2015加拿大国际高中数学竞赛试题I

TheCENTREforEDUCATIONinMATHEMATICSandCOMPUTINGcemc.uwaterloo.caCanadianIntermediateMathematicsContestWednesday,November25,2015(inNorthAmericaandSouthAmerica)Thursday,November26,2015(outsideofNorthAmericaandSouthAmerica)Time:2hoursc2015UniversityofWaterlooCalculatorsareallowed,withthefollowingrestriction:youmaynotuseadevicethathasinternetaccess,thatcancommunicatewithotherdevices,orthatcontainspreviouslystoredinformation.Forexample,youmaynotuseasmartphoneoratablet.Donotopenthisbookletuntilinstructedtodoso.Therearetwopartstothispaper.Thequestionsineachpartarearrangedroughlyinorderofincreasingdiculty.TheearlyproblemsinPartBarelikelyeasierthanthelaterproblemsinPartA.PARTA1.Thispartconsistsofsixquestions,eachworth5marks.2.Entertheanswerintheappropriateboxintheanswerbooklet.Forthesequestions,fullmarkswillbegivenforacorrectanswerwhichisplacedinthebox.Partmarkswillbeawardedonlyifrelevantworkisshowninthespaceprovidedintheanswerbooklet.PARTB1.Thispartconsistsofthreequestions,eachworth10marks.2.Finishedsolutionsmustbewrittenintheappropriatelocationintheanswerbooklet.Roughworkshouldbedoneseparately.Ifyourequireextrapagesforyournishedsolutions,theywillbesuppliedbyyoursupervisingteacher.Insertthesepagesintoyouranswerbooklet.Besuretowriteyourname,schoolnameandquestionnumberonanyinsertedpages.3.Marksareawardedforcompleteness,clarity,andstyleofpresentation.Acorrectsolution,poorlypresented,willnotearnfullmarks.Atthecompletionofthecontest,insertyourstudentinformationforminsideyouranswerbooklet.Donotdiscusstheproblemsorsolutionsfromthiscontestonlineforthenext48hours.Thename,grade,schoolandlocation,andscorerangeofsometop-scoringstudentswillbepublishedontheWebsite,cemc.uwaterloo.ca.Inaddition,thename,grade,schoolandlocation,andscoreofsomestudentsmaybesharedwithothermathematicalorganizationsforotherrecognitionopportunities.CanadianIntermediateMathematicsContestNOTE:1.Pleasereadtheinstructionsonthefrontcoverofthisbooklet.2.Writesolutionsintheanswerbookletprovided.3.Expresscalculationsandanswersasexactnumberssuchas+1andp2,etc.,ratherthanas4:14:::or1:41:::,exceptwhereotherwiseindicated.4.Whilecalculatorsmaybeusedfornumericalcalculations,othermathematicalstepsmustbeshownandjustiedinyourwrittensolutionsandspecicmarksmaybeallocatedforthesesteps.Forexample,whileyourcalculatormightbeabletondthex-interceptsofthegraphofanequationlikey=x3x,youshouldshowthealgebraicstepsthatyouusedtondthesenumbers,ratherthansimplywritingthesenumbersdown.5.Diagramsarenotdrawntoscale.Theyareintendedasaidsonly.6.NostudentmaywriteboththeCanadianSeniorMathematicsContestandtheCanadianIntermediateMathematicsContestinthesameyear.PARTAForeachquestioninPartA,fullmarkswillbegivenforacorrectanswerwhichisplacedinthebox.Partmarkswillbeawardedonlyifrelevantworkisshowninthespaceprovidedintheanswerbooklet.1.Stephaniehas1000eggstopackintocartonsof12eggs.Whilepackingtheeggs,shebreaksneggs.Theunbrokeneggscompletelyllacollectionofcartonswithnoeggsleftover.Ifn12,whatisthevalueofn?2.Inthediagram,ABCDisasquarewithsidelength4.PointsP,Q,R,andSarethemidpointsofthesidesofthesquare,asshown.Whatistheareaoftheshadedregion?PABQCSDR3.Inthediagram,linesegmentsAB,CDandEFareparallel,andpointsAandElieonCGandCH,respectively.If\GAB=100,\CEF=30,and\ACE=x,whatisthevalueofx?DBFCGAEH30º100ºxº4.If12x=4y+2,determinethevalueoftheexpression6y18x+7.5.Determinethelargestpositiveintegernwithn500forwhich6048(28n)isaperfectcube(thatis,itisequaltom3forsomepositiveintegerm).6.Atotalof2015tickets,numbered1;2;3;4;:::;2014;2015,areplacedinanemptybag.Aleremovesticketafromthebag.Bernicethenremovesticketbfromthebag.Finally,Charlieremovesticketcfromthebag.Theynoticethatabcanda+b+c=2018.Inhowmanywayscouldthishappen?PARTBForeachquestioninPartB,yoursolutionmustbewellorganizedandcontainwordsofexplanationorjustication.Marksareawardedforcompleteness,clarity,andstyleofpresentation.Acorrectsolution,poorlypresented,willnotearnfullmarks.1.AtCornthwaiteH.S.,manystudentsenrollinanafter-schoolartsprogram.Theprogramoersadramaclassandamusicclass.Eachstudentenrolledintheprogramisinoneclassorbothclasses.(a)Thisyear,41studentsareinthedramaclassand28studentsareinthemusicclass.If15studentsareinbothclasses,howmanystudentsareenrolledintheprogram?(b)In2014,atotalof80studentsenrolledintheprogram.If3x5studentswereinthedramaclass,6x+13studentswereinthemusicclass,andxstudentswereinbothclasses,determinethevalueofx.(c)In2013,halfofthestudentsinthedramaclasswereinbothclassesandone-quarterofthestudentsinthemusicclasswereinbothclasses.AtotalofNstudentsenrolledintheprogramin2013.IfNisbetween91and99,inclusive,determinethevalueofN.2.Alistair,Conrad,Emma,andSalmacompeteinathree-sportrace.Theyeachswim2km,thenbike40km,andnallyrun10km.Also,theyeachswitchinstantlyfromswimmingtobikingandfrombikingtorunning.(a)Emmahascompleted113ofthetotaldistanceoftherace.Howmanykilometershasshetravelled?(b)Conradbegantheraceat8:00a.m.andcompletedtheswimmingportionin30minutes.Conradbiked12timesasfastasheswam,andran3timesasfastasheswam.Atwhattimedidhenis