美赛历年题目.docx

美赛历年题目.docx

《美赛历年题目.docx》由会员分享，可在线阅读，更多相关《美赛历年题目.docx（10页珍藏版）》请在冰豆网上搜索。

美赛历年题目

具体可以登录官网看

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