SEAMO 2018 Paper F PDF

 

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

fill in your

Rub out any mistakes completely.

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 * 1

integral roots, find a + p.

Find the value of x3 *

(A) 3 r (B) s

x

(A)

).

(c) 13

16s

(B) t70

(c)

Evaluate logo6 abc 1

(A) (B)

+

0

2

None of the above

3. Given

that

y=

2 sina

l-*cosa*sina'

find the value of

.

'1,-cosa*sina L

(A)

* sina

1 v

(B) 7-v

(c)

v

(D) Lty (E)

x*L x2-x*1, xr+Z =-rr-*+l-

logo abc'

(c)

1

(D) (E)

r that satisfies

(A) -2 (B) -1

-1

(c)

4

Find the value of

None of the above

SEAMO 2018 Paper F O TCIMO

(D) (E)

0 1

None of the above

and

positive

(D) t7 (E) None of the above

180

(D) 1eB (E) None of the above 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) (B)

(c)

11,21

[,,;

]

[-;,']

(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) (B)

(c) (D) (E)

are

It

T. FC

is known that f (x,) =

Then f (x ) is

ffi,

n

G

N.

(A) strictly decreasing. (B) non-strictly decreasing.

\tr 2

,6 T

(C) monotonic decreasing. (D) strictly increasing. (E) None of the above

,6

T

'16 T

A network of domestic flights is set up in the country of Utopia. A city is

None of the above

directly connected to at most 3 others

and there is at most 1 transit flight from one city to another. What is the

3 new,classrooms were opened, the average class size reduced by 10.

7. When

maximum number of cities in Utopia?

When 6 new classrooms were opened,

the average class size reduced by Find the total number of students.

(A) (B) (c)

320 360 400

(D) (E)

480 None of the above

(A) e (B) 10

16.

(c)

11

(D) 12 (E) 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)

L2,

of x2 * ax * b = 0 ond a and b are roots

*

I

(c)

6cosa -20tan3 a*6sinsa 1 - cosz a * 1,Ssina a * tan6 a

-

15 sinz a

L5 cosa

a

tan6

r:

(A) (

6sina-20tan3 a*6cossa 1

q.

a2b2

B) a c

(C) (D)

Zabc

(E)

None of the above

Zabcd

6tana-15tan3 at6tansa

(D)

t-

(E)

None of the above

20tan2 a

*

20 tana a

In the figure below, zBAD zBAA

is

6tana -20tan3 a*6tansa 1 - 15 tan2 a t 15tana a -tan6 a

(B)

-

tan6 a

90o and

14. Find the value of x that satisfies

: LDAA :60o.

x-2+Zlr3+ (A) (B)

(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

4 s 6

x-11.+4l1i-3=5

(D) 7 (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 ,+ zr* zr* 24+ '*

getting heads or tails once. The probability of getting heads once is

the sa me as the proba bility of getting heads twice. Find the

(A)

23

(B)

33

(D) (E)

, |

(c)

(A)

(c)

1.

(B)

probability of getting heads thrice.

32

64

m* '

3

2

(D)

2

(E)

None of the above

40 243 47

18,

256

None of the above

It is known that the tangent from a point m to the circle x2 + y2 - 1 is pointto m to xz +y2

also the tangent, 3. Then

perpendicular

from the

the locus of m

iS

-

a circle of radius

16. Find the remainder upon dividing the

following by 1010'

+ 1010' +

101ot

+ ...+

(A) 4 (B) s (c) 6,

(D) (E)

(A) 1 (B) 2

7.

L01o'o

(c)

3

(D)

4

(E)

None of the above

I

3

None of the above

SEAMO 2018 Paper F O TCIMO

 

19. Given

(-

t

L=rfz L=rfz

-

zk +

t)(zn-

and

s1 T \_ 'n- Lk k=L TL

k+

t)

QUESTTONS 21 TO 2s ARE WORTH 6 MARKS EACH

2L. A number sequence is such that Each

term is divisible by 3 and

ea,ch

term is the difference between

a

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)

22. Given that x, and x, are real roots of

x2-4mx*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.

, it is known

ln

LABC

BC

= 13 and AC:15. P

.4C

such that the incircles of

that

,48

-

12,

is a point on 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

F

SEN IOR Candidate Name

Country

Candidate No. ..,........

Level ........

School

Date

Free Response

Questions 1 to 20

ABCDE

15

ABCDE: -l -t ABCDE : : -i ABCDE : E - ABCDE : : -i ABCDE -ABCDE - : ABCDE ABCDE ABCDE : : E ABCDE ABCDE ABCDE ABCDE ABCDE

16

U

3

4 5 o 7 B

tr tr tr 12 13 14

17

@

tr

E

Question -l :

:

21

E :

-l

:

:

:

:

Question 22

Question 23

INSTRUCTIONS Use a 28 or B PENCIL. Rub out COMPLETELY any answer you wish to change usrng an eraser.

Questions 1 to 20 Mark only ONE letter for each question. Colour in the box f ully. example, if you think "A" is the best answer to the question, mark your ansvJer sheet like this: For.

-I

-I

Question 24

Question 25

I

Ar

BCDE

Questions 21 to 25

Fill in the boxes with the right

answer to the question. OR

ABCDE ABCDE

Fill in the diagrams with the right answer to the question.

ABCDE B A CDE ABCDE

Diagrams are only provided where applicable.

Do not vnte rnside this box For EXAMINER use only

ANSWER SHEET

SEAMO Paper F 2018 O TCIMO

 

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