美赛历年题目Word文档格式.docx

美赛历年题目Word文档格式.docx

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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

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?

Themulti-nationalresearchteamconcludedthatthedisease:

•Appearstospreadthroughcontactwithbodilyfluidsofaninfectedperson

•Theelderlyandchildrenaremorelikelytodieifinfected

•Anearbyislandisstartingtoshowsimilarsignsofinfection

•Oneoftheresearchersthatreturnedtoyourcapitalappearsinfected

Requirement4:

Howdoestheadditionalinformationabovechange/modifyyourmodel?

Requirement5:

Writeaone-pagesynopsisofyourfindingsforyourlocalnon-technicalnewsoutlet.

2013MCMProblems

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. 

Water,Water,Everywhere

Freshwateristhelimitingconstraintfordevelopmentinmuchoftheworld.Buildamathematicalmodelfordetermininganeffective,feasible,andcost-efficientwaterstrategyfor2013tomeettheprojectedwaterneedsof[pickonecountryfromthelistbelow]in2025,andidentifythebestwaterstrategy.Inparticular,yourmathematicalmodelmustaddressstorageandmovement;

de-salinization;

andconservation.Ifpossible,useyourmodeltodiscusstheeconomic,physical,andenvironmentalimplicationsofyourstrategy.Provideanon-technicalpositionpapertog