Problems And Solutions SEAMO PAPER E 2016

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13. Find the remainder when \(55^{2015} + 17\) is divided by \(8\).

(A) 0
(B) 1
(C) 3
(D) 5
(E) 7

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14. The sum of a positive integer and 100 is a square number. The sum of same positive integer and 168 is also a square number. Find the integer.
(A) 100
(B) 121
(C) 144
(D) 156
(E) None of the above

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15. The area of rectangle ABCD is is 35 cm² . Points E and F divide side AB into 3 equal segments. G is the midpoint of side CD. Find the area of green triangle GHI in cm².

(A) 1.5
(B) 2
(C) 2.5
(D) 3
(E) None of the above

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16. Given that \(m+\frac{1}{m}=4\), find the value of

\(m^4+\frac{1}{m^4}\)

(A) 169
(B) 181
(C) 194
(D) 196
(E) None of the above

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17.  Find the integer part of

\(\frac{1}{\frac{1}{2010}+\frac{1}{2011}+\frac{1}{2012}+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}+\frac{1}{2016}}\)

(A) 270
(B) 271
(C) 287
(D) 288
(E) None of the above

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18. An integer is chosen from the set \(\{1, 2, 3, … , 499, 500\}\). The probability that this integer is divisible by 9 or 11 is \(\frac{m}{n}\), in its lowest terms. Find the value of \(m + n\).

(A) 110
(B) 111
(C) 117
(D) 118
(E) 119

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19. It is given \(x^2+2x=3\), find the value of

\(x^4+7x^3+8x^2-13x+12\)

(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

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20. The circumference of a circular lake is 30 km. Paul and Mary cycles around the lake in opposite directions, both starting from the same point. Paul’s speed is 250 m/min, but he rests for 5 minutes after every 55 minutes of cycling. Mary’s speed is 200 m/min, but she rests for 10 minutes after every 1h of cycling. Find the time taken, in hours, for them to meet for the first time.

(A)\(1\frac{7}{12}\)
(B)\(1\frac{1}{28}\)
(C)\(2\frac{7}{12}\)
(D)\(2\frac{1}{14}\)
(E)\(2\frac{3}{28}\)

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21. Fill in each circle with one number from 1 to 10 such that the sum of any five adjacent numbers is the minimum.

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22. Suppose \(m\) and \(n\) are two positive integers such that \(m > n\) . Given \(m + n = 28\) and \(m² + n² = 400\), find the value of \(m² – n²\).

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23. Evaluate the following

\(\left(\frac{1}{2}+\frac{1}{3}+…+\frac{1}{30}\right)+\left(\frac{2}{3}+\frac{2}{4}+…+\frac{2}{30}\right)+\left(\frac{3}{4}+\frac{3}{5}+…+\frac{3}{30}\right)+…+\left(\frac{28}{29}+\frac{28}{30}\right)+\left(\frac{29}{30}\right)\)

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24. The radius of the 2 identical circles is 3 cm. They are inscribed in a larger circle of radius 9 cm, as shown in the figure below. Find the area of blue shaded region, in terms of π.

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25. How many ways are there to colour a \(10×10\) grid using two colours, such that each \(2×2\) grid contains exactly 2 squares of each colour?

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