Problems And Solutions SEAMO PAPER D 2020

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17. Find the number of positive integers 𝑛 less than or equal to 2020 such that

\(⌊\frac{𝑛}{2}⌋ + ⌊\frac{n}{3}⌋ + ⌊\frac{𝑛}{6}⌋ = 𝑛\)

⌊𝑥⌋ denotes the greatest integer less than or equal to \(𝑥\).
(A) 240
(B) 312
(C) 330
(D) 336
(E) None of the above

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Solution

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18. A circle is inscribed in a triangle of side lengths 3,4 and 5.
What is the radius of the circle?

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

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Solution

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19. For \(1 ≤ 𝑥 ≤ 10\), let

\(𝑓(𝑥) = |𝑥 − 2| − |𝑥 − 3| + |2𝑥 − 12|\)

Evaluate the sum of the minimum and the maximum values of \(𝑓(𝑥)\).
(A) \(\frac{15}{2}\)
(B) \(8\)
(C) \(\frac{19}{2}\)
(D) \(10\)
(E) None of the above

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Solution

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20. How many positive integers 𝑛 less than 100 are there such that the expression below is a fraction in its lowest terms?

\(\frac{𝑛^2 + 4𝑛 + 6}{𝑛 + 3}\)

(A) 32
(B) 51
(C) 66
(D) 90
(E) None of the above

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Solution

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21. \(𝐴𝐵𝐶\) is a right-angled triangle where \(∠𝐴𝐶𝐵 = 90°\). A circle with radius \(𝑟\) is inscribed inside \(Δ𝐴𝐵𝐶\).
Given that \(𝐴𝐵 = 27\) and the area of \(Δ𝐴𝐵𝐶\) is \(90\), find the area of the shaded region.
Leave your answer is terms of 𝜋.

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Solution

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22. Suppose \(𝑎 , 𝑏\) and \(𝑐\) are prime numbers such that their product is \(31\) times their sum. Evaluate \((𝑎 + 1)(𝑏 + 1)(𝑐 + 1)\).

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Solution

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23. Let \(𝑓(𝑥)\) be a polynomial. When divided by \(𝑥 + 2\) and \(𝑥 + 3\) ,
\(𝑓(𝑥)\) gives a remainder of \(3\) and \(2\), respectively.
What is the remainder when \(𝑓(𝑥)\) is divided by \((𝑥 + 2)(𝑥 + 3)\)?

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Solution

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24. In triangle \(Δ𝐴𝐵𝐶\) , the perpendicular bisector of 𝐵𝐶 intersects the segment \(𝐴𝐶\) and extension of \(𝐵𝐴\) at \(𝑀\) and \(𝑁\) , respectively.
Given that the circumscribed circle of \(Δ𝐴𝐵𝐶\) has a radius of \(6\) and centre \(𝑂\),
find the value of \(𝑂𝑀 . 𝑂𝑁\).

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Solution

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25. Find the number of ordered pairs of integers \((𝑥, 𝑦)\) such that

\(𝑥 + 𝑦 + 𝑥𝑦 = 𝑥^2 + 𝑦^2\)

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Solution

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SEAMO PAPER D 2019 [PROBLEM And SOLUTION]