Problems And Solutions SEAMO PAPER F 2021 - BorneoMath

6. How many distinct ways are there to arrange 4 girls and 10 boys to dance in a circle such that there are at least 2 boys in between any two adjacent girls?
(A) \(\frac{1 0! × 5!}{2!}\)
(B) \(\frac{1 0! × 4!}{2!}\)
(C) \(\frac{9 ! × 5!}{2!}\)
(D) \(\frac{9 ! × 4!}{2!}\)
(E) None of the above

Banyak cara menyusun melingkar \(4\) orang perempuan adalah \((4-1)!=3!\)

Diantara 2 perempuan akan disisipkan minimal 2 laki-laki, karna ada 4 tempat kita misalkan \(a, b, c\) dan \(d\) dimana \( a+b+c+d=10, a,b,c,d>1\)

misalkan \(x+1=a, y+1=b, w+1=c\) dan \(z+1=b\) maka diperoleh persamaan:\((x+1)+(y+1)+(w+1)+(z+1)=10, x,y,w,z\) bilangan asli
\(x+y+w+z=6\)

Dengan menggunakan teorema starbars, banyak solusi \((x,y,w,z)\) adalah

\({6-1\choose {4-1}}={5\choose 3}=\frac{5!}{3!·2!}\)

Banyak cara menyusun \(10\) laki-laki adalah \(10!\)

Jadi banya cara susunan adalah \(10!×3!×\frac{5!}{3!·2!}=\frac{10!×5!}{2!}\)

7. Suppose \(𝑎 , 𝑏\) and \(𝑐\) are positive real numbers satisfying

\(𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 + 𝑎^2 = 45\)
\(𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 + 𝑏^2 = 50\)
\(𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 + 𝑐^2 = 90\)

Evaluate \(𝑎 + 𝑏 + 𝑐\).
(A) 8
(B) 9
(C) 10
(D) 11
(E) None of the above

\((𝑏 + 𝑎)(𝑎 + 𝑐) = 45\)
\((𝑎 + 𝑏)(𝑏 + 𝑐) = 50\)
\((𝑏 + 𝑐)(𝑎 + 𝑐) = 90\)
Nilai-nilai yang memenuhi \(𝑎 + 𝑏 = 5, 𝑎 + 𝑐 = 9\) dan \(𝑏 + 𝑐 = 10\)
Jumlahkan ketiga persamaan diperoleh \(2(𝑎 + 𝑏 + 𝑐) = 24 ⟹ 𝑎 + 𝑏 + 𝑐 = 12\)

8. Given that

\((cos 27° + cos 99° + cos 117° + cos 189°)^2 =\frac{𝑚}{𝑛}\)

where \(𝑚\) and \(𝑛\) are relatively prime.
Evaluate \(𝑚 + 𝑛\).
(A) 3
(B) 4
(C) 7
(D) 9
(E) None of the above

9. How many positive integers \(𝑛\) satisfy the condition:
\(1 + 2 + ⋯ + 𝑛\) is a factor of \(6𝑛\)?
(A) 3
(B) 4
(C) 5
(D) 6
(E) None of the above

\(\frac{6𝑛}{1 + 2 + 3 + ⋯ + 𝑛}= 𝑘, 𝑘 ∈ 𝐵\)
\(\frac{6𝑛}{\frac{𝑛(𝑛 + 1)}{2}}= 𝑘\)
\(\frac{12𝑛}{𝑛(𝑛 + 1)}= 𝑘\)
\(\frac{12}{(𝑛 + 1)}= 𝑘\)
Diperoleh \(𝑛 + 1 = \{2, 3, 4, 6, 12\}\) jadi banyaknya ada \(5\) bilangan

10. In trapezium \(𝐴𝐵𝐶𝐷 , 𝐴𝐵\) is parallel to \(𝐶𝐷, 𝐴𝐵 = 36, 𝐵𝐶 = 15\) and \(𝐷𝐴 = 12. 𝑃\) is a point on \(𝐴𝐵\) such that a circle with centre \(𝑃\) is tangent to both \(𝐵𝐶\) and \(𝐴𝐷\).
Evaluate \(𝐴𝑃^2 + 𝐵𝑃^2\).

(A) 656
(B) 689
(C) 697
(D) 765
(E) None of the above

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