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Unformatted text preview: \Ilestpac AUSTRALIAN MATHEMATICS COMPETITION
FOR THE WESTPAC AWARDS AN ACTIVITY OF THE AUSTRALIAN MATHEMATICS TRUST WEDNESDAY 25 JULY 2007 SENIOR DIVISION COMPETITION PAPER AUSTRALIAN SCHOOL YEARS 11 AND 12
TIME ALLOWED: 75 MINUTES INSTRUCTIONS AND INFORMATION GENERAL . Do not open the booklet until told to do soggy your teacher.
. NO calculators, slide rules, log tables, maths/glstencils, mobile phones or other calculating aids are permitted. Scribbling paper, graph paper/ruler and compasses are permitted, but are not essential. . 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. . 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. . 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 I'S 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. Senior Division Questions 1 to 10, 3 marks each . 2(5.61 — 4.5) equals (A) 3.1 (B) 10.48 (C) 2 i (D) 2.22 (E) 6.72 . If 2” + 2” = 2’”, then (A)n+n:m (B)n+1=m (C)4n=m (D)m+1=n (E)n2=m . PQR is a straight line. The value of IL’
is (A) 30 (B) 45 (C) 50
D 60 E 150
< > < > P R
. Of the following, which is the largest fraction?
7 3 6 4 1
A — — — D * —
< > 15 (B) 7 (c) 11 < > 9 (E) 2 . Nicky started a mobile phone call at 10:57 am. The charge for the call was 89 cents
per minute and the total cost for the call was $6.23. Nicky’s call ended at (A) 11:27am (B) 11:14am (C) 11:04am (D) 11:46am 11:05 am . The straight lines with equations 2m+y = q and y = x —p meet at the point (2, The value of p + q is . PQR is an equilateral triangle, QS S2 and QT divide ZPQR into three equal parts. The size of ZQTS. in degrees, is (A) 40 (B) 70 80
(D) 90 (E) 100“ . Jane’s age is a prime number. Andy’s age has 8 factors and he is one year older than Jane. Of the following numbers, which could be the sum of their ages? p4)27 (3)39 (ca 75 U))87 (E)107 . PQRS is a parallelogram and T lies on PQ such that PT : TQ = 3 : 2. The
ratio of the area of PTBS to the area P T Q
SZ{;///////\\\\z{ (ﬁPQRsa
(A)L2 (B)23 (C)34
an4:5 (E)56 10. Five positive integers have a mean of 5, a median of :3 and just one mode of 8.
What is the difference between the largest and the smallest integers in the set?
(A) 4 (B) 5 (C) 6 (D) T 8
Questions 11 to 20, 4 marks each
11. Dad ﬁlled his sprayer with 8 litres of water. He then added 16 drops of insecti—
cide instead of the recommended dosage of 32 drops. After using 2 litres of the
spray, he realised his mistake, reﬁlled the sprayer with another 2 litres of water
and added a sufﬁcient number of drops of insecticide to reach the recommended
concentration.The number of extra drops that dad needed to add was
(A) 20 (B) 12 (C) 8 (D) 16 24
12. The game of Four T0qu is played on a 4 X 4 grid. 2
When completed, each of the numbers 1, 2, 3 and
4 occurs in each row and column of the 4 X 4 grid _ 1
and also in each 2 X 2 corner of the grid. ‘ 1 3 i
When the grid shown is completed, the sum of the 4
four numbers in the corners of the 4 X 4 grid is $
(A) 13 (B) 11 (C) 15 (D) 12 10 13. 14. 15. 16. 17. 18. SS Holly writes down all the two—digit numbers which can be formed using the digits
1. 3, 7 and 9 (including 11, 33, 77 and 99). Warren selects one of these numbers at random. What is the probability that it is prime? 5 (B) 3 (c) 19—6 11
16 (D) (E) 2 Two rectangular garden beds have a combined area of 40 n12. The larger bed has
twice the perimeter of the smaller and the larger side of the smaller bed is equal to
the smaller side of the larger bed. If the two beds are not similar, and if all edges
are a whole number of metres, what is the length, in metres, of the longer side of the larger bed? I take a two—digit positive number and add to it the number obtained by reversing
the digits. For how many two—digit numbers is the result of this process a perfect square? (A) 1 (B) 3 (C) 5 (D) 8 (E) 10 Ann, Bill and Carol sit on a row of 6 seats. If no two of them sit next to each other, in how many different ways can they be seated? (A) 12 (B) 24 (C) 18 (D) 36
The number of integer solutions of the equation (3:2 — 32: + 1W1 :1
is
(A) 1 (B) 2 (C) 3 (D) 4
Ava and Lucy both jog at 8km / 11 along a straight path with Lucy staying 12m be—
hind Ava. Elizabeth jogs at 6km / h along
a straight path which meets the ﬁrst path
at right—angles at P. When Elizabeth is
at P she is the same distance from Ava as
from Lucy. (E) 48 A
[11
V
Cr! (Elizabeth I“ [A When Ava was ﬁrst 50m from P, how far, in metres, was Elizabeth from P? (A) 40 (B) 42 (C) 44 (D) 46 (E) 48 S4 19. On a 3 X 5 chessboard, a counter can move one square at a time along a row or a
column, but not along any diagonal. Starting from some squares, it can visit each
of the other 14 squares exactly once, without returning to its starting square. Of
the 15 squares, how many could be the counter’s starting square? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 20. The inscribed circle of an equilateral triangle
has radius 1 unit. A smaller circle is tangent
to this circle and to two sides of the triangle as
shown. The radius of this smaller circle is g 1 1 \/§ 1
A — B — C ——
< ) 3 < > 2 < > 6
«g —1 1 ‘L
03) (E) —
2 5
Questions 21 to 25, 5 marks each 21. There are four lifts in a building. Each makes three stops, which do not have to
be on consecutive ﬂoors or include the ground floor. For any two ﬂoors, there is
at least one lift which stops on both of them. What is the maximum number of
ﬂoors that this building can have? (A) 4 (B) 5 (C) 6 (D) 7 12 22. A bee can ﬂy or walk only in a straight line between any two corners on the inside
of a cubic box of edge length 1. The bee managed to move so that it visited every
corner of the box without passing through the same point twice in the air or on
the wall of the box. The largest possible length of such a path is
(A) 2+5ﬁ (B) 1+6\/§ (o) N5 (D) ﬁ+6¢§ (E) 4\/§+3\/§ 23. PQR is an equilateral triangle with side length 2. S is the midpoint of QR and T
and U are points on PR and PQ respec—
tively such that STX U is a square. The area of this square is (A)6—3\/§ 035—22” (0):
(D)? <E>1+2ﬁ S 5
M? + ()3: + c are there with the property that, for all 24. How many functions f x 96,1195) >< f(—:v) = f($2)?
(A) 4 (B) 6 (C) 8 (D) 10 (E) 12 25. Let + 1)2007 = a+b\/2, where a and b are integers. The highest common factor
of b and 81 is  (A) 1 (B) 3 (o) 9 (D) 27 (E) 81 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. A rectangular area measuring 3 units by 6 units on a wall is to be covered with 9
tiles each measuring 1 unit by 2 units. In how many ways can this be done? 27. There are 42 points P1, P2, P3, . . . , P42, placed in order on a straight line so that
1
each distance from P, to P,+1 is —, where 1 g 2' g 41. What is the sum of the z
distances between every pair of these points? 28. A lucky number is a positive integer which is 19 times the sum of its digits. How
many different lucky numbers are there? 29. On my calculator screen the number 2659 can be read upside down as 6592. The
digits that can be read upside down are 0, 1, 2, 5, 6, 8, 9 and are read as 0, 1, 2, 5,
9, 8, 6 respectively. Starting with 1, the ﬁfth number that can be read upside down
is 8 and the ﬁfteenth is 21. What are the last three digits of the 2007th number that can be read upside down? 30. Consider the solutions (95, y, z, u) of the system of equations
:5 + y = 3(2 + u)
x + z = 4(y + u)
a: + u = 5(y + z) where :13, y, z and u are positive integers. What is the smallest value that :5 can
have? ...
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This note was uploaded on 09/27/2010 for the course 456 9852 taught by Professor Chaohue during the Spring '10 term at Mackenzie.
 Spring '10
 ChaoHue

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