2014美国大学生数学建模竞赛题目(含翻译)

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2014年美赛C题翻译

This will take some skilled data extraction and modeling efforts to obtain the correct set of nodes (the Erdös coauthors) and their links (connections with one another as coauthors). 这需要熟练数据提取并在建模上下功夫，以便得到正确的节点和边
Once built, analyze the properties of this network. 建完后分析网络性能(Again, do not include Erdös --- he is the most infnodes in the network. In this case, it’s co-authorship with him that builds the network, but he is not part of the network or the analysis.)
One of the techniques to determine influence of academic research is to build and measure properties of citation or co-author networks.
学术研究的技术来确定影响之一是构建和引文或合著网络的度量属性。
Google Scholar is also a good data tool to use for network influence or impact data collection and analysis.
谷歌学术搜索也是一个好的数据工具用于网络数据收集和分析影响或影响。

美国大学生数学建模比赛2014年B题

Team # 26254
Page 2 oon ............................................................................................................................................................. 3 2. The AHP .................................................................................................................................................................. 3 2.1 The hierarchical structure establishment ....................................................................................................... 4 2.2 Constructing the AHP pair-wise comparison matrix...................................................................................... 4 2.3 Calculate the eigenvalues and eigenvectors and check consistency .............................................................. 5 2.4 Calculate the combination weights vector ..................................................................................................... 6 3. Choosing Best All Time Baseball College Coach via AHP and Fuzzy Comprehensive Evaluation ....................... 6 3.1 Factor analysis and hierarchy relation construction....................................................................................... 7 3.2 Fuzzy comprehensive evaluation ................................................................................................................... 8 3.3 calculating the eigenvectors and eigenvalues ................................................................................................ 9 3.3.1 Construct the pair-wise comparison matrix ........................................................................................ 9 3.3.2 Construct the comparison matrix of the alternatives to the criteria hierarchy .................................. 10 3.4 Ranking the coaches .....................................................................................................................................11 4. Evaluate the performance of other two sports coaches, basketball and football.................................................... 13 5. Discuss the generality of the proposed method for Choosing Best All Time College Coach ................................ 14 6. The strengths and weaknesses of the proposed method to solve the problem ....................................................... 14 7. Conclusions ........................................................................................................................................................... 15

2014年美国大学生数学建模竞赛MCM A题二等奖

For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________
A
Simulations of a Multi-Lane Traffic Model Using Cellular Automata Concerning the Overtaking Effect
Summary
The prosperity of modern industrialized world is largely depend on today’s sophisticated road networks. The rapid growth of vehicle number often exceeds the capacity of existing road [1,2]. Thus the effective utilization of road capacity is indispensable in traffic flow control. Cellular automaton model (CA model) is an very practical model in stimulating traffic flow behavior. This paper aims at building a CA model for multi-lane traffic using right-most overtaking law. The first CA model is the well-known NaSch model[9] Knospe studied a two-lane model focus on the density dependence of lane changes[10,11]. A numerical approach is performed by Daoudia and Moussa to stimulate the 3-lane traffic flow[12]. Basing upon the previous done by [1,9,10,12], we put forward an extended CA model using right-most overtaking rule. By a detailed investigation on overtaking process, we obtain the least safe distance under different speed limits and traffic flow densities, due to the limitation of least safe distance, we put forward incentive and safety criteria of overtaking behavior for particles on different lanes in our CA model. Simulation of our model is performed for three-lane case, and the result shows that our right-most overtaking rule behaves asymmetrically such that the right-most lane firstly reach at the “critical density” where the traffic flow reaches its peak point. And despite the special behavior of our own model, our results indicate a robust behavior of traffic flow such that when traffic density is roughly 0.1, the traffic flow arrives at its peak value, and with an increase of particle density, the phase transition occurs such that the traffic jams and “stop-and-go” phenomena happens. This result is of significance in conducting the daily traffic flow in our real world. Keywords: traffic flow, cellular automaton, right-most overtaking, simulation

2014美国数学竞赛AMC10A、10B试题及答案

2014 AMC 10B Problems Problem 1 Leah has 13 coins^ all of which are pennies and nickels. If 5h已 had one more nickel than she has now, then she would h^ve the same number of pennies and nickels, lr )匚ents, how much are Leah's coins worth?(A) 33 (B) 35 (C) 37 (D) 39 (E) 41Problem 223 4 2s 2-彳 + 2-^ Problem 3Randy drove the first third of his trip on a gravel road^ the next 20 miles on pavement, and ths remaining □ne -fifth on a dirt road . In miles how long was Randy's tripi?(A) 30 (B)等 (C)粤(D) 40 (E)竽Problem 4Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is. how many times as expensive as a banana?m 吨了(A) | (B)彩 (C) f (D) 2 (E)-Problem 5Doug constructs a square window using 8 equal-size panes of glass, as shown ・ The ratio of the height to width ft3『 each pane is 5 : 2f and the borders around and between the panes are 2 inches wide, in inches, what 谄 the sid& length of th© square wind 口w?(A) 26 (B) 28 (C) 30 (D) 32 (E) 34Problem 6Orvin went to the store with just enough rrnoney to buy 30 balloons. When h@ arrived^ he discovered that the store had a special sale on b a I loons ：: buy 1 balloon at the regular p 『i 匚已 and get a second at g off the regular price. What is the greatest number of balloons Orvin could buy?(A) 33 (B) 34 (C) 36 (D) 38 (E) 39what im(A) 16 (B) 24 (C) 32 (D) 48 (E) 64Problem 7Suppose A> U > 0 and A is 近% greater than B. what is £?⑷100(#) (B) 1 叫字)(C) 100(字)(D) 1 叫乎)(E) 100(和SolutionProblem 8A truck tr^ve>l5 f feet every t seconds. There ars 3 feet in a yard i How many yards dDss the truck travel6in 3 rninute-s?b frn 血m10i 1 口⑷廖(X (①〒⑴)石㈣〒Solutionproblem 9For real numbers w and z,丄+丄予二于=2014,ut ww + zWhat is ----------- ?w ― z、一1 J 1(A) - 2014 (B)艄(C) —(D) 1 (E) 2014Problem 10In the addition shown below A. [3. C.and D are distinct digits. How many different values are po^ible ForD?ABBCB+ BCADADI3DDD(A) 2 (E) 4 (C) 7 (D) 8 (E) 9Problem 1111. For the consume^ a single discount of n%is more adv^ntagsous than any of the following discounts:(1)two ^ucces^ive 15% discounts(2)three successive 10% discounts(3) a 25f/J discount Fellow sd by s 5xu discountWhat is the- smallest possibls positive intsgsr valus uf n?(A) 27 (B) 2S (C) 29 (D) 31 (E) 33Problem 1212. The largest divisor of 2, 014h 000* 000 is rtsslf. What is its fifth largest divisor?(A) 125t875.000 (B) 20L400. 000 (C) 251, 750.000 (D) 402： 800.000 (E) 503.500.000Problem 13Sin regular hexagons surround 召regular hexagon of ^ide length 1 as shown. What is the ar?a of A J4BC?(A) 2V5 {B) 3V3 (C) 1+3辺(D) 2 + 2“$(E) 3+2\/3Problem 14Oariica drove her new car on a trip for a whole number of hour勺averaging 55 miles per hour. At the beginning of the trip」abc miles was displayed on thm Dclannstsr^ where abc is 耳3-digit number with ti > 1 and « + 6 —c < 7. At the end of the trip，the odometer showed eba miles. What is(i~ + fc2+ c~ ?(A) 26 (B) 27 (C) 36 (D) 37 (E) 41Problem 15In rectangle 打<7 = 2GR and points E3nd F lie on AF£□ that ED and FD tnsect Z.ADCas shown, what is the ratio of the area of ADEF to the ares of rectangle ABCD?S)晋(B)^ ©瞬㈣笫Problem 16Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the sarme value? (A)吉(B)12(C) I {D)島㈣|Problem 17What is thm greatest power of 2 t hat if a fmutor of 101— 4U：，01?Problem 18(A) 21DO2(B) 21003(C) 21004(D) 21C05(E) 21D0C A list of 11 positive! integers has a mean of 10；a median of 9j and a uiniqus> mads af 8- What is the largest passible value of an integer in the list?(A) 24 {B) 30 (C) 31 (D) 33 (E) 35Problem 19Two ccrncentric circles have radii 1 and 2. Two points an the outer circle are chosen independently and uniformly at random. What iw the probabi^it/ that the chord joining the two points in tersects the inner circle?(A) g (B) i (C)匕詳(D) i (E) +Problem 20For how many integers 工is the number —51J:2 + 50 negative?(A) 8 (B) 10 (C) 12 (D) 14(E) 16Problem 21Trapezoid ABCD has parallel sides AB af length 33 and CD of length 21. The other two sides are of lengths 1(' and 14. The angles at A and B are acute. What js the Imn gth of the shorter diagonal of ABCD?{A) 10?6 (B) 25 (C) S\/10 (D) 18“(E) 26SolutiariProblem 22Eight semicircles line the insid目af a ^qu^re with 吕id日length 2 as shown. What is the radium Q F the circle tangent tc ell of these semicircles?A sphere is inscribed in a truncated right circular cons as shown. The volume of thm truncated cone is twice that of the sphere. What is the ratro of the radius af the bottam base of the truncated cone to the radius of the tap base of the truncated cone?Problem 23(A) | ©近(D)2 (E)^^ Problem 24The numbers 1, 2} 3}斗』5 are to be arranged in a circle. An arrangement rs bad if rt is not true that fcr every n from 1 to 15 one can find a subset of the numberm that appear consecutively on the circle that sum to J ?. Arrangements that differ only by a rotation or a reflection mne considered the same. Haw many different bad arrangements are there?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problem 25In a small pond there are eleven lily pads in a row labeled 0 through 10, A frog is sitting on pad 1. When the frog is on pad N } 0 < N < 10, it 呷illju 叩 to pad N-l with probability — and to pad N +1 with probability 1 - Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake, If the frog reaches pad 10 it will exit the pond> never to return, what is the probability that the frog will escape being eaten by the snake?| (B)曙(C)磊(D)纟(E) \答案：1. C2. E3. E4. B5.A6. C7. A8. E9. A10. C11. C12. C13. B14. D15. A16. B17. D18. E19. D20. C21. B22. B23. E24. B25. Cwwt 冷10诲+ * +矿叮(A) 3 (B) & (C) y (D)爭 (E) 170Problem 2Roy's 亡吕t m吕t 百 of a can cf eat food ev&ry morning and y of a ean of eat food every evening. Before 怡目ding hiis 匚 a )t □“ Mlonday morn in Rciy ope me d a bck containing 6 cans of eat feotl*. On what day of the wisek dbd the 匚已t finish eating all the cat food in the box? (A) Tuesday (U) Wednesday (C) Thur 吕d 暫(U) Friday (E) Saturday Problem 3 Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for 32,501 each. In the aftemaon she sells two thirds of what she has left 3 and bscauss they are not freshshe charges only half priice. Tn the late aftern 口口n shethe r^mainiiing loaves -at B dollar leach. Each loaf 匚口百上鼻S (J_75 farher to make, In dollarSj what is her profit for the day? (A) 24 (B) 36 (C) 44 (D) 4S (E) 52Problem 4Walking down Jane Street, Ralph passed four houses in 吕 row, each painted a different 匚olcr- He passed th® orange house before the red housSi and the passed the blue house before ths yellow house. Ths blue house was not n&^t to t.h& ysillow house. How many orderings of the colored fhouses 囂「白 possible?(A) 2 (B) 3 (C) 4 (D) 5 (E) 6Problem 5On an 3l^eb rs quiz d 10% of Elis students scored 70 point 勺 35% scored 80 points^ 30% scored 90 pointSj and the rest scored 100 points. What is the differsincc between ths mean and median score cf ths students' scares on this quis?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Problerri 6SuppoBis that o 匚OWE ; give 6- gallons of milk in n days. At thus rate, how marTpr gallanE 口f nnilk will d cows give in c days?Probfem 7IN 口 r^zSrd 『自 £l nurfib&f£ ir. 些、ci f and b S-Stiisfy J : < LL <L b. Hdw marhy Gf th€i Follow in g id 白口u£ lit i 自弓 mustba true?(I) H + 空 V ci + b(II) Ji — y < a — 6(III) xy < ab(A) 0 (B) 1 (C) 2 (D) 3 (E) 4(C)cibdc (D) bcdeProblem 8 Which of the following niumbers Is a perfect square?Problem 9(A) 14115!~^T~(D)17fl81~2~18?193~2~Tfie two legs of a right trimngl已which are altitudes, ha\/e lengths 2\/3 and 6. How long is the third altitude of the triangte?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Problem 10Five positive consecutive integers starting with a have average b. what is the average of 5 consecutive integers that start with b?(A) d + 3 (B) ti + 4 (C) a 4^5 (D) u 4- 6 {E) <i + 7Problem 11A cusEomer who intends to purchase art appliance has three coupon5^ only ore of which may be uw总d;Coupon 1: .10/( off the listed price if the listed price rs at least S50Coupon 2. $20 off the listed price if the listed pnce is at Itfast 1100coupon 3： 18*X off the amount by which the listed price e^teeds $100For which of the following listed prices will coupari 1 offer a greater price reduction than either coupon 2 or coupon 3?(A) S179.95 (B) S199.95 (C) S219.95 (D) $239.95 (E) $259.95Problem 12A regular hexagon has side length 6. Ccngruerit arcs with radius 3 are drawn with the center at each of the vertical, creating circular sac tors 启弓shown. The region inside the ha^agon but outside the sectors is shaded as shown Whe t is thm^rea of the shaded regian?(A)幫価—跖(B) 27V3-&r (C) 54?5-18TT(D) &4^3-12^(E) 伽Problem 13Equilateral A>1R(7 hms side length 1』and squmrms AB DE, BOH I, CAFG bs outbids the triangle. What is the area of hexagon DEFGH I?Problem 14Ths y-intencBptSj P目nd Q, of two perp&ndicjldr lines intersecting mt the point j4(6h 8)have m sunn of zero. What is the area of AAPQ?(A) 45 (B) 4S (c) 54 (D) M (E) 72Problem 15Oavidl drives from his home to the airport to cat匚h a flight. He drives 35 milES in th^ first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?(A) 140 (B) 175 (C) 210 (D) 245 (E) 280Problem 16Fn rectangle A拜C0 A/J =L OC7 = 2』and points E, F、and <7 are midpoints of f 门』-and respectively・Point H is the midpoint of GE. What is the arsa of the shaded region?12 + 3>/3-4-(C) 3 + v5 (D)(E) 6/ Z?⑷吉㈣兽©害 (D)卷(E) 111 / 12Problem 17Three fair six-sided dn^e are rolled. What 祐 the prob ability that the values shown on two of the dice sum to the value shown an the rematning die?(A) j (B)第(C)籟(D)备(E) | Problem 18A square in the coordinate plane has vertices whose y-co ordinates are 0』1』4, and 5. What is the area of thm square?(A) 16 (B) 17 (C) 25 (D) 26 (E) 27 Problem 19Four cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portion of XV contdin^d in th© cub@ with edge length 3?(A)警 (B) 2^/3 (C)警 (D) 4 ㈣ 3闪Problem 20The product (8)(888 . .. 8), where ttie second factor has k digits, is an integer whose digits have a sum of 1000. What is fc?(A) 901 (B) 911 (C) 919 (D) 991 (E) 999Problem 21Positive integers a and b are such that the grsplis of = EAX + & and y = 驻+ b intersect this ir-a^is at ths same point. What is the sum erf all possible jr-coondinat&s of ths SB points of intersection?⑷-20 (B) -18 (C) 一苗(D) -12 (E) —8Problem 22[n r直亡tanglm AnC'D r AB= 20 and BG =10. Let £? be a point Ort CD such that Z.CBE= 15°. What is AE?(A) (B) IO A/3(C) 18 (D) 11 辺(E) 20Problem 23A rectmngubr piece of paper whose length is V3 times the width has area A, The paper is divided into three equal sections along the opposite I eng th Sj artd then a dotted fine is drawn from the first divider to the 亡end divider cn the opposite side shown. The paper 左then fbldEd flat along thiim dotted line to cnemtm a. new whmp日with area /?. What is the ratio J? : A?(A) 1 : 2 (B) 3 : &Problem 24A ssquanc^ cf natural nuimber^ i£cori£trut?ted by listing the first 4H then skipping one^ fci sting the newt 5, skipping 2. listing 6』skipping 3d andj on the nth itEration, listing n + 3 and skipping n. The sequence bsgins L 2h 3h 4, 6, 7. 8. 9.10. 13. What is ths 500- QOOth number in the sequence?(A) 996,506 (B) 996507 (C) 996508 (D) 996509 (E) 996510Problem 25The number 5B67is between 22011 and 2Z014r now many pairs or integers ^re there such thmt(C) 2 : 3 (D) 3 : 4 (E) 4 : 5Problem 201 < m < 2012 and(A) 278 (B) 279 (C) 280 (D) 281 (E) 282答案：1.C 2.D 3.D 4.B 5.E 6.B 7.B 8.B 9.D 10.C 11.C 12.C 13.C 14.C 15.B 16.B 17.D 18.E 19.D 20.B 21.C 22.A 23.C 24.B 25.A。

2014美赛题目

PROBLEM A: The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- orover-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with a simple change of orientation, or would additional requirements be needed.Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system –either part of the road network or imbedded in the design of all vehicles using the roadway –to what extent would this change the results of your earlier analysis?PROBLEM B: College Coaching LegendsSports Illustrated, a magazine for sports enthusiasts, is looking for the “best all time college coach”male or female for the previous century. Build a mathematical model to choose the best college coach or coaches (past or present) from among either male or female coaches in such sports as college hockey or field hockey, football, baseball or softball, basketball, or soccer.Does it make a difference which time line horizon that you use in your analysis, i.e., does coaching in 1913 differ from coaching in 2013? Clearly articulate your metrics for assessment. Discuss how your model can be applied in general across both genders and all possible sports. Present your model’s top 5 coaches in each of 3 different sports.In addition to the MCM format and requirements, prepare a 1-2 page article for Sports Illustrated that explains your results and includes a non-technical explanation of your mathematical model that sports fans will understand.。

2014美国数学竞赛AMC10A、10B试题及答案

2014 AMC 10B Problems Problem 1 Leah his 13 coins all Cif Vzhtch are penhiss 占rd r 亡kels . It ske had 亡fim rTicre ritkel than che na5 r aw d ^hefl she would navi the $arrii@ number or pemi&& and Enhow much are Leah 1^ coins 询crthT (A)瞬 (B) .^5(G) 37 (T>) 3f> (E) 41 Problem 22-J +2-^?Problem 3R^ndy dra^s tha first third or his trip an a gravel void, the nsxt 20 miFaf on pdvamsnt, and tha retridinirig one-fifth an a dirt raadL In how long Pandy 1! trip?Problem 4Susi'A pays for 4 mufhin? emd 3 ban^rei5i Calvin twee a? much 口ayring Kor 2 nuuffris ^nd 16 h^nsflas,A muffin 乜 h 口w many t mos as Q^ponsi^e 3^ 口 banana? (A) | (n) | (c) J (D) 2 (E) y(A) 16(B) 24 {C) 32 (r>) 48 (E) 54400 (B)罟 (C >5(A) - 2014 （E ）M4 (C)磊(D) 1 (E) 2014Problem 5daug construets J squara 碘kido 站 u&irtq & ^qujl-fl2B panas of glass^ as shown. Ttia ratio of the height XQ width tor each pane i? ： 2, -and the borcers aro 」nd sne between the pane? are 2 inches wide In inches, what Is tie side length of the squim 「9 win dow?(A) 2fi (B) 28 (C) M (D) 32 (E) 34Problem 6Orvin writ to the stoTB with just snDiigh muhBy to buy 30 ball 口口口弓，When he airivad 〉he disc^overed th/ the h^d A spacial on bd'looni; buy 1 b 訓loan At the ragular pries and get t Escond it 壬 off th« regular p 科亡o. /Vhat is the gr&at*t? F-Hjnrber of bal □亡勺r ：s Orvin cniuld bu^?(A) 33 (B) 34 (C) 36 (D)典 (E)曲 Problem 7Scppo^s A > D > 0 dnd A is u:% grt=dl«i Lhan B What 旨 x7J _ RJ _L R d -L X? a _ ft J W 伽土泸j (B) 1叫爭)(C) lOG(i-) (D) 100(^) M 100(却Sclut anProblem 8 代 truck travels — feet every t seccnds. There are ^eet h a yard How nanv yaiTls does the truck travel Oin 3 minutes?Z.clut anProblem 9F 亡「real rumbe-'S UJ andi±| = 2014it £(A )w^of (H)竽（C ）7 （D ）7 冋丫TVnat isProblem 10[n ihc adch：ian shawn belcrw .1^ B.C, ond D zine istinct ciqrts. Haw man/ dFttGrent value-z 3r e pas^ibls fcrZ??ABECB+ BCADADBDDD(A) 2 (B) 4 (C)7 (D) 8 (E) 9Problem 1111. For the consurriw- A diicouni uF HA is more idv^r td.w-iii thmn AH/ of tha following dif:□ jn：£：< 1 ) twd 5LCIZE5£」TE 15"足(115COUnti(2) t n ee sui utfMw I〔瞋disCLunli<3> m 2合冥discocrit fallowed by a S% d scuuntWhat i£the smallest pa^^ble pasitice integer valLe t4 ri?(A) 27 (B) 28 (C) 29 (D) 31 (E) 33Problem 12IB. The hrgast div也or of Z014.00Ct<XX) ; itself u its fifth largest divisor?(A) 125.番氐000 (B) 201+ 900* DOO (C) 35l t750.000 (D) W, S00, DOG (E) 5OJ, 500” 000Problem 13Six rugula kti^dgoi ib sur tcui id a _egu ch of siUt t?iig th 1 TZ shown. Whdi 0、C is area of A.4J?C A(A) ?\/3 (B) (C) 1 + Sx/2 (D)2+2jA (E) 3 + 2亦Problem 14droi/e her new c^r on 右trip for a \wno:s ni_mber of hourc, ^v^r^girg 55 milet per hour. At the^Agirning nf ths trip, mlz WA^ rrisplaypd nn thp rdnnprRr, A/hPr=可hr is a3-rigir numtipr wit+i /i > 1 and « + ?J + n < 7 AI the end of itie trip, tre JchmEtFt shc^ea 伽tniiss WMt iE J + M + r" ?(A) 26 (B) 27(C) 36 (D> 笄(£} 41Problem 15Tn rur ijlw DC = 2CB and points E and F lie on AB n th^t ED and FJ? trisect ZADCas shown. What is the ratio nf the area nF Z^DEF tn the area of rectangle ABCD T(A)晋(B)罟(C)攀(D) 1 (E)晋Problem 16Nv fan-s\> ; ded dice are relied. Wnal II die probab) itv that a: (east three of the four dies show t issame⑷籟g ◎箱囲Problem 17^hat is the g-eatest 口亡吨r at 2 that ts a factor of >- l^1?(A) 2'1102(B) 2KIU?l(C) 2LmM(D) 2"J,)S(E) 2山阳Problem 18A Ii3t □' 11 positive integers has a mean of 10』a mEdiari of 瞬and a unicue mode of 8.列h日t is the langes t possible value of an inteqe^ in the Ktl(A) 24 (B)加(C) 31 (D) 33 (E) 35Problem 丄9T IAO con匚wrt|-,c c neies r^dil and 2. Two pcints cn the outer circle ai'e chosen inde^enderit y ard uniformly at randoin. What iu the probability lhat ths chord joiriing the two points ntere&ctc the irirer cin cle?⑴！9)字(D>3(E)|Problem 20Fdf how nsn/ int =g?r5 -r is the number r1—+ SO 师第Tiw?(A) y (B) 10 (C) 12 (D) 14 (E) IdProbfem 211 rap^zold AECD has parallel sices AB of lengTh 33 and CD of length 21 I ne 匚ther rwo sid=s are of lengths 1U and ll. The mn glmw at A mnd B mre 己二“土日.Whmt『w ths length of the shorter di^igonsl of ABCDl(A) 1fl"5 (B) 25 (C) fiTin (D) 1 用迈(E) 2fiSolutionProblem 22Ciqhr 弓=厂,匕 rim:；li-i? the irsice nf a 三、“* 讨i「h 5 re =- gti- 2 mm shz./■.■■-i. i= t-|p ra dii.^ 十the : re I? tan^nr to mil of the?e semictrclss?(A)气匹(B) ^2 (C)^±l (D)军(E)^Problem 23A ipliuid 5 ri^Lilibud In a t uriLdLyd right cilLUldi uui it ihuwhi・ ill匕vuluum uT the truiiLditsdl cune I、twice that of "lhe sphere. What 石the ratio of the radios nF th= bottom hasp of "the truncatprl 匚匚no to the radius of the top oase D F the truncated cone?Problem 2斗The nunber? L 2d 3』4, 5 are ;□ be arrangeG in a circle. An aiTangemer^t is 匕ed f it b riot true that for every n from 1 to 15 one can find a subset of the numbers that appear consscutiMel/ on the crirde that wum to n.Arran gemerts that diffe r arl/ by a ratation or a refection are cjnsidEned the same. How many difF&rort □rr-angemont; aro th&ro?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Problem 25In a smell pond t*iene are eleLer lily pads ri a 口许libelee 0 through 10L b frog is sitting on pad 1 //hen the frog is on pad N)U < N < 1© it wilf jump tc pad JV -1 with probability 百and to pad /V +1 with probability 1 - Each jump is independent of the previa us jumps, If the frog reaches pad 0 it 嘶II be eaten by a patientl/ waiting snake. If tne freg reaches pad =0 it w II ezit the pend,, ne\jer to Tturn. jvhat is the probability that the frog will escape being oaten by ttie snake?吨但熄(唸(D圧砒答案：1. C2. E(c) V3 (DJ 2 (E)汇兰633. E4. B5. A6. C7. A8. E9. A10. C11. C12. C13. B14. D15. A16. B17. D18. E19. D20. C21. B22. B23. E24. B25. CProblem 12C 17(1(A) 3 (B) 8 (C) —(D) —(E) 170Problem 21亠IRoi/'s c at eats —of a cai of c^t feed evs-r^ nnorniriq and 了口f t匚耳n 口f cat fccri ever^ pypnifiq.日戶fuirs ife!i?diFiij 1右&L on Munday rnarniB iy^ Ruy npi=*H^d 划；bux itsntdiHiiriy 6 &di is 酎u^st luud^・Di i A h dI dsy uf tliw wttfk did thy c di finish 邑^Hng dll thid edit Food m Ihw btM?(A) Tuesday (B) V^Wric&day (C) ThuTsday (D) Fridas" (R) SaturdayProblem 3Bndq=t bafkes 斗吕a = \jes ot bread for her bakery- SH B sell= half of t hem in t-~ie merninq tor each. I rnthe afternoon she s：三||生tv/o tkirids Df v/hat eke nas left^ and because they are not fresh^ 三hm charges only ha if price in rhp ate aftfimoan she selk rhe r&imiining loav 已弓at a dcxlsr each. Each loaf costs SO. 7F> for h ar to iriAkm・in dollars H wh<3t ig hor praHt for the >dlay?(A) 21 (B)眺(C) 44 (D> 犒(K) 52Problem 4lAijIking down Jane stnget, Ra^ph ft ur dailies In 2 row* aach pointed m dirfisrent color. Heihu riDij_.i=»bufufL ihti r^o rtou^ti, and F IE pmaPiJ thu Lluu hou^u teforb th® yJlu^1 hDu!=u. Thu blu^jhu Jsy *3=rtet tu 七’im yellow hu 卜uvf hiai iy urd&rir»^b o f th«L ul 口Itfd liacl士弋0 Trx posslb w *(A) 2 (B) 3(C) 4 (D) 5 {E) 6Problem 5On I jlc^bTd qui2, LC/JJi af studtritb 70 pOiritL. )5坏沁」3 80 pan ：£, 30'兀 tLcrud 90 puinti. jnd the ro?t scored 100 points, what lg the dlfi^rsnc& batman the maar and median sc ora or ths Student^' 2匚ure5 u” this quiz^(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Pnobl^m ES jppose that a cow5 give b gallons of milk in 栏 dav s ' At this rate, how man/ qaJIans of rmlk will d cows qive n r si.ai/E?Pnoblem 7Man zero raal numb erg# ci … andl ii 亍曰 ti^Fy _r V 口 and y V b How m-any of the following inequalities EU 五 tbe t 『ue? (I) jr + y < ^ + 6(IT) yr — ir <(/ — />(III)刊 < “(A) (] (H> 1 (C) 2 (n) {E )4Problem 8whicn of ths following numtiars is a perfect squara? (C)空空(D)辿 rl (E)(A)呼an 1M161 HiM7! P (D ) (E) 1 KT19JProblem 9The two cf a right irjangle, which are atiitud^ hau? lengths 2\/3 and G. How long is the third昌hitudw of the tr dangle?(A) 1 (R> 2 (C) 3 (D) 4 (E) 5Problem 10Five positive consecutive irn卸日巧startrig with(i have average b、What is the average of 5 corsecutive integers that stare with(A) u + 3 (B) u+4 (C)«+5 (D) u+& (E)a + 7Problem 11A custumsr whe intends to purchase an 日匚has three： cgupons. only one of 宵hHh may used;Coupon 1: 10% off Hhe listed price if the listed price is at least 830CHijp in 2： S2fl \\~f iS'H i price f rli& pfir? S < If 礼祁00Coiipnn m：nff tfie ^niciJITt fcy wh rh thp I F T PC! prirp PK「PP*卜匚「Aihizh of the foliating I tted pnces will caupan 1 offer a greater priem radLiction than coupon 2 ar ccupon 3?(A) $179』5 (B) $190.95 (C) S219.95 (D) S23D D5 (E) S2M.^jProblem 12h rsgiiFar hsitagon hs= z r?ngth □. Congrucnl arcs w th radi 3 z徑drawn wsth tha center ac ^ach of the verti匚匚resting circular sectors as shown, rhe reg on iiisidE the hexagon but outside the sectn^s is shsdAd as shown v>Fi*『；s「F IB吕『吕n o>r the shaded re-gion?"(A) 271/3 - S TJ(B) 27?3-6TU(C) &4S/5-18JT(D) 54/3 —12工(E)丄08爲-陥Problem 13Equilateral △ABU has fide length 1』snd squares AB[>E. t CAF<^is cuUida Vie triarnjie,& th-= am4 oF 卜口n r^EFCi HF?(A) 严 (D)专(C)3+v^ (D)今超(E) 0Problem 14rhe ij-intarcapts. J P arid Q f of two perpondiculBr knes intarsectiinq at ths point A( G, 8) ti.avs a ^um of i&^o Wh竜t is th© 目「吕目of △A J PQ T(A) 45 (B) 48 (C) 54 (D) 60 (E) 72Prablem 15D.3Vin nl-ix/RS from his l^cnE tc thm sirpnrr ―口ratch a fliqhr. He dr\i^^ :i5 in the fir=；r hour,, but nerizRT Ihd L 1 it: will tiu 1 t uur IdLe ir ht: LunLinuE^ 口L l和in J. hit: hin spuuLl Ly 1G rrilt?i pu，i uur fur(ha rest of the to lhe airport mrid arrivs^ 30 minuw^ early, How many miss is the airport 耐onn hit hom»?(A) 140 (B) 175 (C) 210 (D) 245 (E) 260Problem 16In r»ctangl»AHC^rt, AI3 —1』F3C7 ■ 2}and points E, F,and G are nnidponts of HC7, CD,, and AD r rEipeetivtl^. F^clnt II s th* imdpaint of GE.What s cht area of the shaded region fD L L C2窗⑷召咗⑹春⑼容®lProblem 17Th res *air s M-sided dice are ■ol ed ・内h 曰t s the prot ability that the values shown on two nf t 卜旧dice SUT to the value shown on the remaining dis?i n 7 q 空S讣㈣箱(C)36 (D暢(瞻Problem 18A sqja^e in the coord nate plane has vertices whose ir-coorciiriares are 0, L 4」and 5 what Is ihe ansa of the square?(A) 16 (B) 17 (C) 2& (D) 26 (E) 27Problem 19Four cubss with edge lengths L 2, 3f and 4 are stacked as shown. Thwt is the length of the poftcn of .Y V eontoH&d H tbid cubo with adgo length 3?(A)響(B) 2v/3 (C)警(D) 4 (E) 3迈Problem 20rre aroduct ；&){ES8 一 -. inhere Uia second rector ha? k uigt® IM an integer 罰口證zlgitz hjvm 扌surr- ct 1000. Wiiat is it?(A) iW)l (D) <>11 (C) 919 (D) 901 (E) 999Problem 21PositivB integers a arid b are such that ths graphs d y = ax +5 jnd j/ = Jr + A intersec: ths z~axis mt the sdiTfi point. Wl idt is the S^ITII□ f dll fjnsziibl匕X"Luurdin.diteb of Ihe^e poihts J T intersHcticn?(A) - 20 (B) -18 (C) - 15 (D) -12 (E) - &Problem 22in rectangle A BCD, AB= 20 mnd HC= 10, Let £? he pciint on C7D such thmt ZCJ3/17 = 15°. w|i3t is AE1 (A)卑岁(B) 30^ (C) 18 (D) 11^/3 (E) 20Problem 23A rsctang」ln「pne^s D F P a P B" whose length is times the width has =raa A,Th© paper is divided inta three equal sections alonq the 口pposite enqths-, and then a dotted line is drawn from the first divider to the secorid divider on the? o-Dpcsiie 宜ick 35 shsv^in. The paper is then fcldsd flat slong this dotted fine to create d new slape wtth arsaB Wha t is the ratio D: .4?Problem 24A ■■querce of naturBi numbers is ranitrucked hy listing the frit 4. lii 曰i skipping one, listing ll it rit^：d 3 sluppinci 2. Iistifi^ 6, shipping 3,吕nd 」cn the retn iteratorij listing M +3 and skipping a. "■ he seq 」三"匕巳 begHs l n L?n 3,丄 G, 7. K. 9,10n 13. What is the 500* 000th nutrrber in the “quencm 亍(A) 996 f>ft6 (B) ^^7 (C)怖5俯 (D) 996509 (E) 99651C1 Problem 25T TB n limber 5*"丁 © botwoon 2?111 and 2'Q14. How manv p^lrs of integers (rn. JA ' are th are ^uch tnat 1 兰 ru 兰 jMJl 空 ai id5* < 2fa < 2*1*2 < 5rt+1 ?答案：1.C 2.D 3.D 4.B 5.E 6.B 7.B 8.B 9.D 10.C 11.C 12.C 13.C 14.C 15.B 16.B 17.D 18.E 19.D 20.B 21.C 22.A25.A(D) 3 "(A) 27B (B) 2TO (C) 2S0 (D) 281 (E) 2&2。

历届美国数学建模竞赛赛题(汉语版)

历届美国数学建模竞赛赛题, 1985－2006AMCM1985问题-A 动物群体的管理AMCM1985问题-B 战购物资储备的管理AMCM1986问题-A 水道测量数据AMCM1986问题-B 应急设施的位置AMCM1987问题-A 盐的存贮AMCM1987问题-B 停车场AMCM1988问题-A 确定毒品走私船的位置AMCM1988问题-B 两辆铁路平板车的装货问题AMCM1989问题-A 蠓的分类AMCM1989问题-B 飞机排队AMCM1990问题-A 药物在脑内的分布AMCM1990问题-B 扫雪问题AMCM1991问题-A 估计水塔的水流量AMCM1992问题-A 空中交通控制雷达的功率问题AMCM1992问题-B 应急电力修复系统的修复计划AMCM1993问题-A 加速餐厅剩菜堆肥的生成AMCM1993问题-B 倒煤台的操作方案AMCM1994问题-A 住宅的保温AMCM1994问题-B 计算机网络的最短传输时间AMCM1995问题-A 单一螺旋线AMCM1995问题-B A1uacha Balaclava学院AMCM1996问题-A 噪音场中潜艇的探测AMCM1996问题-B 竞赛评判问题AMCM1997问题-A Velociraptor(疾走龙属)问题AMCM1997问题-B为取得富有成果的讨论怎样搭配与会成员AMCM1998问题-A 磁共振成像扫描仪AMCM1998问题-B 成绩给分的通胀AMCM1999问题-A 大碰撞AMCM1999问题-B “非法”聚会AMCM1999问题- C 大地污染AMCM2000问题-A空间交通管制AMCM2000问题-B: 无线电信道分配AMCM2000问题-C：大象群落的兴衰AMCM2001问题- A: 选择自行车车轮AMCM2001问题-B：逃避飓风怒吼（一场恶风…）AMCM2001问题-C我们的水系-不确定的前景AMCM2002问题-A风和喷水池AMCM2002问题-B航空公司超员订票AMCM2002问题-C蜥蜴问题AMCM2003问题-A: 特技演员AMCM2003问题-C航空行李的扫描对策AMCM2004问题-A：指纹是独一无二的吗？AMCM2004问题-B：更快的快通系统AMCM2004问题-C：安全与否？AMCM2005问题-A：.水灾计划AMCM2005问题-B：TollboothsAMCM2005问题-C：.Nonrenewable ResourcesAMCM2006问题-A：用于灌溉的自动洒水器的安置和移动调度AMCM2006问题-B：通过机场的轮椅AMCM2006问题-C：在与HIV/爱滋病的战斗中的交易AMCM85问题-A 动物群体的管理在一个资源有限，即有限的食物、空间、水等等的环境里发现天然存在的动物群体。

2014美国数学建模-B题paper-30680

A Networks and Machine Learning Approach toDetermine the Best College Coaches of the20th-21st CenturiesTian-Shun Allan Jiang,Zachary T Polizzi,Christopher Qian YuanMentor:Dr.Dan TeagueThe North Carolina School of Science and Mathematics∗February10,2014Team#30680Page2of18Contents1Problem Statement3 2Planned Approach3 3Assumptions3 4Data Sources and Collection44.1College Football (5)4.2Men’s College Basketball (5)4.3College Baseball (5)5Network-based Model for Team Ranking65.1Building the Network (6)5.2Analyzing the Network (6)5.2.1Degree Centrality (6)5.2.2Betweenness and Closeness Centrality (7)5.2.3Eigenvector Centrality (8)6Separating the Coach Eﬀect106.1When is Coach Skill Important? (11)6.2Margin of Win Probability (12)6.3Optimizing the Probability Function (13)6.3.1Genetic Algorithm (13)6.3.2Nelder-Mead Method (14)6.3.3Powell’s Method (14)7Ranking Coaches157.1Top Coaches of the Last100Years (15)8Testing our Model158.1Sensitivity Analysis (15)8.2Strengths (16)8.3Weaknesses (16)9Conclusions17 10Acknowledgments172Team#30680Page3of181Problem StatementCollege sport coaches often achieve widespread recognition.Coaches like Nick Saban in football and Mike Krzyzewski in basketball repeatedly lead their schools to national championships.Because coaches inﬂuence both the per-formance and reputation of the teams they lead,a question of great concern to universities,players,and fans alike is:Who is the best coach in a given sport? Sports Illustrated,a magazine for sports enthusiasts,has asked us toﬁnd the best all-time college coaches for the previous century.We are tasked with creat-ing a model that can be applied in general across both genders and all possible sports at the college-level.The solution proposed within this paper will oﬀer an insight to these problems and will objectively determine the topﬁve coaches of all time in the sports of baseball,men’s basketball,and football.2Planned ApproachOur objective is to rank the top5coaches in each of3diﬀerent college-level sports.We need to determine which metrics reﬂect most accurately the ranking of coaches within the last100years.To determine the most eﬀective ranking system,we will proceed as follows:1.Create a network-based model to visualize all college sports teams,theteams won/lost against,and the margin of win/loss.Each network de-scribes the games of one sport over a single year.2.Analyze various properties of the network in order to calculate the skill ofeach team.3.Develop a means by which to decouple the eﬀect of the coach from theteam performance.4.Create a model that,given the player and coach skills for every team,canpredict the probability of the occurrence of a speciﬁc network of a)wins and losses and b)the point margin with which a win or loss occurred.5.Utilize an optimization algorithm to maximize the probability that thecoach skill matrix,once plugged into our model,generates the network of wins/losses and margins described in(1).6.Analyze the results of the optimization algorithm for each year to deter-mine an overall ranking for all coaches across history.3AssumptionsDue to limited data about the coaching habits of all coaches at all teams over the last century in various collegiate sports,we use the following assumptions to3Team#30680Page4of18 complete our model.These simplifying assumptions will be used in our report and can be replaced with more reliable data when it becomes available.•The skill level of a coach is ultimately expressed through his/her team’s wins over another and the margin by which they win.This assumes thata team must win to a certain degree for their coach to be good.Even ifthe coach signiﬁcantly ampliﬁes the skills of his/her players,he/she still cannot be considered“good”if the team wins no games.•The skills of teams are constant throughout any given year(ex:No players are injured in the middle of a season).This assumption will allow us to compare a team’s games from any point in the season to any other point in the season.In reality,changing player skills throughout the season make it more diﬃcult to determine the eﬀect of the coach on a game.•Winning k games against a good team improves team skill more than winning k games against an average team.This assumption is intuitive and allows us to use the eigenvector centrality metric as a measure of total team skill.•The skill of a team is a function of the skill of the players and the skill of the coach.We assume that the skill of a coach is multiplicative over the skill of the players.That is:T s=C s·P s where T s is the skill of the team,C s is the skill of the coach,and P s is a measure of the skill of the players.Making coach skill multiplicative over player skill assumes that the coach has the same eﬀect on each player.This assumption is important because it simpliﬁes the relationship between player and coach skill to a point where we can easily optimize coach skill vectors.•The eﬀect of coach skill is only large when the diﬀerence between player skill is small.For example,if team A has the best players in the conference and team B has the worst,it is likely that even the best coach would not be able to,in the short run,bring about wins over team A.However, if two teams are similarly matched in players,a more-skilled coach will make advantageous plays that lead to his/her team winning more often than not.•When player skills between two teams are similarly matched,coach skill is the only factor that determines the team that wins and the margin by which they win by.By making this assumption,we do not have to account for any other factors.4Data Sources and CollectionSince our model requires as an input the results of all the games played in a season of a particular sport,weﬁrst set out to collect this data.Since we were unable to identify a single resource that had all of the data that we required,we4Team#30680Page5of18 found a number of diﬀerent websites,each with a portion of the requisite data. For each of these websites,we created a customized program to scrape the data from the relevant webpages.Once we gathered all the data from our sources,we processed it to standardize the formatting.We then aimed to merge the data gathered from each source into a useable format.For example,we gathered basketball game results from one source,and data identifying team coaches from another.To merge them and show the game data for a speciﬁc coach,we attempted to match on commonﬁelds(ex.“Team Name”).Often,however,the data from each source did not match exactly(ex.“Florida State”vs“Florida St.”).In these situations,we had to manually create a matching table that would allow our program to merge the data sources.Although we are seeking to identify the best college coach for each sport of interest for the last century,it should be noted that many current college sports did not exist a century ago.The National Collegiate Athletic Association (NCAA),the current managing body for nearly all college athletics,was only oﬃcially established in1906and theﬁrst NCAA national championship took place in1921,7years short of a century ago.Although some college sports were independently managed before being brought into the NCAA,it is often diﬃcult to gather accurate data for this time.4.1College FootballOne of the earliest college sports,College Football has been popular since its inception in the1800’s.The data that we collected ranges from1869to the present,and includes the results andﬁnal scores of every game played between Division1men’s college football teams(or the equivalent before the inception of NCAA)[2].Additionally,we have gathered data listing the coach of each team for every year we have collected game data[4],and combined the data in order to match the coach with his/her complete game record for every year that data was available.4.2Men’s College BasketballThe data that we gathered for Men’s College Basketball ranges from the sea-son of theﬁrst NCAA Men’s Basketball championship in1939to the present. Similarly to College Football,we gathered data on the result andﬁnal scores of each game in the season and inﬁnals[2].Combining this with another source of coach names for each team and year generated the game record for each coach for each season[4].4.3College BaseballAlthough College Baseball has historically had limited popularity,interest in the sport has grown greatly in the past decades with improved media coverage and collegiate spending on the sport.The game result data that we collected5Team#30680Page6of18 ranges from1949to the present,and was merged with coach data for the same time period.5Network-based Model for Team Ranking Through examination of all games played for a speciﬁc year we can accurately rank teams for that year.By creating a network of teams and games played, we can not only analyze the number of wins and losses each team had,but can also break down each win/loss with regard to the opponent’s skill.5.1Building the NetworkWe made use of a weighted digraph to represent all games played in a single year.Each node in the graph represents a single college sports team.If team A wins over team B,a directed edge with a weight of1will be drawn from A pointing towards B.Each additional time A wins over B,the weight of the edge will be increased by1.If B beats A,an edge with the same information is drawn in the opposing direction.Additionally,a list containing the margin of win/loss for each game is associated with the edge.For example,if A beat B twice with score:64−60,55−40,an edge with weight two is constructed and the winning margin list4,15is associated with the edge.Since each graph represents a single season of a speciﬁc sport,and we are interested in analyzing a century of data about three diﬀerent sports,we have created a program to automate the creation of the nearly300graphs used to model this system.The program Gephi was used to visualize and manipulate the generated graphs. 5.2Analyzing the NetworkWe are next interested in calculating the skill of each team based on the graphs generated in the previous section.To do this,we will use the concept of central-ity to investigate the properties of the nodes and their connections.Centrality is a measure of the relative importance of a speciﬁc node on a graph based on the connections to and from that node.There are a number of ways to calculate centrality,but the four main measures of centrality are degree,betweenness, closeness,and eigenvector centrality.5.2.1Degree CentralityDegree centrality is the simplest centrality measure,and is simply the total number of edges connecting to a speciﬁc node.For a directional graph,indegree is the number of edges directed into the node,while outdegree is the number of edges directed away from the node.Since in our network,edges directed inward are losses and edges directed outwards are wins,indegree represents the total number of losses and outdegree measures the total number of wins.Logically,therefore,outdegreeeindegreee represents the winlossratio of the team.This ratiois often used as a metric of the skill of a team;however,there are several6Team#30680Page7of18Figure1:A complete network for the2009-2010NCAA Div.I basketball season. Each node represents a team,and each edge represents a game between the two teams.Note that,since teams play other teams in their conference most often, many teams have clustered into one of the32NCAA Div.1Conferences. weaknesses to this metric.The most prominent of these weaknesses arises from the fact that,since not every team plays every other team over the course of the season,some teams will naturally play more diﬃcult teams while others will play less diﬃcult teams.This is exaggerated by the fact that many college sports are arranged into conferences,with some conferences containing mostly highly-ranked teams and others containing mostly low-ranked teams.Therefore, win/loss percentage often exaggerates the skill of teams in weaker conferences while failing to highlight teams in more diﬃcult conferences.5.2.2Betweenness and Closeness CentralityBetweenness centrality is deﬁned as a measure of how often a speciﬁc node acts as a bridge along the shortest path between two other nodes in the graph. Although a very useful metric in,for example,social networks,betweenness centrality is less relevant in our graphs as the distance between nodes is based on the game schedule and conference layout,and not on team skill.Similarly, closeness centrality is a measure of the average distance of a speciﬁc node to7Team#30680Page8of18 another node in the graph-also not particularly relevant in our graphs because distance between nodes is not related to team skills.5.2.3Eigenvector CentralityEigenvector centrality is a measure of the inﬂuence of a node in a network based on its connections to other nodes.However,instead of each connection to another node having aﬁxed contribution to the centrality rating(e.g.de-gree centrality),the contribution of each connection in eigenvector centrality is proportional to the eigenvector centrality of the node being connected to. Therefore,connections to high-ranked nodes will have a greater inﬂuence on the ranking of a node than connections to low-ranking nodes.When applied to our graph,the metric of eigenvector centrality will assign a higher ranking to teams that win over other high-ranking teams,while winning over lower-ranking nodes has a lesser contribution.This is important because it addresses the main limitation over degree centrality or win/loss percentage,where winning over many low-ranked teams can give a team a high rank.If we let G represent a graph with nodes N,and let A=(a n,t)be an adjacency matrix where a n,t=1if node n is connected to node t and a n,t=0 otherwise.If we deﬁne x a as the eigenvector centrality score of node a,then the eigenvector centrality score of node n is given by:x n=1λt∈M(n)x t=1λt∈Ga n,t x t(1)whereλrepresents a constant and M(n)represents the set of neighbors of node n.If we convert this equation into vector notation,weﬁnd that this equation is identical to the eigenvector equation:Ax=λx(2) If we place the restriction that the ranking of each node must be positive, weﬁnd that there is a unique solution for the eigenvector x,where the n th component of x represents the ranking of node n.There are multiple diﬀerent methods of calculating x;most of them are iterative methods that converge on a ﬁnal value of x after numerous iterations.One interesting and intuitive method of calculating the eigenvector x is highlighted below.It has been shown that the eigenvector x is proportional to the row sums of a matrix S formed by the following equation[6,9]:S=A+λ−1A2+λ−2A3+...+λn−1A n+ (3)where A is the adjacency matrix of the network andλis a constant(the principle eigenvalue).We know that the powers of an adjacency matrix describe the number of walks of a certain length from node to node.The power of the eigenvalue(x)describes some function of length.Therefore,S and the8Team#30680Page9of18 eigenvector centrality matrix both describe the number of walks of all lengths weighted inversely by the length of the walk.This explanation is an intuitive way to describe the eigenvector centrality metric.We utilized NetworkX(a Python library)to calculate the eigenvector centrality measure for our sports game networks.We can apply eigenvector centrality in the context of this problem because it takes into account both the number of wins and losses and whether those wins and losses were against“good”or“bad”teams.If we have the following graph:A→B→C and know that C is a good team,it follows that A is also a good team because they beat a team who then went on to beat C.This is an example of the kind of interaction that the metric of eigenvector centrality takes into account.Calculating this metric over the entire yearly graph,we can create a list of teams ranked by eigenvector centrality that is quite accurate. Below is a table of top ranks from eigenvector centrality compared to the AP and USA Today polls for a random sample of our data,the2009-2010NCAA Division I Mens Basketball season.It shows that eigenvector centrality creates an accurate ranking of college basketball teams.The italicized entries are ones that appear in the top ten of both eigenvector centrality ranking and one of the AP and USA Today polls.Rank Eigenvector Centrality AP Poll USA Today Poll 1Duke Kansas Kansas2West Virginia Michigan St.Michigan St.3Kansas Texas Texas4Syracuse Kentucky North Carolina5Purdue Villanova Kentucky6Georgetown North Carolina Villanova7Ohio St.Purdue Purdue8Washington West Virginia Duke9Kentucky Duke West Virginia10Kansas St.Tennessee ButlerAs seen in the table above,six out of the top ten teams as determined by eigenvector centrality are also found on the top ten rankings list of popular polls such as AP and USA Today.We can see that the metric we have created using a networks-based model creates results that aﬃrms the results of commonly-accepted rankings.Our team-ranking model has a clear,easy-to-understand basis in networks-based centrality measures and gives reasonably accurate re-sults.It should be noted that we chose this approach to ranking teams over a much simpler approach such as simply gathering the AP rankings for vari-ous reasons,one of which is that there are not reliable sources of college sport ranking data that cover the entire history of the sports we are interested in. Therefore,by calculating the rankings ourselves,we can analyze a wider range of historical data.Below is a graph that visualizes the eigenvector centrality values for all games played in the2010-2011NCAA Division I Mens Football tournament.9Team#30680Page10of18 Larger and darker nodes represent teams that have high eigenvector centrality values,while smaller and lighter nodes represent teams that have low eigenvector centrality values.The large nodes therefore represent the best teams in the 2010-2011season.Figure2:A complete network for the2012-2013NCAA Div.I Men’s Basketball season.The size and darkness of each nodes represents its relative eigenvector centrality value.Again,note the clustering of teams into NCAA conferences. 6Separating the Coach EﬀectThe model we created in the previous section works well forﬁnding the relative skills of teams for any given year.However,in order to rank the coaches,it is necessary to decouple the coach skill from the overall team skill.Let us assume that the overall team skill is a function of two main factors,coach skill and player skill.Speciﬁcally,if C s is the coach skill,P s is the player skill,and T s is10Team #30680Page 11of 18the team skill,we hypothesize thatT s =C s ·P s ,(4)as C s of any particular team could be thought of as a multiplier on the player skill P s ,which results in team skill T s .Although the relationship between these factors may be more complex in real life,this relationship gives us reasonable results and works well with our model.6.1When is Coach Skill Important?We will now make a key assumption regarding player skill and coach skill.In order to separate the eﬀects of these two factors on the overall team skill,we must deﬁne some diﬀerence in eﬀect between the two.That is,the player skill will inﬂuence the team skill in some fundamentally diﬀerent way from the coach skill.Think again to a game played between two arbitrary teams A and B .There are two main cases to be considered:Case one:Player skills diﬀer signiﬁcantly:Without loss of generality,assume that P (A )>>P (B ),where P (x )is a function returning the player skills of any given team x .It is clear that A winning the game is a likely outcome.We can draw a plot approximating the probability of winning by a certain margin,which is shown in Figure 3.Margin of WinProbabilityFigure 3:A has a high chance of winning when its players are more skilled.Because the player skills are very imbalanced,the coach skill will likely not change the outcome of the game.Even if B has an excellent coach,the eﬀect of the coach’s skill will not be enough to make B ’s win likely.Case two:Player skills approximately equal:If the player skills of the two teams are approximately evenly matched,the coach skill has a much higher likelihood of impacting the outcome of the game.When the player skills are11Team #30680Page 12of 18similar for both teams,the Gaussian curve looks like the one shown in Figure 4.In this situation,the coach has a much greater inﬂuene on the outcome of the game -crucial calls of time-outs,player substitutions,and strategies can make or break an otherwise evenly matched game.Therefore,if the coach skills are unequal,causing the Gaussian curve is shifted even slightly,one team will have a higher chance of winning (even if the margin of win will likely be small).Margin of WinProbabilityFigure 4:Neither A nor B are more likely to win when player skills are the same (if player skill is the only factor considered).With the assumptions regarding the eﬀect of coach skill given a diﬀerence in player skills,we can say that the eﬀect of a coach can be expressed as:(C A −C B )· 11+α|P A −P B |(5)Where C A is the coach skill of team A ,C B is the coach skill of team B ,P A is the player skill of team A ,P B is the player skill of team B ,and αis some scalar constant.With this expression,the coach eﬀect is diminished if the diﬀerence in player skills is large,and coach eﬀect is fully present when players have equal skill.6.2Margin of Win ProbabilityNow we wish to use the coach eﬀect expression to create a function giving the probability that team A will beat team B by a margin of x points.A negative value of x means that team B beat team A .The probability that A beats B by x points is:K ·e −1E (C ·player eﬀect +D ·coach eﬀect −margin ) 2(6)where C,D,E are constant weights,player eﬀect is P A −P B ,coach eﬀect is given by Equation 5,and margin is x .12Team#30680Page13of18This probability is maximized whenC·player eﬀect+D·coach eﬀect=margin.This accurately models our situation,as it is more likely that team A wins by a margin equal to their combined coach and team eﬀects over team B.Since team skill is comprised of player skill and coach skill,we may calculate a given team’s player skill using their team skill and coach skill.Thus,the probability that team A beats team B by margin x can be determined solely using the coach skills of the respective teams and their eigenvector centrality measures.6.3Optimizing the Probability FunctionWe want to assign all the coaches various skill levels to maximize the likelihood that the given historical game data occurred.To do this,we maximize the probability function described in Equation6over all games from historical data byﬁnding an optimal value for the coach skill vectors C A and C B.Formally, the probability that the historical data occurred in a given year isall games K·e−1E(C·player eﬀect+D·coach eﬀect−margin)2.(7)After some algebra,we notice that maximizing this value is equivalent to minimizing the value of the cost function J,whereJ(C s)=all games(C·player eﬀect+D·coach eﬀect−margin)2(8)Because P(A beats B by x)is a nonlinear function of four variables for each edge in our network,and because we must iterate over all edges,calculus and linear algebra techniques are not applicable.We will investigate three techniques (Genetic Algorithm,Nelder-Mead Search,and Powell Search)toﬁnd the global maximum of our probability function.6.3.1Genetic AlgorithmAtﬁrst,our team set out to implement a Genetic Algorithm to create the coach skill and player skill vectors that would maximize the probability of the win/loss margins occurring.We created a program that would initialize1000random coach skill and player skill vectors.The probability function was calculated for each pair of vectors,and then the steps of the Genetic Algorithm were ran (carry over the“mostﬁt”solution to the next generation,cross random elements of the coach skill vectors with each other,and mutate a certain percentage of the data randomly).However,our genetic algorithm took a very long time to converge and did not produce the optimal values.Therefore,we decided to forgo optimization with genetic algorithm methods.13Team#30680Page14of186.3.2Nelder-Mead MethodWe wanted to attempt optimization with a technique that would iterate over the function instead of mutating and crossing over.The Nelder-Mead method starts with a randomly initialized coach skills vector C s and uses a simplex to tweak the values of C s to improve the value of a function for the next iteration[7]. However,running Nelder-Mead found local extrema which barely increased the probability of the historical data occurring,so we excluded it from this report.6.3.3Powell’s MethodA more eﬃcient method ofﬁnding minima is Powell’s Method.This algorithm works by initializing a random coach skills vector C s,and uses bi-directional search methods along several search vectors toﬁnd the optimal coach skills.A detailed explanation of the mathematical basis for Powell’s method can be found in Powell’s paper on the algorithm[8].We found that Powell’s method was several times faster than the Nelder-Mead Method and produced reasonable results for the minimization of our probability function.Therefore,our team decided to use Powell’s method as the main algorithm to determine the coach skills vector.We implemented this algorithm in Python and ran it across every edge in our network for each year that we had data.It signiﬁcantly lowered our cost function J over several thousand iterations.Rank1962200020051John Wooden Lute Olson Jim Boeheim2Forrest Twogood John Wooden Roy Williams3LaDell Anderson Jerry Dunn Thad Matta The table above shows the results of running Powell’s method until the probability function shown in Equation6is optimized,for three widely separated arbitrary years.We have chosen to show the top three coaches per year for the purposes of conciseness.We will additionally highlight the performance of our top three three outstanding coaches.John Wooden-UCLA:John Wooden built one of the’greatest dynasties in all of sports at UCLA’,winning10NCAA Division I Basketball tournaments and leading an unmatched streak of seven tournaments in a row from1967to 1973[1].He won88straight games during one stretchJim Boeheim-Syracuse:Boeheim has led Syracuse to the NCAA Tour-nament28of the37years that he has been coaching the team[3].He is second only to Mike Krzyzewsky of Duke in total wins.He consistently performs even when his players vary-he is the only head coach in NCAA history to lead a school to fourﬁnal four appearances in four separate decades.Roy Williams-North Carolina:Williams is currently the head of the basketball program at North Carolina where he is sixth all-time in the NCAA for winning percentage[5].He performs impressively no matter who his players are-he is one of two coaches in history to have led two diﬀerent teams to the Final Four at least three times each.14Team#30680Page15of187Ranking CoachesKnowing that we are only concerned withﬁnding the topﬁve coaches per sport, we decided to only consider theﬁve highest-ranked coaches for each year.To calculate the overall ranking of a coach over all possible years,we considered the number of years coached and the frequency which the coach appeared in the yearly topﬁve list.That is:C v=N aN c(9)Where C v is the overall value assigned to a certain coach,N a is the number of times a coach appears in yearly topﬁve coach lists,and N c is the number of years that the coach has been active.This method of measuring overall coach skill is especially strong because we can account for instances where coaches change teams.7.1Top Coaches of the Last100YearsAfter optimizing the coach skill vectors for each year,taking the topﬁve,and ranking the coaches based on the number of times they appeared in the topﬁve list,we arrived at the following table.This is our deﬁnitive ranking of the top ﬁve coaches for the last100years,and their associated career-history ranking: Rank Mens Basketball Mens Football Mens Baseball 1John Wooden-0.28Glenn Warner-0.24Mark Marquess-0.27 2Lute Olson-0.26Bobby Bowden-0.23Augie Garrido-0.24 3Jim Boeheim-0.24Jim Grobe-0.18Tom Chandler-0.22 4Gregg Marshall-.23Bob Stoops-0.17Richard Jones-0.19 5Jamie Dixon-.21Bill Peterson-0.16Bill Walkenbach-0.168Testing our Model8.1Sensitivity AnalysisA requirement of any good model is that it must be tolerant to a small amount of error in its inputs.In our model,possible sources of error could include im-properly recorded game results,incorrectﬁnal scores,or entirely missing games. These sources of error could cause a badly written algorithm to return incorrect results.To test the sensitivity of our model to these sources of error,we decided to create intentional small sources of error in the data and compare the results to the original,unmodiﬁed results.Theﬁrst intentional source of error that we incorporated into our model was the deletion of a game,speciﬁcally a regular-season win for Alabama(the team with the top-ranked coach in1975)over Providence with a score of67to 60.We expected that the skill value of the coach of the Alabama team would15。

2014美赛A题翻译

A 题：除了超车以外都要靠右行驶的规则（靠右行驶规则）在某些国家，开车时规定要靠右行驶（如美国，中国以及除英国，澳大利亚和前英国殖民地的大多数国家）多车道高速公路往往会制定一个规则，要求司机在驾驶时靠最右车道行驶，除非在超车的时候可以行驶到左一个车道超车，超车结束后回归到原行驶车道。

建立一个数学模型，分析靠右行驶规则在交通畅通时和在交通拥挤的
情况下的性能表现。

你可以在交通流和安全之间，在限速中扮演的角色之间（也就是说限速过低或者过高）,和/或其他可能不会在这个问题中明确提出的因素之间权衡。

这是有效地改善交通流通性的规则吗？如果不是,提出并分析不同的方案选择(包括可能根本不存在的规则),这些方案可能改善交通流,安全性,和/或其他你认为重要的因素。

在靠左边驾驶汽车的国家中,讨论您的解决方案是否可以用一个简单的方向改变就可以转换,或者需要额外的要求。

最后,如上所述的规则依赖人类合规性判断。

如果相同道路的车辆运输是完全的控制在一个智能系统下,这个智能系统或者是道路网络的一部分或者是嵌入在道路中所有车辆的设计中-这在多大程度上改变你之前分析的结果?数学中国（w w w .m a d i o .n e t ）。

【VIP专享】2010 -2014MCM Problems建模竞赛美赛题目中英文双语翻译版

2010 MCM ProblemsPROBLEM A: The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?MCM 2010 A题：解释棒球棒上的“最佳击球点”每一个棒球手都知道在棒球棒比较粗的部分有一个击球点，这里可以把打击球的力量最大程度地转移到球上。

2014年美赛B题翻译

一本为体育爱好者的杂志《体育画报》正在为上世纪寻找“最佳全职大学教练”。

建立一个数学模型，从大学曲棍球和陆上曲棍球,足球,棒球或垒球、篮球、足球的男教练或者女教练中，选择现在或者过去的最佳大学教练。

当你在分析中采用不同的时间点时，会有不同吗？例如：1913年的教练是否与2013年的不同？
清晰的表达你的评价指标。

讨论你的模型怎样用于男、女教练和所有可能的运动。

介绍一下你模型中，在3种不同运动中的前五名的教练。

除了MCM的格式和要求，为《体育画报》准备一份1-2页的文章来解释你的结果，这篇文章需要包括你的数学模型的非技术性解释，从而使体育迷们理解。

2014美赛ICM赛题翻译

2014 ICM Problem使用网络来测量的影响和冲击Using Networks to Measure Influence and Impact确定学术研究的影响力的一种方法是构建和测量引文或合著者网络的属性.共同创作的文章通常意味着研究者之间的影响力有了重要的连接。

其中最有名的学术合著者是20世纪的数学家保罗·埃尔德什（Paul Erdös）他拥有超过500共同作者，并发表了超过1400的技术研究论文。

说埃尔德什（Erdös）是具有学科交叉特点的网络科学（science of networks）这一新兴研究的奠基人之一或许具有讽刺意味，或者也不。

特别是1959年，他和阿尔弗雷德的共同撰写的论文“关于随机图”（“On Random Graphs”）的发表，使得埃尔德什的作为合作者的角色在数学领域变得十分重要以至于数学家们经常会通过分析埃尔德什数量惊人的合作者来衡量自己与埃尔德什联系的紧密程度（their closeness to Erdös）。

(see the website http://www.oakla /enp/ )保罗.埃尔德什作为一个天才的数学家，天才的问题解决者，和著名合作者的不寻常和令人着迷的故事公布也在了许多书籍和在线网站上。

(例如,/Biographies/Erdos.html ).也许，他的生活方式就是经常和他的合作者待在一起或者住在一起。

或者把钱给他的学生作为解决问题的奖励，从而使他的合作蓬勃发展，并帮助他在数学的几个领域里建立了具有惊人影响力的网络。

为了衡量诸如埃尔德什等人产生的影响力，已经有了一些基于网络评价的工具（network-based evaluation tools）。

即是利用共同作者和引文数据来确定学者，出版物和学术期刊的影响因子，比如：科学文献索引（Science Citation Index，SCI的），H- factor（一种评价学术成就的新方法）Impact factor （期刊影响因子，SCI），Eigenfactor等等。

2014年美赛A题

2014年美国数学建模比赛试题、翻译和分析(终稿)

ROBLEM A: The Keep-Right-Except-To-Pass Rule问题A：除非超车否则靠右行驶的交通规则In countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.在一些汽车靠右行驶的国家（比如美国，中国等等），多车道的高速公路常常遵循以下原则：司机必须在最右侧驾驶，除非他们正在超车，超车时必须先移到左侧车道在超车后再返回。

Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this rule effective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.建立数学模型来分析这条规则在低负荷和高负荷状态下的交通路况的表现。

(完整版)2014大学生数学建模美赛B题数据全

1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956
29 BAA
WSC
30 BAA
WSC
31 BAA
WSC
32 NBA
三
33 NBA
BOS
34 NBA
BOS
35 NBA
BOS
36 NBA
BOS
37 NBA
BOS
38 NBA
BOS
39 NBA
BOS
34 0.585
56
21
35 0.375
82
39
43 0.476
82
26
56 0.317
16
8
8
0.5
70
12
58 0.171
Stan Albeck Stan Albeck Stan Albeck Stan Albeck Bob Bass
1979 1980 1981 1982 1983
K.C. Jones
41 NBA
CAP
46 ABA
SAA
40 NBA
NOJ
42 NBA
WSB
47 ABA
SAA
43 NBA
WSB
42 NBA
NOJ
43 NBA
NOJ
44 NBA
NOJ
51 NBA
SAS
42 NBA
DET
79
5722 0.722806020
0.75
80
58
22 0.725
80
59
21 0.738
80
62
18 0.775
Coach Season

2014年数学建模美赛题目原文及翻译

2014年数学建模美赛题目原文及翻译作者：Ternence Zhang转载注明出处：MCM原题PDF：PROBLEM A: The Keep-Right-Except-To-Pass RuleIn countries where driving automobiles on the right is the rule (that is, USA, China and most other countries except for Great Britain, Australia, and some former British colonies), multi-lane freeways often employ a rule that requires drivers to drive in the right-most lane unless they are passing another vehicle, in which case they move one lane to the left, pass, and return to their former travel lane.Build and analyze a mathematical model to analyze the performance of this rule in light and heavy traffic. You may wish to examine tradeoffs between traffic flow and safety, the role of under- or over-posted speed limits (that is, speed limits that are too low or too high), and/or other factors that may not be explicitly called out in this problem statement. Is this ruleeffective in promoting better traffic flow? If not, suggest and analyze alternatives (to include possibly no rule of this kind at all) that might promote greater traffic flow, safety, and/or other factors that you deem important.In countries where driving automobiles on the left is the norm, argue whether or not your solution can be carried over with a simple change of orientation, or would additional requirements be needed.Lastly, the rule as stated above relies upon human judgment for compliance. If vehicle transportation on the same roadway was fully under the control of an intelligent system –either part of the road network or imbedded in the design of all vehicles using the roadway –to what extent would this change the results of your earlier analysis?问题A：车辆右行在一些规定汽车靠右行驶的国家（即美国，中国和其他大多数国家，除了英国，澳大利亚和一些前英国殖民地），多车道的高速公路经常使用这样一条规则：要求司机开车时在最右侧车道行驶，除了在超车的情况下，他们应移动到左侧相邻的车道，超车，然后恢复到原来的行驶车道（即最右车道）。

2014年美国数学建模比赛B题(最佳教练)

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Summary
2014 Mathematical Contest in Modeling (MCM) Summary Sheet
This paper mainly sets a model of selecting five “best all time college coach” in basketball, football and hockey respectively. The model is separated to six sub-problems and its solution in our paper:1)Selecting the best coach in one NCAA basketball season; 2)Screening the college coach at top-20 to top-30 in the past 100 years; 3)Further analysis to these selected coaches, and ranking the top-5; 4)Applying this model to football and hockey; 5)Gender impact analysis and time impact analysis; 6)Analyzing the reasons that some famous coach selected by some magazines and media are not on our ranking list. For sub-problem1:Identify four indicators, and confirm the weight of each indicator by Analytic Hierarchy Process, then transformer the indicator data into scores and add these scores multiplied by the weight of itself to get the final scores. The best coach in one season is the one with highest score. For sub-problem2: Use the method above to score for all of the coaches in the past 100 years , then select the top-20.For purpose of reducing the influence of subjective factors, we use Principal Component Analysis to get another ranking list, also, we select the top-20.And union these two top-20. For sub-problem3:Surveying these coaches in the union to learn how many NBA players they've respectively brought up. And regarding this as an indicator, then dividing all indicators into Experience,Leading ability and Ability of player cultivation three aspects, and using Multilevel Hierarchical Analysis to weight them. Next, score for each person by the Fuzzy Criterion of Composite and rank according to the scores. With the purpose of reducing the influence of subjective factors, using Grey Correlation Analysis to calculate the grey correlation degree between “perfect coach” and these coaches. After that, using t-test to judge whether the two results with significant differences. The conclusion obtained: There are no significant difference. Namely, this method is generality. For sub-problem4:Fine-tuning part of indicators and weights, then apply this model in football and hockey. For sub-problem5:As One-way Analysis of Variance used, and the conclusion obtained: gender doesn’t affect the overall score or the ranking of coaches, but female coach has unique advantages in women's basketball league. And the time has no significant effect on the comprehensive score ,but the most excellent coach were in the 70s. Combined with the history of NCAA development, we think that it is necessary to improve the weight of the indicator---champions that gotten recently, making this model better. .For sub-problem6: Comparing the data of their career with our standards to find the reasons.