Problems And Solutions SEAMO PAPER D 2020 - BorneoMath

9. Evaluate \(\sqrt{31\times 30\times 29\times 28 + 1}\)
(A) 800
(B) 810
(C) 812
(D) 869
(E) None of the above

Belum tersedia

10. Find the total number of integers in the sequence 20, 22, 24, … , 2020 that are divisible by 3 but not 5.
(A) 158
(B) 225
(C) 252
(D) 270
(E) None of the above

Bilangan yang habis dibagi \(3 : \{24, 30, 36, …, 2016\}\) banyaknya ada
\(\frac{2016−24}{6}+ 1 = 333\)
Bilangan yang habis dibagi \(15 : \{30, 60, 90, ….2010\}\) bnyaknya ada
\(\frac{2010−30}{30}+ 1 = 67\)
Jadi banyak bilangan yang habis dibagi \(3\) tetapi tidak habis dibagi \(5\) adalah \(333 – 67 = 266\) bilangan

11. \(𝑎_1, 𝑎_2, 𝑎_3, …\) is an arithmetic series with common difference \(1\).
For \(𝑎_1 + 𝑎_2 + 𝑎_3 + ⋯ + 𝑎_{98} = 137\), find the
value of \(𝑎_2 + 𝑎_4 + 𝑎_6 + ⋯ + 𝑎_{98}\).
(A) 89
(B) 91
(C) 93
(D) 97
(E) None of the above

Misalkan
\(𝑎_1 = 𝑎\)
\(𝑎_1 + 𝑎_2 + 𝑎_3 + ⋯ + 𝑎_{98} = 137\)
\(⇒𝑎 + (𝑎 + 1) + (𝑎 + 2) + ⋯ + (𝑎 + 97) = 137\)
\(⇒\frac{(𝑎 + 𝑎 + 97)98}{2}= 137\)
\(⇒(2𝑎 + 97)49 = 137\)
\(⇒98𝑎 = 137 − 97(49)\)
Selanjutnya
\(𝑎_2 + 𝑎_4 + 𝑎_6 + ⋯ + 𝑎_{98} = (𝑎 + 1) + (𝑎 + 3) + (𝑎 + 5) + ⋯ + (𝑎 + 97)\)
\(=\frac{(𝑎+1+𝑎+97)49}{2}\)
\(=\frac{(2𝑎+98)49}{2}\)
\(=\frac{98𝑎+98(49))}{2}\)
\(=\frac{137-97(49)+98(49)}{2}=\frac{137+49}{2}=\frac{186}{2}=93\)

12. Let 𝑎 and 𝑏 be positive real numbers such that
\(\frac{1}{𝑎}+\frac{1}{𝑏}=\frac{5}{𝑎 − 𝑏}\)
Evaluate
\((\frac{𝑎}{𝑏}+\frac{b}{a})^2\)
(A) \(29\)
(B) \(16\sqrt 5\)
(C) \(25\sqrt 2\)
(D) \(36\)
(E) None of the above

\(\frac{1}{𝑎}+\frac{1}{𝑏}=\frac{5}{𝑎 − 𝑏}\)
\(⇒\frac{𝑎 + 𝑏}{𝑎𝑏}=\frac{5}{𝑎 − 𝑏}\)
\(⇒\frac{𝑎^2 − 𝑏^2}{𝑎𝑏}= 5\)
\(⇒ \frac{𝑎}{𝑏} – \frac{𝑏}{𝑎}= 5\)
\(⇒ (\frac{𝑎}{𝑏}−\frac{𝑏}{𝑎})^2=\frac{𝑎^2}{𝑏^2} +\frac{𝑏^2}{𝑎^2} − 2 = 25\)
\(⇒ \frac{𝑎^2}{𝑏^2} +\frac{𝑏^2}{𝑎^2} = 27\)
Selanjutnya
\((\frac{𝑎}{𝑏}+\frac{𝑏}{𝑎})^2=\frac{𝑎^2}{𝑏^2} +\frac{𝑏^2}{𝑎^2} + 2 = 27 + 2 = 29\)

13. Evaluate

\(\sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + 2\sqrt 1+ ⋯}}}\)

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

\(\sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + 2\sqrt 1+ ⋯}}}=a, a>0\)
\(⇒\sqrt{1 + 2𝑎} = 𝑎\)
\(⇒1 + 2𝑎 = 𝑎^2\)
\(⇒𝑎^2 − 2𝑎 = 1\)
\(⇒𝑎^2 − 2𝑎 + 1 = 1 + 1\)
\(⇒(𝑎 − 1)^2 = 2\)
\(⇒𝑎 − 1 = \sqrt 2\)
\(⇒𝑎 = \sqrt 2 + 1\)

14. How many integers between 0 and \(10^4\) are there that have their sum of digits equal to 5?
(A) 52
(B) 54
(C) 56
(D) 58
(E) None of the above

Belum tersedia

15. Two circles with centers \(𝑀\) and \(𝑁\) are tangent to a line \(ℓ\) at \(𝐴\) and \(𝐵\) ,
respectively, as shown in the figure below.
Given that \(𝐴𝐵 = 25 , 𝑀𝑁 = 24\) and quadrilateral \(𝐴𝐵𝑁𝑀\) has an area of \(300\), evaluate the product of the radii of the two circles.

(A) 120
(B) 144
(C) 150
(D) 176
(E) None of the above

Belum tersedia

16. Using the digits 1,2,3,4 and 5, without repeat, one can form distinct 5-digit numbers and arrange them in ascending order. What is the \(100^{th}\) number?
(A) 45132
(B) 51243
(C) 52314
(D) 54123
(E) None of the above

Belum tersedia

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