2017滑铁卢竞赛试题

www.linstitute.netTheCENTREforEDUCATIONinMATHEMATICSandCOMPUTINGcemc.uwaterloo.caEuclidContestThursday,April6,2017(inNorthAmericaandSouthAmerica)Friday,April7,2017(outsideofNorthAmericaandSouthAmerica)Time:212hoursc2017UniversityofWaterloo󰀄Donotopenthisbookletuntilinstructedtodoso.Eachquestionisworth10marksNumberofquestions:10Calculatorsareallowed,withthefollowingrestriction:youmaynotuseadevicethathasinternetaccess,thatcancommunicatewithotherdevices,orthatcontainspreviouslystoredinformation.Forexample,youmaynotuseasmartphoneoratablet.Partsofeachquestioncanbeoftwotypes:1.SHORTANSWERpartsindicatedby•worth3markseach•fullmarksgivenforacorrectanswerwhichisplacedinthebox•partmarksawardedonlyifrelevantworkisshowninthespaceprovided2.FULLSOLUTIONpartsindicatedby••••worththeremainderofthe10marksforthequestionmustbewrittenintheappropriatelocationintheanswerbookletmarksawardedforcompleteness,clarity,andstyleofpresentationacorrectsolutionpoorlypresentedwillnotearnfullmarksWRITEALLANSWERSINTHEANSWERBOOKLETPROVIDED.•Extrapaperforyourfinishedsolutionssuppliedbyyoursupervisingteachermustbeinsertedintoyouranswerbooklet.Writeyourname,schoolname,andquestionnumberonanyinsertedpages.•Expressanswersassimplifiedexactnumbersexceptwhereotherwiseindicated.√Forexample,π+1and1−2aresimplifiedexactnumbers.Donotdiscusstheproblemsorsolutionsfromthiscontestonlineforthenext48hours.Thename,grade,schoolandlocation,andscorerangeofsometop-scoringstudentswillbepublishedonourwebsite,cemc.uwaterloo.ca.Inaddition,thename,grade,schoolandlocation,andscoreofsometop-scoringstudentsmaybesharedwithothermathematicalorganizationsforotherrecognitionopportunities.www.linstitute.netNOTE:1.Pleasereadtheinstructionsonthefrontcoverofthisbooklet.2.Writeallanswersintheanswerbookletprovided.3.4.Forquestionsmarked,placeyouranswerintheappropriateboxintheanswerbookletandshowyourwork.Forquestionsmarked,provideawell-organizedsolutionintheanswerbooklet.Usemathematicalstatementsandwordstoexplainallofthestepsofyoursolution.Workoutsomedetailsinroughonaseparatepieceofpaperbeforewritingyourfinishedsolution.Diagramsarenotdrawntoscale.Theyareintendedasaidsonly.Whilecalculatorsmaybeusedfornumericalcalculations,othermathematicalstepsmustbeshownandjustifiedinyourwrittensolutionsandspecificmarksmaybeallocatedforthesesteps.Forexample,whileyourcalculatormightbeabletofindthex-interceptsofthegraphofanequationlikey=x3−x,youshouldshowthealgebraicstepsthatyouusedtofindthesenumbers,ratherthansimplywritingthesenumbersdown.5.6.ANoteaboutBubbling

Pleasemakesurethatyouhavecorrectlycodedyourname,dateofbirthandgradeontheStudentInformationForm,andthatyouhaveansweredthequestionabouteligibility.1.

(a)Thereisonepair(a,b)ofpositiveintegersforwhich5a+3b=19.

Whatarethevaluesofaandb?

(b)Howmanypositiveintegersnsatisfy5<2n<2017?

(c)Jimmybought600Eurosattherateof1Euroequals$1.50.Hethenconverted

his600Eurosbackintodollarsattherateof$1.00equals0.75Euros.HowmanyfewerdollarsdidJimmyhaveafterthesetwotransactionsthanhehadbeforethesetwotransactions?

2.

(a)Whatareallvaluesofxforwhichx=0andx=1and(b)Inamagicsquare,thenumbersineach

row,thenumbersineachcolumn,andthenumbersoneachdiagonalhavethesamesum.Inthemagicsquareshown,whatarethevaluesofa,bandc?

511

=+?

x(x−1)xx−1

0c

204−12

a

b

(c)(i)Forwhatpositiveintegernis1002−n2=9559?

(ii)Determineonepair(a,b)ofpositiveintegersforwhicha>1andb>1and

ab=9559.

www.linstitute.net3.

(a)Inthediagram,󰀈ABCisright-angledatB

and󰀈ACDisright-angledatA.Also,AB=3,BC=4,andCD=13.WhatistheareaofquadrilateralABCD?

B4C

3A

13

(b)ThreeidenticalrectanglesPQRS,WTUV

andXWVYarearranged,asshown,sothatRSliesalongTX.Theperimeterofeachofthethreerectanglesis21cm.Whatistheperimeterofthewholeshape?

DPW

SX

QT

RUVY(c)Oneofthefacesofarectangularprismhasarea27cm2.Anotherfacehasarea

32cm2.Ifthevolumeoftheprismis144cm3,determinethesurfaceareaoftheprismincm2.(a)Theequationsy=a(x−2)(x+4)andy=2(x−h)2+krepresentthesame

parabola.Whatarethevaluesofa,handk?(b)Inanarithmeticsequencewith5terms,thesumofthesquaresofthefirst3terms

equalsthesumofthesquaresofthelast2terms.Ifthefirsttermis5,determineallpossiblevaluesofthefifthterm.

(Anarithmeticsequenceisasequenceinwhicheachtermafterthefirstisobtainedfromtheprevioustermbyaddingaconstant.Forexample,3,5,7,9,11isanarithmeticsequencewithfiveterms.)

4.

5.

(a)Danwasborninayearbetween1300and1400.Stevewasborninayearbetween

1400and1500.EachwasbornonApril6inayearthatisaperfectsquare.Eachlivedfor110years.InwhatyearwhiletheywerebothaliveweretheiragesbothperfectsquaresonApril7?(b)DetermineallvaluesofkforwhichthepointsA(1,2),B(11,2)andC(k,6)form

aright-angledtriangle.

6.

(a)Thediagramshowstwohillsthatmeet

atO.Onehillmakesa30◦anglewiththehorizontalandtheotherhillmakesa45◦anglewiththehorizontal.PointsAandBareonthehillssothatOA=OB=20m.VerticalpolesBDandACareconnectedbyastraightcableCD.IfAC=6m,whatisthelengthofBDforwhichCDisasshortaspossible?

CA30°DBO45°(b)Ifcosθ=tanθ,determineallpossiblevaluesofsinθ,givingyouranswer(s)as

simplifiedexactnumbers.

www.linstitute.net7.

(a)Linhisdrivingat60km/honalongstraighthighwayparalleltoatraintrack.

Every10minutes,sheispassedbyatraintravellinginthesamedirectionassheis.Thesetrainsdepartfromthestationbehindherevery3minutesandalltravelatthesameconstantspeed.Whatistheconstantspeedofthetrains,inkm/h?(b)Determineallpairs(a,b)ofrealnumbersthatsatisfythefollowingsystemof

equations:

√√

a+b=8

log10a+log10b=2

Giveyouranswer(s)aspairsofsimplifiedexactnumbers.

8.

(a)Inthediagram,linesegmentsACandDF

aretangenttothecircleatBandE,respectively.Also,AFintersectsthecircleatPandR,andintersectsBEatQ,asshown.If∠CAF=35◦,∠DFA=30◦,and∠FPE=25◦,determinethemeasureof∠PEQ.

APDBQERC(b)Inthediagram,ABCDandPNCDare

squaresofsidelength2,andPNCDisperpendiculartoABCD.PointMischosenonthesamesideofPNCDasABsothat󰀈PMNisparalleltoABCD,sothat∠PMN=90◦,andsothatPM=MN.DeterminethevolumeoftheconvexsolidABCDPMN.

FP

MN

DA

BC

www.linstitute.net9.

Apermutationofalistofnumbersisanorderedarrangementofthenumbersinthatlist.Forexample,3,2,4,1,6,5isapermutationof1,2,3,4,5,6.Wecanwritethispermutationasa1,a2,a3,a4,a5,a6,wherea1=3,a2=2,a3=4,a4=1,a5=6,anda6=5.

(a)Determinetheaveragevalueof

|a1−a2|+|a3−a4|

overallpermutationsa1,a2,a3,a4of1,2,3,4.(b)Determinetheaveragevalueof

a1−a2+a3−a4+a5−a6+a7

overallpermutationsa1,a2,a3,a4,a5,a6,a7of1,2,3,4,5,6,7.(c)Determinetheaveragevalueof

|a1−a2|+|a3−a4|+···+|a197−a198|+|a199−a200|

(∗)

overallpermutationsa1,a2,a3,...,a199,a200of1,2,3,4,...,199,200.(Thesumlabelled(∗)contains100termsoftheform|a2k−1−a2k|.)

10.

ConsiderasetSthatcontainsm≥4elements,eachofwhichisapositiveintegerandnotwoofwhichareequal.WecallSboringifitcontainsfourdistinctintegersa,b,c,dsuchthata+b=c+d.WecallSexcitingifitisnotboring.Forexample,{2,4,6,8,10}isboringsince4+8=2+10.Also,{1,5,10,25,50}isexciting.(a)Findanexcitingsubsetof{1,2,3,4,5,6,7,8}thatcontainsexactly5elements.(b)Provethat,ifSisanexcitingsetofm≥4positiveintegers,thenScontainsan

m2−m

integergreaterthanorequalto.

4(c)Definerem(a,b)tobetheremainderwhenthepositiveintegeraisdividedbythe

positiveintegerb.Forexample,rem(10,7)=3,rem(20,5)=0,andrem(3,4)=3.

Letnbeapositiveintegerwithn≥10.Foreachpositiveintegerkwith1≤k≤n,definexk=2n·rem(k2,n)+k.Determine,withproof,allpositiveintegersn≥10forwhichtheset{x1,x2,...,xn−1,xn}ofnintegersisexciting.