Asian Science And Math Olympiad (ASMO) 2019 For Grade 9

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11. In the following diagram, ABCD is a square, and E is the center of the square ABCD. Q is a point on a semi-circle with diameter AD. P is a point on a semi-circle with diameter AB. Moreover, P, A and Q are collinear. If AP=46 units and AQ=14 units, then determine length of AE.

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12. Determine the largest integer \(x\) such that both \(x + 224\) and \(x + 496\) are perfect squares.

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13. Determine how many three-digit positive integers where the product of the digits is equal to 20.

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14. Let \(a\) and \(b\) be positive integers such that \(\frac{100}{151}<\frac{b}{a}<\frac{ 200}{251}\). Determine the minimum value of \(a\).

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15. Determine the number of pairs of positive integers \((x, y)\) which satisfy the equation \(2x + 3y = 2007\) .

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16. Determine the value of \(2019^2+ 2018^2 -1991^2 -1992^2\)

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17. Figure shows a semicircle is inscribed in a quarter circle. Determine the fraction of the black semicircle inside the quarter circle.

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18. A list of integers contains two square numbers, two prime numbers, and two cube numbers. Determine the smallest number of integers that could be in the list.

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19. If \(y=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4} + … +\frac{20}{1+200^2+200^4}\), determine the value of \(y×80402\)

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20. Consider two positive integers \(a\) and \(b\) such that \(a^a b^b\) is divisible by 2000. Determine the least possible value of the product \(ab\).

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21. In the sum shown, different shapes represent different digits. Determine the digit which the square, circle and triangle represent.

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22. Consider the simultaneous equations

\(ab+ac=255\)
\(ac-bc=224\)

Determine the number of ordered triples of positive integers \((a, b, c)\) that satisfy the above system of equations.

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23. A convex octagon inscribed in a circle has four consecutive sides of length 2 units and four consecutive sides of length of 2 units. Determine the area of the octagon.

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24. The 9-digit positive integer N with digit pattern ABCABCBBB is divisible by every integer from 1 to 17 inclusive. The digits A, B and C are distinct. Determine the values of A, B and C.

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25. The function \(E(n)\) is defined for each positive integer \(n\) to be the sum of the even digits of \(n\). For example, \(E(5681) = 6 + 8 = 14\). What is the value of

\(E(1) + E(2) + · · · + E(100)?\)

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Solution

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