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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
6 a u g u s t
2 0 0 9
intermediate 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 9 a n d 1 0
t i m e a l l o w e d : 7 5 m i n u t e s
Intermediate Division
Questions 1 to 10, 3 marks each
1.
(2000 + 9) + (2000
− 9) equals
(A) 4000
(B) 4009
(C) 200
(D) 2000
(E) 5000
2.
In the diagram, x equals
(A) 140
(B) 122
(C) 80
(D) 90
(E) 98
... ...... ... ...... ...... ... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...x
◦
140
◦
122
◦
3.
The value of
1
2
×
2
3
×
3
4
×
4
5
is
(A)
1
5
(B)
5
7
(C)
1
6
(D)
1
15
(E)
1
60
4.
Which of the following has the largest value?
(A)
1
3
(B)
1
3
+
1
3
(C)
1
3
×
1
3
(D)
1
3
−
1
3
(E)
1
3
÷
1
3
5.
Which of the following values can replace the box so that
0.1
× 0.2 × 0.3 × 0.4 ×
= 0.12 ?
(A) 500
(B) 50
(C) 5
(D) 0.5
(E) 0.05
6.
If 3
k
= 9
30
then k equals
I 2
7.
(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
8.
The point (k, 17) lies on the line joining the points (1, 5) and (4, 11). The value of
k is
(A) 37
(B) 14
(C) 8
(D) 6
(E) 7
9.
Paperback books cost $5 each and hardcover books cost $7 each. I spend exactly
$86 on books of these two types. What is the maximum number of books that I
could have bought?
(A) 10
(B) 14
(C) 16
(D) 18
(E) 20
10.
The figure P QRS is a rectangle
divided into 10 squares as shown.
The perimeter of this rectangle
is 21 centimetres.
In
centime-tres, what is the perimeter of each
square ?
(A) 2.1
(B) 3
(C) 6
(D) 8.4
(E) 12
P
R
S
Q
Questions 11 to 20, 4 marks each
11.
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?
I 3
12.
In the diagram, triangles P QT ,
QT S and QRS are isosceles and
�
P QR is a right angle. Angles P QT
and RQS are 2x
◦
and angle QT S is
5x
◦
. The value of x is
(A) 10
(B) 12
(C) 14
(D) 15
(E) 20
... .... .... ... .... .... .... ... .... ... .... .... ... .... .... ... .... .... .... ... .... .... ... .... .... ... .... .... ... .... .... .... ... .... .... ... .... .... .... ... .... .... ... .... ... .... .... .... .... ... .... ... .... .... ... .... .... ... .... .... ... .... .... .... ... .... .... ... .... .... ... .... .... .... ... .... ... .... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ... ...... ...... ...... ... ... ... ...... ...P
R
T
Q
S
2x
◦
2x
◦
5x
◦
13.
This pattern consists of squares. Areas,
in square units, of four of the squares
are shown. Given that X and Y are
also squares, their areas are
respec-tively
(A) 16, 25
(B) 16, 36
(C) 25, 36
(D) 25, 64
(E) 25, 100
1 1
4
9
X
Y
14.
Two numbers P and Q are such that P is 40% greater than Q. The ratio P : Q is
(A) 40 : 1
(B) 5 : 7
(C) 5 : 3
(D) 5 : 2
(E) 7 : 5
15.
What is the last digit of 6
× 8
2009
?
(A) 0
(B) 2
(C) 4
(D) 6
(E) 8
16.
Two dice are each numbered from 1 to 6, but are biased so that each is twice as
likely to land on any of the even numbers as on any of the odd numbers. The two
dice are rolled and the numbers multiplied together. What is the probability that
the resulting product is odd?
(A)
1
9
(B)
2
9
(C)
1
3
(D)
4
9
(E)
2
3
17.
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
I 4
18.
P QRS is a rectangle. X is halfway along
P Q, Y is a third of the way along QR
and Z is a quarter of the way along RS.
What fraction of the area of P QRS is
represented by the quadrilateral XY ZS?
(A)
1
2
(B)
7
12
(C)
2
3
(D)
3
4
(E)
3
5
P
S
R
Q
X
Y
Z
... ...... ... ...... ...... ...... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ..19.
The difference between a positive fraction and its reciprocal is
9
20
. The sum of the
fraction and its reciprocal is
(A)
41
40
(B)
20
9
(C)
25
16
(D)
41
20
(E) 5
20.
A circular bottle with dimensions, in centimetres, is shown partially filled with
water (figure 1).
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ...... ... ... ... ... .... .... .... .... .... .... ... .... .... ... ... .... ... ... ... .... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... .... ... ... .... .... .... ... .... ... ... ... ... ...... ... ... ... ... .... .... .... .... .... .... ... .... .... ... ... .... ... ... ... .... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... .... ... ... .... .... .... ... .... ... ... ... ... ...... ... ... ... .... .... .... .... .... .... .... ... .... .... ... ... .... ... ... ... .... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... .... ... ... .... .... ... .... .... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
4
2
2
3
1
�
�
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... .... .... .... ... .... .... ... .... ... .... ... .... ... ... ... ... .... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... .... ... ... .... .... ... .... .... .... ... ... ... ... ... ......... ... ... ... .... .... .... ... .... .... ... .... ... .... ... .... ... ... ... ... .... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... .... ... ... .... ... .... .... .... .... ... ... ... ... ...... ... ... ... .... .... .... .... ... .... ... .... ... .... ... .... ... ... ... ... .... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... .... ... ... .... ... .... .... .... .... ... ... ... ...h
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .�
�
figure 1
figure 2
The bottle is sealed and then turned upside down (figure 2). The height h, in
centimetres, of the air in the upturned bottle is
(A) 2
(B) 2
1
3
(C) 2
7
25
(D) 2
1
5
(E) 2
3
25
I 5
Questions 21 to 25, 5 marks each
21.
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
22.
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
23.
P QRS is a square. T and U are
mid-points of the sides P S and P Q
respec-tively. T Q and SU intersect at V .What
fraction of the area of the square is the
area of quadrilateral QV SR?
(A)
1
2
(B)
5
8
(C)
2
3
(D)
3
4
(E)
5
9
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ...... ...... ... ...... ...... ...... ...... ...... ...... ...... ......P
R
Q
S
T
U
V
I 6
24.
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
25.
Which of the following cannot be the last digit of the sum of the squares of seven
consecutive numbers?
(A) 3
(B) 5
(C) 6
(D) 7
(E) 8
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.
What is the smallest positive integer which, when divided by each of 2, 3, 4, 5, 6
and 7, will give in each case a remainder that is one less than the divisor?
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 magician deposits the same number of rabbits (at least one) at each of five
houses. To get to the first house he crosses a magic river once, and to get to any
house from another, he also crosses a magic river once. Each time he crosses a
magic river, the number of rabbits he has doubles. He has no rabbits left when he
leaves the fifth house. What is the minimum number of rabbits he could have at
the start?
I 7
29.
Consider this sequence of patterns made from hexagons.
...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ......... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
