国际竞赛题英文版.docx

国际竞赛题英文版.docx

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国际竞赛题英文版

Contents

1996AmericanMCMProblems1

ProblemA1

ProblemB1

1997AmericanMCMProblems1

ProblemATheVelociraptorProblem1

Part11

Part22

ProblemBMixWellForFruitfulDiscussions2

1998AmericanMCMProblems2

ProblemAMRIScanners2

Problem3

DataSets3

Summary3

ProblemBGradeInflation3

Background3

Problem3

DataSets4

MathematicalContestinModeling1999Problems4

ProblemB-UnlawfulAssembly4

2000MathematicalContestinModeling4

ProblemAAirtrafficControl4

ProblemBRadioChannelAssignments5

2001MathematicalContestinModeling（MCM）5

ProblemA:

ChoosingaBicycleWheel5

ProblemB:

EscapingaHurricane'sWrath（AnIllWind...）6

2002ContestProblems7

ProblemAWindandWaterspray7

ProblemBAirlineOverbooking7

2003ContestProblems7

ProblemATheStuntPerson7

ProblemBGammaKnifeTreatmentPlanning8

2004MCM8

PROBLEMA:

AreFingerprintsUnique?

8

PROBLEMB:

AFasterQuickPassSystem8

ProblemC ToBeSecureorNottoBe?

9

2005MCMProblem11

2006年美国大学生数学建模竞赛（MCM）和交叉学科建模竞赛（ICM）赛题12

PROBLEMA:

PositioningandMovingSprinklerSystemsforIrrigation12

PROBLEMB:

WheelChairAccessatAirports13

PROBLEMC:

Trade-offsinthefightagainstHIV/AIDS13

2007年美国大学生数学建模竞赛（MCM）和交叉学科建模竞赛（ICM）赛题PROBLEMA:

Gerrymandering19

PROBLEMB:

TheAirplaneSeatingProblem19

ProblemCOrganTransplant:

TheKidneyExchange19

2008年美国大学生数学建模竞赛（MCM）和交叉学科建模竞赛（ICM）赛题23

PROBLEMA:

TakeaBath23

PROBLEMB:

CreatingSudokupuzzles23

2008ICMProblemCFindingtheGoodinHealthCareSystems23

2009年美国大学生数学建模竞赛（MCM）和交叉学科建模竞赛（ICM）赛题24

2009MCMA:

DesigningaTrafficCircle24

2009MCMB:

EnergyandtheCellPhone25

2009ICM:

CreatingFoodSystems:

Re-BalancingHuman-InfluencedEcosystems26

2010年美国大学生数学建模竞赛（MCM）和交叉学科建模竞赛（ICM）赛题26

2010MCMA:

TheSweetSpot26

2010MCMB:

Criminology26

2010ICM:

TheGreatPacificOceanGarbagePatch27

2011年美国大学生数学建模竞赛（MCM）和交叉学科建模竞赛（ICM）赛题28

PROBLEMA:

SnowboardCourse28

PROBLEMB:

RepeaterCoordination28

PROBLEMC:

Howenvironmentallyandeconomicallysoundareelectricvehicles?

Istheirwidespreadusefeasibleandpractical?

29

1996AmericanMCMProblems

ProblemA

Theworld'soceanscontainanambientnoisefield.Seismicdisturbances,surfaceshipping,andmarinemammalsaresourcesthat,indifferentfrequencyranges,contributetothisfield.Wewishtoconsiderhowthisambientnoisemightbeusedtodetectlargemovingobjects,e.g.,submarineslocatedbelowtheoceansurface.Assumingthatasubmarinemakesnointrinsicnoise,developamethodfordetectingthepresenceofamovingsubmarine,itssize,anditsdirectionoftravel,usingonlyinformationobtainedbymeasuringchangestotheambientnoisefield.Beginwithnoiseatonefixedfrequencyandamplitude.

ProblemB

WhendeterminingthewinnerofacompetitionliketheMathematicalContestinModeling,therearegenerallyalargenumberofpaperstojudge.Let'ssaythereareP=100papers.AgroupofJjudgesiscollectedtoaccomplishthejudging.Fundingforthecontestconstainsboththenumberofjudgesthatcanbeobtainedandamountoftimethattheycanjudge.ForeampleifP=100,thenJ=8istypical.Ideally,eachjudgewouldreadpaperandrank-orderthem,buttherearetoomanypapersforthis.Instead,therewillbeanumberofscreeningroundsinwhicheachjudgewillreadsomenumberofpapersandgivethemscores.Thensomeselectionschemeisusedtoreducethenumberofpapersunderconsideration:

Ifthepapersarerank-ordered,thenthebottom30%thateachjudgerank-orderscouldberejected.Alternatively,ifthejudgesdonotrank-order,butinsteadgivethemnumericalscore（say,from1to100）,thenallpapersbelowsomecut-offlevelcouldberejected.

Thenewpoolofpapersisthenpassedbacktothejudges,andtheprocessisrepeated.AconcernisthenthetotalnumberofpapersthatjudgereadsmustbesubstantiallylessthanP.TheprocessisstoppedwhenthereareonlyWpapersleft.Therearethewinners.TypicallyforP=100,W=3.

Yourtaskistodetermineaselectionscheme,usingacombinationofrank-ordering,numericalscoring,andothermethods,bywhichthefinalWpaperswillincludeonlypapersfromamongthe"best"2Wpapers.（By"best",weassumethatthereisanabsoluterank-orderingtowhichalljudgeswouldagree.）Forexample,thetopthreepapers.Amongallsuchmethods,theonethatrequiredeachjudgetoreadtheleastnumberofpapersisdesired.

Notethepossibilityofsystematicbiasinanumericalscoringscheme.Forexample,foraspecificcollectionofpapers,onejudgecouldaverage70points,whileanothercouldaverage80points.Howwouldyouscaleyourschemetoaccommodateforchangesinthecontestparameters（P,J,andW）?

1997AmericanMCMProblems　

ProblemATheVelociraptorProblem

Thevelociraptor,Velociraptormongoliensis,wasapredatorydinosaurthatlivedduringthelateCretaceousperiod,approximately75millionyearsago.Paleontologiststhinkthatitwasaverytenacioushunter,andmayhavehuntedinpairsorlargerpacks.Unfortunately,thereisnowaytoobserveitshuntingbehaviorinthewildascanbedonewithmodernmammalianpredators.Agroupofpaleontologistshasapproachedyourteamandaskedforhelpinmodelingthehuntingbehaviorofthevelociraptor.Theyhopetocompareyourresultswithfielddatareportedbybiologistsstudyingthebehaviorsoflions,tigers,andsimilarpredatoryanimals.

Theaverageadultvelociraptorwas3meterslongwithahipheightof0.5metersandanapproximatemassof45kg.Itisestimatedthattheanimalcouldrunextremelyfastatspeedof60km/hrforabout15seconds.Aftertheinitialburstofspeed,theanimalneededtostopandrecoverfromabuildupoflacticacidinitsmuscles.

SupposethatvelociraptorpreyedonThescelosaurusneglectus,aherbivorousbipedapproximatelythesamesizeastheVelociraptor.AbiomachanicalanalysisofafossilizedThescelosaurusindicatesthatitcouldrunataspeedofabout50km/hr.forlongperiodoftime.

Part1

Assumingthevelociraptorisasolitaryhunter,designamathematicalmodelthatdescribeahuntingstrategyforasinglevelociraptorstalkingandchasingasingleThescelosaurusaswellastheevasivestrategyoftheprey.AssumethattheThescelosauruscanalwaysdetectthevelociraptorwhenitcomeswithin15meters.butmaydetectthepredatoratevengreaterranges（upto50metersdependinguponthehabitatandweatherconditions.Additionally,duetoitsphysicalstructureandstrength,thevelociraptorhasalimitedturningradiuswhenrunningatfullspeed.Thisradiusisestimatedtobethreetimestheanimal'shipheight.Ontheotherhand,theThescelosaurusisextremelyagileandhasaturningradiusof0.5meters.

Part2

Assumingmorerealisticallythatthevelociraptorhuntedinpairs,designanewmodelthatdescribesahuntingstrategyfortwovelociraptorstalkingandchasingasingleThescelosaurusaswellastheevasivestrategyoftheprey.UsetheotherassumptionsandlimitationsgiveninPart1.

ProblemBMixWellForFruitfulDiscussions

Smallgroupmeetingforthediscussionsofimportantissues,particularlong-rangeplanning,aregainingpopularity.Itisbelievedthatlargegroupsdiscourageproductivediscussionandthatadominantpersonalitywillusuallycontrolanddirectthediscussion.Thus,incorporate　boardmeetingstheboardwillmeetinsmallgroupstodiscussissuesbeforemeetingasawhole,thesesmallergroupsstilltuntheriskofcontrolbyadominantpersonality.Inanattempttoreducethisdangeritiscommontoscheduleseveralsessionswithadifferentmixofpeopleineachgroup.

AmeetingofAnTostalCorporationwillbeattendedby29BoardMembersofwhichninearein-housemembers（i.e.,corporateemployees）.Themeetingistobeanall-dayaffairwiththreesessionsscheduledforthemorningandfourfortheafternoon.Eachsessionwilltake45minutes,beginningonthehourfrom9:

00A.M.to4:

00P.M.,withlunchscheduledatnoon.Eachmorningsessionwillconsistofsixdiscussiongroupswitheachdiscussiongroupledbyoneofthecorporation'ssixseniorofficers.Noneoftheseofficersareboardmembers.Thuseachseniorofficerswillnotbeinvolvedintheafternoonsessionsandeachofthesesessionswillconsistofonlyfourdifferentdiscussiongroups.

Thepresidentofthecorporationwantsalistofboard-memberassignmenttodiscussiongroupsforeachofthesevensessions.Theassignmentsshouldachieveasmuch　ofamixofthemembersasmuchaspossible.Theidealassignmentwouldhaveeachboardmemberinadiscussiongroupthesamenumberoftimeswhileminimizingcommonmembershipofgroupsforthedifferentsessions.

Theassignmentshouldalsosatisfythefollowingcriteria:

1.Forthemorningsessions,noboardmembershouldbeinthesameseniorofficer'sdiscussiongrouptwice.

2.Nodiscussiongroupshouldcontainadisproportionatenumberofin-housemembers.

Givealistofassignmentsformembers1-9and10-29andofficers1-6.Indicatehowwellthecriteriainthepreviousparagraphsaremet.Sinceitispossiblethatsomeboardmemberswillcancelatthelastminuteorthatsomenotscheduledwillshowup,analgorithmthatthesecretarycouldusetoadjusttheassignmentswithanhour'snoticewouldbeappreciated.Itwouldbeidealifthealgorithmcouldalsobeusedtomakeassignmentsforfuturemeetingsinvolvingdifferentlevelsofparticipationforeachtypeofattendee.

1998AmericanMCMProblems

ProblemAMRIScanners

IndustrialmedicaldiagnosticmachinesknownasMagneticResonanceImager（MRI）scanathree-dimensionalobjectsuchasabrain,anddelivertheirresultsintheformofathree-dimensionalarrayofpixel.Eachpixelconsist