美国数模竞赛题目历届.docx

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