2008 中學高級卷 英文試題(2008 Senior English Paper)

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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

31 J u ly

2 0 0 8

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 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

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

tHE ansWEr sHEEt

1. Use only lead pencil.

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colouring the circle matching your answer.

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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.

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Senior Division

Questions 1 to 10, 3 marks each

1.

The value of 8002

− 2008 is

(A) 200

(B) 8

(C) 6006

(D) 1060

(E) 5994

2.

The difference between

1

20

and

2

10

is

(A) 0

(B)

1

10

(C)

3

5

(D)

3

10

(E)

3

20

3.

In the diagram,

x equals

... ... ... ... ... ... ..._... ..._... ..._... ...

x

◦

100

◦

110

◦

80

◦

(A) 100

(B) 110

(C) 120

(D) 130

(E) 140

4.

The value of

200

× 8

200

÷ 8

is

(A) 1

(B) 8

(C) 16

(D) 64

(E) 200

5.

The smallest value that

x

2

− 4x + 3 can have is

(A)

−1

(B)

−3

(C) 1

(D) 3

(E) 2

6.

$3 is shared between two people. One gets 50 cents more than the other. The ratio

of the larger share to the smaller share is

S 2

7.

When 1000

2008

is written as a numeral, the number of digits written is

(A) 2009

(B) 6024

(C) 6025

(D) 8032

(E) 2012

8.

A semicircle is drawn on one side of an

equilateral triangle. The ratio of the area

of the semicircle to the area of the triangle

is

(A) 1 : 1

(B)

π : 2

√

3

(C)

π :

√

3

(D)

√

3 :

π

(E) 3 :

π

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

9.

Given that cos

x = 0.5 and 0

◦

< x < 90

◦

, which of the following has the greatest

value?

(A) cos

2

x

(B) cos

x

(C) 0

.75

(D) sin

x

(E) tan

x

10.

A fishtank with base 100 cm by 200 cm and depth 100 cm contains water to a depth

of 50 cm. A solid metal rectangular prism with dimensions 80 cm by 100 cm by

60 cm is then submerged in the tank with an 80 cm by 100 cm face on the bottom.

..._... ..._... ... ..._... ..._... ... ..._... ..._... ...

6

?

_... ..._... ... ... ..._... ..._... ... ..._... ..._... ... .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . ..._... ... ..._... ... ...

100

100

200

50

100

60

80

The depth of water, in centimetres, above the prism is then

(A) 12

(B) 14

(C) 16

(D) 18

(E) 20

Questions 11 to 20, 4 marks each

11.

Which of the following numbers is the largest?

(A) 2

500

(B) 3

400

(C) 4

300

(D) 5

200

(E) 6

100

12.

A normal die is thrown 100 times. The sum of the numbers obtained will most

likely be

S 3

13.

What is the smallest whole number which gives a square number when multiplied

by 2008?

(A) 2

(B) 4

(C) 251

(D) 502

(E) 2008

14.

A cross is made up of five squares, each with side length 1 unit. Two cuts are

made, the first from

X to Y and the second from Z to T , so that ZT X is a right

angle. The three pieces are then arranged to form a rectangle.

... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ..._... ... ... ..._... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ..._... ... ... ..._... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ..._... ...

Z

Y

X

T

I

II

III

.... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ..._... ..._... ..._...... ..._... ... ... .... ... ... ... ... .... ... ... ... ... .... ..._... ..._... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

II

III

I

What is the ratio of the length to the width of the rectangle?

(A) 3 : 1

(B)

√

10 : 1

(C) 2 : 1

(D) 2

√

3 : 1

(E) 5 : 2

15.

A function is said to be a toggle function on (

p, q, r) if f(p) = q, f(q) = r and

f(r) = p.

The function

f(x) = ax

2

+

bx + c is a toggle function on (1, 2, 3).

What is the value of

c?

(A)

−2

(B) 0

(C) 3

(D) 9

(E) 14

16.

Two conical rollers with

perpendicu-lar axes touch on a line that is 30

◦

to

the axis of the smaller roller and 60

◦

to the axis of the larger roller. If the

larger roller makes 1 revolution per

sec-ond and there is no slipping, how many

revolutions per second does the smaller

roller make?

(A)

1

2

(B) 1

(C)

√

2

(D)

√

3

(E) 2

.... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ..._... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . ... ... ... ... ... ... ... ... ... ... ... .... .... ... ... ... ... ... ... ... ... ... ..._... ... ..._... ... ... ... ... ... ... ..._... ..._...... ... ... ... . ..._... ..._... ..._... ..._... ..._... ..._... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ..._... ..._... ..._... ..._... ..._... ..._... ..._... ..._... ... ... ... ... .... .... .... .. ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

60

◦

30

◦

S 4

17.

Consider the set

X = {1, 2, 3, 4, 5, 6}.

How many subsets of

X, with at least one element, do not contain two consecutive

integers?

(A) 16

(B) 18

(C) 20

(D) 21

(E) 24

18.

Farmer Taylor of Burra has two tanks. Water from the roof of his farmhouse is

collected in a 100 kL tank and water from the roof of his barn is collected in a

25 kL tank. The collecting area of his farmhouse roof is 200 square metres while

that of his barn is 80 square metres. Currently, there are 35 kL in the farmhouse

tank and 13 kL in the barn tank.

Rain is forecast and he wants to collect as much water as possible. He should:

(A) empty the barn tank into the farmhouse tank

(B) fill the barn tank from the farmhouse tank

(C) pump 10 kL from the farmhouse tank into the barn tank

(D) pump 10 kL from the barn tank into the farmhouse tank

(E) do nothing

19.

A sequence

{u

1

, u

2

, . . . , u

n

} of real numbers is defined by

u

1

=

√

2

,

u

2

=

π,

u

n

=

u

n−1

− u

n−2

for

n ≥ 3.

What is

u

₂₀₀₈

?

(A)

−

√

2

(B) 2008(

√

2

− 2008π)

(C) 1003

√

2

− 1004π

(D)

π

(E)

√

2

20.

In the diagram,

RU is equal in length

to

ST . What is the ratio of the area of

QRU to the area of QST ?

(A)

√

3 : 1

(B) 2 : 1

(C)

√

6 : 1

(D)

√

3 : 2

(E)

√

6 : 2

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

45

◦

30

◦

U

T

Q

R

S

Questions 21 to 25, 5 marks each

21.

P , Q, R, S and T are consecutive vertices of a regular polygon. When extended,

the lines

P Q and T S meet at U with



_{QUS = 160}

◦

_{. How many sides has the}

polygon?

S 5

22.

How many numbers from 1, 2, 3, 4,

. . ., 2008 have a cubic number other than 1 as

a factor?

(A) 346

(B) 336

(C) 347

(D) 251

(E) 393

23.

The numbers 828 and 313 are 3-digit palindromes where 828

− 313 = 515, which

is also a palindrome. How many pairs (

a, b) of 3-digit palindromes are there with

a > b and with a − b also a 3-digit palindrome?

(A) 1972

(B) 1980

(C) 1988

(D) 1996

(E) 2008

24.

The centres of all faces of a cube are joined to form an octahedron. The centres

of all faces of this octahedron are now joined to form a smaller cube. What is the

ratio of an edge of the smaller cube to an edge of the original cube?

(A) 1 :

√

2

(B) 1 :

√

3

(C) 1 : 2

(D) 1 : 3

(E) 1 : 4

25.

In the figure, all line segments are

par-allel to one of the sides of the

equi-lateral triangle

P QR which has side

length 1 unit. How long should

P X

be to maximise the smallest of the ten

areas defined?

(A)

1

3

(B)

4

−

√

2

14

(C)

1

4

(D)

1

5

(E)

1

√

10

... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ..._... ... ... ...

P

Q

R

X

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.

All possible straight lines joining the vertices of a cube with mid-points of its edges

S 6

27.

Let us call a sum of integers

cool if the first and last terms are 1 and each term

differs from its neighbours by at most 1. For example, the sum 1 + 2 + 3 + 4 + 3 +

2 + 3 + 3 + 3 + 2 + 3 + 3 + 2 + 1 is cool.

How many terms does it take to write 2008 as a cool sum if we use no more terms

than necessary?

28.

The positive integers

x and y satisfy

3

x

2

− 8y

2

+ 3

x

2

y

2

= 2008

.

What is the value of

xy?

29.

A point

O is inside an equilateral triangle

P QR and the perpendiculars OL, OM and

ON are drawn to the sides P Q, QR and RP

respectively.

The ratios of lengths of the perpendiculars

OL : OM : ON is 1 : 2 : 3.

If

area of

LONP

area of

P QR

=

a

b

, where a and b are

integers with no common factors, what is the

value of

a + b?

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

R

P

Q

L

M

N

O

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

30.

What is the smallest value that



49 +

a

2

− 7

√

2

a +



a

2

₊

_b

2

₋

√

₂

_{ab +}

√

_{50 +}

_b

2

_{− 10b}

can have for positive real numbers

a and b?