美赛历年试题.docx

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