Galois滑铁卢数学竞赛(Grade 10).docx

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1、Galois滑铁卢数学竞赛(Grade10)2007GaloisContest(Grade10)Wednesday,April18,20071.Jimshopsatastrangefruitstore.Insteadofputtingpricesoneachitem,themathematicalstoreownerwillanswerquestionsaboutcombinationsofitems.(a)InAisle1,Jimreceivesthefollowinganswerstohisquestions:JimsQuestionAnswerWhatisthesumoftheprice

2、sofanAppleandaCherry?62centsWhatisthesumofthepricesofaBananaandaCherry?66centsWhatisdi?erencebetweenthepricesofanAppleandaBanana?Whichhasahigherprice?Explainhowyouobtainedyouranswer.(b)InAisle2,Jimreceivesthefollowinganswerstohisquestions:JimsQuestionAnswerWhatisthesumofthepricesofaMangoandaNectarin

3、e?60centsWhatisthesumofthepricesofaPearandaNectarine?60centsWhatisthesumofthepricesofaMangoandaPear?68centsWhatisthepriceofaPear?Explainhowyouobtainedyouranswer.(c)InAisle3,Jimreceivesthefollowinganswerstohisquestions:JimsQuestionAnswerWhatisthesumofthepricesofaTangerineandaLemon?60centsHowmuchmored

4、oesaTangerinecostthanaGrapefruit?6centsWhatisthesumofthepricesofGrapefruit,aTangerineandaLemon?94centsWhatisthepriceofaLemon?Explainhowyouobtainedyouranswer.2.(a)Inthediagram,whatistheperimeterofthesectorofthecirclewithradius12?Explainhowyouobtainedyouranswer. (b)Twosectorsofacircleofradius12areplac

5、edsidebyside,asshown.Determinetheareaof?gureABCD.Explainhowyouobtainedyouranswer.A(c)Inthediagram,AOBisasectorofacirclewithAOB=60?.OYisdrawnperpendiculartoABandintersectsABatX.WhatisthelengthofXY?Explainhowyouobtainedyouranswer.AOBXY1212(d)Seeover.2007GaloisContestPage2(d)Twosectorsofacircleofradius

6、12overlapasshown.Determinetheareaoftheshadedregion.Explainhowyouobtainedyouranswer.R3.(a)Eachfaceofa5by5by5woodencubeisdividedinto1by1squares.Eachsquareispaintedblackorwhite,asshown.Next,thecubeiscutinto1by1by1cubes.Howmanyofthesecubeshaveatleasttwopaintedfaces?Explainhowyouobtainedyouranswer.(b)A(2

7、k+1)by(2k+1)by(2k+1)cube,wherekisainthesamemannerasthe5by5by5cubewithwhitesquaresinthecorners.Again,thecubeiscutinto1by1by1cubes.i.Intermsofk,howmanyofthesecubeshaveexactlytwowhitefaces?Explainhowyouobtainedyouranswer.ii.Provethatthereisnovalueofkforwhichthenumberofcubeshavingatleasttwowhitefacesis2

8、006.4.Jillhasacontainerofsmallcylindricalrodsinsixdi?erentcolours.Eachcolourofrodhasadi?erentlengthassummarizedinthechart.ColourLengthGreen3cmPink4cmYellow5cmBlack7cmViolet8cmRed9cmTheserodscanbeattachedtogethertoformapole.Thereare2waystochooseasetofyellowandgreenrodsthatwillformapole29cminlength:8g

9、reenrodsand1yellowrodOR3greenrodsand4yellowrods.(a)Howmanydi?erentsetsofyellowandgreenrodscanbechosenthatwillformapole62cmlong?Explainhowyouobtainedyouranswer.(b)Amongthegreen,yellow,blackandredrods,?nd,withjusti?cation,twocoloursforwhichitisimpossibletomakeapole62cminlengthusingonlyrodsofthosetwocolours.(c)Ifatleast81rodsofeachofthecoloursgreen,pink,violet,andredmustbeused,howmanydi?erentsetsofrodsofthesefourcolourscanbechosenthatwillformapole2007cminlength?Explainhowyougotyouranswer.