美赛历年题目.docx
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美赛历年题目
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2014MCMProblems
PROBLEMA:
TheKeep-Right-Except-To-PassRule
Incountrieswheredrivingautomobilesontherightistherule(thatis,USA,ChinaandmostothercountriesexceptforGreatBritain,Australia,andsomeformerBritishcolonies),multi-lanefreewaysoftenemploy arulethatrequiresdriverstodriveintheright-mostlaneunlesstheyarepassinganothervehicle,inwhichcasetheymoveonelanetotheleft,pass,andreturntotheirformertravellane.
Buildandanalyzeamathematicalmodeltoanalyzetheperformanceofthisruleinlightandheavytraffic.Youmaywishtoexaminetradeoffsbetweentrafficflowandsafety,theroleofunder-orover-postedspeedlimits(thatis,speedlimitsthataretoolowortoohigh),and/orotherfactorsthatmaynotbeexplicitlycalledoutinthisproblemstatement.Isthisruleeffectiveinpromotingbettertrafficflow?
Ifnot,suggestandanalyzealternatives(toincludepossiblynoruleofthiskindatall)thatmightpromotegreatertrafficflow,safety,and/orotherfactorsthatyoudeemimportant.
Incountrieswheredrivingautomobilesontheleftisthenorm,arguewhetherornotyoursolutioncanbecarriedoverwithasimplechangeoforientation,orwouldadditionalrequirementsbeneeded.
Lastly,theruleasstatedabovereliesuponhumanjudgmentforcompliance.Ifvehicletransportationonthesameroadwaywasfullyunderthecontrolofanintelligentsystem–eitherpartoftheroadnetworkorimbeddedinthedesignofallvehiclesusingtheroadway–towhatextentwouldthischangetheresultsofyourearlieranalysis?
PROBLEMB:
CollegeCoachingLegends
SportsIllustrated,amagazineforsportsenthusiasts,islookingforthe“bestalltimecollegecoach”maleorfemaleforthepreviouscentury.Buildamathematicalmodeltochoosethe best collegecoachorcoaches(pastorpresent)fromamongeithermaleorfemalecoachesinsuchsportsascollegehockeyorfieldhockey,football,baseballorsoftball,basketball,orsoccer.Doesitmakeadifferencewhichtimelinehorizonthatyouuseinyouranalysis,i.e.,doescoachingin1913differfromcoachingin2013?
Clearlyarticulateyourmetricsforassessment.Discusshowyourmodelcanbeappliedingeneralacrossbothgendersandallpossiblesports.Presentyourmodel’stop5coachesineachof3differentsports.
InadditiontotheMCMformatandrequirements,preparea1-2pagearticlefor SportsIllustrated thatexplainsyourresultsandincludesanon-technicalexplanationofyourmathematicalmodelthat sportsfans willunderstand.
ProblemA
Problem:
UnloadingCommuterTrains
TrainsarriveoftenatacentralStation,thenexusformanycommutertrainsfromsuburbsoflargercitiesona“commuter”line.Mosttrainsarelong(perhaps10ormorecarslong).Thedistanceapassengerhastowalktoexitthetrainareaisquitelong.Eachtraincarhasonlytwoexits,oneneareachendsothatthecarscancarryasmanypeopleaspossible.Eachtraincarhasacenteraisleandtherearetwoseatsononesideandthreeseatsontheotherforeachrowofseats.
Toexitatypicalstationofinterest,passengersmustexitthecar,andthenmaketheirwaytoastairwaytogettothenextleveltoexitthestation.Usuallythesetrainsarecrowdedsothereisa“fan”ofpassengersfromthetraintryingtogetupthestairway.Thestairwaycouldaccommodatetwocolumnsofpeopleexitingtothetopofthestairs.
Mostcommutertrainplatformshavetwotracksadjacenttotheplatform.Intheworstcase,iftwofullyoccupiedtrainsarrivedatthesametime,itmighttakealongtimeforallthepassengerstogetuptothemainlevelofthestation.
Buildamathematicalmodeltoestimatetheamountoftimeforapassengertoreachthestreetlevelofthestationtoexitthecomplex.Assumethereare n carstoatrain,eachcarhaslengthd.Thelengthoftheplatformis p,andthenumberofstairsineachstaircaseis q.
Useyourmodeltospecificallyoptimize(minimize)thetimetraveledtoreachstreetleveltoexitastationforthefollowing:
Requirement1. Onefullyoccupiedtrain’spassengerstoexitthetrain,andascendthestairstoreachthestreetaccesslevelofthestation
Requirement2. Twofullyoccupiedtrains’passengers(allpassengersexitontoacommonplatform)toexitthetrains,andascendthestairstoreachthestreetaccesslevelofthestation.
Requirement3. Ifyoucouldredesignthelocationofthestairwaysalongtheplatform,whereshouldthesestairwaysbeplacedtominimizethetimeforoneortwotrains’passengerstoexitthestation?
Requirement4. Howdoesthetimetostreetlevelvarywiththenumbersofstairwaysthatonebuilds?
Requirement5. Howdoesthetimevaryifthestairwayscanaccommodate kpeople, k anintegergreaterthanone?
InadditiontotheHiMCMformat,prepareashortnon-technicalarticletothedirectoroftransportationexplainingwhytheyshouldadoptyourmodeltoimproveexitingastation.
ProblemB
Problem:
TheNextPlague?
In2014,theworldsawtheinfectiousEbolavirusspreadinginwesternAfrica.Throughouthumanhistory,epidemicshavecomeandgonewithsomeinfectingand/orkillingthousandsandlastingforyearsandotherstakinglessofahumantoll.Somebelievetheseeventsarejustnature’swayofcontrollingthegrowthofaspecieswhileothersthinktheycouldbeaconspiracyordeliberateacttocauseharm.Thisproblemwillmostlikelycomedowntohowtoexpend(ornotexpend)scarceresources(doctors,containmentfacilities,money,research,serums,etc…)todealwithacrisis.
Situation:
AroutinehumanitarianmissiononanislandinIndonesiareportedasmallvillagewherealmosthalfofits300inhabitantsareshowingsimilarsymptoms.Inthepastweek,15ofthe“infected”havedied.Thisvillageisknowntotradewithnearbyvillagesandotherislands.Yourmodelingteamworksforamajorcenterofdiseasecontrolinthecapitalof your country(orifyouprefer,fortheInternationalWorldHealthOrganization).
Requirement1:
Developamathematicalmodel(s)thatperformsthefollowingfunctionsaswellashow/whentobestallocatethesescarceresourcesand…
•Determinesandclassifiesthetypeandseverityofthespreadofthedisease
•Determinesifanepidemiciscontainedornot
•Triggersappropriatemeasures(whentotreat,whentotransportvictims,whentorestrictmovement,whentoletadiseaserunitscourse,etc…)tocontainadiseaseNote:
Whileyoumaywanttostartwiththewell-known“SIR”familyofmodelsforpartsofthisproblem,considerothers,modificationstotheSIR,multiplemodels,orcreatingyourown.
Requirement2:
Basedontheinformationgiven,yourmodel,andtheassumptionsyourteamhasmade,whatinitialrecommendationsdoesyourteamhaveforyourcountry’scenterfordiseasecontrol?
(Give3-5recommendationswithjustifications)
AdditionalSituationalInformation:
Amulti-nationalresearchteamjustreturnedtoyourcountry’scapitalafterspending7daysgatheringinformationintheinfectedvillage.
Requirement3:
Youcanaskthemupto3questionstoimproveyourmodel.Whatwouldyouaskandwhy?
AdditionalSituationalInformation:
Themulti-nationalresearchteamconcludedthatthedisease:
•Appearstospreadthroughcontactwithbodilyfluidsofaninfectedperson
•Theelderlyandchildrenaremorelikelytodieifinfected
•Anearbyislandisstartingtoshowsimilarsignsofinfection
•Oneoftheresearchersthatreturnedtoyourcapitalappearsinfected
Requirement4:
Howdoestheadditionalinformationabovechange/modifyyourmodel?
Requirement5:
Writeaone-pagesynopsisofyourfindingsforyourlocalnon-technicalnewsoutlet.
2013MCMProblems
PROBLEMA:
TheUltimateBrowniePan
Whenbakinginarectangularpanheatisconcentratedinthe4cornersandtheproductgetsovercookedatthecorners(andtoalesserextentattheedges).Inaroundpantheheatisdistributedevenlyovertheentireouteredgeandtheproductisnotovercookedattheedges.However,sincemostovensarerectangularinshapeusingroundpansisnotefficientwithrespecttousingthespaceinanoven.Developamodeltoshowthedistributionofheatacrosstheouteredgeofapanforpansofdifferentshapes-rectangulartocircularandothershapesinbetween.
Assume
1.Awidthtolengthratioof W/Lfortheovenwhichisrectangularinshape.
2.Eachpanmusthaveanareaof A.
3.Initiallytworacksintheoven,evenlyspaced.
Developamodelthatcanbeusedtoselectthebesttypeofpan(shape)underthefollowingconditions:
1.Maximizenumberofpansthatcanfitintheoven(N)
2.Maximizeevendistributionofheat(H)forthepan
3.Optimizeacombinationofconditions
(1)and
(2)whereweightspand(1- p)areassignedtoillustratehowtheresultsvarywithdifferentvaluesof W/L and p.
InadditiontoyourMCMformattedsolution,prepareaonetotwopageadvertisingsheetforthenewBrownieGourmetMagazinehighlightingyourdesignandresults.
PROBLEMB:
Water,Water,Everywhere
Freshwateristhelimitingconstraintfordevelopmentinmuchoftheworld.Buildamathematicalmodelfordetermininganeffective,feasible,andcost-efficientwaterstrategyfor2013tomeettheprojectedwaterneedsof[pickonecountryfromthelistbelow]in2025,andidentifythebestwaterstrategy.Inparticular,yourmathematicalmodelmustaddressstorageandmovement;de-salinization;andconservation.Ifpossible,useyourmodeltodiscusstheeconomic,physical,andenvironmentalimplicationsofyourstrategy.Provideanon-technicalpositionpapertog