2010滑铁卢竞赛试题

www.linstitute.netCanadianMathematicsCompetitionAnactivityoftheCentreforEducationinMathematicsandComputing,UniversityofWaterloo,Waterloo,OntarioEuclidContestWednesday,April7,2010c2010CentreforEducationinMathematicsandComputingTime:21󰀂2hoursCalculatorsarepermitted,providedtheyarenon-programmableandwithoutgraphicdisplays.Donotopenthisbookletuntilinstructedtodoso.Thepaperconsistsof10questions,eachworth10marks.Partsofeachquestioncanbeoftwotypes.SHORTANSWERpartsareworth3markseach.FULLSOLUTIONpartsareworththeremainderofthe10marksforthequestion.InstructionsforSHORTANSWERparts:1.SHORTANSWERpartsareindicatedlikethis:.2.Entertheanswerintheappropriateboxintheanswerbooklet.Forthesequestions,fullmarkswillbegivenforacorrectanswerwhichisplacedinthebox.Partmarkswillbeawardedonlyifrelevantworkisshowninthespaceprovidedintheanswerbooklet.InstructionsforFULLSOLUTIONparts:1.FULLSOLUTIONpartsareindicatedlikethis:.2.Finishedsolutionsmustbewrittenintheappropriatelocationintheanswerbooklet.Roughworkshouldbedoneseparately.Ifyourequireextrapagesforyourfinishedsolutions,foolscapwillbesuppliedbyyoursupervisingteacher.Insertthesepagesintoyouranswerbooklet.Besuretowriteyourname,schoolnameandquestionnumberonanyinsertedpages.3.Marksareawardedforcompleteness,clarity,andstyleofpresentation.Acorrectsolutionpoorlypresentedwillnotearnfullmarks.NOTE:AtthecompletionoftheContest,inserttheinformationsheetinsidetheanswerbooklet.Thenamesofsometop-scoringstudentswillbepublishedintheEuclidResultsonourWebsite,http://www.cemc.uwaterloo.ca.www.linstitute.netNOTES:1.2.3.Pleasereadtheinstructionsonthefrontcoverofthisbooklet.Writeallanswersintheanswerbookletprovided.Forquestionsmarked“”,fullmarkswillbegivenforacorrectanswerplacedintheappropriateboxintheanswerbooklet.Ifanincorrectanswerisgiven,marksmaybegivenforworkshown.Studentsarestronglyencouragedtoshowtheirwork.Allcalculationsandanswersshouldbeexpressedasexactnumberssuchas√4π,2+7,etc.,ratherthanas12.566...or4.6...,exceptwhereotherwiseindicated.4.ANoteaboutBubbling

Pleasemakesurethatyouhavecorrectlycodedyourname,dateofbirth,gradeandsex,ontheStudentInformationForm,andthatyouhaveansweredthequestionabouteligibility.ANoteaboutWritingSolutionsForeachproblemmarked“

”,afullsolutionisrequired.

Thesolutionsthat

youprovideintheanswerbookletshouldbewellorganizedandcontainmathematicalstatementsandwordsofexplanationwhenappropriate.Workingoutsomeofthedetailsinroughonaseparatepieceofpaperbeforewritingyourfinishedsolutionisagoodidea.Yourfinalsolutionshouldbewrittensothatthemarkercanunderstandyourapproachtotheproblemandallofthemathematicalstepsofyoursolution.1.

(a)If3x=27,whatisthevalueof3x+2?(b)If2531359x=2731459,whatisthevalueofx?(c)TriangleABCisenclosedbythelines

y=x+2,y=−12x+2andthex-axis.Determinetheareaof󰀄ABC.

yBx

A2.

OC(a)Mariahasaredpackage,agreenpackage,andabluepackage.

Thesumofthemassesofthethreepackagesis60kg.

Thesumofthemassesoftheredandgreenpackagesis25kg.Thesumofthemassesofthegreenandbluepackagesis50kg.Whatisthemassofthegreenpackage,inkg?

(b)Apalindromeisapositiveintegerthatisthesamewhenreadforwardsor

backwards.Forexample,151isapalindrome.Whatisthelargestpalindromelessthan200thatisthesumofthreeconsecutiveintegers?(c)If(x+1)(x−1)=8,determinethenumericalvalueof(x2+x)(x2−x).

www.linstitute.net3.

(a)Beathebeesetsoutfromherhive,H,and

fliessouthfor1hourtoafield,F.Shespends30minutesinthefield,andthenflies45minuteswesttoagarden,G.Afterspending1hourinthegarden,shefliesbacktoherhivealongastraightlineroute.Beaalwaysfliesatthesameconstantspeed.Whatisthetotallengthoftime,inminutes,thatsheisawayfromherhive?(b)Inthediagram,pointsP(p,4),B(10,0),and

O(0,0)areshown.If󰀄OPBisright-angledatP,determineallpossiblevaluesofp.

H

G

yP (p, 4)F

O4.

Bx

(a)Thurkaboughtsomestuffedgoatsandsometoyhelicopters.Shepaidatotal

of$201.Shedidnotbuypartialgoatsorpartialhelicopters.Eachstuffedgoatcost$19andeachtoyhelicoptercost$17.Howmanyofeachdidshebuy?(b)Determineallrealvaluesofxforwhich(x+8)4=(2x+16)2.

5.

(a)Iff(x)=2x+1andg(f(x))=4x2+1,determineanexpressionforg(x).(b)Ageometricsequencehas20terms.

Thesumofitsfirsttwotermsis40.Thesumofitsfirstthreetermsis76.Thesumofitsfirstfourtermsis130.

Determinehowmanyofthetermsinthesequenceareintegers.

(Ageometricsequenceisasequenceinwhicheachtermafterthefirstisobtainedfromtheprevioustermbymultiplyingitbyaconstant.Forexample,3,6,12isageometricsequencewiththreeterms.)

6.

(a)Asnail’sshellisformedfromsixtriangular

sections,asshown.Eachtrianglehasinterioranglesof30◦,60◦and90◦.IfABhasalengthof1cm,whatisthelengthofAH,incm?

FGH303030ED30C3030A1BB

(b)InrectangleABCD,pointEisonsideDC.

LinesegmentsAEandBDareperpendicularandintersectatF.IfAF=4andDF=2,determinetheareaofquadrilateralBCEF.

A

FD

E

C

www.linstitute.net32x27.(a)Determineallrealvaluesofxforwhich3

(x−1)

9=27.

(b)Determineallpoints(x,y)wherethetwocurvesy=log10(x4)andy=(log10x)3

intersect.

8.

(a)Oi-Lamtossesthreefaircoinsandremovesallofthecoinsthatcomeupheads.

Georgethentossesthecoinsthatremain,ifany.DeterminetheprobabilitythatGeorgetossesexactlyonehead.(b)Inthediagram,pointsB,P,Q,andClie

onlinesegmentAD.Thesemi-circlewithdiameterAChascentrePandthesemi-circlewithdiameterBDhascentreQ.Thetwosemi-circlesintersectatR.If∠PRQ=40◦,determinethemeasureof∠ARD.

9.

R

A

BPQ

CD

(a)(i)Ifθisananglewhosemeasureisnotanintegermultipleof90◦,provethat

cotθ−cot2θ=

1sin2θ

(ii)Rossstartswithanangleofmeasure8◦anddoublesit10timesuntilhe

obtains8192◦.Hethenaddsupthereciprocalsofthesinesofthese11angles.Thatis,hecalculates

S=

11111

+++···++sin8◦sin16◦sin32◦sin4096◦sin8192◦Determine,withoutusingacalculator,themeasureoftheacuteangleαso1

thatS=.

sinα

(b)In󰀄ABC,BC=a,AC=b,AB=c,anda<12(b+c).

1

Provethat∠BAC<2(∠ABC+∠ACB).

10.

Foreachpositiveintegern,letT(n)bethenumberoftriangleswithintegersidelengths,positivearea,andperimetern.Forexample,T(6)=1sincetheonlysuchtrianglewithaperimeterof6hassidelengths2,2and2.(a)DeterminethevaluesofT(10),T(11)andT(12).

(b)Ifmisapositiveintegerwithm≥3,provethatT(2m)=T(2m−3).(c)DeterminethesmallestpositiveintegernsuchthatT(n)>2010.