美国国际大学生数学建模竞赛MCM+ICM题目+翻译.docx
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美国国际大学生数学建模竞赛MCM+ICM题目+翻译.docx
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美国国际大学生数学建模竞赛MCM+ICM题目+翻译
IMPORTANTCHANGETOCONTESTRULESFORMCM/ICM2012:
Teams(StudentorAdvisor)arenowrequiredtosubmitanelectroniccopy(summarysheetandsolution)oftheirsolutionpaperbyemailtosolutions@. YouremailMUSTbereceivedatCOMAPbythesubmissiondeadlineof8:
00PMEST,February13,2012.TeamsarefreetochoosebetweenMCMProblemA,MCMProblemBorICMProblemC.
COMAPMirrorSite:
Formorein:
MCM:
TheMathematicalContestinModeling
ICM:
TheInterdisciplinaryContestinModeling
2012ContestProblems
MCMPROBLEMS
PROBLEMA:
TheLeavesofaTree
"Howmuchdotheleavesonatreeweigh?
"Howmightoneestimatetheactualweightoftheleaves(orforthatmatteranyotherpartsofthetree)?
Howmightoneclassifyleaves?
Buildamathematicalmodeltodescribeandclassifyleaves.Considerandanswerthefollowing:
•Whydoleaveshavethevariousshapesthattheyhave?
•Dotheshapes“minimize”overlappingindividualshadowsthatarecast,soastomaximizeexposure?
Doesthedistributionofleaveswithinthe“volume”ofthetreeanditsbrancheseffecttheshape?
•Speakingofprofiles,isleafshape(generalcharacteristics)relatedtotreeprofile/branchingstructure?
•Howwouldyouestimatetheleafmassofatree?
Isthereacorrelationbetweentheleafmassandthesizecharacteristicsofthetree(height,mass,volumedefinedbytheprofile)?
Inadditiontoyouronepagesummarysheetprepareaonepagelettertoaneditorofascientificjournaloutliningyourkeyfindings.
2012美赛A题:
一棵树的叶子
(数学中国翻译)
“一棵树的叶子有多重?
”怎么能估计树的叶子(或者树的任何其它部分)的实际重量?
怎样对叶子进行分类?
建立一个数学模型来对叶子进行描述和分类。
模型要考虑和回答下面的问题:
•为什么叶子具有各种形状?
•叶子之间是要将相互重叠的部分最小化,以便可以最大限度的接触到阳光吗?
树叶的分布以及树干和枝杈的体积影响叶子的形状吗?
就轮廓来讲,叶形(一般特征)是和树的轮廓以及分枝结构有关吗?
•你将如何估计一棵树的叶子质量?
叶子的质量和树的尺寸特征(包括和外形轮廓有关的高度、质量、体积)有联系吗?
除了你的一页摘要以外,给科学杂志的编辑写一封信,阐述你的主要发现。
PROBLEMB:
CampingalongtheBigLongRiver
VisitorstotheBigLongRiver(225miles)canenjoyscenicviewsandexcitingwhitewaterrapids.Theriverisinaccessibletohikers,sotheonlywaytoenjoyitistotakearivertripthatrequiresseveraldaysofcamping.RivertripsallstartatFirstLaunchandexittheriveratFinalExit,225milesdownstream.Passengerstakeeitheroar-poweredrubberrafts,whichtravelonaverage4mphormotorizedboats,whichtravelonaverage8mph.Thetripsrangefrom6to18nightsofcampingontheriver,starttofinish..Thegovernmentagencyresponsibleformanagingthisriverwantseverytriptoenjoyawildernessexperience,withminimalcontactwithothergroupsofboatsontheriver.Currently,XtripstraveldowntheBigLongRivereachyearduringasixmonthperiod(therestoftheyearitistoocoldforrivertrips).ThereareYcampsitesontheBigLongRiver,distributedfairlyuniformlythroughouttherivercorridor.Giventheriseinpopularityofriverrafting,theparkmanagershavebeenaskedtoallowmoretripstotraveldowntheriver.Theywanttodeterminehowtheymightscheduleanoptimalmixoftrips,ofvaryingduration(measuredinnightsontheriver)andpropulsion(motororoar)thatwillutilizethecampsitesinthebestwaypossible.Inotherwords,howmanymoreboattripscouldbeaddedtotheBigLongRiver’sraftingseason?
Therivermanagershavehiredyoutoadvisethemonwaysinwhichtodevelopthebestscheduleandonwaysinwhichtodeterminethecarryingcapacityoftheriver,rememberingthatnotwosetsofcamperscanoccupythesamesiteatthesametime.Inadditiontoyouronepagesummarysheet,prepareaonepagememotothemanagersoftheriverdescribingyourkeyfindings.
问题B:
沿长江大露营到BigLongRiver(225英里)游玩的游客可以享受那里的风景和振奋人心的急流。
远足者没法到达这条河,唯一去的办法是漂流过去。
这需要几天的露营。
河流旅行始于FirstLaunch,在FinalExit结束,共225英里的顺流。
旅客可以选择依靠船桨来前进的橡皮筏,它的速度是4英里每小时,或者选择8英里每小时的摩托船。
旅行从开始到结束包括大约6到18个晚上的河中的露营。
负责管理这条河的政府部门希望让每次旅行都能尽情享受野外经历,同时能尽量少的与河中其他的船只相遇。
当前,每年经过BigLong河的游客有X组,这些漂流都在一个以6个月长短的时期内进行,一年中的其他月份非常冷,不会有漂流。
在BigLong上有Y处露营地点,平均分布于河廊。
随着漂流人数的增加,管理者被要求应该允许让更多的船只漂流。
他们要决定如何来安排最优的方案:
包括旅行时间(以在河上的夜晚数计算)、选择哪种船(摩托还是桨船),从而能够最好地利用河中的露营地。
换句话说,BigLongRiver在漂流季节还能增加多少漂流旅行数?
管理者希望你能给他们最好的建议,告诉他们如何决定河流的容纳量,记住任两组旅行队都不能同时占据河中的露营地。
此外,在你的摘要表一页,准备一页给管理者的备忘录,用来描述你的关键发现。
ICMPROBLEM
PROBLEMC:
ModelingforCrimeBusting
ClickthetitlebelowtodownloadaZIPfilecontainingthe2012ICMProblem.
YourICMsubmissionshouldconsistofa1pageSummarySheetandyoursolutioncannotexceed20pagesforamaximumof21pages.
2012ICMProblem
ModelingforCrimeBusting
Yourorganization,theIntergalacticCrimeModelers(ICM),isinvestigatingaconspiracytocommita
criminalact.Theinvestigatorsarehighlyconfidenttheyknowseveralmembersoftheconspiracy,but
hopetoidentifytheothermembersandtheleadersbeforetheymakearrests.Theconspiratorsand
thepossiblesuspectedconspiratorsallworkforthesamecompanyinalargeofficecomplex.The
companyhasbeengrowingfastandmakinganameforitselfindevelopingandmarketingcomputer
softwareforbanksandcreditcardcompanies.ICMhasrecentlyfoundasmallsetofmessagesfroma
groupof82workersinthecompanythattheybelievewillhelpthemfindthemostlikelycandidatesfor
theunidentifiedco‐conspiratorsandunknownleaders.Sincethemessagetrafficisforalltheoffice
workersinthecompany,itisverylikelythatsome(maybemany)oftheidentifiedcommunicatorsinthe
messagetrafficarenotinvolvedintheconspiracy.Infact,theyarecertainthattheyknowsomepeople
whoarenotintheconspiracy.Thegoalofthemodelingeffortwillbetoidentifypeopleintheoffice
complexwhoarethemostlikelyconspirators.AprioritylistwouldbeidealsoICMcouldinvestigate,
placeundersurveillance,and/orinterrogatethemostlikelycandidates.Adiscriminatelineseparating
conspiratorsfromnon‐conspiratorswouldalsobehelpfultodistinctlycategorizethepeopleineach
group.ItwouldalsobehelpfultotheDA’sofficeifthemodelnominatedtheconspiracyleaders.
Beforethedataofthecurrentcasearegiventoyourcrimemodelingteam,yoursupervisorgivesyou
thefollowingscenario(calledInvestigationEZ)thatsheworkedonafewyearsagoinanothercity.Even
thoughsheisveryproudofherworkontheEZcase,shesaysitisjustaverysmall,simpleexample,but
itmayhelpyouunderstandyourtask.Herdatafollow:
Thetenpeopleshewasconsideringasconspiratorswere:
Anne#,Bob,Carol,Dave*,Ellen,Fred,
George*,Harry,Inez,andJaye#.(*indicatespriorknownconspirators,#indicatepriorknownnonconspirators)
Chronologyofthe28messagesthatshehadforhercasewithacodenumberforthetopicofeach
messagethatsheassignedbasedonheranalysisofthemessage:
AnnetoBob:
Whywereyoulatetoday?
(1)
BobtoCarol:
ThatdarnAnnealwayswatchesme.Iwasn'tlate.
(1)
CaroltoDave:
AnneandBobarefightingagainoverBob'stardiness.
(1)
DavetoEllen:
Ineedtoseeyouthismorning.Whencanyoucomeby?
Bringthebudgetfiles.
(2)
DavetoFred:
Icancomebyandseeyouanytimetoday.Letmeknowwhenitisagoodtime.ShouldI
bringthebudgetfiles?
(2)
DavetoGeorge:
Iwillseeyoulater‐‐‐lotstotalkabout.Ihopetheothersareready.Itisimportantto
getthisright.(3)
HarrytoGeorge:
Youseemstressed.Whatisgoingon?
Ourbudgetwillbefine.
(2)(4)
IneztoGeorge:
Iamrealtiredtoday.Howareyoudoing?
(5)
JayetoInez:
Notmuchgoingontoday.Wannagotolunchtoday?
(5)
IneztoJaye:
Goodthingitisquiet.Iamexhausted.Can'tdolunchtoday‐‐‐sorry!
(5)
GeorgetoDave:
Timetotalk‐‐‐now!
(3)
JayetoAnne:
Canyougotolunchtoday?
(5)
DavetoGeorge:
Ican't.OnmywaytoseeFred.(3)
GeorgetoDave:
Gethereafterthat.(3)
AnnetoCarol:
WhoissupposedtowatchBob?
Heisgoofingoffallthetime.
(1)
CaroltoAnne:
Leavehimalone.HeisworkingwellwithGeorgeandDave.
(1)
GeorgetoDave:
Thisisimportant.DarnFred.HowaboutEllen?
(3)
EllentoGeorge:
HaveyoutalkedwithDave?
(3)
GeorgetoEllen:
Notyet.Didyou?
(3)
BobtoAnne:
Iwasn'tlate.Andjustsoyouknow‐‐‐Iamworkingthroughlunch.
(1)
BobtoDave:
TellthemIwasn'tlate.Youknowme.
(1)
EllentoCarol:
GetwithAnneandfigureoutthebudgetmeetingschedulefornextweekandhelpme
calmGeorge.
(2)
HarrytoDave:
DidyounoticethatGeorgeisstressedoutagaintoday?
(4)
DavetoGeorge:
DarnHarrythinksyouarestressed.Don'tgethimworriedorhewillbenosingaround.
(4)
GeorgetoHarry:
Justworkinglateandhavingproblemsathome.Iwillbefine.(4)
EllentoHarry:
WoulditbeOK,ifImissthemeetingtoday?
Fredwillbethereandheknowsthe
budgetbetterthanIdo.
(2)
HarrytoFred:
Ithinknextyear'sbudgetisstressingoutafewpeople.Maybeweshouldtaketimeto
reassurepeopletoday.
(2)(4)
FredtoHarry:
Ithinkourbudgetisprettyhealthy.Idon'tseeanythingtostressover.
(2)
ENDofMESSAGETRAFFIC
Yoursupervisorpointsoutsthatsheassignedandcodedonly5differenttopicsofmessages:
1)Bob's
tardiness,2)thebudget,3)importantunknownissuebutassumedtobepartofconspiracy,4)George's
stress,and5)lunchandothersocialissues.Asseeninthemessagecoding,somemessageshadtwo
topicsassignedbecauseofthecontentofthemessages.
Thewayyoursupervisoranalyzedhersituationwaswithanetworkthatshowedthecommunication
linksandthetypesofmessages.Thefollowingfigureisamodelofthemessagenetworkthatresulted
withthecodeforthetypesofmessagesannotatedonthenetworkgraph.
Figure1:
NetworkofmessagesfromEZCase
YoursupervisorpointsoutthatinadditiontoknownconspiratorsGeorgeandDave,EllenandCarol
wereindictedforthecons
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