美賽歷年試題翻譯.doc

MCM1998問(wèn)題-A; ; x) e. a$ c6 K) T) _磁共振成像掃描儀引言 用于工業(yè)和醫(yī)療的磁共振成像掃描儀診斷機(jī)對(duì)像腦那樣的三維物體進(jìn)行掃描，并把掃描的結(jié)果以三維像素陣列的形式傳送之。每個(gè)像素由一個(gè)指示其顏色或灰度的數(shù)構(gòu)成，它對(duì)像素所在位置處的被掃描物體的一個(gè)小區(qū)域中含水量(濃度)的度量進(jìn)行編碼。例如，0能以黑色來(lái)描繪出高含水量(腦室、血管)，128能以灰色來(lái)描繪出中等含水量(腦核和灰質(zhì))，而255以白色來(lái)描繪出低含水量(組成有髓體軸的富含脂類白質(zhì))。這類磁共振成像掃描儀還包括能在屏幕畫出通過(guò)該三維像素陣列的平行或垂直片(與三個(gè)笛卡爾坐標(biāo)軸平行的平片)的設(shè)備能夠描繪出斜的平片的算法是專賣的。眼下的算法利用了角度及可供使用的參數(shù)選擇而受到限制，算法的執(zhí)行也有賴于大量使用專用的工作站；在切片之前缺少在畫面上作點(diǎn)的輸入能力；從而使原始像素間明晰的邊界變得模糊。 能在個(gè)人計(jì)算機(jī)上實(shí)現(xiàn)的更為準(zhǔn)確可靠的、靈活的算法對(duì)于以下幾方面來(lái)說(shuō)將是極為有用的： 設(shè)計(jì)盡可能少的介入處理； 校準(zhǔn)磁共振成像掃描儀； 研究諸如動(dòng)物研究中尸體解剖組織部分那樣的在空間中斜向的結(jié)構(gòu)； 能作出以任意角度和由黑白固線組成的腦圖譜相交的截面。 為設(shè)計(jì)這樣的算法，就要能存取任意像素的值和位置,不僅僅是由掃描儀收集到的原始數(shù)據(jù)。問(wèn)題設(shè)計(jì)并測(cè)試能產(chǎn)生與三維陣列在空間任意指向的平面的截面部分的算法，并盡可能保持原始的灰度值。數(shù)據(jù)集 典型的數(shù)據(jù)集由表示物體在位置處的濃度的由數(shù)A(i，j，k)構(gòu)成的三維陣列A典型的情形，A(i，j，k)的取值范圍為0到255在大多數(shù)應(yīng)用中，該數(shù)據(jù)集是相當(dāng)大的。 參賽隊(duì)要設(shè)計(jì)用以測(cè)試井論證其算法的數(shù)據(jù)集。數(shù)據(jù)集應(yīng)能反映大概是有診斷意義的情況。參賽隊(duì)還應(yīng)敘述限制其算法有效性的數(shù)據(jù)集的特征。8 V( ?% |* b( S/ V; C3 c總結(jié)算法一定要生成由空間一平面與三維陣列相交出的切片部分的圖象。這種平面在空間可以釘任意的指向和位置(該平面可能會(huì)漏掉一些或全部數(shù)據(jù)點(diǎn))。算法的結(jié)果應(yīng)該是所掃描的物體在所選平面上的濃度的一個(gè)模型。AMCM98問(wèn)題-B 成績(jī)給分的通脹背景 一些行政領(lǐng)導(dǎo)很為A Better class(ABL)學(xué)院的成績(jī)給分擔(dān)擾。平均說(shuō)來(lái)，ABC學(xué)院的教師一直在給高分(現(xiàn)在結(jié)出的平均成績(jī)分?jǐn)?shù)為A-)，從而不可能區(qū)分好學(xué)生和中等水平的學(xué)生。金額很大的獎(jiǎng)學(xué)金只能資助班上前10%的學(xué)生，因而要對(duì)班上的學(xué)生排名次。 院長(zhǎng)有一想法：把班上每個(gè)學(xué)生和其他學(xué)生進(jìn)行比較，并用比較獲得的信息來(lái)排名次。例如，若某個(gè)學(xué)生得分為A而全班學(xué)生都得A，那么這個(gè)學(xué)生只能屬于這個(gè)班上的“平均水平”。另一方面，如果班上只有一個(gè)學(xué)生得A，那么這個(gè)學(xué)生顯然在“平均水平之上”，結(jié)合幾門課中得到的比較信息就能把全校的學(xué)生按十分位數(shù)排名次(前l(fā)0%。次l0%，等等)。問(wèn)題 假定給出的成績(jī)記分為(A+，A，A-,B+,）院長(zhǎng)的想法能否實(shí)現(xiàn)? 假定給出的成績(jī)記分只有(A、B，C）院長(zhǎng)的想法能否實(shí)現(xiàn)?有沒(méi)有其他能給出名次排列的方案?一種擔(dān)心是，一個(gè)班級(jí)的成績(jī)記分可能會(huì)改變?cè)S多學(xué)生的十分位數(shù)的排名次?？赡艹霈F(xiàn)這種情況嗎?數(shù)據(jù)集參賽隊(duì)要設(shè)計(jì)用以測(cè)試并論證其算法的數(shù)據(jù)集，參賽隊(duì)還應(yīng)敘述限制其算法有效性的數(shù)據(jù)集的特征。AMCM99問(wèn)題-A 大碰撞 L, g+ T: H) a) J$ f $ W+ B9 + s7 Z1 E( eNASA(航空航天管理局)常常考慮這樣一個(gè)問(wèn)題：一顆較大的小行星與地球的碰撞將會(huì)產(chǎn)生怎樣的后果。6 9 H0 Q1 l5 ( ?9 作為這個(gè)問(wèn)題的一部分，要求你們討論這顆小行星該到地球南極所造成的后果，有人認(rèn)為其后果將與該行星撞到地球其它區(qū)域的后果有很大的不同。/ fU3 S8 |) Q: 0 s1 U6 P7 g 你們可以假設(shè)這顆小行星的直徑大約為l000米，并且立接撞在南極點(diǎn)處。 c: T5 r7 D. G J) D+ k1 V 你們應(yīng)當(dāng)給出這樣一次碰撞的后果的估計(jì)，特別地，NASA希望得到由這次碰撞所造成的人員傷亡的地區(qū)及數(shù)量的估計(jì)，還希望得到關(guān)于對(duì)南半球的大洋區(qū)域的農(nóng)作物生長(zhǎng)的危害的估計(jì)，以及由于南極冰層大規(guī)模融化引起的沿海洪水的一個(gè)估計(jì)。AMCM99問(wèn)題-B6 # b+ ?+ 8 n6 J9 i j“非法”聚會(huì) 許多公共設(shè)施的房間都柯一種標(biāo)有人數(shù)的記號(hào)，當(dāng)房間中人數(shù)超過(guò)記號(hào)上人數(shù)時(shí)就視為“非法”，該數(shù)目可假定是以緊急情況下從房屋出口逃出的人數(shù)為基準(zhǔn)確定的，類似地，電梯及其它設(shè)施經(jīng)常有一個(gè)“最大容量”。 建立數(shù)學(xué)模型以確定標(biāo)上多大人數(shù)值才是“合法容量”，作為求解的一部分要討論若干準(zhǔn)則(并非在火災(zāi)或其它緊急情況下的公共安全)決定出房屋或空間)達(dá)到“非法”聚會(huì)的人數(shù)，而且，在所建模型中要考慮幾種不同的房屋結(jié)構(gòu)，例如，像咖啡屋(擁有桌和椅子)那樣具有可移動(dòng)家俱的房子，具有成排椅子和走廊的演訓(xùn)廳等，你還可以對(duì)各種不同情形進(jìn)行比較與對(duì)比，例如：電梯，演講廳，游泳池，咖啡屋或健身房等。 收集搖滾音樂(lè)會(huì)或足球比賽的相關(guān)資料也許會(huì)為你提供一些特殊的信息。將所建模型用于你所在學(xué)院(或附近城鎮(zhèn))的一個(gè)或多個(gè)公共設(shè)施中，如果該類設(shè)施已標(biāo)有“合法”人數(shù)的話，請(qǐng)將模型所得結(jié)果與之比較。如果得到使用，你的模型可能部分受到利益驅(qū)動(dòng)下要增加容量之觀點(diǎn)的挑戰(zhàn)，為當(dāng)?shù)貓?bào)刊撰寫一篇文章以捍衛(wèi)模型所給的分析。AMCM1999問(wèn)題-C 大地污染9 E. Z2 _6 j j9 P 背景 若干實(shí)踐中重要但理論上困難的數(shù)學(xué)問(wèn)題與污染的評(píng)估有關(guān)。這種問(wèn)題之一就是根據(jù)只是在被懷疑為已污染地區(qū)的周圍而不必直接在該地區(qū)中測(cè)得的很少的測(cè)量數(shù)據(jù)來(lái)導(dǎo)出不易進(jìn)入的地下的滲漏污染物的位置和數(shù)量、以及污染源的精確估計(jì)。% 4 V$ 5 l3 N5 u 例子 數(shù)據(jù)可通過(guò)/data.xis 找到。3 M5 N. y1 G. D R+ D 該數(shù)據(jù)集（一種電子表格文件an Excel file），它能卸載到大多數(shù)電子數(shù)據(jù)表（spreadsheets）展示了從1990到1997在10個(gè)監(jiān)測(cè)井處地下水中污染物的數(shù)量數(shù)據(jù)。單位是微克（mg/l）。8個(gè)測(cè)井的位置和高度是已知的并在下表給出。頭兩個(gè)數(shù)是在一張地圖的直角格點(diǎn)上井的位置的坐標(biāo)。第三個(gè)數(shù)是井中水面高出平均海平面的高度（以英尺計(jì)）。! B( x1 A F7 y5 ?- Z: ! k0 z( P! N8 c ( D9 a i* b7 ; J5 - U( 5 s4 d. B1 K; H 1 l9 x% X- F( _) y, T; p D3 r; ; t( z7 y- m; c q6 o0 H8 : i: S 井號(hào) y+ |) ?( # o S* l; B0 V$ v2 R8 ? x坐標(biāo)（英尺計(jì)）# V C3 n; B( o$ 5 Y , V a# m/ b0 g6 M y坐標(biāo)（英尺計(jì)）5 U1 G; P4 r- d8 H ( m, I3 H: S0 F2 W- / v* 9 R F 高度 （英尺計(jì)）! W* B* % G3 ?2 M8 l$ p & Y4 D; x1 z( # L: # M 0 4 F% Z7 a2 Z7 s) L6 z _ I MW-14 M2 S5 N# h l e $ H0 E0 I3 E8 u* E8 4187.5, W! q7 E6 % h6 C# a3 D8 l ! v+ F Y4 M7 P4 h, _. O- P8 a 6375.03 X/ Y9 8 f G5 O: A5 e+ Z# Y g2 | % Q4 F$ G, w, V/ # b1 G 1482.23+ d o5 i! v1 a2 p j$ m& e7 Z8 y* B, o2 j: / W$ M MW-3 . i9 Z * K! D9 - p! u$ x # J3 w- s) 7 e6 : 7 I8 / I; ?) E 9062.52 f/ ?, z T$ R6 z. L 3 w* V: c b* I5 o8 _% _5 G- K 4375.0& U2 V7 n2 C n4 I / K9 : X8 e4 |% _* ? 1387.92( F3 $ B( |) w7 0 N# w - 2 ?4 7 k V- F % V J; D- W9 s% 9 0 V6 J # b MW-7/ ; b9 U9 O8 ? ) & X/ l1 Z. F1 8 h 7625.0& V+ L, Z7 w T0 i1 O l E+ $ | & 8 S9 a1 l, e; - 4 X1 w 5812.5 g0 A7 v/ Q0 G ?% y8 o/ X- v8 v $ B, ! r% Y/ k6 7 d 1400.193 T6 o, H p3 : 8 e M ?5 F$ # J: Q, B. R% F* f R8 F6 E# n% . D MW-93 t7 : S3 b1 c: M7 h7 % z3 l1 t- 7 I; C& k& J 9125.0& R) z t& Y/ G2 x3 G1 L 3 m$ j# i6 f: H! j 4000.0* ; T+ F# K$ U6 y, E / C! _. O; f3 l. W/ Z7 k 1384.53( P3 O5 S$ u k. X4 b Y/ p; H- p$ w# j P4 j : K F. b& ) o! k! J; x; s2 2 | MW-111 r6 k, f9 o8 K v x0 ! l* V+ 8 x8 m$ I4 : _) a 9062.5 W* X* 3 r N/ C; * e! F& h) l+ y: c( O0 w( G 5187.5& ?5 e: h/ Z9 z& | ) ?8 f8 I4 s1 v K8 E$ k 1394.26# o# h0 E- N+ L& M O& r& v% _! 7 d+ m% H 4 W M, 2 E3 J# I V4 f; B y MW-12( : Y! ) , ; P3 k4 1 X: R S( j, w x7 h: l) ; U0 9062.5( m7 p! r$ M0 A% e- G7 S 5 y7 z0 x2 Q/ $ E$ u 4562.5, 7 z/ s; C i _1 G+ |$ s5 3 X4 t o* f0 H 1388.94! a# z# J- 1 N, |4 2 W 0 $ Z: p) _4 1 t3 I & q$ Y+ w0 v( h1 m MW-13, y& I0 ?- ?+ ?0 m8 M( e ? - w7 B. D: M: j h+ U, q 9062.5& p, j$ h- c: d! w+ e% n( B, l H / Q/ Y# P* C 5000.0! i Q9 T4 x# f7 1 B % F# T- E% 9 L. l2 U( 1394.251 R! A8 3 M: ! l; v- b, / W, l & n7 W( t: F. # h! N 5 p, 5 D+ - M1 G# L; N6 N MW-14+ v5 F$ w) I4 k* W! S0 ? $ G. j j3 J0 O! c* D; r! x y 4750.03 K: N- a% X6 I& f, H 0 q( I5 S/ _( U3 m+ E5 u( 1 C 2562.5! Y, o: d* ? ?+ f6 W, ( O5 J5 9 R L$ w5 o4 u+ h 1412.00. s3 v7 t/ X& B3 r6 T1 數(shù)據(jù)集中另兩個(gè)井（MW-27和MW-33）的位置和高度不同。在該數(shù)據(jù)集中你還會(huì)看到數(shù)字后面的字母T(Top),M(Middle)或B(Bottom)，它們分別表示測(cè)量是在井的含水層的頂部、中部和底部進(jìn)行的。因此，MW-7B和MWffice:smarttags /-7M是來(lái)自同一個(gè)井，但分別是底部和中部的測(cè)量。此外，其它的測(cè)量數(shù)據(jù)表明水有流向該區(qū)域中的MW-9號(hào)井的趨勢(shì)。+ 8 o; B2 A6 G, 3 g4 T 問(wèn)題一 試建立一個(gè)數(shù)學(xué)模型來(lái)決定在由該數(shù)據(jù)集來(lái)表示的區(qū)域和時(shí)間里是否由任何新的污染物產(chǎn)生。若有，試識(shí)別新的污染物并估計(jì)它們的污染源的位置和時(shí)間。% R7 T/ k6 % f4 u& 問(wèn)題二 在收集任何數(shù)據(jù)之前，會(huì)提出下列問(wèn)題：是否擬議中的數(shù)據(jù)類型和模型能給出關(guān)于污染物所在的位置和數(shù)量的我們想要的估計(jì)。液態(tài)的化學(xué)物質(zhì)會(huì)從埋置在均勻的土壤的儲(chǔ)藏中的一個(gè)儲(chǔ)藏罐中滲漏。因?yàn)槿粢谠S多大罐的下面去探測(cè)的費(fèi)用會(huì)過(guò)分昂貴而且危險(xiǎn)，所以只能在儲(chǔ)藏設(shè)備的邊緣地區(qū)附近或在看來(lái)更合適的地區(qū)的表面進(jìn)行測(cè)量。試決定只是在整個(gè)儲(chǔ)藏罐的邊界的外面或表面進(jìn)行什么樣類型的測(cè)量以及測(cè)量數(shù)目可以用于一個(gè)數(shù)學(xué)模型以決定上樓是否發(fā)生，何時(shí)發(fā)生，何處（從哪個(gè)罐）發(fā)生，以及滲漏多少液體。2000 MCM ProblemsProblem AAir traffic ControlTo improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA r traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA has posed the following problemsRequirement A: Given two airplanes flying in space, when should the air traffic controller ld the air traffic controller consider the objects to be too close and to require intervention? Requirement B: An airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of we measure how complex it is from an air traffic workload perspective? To what extent is complexity determined by the number of aircraft simultaneously passing through that sector (1) at any one instant? (2) During any given interval of time? (3) During particular time of day? How does the number of potential conflicts arising during those periods affect complexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this complexity? In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions問(wèn)題A: 空間交通管制$ m+ p% g/ d/ s/ d; Z0 W為加強(qiáng)安全并減少空中交通指揮員的工作量，聯(lián)邦航空局(FAA)考慮對(duì)空中交通管制系統(tǒng)添加軟件，以便自動(dòng)探測(cè)飛行器飛行路線可能的沖突，并提醒指揮員。為完成此項(xiàng)工作，F(xiàn)AA的分析員提出了下列問(wèn)題。8 s- C& h4 z3 , # 要求A: 對(duì)于給定的兩架空中飛行的飛機(jī)，空中交通指揮員應(yīng)在什么時(shí)候把該目標(biāo)視為太靠近，并予以干預(yù)。6 R8 b4 n$ T0 H, p要求B: 空間扇形是指某個(gè)空中交通指揮員所控制的三維空間部分。給定任意一個(gè)空間扇形，我們?cè)鯓訌目罩薪煌üぷ髁康姆轿粊?lái)估量它是否復(fù)雜？當(dāng)幾個(gè)飛行器同時(shí)通過(guò)該扇形時(shí),在下面情形所確定的復(fù)雜性會(huì)達(dá)到什么程度：（1）在任一時(shí)刻？（2）在任意給定的時(shí)間范圍內(nèi)？（3）在一天的特別時(shí)間內(nèi)？在此期間可能出現(xiàn)的沖突總數(shù)是怎樣影響著復(fù)雜性來(lái)的？3 m k1 C1 nw+ + _提出所添加的軟件工具對(duì)于自動(dòng)預(yù)告沖突并提醒指揮員，這是否會(huì)減少或增加此種復(fù)雜性？* S5 1 U1 c+ G4 l8 I在作出你的報(bào)告方案的同時(shí)，寫出概述（不多于二頁(yè)）使FAA分析員能提交給FAA當(dāng)局Jane Garvey ，并對(duì)你的結(jié)論進(jìn)行答辯。Problem BRadio Channel AssignmentsWe seek to model the assignment of radio channels to a symmetric network of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grid (honeycomb-style), as shown in Figure 1, where a transmitter is located at the center of each hexagon. An interval of the frequency spectrum is to be allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3, . . Each transmitter will be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided. Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assign channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span. Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interferenceRequirement A: There are several constraints on frequency assignments. First, no two transmitters within distance of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these constraints, what can we say about the span in, Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions. Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance differ by at least some given integer k, while those at distance at most must still differ by at least one. What can we say about the span and about efficient strategies for designing assignments, as a function of k? Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider? Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings問(wèn)題B: 無(wú)線電信道分配: A1 B0 uwj1 Z& z1 p4 j0 w) W8 V我們尋找無(wú)線電信道配置模型.在一個(gè)大的平面區(qū)域上設(shè)置一個(gè)傳送站的均衡網(wǎng)絡(luò),以避免干擾.一個(gè)基本的方法是將此區(qū)域分成正六邊形的格子(蜂窩狀),如圖1.傳送站安置在每個(gè)正六邊形的中心點(diǎn).! ( + U4 d W容許頻率波譜的一個(gè)區(qū)間作為各傳送站的頻率.將這一區(qū)間規(guī)則地分割成一些空間信道,用整數(shù)1,2,3,來(lái)表示.每一個(gè)傳送站將被配置一正整數(shù)信道.同一信道可以在許多局部地區(qū)使用,前提是相鄰近的傳送站不相互干擾. 根據(jù)某些限制設(shè)定的信道需要一定的頻率波譜,我們的目標(biāo)是極小化頻率波譜的這個(gè)區(qū)間寬度.這可以用跨度這一概念.跨度是某一個(gè)局部區(qū)域上使用的最大信道在一切滿足限制的配置中的最小值.在一個(gè)獲得一定跨度的配置中不要求小於跨度的每一信道都被使用.2 F2 k2 ( M. v令s為一個(gè)正六邊形的一側(cè)的長(zhǎng)度.我們集中考慮存在兩種干擾水平的一種情況.6 h/ v- S, f9 u$ g7 w r: E要求A: 頻率配置有幾個(gè)限制,第一,相互靠近的兩個(gè)傳送站不能配給同一信道.第二,由於波譜的傳播,相互距離在2s內(nèi)的傳送站必須不配給相同或相鄰的信道,它們至少差2.在這些限制下,關(guān)于跨度能說(shuō)些什么.% G7 x% A3 : C1 f要求B: 假定前述圖1中的格子在各方向延伸到任意遠(yuǎn),回答要求A.) l0 & s: W+ p% Co5 j2 U) q5 要求C: 在下述假定下,重復(fù)要求A和B.更一般地假定相互靠近的傳送站的信道至少差一個(gè)給定的整數(shù)k,同時(shí)那些隔開(kāi)一點(diǎn)的保持至少差1.關(guān)于跨度和關(guān)于設(shè)計(jì)配置的有效策略作為k的一個(gè)函數(shù)能說(shuō)點(diǎn)什么.; H: & ( G+ f要求D: 考慮問(wèn)題的一般化,比如各種干擾水平,或不規(guī)則的傳送站布局.其他什么因素在考慮中是重要的.7 z& o% C1 V! T3 F- f要求E: 寫一篇短文(不超過(guò)兩頁(yè))給地方報(bào)紙,闡述你的發(fā)現(xiàn).% d. A, u6 Nu9 WU+ tCPrepare a darting plan for parks of various sizes (30025,000 elephants), with slightly different survival rates and transportation possibilities. % s% LY& w: ?: ) f+ e + T7 U+ b( Z6 k6 ?, F ; Q/ B5 H Z- b: # G大象群落的興衰! m. b$ Y( x, 歸根到底，如果象群對(duì)于棲息地造成不盡人意的影響，就要考慮對(duì)它們的驅(qū)除，即使是運(yùn)用淘汰法則。國(guó)家地理雜志（地球年鑒）1999年12月, kX. w1 Z e7 B9 F在位于南非的一個(gè)巨大的國(guó)家公園里，棲息著近乎11000只象。管理策略要求一個(gè)健康的環(huán)境以便維持11000只象的穩(wěn)定群落。公園的管理員們逐年統(tǒng)計(jì)象的總數(shù)。在過(guò)去的20年間，整個(gè)群落經(jīng)受驅(qū)除得以保持其總數(shù)盡量接近11000只。這個(gè)過(guò)程涉及槍殺（對(duì)于大部分）和每年轉(zhuǎn)移近乎600到800只象到異地。# L+ A7 G2 B% X6 k近年來(lái)，公眾抗議槍殺這些象。此外，即使每年轉(zhuǎn)移少量的象也是不可能了。然而，一種避孕注射法開(kāi)發(fā)成功，它可以在兩年期間內(nèi)阻止一只成熟的母象受孕。2 R- W qz . V) ) S下面是一些關(guān)于這個(gè)公園內(nèi)象的信息：2 C, q1 O; x8 u% * C很少發(fā)生象本身移入移出該公園的事。* 1 g3 b* $ U性別比非常接近1:1，而且采取控制措施力求維持均衡。9 3 F+ , C1 i7 y6 J% n$ O* _新生幼象的性別比也是1:1左右。雙胞胎的機(jī)會(huì)接近于1.35%。9 g% _7 K P5 Ok% m7 X- f+ X2 母象在10歲和12歲之間第一次懷孕，平均每3.5年產(chǎn)下一個(gè)崽兒，直到60歲左右為止。懷孕期約為22個(gè)月。! Ov2 C6 l1 B0 V) h7 . Z1 b避孕注射使一只母象每個(gè)月發(fā)情（但不懷孕）。象通常在3.5年內(nèi)僅求偶一次，所以，上述按月周期能夠引起附加的反應(yīng)。7 x* w: e/ / o& l8 Z0 P一只母象可以每年注射而沒(méi)有任何有害的影響。一只成熟的母象在上次注射后兩年內(nèi)將不能懷孕。( K, , r# R1 Q新生幼象中的70%到80%活到一歲，其后，存活率非常高（超過(guò)95%）并且在各年齡段一致，直到60歲左右；假定象死于70歲之前是恰當(dāng)?shù)摹Ｔ谶@個(gè)公園內(nèi)沒(méi)有狩獵，偷獵也是微乎其微。4 X* N# g6 B* d& Y1 l% |公園管理部門有一個(gè)粗略的數(shù)據(jù)文件，其中列出近兩年內(nèi)由這個(gè)地區(qū)運(yùn)出的象的大致年齡和性別。這組數(shù)據(jù)可在網(wǎng)站/icm/icm2000data.xls上找到。可惜的是，沒(méi)有關(guān)于在這個(gè)公園內(nèi)被射殺和留下來(lái)的象的可用數(shù)據(jù)。你的全部任務(wù)是發(fā)展和利用模型來(lái)研究避孕注射會(huì)如何用于控制象的數(shù)量。特別是：1 i) G: - & ie0 y. P任務(wù)1：發(fā)展和利用一個(gè)模型來(lái)推測(cè)年齡在2歲到60歲之間象的合理存活率。并且推測(cè)這個(gè)大象群落的當(dāng)前年齡結(jié)構(gòu)。4 N4 y2 I+ * |. , V0 N1 8 r任務(wù)2：估計(jì)每年有多少只母象需要避孕注射以保持這個(gè)群落固定在11000只象左右。說(shuō)明被處理數(shù)據(jù)的不確定性如何影響你的估計(jì)。試加評(píng)論這個(gè)群落年齡結(jié)構(gòu)的任何改變以及會(huì)如何影響旅游者。（你或許要前瞻30-60年左右。）8 1 n6 ! z1 v0 $ E任務(wù)3：假如每年轉(zhuǎn)移50至300只象是可行的，這會(huì)怎樣減少承受避孕注射的象只數(shù)量？試加評(píng)定避孕注射和轉(zhuǎn)移之間的折衷辦法。& F8 q6 u+ w$ j2 m任務(wù)4：若干反對(duì)避孕注射的人提出疑問(wèn)，如果發(fā)生一場(chǎng)大量象只的突然滅絕（由于疾病或不受控制的偷獵），即使立即停止避孕注射，這個(gè)群落重新壯大的能力也會(huì)受到嚴(yán)重阻礙。對(duì)這個(gè)顧慮進(jìn)行研究并作出回應(yīng)。5 ; 0 w& t$ E5 Q5 ?4 Q. q* S8 z9 D9 任務(wù)5：這個(gè)公園的管理部門不相信建模。他們特別表示，由于缺少完整的數(shù)據(jù)，任何通過(guò)模型來(lái)引導(dǎo)他們作出決定的嘗試都構(gòu)成一種愚弄。除了你的技術(shù)報(bào)告之外，請(qǐng)附上一份字斟句酌寫給公園管理部門的報(bào)告（最多三頁(yè)），對(duì)于他們的疑慮作出回應(yīng)并且給予勸告。還要提出一些辦法來(lái)增加公園管理部門對(duì)于你的模型和結(jié)論的信賴程度。! V. ?, R# h; - c x; d; V任務(wù)6：如果你的模型有效，南非的其他大象公園會(huì)樂(lè)于采用它。請(qǐng)為各種規(guī)模的公園（300至25000只象）準(zhǔn)備一項(xiàng)避孕注射計(jì)劃，同時(shí)帶有略微不同的存活率和轉(zhuǎn)運(yùn)可能性。 2001 MCM ProblemsProblem A: Choosing a Bicycle Wheel Cyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right. The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear. Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades starting from zero percent to ten percent in one percent increments. (Road grade is defined to be the ratio of the total rise of a hill divided by the length of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportional to the road grade. A rider will lose about 8 kph for a five percent grade over 100 meters. Task 2. Provide an example of how the table could be used for a specific time trial course. Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision. Problem B: Escaping a Hurricanes Wrath (An Ill Wind.)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare. The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned. A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.) The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed: 1. Under what conditions does the plan for turning the two coastal-bound lanes of I-26 into two lanes of Columbia-bound traffic, essentially turning the entire I-26 into one-way traffic, significantly improve evacuation traffic flow? 2. In 1999, the simultaneous evacuation of the states