2016滑铁卢竞赛试题 6页

www.linstitute.netTheCENTREforEDUCATIONinMATHEMATICSandCOMPUTINGcemc.uwaterloo.caEuclidContestTuesday,April12,2016(inNorthAmericaandSouthAmerica)Wednesday,April13,2016(outsideofNorthAmericaandSouthAmerica)Time:212hours c2016UniversityofWaterlooDonotopenthisbookletuntilinstructedtodoso.Numberofquestions:10Eachquestionisworth10marksCalculatorsareallowed,withthefollowingrestriction:youmaynotuseadevice thathasinternetaccess,thatcancommunicatewithotherdevices,orthatcontains previouslystoredinformation.Forexample,youmaynotuseasmartphoneora tablet.Partsofeachquestioncanbeoftwotypes:1.SHORTANSWERpartsindicatedby•worth3markseach•fullmarksgivenforacorrectanswerwhichisplacedinthebox•partmarksawardedonlyifrelevantworkisshowninthespaceprovided2.FULLSOLUTIONpartsindicatedby•worththeremainderofthe10marksforthequestion•mustbewrittenintheappropriatelocationintheanswerbooklet •marksawardedforcompleteness,clarity,andstyleofpresentation•acorrectsolutionpoorlypresentedwillnotearnfullmarksWRITEALLANSWERSINTHEANSWERBOOKLETPROVIDED.•Extrapaperforyourfinishedsolutionssuppliedbyyoursupervisingteachermustbeinsertedintoyouranswerbooklet.Writeyourname,schoolname,andquestionnumberonanyinsertedpages. 2,etc.,rather•Expresscalculationsandanswersasexactnumberssuchasπ+1and√ thanas4.14...or1.41...,exceptwhereotherwiseindicated.Donotdiscusstheproblemsorsolutionsfromthiscontestonlineforthenext48hours.\nThename,grade,schoolandlocation,andscorerangeofsometop-scoringstudentswillbepublishedonourwebsite,cemc.uwaterloo.ca.Inaddition,thename,grade,schoolandlocation,andscoreofsometop-scoringstudentsmaybesharedwithothermathematicalorganizationsforotherrecognitionopportunities.\nwww.linstitute.netNOTE:1.Pleasereadtheinstructionsonthefrontcoverofthisbooklet.2.Writeallanswersintheanswerbookletprovided.3.Forquestionsmarked,placeyouranswerintheappropriateboxintheanswerbookletandshowyourwork.4.Forquestionsmarked,provideawell-organizedsolutionintheanswerbooklet. Usemathematicalstatementsandwordstoexplainallofthestepsofyoursolution. Workoutsomedetailsinroughonaseparatepieceofpaperbeforewritingyourfinished solution.5.Diagramsarenotdrawntoscale.Theyareintendedasaidsonly.6.Whilecalculatorsmaybeusedfornumericalcalculations,othermathematicalstepsmust beshownandjustifiedinyourwrittensolutionsandspecificmarksmaybeallocatedforthesesteps.Forexample,whileyourcalculatormightbeabletofindthex-intercepts ofthegraphofanequationlikey=x3−x,youshouldshowthealgebraicstepsthatyouusedtofindthesenumbers,ratherthansimplywritingthesenumbersdown.ANoteaboutBubblingPleasemakesurethatyouhavecorrectlycodedyourname,dateofbirthandgradeonthe StudentInformationForm,andthatyouhaveansweredthequestionabouteligibility.1.(a)Whatistheaverageoftheintegers5,15,25,35,45,55?(b)Ifx2=2016,whatisthevalueof(x+2)(x−2)?(c)Inthediagram,pointsP(7,5),Q(a,2a),andR(12,30)lieonastraightline.Determineythevalueofa. R(12,30)Q(a,2a)P(7,5)x2.(a)Whatareallvaluesofnforwhichn9=25n?(b)Whatareallvaluesofxforwhich(x−3)(x−2)=6?(c)AtWillard’sGroceryStore,thecostof2applesisthesameasthecostof 3bananas.Rossbuys6applesand12bananasforatotalcostof$6.30.Determine thecostof1apple.\nwww.linstitute.net3.(a)Inthediagram,pointBisonAC,pointFisonDB,andpointGisonEB.DFq˚r˚p˚ABs˚Gt˚Eu˚CWhatisthevalueofp+q+r+s+t+u?(b)Letnbetheintegerequalto1020−20.Whatisthesumofthedigitsofn?(c)Aparabolaintersectsthex-axisatP(2,0)andQ(8,0).ThevertexoftheparabolaisatV,whichisbelowthex-axis.Iftheareaof4VPQis12,determinethe coordinatesofV.4.(a)Determineallanglesθwith0◦≤θ≤180◦andsin2θ+2cos2θ=74.(b)Thesumoftheradiioftwocirclesis10cm.Thecircumferenceofthelarger circleis3cmgreaterthanthecircumferenceofthesmallercircle.Determinethe differencebetweentheareaofthelargercircleandtheareaofthesmallercircle.5.(a)Charlotte’sConvenienceCentrebuysacalculatorfor$p(wherep>0),raisesits pricebyn%,thenreducesthisnewpriceby20%.Ifthefinalpriceis20%higher than$p,whatisthevalueofn?(b)Afunctionfisdefinedsothatifnisanoddinteger,thenf(n)=n−1andifnis aneveninteger,thenf(n)=n2−1.Forexample,ifn=15,thenf(n)=14andifn=−6,thenf(n)=35,since15isanoddintegerand−6isaneveninteger. Determineallintegersnforwhichf(f(n))=3.6.(a)Whatisthesmallestpositiveintegerxforwhichintegery?132=x10yforsomepositive(b)Determineallpossiblevaluesfortheareaofaright-angledtrianglewithoneside lengthequalto60andwiththepropertythatitssidelengthsformanarithmetic sequence.(Anarithmeticsequenceisasequenceinwhicheachtermafterthefirstisobtained fromtheprevioustermbyaddingaconstant.Forexample,3,5,7,9arethefirst fourtermsofanarithmeticsequence.)\nwww.linstitute.net7.(a)AmritaandZhangcrossalakeinastraightlinewiththehelpofaone-seatkayak.Eachcanpaddlethekayakat7km/handswimat2km/h.Theystartfromthe samepointatthesametimewithAmritapaddlingandZhangswimming.Aftera while,Amritastopsthekayakandimmediatelystartsswimming.Uponreaching thekayak(whichhasnotmovedsinceAmritastartedswimming),Zhanggetsin andimmediatelystartspaddling.Theyarriveonthefarsideofthelakeatthe sametime,90minutesaftertheybegan.Determinetheamountoftimeduring these90minutesthatthekayakwasnotbeingpaddled.(b)Determineallpairs(x,y)ofrealnumbersthatsatisfythesystemofequationsx2
2+y−2x1y2+x−y
5=0=08.(a)Inthediagram,ABCDisaparallelogram.PointEisonDCwithAEperpendicularABtoDC,andpointFisonCBwithAFperpendiculartoCB.IfAE=20,AF=32,3220andcos(∠EAF)=13,determinetheexact3,determinetheexactvalueoftheareaofquadrilateralAECF.DECF(b)Determineallrealnumbersx>0forwhichlog4x−logx16=76−logx89.(a)ThestringAAABBBAABBisastringoftenletters,eachofwhichisAorB, thatdoesnotincludetheconsecutivelettersABBA.ThestringAAABBAAABBisastringoftenletters,eachofwhichisAorB, thatdoesincludetheconsecutivelettersABBA.Determine,withjustification,thetotalnumberofstringsoftenletters,eachofwhichisAorB,thatdonotincludetheconsecutivelettersABBA.(b)Inthediagram,ABCDisasquare.PointsEandFarechosenonACsothat∠EDF=45◦.IfAE=x,EF=y,andFC=z,provethaty2=x2+z2.ABxEyF45˚zDC\nwww.linstitute.net10.Letkbeapositiveintegerwithk≥2.Twobagseachcontainkballs,labelledwith thepositiveintegersfrom1tok.Andr´eremovesoneballfromeachbag.(Ineachbag,eachballisequallylikelytobechosen.)DefineP(k)tobetheprobabilitythat theproductofthenumbersonthetwoballsthathechoosesisdivisiblebyk.(a)CalculateP(10).(b)Determine,withjustification,apolynomialf(n)forwhich•P(n)≥f(n)n2forallpositiveintegersnwithn≥2,and•P(n)=f(n)n2forinfinitelymanypositiveintegersnwithn≥2.(Apolynomialf(x)isanalgebraicexpressionoftheform f(x)=amxm+am−1xm−1+···+a1x+a0forsomeintegerm≥0andforsomerealnumbersam,am−1,...,a1,a0.)2016(c)ProvethereexistsapositiveintegermforwhichP(m)>.m