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2009 中學高級卷 英文試題(2009 Senior English Paper)

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(2)

A

u s t r A l i A n

M

At h e M At i c s

c

o M p e t i t i o n

a n

a c t i v i t y

o f

t h e

a u s t r a l i a n

m a t h e m a t i c s

t r u s t

t h u r s d ay

6 a u g u s t

2 0 0 9

senior Division Competition paper

instruCtions anD information

GeneraL

1. Do not open the booklet until told to do so by your teacher.

2. NO calculators, slide rules, log tables, maths stencils, mobile phones or other calculating aids are

permitted. Scribbling paper, graph paper, ruler and compasses are permitted, but are not essential.

3. Diagrams are NOT drawn to scale. They are intended only as aids.

4. There are 25 multiple-choice questions, each with 5 possible answers given and 5 questions that

require a whole number answer between 0 and 999. The questions generally get harder as you

work through the paper. There is no penalty for an incorrect response.

5. This is a competition not a test; do not expect to answer all questions. You are only competing

against your own year in your own State or Region so different years doing the same paper

are not compared.

6. Read the instructions on the

answer sheet carefully. Ensure your name, school name and school

year are filled in. It is your responsibility that the Answer Sheet is correctly coded.

7. When your teacher gives the signal, begin working on the problems.

tHe ansWer sHeet

1. Use only lead pencil.

2. Record your answers on the reverse of the Answer Sheet (not on the question paper) by FULLY

colouring the circle matching your answer.

3. Your Answer Sheet will be read by a machine. The machine will see all markings even if they are

in the wrong places, so please be careful not to doodle or write anything extra on the Answer

Sheet. If you want to change an answer or remove any marks, use a plastic eraser and be sure to

remove all marks and smudges.

inteGritY of tHe Competition

The AMC reserves the right to re-examine students before deciding whether to grant official status

to their score.

a u s t r a l i a n s c h o o l y e a r s 1 1 a n d 1 2

t i m e a l l o w e d : 7 5 m i n u t e s

(3)

Senior Division

Questions 1 to 10, 3 marks each

1.

The value of (2009 + 9)

− (2009 − 9) is

(A) 4000

(B) 2018

(C) 3982

(D) 0

(E) 18

2.

In the diagram, x equals

(A) 140

(B) 122

(C) 80

(D) 90

(E) 98

... ...... ... ...... ...... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

x

140

122

3.

The graph of y = kx passes through the point (

−2, −1). The value of k is

(A) 2

(B)

−2

(C) 4

(D)

1

2

(E)

1

2

4.

The value of (0.6)

−2

is

(A)

−0.36

(B) 0.036

(C)

9

25

(D)

25

9

(E) 3.6

5.

(x

− y) − 2(y − z) + 3(z − x) equals

(A)

−2x − 3y + 5z

(B)

−2x − 3y − z

(C) 4x + y

− z

(D) 4x + 3y

− z

(E) 2x + 3y

− 5z

6.

On a string of beads, the largest bead is in the centre and the smallest beads are

on the ends. The size of the beads increases from the ends to the centre as shown

in the diagram.

... ... ... ... ... ... ... ... .... .... .... ... .... .... ... ... .... ... ... ... ... ... ... ... .... .... .... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... .... .... .... ... ... ... ... ... ... ... .... ... ... .... .... .... ... ... ... ... ...... ... ... ... ... ... ... ... .... .... .... ... .... .... ... ... ... ... ... ... ... .... ... .... .... ... ... ... ... ... ... ... ... ... ... ... ... .... .... ... ... .... ... ... ... ... ... .... ... .... ... ... ... ... ...... ...... ... ... ... ... ... .... .... ... .... ... ... ... ... ... .... ... ... .... ... ... ... ... ...... ... ... ... ... ... .... .... .... ... ... ... ... ... .... .... ... ... ... ... ... ... ... ... ... .... .... .... ... ... ... ... ... .... .... ... ... ... ... ... ... ... .... .... ... ... ... ... .... ... ... ...... ...... .... .... ... .... ... ... ... .... .... ... ...... ... .... ... ... ... .... ... ...... ... .... ... ... ... .... ... ...... ... ...

The smallest beads cost $1 each, the next smallest beads cost $2 each, the next

smallest $3 each, and so on. How much change from $200 would there be for the

beads on a string with 25 such beads?

(4)

S 2

7.

If a

∗ b = a +

1

b

for every pair a, b of positive numbers, the value of 1

∗ (2 ∗ 3) is

(A)

10

3

(B)

10

7

(C)

11

6

(D)

9

2

(E)

3

10

8.

The graph of y = ax

2

+ bx + c is shown,

with its vertex on the y-axis. Which of the

following statements must be true?

(A) a + b + c = 0

(B) a + b

− c < 0

(C)

−a + b − c > 0

(D) a + b + c < 0

(E) there is not enough information

......

... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ....

x

y

9.

In a school of 1000 students, 570 are girls. One-quarter of the students travel to

school by bus and 313 boys do not go by bus. How many girls travel to school by

bus?

(A) 7

(B) 63

(C) 153

(D) 180

(E) 133

10.

A box in the dressing shed of a sporting team contains 6 green and 3 red caps. The

probability that the first 2 caps taken at random from the box will be the same

colour is

(A)

1

2

(B)

5

12

(C)

2

3

(D)

3

4

(E)

2

9

Questions 11 to 20, 4 marks each

11.

QRST is a square with T at (1, 0) and

S at (2, 0). Which of the following is an

equation of the line through the origin

which bisects the area of the square?

(A) y =

1

2

x

(B) y =

1

3

x

(C) y =

2

3

x

(D) y = 2x

(E) y = 3x

... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ...

x

y

O

Q

R

S

T

12.

A rectangle P QRS has P Q = 2x cm

and P S = x cm. The diagonals P R and

QS meet at T . X lies on RS so that

QX divides the pentagon P QRST into

two sections of equal area. The length,

in centimetres, of RX is

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .. . .. .. .. .. . .. .. .. .. .. . .. .. .. . .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. . . .. .. . .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. . .. .. . .. .. .. .. .. . .. . . .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. . .. . . .. .. .. .. . .. .. .. .. . .. .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . . .. . .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. . .. . .. .. .. .. .. . . . .. .. .. . .. .. .. .. .. . .. .. .. .. .. . . . .. .. .. . .. .. .. . .. . .. .. .. .. .. . .. .. .. . .. .. .. . .. .. .. .. .. .. . .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. . .. .. .. . .. .. .. .. .. . .. .. . .. .. . . .. .. .. .. . .. .. . .. .. . .. .. .. .. . .. .. . .. .. .. .. .. .. . .. . .. .. . .. .. .. . .. . .. . . .. .. . .. . .. . .. .. . .. . .. .. .. .. .. . .. .. . . ...

P

Q

R

S

T

X

2x

x

(A)

x

2

(B) x

(C)

5x

4

(D)

3x

2

(E)

3x

4

(5)

S 3

13.

The solution to the equation 5

x

− 5

x

−2

= 120

5 is a rational number of the form

a

b

, where b

�= 0 and a and b are positive and have no common factors. What is the

value of a + b?

(A) 3

(B) 5

(C) 7

(D) 9

(E) 11

14.

How many points (x, y) on the circle x

2

+ y

2

= 50 are such that at least one of the

coordinates x, y is an integer?

(A) 16

(B) 30

(C) 48

(D) 60

(E) 100

15.

An eyebrow is an arrangement of the numbers 1, 2, 3, 4 and 5 such that the second

and fourth numbers are each bigger than both their immediate neighbours. For

example, (1, 3, 2, 5, 4) is an eyebrow and (1, 3, 4, 5, 2) is not.

The number of eyebrows is

(A) 16

(B) 12

(C) 15

(D) 24

(E) 18

16.

The sum of the positive solutions to the equation (x

2

− x)

2

= 18(x

2

− x) − 72 is

(A) 5

(B) 7

(C) 8

(D) 9

(E) 18

17.

On a clock face, what is the size, in degrees, of the acute angle between the line

joining the 5 and the 9 and the line joining the 3 and the 8?

(A) 15

(B) 22

1

2

(C) 30

(D) 45

(E) 60

18.

A positive fraction is added to its reciprocal. The sum is

x

60

in lowest terms, where

x is an integer. The number of possible values of x is

(6)

S 4

19.

In

�P QT , P Q = 10 cm, QT = 5 cm

and

P QT = 60

.

P W , P Y and T Q are tangents to

the circle with centre S at W , Y and

V respectively.

The radius of the circle, in

centime-tres, is

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... .... .... .... ... .... ... .... ... .... ... .... ... .... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... .... ... ... .... .... .... .... .... ... ... .... ... ... ... ... ... ... ... ...... ...... ...... ...

P

Q W

T

V

.

Y

S

(A)

5

3

2 +

3

(B)

5(3

3)

2

(C)

5

1 +

3

(D)

5

3

2

(E)

25

3

6

20.

I bought a map of Australia, unfolded it and marked eight places I wanted to visit.

I then refolded the map and placed it back on the table as it was. In what order

are my marks stacked from top to bottom?

(A) RTYQKAWP

(B) YKRAWTPQ

(C) RTQYKAWP

(D) YKTPRAWQ

(E) YKWARTPQ

Questions 21 to 25, 5 marks each

21.

A palindromic number is a ‘symmetrical’ number which reads the same forwards

as backwards. For example, 55, 101 and 8668 are palindromic numbers.

There are 90 four-digit palindromic numbers.

How many of these four-digit palindromic numbers are divisible by 7?

(A) 7

(B) 9

(C) 14

(D) 18

(E) 21

22.

What is the area, in square centimetres,

of the parallelogram that would fit snugly

around 6 circles, each of radius 3 cm, as

shown in the diagram?

... ... ... ... .... ... .... .... ... ... ... ... ... ...... ......... ... ... ... ... ... ... .... .... ... .... ... .... ... .... ... ... ... .. ...... ... ... ...... ......... ... ... ... ... ... ... ... .... .... .... ... .... .... ... ... ... ... ... . ......... ... ... ... ...... ......... ... ... ... ... ... ... .... .... .... .... .... .... ... ... ... ... ... . ... ... ... ... .... ... .... .... ... ... ... ... ... ...... ......... ... ... ... ... ... ... ... .... .... .... ... .... .... ... ... ... ... ... . ......... ... ... ... ...... ......... ... ... ... ... ... ... ... .... .... .... ... .... .... ... ... ... ... ... . ......... ... ... ... ...... ......... ... ... ... ... ... ... ... .... .... .... ... .... .... ... ... ... .... ... ... ...... ...... ... ...... ... ... ...... ... ...... ...... ... ...... ... ...... ...... ...

(A) 108

(B) 8(4 + 3

3)

(C) 15(2 +

3)

(D) 12(9 + 5

3)

(E) 216

(7)

S 5

23.

In 3009, King Warren of Australia suspects the Earls of Akaroa, Bairnsdale,

Clare-mont, Darlinghurst, Erina and Frankston are plotting a conspiracy against him.

He questions each in private and they tell him:

Akaroa: Frankston is loyal but Erina is a traitor.

Bairnsdale: Akaroa is loyal.

Claremont: Frankston is loyal but Bairnsdale is a traitor.

Darlinghurst: Claremont is loyal but Bairnsdale is a traitor.

Erina: Darlinghurst is a traitor.

Frankston: Akaroa is loyal.

Each traitor knows who the other traitors are, but will always give false

informa-tion, accusing loyalists of being traitors and vice versa. Each loyalist tells the truth

as he knows it, so his information on traitors can be trusted, but he may be wrong

about those he claims to be loyal.

How many traitors are there?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5

24.

Four circles of radius 1 cm are drawn with their centres at the four vertices of

a square with side length 1 cm. The area, in square centimetres, of the region

overlapped by all four circles is

(A) 2

3

− π

(B) π

2

(C) 1 +

π

3

3

(D) π

− 2

2

(E)

π

− 3 −

3

2

25.

Let f(x) =

x + 6

x

and f

n

(x) = f(f(

· · · (f(x)) · · ·)) be the n-fold composite of f.

For example, f

2

(x) =

x+6

x

+ 6

x+6

x

=

7x + 6

x + 6

and f

3

(x) =

7x+6

x+6

+ 6

7x+6

x+6

=

13x + 42

7x + 6

.

Let S be the complete set of real solutions of the equation f

n

(x) = x. The number

of elements in S is

(A) 2

(B) 2n

(C) 2

n

(D) 1

(E) infinite

For questions 26 to 30, shade the answer as an integer from 0 to 999 in

the space provided on the answer sheet.

Question 26 is 6 marks, question 27 is 7 marks, question 28 is 8 marks,

question 29 is 9 marks and question 30 is 10 marks.

26.

The reciprocals of 4 positive integers add up to

19

20

. Three of these integers are in

(8)

S 6

27.

We say a number is ascending if its digits are strictly increasing. For example, 189

and 3468 are ascending while 142 and 466 are not. For which ascending 3-digit

number n (between 100 and 999) is 6n also ascending?

28.

A regular octahedron has edges of

length 6 cm. If d cm is the shortest

distance from the centre of one face to

the centre of the opposite face

mea-sured around the surface of the

octa-hedron, what is the value of d

2

?

.... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ...... ...... ...... ... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... ... ... . ... ... .

start

end

29.

The country of Big Wally has a railway which runs in a loop 1080 km long. Three

companies, A, B and C run trains on the track and plan to build stations. Company

A will build three stations, equally spaced at 360 km intervals. Company B will

build four stations at 270 km intervals and Company C will build five stations at

216 km intervals.

... ... ... ... ... ...... ...... ...... ... ...... ... ...... ...... ...... ...... ... ... ... ... ... ... ...

B

B

B

B

A

A

A

C

C

C

C

C

The government tells them to space their stations so that the longest distance

between consecutive stations is as small as possible. What is this distance in

kilometres?

30.

A trapezium ABCD has AD

� BC and a point E is chosen on the base AD so

that the line segments BE and CE divide the trapezium into three right-angled

triangles. These three triangles are similar, but no two are congruent. In common

units, all the triangles’ side lengths are integers. The length of AD is 2009. What

is the length of BC?

(9)

senior Division

Competition paper

©AMT P

ublishing2009AMTTliMiTedAcn083 950 341

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