SEAMO PAPER F 2019 [PROBLEM And SOLUTION]

---

11.  \(M\) and \(n\) are positive integers such that \(M = n^2 + 15n + 26\) is a perfect square.
Find the value of \(n\).
(A) 19
(B) 23
(C) 29
(D) 31
(E) None of the above

---

Solution

---

12. The right rectangular prism shown below has sides of length 3, 4, and 5. The centres of its faces are joined to form an octahedron.
Find the volume of this octahedron.

(A) 10
(B) 11
(C) 12
(D) 14
(E) None of the above

---

Solution

---

13. ABCDE is a regular pentagon. A circle can be drawn through the pentagon such that it is tangent to CD at D and to AB at A.
What is the angle formed by the minor arc AD?

(A) 108°
(B) 120°
(C) 135°
(D) 148°
(E) None of the above

---

Solution

---

14 Find the value of \(n\) given that

\((1 + 10)(1 + 10^2)(1 + 10^4)…(1+10^{2^{n}})
= 1 + 10 + 10^2 + 10^3 + ⋯+ 10^{255}\)

(A) 5
(B) 6
(C) 7
(D) 8
(E) None of the above

---

Solution

---

15. ABCD is a square with AB = m, and AEFG is a rectangle, such that E lies on side BC and D lies on side FG. Given that AE = n, what is the length of side EF?

(A) \(\frac{m^2}{n}\)
(B) \(\frac{n^2}{m}\)
(C) \(\frac{\sqrt{2}m^2}{n}\)
(D) \(\frac{\sqrt{2}n^2}{m}\)
(E) None of the above

---

Solution

---

16. Suppose a, b and c are real numbers greater than 1, evaluate

\(\frac{1}{1+\log_{a^2b}(\frac{c}{a})}+\frac{1}{1+\log_{b^2c}(\frac{a}{b})}+\frac{1}{1+\log_{c^2a}(\frac{b}{c})}\)

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

---

Solution

---

17. Find the least possible value of

\(f(x)=\frac{8}{1+cos\;x}+\frac{18}{1-cos\;x}\)

where \(x\) can be any real number for
which \(f(x)\) is defined.
(A) 8
(B) 13
(C) 18
(D) 25
(E) None of the above

---

Solution

---

18. Let \(ΔABC\) be an equilateral triangle inscribed in a circle. \(M\) is a point on the arc \(BC\) as shown below.
Given that \(k ∙ MA = MB + MC\), find the
value of \(k\).

(A) \(\frac{1}{2}\)
(B) \(\frac{2}{3}\)
(C) \(1\)
(D) \(\frac{3}{2}\)
(E) None of the above

---

Solution

---

19. Suppose \(a, b, c\) and \(d\) are real numbers such that

\(a+b+c+d=0\)

and

\(abcd(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d})=4\)

Find the value of \(a^3+b^3+c^3+d^3\).

(A) -4
(B) 8
(C) -12
(D) 16
(E) None of the above

---

Solution

---

20. Determine the number of fractions in
the pattern below that are expressed
in lowest terms.

\(\frac{1}{2019}, \frac{2}{2018}, \frac{3}{2017}, … , \frac{1010}{1010}\)

(A) 400
(B) 500
(C) 600
(D) 700
(E) None of the above

---

Solution

---

21. Simplify

\(\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+… +\frac{2019}{2017!+2018!+2019!}\)

---

Solution

---

22. Three girls \(A, B\) and \(C\), and seven boys are to be lined up in a row.
Let \(n\) be the number of ways to arrange them such that \(B\) must lie between \(A\) and \(C\) , and \(A\) and \(B\) must be separated by exactly 4 boys.
Then, determine \(⌊\frac{n}{5!}⌋\).

---

Solution

---

23. Determine the number of 3-element subsets of the set \(A=\{1, 2, 3,…, 99\}\) for which the sum of the three elements is a multiple of 3.

---

Solution

---

24. In \(ΔABC\) with sides \(a, b\) and \(c\) , it is known that \(b + c = 6\) and its area \(S = a^2 − (b − c)^2\).
Find the possible largest value of \(S\).

---

Solution

---

25. Let \(x\) be a positive real number.
Suppose

\(m=⌊log_{10}x⌋\)

and

\(n=⌊log_{10}\frac{100}{x}⌋\)

\(⌊z⌋\) denotes the largest integer less than or equal to \(z\).
Find the least possible value of \(4m^2-3n^2\).

---

Solution

---

Baca juga SEAMO PAPER C 2019 [PROBLEM And SOLUTION]

---