[PDF] 2012 Australian Mathematics Competition AMC Junior Years 7 and 8.pdf | Plain Text

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 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. AUSTRALIAN SCHOOL YEARS 7 AND 8 TIME ALLOWED: 75 MINUTES ©AMT P ublishing 2012 AMTT liMiTed Acn 0 8 3 9 5 0 3 41 A u s t r Al i An M At h e M At i c s c o M p e t i t i o n An AcTi v i Ty o f Th e A u s Tr A l i An M A T h eM A T i c s Tr u s T THURSDAY 2 AUGUST 2012 NAME

JuniorDivi sion Quest io ns 1to 10, 3 m arks each 1 . The value of99 −2+ 1+ 102 is (A) 0 (B)100 (C)198 (D)200 (E)202 2. The size, indegrees, of 6Qis .............................................................................................................................................................................................................................................................................................................................................................................................................. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ◦ Q R 45 ◦ P (A) 40 (B)55 (C)60 (D)80 (E)90 3. Yesterda yit rained continuously from9:45amuntil 3:10 pm. For how long didit rain? (A) 3hours 25min utes (B)3hours 35min utes (C)5hours 25min utes (D) 6hours 25min utes (E)6hours 35min utes 4. The value of8× 3.1 is (A) 11.1 (B)16.8 (C)8.31 (D)24.1 (E)24.8 5. The change you should receivefrom a$20 note after paying abill of$9.45 is (A) $10.55 (B)$10.45 (C)$11.55 (D)$9.55 (E)$10.65 6. Three-fifths ofanum ber is48. What isthe num ber? (A) 54 (B)60 (C)64 (D)80 (E)84 7. Whic hof the follo wing isclosest to100? (A) 99+2.01 (B)98+3.011 (C)97+4.0111 (D)101−1.01 (E)102−2.011

J2 8. The adjacen tsides ofthe decagon shown meet atrigh tangles andalldimensions are inmetres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................................................................... ........................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7 8 6 16 What isthe perimeter, inmetres, ofthis decagon? (A) 45 (B)60 (C)34 (D)90 (E)cannot be calculated 9. If 5 9of the children inachoir areboys and therest aregirls, theratio ofboys to girls is (A) 4:9 (B)4:5 (C)5:4 (D)9:4 (E)5:9 1 0. By what num ber must 6be divided toobtain 1 3as aresult? (A) 18 (B)1 2 (C) 1 18 (D) 2 (E)9 Quest io ns 11 to 20, 4 m arks each 1 1. In the diagram, thesizes ofthree angles are giv en. Find thevalue ofx. (A) 90 (B)95 (C)100 (D) 110 (E)120 .............................................................................................................................................................................................................................................................................. . .................................................................................. .................................................................................................................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30◦ 50◦ 40◦ x◦ 1 2. A jar ofmixed lolliescontains 100gof jellyb eans, 30gof licorice bulletsand20g of bilb ybears. Extra bilbybears areadded tomak ethe mix 50% bilbybears by w eigh t.Ho wman ygrams ofbilb ybears areadded? (A) 20 (B)30 (C)60 (D)110 (E)600

J3 13. A square pieceofpap erisfolded inhalf. Theresulting rectangle hasaperimeter of 18 cm. What isthe area, insquare centimetres, ofthe original square? (A) 9 (B)16 (C)36 (D)81 (E)144 14. If 750 ×45 =p, then 750×44 equals (A) p− 45 (B)p− 750 (C)p− 1 (D)44p (E)750p 15. The gridshown ispart ofacross-n umber puzzle. ........................................................................................................................................................................................................................................................... . .................................................................................................................................... ............................................................................................. . ...................................................................................................................................................................................................................................... . ........................................................................... .............................................................................................................................................. . 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.. . . . . . . .. . . . . . .. . . . . .. . . . .. . . .. . .. . ... . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . . .. . .. . ... . . . . . ... 1 23 6 7 11 12 13 16 17 20 2122 2 C lues 16 across isthe reverse of2do wn 1 do wn isthe sum of16 across and2do wn 7 do wn isthe sum ofthe digits in16 across What is7do wn? (A) 11 (B)12 (C)13 (D)14 (E)15 16. Ican ride my bik e3 times asfast asTed can jog. Ted starts 40min utes before me and then Ichase him. Howlong does ittak eme tocatc hTed? (A) 20min (B)30min (C)40min (D)50min (E)60min 17. Fiv eto wns arejoined by roads, assho wn inthe diagram. . ............................................................................................................................................................................................................................. . ............................................................................................................................................................... ................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................... ............................................................................................................................................................................. P Q R S T Ho wman yways are there oftra velling fromtown Pto town Tifno town can be visited morethanonce? (A) 3 (B)5 (C)6 (D)7 (E)9

J4 18. What arethe last three digits of7777 ×9999? (A) 223 (B)233 (C)333 (D)323 (E)343 19. In how man yways can 52be written asthe sum ofthree prime num bers? (A) 1 (B)2 (C)3 (D)4 (E)5 20. Four poin tsP,Q ,R and Sare suchthat PQ =10, QR = 30, RS = 15 and P S = m.If m isan integer andnothree ofthese poin tslie on astraigh tline, what is the num ber ofpossible values ofm? (A) 5 (B)49 (C)50 (D)54 (E)55 Quest io ns 21 to 25, 5 m arks each 2 1. A courier companyhas motorbik eswhic hcan travel 300 km starting withafull tank. Twocouriers, AnnaandBrian, setofffrom thedepottogether todeliv er a letter toConnor’s house.Theonly refuelling iswhen theystopforAnna to transfer somefuelfrom hertank toBrian’s tank.Shethen returns tothe depot while Brian keeps going, deliversthe letter andreturns tothe depot. What isthe greatest distance thatConnor’s housecouldbe from thedepot? (A) 180km (B)200km (C)225km (D)250km (E)300km 22. The square PQ RS has sides of3metres. Thepoin tsX and Ydivide PQ into 3 equal parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................................................. ................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................................................................................................................................................................................................................................................................... .... . .............................................................................................................................................................................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . PX Y Q R S Z Find thearea, insquare metres, of4X YZ. (A) 3 8 (B) 1 2 (C) 3 16 (D) 1 3 (E) 1 4

J5 23. The product ofthree consecutiv eodd num bers is226 737. What isthe middle n um ber? (A) 57 (B)59 (C)61 (D)63 (E)65 24. A Me eker num ber isa7-digit num ber ofthe form pqrstup ,where p× q= 10r+ s and s× t= 10u+ pand none ofthe digits arezero. For example, 6742 816 isa Meek ernum ber. The value ofsin the largest Meekernum ber is (A) 2 (B)3 (C)5 (D)7 (E)8 25. Four positiv ein tegers arearranged ina2× 2table. For eac hro w and column ofthe table, theproduct ofthe tw o num bers inthis row or column iscalculated. When all four suchpro ducts areadded together, theresult is1001. What isthe largest p ossible sumoftw o num bers inthe table thatareneither inthe same row nor in the same column? (A) 33 (B)77 (C)91 (D)143 (E)500 For quest io ns 26 to 30, shade the answ eras an in teger from 0to 999 in t he space provided on the answ ersheet. Quest io n 26 is 6 m arks, questio n 27 is 7 m arks, questio n 28 is 8 m arks, quest io n 29 is 9 m arks andquest io n 30 is 10 m arks. 26. This cubehas adifferen twhole num ber on eachface, andhastheprop ertythat whic heverpair ofopp osite faces ischosen, thetw o num bers multiply togivethe same result. What isthe smallest possible totalofall 6num bers onthe cube? 27. Ho wman yfour-digit num bers containing nozeros have the prop ertythat whenev er an yone ofits four digits isremo ved, the resulting three-digit num ber isdivisible b y 3?

J6 28. A rhom bus-shap edtile isformed by joining tw o equilateral trianglestogether. Three ofthese tilesarecombined edgetoedge toform avariet yof shap esas in the example given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................................................................... . ........................................................................... . ................................................... ........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ho wman ydifferen tshap escan be formed? (Shapeswhic hare reflections orrota- tions ofother shapesare not considered different.) 29. W arren hasastrip ofpap er10 metres long.Hewishes tocut from itas man y pieces aspossible, notnecessarily usingallthe paper,with eachpiece ofpap era whole num ber ofcen timetres long.Thesecond piecemust be 10 cm longer than the first, thethird 10cm longer thanthesecond andsoon. What isthe length, in cen timetres, ofthe largest possible piece? 30. Terry hasinven ted anew way to extend listsofnum bers. To Ter ryfy alist such as [1, 8]he creates tw o lists [2,9]and [3,10], where eachterm isone more than the corresp onding terminthe previous list,andthen joins thethree liststogether to give[1, 8,2,9,3,10]. Ifhe starts withalist containing onenum ber [0] and rep eatedly Ter ryfies ithe creates thelist [0 ,1 ,2 ,1 ,2 ,3 ,2 ,3 ,4 ,1 ,2 ,3 ,2 ,3 ,4 ,3 ,4 ,5 ,2 ,3 ,4 ,... ]. What isthe 2012th num ber inthis Ter ryfic list?

©AMT P ublishing 2012 AMTT liMiTed Acn 0 8 3 9 5 0 3 41 AMC SOLUTIONS AND STATISTICS This book provides a record of the Australian Mathematics Competition Junior, Intermediate and Senior questions, solutions and statistics for 2012. It also includes details of medallists and prizewinners and provides statistical information on levels of Australian response rates and other information. AMC Solutions and Statistics 2012 (available early 2013) can be ordered online from the Australian Mathematics Trust website where a wide range of other products are also available. www.amt.edu.au