美赛历年试题.docx
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美赛历年试题.docx
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美赛历年试题
2009年美国大学生数学建模竞赛MCM、ICM试题
2009MCMA:
DesigningaTrafficCircle
Manycitiesandcommunitieshavetrafficcircles—fromlargeoneswithmanylanesinthecircle(suchasattheArcdeTriompheinParisandtheVictoryMonumentinBangkok)tosmalloneswithoneortwolanesinthecircle.Someofthesetrafficcirclespositionastopsignorayieldsignoneveryincomingroadthatgivesprioritytotrafficalreadyinthecircle;somepositionayieldsigninthecircleateachincomingroadtogiveprioritytoincomingtraffic;andsomepositionatrafficlightoneachincomingroad(withnorightturnallowedonaredlight).Otherdesignsmayalsobepossible.
Thegoalofthisproblemistouseamodeltodeterminehowbesttocontroltrafficflowin,around,andoutofacircle.Stateclearlytheobjective(s)youuseinyourmodelformakingtheoptimalchoiceaswellasthefactorsthataffectthischoice.IncludeaTechnicalSummaryofnotmorethantwodouble-spacedpagesthatexplainstoaTrafficEngineerhowtouseyourmodeltohelpchoosetheappropriateflow-controlmethodforanyspecifictrafficcircle.Thatis,summarizetheconditionsunderwhicheachtypeoftraffic-controlmethodshouldbeused.Whentrafficlightsarerecommended,explainamethodfordetermininghowmanysecondseachlightshouldremaingreen(whichmayvaryaccordingtothetimeofdayandotherfactors).Illustratehowyourmodelworkswithspecificexamples.
2009MCMB:
EnergyandtheCellPhone
Thisquestioninvolvesthe“energy”consequencesofthecellphonerevolution.Cellphoneusageismushrooming涌现,andmanypeopleareusingcellphonesandgivinguptheirlandlinetelephones.Whatistheconsequenceofthisintermsofelectricityuse?
Everycellphonecomeswithabatteryandarecharger.充电器
Requirement1
ConsiderthecurrentUS,acountryofabout300millionpeople.EstimatefromavailabledatathenumberHofhouseholds,withmmemberseach,thatinthepastwereservicedbylandlines.Now,supposethatallthelandlinesarereplacedbycellphones;thatis,eachofthemmembersofthehouseholdhasacellphone.Modeltheconsequencesofthischangeforelectricityutilization使用inthecurrentUS,bothduringthetransitionandduringthesteadystate.Theanalysisshouldtakeintoaccounttheneedforcharging索价thebatteriesofthecellphones,aswellasthefactthatcellphonesdonotlastaslongaslandlinephones(forexample,thecellphonesgetlostandbreak).
Requirement2
Considerasecond“PseudoUS”—acountryofabout300millionpeoplewithaboutthesameeconomicstatusasthecurrentUS.However,thisemerging新兴的countryhasneitherlandlinesnorcellphones.Whatistheoptimal最理想的wayofprovidingphoneservicetothiscountryfromanenergyperspective观点?
Ofcourse,cellphoneshavemanysocialconsequencesandusesthatlandlinephonesdonotallow.Adiscussionofthebroad明显的andhidden隐藏的consequencesofhavingonlylandlines,onlycellphones,oramixtureofthetwoiswelcomed.
Requirement3
Cellphonesperiodically定期的needtoberecharged.充电However,manypeoplealwayskeeptheirrechargerplugged塞in.Additionally,manypeoplechargetheirphoneseverynight,whethertheyneedtoberechargedornot.ModeltheenergycostsofthiswastefulpracticeforaPseudoUSbaseduponyouranswertoRequirement2.AssumethatthePseudoUSsupplieselectricityfromoil.Interpret解释yourresultsintermsof就……而言barrels桶ofoil.
Requirement4
Estimatesvaryontheamountofenergythatisusedbyvariousrechargertypes(TV,DVR,computerperipherals,andsoforth)whenleftpluggedinbutnotchargingthedevice.充电器插着单没用来给手机充电UseaccuratedatatomodeltheenergywastedbythecurrentUSintermsofbarrelsofoilperday.
Requirement5
Nowconsiderpopulationandeconomicgrowthoverthenext50years.HowmightatypicalPseudoUSgrow?
Foreach10yearsforthenext50years,predicttheenergyneedsforprovidingphoneservicebaseduponyouranalysisinthefirstthreerequirements.Again,assumeelectricityisprovidedfromoil.Interpretyourpredictionsintermofbarrelsofoil.
2008年美国大学生数学建模竞赛MCM、ICM试题
2008MCMA:
TakeaBath
Considertheeffectsonlandfromthemeltingofthenorthpolaricecapduetothepredictedincreaseinglobaltemperatures.Specifically,modeltheeffectsonthecoastofFloridaeverytenyearsforthenext50yearsduetothemelting,withparticularattentiongiventolargemetropolitanareas.Proposeappropriateresponsestodealwiththis.Acarefuldiscussionofthedatausedisanimportantpartoftheanswer.
2008MCMB:
CreatingSudoku数独Puzzles
Developanalgorithm演算法toconstruct构造Sudokupuzzles困惑ofvaryingdifficulty.Developmetrics公尺的todefineadifficultylevel.Thealgorithmandmetricsshouldbeextensible可扩充的toavaryingnumberofdifficultylevels.Youshouldillustrate阐明thealgorithmwithatleast4difficultylevels.Youralgorithmshouldguaranteeauniquesolution.Analyzethecomplexity复杂度ofyouralgorithm.Yourobjective目标shouldbetominimizethecomplexityofthealgorithmandmeettheaboverequirements.
2007年美国大学生数学建模竞赛MCM、ICM试题
2007MCMA:
Gerrymandering选区不公正的重新划分
GerrymanderingTheUnitedStatesConstitution宪法providesthattheHouseofRepresentativesshallbecomposedof由什么组成somenumber(currently435)ofindividualswhoareelectedfromeachstateinproportion比例tothestate’spopulationrelative相应的,成比例的tothatofthecountryasawhole.Whilethisprovidesawayofdetermininghowmanyrepresentatives州议员eachstatewillhave,itsaysnothingabouthowthedistrict行政区,辖区representedbyaparticular特定的representativeshallbedeterminedgeographically.Thisoversight疏忽出错hasledtoegregious极坏的(atleastsomepeoplethinkso,usuallynottheincumbent)districtshapes情况thatlook“unnatural”bysomestandards.标准
Hencethefollowingquestion:
Supposeyouweregiventheopportunitytodrawcongressional美国国会的districtsforastate.Howwouldyoudosoasapurely纯粹的“baseline”基准线exercisetocreatethe“simplest”shapesforallthedistrictsinastate?
Therulesincludeonlythateachdistrictinthestatemustcontainthesamepopulation.Thedefinitionof“simple”isuptoyou;butyouneedtomakeaconvincing令人信服的argument论据tovoters选举人inthestatethatyoursolutionisfair公平的.Asanapplication应用ofyourmethod,draw草拟制定geographically地理上的simplecongressionaldistrictsforthestateofNewYork.
2007MCMB:
TheAirplaneSeatingProblem
Airlinesarefreetoseatpassengerswaitingtoboard上飞机anaircraftinanyorderwhatsoever无论什么.Ithasbecomecustomary习惯上的toseatpassengerswithspecialneedsfirst,followedbyfirst-class头等的passengers(whositatthefrontoftheplane).Thencoachandbusiness-classpassengersareseatedbygroupsofrows,beginningwiththerowatthebackoftheplaneandproceeding行进forward.
Apartfromconsideration考虑的事ofthepassengers’waittime,fromtheairline’spointofview,timeismoney,andboardingtimeisbestminimized.Theplanemakesmoneyfortheairlineonlywhenitisinmotion,运转andlongboardingtimeslimitthenumberoftrips航行thataplanecanmakeinaday.
Thedevelopmentoflargerplanes,suchastheAirbusA380(800passengers),accentuate更突出theproblemofminimizingboarding(anddeboarding)time.
Devise想出andcompareprocedures步骤forboardinganddeboardingplaneswithvaryingnumbersofpassengers:
small(85–210),midsize(210–330),andlarge(450–800).
Prepareanexecutive执行的summary,nottoexceed超出twosingle-spacedpages,inwhichyousetoutyourconclusionstoanaudienceofairlineexecutives,gateagents,代理人andflightcrews.工作人员
AnarticleappearedintheNYTimesNov14,2006addressingprocedurescurrently当前beingfollowedandtheimportancetotheairlineoffindingbettersolutions.Thearticlecanbeseenat:
2006年美国大学生数学建模竞赛MCM、ICM试题
2006MCMA:
PositioningandMovingSprinkler洒水器SystemsforIrrigation灌溉
Thereareawidevarietyoftechniquesavailableforirrigatingafield.Thetechnologiesrangefromadvanceddrip洒下systemstoperiodic周期性的flooding.Oneofthesystemsthatisusedonsmallerranchesistheuseof“handmove”irrigationsystems.Lightweight轻量的aluminum铝pipes输送管withsprinkler洒水器headsareputinplaceacrossfields,andtheyaremovedbyhandatperiodic周期性的intervals距离toinsurethatthewholefieldreceivesanadequateamountofwater.Thistypeofirrigationsystemischeaperandeasiertomaintain维修thanothersystems.Itisalsoflexible,allowingforuseonawidevarietyoffieldsandcrops.Thedisadvantageisthatitrequiresagreatdealoftimeandefforttomoveandsetuptheequipmentatregularintervals.距离
Giventhatthistypeofirrigationsystemistobeused,howcanitbeconfigured安装tominimizetheamountoftimerequiredtoirrigateafieldthatis80metersby30meters?
Forthistaskyouareaskedtofindanalgorithm算法todeterminehowtoirrigatetherectangular矩形的fieldthatminimizestheamountoftimerequiredbyarancher农场工人tomaintaintheirrigationsystem.Onepipe输送管setisusedinthefield.Youshoulddeterminethenumberofsprinklers洒水器andthespacingbetweensprinklers,andyoushouldfindaschedule计划表tomovethepipes,includingwheretomovethem.
Apipesetconsistsofanumberofpipesthatcanbeconnectedtogetherinastraightline.Eachpipehasa10cminnerdiameter直径withrotating转动spray喷雾器nozzles管嘴thathavea0.6cminnerdiameter.Whenputtogethertheresultingpipeis20meterslong.Atthewatersource,thepressureis420Kilo-Pascal’sandhasaflowrateof150litersperminute.Nopartofthefieldshouldreceivemorethan0.75cmperhourofwater,andeachpartofthefieldshouldreceiveatlea
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