TERRY CHEW INSTITUTE OF MATHEMATICAL COLYMPIADS ân TCIÀACO A2 ) 3 S EA M O DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED Southeast Asian STUDENT''''S NAME Mathematical Olympiads m Read the instructions on th[.]
Trang 21 Let ! be a number such that ! + #
$ = 6
Find the value of !'+ #
$ ( (A) 165
(B) 170
(C) 180
(D) 198
(E) None of the above
4 If ) and * are distinctive primes and
!+− )! + * = 0 has distinctive positive integral roots, find + /
(A) 3 (B) 5 (C) 13 (D) 17 (E) None of the above
5 Find the value of ! that satisfies
!+− ! + 1
!++ 2 +
! + 1
!+− ! + 1= 1 (A) -2
(B) -1 (C) 0 (D) 1 (E) None of the above
2 Evaluate #
23456789 + #
2346:789 + #
2345:789 (A) -1
(B) 0
(C) 1
(D) 2
(E) None of the above
3 Given that
; = 2 sin
1 + cos + sin ;
find the value of
1 − cos + sin
1 + sin
(A) 1
;
(B) 1 − ;
(C) ;
(D) 1 + ;
(E) None of the above
QUESTIONS 1 TO 10 ARE WORTH
3 MARKS EACH
Trang 36 Two circles with centres D and E are
externally tangent at point F
GH is tangent to the two circles Given
that the radii of the two circles are
2 cm and 3 cm, respectively, find IJ
KJ
(A) √3
2
(B) √53
(C) √5
2
(D) √6
3
(E) None of the above
8 Given that O+ + P++ Q+ = 1, find the range of OP + PQ + QO
(A) [1, 2]
(B) U0,1
2 V (C) U−1
2, 1V (D) [0, 1]
(E) None of the above
9 It is known that W(!Y) = Y[#
+Y[#, \ ∈ ^
Then W(!Y) is
(A) strictly decreasing
(B) non-strictly decreasing
(C) monotonic decreasing
(D) strictly increasing
(E) None of the above
10 A network of domestic flights is set up
in the country of Utopia A city is directly connected to at most 3 others and there is at most 1 transit flight from one city to another What is the maximum number of cities in Utopia?
(A) 9 (B) 10 (C) 11 (D) 12 (E) None of the above
7 When 3 new classrooms were opened,
the average class size reduced by 10
When 6 new classrooms were opened,
the average class size reduced by 16
Find the total number of students
(A) 320
(B) 360
(C) 400
(D) 480
(E) None of the above
Trang 411 tan 6 can also be expressed as
(A) 1 − 15 tan6 tan − 20 tan+ + 15 tan' + 6 tanb − tana.c.
(B) 1 − 15 sin6 sin − 20 tan+ + 15 cos' + 6 cosb + tana.c.
(C) 1 − cos6 cos − 20 tan+ + 15 sin' + 6 sinb + tanac..
(D) 1 − 20 tan6 tan − 15 tan+ + 20 tan' + 6 tanb − tana.c.
(E) None of the above
13 Given that OPQd ≠ 0, Q and d are roots
of !++ O! + P = 0 and O and P are roots
of !++ Q! + d = 0, then (O + P + Q + d)+
is
(A) O+P+ (B) O+Q+ (C) 2OPQ (D) 2OPQd (E) None of the above
12 In the figure below, ∠EDG = 90° and
∠EDDi = ∠GDDi = 60°
Find the length of DFi
(A) √15
(B) √17
(C) √23
(D) √31
(E) None of the above
14 Find the value of ! that satisfies
k! − 2 + 2√! − 3 + k! + 1 + 4√! − 3 = 5
(A) 4 (B) 5 (C) 6 (D) 7 (E) None of the above
QUESTIONS 11 TO 20 ARE WORTH
4 MARKS EACH
Trang 515 A biased coin is tossed 5 times
There is a non-zero probability of
getting heads or tails once The
probability of getting heads once is
the same as the probability of
getting heads twice Find the
probability of getting heads thrice
(A) 23
32
(B) 33
64
(C) 40
243
(D) 47
256
(E) None of the above
17 Evaluate
1
2+
2+
2++3+
2' +4+
2b+ ⋯ +\+
2Y+ ⋯
(A) 12 (B) 1
(C) 3
2 (D) 2 (E) None of the above
18 It is known that the tangent from a point n to the circle !++ ;+ = 1 is perpendicular to the tangent, also from the point n to !++ ;+ = 3 Then the locus of n is a circle of radius
(A) 1 (B) 2 (C) 3 (D) 4 (E) None of the above
16 Find the remainder upon dividing the
following by 7
10#o p
+ 10#o q
+ 10#o (
+ ⋯ + 10#o pr
(A) 4
(B) 5
(C) 6
(D) 3
(E) None of the above
Trang 619 Given
sY = t(2\ − 2u + 1)(2\ − u + 1)u
Y
vw#
and
xY = t1
u Y
vw#
then
(A) sY > xY
(B) sY ≥ xY
(C) sY = 2xY
(D) sY = xY
(E) None of the above
21 A number sequence is such that Each term is divisible by 3 and each term is the difference between a square number and 1, as shown below
3, 15, 24, 48, 63, …
Find the remainder when the 2018}~ term is divided by 1000
22 Given that !# and !+ are real roots of
2!+− 4n! + 2n++ 3n − 2 = 0,
find the value of n so that the value
of !#++ !++ is minimum.
20 A circle of radius 17 is inscribed in
triangle DEF such that it is tangent to
side DE at point It is known D =
19 , E = 21 , find the perimeter of
∆DEF
(A) 183
(B) 184
(C) 185
(D) 186
(E) None of the above
23 Find the value of \ in
133a+ 110a+ 84a+ 27a = \+
24 Suppose
O++ P+ = 2019Q+,
where a, b and c are sides of a triangle and a, b, g are its angles Find the value of Å3} Ç
Å3} É[Å3} Ñ
QUESTIONS 21 TO 25 ARE WORTH
6 MARKS EACH
Trang 725 In ∆DEF , it is known that DE = 12 ,
EF = 13 and DF = 15 is a point on
DF such that the incircles of ∆DE
and ∆EF have equal radii
Let n and \ be positive relative
prime integers such that ÖÜ
JÜ = á
Y
Find n + \
End of Paper
Trang 8SEAMO 2018
Paper F – Answers Multiple-Choice Questions
Questions 1 to 10 carry 3 marks each
Questions 11 to 20 carry 4 marks each
Free-Response Questions
Questions 21 to 25 carry 6 marks each