注意:
允許學生個人、非營利性的圖書館或公立學校合理使用
本基金會網站所提供之各項試題及其解答。可直接下載
而不須申請。
重版、系統地複製或大量重製這些資料的任何部分,必
須獲得財團法人臺北市九章數學教育基金會的授權許
可。
申請此項授權請電郵
ccmp@seed.net.tw
Notice:
Individual students, nonprofit libraries, or schools are
permitted to make fair use of the papers and its
solutions. Republication, systematic copying, or
multiple reproduction of any part of this material is
permitted only under license from the Chiuchang
Mathematics Foundation.
Requests for such permission should be made by
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
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?
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
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
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
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
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
?
.... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ...... ...... ...... ... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... . ... ... ... ... . ... ... .