Problem And Solution SEAMO 2017 Paper E

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9. Simplify \(\sqrt{21+12\sqrt 3}-\sqrt{21-12\sqrt 3}\)

(A) 2
(B) 3
(C) 4
(D) 5
(E) None of the above

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Solution

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10. Find the largest number \(n\) such that there is only one integer \(k\) that satisfies

\(\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13}\)

(A) 97
(B) 112
(C) 114
(D) 198
(E) None of the above

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Solution

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11. Given that

\([m]=\frac{\sqrt 3-\sqrt 2}{\sqrt 3+\sqrt 2}\), \([n]= \frac{\sqrt 3+\sqrt 2}{\sqrt 3-\sqrt 2}\)

Find the value of \(\frac{n}{m^2} + \frac{m}{n^2}\)

(A) 940
(B) 950
(C) 960
(D) 970
(E) 980

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Solution

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12. A bus is supposed to ferry two groups of tourists to a theme park. For the first part of the journey, Group B begins the journey with walking while Group A takes the bus for the first part of the journey. It is known that both groups arrive at the theme park at the same time. It is given that the walking speed is 4 km/h. The speed of the bus is 40 km/h when it is a loaded and 50 km/h when it is empty. Suppose the ratio of the distance by walking to the distance by bus is k. Find the value of k.
(A) \(\frac{1}{5}\)
(B) \(\frac{1}{6}\)
(C) \(\frac{1}{7}\)
(D) \(\frac{1}{8}\)
(E) \(\frac{1}{9}\)

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Solution

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13. It is known that \(\log_8 a + \log_4 b^2 = 3\) and \(\log_8 b + \log_4 a^2 = 5\).

Find the value of \(ab\).

(A) 8
(B) 16
(C) 32
(D) 64
(E) 128

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Solution

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14. 4 identical circles are circumscribed by a square of area 64 cm². A smaller circle is placed in such a way it is tangential to the other 4 circles. Find the diameter of the small circle.

(A) \(2(\sqrt 2 – 1)\)
(B) \(3(\sqrt 2 – 1)\)
(C) \(4(\sqrt 2 – 1)\)
(D) \(3(\sqrt 3 – 1)\)
(E) None of the above

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Solution

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15. Find the remainder when

\(1^5 + 2^5 + 3^5 + 4^5 + … + 99^5 + 100^5\)

is divided by 4.

(A) 0
(B) 1
(C) 2
(D) 3
(E) None of these

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Solution

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16. Evaluate

\(\frac{1}{a-b} + \frac{1}{a+b} + \frac{2a}{a^2+b^2} + \frac{4a^3}{a^4+b^4}\)

(A) \(\frac{8a^7}{a^8 + b^8}\)
(B) \(\frac{8b^7}{a^8 + b^8}\)
(C) \(\frac{8a^7}{a^8 – b^8}\)
(D) \(\frac{8ab}{a^8 – b^8}\)
(E) None of the above

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Solution

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17. Let \(n_1, n_2, n_3, …, n_{2017}\) be positive integer such that

\(x=(n_1 + n_2 + … + n_{2016})(n_2 + n_3 + … + n_{2017})\)
\(y=(n_1 + n_2 + … + n_{2017})(n_2 + n_3 + … + n_{2016})\)

which of the followings is true?

(A) \(x=y\)
(B) \(x>y\)
(C) \(x<y\)
(D) \(x-n<y-n_{2017}\)
(E) None of the above

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Solution

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18. A small circle with radius \(r\) and centre \(D\) is tangential to \(3\) identical semicircles with centres \(A, B, C\) and radius \(R\). Find \(R:r\).

(A) 2 :1
(B) 3 :1
(C) 5 :2
(D) 7 :5
(E) None of the above

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Solution

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