that $$xy + yz + xz = 6$$ . Determine the least possible value of $$x^2 + y^2 + z^2$$.

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8. The internal bisector of angle A of triangle ABC meets BC at D. If AB=6 units, AC=9 units and BC=10 units, determine the length BD. not yet available

9. Jason spent 100 dollars to get 100 toy animals. Jason bought at least one ant, one bat and one cat, and did not buy any other toys. Given the cost of an ant, a bat and a cat are 0.5 dollar, 3 dollars and 10 dollars respectively. Determine how many of each toy did Jason buy.

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10. Terry colours all the small squares that lie on the two longest diagonals of a square grid. He colours 2021 small squares. Determine the size of the square grid.

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