Problem And Solution SEAMO 2017 Paper F

The Southeast Asia Mathematical Olympiad (SEAMO) is an international Math Olympiad competition that originated in Singapore and was founded by Mr. Terry Chew in 2016 in 8 Southeast Asian Countries. Since then, it is growing its popularity around the world. In 2019 it was recognized by 18 countries. In 2020 total number of participating countries increased to 22, including students from Indonesia, Brazil, China, Newzealand, and Taiwan students enrolled in SEAMO 2022-23.

Problem and Solution SEAMO 2017 paper E. Soal ini bersumber dari seamo-official.org

1. The radii of circles with centres \(O_1\) and \(O_2\) are 7 and 14 respectively. A circle with centre \(O_3\) circumscribes the 2 circles. Find the radius of the circle with centre \(O_4\).

(A) 3
(B) 4
(C) 5
(D) 6
(E) None of the above

2. Find the value of

\(\frac{1}{4033}+\frac{2×2016}{2016^2+2017^2}+\frac{4×2016^3}{2016^4+2017^4}-\frac{8×2016^7}{2016^4+2017^4}\).

(A) \(1\)
(B) \(\frac{2017^{16}-2016^{16}}{16}\)
(C) \(\frac{1}{16}\)
(D) \(-\frac{1}{4033}\)
(E) \(\frac{2017^{8}-2016^{8}}{16}\)

3. When \(a\) is divided by \(5\), the remainder is \(1\). When \(b\) is divided by \(5\), the remainder is \(4\). If \(3a>b\), what is the remainder when \(3a-b\) is divided by \(5\)?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

4. If \((x − 1)(x − 4)(x + 3)(x − 8) + P\) is a perfect square, then \(P\) is
(A) 24
(B) 32
(C) 98
(D) 196
(E) 256

5. Given that \((x_1,y_1)\) and \((x_2,y_2)\) are the solution to the equations

\(\log_5 x+\log_{27} y=4\)
\(\log_x 5+\log_y {27}=1\)

find \(\log_{15}(x_1x_2y_1y_2)\)

(A) 1
(B) 2
(C) \(2+\log_{15} 16\)
(D) \(2+\log_{15} 32\)
(E) 6

6. Find the minimum value of

\(4(a^2+b^2+c^2) – (a+b+c)^2\)

where \(a, b\) and \(c\) are unique integers.
(A) 1
(B) 3
(C) 8
(D) 11
(E) 20

7. Given that \(a>0\) and \(b>0\), find the smallest possible value of

\(a^2 \sec^2 θ + b^2 cosec^2 θ\)

(A) \(a+b\)
(B) \((a+b)^2\)
(C) \(a^2+b^2\)
(D) \(ab\)
(E) None of the above

8. Let \(y = (15 − x)(17 − x)(15 + x)(17 + x)\) , where \(x\) is areal number.
Find the minimum value of \(y\)
(A) −1016
(B) −1018
(C) −1020
(D) −1022
(E) −1024