PAPER
SEAMO
Southeast Asian
M athematica I
Olympiads
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED.
STUDENT'S NAME:
Read the instructions on the ANSWER SHEET and
NAME, CANDIDATE NUMBER and SCHOOL.
Use a 28 or B pencil.
Do NOT use a pen
Rub out any mistakes completely.
fill in your
You must complete all the questions within 90 MINUTES.
SENIOR
Mark only ONE answer for each question.
Marks are NOT deducted for incorrect answers.
SECTION A
Questions
1
to 10 are worth
THREE (3) marks each.
SECTION B
Questions 11 to 20 are worth FOUR (4) marks each.
SECTION
C
Questions 21 to 25 are worth SIX (5) marks each.
You are NOT allowed to use a calculator.
QUESTTONS I TO 10 ARE WORTH
3 MARKS EACH
1.
x2
- px* q = 0 has distinctive
Let x be a number such that x * 1x
integral roots, find a + p.
Find the value of x3 *
(A) 3
(B) rs
).
(A) 16s
(B) t70
(c) 13
(D) 1eB
(E) None of the above
Evaluate
logo6 abc
1
(A)
(B)
+
(D)
(E)
1
(c)
2
(D) 1
(E) None of the above
that
2 sina
y= l-*cosa*sina'
find the value of
.
'1,-cosa*sina
L
(A)
* sina
1
v
(B) 7-v
(c) v
(D) Lty
(E)
x2-x*1, x*L
xr+Z =-rr-*+l-
logo" abc'
0
None of the above
3. Given
r that satisfies
(A) -2
(B) -1
-1
(c)
4
Find the value of
None of the above
SEAMO 2018 Paper F O TCIMO
0
and
positive
(D) t7
(E) None of the above
(c) 180
2.
If p and q are distinctive pdmes
6. Two circles with centres,4 and B are
externally tangent at point
Given that a2 + b2 +
c2
:
1, find'the
range of ab * bc * ca.
C.
(A)
11,21
(B)
[,,;
]
(c)
[-;,']
(D) [0,1]
(E) None of the above
DF is tangent to the two circles. Given
that the radii of the two circles
2 cm and 3 cm, respectively, find
(A)
,6
ffi,
n
G
N.
(A) strictly decreasing.
(B) non-strictly decreasing.
(C) monotonic decreasing.
(D) strictly increasing.
(E) None of the above
T
'16
T
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
None of the above
3 new,classrooms were opened,
the average class size reduced by
is known that f (x,) =
Then f (x") is
,6
(c)
7. When
T.
FC
2
T
(E)
It
\tr
(B)
(D)
are
maximum number of cities in Utopia?
10.
When 6 new classrooms were opened,
the average class size reduced by
Find the total number of students.
(A)
(B)
(c)
(D)
(E)
(A) e
(B) 10
16.
(c) 11
(D) 12
(E) None of the above
320
360
400
480
None of the above
2
SEAMO 2018 Paper F O TCIMO
13. Given that abcd+ 0, c and d are roots
QUESTTONS 11 TO 20 ARE WORTH
4 MARKS EACH
tt.
of x2 *cxtd=0, then
tan6a can also be expressed as
(A)
(B)
(c)
(D)
(E)
L2,
of x2 * ax * b = 0 ond a and b are roots
is
6tana -20tan3 a*6tansa
1 - 15 tan2 a t 15tana a -tan6 a
-
15 sinz a
*
L5 cosa
a
I
tan6
r:'
(A)
a2b2
B) a'c' ''
(C) Zabc
(D) Zabcd
(E) None of the above
(
6sina-20tan3 a*6cossa
1
q.
6cosa -20tan3 a*6sinsa
1 - cosz a * 1,Ssina a * tan6 a
6tana-15tan3 at6tansa
t-
20tan2 a
* 20 tana a -
tan6 a
None of the above
In the figure below, zBAD zBAA' : LDAA':60o.
90o and
14. Find the value of x that satisfies
x-2+Zlr3+
x-11.+4l1i-3=5
(A) 4
(B) s
(c)
L,'
B
Find the length of AC'.
(A) tr
(B) 'm
(C) ,lE
(D) ,lT
None of the above
SEAMO 2018 Paper F O TCIMO
6
7
(D)
(E) None of the above
iD
)-----
A
(E)
(a+btc*d)2
15.
A
biased coin is tossed 5 times. 17. Evaluate
There is a non-zero probability of
IZ23z4zn2
getting heads or tails once. The
,+ zr* zr* 24+"'*
probability of getting heads once is
the
me as the proba bility of
getting heads twice. Find the
1.
(A)
sa
,
|
(B)
probability of getting heads thrice.
3
(c)
(A)
(B)
33
64
(c)
2
23
32
m*"'
(D)
2
(E)
None of the above
40
243
(D)
47
18,
256
(E)
None of the above
It is known that the tangent from a
point m to the circle x2 + y2 - 1 is
perpendicular to the tangent, also
from the point m to xz +y2 - 3. Then
the locus of m
iS
a circle of radius
16. Find the remainder upon dividing the
following by
1010'
(A) 1
(B) 2
7.
+ 1010' +
101ot
+ ...+
(c)
L01o'o
(D) 4
(A) 4
(B) s
(c) 6,
(D)
(E)
3
(E) None of the above
I
3
None of the above
4
SEAMO 2018 Paper F O TCIMO
19. Given
(-
t
*L=rfz"
QUESTTONS 21 TO 2s ARE WORTH
6 MARKS EACH
-
zk +
t)(zn-
k+
t)
2L. A number sequence is such that
term is divisible by 3 and
ea,ch term is the difference between
Each
and
s1
\_
'n- Lk
k=L
TL
T
square number and 1, as shown below
then
3, 15,
24,49,63,
...
Find the remainder when the ZOtgtr
term is divided by 1000.
(A) Sn ) Tn
(B) S?.2T7.
(C) sn:ZTn
(D) Sn: Tn
(E)
a
22. Given that x, and x, are real roots of
2x2-4mx*2m2+3m-Z -0,
None of the above
find the value of m so that the value
of xl + x| is minimum.
20.
A circle of radius
is inscribed in
triangle ABC such that it is tangent to
side .48 at point P. It is known AP - 23. Find the value of n in
L9 , PB - 2L, find the perimeter of
133s + l-1os + 84s + 2Ts = nz
L7
AABC.
24.
Suppose
a2 + b2
where
:20L9c2,
at b and c are sides of
triangle add cr, B, y or€ its angles.
(A)
(B)
(c)
(D)
(E)
183
LB4
185
186
None of the above
SEAMO 2018 Paper F O TCIMO
Find the value
,tol/ . -.
ofcotq.+cot
p
25.
ln LABC , it is known that ,48 - 12,
BC = 13 and AC:15. P is a point on
.4C such that the incircles of LABP
and LBCP have equal radii.
Let m and
n be positive relative
prime integers such
that #:+.
Find m + n.
End of Paper
SEAMO 2018 Paper F O TCIMO
PAPER
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SEN IOR
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Candidate No. ..,........
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School
Date
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Questions 1 to 20
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7
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12
13
14
15
16
17
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tr
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ABCDE:
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ABCDE
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ABCDE
:
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- ABCDE
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ABCDE:
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ABCDE
ABCDE
ABCDE
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:
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ABCDE
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U
ABCDE
Question
21
E
Use a 28 or B PENCIL.
Rub out COMPLETELY any
answer you wish to change
usrng an eraser.
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:
INSTRUCTIONS
:
Question 22
:
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:
:
:
:
Questions 1 to 20
Mark only ONE letter for each
question.
Question 23
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For. example, if you think "A" is the best
answer to the question, mark your
ansvJer sheet like this:
Question 24
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Question 25
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Questions 21 to 25
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right answer to the question.
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SEAMO Paper F 2018 O TCIMO
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