2010PascalContest滑铁卢竞赛题

CanadianMathematicsCompetitionAnactivityoftheCentreforEducationinMathematicsandComputing,UniversityofWaterloo,Waterloo,OntarioPascalContest(Grade9)Thursday,February25,2010Time:60minutesCalculatorsarepermittedInstructions©2009CentreforEducationinMathematicsandComputing1.DonotopentheContestbookletuntilyouaretoldtodoso.2.Youmayuserulers,compassesandpaperforroughwork.3.Besurethatyouunderstandthecodingsystemforyourresponseform.Ifyouarenotsure,askyourteachertoclarifyit.Allcodingmustbedonewithapencil,preferablyHB.Fillincirclescompletely.4.Onyourresponseform,printyourschoolname,city/town,andprovinceintheboxintheupperleftcorner.5.Becertainthatyoucodeyourname,age,sex,grade,andtheContestyouarewritingintheresponseform.Onlythosewhodosocanbecountedasofficialcontestants.6.Thisisamultiple-choicetest.EachquestionisfollowedbyfivepossibleanswersmarkedA,B,C,D,andE.Onlyoneoftheseiscorrect.Aftermakingyourchoice,fillintheappropriatecircleontheresponseform.7.Scoring:Eachcorrectanswerisworth5inPartA,6inPartB,and8inPartC.Thereisnopenaltyforanincorrectanswer.Eachunansweredquestionisworth2,toamaximumof10unansweredquestions.8.Diagramsarenotdrawntoscale.Theyareintendedasaidsonly.9.Whenyoursupervisortellsyoutobegin,youwillhavesixtyminutesofworkingtime.Thenamesofsometop-scoringstudentswillbepublishedinthePCFResultsonourWebsite,http://www.cemc.uwaterloo.ca.Scoring:

Thereisnopenaltyforanincorrectanswer.

Eachunansweredquestionisworth2,toamaximumof10unansweredquestions.

PartA:Eachcorrectanswerisworth5.1.

Whichofthefollowingisclosestinvalueto$1.00?(A)$0.502.

Thevalueof(A)53.

(B)$0.90

(C)$0.95

(D)$1.01

(E)$1.15

(20−16)×(12+8)

is

4(B)9(C)20

(D)44(E)56

Tomakepizzadough,Lucamixes50mLofmilkforevery250mLofflour.How

muchmilkdoeshemixwith750mLofflour?(A)100mL

(B)125mL

(C)150mL

(D)200mL

(E)250mL

4.

Oneofthefollowing8figuresisrandomlychosen.Whatistheprobabilitythatthechosenfigureisatriangle?

(A)5.

If

38(B)

34(C)

18(D)

12(E)

13111+=,thenthenumberthatreplacesthe󰀂tomaketheequationtrueis918󰀂(A)2(B)3(C)6(D)9(E)18Squaresofsidelength1arearrangedtoformthefigureshown.Whatistheperimeterofthefigure?(A)12(D)24

√(B)16(E)2633+33+33is(B)9

(C)27

(D)81

(E)243

(C)20

6.

7.Thevalueof(A)3

8.

Inthediagram,thepointsareequallyspacedonthenumberline.WhatnumberisrepresentedbypointP?

7.46(D)7.519.

(E)7.52

PThenineinteriorintersectionpointsona4by4gridofsquaresareshown.Howmanyinteriorintersectionpointsarethereona12by12gridofsquares?(A)100(D)144

(B)121(E)169

(C)132

7.62(A)7.48(B)7.49(C)7.50

10.Thediagramshowsacirclegraphwhichshowsthe

amountofhomeworkdoneeachdaybyMr.Auckland’sGrade9class.Basedonthecirclegraph,whatpercentageofstudentsdoatleastonehourofhomeworkperday?

(A)25%(D)67%

(B)33%(E)75%

(C)50%

Hours of homework per day

More than 2Less than 1

1 to 2

PartB:Eachcorrectanswerisworth6.

11.Severalthree-leggedtablesandfour-leggedtableshaveatotalof23legs.Ifthereis

morethanonetableofeachtype,whatisthenumberofthree-leggedtables?

(A)6

(B)7

(C)3

(D)4

(E)5

12.Twelve1by1squaresformarectangle,asshown.What

isthetotalareaoftheshadedregions?

(A)8(D)11

(B)9(E)12

(C)10

13.Thereare400studentsatCayleyH.S.,wheretheratioofboystogirlsis3:2.There

are600studentsatFermatC.I.,wheretheratioofboystogirlsis2:3.Whenconsideringallthestudentsfrombothschools,whatistheratioofboystogirls?

(A)2:3

(B)12:13

(C)1:1

(D)6:5

(E)3:2

14.Thenumberednetshownisfoldedtoformacube.What

istheproductofthenumbersonthefourfacessharinganedgewiththefacenumbered1?

(A)120(D)240

(B)144(E)360

(C)180

4

1356

2

15.If10%ofsist,thensequals

(A)0.1t

(B)0.9t

(C)9t

(D)10t

(E)90t

16.Fouridenticalsquaresarecutfromthecornersofthe

rectangularsheetofcardboardshown.Thissheetisthenfoldedalongthedottedlinesandtapedtomakeaboxwithanopentop.Thebaseoftheboxmeasures5cmby4cm.Thevolumeoftheboxis60cm3.Whatwastheareaoftheoriginalsheetofcardboard?

(A)56cm2(D)180cm2

(B)110cm2(E)210cm2

(C)156cm2

5 cm4 cm17.Inthediagram,PWisparalleltoQX,SandTlieon

QX,andUandVarethepointsofintersectionofPWwithSRandTR,respectively.If∠SUV=120◦and∠VTX=112◦,whatisthemeasureof∠URV?

(A)52◦(D)64◦

(B)56◦(E)68◦

QSR(C)60◦

PUVWTX

1

18.ThegastankinCatherine’scaris8full.When30litresofgasareadded,thetank

becomes34full.IfthegascostsCatherine$1.38perlitre,howmuchwillitcosthertofilltheremainingquarterofthetank?

(A)$8.80(B)$13.80(C)$16.56(D)$24.84(E)$41.40

19.Inthediagram,pointsU,V,W,X,Y,andZlieona

straightlinewithUV=VW=WX=XY=YZ=5.SemicircleswithdiametersUZ,UV,VW,WX,XY,andYZcreatetheshapeshown.Whatistheareaoftheshadedregion?

(A)(D)

325π4625π4(B)(E)

375π4625π2(C)

325π2UVWXYZ

20.Theoddintegersfrom5to21areusedtobuilda3by3

magicsquare.(Inamagicsquare,thenumbersineachrow,thenumbersineachcolumn,andthenumbersoneachdiagonalhavethesamesum.)If5,9and17areplacedasshown,whatisthevalueofx?

(A)7(D)15

(B)11(E)19

(C)13

59x

17

PartC:Eachcorrectanswerisworth8.

21.Inthediagram,eachofthefiveboxesistocontaina

number.Eachnumberinashadedboxmustbetheaverageofthenumberintheboxtotheleftofitandthenumberintheboxtotherightofit.Whatisthevalueofx?

(A)28(D)32

(B)30(E)34

(C)31

826x22.RhombusPQRSisinscribedinrectangleJKLM,as

shown.(Arhombusisaquadrilateralwithfourequalsidelengths.)SegmentsPZandXRareparalleltoJM.SegmentsQWandYSareparalleltoJK.IfJP=39,JS=52,andKQ=25,whatistheperimeterofrectangleWXYZ?

(A)48(D)96

(B)58(E)108

(C)84

J

W

P

X

KQ

SM

Z

YR

L

23.TheproductofNconsecutivefour-digitpositiveintegersisdivisibleby20102.What

istheleastpossiblevalueofN?

(A)5

(B)12

(C)10

(D)6

(E)7

24.Asequenceconsistsof2010terms.Eachtermafterthefirstis1largerthanthe

previousterm.Thesumofthe2010termsis5307.Wheneverysecondtermisaddedup,startingwiththefirsttermandendingwiththesecondlastterm,thesumis

(A)2155

(B)2153

(C)2151

(D)2149

(E)2147

25.SixsoccerteamsarecompetinginatournamentinWaterloo.Everyteamistoplay

threegames,eachagainstadifferentteam.(Notethatnoteverypairofteamsplaysagametogether.)Judeneisinchargeofpairinguptheteamstocreateascheduleofgamesthatwillbeplayed.Ignoringtheorderandtimesofthegames,howmanydifferentschedulesarepossible?

(A)90

(B)100

(C)80

(D)60

(E)70

2010PascalContest(English)TheCENTREforEDUCATIONinMATHEMATICSandCOMPUTINGForstudents...Thankyouforwritingthe2010PascalContest!In2009,morethan84000studentsaroundtheworldregisteredtowritethePascal,CayleyandFermatContests.CheckouttheCEMC’sgrouponFacebook,called“WhoisTheMathiest?”.EncourageyourteachertoregisteryoufortheFryerContestwhichwillbewrittenonApril9,2010.Visitourwebsitewww.cemc.uwaterloo.catofind•MoreinformationabouttheFryerContest•Freecopiesofpastcontests•Workshopstohelpyouprepareforfuturecontests•InformationaboutourpublicationsformathematicsenrichmentandcontestpreparationForteachers...Visitourwebsitewww.cemc.uwaterloo.cato•RegisteryourstudentsfortheFryer,GaloisandHypatiaContestswhichwillbewrittenonApril9,2010•Learnaboutworkshopsandresourcesweofferforteachers•Findyourschoolresults