美国国际大学生数学建模竞赛MCM+ICM题目+翻译.docx

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