Asian Science And Math Olympiad (ASMO) 2019 For Grade 11

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11. There are 88 numbers \(x_1, x_2 , x_3,…, x_{88}\) and each of them is either equal to -1 or -3. Given
that \(x_1^2+x_2^2+x_3^2+…+x_{88}^2=280\), determine the value of \(x_1^4+x_2^4+x_3^4+…+x_{88}^4\)

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Solution

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12. Solve the value of \(x\) and \(y\) by considering the following equations, \((x+y)(x^2-y^2)=4\) and \((x-y)(x^2+y^2)=\frac{5}{2}\)

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Solution

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13. We are given a common external tangent \(t\) to circles  \(c_1 (O_1 , r_1 )\) and  \(c_2 (O_2 , r_2 )\) , which have no common point and lie in the same half-plane defined by \(t\). Let d be the distance between the tangent points of circles \(c_1\) and \(c_2\) with \(t\). Determine the smallest possible length of a broken line \(AXB\) (i.e., the union of line segments \(AX\) and \(XB\)), such that \(A\) lies on \(c_1 , B\) lies on \(c_2\) , and \(X\) lies on \(t\).

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Solution

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14. Determine in how many ways one can assign numbers of the set \(\{1, 2, …, 8\}\) to the vertices of a cube \(ABCDEFGH\), such that the sum of any two numbers at vertices with a common edge is always an odd number.

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15. Determine the largest integer for which each pair of consecutive digits is a square.

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16. Suppose the quadratic equation \(x^2 – 2007x + b = 0\) with \(a\) real parameter \(b\) has two positive integer roots. Find the maximum value of \(b\).

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Solution

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17. The points \(A, B\) and \(C\) are the centres of three circles. Each circle touches the other two circles, and each centre lies outside the other two circles. The sides of the triangle \(ABC\) have lengths 15 units, 18 units and 24 units. Determine the radius of the three circles.

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Solution

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18. Let \(x\) and \(y\) be positive real numbers. Determine the smallest possible value of
\(xy+\frac{108}{x}+\frac{16}{y}\)

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19. How many 4-digit integers can be formed with the set of digits \(\{0, 1, 2, 3, 4, 5\}\) such that no digit is repeated and the resulting integer is a multiple of 3?

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20. Determine all the real numbers x for which:

\((x^2 – 7x + 11)^{x^2-13x+42}=1\)

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21. Solve for the radius \(r\) in terms of \(a, b, c\).

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22. Determine all possible triples of these integers if these three positive integers have sum 25 and product 360.

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23. Determine the number of nine-digit integers of the form \(‘pqrpqrpqr’\) which are multiples of 24, provided that \(p, q\) and \(r\) do need not to be different.

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24. Let \(ABC\) be a triangle with \(AB=AC\). The angle bisectors of \(A\) and \(B\) meet the sides \(BC\) and \(AC\) in \(D\) and \(E\), respectively. Let \(K\) be the in center of triangle \(ADC\). Suppose that \(∠BEK = 45°\).
Determine all possible values of \(∠BAC\) .

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Solution

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25. Determine the total number of 6-digit numbers which satisfy the following two conditions:

(i) the digits of each number are all from the set \(\{1, 2, 3, 4, 5\}\)
(ii) any digit that appears in the number appears at least twice.

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Solution

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