Asian Science And Math Olympiad (ASMO) 2019 For Grade 10

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11. Let \(a\) and \(b\) be the roots of \(x^2 + x\sin α +1=0\) while \(c\) and \(d\) are the roots of the equation. \(x^2 + x\cos α – 1 = 0\) Determine the value of \(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}\).

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Solution

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12. Nick selects an integer, multiplies it by 4 then subtracts 30. He then multiplies his answer by 2 and finally subtracts 10. His answer is a two-digit number. Determine the largest integer he could select.

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Solution

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13. Determine the value of the expression

\(10\sqrt{10\sqrt{10\sqrt{10\sqrt{…}}}}\).

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Solution

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14. Consider three-digit integers N with the two properties:
(a) no digit of N is exactly divisible by 2, 3 or 5.
(b) N is not exactly divisible by 2, 3 or 5.
Determine how many such integers N are there.

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Solution

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15. When a conical bottle rests on its flat base, the water in the bottle is 8 cm from its vertex. When the same conical flask bottle is turned upside down, the water level is 2 cm from its base. Determine the height of the bottle. Express the answer in surd form.

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Solution

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16. Alicia and Brendon can complete a task in 3 hours. Brendon and Cathy can complete a task in 6 hours. Cathy and Alicia can complete a task in 4 hours. Determine how long will the task be completed if Alicia, Brendon and Cathy work together. Assume each person works at a constant rate, whether working alone or working with others.

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Solution

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17. Mary starts in the small square shown shaded on the grid, and makes a sequence of moves. Each move is to a neighbouring small square. Two small squares are neighbouring if they have an edge in common. She may visit a square more than once. Mary makes four moves. In how many different small squares could Mary finish?

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Solution

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18. In the diagram, O is the center of the square, OA=OC=2, AB=CD=4, CD is perpendicular to OC, which is perpendicular to OA, which in turn is perpendicular to AB. The square has area 64 cm².

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(a) Find the area of trapezoid ABCO.
(b) Find the area of quadrilateral BCDE.

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Solution

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19. Determine all the integers \(a, b, c, d\) satisfying the following two relations:
(i) \(ab+cd = a + b + c + d + 3.\)
(ii) \(1≤ a ≤ b ≤ c ≤ d\);

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Solution

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20. Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order onto a spreadsheet, which recalculated the class average after each score was entered, Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71, 76, 80, 82, and 91. What was the fourth and last score Mrs. Walter entered?

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Solution

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21. Determine all the possible solutions so that the equation \(m^4+8n^2 + 425 = n^4 + 42m^2\) is valid provided that the \(m\) and \(n\) are integers.

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Solution

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22. Based on the diagram below, find the length \(AB\).

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Solution

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23. Find the smallest positive integer \(n\) such that \(\frac{n}{2}\) is a perfect square, \(\frac{n}{3}\) is a perfect cube, and \(\frac{n}{5}\) is a perfect fifth power.

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24. Determine all positive integers \(m, n,\) and primes \(p ≥ 5\) such that \(m(4m^2 + m + 12)=3(p^n – 1)\)

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Solution

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25. Let us consider a trapezoid with sides of lengths \(3, 3, 3, k\), with positive integer \(k\). Determine the maximum area of such a trapezoid.

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Solution

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