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
e-mailing Mr. Wen-Hsien SUN ccmp@seed.net.tw
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 day 2 au g u s t 2 01 2
N a m e
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 entered. It is your responsibility to correctly code your answer sheet.
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 scanned. The optical scanner will attempt to read 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 AMT 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.
The value of 8
× 3.3 is
(A) 24.24
(B) 24.4
(C) 25.4
(D) 26.24
(E) 26.4
2.
Sally has $20 of her pocket money left after 3 weeks, having spent just $1 on a
drink. How much pocket money does she get each week?
(A) $5
(B) $7
(C) $9
(D) $20
(E) $21
3.
In the diagram, the size of
6
P QR is
(A) 40
◦
(B) 50
◦
(C) 60
◦
(D) 70
◦
(E) 80
◦
... ... ... ... ... ... ... ... ...... ...... ... ...... ...... ...... ......120
◦
130
◦
P
R
Q
4.
Three-fifths of a number is 48. What is the number?
(A) 54
(B) 60
(C) 64
(D) 80
(E) 84
5.
By what number must 6 be divided to obtain
1
3
as a result?
(A) 18
(B)
1
2
(C)
1
18
(D) 2
(E) 9
6.
The average of the five numbers x, 1,
1
2
,
1
3
and
1
4
is 1. The value of x is
(A)
1
5
(B)
2
3
(C)
11
5
(D)
25
12
(E)
35
12
I 2
7.
In the diagram, P QRS is a square.
The value of x is
(A) 45
(B) 60
(C) 67.5
(D) 75
(E) 82.5
...... ...... ...... ...... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ...x
◦
x
◦
P
R
Q
S
8.
Five positive integers have a mean of 10, a median of 10 and only one mode, which
is 12. What is the difference between the largest and smallest of these numbers?
(A) 3
(B) 5
(C) 6
(D) 7
(E) 8
9.
If 750
× 45 = p, then 750 × 44 equals
(A) p
− 45
(B) p
− 750
(C) p
− 1
(D) 44p
(E) 750p
10.
I can ride my bike 3 times as fast as Ted can jog. Ted starts 40 minutes before me
and then I chase him. How long does it take me to catch Ted?
(A) 20 min
(B) 30 min
(C) 40 min
(D) 50 min
(E) 60 min
Questions 11 to 20, 4 marks each
11.
If p% of q is k, then q% of p is
(A)
k
100
(B)
pq
200
(C)
pk
100
(D)
qk
100
(E) k
12.
On one side of each of the five coins below there is a number and on the other side
there is a shape.
... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... ... ... ... ... ... ... ...... ...... ...2
3
4
...... ... ... ... ... ... ... ... ... ... ... ......P
P
Q
R
S
T
Peter is told that if there is a triangle on one side of a coin then there is an even
number on the other. Which of the following is the fewest coins that Peter can
turn over from the five to check this?
I 3
13.
The architecture of Federation Square in Melbourne is based on frames as shown
in which a large triangle is subdivided into 5 identical triangles, each similar to the
large triangle.
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...If the shortest side of one of the smallest triangles is 1 m, how many metres of
framing are required to construct the whole shape?
(A) 20
(B) 8 + 4
√
5
(C) 10 + 4
√
5
(D) 12 + 4
√
5
(E) 15 + 5
√
5
14.
If a : b = 3 : 2 and a + 3b = 27, what is the value of a + b?
(A) 5
(B) 9
(C) 13
(D) 15
(E) 21
15.
This sheriff’s badge has ten equal sides, five 60
◦
angles and five equal reflex angles.
...... ... ... ... ...... ... ... ... ...... ...... ... ......... ... ... ... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... ... ... ...
SHERIFF
60
◦
x
◦
...... ... ...The value of x is
(A) 108
(B) 132
(C) 135
(D) 138
(E) 140
16.
The shape shown is formed from four identical arcs, each a quarter of the
circum-ference of a circle of radius 5 cm.
... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ... ... ...... ...... ...... ...... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .. .... .. .... .. ... . ... .. ... ... ... ... ...
What is the area, in square centimetres, of the shape?
(A) 50
(B)
25π
2
(C) 25π
− 25
(D) 100
−
25π
I 4
17.
The number 2012
× 2013 × 2014 + 2013 is the cube of
(A) 2012
(B) 2013
(C) 2014
(D) 2112
(E) 2113
18.
A partition of a positive integer is a way of writing the integer as a sum of at least
two positive integers. For example, the partitions of 4 are:
3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1.
How many partitions of 7 are there?
(A) 11
(B) 12
(C) 13
(D) 14
(E) 15
19.
These five shapes all have the same area. Which one has the largest perimeter?
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .
(A)
a
a
a
a
b
b
(B)
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ............ ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...c
c
4c
4c
(C)
... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... .... .... .... ... ... ... ... ... ... ...d
(D)
...✛
✲
e
2e
(E)
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...20.
Pippa made a litre of drink from apple juice and water in the ratio of 1 : 2. She
found the taste too strong so she made a litre again in the ratio 1 : 3, but found
this too weak. So she thought if she combined these two mixtures, it should be
about right. What is the ratio of apple juice to water in this new mixture?
(A) 2 : 5
(B) 2 : 7
(C) 5 : 12
(D) 7 : 17
(E) 7 : 24
Questions 21 to 25, 5 marks each
21.
A courier company has motorbikes that can travel 300 km starting with a full tank.
Two couriers, Anna and Brian, set off from the depot together to deliver a letter to
Connor’s house. The only refuelling is when they stop for Anna to transfer some
fuel from her tank to Brian’s tank. She then returns to the depot while Brian keeps
going, delivers the letter and returns to the depot. What is the greatest distance
that Connor’s house could be from the depot?
I 5
22.
QRST is a trapezium in which QR
k T S and QR : ST = 2 : 3.
... ... ... ... ... ... ... ...... ...... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ....
Q
R
S
T
X
... . ... .If the area of
4XST is 18 square units, what is the area, in square units, of
4RT S?
(A) 24
(B) 28
(C) 30
(D) 34
(E) 36
23.
If abc + ab + bc + ca + a + b + c = 104, and a, b and c are positive integers, then
a
2
+ b
2
+ c
2
is equal to
(A) 49
(B) 51
(C) 54
(D) 56
(E) 60
24.
A teacher has a class of twelve students. She thinks it would be a nice idea if they
change desks every day, so she has painted arrows on the floor from desk to desk.
Each desk has one arrow going to it and another going from it. Each morning,
the students pick up their books and move to the desk indicated by the arrow.
By choosing her arrows carefully, the teacher has arranged it so that the longest
possible time will pass before all the students are back in their original desks at
the same time. How many days is that?
(A) 30
(B) 35
(C) 42
(D) 60
(E) 72
25.
The number 33
33
can be expressed as the sum of 33 consecutive odd numbers. The
largest of these odd numbers is
(A) 33
32
+ 32
(B) 33
31
+ 32
(C) 33
32
− 32
(D) 33
31
− 32
(E) 33
32
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.
I 6
26.
Slim took a long road trip across Australia over a number of days (more than 1).
When he arrived at his destination, he noted that he had travelled exactly 2012
kilometres. On the first day he travelled a whole number of kilometres and each
subsequent day he travelled one more kilometre than the day before. What is the
largest distance, in kilometres, that he could have travelled on the first day?
27.
Five consecutive positive integers, p, q, r, s and t, each less than 10 000, produce
a sum which is a perfect square, while the sum q + r + s is a perfect cube. What
is the value of
√
p + q + r + s + t?
28.
A quadrilateral with sides 15, 15, 15 and 20 is drawn with each vertex on a circle.
Around this circle a square is drawn, with each side tangent to the circle. What is
the area, in square units, of this square?
29.
In the grid shown, we need to fill in the squares with numbers so that the number
in every square, except for the corner ones, is the average of its neighbours. The
edge squares have three neighbours, the others four.
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
