美国数模竞赛题目历届.docx
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美国数模竞赛题目历届.docx
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美国数模竞赛题目历届
2008MCMProblems
PROBLEMA:
TakeaBath
Considertheeffectsonlandfromthemeltingofthenorthpolaricecapduetothepredictedincreaseinglobaltemperatures.Specifically,modeltheeffectsonthecoastofFloridaeverytenyearsforthenext50yearsduetothemelting,withparticularattentiongiventolargemetropolitanareas.Proposeappropriateresponsestodealwiththis.Acarefuldiscussionofthedatausedisanimportantpartoftheanswer.
PROBLEMB:
CreatingSudokuPuzzles
DevelopanalgorithmtoconstructSudokupuzzlesofvaryingdifficulty.Developmetricstodefineadifficultylevel.Thealgorithmandmetricsshouldbeextensibletoavaryingnumberofdifficultylevels.Youshouldillustratethealgorithmwithatleast4difficultylevels.Youralgorithmshouldguaranteeauniquesolution.Analyzethecomplexityofyouralgorithm.Yourobjectiveshouldbetominimizethecomplexityofthealgorithmandmeettheaboverequirements.
2007MCMProblems
PROBLEMA:
Gerrymandering
TheUnitedStatesConstitutionprovidesthattheHouseofRepresentativesshallbecomposedofsomenumber(currently435)ofindividualswhoareelectedfromeachstateinproportiontothestate’spopulationrelativetothatofthecountryasawhole.Whilethisprovidesawayofdetermininghowmanyrepresentativeseachstatewillhave,itsaysnothingabouthowthedistrictrepresentedbyaparticularrepresentativeshallbedeterminedgeographically.Thisoversighthasledtoegregious(atleastsomepeoplethinkso,usuallynottheincumbent)districtshapesthatlook“unnatural”bysomestandards.
Hencethefollowingquestion:
Supposeyouweregiventheopportunitytodrawcongressionaldistrictsforastate.Howwouldyoudosoasapurely“baseline”exercisetocreatethe“simplest”shapesforallthedistrictsinastate?
Therulesincludeonlythateachdistrictinthestatemustcontainthesamepopulation.Thedefinitionof“simple”isuptoyou;butyouneedtomakeaconvincingargumenttovotersinthestatethatyoursolutionisfair.Asanapplicationofyourmethod,drawgeographicallysimplecongressionaldistrictsforthestateofNewYork.
PROBLEMB:
TheAirplaneSeatingProblem
Airlinesarefreetoseatpassengerswaitingtoboardanaircraftinanyorderwhatsoever.Ithasbecomecustomarytoseatpassengerswithspecialneedsfirst,followedbyfirst-classpassengers(whositatthefrontoftheplane).Thencoachandbusiness-classpassengersareseatedbygroupsofrows,beginningwiththerowatthebackoftheplaneandproceedingforward.
Apartfromconsiderationofthepassengers’waittime,fromtheairline’spointofview,timeismoney,andboardingtimeisbestminimized.Theplanemakesmoneyfortheairlineonlywhenitisinmotion,andlongboardingtimeslimitthenumberoftripsthataplanecanmakeinaday.
Thedevelopmentoflargerplanes,suchastheAirbusA380(800passengers),accentuatetheproblemofminimizingboarding(anddeboarding)time.
Deviseandcompareproceduresforboardinganddeboardingplaneswithvaryingnumbersofpassengers:
small(85–210),midsize(210–330),andlarge(450–800).
Prepareanexecutivesummary,nottoexceedtwosingle-spacedpages,inwhichyousetoutyourconclusionstoanaudienceofairlineexecutives,gateagents,andflightcrews.
Note:
The2pageexecutivesummaryistobeincludedINADDITIONtothereportsrequiredbythecontestguidelines.
AnarticleappearedintheNYTimesNov14,2006addressingprocedurescurrentlybeingfollowedandtheimportancetotheairlineoffindingbettersolutions.Thearticlecanbeseenat:
2006MCMProblems
PROBLEMA:
PositioningandMovingSprinklerSystemsforIrrigation
Thereareawidevarietyoftechniquesavailableforirrigatingafield.Thetechnologiesrangefromadvanceddripsystemstoperiodicflooding.Oneofthesystemsthatisusedonsmallerranchesistheuseof"handmove"irrigationsystems.Lightweightaluminumpipeswithsprinklerheadsareputinplaceacrossfields,andtheyaremovedbyhandatperiodicintervalstoinsurethatthewholefieldreceivesanadequateamountofwater.Thistypeofirrigationsystemischeaperandeasiertomaintainthanothersystems.Itisalsoflexible,allowingforuseonawidevarietyoffieldsandcrops.Thedisadvantageisthatitrequiresagreatdealoftimeandefforttomoveandsetuptheequipmentatregularintervals.
Giventhatthistypeofirrigationsystemistobeused,howcanitbeconfiguredtominimizetheamountoftimerequiredtoirrigateafieldthatis80metersby30meters?
Forthistaskyouareaskedtofindanalgorithmtodeterminehowtoirrigatetherectangularfieldthatminimizestheamountoftimerequiredbyaranchertomaintaintheirrigationsystem.Onepipesetisusedinthefield.Youshoulddeterminethenumberofsprinklersandthespacingbetweensprinklers,andyoushouldfindascheduletomovethepipes,includingwheretomovethem.
Apipesetconsistsofanumberofpipesthatcanbeconnectedtogetherinastraightline.Eachpipehasa10cminnerdiameterwithrotatingspraynozzlesthathavea0.6cminnerdiameter.Whenputtogethertheresultingpipeis20meterslong.Atthewatersource,thepressureis420Kilo-Pascal抯andhasaflowrateof150litersperminute.Nopartofthefieldshouldreceivemorethan0.75cmperhourofwater,andeachpartofthefieldshouldreceiveatleast2centimetersofwaterevery4days.Thetotalamountofwatershouldbeappliedasuniformlyaspossible
PROBLEMB:
WheelChairAccessatAirports
Oneofthefrustrationswithairtravelistheneedtoflythroughmultipleairports,andeachstopgenerallyrequireseachtravelertochangetoadifferentairplane.Thiscanbeespeciallydifficultforpeoplewhoarenotabletoeasilywalktoadifferentflight'swaitingarea.Oneofthewaysthatanairlinecanmakethetransitioneasieristoprovideawheelchairandanescorttothosepeoplewhoaskforhelp.Itisgenerallyknownwellinadvancewhichpassengersrequirehelp,butitisnotuncommontoreceivenoticewhenapassengerfirstregistersattheairport.Inrareinstancesanairlinemaynotreceivenoticefromapassengeruntiljustpriortolanding.
Airlinesareunderconstantpressuretokeeptheircostsdown.Wheelchairswearoutandareexpensiveandrequiremaintenance.Thereisalsoacostformakingtheescortsavailable.
2005MCMProblems
PROBLEMA:
FloodPlanning
LakeMurrayincentralSouthCarolinaisformedbyalargeearthendam,whichwascompletedin1930forpowerproduction.Modelthefloodingdownstreamintheeventthereisacatastrophicearthquakethatbreachesthedam.
Twoparticularquestions:
RawlsCreekisayear-roundstreamthatflowsintotheSaludaRiverashortdistancedownriverfromthedam.HowmuchfloodingwilloccurinRawlsCreekfromadamfailure,andhowfarbackwillitextend?
CouldthefloodbesomassivedownstreamthatwaterwouldreachuptotheS.C.StateCapitolBuilding,whichisonahilloverlookingtheCongareeRiver?
PROBLEMB:
Tollbooths
Heavily-traveledtollroadssuchastheGardenStateParkway,Interstate95,andsoforth,aremulti-lanedividedhighwaysthatareinterruptedatintervalsbytollplazas.Becausecollectingtollsisusuallyunpopular,itisdesirabletominimizemotoristannoyancebylimitingtheamountoftrafficdisruptioncausedbythetollplazas.Commonly,amuchlargernumberoftollboothsisprovidedthanthenumberoftravellanesenteringthetollplaza.Uponenteringthetollplaza,theflowofvehiclesfansouttothelargernumberoftollbooths,andwhenleavingthetollplaza,theflowofvehiclesisrequiredtosqueezebackdowntoanumberoftravellanesequaltothenumberoftravellanesbeforethetollplaza.Consequently,whentrafficisheavy,congestionincreasesupondeparturefromthetollplaza.Whentrafficisveryheavy,congestionalsobuildsattheentrytothetollplazabecauseofthetimerequiredforeachvehicletopaythetoll.
Makeamodeltohelpyoudeterminetheoptimalnumberoftollboothstodeployinabarrier-tollplaza.Explicitlyconsiderthescenariowherethereisexactlyonetollboothperincomingtravellane.Underwhatconditionsisthismoreorlesseffectivethanthecurrentpractice?
Notethatthedefinitionof "optimal"isuptoyoutodetermine.
2005ICMProblem
PROBLEMC:
NonrenewableResources
Selectavitalnonrenewableorexhaustibleresource(water,mineral,energy,food,etc.)forwhichyourteamcanfindappropriateworld-widehistoricdataonitsendowment,discovery,annualconsumption,andprice.
Themodelingtasksare:
1.Usingtheendowment,discoveries,andconsumptiondata,modelthedepletionordegradationofthecommodityoveralonghorizonusingresourcemodelingprinciples.
2.Adjustthemodeltoaccountforfutureeconomic,demographic,politicalandenvironmentalfactors. Besuretorevealthedetailsofyourmodel,providevisualizationsofthemodel’soutput,andexplainlimitationsofthemodel.
3.Createafair,practical"harvesting/management"policythatmayincludeeconomicincentivesordisincentives,whichsustaintheusageoveralongperiodoftimewhileavoidingseveredisruptionofconsumption,degradationorrapidexhaustionoftheresource.
4.Developa"security"policythatprotectstheresourceagainsttheft,misuse,disruption,andunnecessarydegradationordestructionoftheresource. Otherissuesthatmayneedtobeaddressedarepoliticalandsecuritymanagementalternativesassociatedwiththesepolicies.
5.Developpoliciestocontrolanyshort-orlong-term"environmentaleffects"oftheharvesti
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