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A u s t rAl i An 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 thursday 31 July 2008 junIor 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 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. australian school years 7 and 8 time allowed: 75 minutes ©amt P u b l i s h i n g 2008 a m t t l i m i t e d a c n 083 950 341 Indicate Quantity Required in Box australian mathematics competition BooKs 2007 amc soLutIons and statIstIcs PrImarY vErsIon – $a35.00 EacH 2007 amc soLutIons and statIstIcs sEcondarY vErsIon – $a35.00 EacH 2007 amc soLutIons and statIstIcs PrImarY and sEcondarY vErsIons – $a57.00 for botH Two books are published each year for the Australian Mathematics Competition for the Westpac Awards, a Primary version for the Middle and Upper Primary divisions and a Secondary version for the Junior, Intermediate and Senior divisions. The books include the questions, full solutions, prize winners, statistics, information on Australian achievement rates, analyses of the statistics as well as discrimination and difficulty factors for each question. The 2007 books will be available early 2008. austraLIan matHEmatIcs comPEtItIon book 1 (1978-1984) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 2 (1985-1991) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 3 (1992-1998) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 3-cd (1992-1998) – $a40.00 EacH austraLIan matHEmatIcs comPEtItIon book 4 (1999-2005) – $a40.00 EacH These four books contain the questions and solutions from the Australian Mathematics Competition for the Westpac Awards for the years indicated. They are an excellent training and learning resource with questions grouped into topics and ranked in order of difficulty. BooKs For Further deVelopment oF mathematical sKills ProbLEms to soL vE In mIddLE scHooL matHEmatIcs – $a50.oo EacH This collection of problems is designed for use with students in Years 5 to 8. Each of the 65 problems is presented ready to be photocopied for classroom use. With each problem there are teacher’s notes and fully worked solutions. Some problems have extension problems presented with the teacher’s notes. The problems are arranged in topics (Number, Counting, Space and Number, Space, Measurement, Time, Logic) and are roughly in order of difficulty within each topic. ProbLEm soLvInG vIa tHE amc – $a40.00 EacH This book uses nearly 150 problems from past AMC papers to demonstrate strategies and techniques for problem solving. The topics selected include Geometry, Motion and Counting Techniques. cHaLLEnGE! (1991–1995) – $a40.00 EacH This book reproduces problems, solutions and extension questions from both Junior (Years 7 and 8) and Intermediate (Years 9 and 10) versions of the Mathematics Challenge for Young Australians, Challenge Stage. It is a valuable resource book for the classroom and the talented student. the above prices are current to 31 december 2008. details of other amt publications are available on the australian mathematics trust’s web site www.amt.canberra.edu.au/amtpub.html. all BooKs can Be ordered online @ www.amt.edu.au/amtpub.htnl a selection oF australian mathematics trust puBlications payment details payment must accompany orders. please allow up to 14 days for delivery. please forward publications to: (print clearly) NAME: ADDRESS: COUNTRY: POSTCODE: Postage and Handling - within Australia, add $A3.00 for the first book and $A1.00 for each additional book - outside Australia, add $A13.00 for the first book and $A5.00 for each additional book TOTAL: Cheque/Bankdraft enclosed for the amount of $A Please charge my Credit Card ( Visa, Mastercard) Amount authorised:$A Cardholder’s Name (as shown on card): Cardholder’s Signature: Expiry Date: Date: Tel (bh): Card Number: All payments (cheques/bankdrafts, etc) should be in Australian currency, made payable to austraLIan matHEmatIcs trust and sent to: australian mathematics trust, university of canberra act 2601, australia. tel: 02 6201 5137 Fax: 02 6201 5052

Junior Division Questions 1 to 10, 3 marks each 1.The value of 2008 + 8002 is (A) 1010 (B) 4004 (C) 10 008 (D) 8910 (E) 10 010 2.Which of the following numbers has the largest value? (A) 2.15 (B) 2.2 (C) 2.08 (D) 2.1 (E) 2.185 3.The perimeter of the figure, in centimetres, is (A) 8 (B) 10 (C) 12 (D) 16 (E) 20 ............. . . . . . . . . . . . . . ............. . . . . . . . . . . . . .. . . . . . . . . . . . . ............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............. 4cm2cm 4cm2cm 4.One half of 1991 2is (A) 951 2(B) 953 4(C) 991 4(D) 991 2(E) 993 4 5.The value ofxis (A) 135 (B) 95 (C) 35 (D) 55 (E) 45 ..................................................................................................... ......................................... ............................................................ .................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 135 ◦ x◦ 6.The value of200×8 200÷8is (A) 1 (B) 8 (C) 16 (D) 64 (E) 200 7.How many squares of any size are there in the diagram? (A) 9 (B) 11 (C) 12 (D) 14 (E) 16 1 1 1 1 1111 22

J2 8.A train left Fassifern at 8:58 am and arrived at Broadmeadow at 9:34 am on the sameday. Thetimetaken,inminutes,was (A) 82 (B) 22 (C) 36 (D) 38 (E) 78 9.The digits 5, 6, 7, 8 and 9 can be arranged to form even five-digit numbers. The tens digit in the largest of these numbers is (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 10.PQRSis a square and pointsEandFare outside the square so thatPQEand QRFare equilateral triangles. The size of EQF, in degrees, is (A) 60 (B) 90 (C) 120 (D) 150 (E) 180 Questions 11 to 20, 4 marks each 11.A rectangle has an area of 72 square centimetres and the length is twice the width. The perimeter, in centimetres, of the rectangle is (A) 34 (B) 36 (C) 42 (D) 48 (E) 54 12.Marbles of three different colours are in a tin and2 5of the marbles are red,1 3are green and the remaining 12 are yellow. The number of marbles in the tin is (A) 30 (B) 45 (C) 54 (D) 60 (E) 90 13.In the diagram, trianglesPQRandLM Nare both equilateral and QS M=20 ◦. What is the val u e ofx? (A) 70 (B) 80 (C) 90 (D) 100 (E) 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... . .. ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... .. ... ... ... . . ... ... ... .. ... ... ... .. ... ... .. ... ... ... .. ... ... ... .. ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................. ..................................................................................................... .............................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L NM PQ R S 20 ◦ x◦

J3 14.At half-time in a soccer match between Newcastle and Melbourne, the score was Newcastle 1, Melbourne 0. Three goals were scored in the second half. Which of the following couldnotbe the result of the match? (A) The match was drawn (B) Newcastle won by 2 goals (C) Melbourne won by 2 goals (D) Newcastle won by 1 goal (E) Newcastle won by 4 goals 15.In how many ways can 12 be written as the sum of two or more different positive whole numbers? (Changing the order of addition does not count as a different way.) (A) 12 (B) 13 (C) 14 (D) 15 (E) 16 16.How many different positive numbers are equal to the product of two odd one-digit numbers? (A) 25 (B) 15 (C) 14 (D) 13 (E) 11 17.The perimeter of this rectangular paddock is 700 m. It is subdivided into six identical paddocks as shown. The perimeter, in metres, of each of the six smaller paddocks is (A) 1161 3(B) 300 (C) 200 (D) 150 (E) 600 18.The student lockers at Euler High School are to be numbered consecutively from 1 to 500 using plastic digits which cost 5 cents each. The total cost of all the digits will be (A) $25 (B) $63.65 (C) $69.50 (D) $69.60 (E) $85

J4 19.In the grid below, the squares are to be filled with the numbers 1, 2, 3 and 4 so that they appear once only in each row, each column and each diagonal. 1 2 3 X Y The largest possible value of X + Y is (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 20.The average of one group of numbers is 4. A second group contains twice as many numbers and has an average of 10. The average of both groups of numbers combined is (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 Questions 21 to 25, 5 marks each 21.A cube with edge length 2 metres is cut up into cubes each with edge length 5 centimetres. If all these cubes were stacked one on top of the other to form a tower, the height of the tower would be (A) 32 km (B) 160 m (C) 1600 m (D) 3.2 km (E) 320 m 22.A number is less than 2008. It is odd, it leaves a remainder of 2 when divided by 3 and a remainder of 4 when divided by 5. What is the sum of the digits of the largest such number? (A) 26 (B) 25 (C) 24 (D) 23 (E) 22 23.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

J5 24.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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 100 20050 10060 80 The depth of water, in centimetres, above the prism is then (A) 12 (B) 14 (C) 16 (D) 18 (E) 20 25.A strip of paper is folded in a line at an anglex ◦to the sides and then folded underneath forming an angle of 20 ◦as shown. ... ............. ............ ... .......... ... . x◦ =⇒ .... ....... ........ ....... ....... ........ ....... ........ ....... ....... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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...... ....... ........ ....... ....... ........ ....... ........ ....... ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 ◦ The value ofxis (A) 60 (B) 65 (C) 70 (D) 75 (E) 80 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 two-digit numberaband its reversalbaare both prime. For example, 13 and 31 are both prime. What is the largest possible sum of these two numbersabandba?

J6 27.Given a regular heptagon (7-sided polygon), how many obtuse-angled triangles are there, where the vertices of each triangle are vertices of the heptagon? 28.A rectangular prism 6 cm by 3 cm by 3 cm is made up by stacking 1 cm by 1 cm by 1 cm cubes. How many rectangular prisms, including cubes, are there whose vertices are vertices of the cubes, and whose edges are parallel to the edges of the original rectangular prism? (Rectangular prisms with the same dimensions but in different positions are different.) 29.Let us call a sum of integerscoo lif 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? 30.A monument has been constructed from identical stone cubes. The views from above, the front f and the side s are shown. f s above view front view side view What is the largest number of stones in the monument consistent with these views? ***