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1985~2014 年美国大学生数学建模竞赛题目集锦 目录 1985 MCM A: Animal Populations ........................................................................ 3 1985 MCM B: Strategic Reserve Management .................................................. 3 1986 MCM A: Hydrographic Data .......................................................................... 4 1986 MCM B: Emergency-Facilities Location .................................................... 4 1987 MCM A: The Salt Storage Problem ............................................................. 5 1987 MCM B: Parking Lot Design ......................................................................... 5 1988 MCM A: The Drug Runner Problem ............................................................ 5 1988 MCM B: Packing Railroad Flatcars ............................................................. 6 1989 MCM A: The Midge Classification Problem.............................................. 6 1989 MCM B: Aircraft Queueing ............................................................................ 6 1990 MCM A: The Brain-Drug Problem ............................................................... 7 1990 MCM B: Snowplow Routing.......................................................................... 8 1991 MCM A: Water Tank Flow .............................................................................. 8 1991 MCM B: The Steiner Tree Problem ............................................................. 8 1992 MCM A: Air-Traffic-Control Radar Power ................................................. 9 1992 MCM B: Emergency Power Restoration.................................................... 9 1993 MCM A: Optimal Composting .................................................................... 11 1993 MCM B: Coal-Tipple Operations................................................................ 12 1994 MCM A: Concrete Slab Floors ................................................................... 12 1994 MCM B: Network Design ............................................................................. 13 1995 MCM A: Helix Construction ........................................................................ 14 1995 MCM B: Faculty Compensation ................................................................. 14 1996 MCM A: Submarine Tracking ..................................................................... 14 1996 MCM B: Paper Judging ................................................................................ 14 1997 MCM A: The Velociraptor Problem ........................................................... 15 1997 MCM B: Mix Well for Fruitful Discussions.............................................. 16 1998 MCM A: MRI Scanners ................................................................................. 17 1998 MCM B: Grade Inflation ............................................................................... 18 1999 MCM A: Deep Impact .................................................................................... 19 1999 MCM B: Unlawful Assembly ....................................................................... 20 2000 MCM A: Air Traffic Control ......................................................................... 20 2000 MCM B: Radio Channel Assignments ..................................................... 21 2001 MCM A: Choosing a Bicycle Wheel .......................................................... 22 2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...) ....................... 23 2002 MCM A: Wind and Waterspray ................................................................... 25 2002 MCM B: Airline Overbooking ..................................................................... 26 2003 MCM A: The Stunt Person .......................................................................... 26 2003 MCM B: Gamma Knife Treatment Planning ........................................... 27 2004 MCM A: Are Fingerprints Unique? ............................................................ 28 2004 MCM B: A Faster QuickPass System ....................................................... 28 2005 MCM A: Flood Planning ............................................................................... 29 2005 MCM B: Tollbooths ....................................................................................... 29

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2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation .. 29 2006 MCM B: Wheel Chair Access at Airports ................................................ 30 2007 MCM A: Gerrymandering............................................................................. 31 2007 MCM B: The Airplane Seating Problem ................................................... 32 2008 MCM A: Take a Bath...................................................................................... 32 2008 MCM B: Creating Sudoku Puzzles ............................................................ 33 2009 MCM A: Designing a Traffic Circle ............................................................ 33 2009 MCM B: Energy and the Cell Phone ......................................................... 33 2010 MCM A: The Sweet Spot .............................................................................. 35 2010 MCM B: Criminology .................................................................................... 35 2011 MCM A: Snowboard Course ....................................................................... 36 2011 MCM B: Repeater Coordination ................................................................. 36 2012 MCM A: The Leaves of a Tree .................................................................... 37 2012 MCM B: Camping along the Big Long River .......................................... 37 2013 MCM A: The Ultimate Brownie Pan .......................................................... 38 2013 MCM B: Water, Water, Everywhere ........................................................... 38 2014 MCM A:The Keep-Right-Except-To-Pass Rule ...................................... 39 2014 MCM B:College Coaching Legends ......................................................... 39

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1985 MCM A: Animal Populations Choose a fish or mammal for which appropriate data are available to model it accurately. Model the animal's natural interactions with its environment by expressing population levels of different groups in terms of the significant parameters of the environment. Then adjust the model to account for harvesting in a form consistent with the actual method by which the animal is harvested. Include any outside constraints imposed by food or space limitations that are supported by the data. Consider the value of the various quantities involved, the number harvested, and the population size itself, in order to devise a numerical quantity that represents the overall value of the harvest. Find a harvesting policy in terms of population size and time that optimizes the value of the harvest over a long period of time. Check that the policy optimizes that value over a realistic range of environmental conditions. 1985 MCM B: Strategic Reserve Management Cobalt, which is not produced in the US, is essential to a number of industries. (Defense accounted for 17% of the cobalt production in 1979.) Most cobalt comes from central Africa, a politically unstable region. The Strategic and Critical Materials Stockpiling Act of 1946 requires a cobalt reserve that will carry the US through a three-year war. The government built up a stockpile in the 1950s, sold most of it off in the early 1970s, and then decided to build it up again in the late 1970s, with a stockpile goal of 85.4 million pounds. About half of this stockpile had been acquired by 1982. Build a mathematical model for managing a stockpile of the strategic metal cobalt. You will need to consider such questions as:

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How big should the stockpile be? ? At what rate should it be acquired? ? What is a reasonable price to pay for the metal? You will also want to consider such questions as:

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At what point should the stockpile be drawn down? ? At what rate should it be drawn down? ? At what price is it reasonable to sell the metal? ? How should it be allocated? Useful Information on Cobalt The government has projected a need ot 25 million pounds of cobalt in 1985.

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The U.S. has about 100 million pounds of proven cobalt deposits. Production becomes economically feasible when the price reaches $22/lb (as occurred in 1981). It takes four years to get operations rolling, and thsn six million pounds per year can be produced. In 1980, 1.2 million pounds of cobalt were recycled, 7% of total consumption. 1986 MCM A: Hydrographic Data The table below gives the depth Z of water in feet for surface points with rectangular coordinates X, Y in yards [table of 14 data points omitted]. The depth measurements were taken at low tide. Your ship has a draft of five feet. What region should you avoid within the rectangle (75,200) x (-50, 150)? X 129.0 140.0 108.5 88.0 185.5 195.0 105.5 157.5 107.5 77.0 162.0 162.0 117.5 Y 7.5 141.5 28.0 147.0 22.5 137.5 85.5 -6.5 -81.0 3.0 -66.5 84.0 -35.5 1986 MCM B: Emergency-Facilities Location The township of Rio Rancho has hitherto not had its own emergency facilities. It has secured funds to erect two emergency facilities in 1986, each of which will combine ambulance, fire, and police services. Figure 1 indicates the demand [figure omitted], or number of emergencies per square block, for 1985. The ―L‖ region in the north is an obstacle, while the rectangle in the south is a part with shallow pond. It takes an emergency vehicle an average of 15 seconds to go one block in the N-S direction and 20 seconds in the E-W Z 4 8 6 8 6 8 8 9 9 8 9 4 9

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direction. Your task is to locate the two facilities so as to minimize the total response time.

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Assume that the demand is concentrated at the center of the block and that the facilities will be located on corners. Assume that the demand is uniformly distributed on the streets bordering each block and that the facilities may be located anywhere on the streets. 1987 MCM A: The Salt Storage Problem

For approximately 15 years, a Midwestern state has stored salt used on roads in the winter in circular domes. Figure 1 shows how salt has been stored in the past. The salt is brought into and removed from the domes by driving front-end loaders up ramps of salt leading into the domes. The salt is piled 25 to 30 ft high, using the buckets on the front-end loaders. Recently, a panel determined that this practice is unsafe. If the front-end loader gets too close to the edge of the salt pile, the salt might shift, and the loader could be thrown against the retaining walls that reinforce the dome. The panel recommended that if the salt is to be piled with the use of the loaders, then the piles should be restricted to a matimum height of 15 ft. Construct a mathematical model for this situation and find a recommended maximum height for salt in the domes. 1987 MCM B: Parking Lot Design The owner of a paved, 100' by 200' , corner parking lot in a New England town hires you to design the layout, that is, to design how the ``lines are to be painted. You realize that squeezing as many cars into the lot as possible leads to right-angle parking with the cars aligned side by side. However, inexperienced drivers have difficulty parking their cars this way, which can give rise to expensive insurance claims. To reduce the likelihood of damage to parked vehicles, the owner might then have to hire expert drivers for ``valet parking. On the other hand, most drivers seem to have little difficulty in parking in one attempt if there is a large enough ``turning radius'' from the access lane. Of course, the wider the access lane, the fewer cars can be accommodated in the lot, leading to less revenue for the parking lot owner. 1988 MCM A: The Drug Runner Problem Two listening posts 5.43 miles apart pick up a brief radio signal. The sensing devices were oriented at 110 degrees and 119 degrees, respectively, when the signal was detected; and they are accurate to within 2 degrees. The signal

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came from a region of active drug exchange, and it is inferred that there is a powerboat waiting for someone to pick up drugs. it is dusk, the weather is calm, and there are no currents. A small helicopter leaves from Post 1 and is able to fly accurately along the 110 degree angle direction. The helicopter's speed is three times the speed of the boat. The helicopter will be heard when it gets within 500 ft of the boat. This helicopter has only one detection device, a searchlight. At 200 ft, it can just illuminate a circular region with a radius of 25 ft.

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Develop an optimal search method for the helicopter. Use a 95% confidence level in your calculations. 1988 MCM B: Packing Railroad Flatcars

Two railroad flatcars are to be loaded with seven types of packing crates. The crates have the same width and height but varying thickness (t, in cm) and weight (w, in kg). Table 1 gives, for each crate, the thickness, weight, and number available [table omitted]. Each car has 10.2 meters of length available for packing the crates (like slices of toast) and can carry up to 40 metric tons. There is a special constraint on the total number of C_5, C_6, and C_7 crates because of a subsequent local trucking restriction: The total space (thickness) occupied by these crates must not exceed 302.7 cm. Load the two flatcars (see Figure 1) so as to minimize the wasted floor space [figure omitted]. 1989 MCM A: The Midge Classification Problem Two species of midges, Af and Apf, have been identified by biologists Grogan and Wirth on the basis of antenna and wing length (see Figure 1). It is important to be able to classify a specimen as Af of Apf, given the antenna and wing length. 1. Given a midge that you know is species Af or Apf, how would you go about classifying it? 2. Apply your method to three specimens with (antenna, wing) lengths (1.24,1.80),(1.28,1.84),(1.40,2.04). 3. Assume that the species is a valuable pollinator and species Apf is a carrier of a debilitating disease. Would you modify your classification scheme and if so, how? 1989 MCM B: Aircraft Queueing A common procedure at airports is to assign aircraft (A/C) to runways on a first-come-first-served basis. That is, as soon as an A/C is ready to leave the gate (―push-back‖), the pilot calls ground control and is added to the queue. Suppose that a control tower has access to a fast online database with the following information for each A/C:

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the time it is scheduled for pushback; the time it actually pushes back; the number of passengers who are scheduled to make a connection at the next stop, as well as the time to make that connection; and the schedule time of arrival at its next stop Assume that there are seven types of A/C with passenger capacities varying from 100 to 400 in steps of 50. Develop and analyze a mathematical model that takes into account both the travelers' and airlines' satisfaction. 1990 MCM A: The Brain-Drug Problem

Researches on brain disorders test the effects of the new medical drugs – for example, dopamine against Parkinson's disease – with intracerebral injections. To this end, they must estimate the size and the sape of the spatial distribution of the drug after the injection, in order to estimate accurately the region of the brain that the drug has affected. The research data consist of the measurements of the amounts of drug in each of 50 cylindrical tissue samples (see Figure 1 and Table 1). Each cylinder has length 0.76 mm and diameter 0.66 mm. The centers of the parallel cylinders lie on a grid with mesh 1mm X 0.76mm X 1mm, so that the sylinders touch one another on their circular bases but not along their sides, as shown in the accompanying figure. The injection was made near the center of the cylinder with the highest scintillation count. Naturally, one expects that there is a drug also between the cylinders and outside the region covered by the samples. Estimate the distribution in the region affected by the drug. One unit represents a scintillation count, or 4.753e-13 mole of dopamine. For example, the table shows that the middle rear sylinder contails 28353 units. Table 1. Amounts of drug in each of 50 cylindrical tissue samples. Rear vertical section 164 480 2091 789 213 Front vertical section 442 7022 23027 21260 1303 1320 14411 28353 20921 3765 414 5158 13138 11731 1715 188 352 681 727 453

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163 712 2137 444 294

324 4055 15531 11431 2061

432 6098 19742 14960 1036

243 1048 4785 3182 258

166 232 335 301 188

1990 MCM B: Snowplow Routing The solid lines of the map (see Figure 1) represent paved two-lane county roads in a snow removal district in Wicomico County, Maryland [figure omitted]. The broken lines are state highways. After a snowfall, two plow-trucks are dispatched from a garage that is about 4 miles west of each of the two points (*) marked on the map. Find an efficient way to use the two trucks to sweep snow from the county roads. The trucks may use the state highways to access the county roads. Assume that the trucks neither break down nor get stuck and that the road intersections require no special plowing techniques. 1991 MCM A: Water Tank Flow Some state water-right agencies require from communities data on the rate of water use, in gallons per hour, and the total amount of water used each day. Many communities do not have equipment to measure the flow of water in or out of the municipal tank. Instead, they can measure only the level of water in the tank, within 0.5% accuracy, every hour. More importantly, whenever the level in the tank drops below some minimum level L, a pump fills the tank up to the maximum level, H; however, there is no measurement of the pump flow either. Thus, one cannot readily relate the level in the tank to the amount of water used while the pump is working, which occurs once or twice per day, for a couple of hours each time. Estimate the flow out of the tank f(t) at all times, even when the pump is working, and estimate the total amount of water used during the day. Table 1 gives real data, from an actual small town, for one day[ table omitted]. The table gives the time, in, since the first measurement, and the level of water in the tank, in hundredths of a foot. For example, after 3316 seconds, the depth of water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and a diameter of 57 feet. Usually, the pump starts filling the tank when the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet. 1991 MCM B: The Steiner Tree Problem The cost for a communication line between two stations is proportional to the length of the line. The cost for conventional minimal spanning trees of a set of stations can often be cut by introducing ―phantom‖ stations and then constructing a new Steiner tree. This device allows costs to be cut by up to

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13.4% (= 1- sqrt(3/4)). Moreover, a network with n stations never requires more than n-2 points to construct the cheapest Steiner tree. Two simple cases are shown in Figure 1. For local networks, it often is necessary to use rectilinear or ―checker-board‖ distances, instead of straight Euclidean lines. Distances in this metric are computed as shown in Figure 2. Suppose you wish to design a minimum costs spanning tree for a local network with 9 stations. Their rectangular coordinates are: a(0,15), b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0) i(10,3). You are restricted to using rectilinear lines. Moreover, all ―phantom‖ stations must be located at lattice points (i.e., the coordinates must be integers). The cost for each line is its length. 1. 2. Find a minimal cost tree for the network. Suppose each stations has a cost w*d^(3/2), where d=degree of the station. If w=1.2, find a minimal cost tree. 3. Try to generalize this problem 1992 MCM A: Air-Traffic-Control Radar Power You are to determine the power to be radiated by an air-traffic-control radar at a major metropolitan airport. The airport authority wants to minimize the power of the radar consistent with safety and cost. The authority is constrained to operate with its existing antennae and receiver circuitry. The only option that they are considering is upgrading the transmitter circuits to make the radar more powerful. The question that you are to answer is what power (in watts) must be released by the radar to ensure detection of standard passenger aircraft at a distance of 100 kilometers. 1992 MCM B: Emergency Power Restoration Power companies serving coastal regions must have emergency response systems for power outages due to storms. Such systems require the input of data that allow the time and cost required for restoration to be estimated and the ―value‖ of the outage judged by objective criteria. In the past, Hypothetical Electric Company (HECO) has been criticized in the media for its lack of a prioritization scheme. You are a consultant to HECO power company. HECO possesses a computerized database with real time access to service calls that currently require the following information:

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time of report, type of requestor,

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estimated number of people affected, and location (x,y). Cre sites are located at coordinates (0,0) and (40,40), where x and y are in miles. The region serviced by HECO is within -65 < x < 60 and -50 < y < 50. The region is largely metropolitan with an excellent road network. Crews must return to their dispatch site only at the beginning and end of shift. Company policy requires that no work be initiated until the storm leaves the area, unless the facility is a commuter railroad or hospital, which may be processed immediately if crews are available. HECO has hired you to develop the objective criteria and schedule the work for the storm restoration requirements listed in Table 1 using their work force described in Table 2. Note that the first call was received at 4:20 A.M. and that the storm left the area at 6:00 A.M. Also note that many outages were not reported until much later in the day. HECO has asked for a technical report for their purposes and an ―executive summary‖ in laymen's terms that can be presented to the media. Further, they would like recommendations for the future. To determine your prioritized scheduling system, you will have to make additional assumptions. Detail those assumptions. In the future, you may desire additional data. If so, detail the information desired. Table 1. Storm restoration requirements. (table incomplete)

Time Location 4:20 5:30 5:35 5:55 6:00 6:05 (13,30) (-10,3) (3,3) (20,5) (-10,5)

Type Business (cable TV) Residential Business (hospital) Business (railroad sys.) Storm leaves area Residential

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# Affected ? 20 240 25 wrkrs, 75,000 commuters

Est Repair Time hrs 6 7 8 5

45

2

6:06

(5,20)

Area*

2000

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Table 2. Crew descriptions. Dispatch locations at (0,0) and (40,40). Crews consist of three trained workers. Crews report to the dispatch location only at the beginning and end of their shifts. One crew is scheduled for duty at all times on jobs assigned to each dispatch location. These crews would normally be performing routine assignments. Until the “storm leaves the area,” They can be dispatched for “emergencies” only. Crews work 8 hour shifts. There are six crew teams available at each location. Crews can work only one overtime shift in a work day and receive time-and-a-half for overtime. 1993 MCM A: Optimal Composting An environmentally conscious institutional cafeteria is recycling customers' uneaten food into compost by means of microorganisms. Each day, the cafeteria blends the leftover food into a slurry, mixes the slurry with crisp salad wastes from the kitchen and a small amount of shredded newspaper, and feeds the resulting mixture to a culture of fungi and soil bacteria, which digest slurry, greens, and papers into usable compost. The crisp green provide pockets of oxygen for the fungi culture, and the paper absorbs excess humidity. At times, however, the fungi culture is unable or unwilling to digest as much of the leftovers as customers leave; the cafeteria does not blame the chef for the fungi culture's lack of appetite. Also, the cafeteria has received offers for the purchase of large quantities of it compost. Therefore, the cafeteria is investigating ways to increase its production of compost. Since it cannot yet afford to build a new composting facility, the cafeteria seeks methods to accelerate the fungi culture's activity, for instance, by optimizing the fungi culture's environment (currently held at about 120 F and 100% humidity), or by optimizing the composition of the moisture fed to the fungi culture, or both. Determine whether any relation exists between the proportions of slurry, greens, and paper in the mixture fed to the fungi culture, and the rate at which the fungi culture composts the mixture. if no relation exists, state so. otherwise, determine what proportions would accelerate the fungi culture's activity. In addition to the technical report following the format prescribed in the contest instructions, provide a one-page nontechnical recommendation for

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implementation for the cafeteria manager. Table 1 shows the composition of various mixtures in pounds of each ingredient kept in separate bins, and the time that it took the fungi to culture to compost the mixtures, from the date fed to the date completely composted [table omitted]. 1993 MCM B: Coal-Tipple Operations The Aspen-Boulder Coal Company runs a loading facility consisting of a large coal tipple. When the coal trains arrive, they are loaded from the tipple. The standard coal train takes 3 hours to load, and the tipple's capacity is 1.5 standard trainloads of coal. Each day, the railroad sends three standard trains to the loading facility, and they arrive at any time between 5 A.M. and 8 P.M. local time. Each of the trains has three engines. If a train arrives and sits idle while waiting to be loaded, the railroad charges a special fee, called a demurrage. The fee is $5,000 per engine per hour. In addition, a high-capacity train arrives once a week every Thursday between 11 A.M. and 1 P.M. This special train has five engines and holds twice as much coal as a standard train. An empty tipple can be loaded directly from the mine to its capacity in six hours by a single loading crew. This crew (and its associated equipment) cost $9,000 per hour. A second crew can be called out to increase the loading rate by conducting an additional tipple-loading operation at the cost of $12,000 per hour. Because of safety requirements, during tipple loading no trains can be loaded. Whenever train loading is interrupted to load the tipple, demurrage charges are in effect. The management of the Coal Company has asked you to determine the expected annual costs of this tipple's loading operations. Your analysis should include the following considerations:

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How often should the second crew be called out? What are the expected monthly demurrage costs? If the standard trains could be scheduled to arrive at precise times, what daily schedule would minimize loading costs? Would a third tipple-loading crew at $12,000 per hour reduce annual operations costs? Can this tipple support a fourth standard train every day? 1994 MCM A: Concrete Slab Floors

The U.S. Dept. of Housing and Urban Development (HUD) is considering constructing dwellings of various sizes, ranging from individual houses to large apartment complexes. A principal concern is to minimize recurring costs to occupants, especially the costs of heating and cooling. The region in which the construction is to take place is temperate, with a moderate variation in temperature throughout the year.

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Through special construction techniques, HUD engineers can build dwellings that do not need to rely on convection- that is, there is no need to rely on opening doors or windows to assist in temperature variation. The dwellings will be single-story, with concrete slab floors as the only foundation. You have been hired as a consultant to analyze the temperature variation in the concrete slab floor to determine if the temperature averaged over the floor surface can be maintained within a prescribed comfort zone throughout the year. If so, what size/shape of slabs will permit this? Part 1, Floor Temperature: Consider the temperature variation in a concrete slab given that the ambient temperature varies daily within the ranges given Table 1. Assume that the high occurs at noon and the low at midnight. Determine if slabs can be designed to maintain a temperature averaged over the floor surface within the prescribed comfort zone considering radiation only. Initially, assume that the heat transfer into the dwelling is through the exposed perimeter of the slab and that the top and bottom of the slabs are insulated. Comment on the appropriateness and sensitivity of these assumptions. If you cannot find a solution that satisfies Table 1, can you find designs that satisfy a Table 1 that you propose? Ambient High Low Temperature 85 60 Comfort Zone 76 65

Part 2, Building Temperature: Analyze the practicality of the initial assumptions and extend the analysis to temperature variation within the single-story dwelling. Can the house be kept within the comfort zone? Part 3, Cost of Construction: Suggest a design that considers HUD's objective of reducing or eliminating heating and cooling costs, considering construction restrictions and costs. 1994 MCM B: Network Design In your company, information is shared among departments on a daily basis. This information includes the previous day's sales statistics and current production guidance. It is important to get this information out as quickly as possible. [Network diagram (with 5 nodes and 7 capacitated edges) omitted.] We are interested in scheduling transfers in an optimal way to minimize the total time it takes to complete them all. This minimum total time is called the makespan. Consider the three following situations for your company: [Three more network diagrams (on roughly 20 nodes each) omitted.]

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1995 MCM A: Helix Construction A small biotechnological company must design, prove, program and test a mathematical algorithm to locate ―in real time‖ all the intersections of a helix and a plane in general positions in space. Design, justify, program and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space. A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus. Theoretical justification of the proposed algorithm is necessary to verify the solution from several points of view, for instance, through mathematical proofs of parts of the algorithm, and through tests of the final program with known examples. Such documentation and tests will be required by government agencies for medical use. 1995 MCM B: Faculty Compensation Aluacha Balaclava College, and undergraduate facility, has just hired a new Provost whose first priority is the institution of a fair and reasonable faculty-compensation plan. She has hired your consulting team to design a compensation system that reflects the following circumstances and principles: [Three paragraphs of details omitted] Design a new pay system, first without cost-of-living increases. Incorporate cost-of-living increases, and then finally, design a transition process for current faculty that will move all salaries towards your system without reducing anyone's salary. The Provost requires a detailed compensation system plan for implementation, as well as a brief, clear, executive summary outlining the model, its assumptions, strengths, weaknesses and expected results, which she can present to the Board and faculty. [A detailed table of current salaries is omitted.] 1996 MCM A: Submarine Tracking The world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large maving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its speed, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude. 1996 MCM B: Paper Judging When determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the

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judging. Funding for the contest contrains both the number of judges that can be obtained and the amount of time they can judge. For example if P=100, then J=8 is typical. Ideally, each judge would read each paper and rank-order them, but there are too many papers for this. Instead, there will be a number of screening rounds in which each judge will read some papers and give them scores. Then some selection scheme is used to reduce the number of papers under consideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orders could be rejected. Alternatively, if the judges do not rank order the papers, but instead give them numerical scores (say, from 1 to 100), then all the papers falling below some cut-off level could be rejected. The new pool of papers is then passed back to the judges, and the process is repeated. A concern is that the total number of papers that each judge reads must be substantially less than P. The process is stopped when there are only W papers left. These are the winners. Typically for P=100, W=3. Your task is to determine a selection scheme, using a combination of rank-ordering, numerical scoring, and other methods, by which the final W papers will include only papers from among the ―best‖ 2W papers. (By ―best,‖ we assume that there is an absolute rank-ordering to which all judges would agree.) For example, the top three papers found by your method will consist entirely of papers from among the ―best‖ six papers. Among all such methods, the one that requires each judge to read the least number of papers is desired. Note the possibility of systematic bias in a numerical scoring scheme. For example, for a specific collection of papers, one judge could average 70 points, while another could average 80 points. How would you scale your scheme to accommodate for changes in the contest parameters (P, J, and W)? 1997 MCM A: The Velociraptor Problem The velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter, and may have hunted in pairs of larger packs. Unfortunately, there is no way to oversee its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached our team and asked for help in modeling the hunting behavior of the velociraptor. They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.

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The average adult velociraptor was 3 meters long with a hip height of 0.5 meters and an approximate mass of 45 kg. It is estimated that the animal could run extremely fast, at speeds of 60 km/hr., for about 15 seconds. After the initial burst of speed, the animal needed to stop and recover from a buildup of lactic acid in its muscles. Suppose the velociraptor preyed on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that it could run at a speed of about 50 km/hr. for long periods of time. Part 1. Assuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect the velociraptor when it comes within 15 meters, but may detect the predator at even greater ranges (up to 50 meters) depending upon the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 meters. Part 2. Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions given in Part 1. 1997 MCM B: Mix Well for Fruitful Discussions Small group meetings for the discussion of important issues, particularly longrange planning, are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus, in corporate board meetings the board will meet in small groups to discuss issue before meeting as a whole. These smaller groups still run the risk of control by a dominant personality. In an attempt to reduce this danger it is common to schedule several sessions with a different mix of people in each group. A meeting of An Tostal Corporation will be attended by 29 Board Members of which nine are in-house members (i.e., corporate employees). The meeting is to be an all-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hours from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each morning session will consist of six discussion groups with each discussion group led by

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one of the corporation's six senior officers. None of these officers are board members. Thus each senior officer will lead three different discussion groups. The senior officers will not be involved in the afternoon sessions and each of these sessions will consist of only four different discussion groups. The president of the corporation wants a list of board-member assignments to discussion groups for each of the seven sessions. The assignments should achieve as much of a mix of the members as much as possible. The ideal assignment would have each board member with each other board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions. The assignments should also satisfy the following criteria: 1. For the morning sessions, no board member should be in the same senior officer's discussion group twice. 2. No discussion group should contain a disproportionate number of in-house members. Give a list of assignments for members 1-9 and 10-29 and officers 1-6. Indicate how well the criteria in the previous paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an hour's notice would be appreciated. It would be ideal if the algorithm could also be used to make assignments for future meetings involving different levels of participation for each type of attendee. 1998 MCM A: MRI Scanners Introduction Industrial and medical diagnostic machines known as Magnetic Resonance Imagers (MRI) scan a three-dimensional object such as a brain, and deliver their results in the form of a three-dimensional array of pixels. Each pixel consists of one number indicating a color or a shade of gray that encodes a measure of water concentration in a small region of the scanned object at the location of the pixel. For instance, 0 can picture high water concentration in black (ventricles, blood vessels), 128 can picture a low water density in white (lipid-right white matter consisting of myelinated axons). Such MRI scanners also include facilities to pictures on a screen any horizontal or vertical slide through the three-dimensional array (slices are parallel to any of the three Cartesian coordinate axes). Algorithms for picturing slices through oblique planes, however, are proprietary. Current algorithms are limited in terms of the angles and parameter options available; are implemented only on heavily used dedicated workstations; lack

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input capabilities for marking points in the picture before slicing; and tend to blur and ―feather out‖ sharp boundaries between the original pixels. A more faithful, flexible algorithm implemented on a personal computer would be useful 1. 2. 3. for planning minimally invasive treatments, for calibrating the MRI machines, for investigating structures oriented obliquely in space, such as post-mortem tissue sections in animal research, 4. for enabling cross-sections at any angle through a brain atlas consisting of black-and-white line drawings. To design such an algorithm, one can access the values and locations of the pixels, but not the initial data gathered by the scanner. Problem Design and test an algorithm that produces sections of three-dimensional arrays by planes in any orientation in space, preserving the original gray-scale values as closely as possible. Data Sets The typical data set consists of a three-dimensional array A of numbers A(i,j,k) which indicates the density A(i,j,k) of the object at the location (x,y,z)_{ijk}. Typically, A(i,j,k) can range from 0 through 255. In most applications, the data set is quite large. Teams should design data sets to test and demonstrate their algorithms. The data sets should reflect conditions likely to be of diagnostic interest. Teams should also characterize data sets that limit the effectiveness of their algorithms. Summary The algorithm must produce a picture of the slice of the three-dimensional array by a plane in space. The plane can have any orientation and any location in space. (The plane can miss some or all data points). The result of the algorithm should be a model of the density of the scanned object over the selected plane. 1998 MCM B: Grade Inflation Background

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Some college administrators are concerned about the grading at A Better Class (ABC) college. On average, the faculty at ABC have been giving out high grades (the average grade now given out is an A-), and it is impossible to distinguish between the good and mediocre students. The terms of a very generous scholarship only allow the top 10% of the students to be funded, so a class ranking is required. The dean had the thought of comparing each student to the other students in each class, and using this information to build up a ranking. For example, if a student obtains an A in a class in which all students obtain an A, then this student is only ―average‖ in this class. On the other hand, if a student obtains the only A is a class, then that student is clearly ―above average.‖ Combining information from several classes might allow students to be placed in deciles (top 10%, next 10%, etc.) across the college. Problem Assuming that the grades given out are (A+, A, A-, B+,…), can the dean's idea be made to work? Assuming that the grades given out are only (A,B,C,…), can the dean's idea be made to work? Can any other schemes produce a desired ranking? A concern is that the grade in a single class could change many student's deciles. Is this possible? Data Sets Teams should design data sets to test and demonstrate their algorithms. Teams should characterize data sets that limit the effectiveness of their algorithms. 1999 MCM A: Deep Impact For some time, the National Aeronautics and Space Administration (NASA) has been considering the consequences of a large asteroid impact on the earth. As part of this effort, your team has been asked to consider the effects of such an impact were the asteroid to land in Antarctica. There are concerns that an impact there could have considerably different consequences than one striking elsewhere on the planet. You are to assume that an asteroid is on the order of 1000 m in diameter, and that it strikes the Antarctic continent directly at the South Pole.

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Your team has been asked to provide an assessment of the impact of such an asteroid. In particular, NASA would like an estimate of the amount and location of likely human casualties from this impact, an estimate of the damage done to the food production regions in the oceans of the southern hemisphere, and an estimate of possible coastal flooding caused by large-scale melting of the Antarctic polar ice sheet. 1999 MCM B: Unlawful Assembly Many public facilities have signs in rooms used for public gatherings which state that it is ―unlawful‖ for the rooms to be occupied by more than a specified number of people. Presumably, this number is based on the speed with which people in the room could be evacuated from the room's exits in case of an emergency. Similarly, elevators and other facilities often have ―maximum capacities‖ posted. Develop a mathematical model for deciding what number to post on such a sign as being the ―lawful capacity‖. As part of your solution discuss criteria, other than public safety in the case of a fire or other emergency, that might govern the number of people considered ―unlawful‖ to occupy the room (or space). Also, for the model that you construct, consider the differences between a room with movable furniture such as a cafeteria (with tables and chairs), a gymnasium, a public swimming pool, and a lecture hall with a pattern of rows and aisles. You may wish to compare and contrast what might be done for a variety of different envirorunents: elevator, lecture hall, swimming pool, cafeteria, or gymnasium. Gatherings such as rock concerts and soccer tournaments may present special conditions. Apply your model to one or more public facilities at your institution (or neighboring town). Compare your results with the stated capacity, if one is posted. If used, your model is likely to be challenged by parties with interests in increasing the capacity. Write an article for the local newspaper defending your analysis. 2000 MCM A: Air Traffic Control To 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 analysit at the FAA has posed the following problems. Requirement A: Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention?

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Requirement B: And airspace sector is the section of three-dimensional airspace that one air traffic controller controls. Given any airspace sector, how do 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 a 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. 2000 MCM B: Radio Channel Assignments We 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 grix (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 alloted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1,2,3, … . Each transmitter wil be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided.

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Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assugn 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 interference. Requirement A: There are several contrainsts on the frequency assignments. First, no two transmitters within distance 4s 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 contraints, what can we say about the span in Figure 1? 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 2s differ by at least some given integer k, while those at distance at most 4s must still differ by at least one. What cna 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. 2001 MCM 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

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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.

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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. 2001 MCM B: Escaping a Hurricane's 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.

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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? In 1999, the simultaneous evacuation of the state's entire coastal region was ordered. Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation, perhaps county-by-county over some time period consistent with the pattern of how hurricanes affect the coast? Several smaller highways besides I-26 extend inland from the coast. Under what conditions would it improve evacuation flow to turn around traffic on these? What effect would it have on evacuation flow to establish more temporary shelters in Columbia, to reduce the traffic leaving Columbia? In 1999, many families leaving the coast brought along their boats, campers, and motor homes. Many drove all of their cars. Under what conditions should there be restrictions on vehicle types or numbers of vehicles brought in order to guarantee timely evacuation?

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2.

3.

4. 5.

6.

It has been suggested that in 1999 some of the coastal residents of Georgia and Florida, who were fleeing the earlier predicted landfalls of Hurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation traffic flow? Clearly identify what measures of performance are used to compare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public. Clearly identify what measures of performance are used to compare strategies. Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public.

2002 MCM A: Wind and Waterspray An ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.

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Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change. 2002 MCM B: Airline Overbooking You're all packed and ready to go on a trip to visit your best friend in New York City. After you check in at the ticket counter, the airline clerk announces that your flight has been overbooked. Passengers need to check in immediately to determine if they still have a seat. Historically, airlines know that only a certain percentage of passengers who have made reservations on a particular flight will actually take that flight. Consequently, most airlines overbook-that is, they take more reservations than the capacity of the aircraft. Occasionally, more passengers will want to take a flight than the capacity of the plane leading to one or more passengers being bumped and thus unable to take the flight for which they had reservations. Airlines deal with bumped passengers in various ways. Some are given nothing, some are booked on later flights on other airlines, and some are given some kind of cash or airline ticket incentive. Consider the overbooking issue in light of the current situation: Less flights by airlines from point A to point B Heightened security at and around airports Passengers' fear Loss of billions of dollars in revenue by airlines to date Build a mathematical model that examines the effects that different overbooking schemes have on the revenue received by an airline company in order to find an optimal overbooking strategy, i.e., the number of people by which an airline should overbook a particular flight so that the company's revenue is maximized. Insure that your model reflects the issues above, and consider alternatives for handling ―bumped‖ passengers. Additionally, write a short memorandum to the airline's CEO summarizing your findings and analysis. 2003 MCM A: The Stunt Person An exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.). Your job is to:

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? ?

determine what size boxes to use determine how many boxes to use ? determine how the boxes will be stacked ? determine if any modifications to the boxes would help ? generalize to different combined weights (stunt person & motorcycle) and different jump heights Note that, in ―Tomorrow Never Dies‖, the James Bond character on a motorcycle jumps over a helicopter. 2003 MCM B: Gamma Knife Treatment Planning Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators. The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a ―shot.‖ Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points. The goal of radiosurgery is to deplete tumor cells while preserving normal structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements. 1. 2. 3. 4. Minimize the dose gradient across the target volume. Match specified isodose contours to the target volumes. Match specified dose-volume constraints of the target and critical organ. Minimize the integral dose to the entire volume of normal tissues or organs. 5. Constrain dose to specified normal tissue points below tolerance doses. 6. Minimize the maximum dose to critical volumes. In gamma unit treatment planning, we have the following constraints:

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1. 2. 3.

Prohibit shots from protruding outside the target. Prohibit shots from overlapping (to avoid hot spots). Cover the target volume with effective dosage as much as possible. But at least 90% of the target volume must be covered by shots. 4. Use as few shots as possible. Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient. 2004 MCM A: Are Fingerprints Unique? It is a commonplace belief that the thumbprint of every human who has ever lived is different. Develop and analyze a model that will allow you to assess the probability that this is true. Compare the odds (that you found in this problem) of misidentification by fingerprint evidence against the odds of misidentification by DNA evidence. 2004 MCM B: A Faster QuickPass System ―QuickPass‖ systems are increasingly appearing to reduce people's time waiting in line, whether it is at tollbooths, amusement parks, or elsewhere. Consider the design of a QuickPass system for an amusement park. The amusement park has experimented by offering QuickPasses for several popular rides as a test. The idea is that for certain popular rides you can go to a kiosk near that ride and insert your daily park entrance ticket, and out will come a slip that states that you can return to that ride at a specific time later. For example, you insert your daily park entrance ticket at 1:15 pm, and the QuickPass states that you can come back between 3:30 and 4:30 pm when you can use your slip to enter a second, and presumably much shorter, line that will get you to the ride faster. To prevent people from obtaining QuickPasses for several rides at once, the QuickPass machines allow you to have only one active QuickPass at a time. You have been hired as one of several competing consultants to improve the operation of QuickPass. Customers have been complaining about some anomalies in the test system. For example, customers observed that in one instance QuickPasses were being offered for a return time as long as 4 hours later. A short time later on the same ride, the QuickPasses were given for times only an hour or so later. In some instances, the lines for people with Quickpasses are nearly as long and slow as the regular lines. The problem then is to propose and test schemes for issuing QuickPasses in order to increase people's enjoyment of the amusement park. Part of the problem is to determine what criteria to use in evaluating alternative schemes. Include in your report a non-technical summary for amusement park

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executives who must choose between alternatives from competing consultants. 2005 MCM A: Flood Planning Lake Murray in central South Carolina is formed by a large earthen dam, which was completed in 1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam. Two particular questions: Rawls Creek is a year-round stream that flows into the Saluda River a short distance downriver from the dam. How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend? Could the flood be so massive downstream that water would reach up to the S.C. State Capitol Building, which is on a hill overlooking the Congaree River? 2005 MCM B: Tollbooths Heavily-traveled toll roads such as the Garden State Parkway, Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas. Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas. Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza. Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza. Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza. When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll. Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza. Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane. Under what conditions is this more or less effective than the current practice? Note that the definition of ―optimal‖ is up to you to determine. 2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation There are a wide variety of techniques available for irrigating a field. The technologies range from advanced drip systems to periodic flooding. One of the systems that is used on smaller ranches is the use of ―hand move‖

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irrigation systems. Lightweight aluminum pipes with sprinkler heads are put in place across fields, and they are moved by hand at periodic intervals to insure that the whole field receives an adequate amount of water. This type of irrigation system is cheaper and easier to maintain than other systems. It is also flexible, allowing for use on a wide variety of fields and crops. The disadvantage is that it requires a great deal of time and effort to move and set up the equipment at regular intervals. Given that this type of irrigation system is to be used, how can it be configured to minimize the amount of time required to irrigate a field that is 80 meters by 30 meters? For this task you are asked to find an algorithm to determine how to irrigate the rectangular field that minimizes the amount of time required by a rancher to maintain the irrigation system. One pipe set is used in the field. You should determine the number of sprinklers and the spacing between sprinklers, and you should find a schedule to move the pipes, including where to move them. A pipe set consists of a number of pipes that can be connected together in a straight line. Each pipe has a 10 cm inner diameter with rotating spray nozzles that have a 0.6 cm inner diameter. When put together the resulting pipe is 20 meters long. At the water source, the pressure is 420 Kilo- Pascal’s and has a flow rate of 150 liters per minute. No part of the field should receive more than 0.75 cm per hour of water, and each part of the field should receive at least 2 centimeters of water every 4 days. The total amount of water should be applied as uniformly as possible. 2006 MCM B: Wheel Chair Access at Airports One of the frustrations with air travel is the need to fly through multiple airports, and each stop generally requires each traveler to change to a different airplane. This can be especially difficult for people who are not able to easily walk to a different flight's waiting area. One of the ways that an airline can make the transition easier is to provide a wheel chair and an escort to those people who ask for help. It is generally known well in advance which passengers require help, but it is not uncommon to receive notice when a passenger first registers at the airport. In rare instances an airline may not receive notice from a passenger until just prior to landing. Airlines are under constant pressure to keep their costs down. Wheel chairs wear out and are expensive and require maintenance. There is also a cost for making the escorts available. Moreover, wheel chairs and their escorts must be constantly moved around the airport so that they are available to people when their flight lands. In some large airports the time required to move across the airport is nontrivial. The wheel chairs must be stored somewhere, but

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space is expensive and severely limited in an airport terminal. Also, wheel chairs left in high traffic areas represent a liability risk as people try to move around them. Finally, one of the biggest costs is the cost of holding a plane if someone must wait for an escort and becomes late for their flight. The latter cost is especially troubling because it can affect the airline's average flight delay which can lead to fewer ticket sales as potential customers may choose to avoid an airline. Epsilon Airlines has decided to ask a third party to help them obtain a detailed analysis of the issues and costs of keeping and maintaining wheel chairs and escorts available for passengers. The airline needs to find a way to schedule the movement of wheel chairs throughout each day in a cost effective way. They also need to find and define the costs for budget planning in both the short and long term. Epsilon Airlines has asked your consultant group to put together a bid to help them solve their problem. Your bid should include an overview and analysis of the situation to help them decide if you fully understand their problem. They require a detailed description of an algorithm that you would like to implement which can determine where the escorts and wheel chairs should be and how they should move throughout each day. The goal is to keep the total costs as low as possible. Your bid is one of many that the airline will consider. You must make a strong case as to why your solution is the best and show that it will be able to handle a wide range of airports under a variety of circumstances. Your bid should also include examples of how the algorithm would work for a large (at least 4 concourses), a medium (at least two concourses), and a small airport (one concourse) under high and low traffic loads. You should determine all potential costs and balance their respective weights. Finally, as populations begin to include a higher percentage of older people who have more time to travel but may require more aid, your report should include projections of potential costs and needs in the future with recommendations to meet future needs. 2007 MCM A: Gerrymandering Gerrymandering The United States Constitution provides that the House of Representatives shall be composed of some number (currently 435) of individuals who are elected from each state in proportion to the state’s population relative to that of the country as a whole. While this provides a way of determining how many representatives each state will have, it says nothing about how the district represented by a particular representative shall be determined geographically. This oversight has led to egregious (at least some

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people think so, usually not the incumbent) district shapes that look ―unnatural‖ by some standards. Hence the following question: Suppose you were given the opportunity to draw congressional districts for a state. How would you do so as a purely ―baseline‖ exercise to create the ―simplest‖ shapes for all the districts in a st ate? The rules include only that each district in the state must contain the same population. The definition of ―simple‖ is up to you; but you need to make a convincing argument to voters in the state that your solution is fair. As an application of your method, draw geographically simple congressional districts for the state of New York. 2007 MCM B: The Airplane Seating Problem Airlines are free to seat passengers waiting to board an aircraft in any order whatsoever. It has become customary to seat passengers with special needs first, followed by first-class passengers (who sit at the front of the plane). Then coach and business-class passengers are seated by groups of rows, beginning with the row at the back of the plane and proceeding forward. Apart from consideration of the passengers’ wait time, from the airline’s point of view, time is money, and boarding time is best minimized. The plane makes money for the airline only when it is in motion, and long boarding times limit the number of trips that a plane can make in a day. The development of larger planes, such as the Airbus A380 (800 passengers), accentuate the problem of minimizing boarding (and deboarding) time. Devise and compare procedures for boarding and deboarding planes with varying numbers of passengers: small (85–210), midsize (210–330), and large (450–800). Prepare an executive summary, not to exceed two single-spaced pages, in which you set out your conclusions to an audience of airline executives, gate agents, and flight crews. An article appeared in the NY Times Nov 14, 2006 addressing procedures currently being followed and the importance to the airline of finding better solutions. The article can be seen at: http://travel2.nytimes.com/2006/11/14/business/14boarding.html

2008 MCM A: Take a Bath

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Consider the effects on land from the melting of the north polar ice cap due to the predicted increase in global temperatures. Specifically, model the effects on the coast of Florida every ten years for the next 50 years due to the melting, with particular attention given to large metropolitan areas. Propose appropriate responses to deal with this. A careful discussion of the data used is an important part of the answer. 2008 MCM B: Creating Sudoku Puzzles Develop an algorithm to construct Sudoku puzzles of varying difficulty. Develop metrics to define a difficulty level. The algorithm and metrics should be extensible to a varying number of difficulty levels. You should illustrate the algorithm with at least 4 difficulty levels. Your algorithm should guarantee a unique solution. Analyze the complexity of your algorithm. Your objective should be to minimize the complexity of the algorithm and meet the above requirements.

2009 MCM A: Designing a Traffic Circle Many cities and communities have traffic circles—from large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (with no right turn allowed on a red light). Other designs may also be possible. The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of traffic-control method should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples. 2009 MCM B: Energy and the Cell Phone This question involves the ―energy‖ consequences of the cell phone revolution. Cell phone usage is mushrooming, and many people are using cell phones

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and giving up their landline telephones. What is the consequence of this in terms of electricity use? Every cell phone comes with a battery and a recharger. Requirement 1 Consider the current US, a country of about 300 million people. Estimate from available data the number H of households, with m members each, that in the past were serviced by landlines. Now, suppose that all the landlines are replaced by cell phones; that is, each of the m members of the household has a cell phone. Model the consequences of this change for electricity utilization in the current US, both during the transition and during the steady state. The analysis should take into account the need for charging the batteries of the cell phones, as well as the fact that cell phones do not last as long as landline phones (for example, the cell phones get lost and break). Requirement 2 Consider a second ―Pseudo US‖—a country of about 300 million people with about the same economic status as the current US. However, this emerging country has neither landlines nor cell phones. What is the optimal way of providing phone service to this country from an energy perspective? Of course, cell phones have many social consequences and uses that landline phones do not allow. A discussion of the broad and hidden consequences of having only landlines, only cell phones, or a mixture of the two is welcomed. Requirement 3 Cell phones periodically need to be recharged. However, many people always keep their recharger plugged in. Additionally, many people charge their phones every night, whether they need to be recharged or not. Model the energy costs of this wasteful practice for a Pseudo US based upon your answer to Requirement 2. Assume that the Pseudo US supplies electricity from oil. Interpret your results in terms of barrels of oil. Requirement 4 Estimates vary on the amount of energy that is used by various recharger types (TV, DVR, computer peripherals, and so forth) when left plugged in but not charging the device. Use accurate data to model the energy wasted by the current US in terms of barrels of oil per day. Requirement 5

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Now consider population and economic growth over the next 50 years. How might a typical Pseudo US grow? For each 10 years for the next 50 years, predict the energy needs for providing phone service based upon your analysis in the first three requirements. Again, assume electricity is provided from oil. Interpret your predictions in term of barrels of oil.

2010 MCM A: The Sweet Spot Explain 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? 2010 MCM B: Criminology In 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a ―center of mass‖ of the locations of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the ―geographical profile‖ of a suspected serial criminal based on the locations of the crimes. Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your

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method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings. In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience. 2011 MCM A: Snowboard Course Determine the shape of a snowboard course (currently known as a ―halfpipe‖) to maximize the production of ―vertical air‖ by a skilled snowboarder. ―Vertical air‖ is the maximum vertical distance above the edge of the halfpipe. Tailor the shape to optimize other possible requirements, such as maximum twist in the air. What tradeoffs may be required to develop a ―practical‖ course? 2011 MCM B: Repeater Coordination The VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by ―repeaters,‖ which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies. In addition to geographical separation, the ―continuous tone -coded squelch system‖ (CTCSS), sometimes nicknamed ―private line‖ (PL), technology can be used to mitigate interference problems. This system associates to each repeater a separate subaudible tone that is transmitted by all users who wish to communicate through that repeater. The repeater responds only to received signals with its specific PL tone. With this system, two nearby repeaters can share the same frequency pair (for receive and transmit); so more repeaters (and hence more users) can be accommodated in a particular area. For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter

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frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available. How does your solution change if there are 10,000 users? Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas. 2012 MCM A: The Leaves of a Tree ―How much do the leaves on a tree weigh?‖ How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:

?

Why do leaves have the various shapes that they have? ? Do the shapes ―minimize‖ overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the ―volume‖ of the tree and its branches effect the shape? ? Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure? ? How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)? In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings. 2012 MCM B: Camping along the Big Long River Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip that requires several days of camping. River trips all start at First Launch and exit the river at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4 mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 nights of camping on the river, start to finish.. The government agency responsible for managing this river wants every trip to enjoy a wilderness experience, with minimal contact with other groups of boats on the river. Currently, X trips travel down the Big Long River each year during a six month period (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park managers have been asked to allow more trips to travel down the river. They want to determine how they might schedule an optimal mix of trips, of varying duration (measured in nights on the river) and propulsion (motor or oar) that will utilize the campsites

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in the best way possible. In other words, how many more boat trips could be added to the Big Long River’s rafting season? The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepare a one page memo to the managers of the river describing your key findings. 2013 MCM A: The Ultimate Brownie Pan When baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven. Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between. Assume 1. A width to length ratio of W/L for the oven which is rectangular in shape. 2. Each pan must have an area of A. 3. Initially two racks in the oven, evenly spaced. Develop a model that can be used to select the best type of pan (shape) under the following conditions: 1. Maximize number of pans that can fit in the oven (N) 2. Maximize even distribution of heat (H) for the pan 3. Optimize a combination of conditions (1) and (2) where weights p and (1- p) are assigned to illustrate how the results vary with different values of W/L and p. In addition to your MCM formatted solution, prepare a one to two page advertising sheet for the new Brownie Gourmet Magazine highlighting your design and results. 2013 MCM B: Water, Water, Everywhere Fresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, and cost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your

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approach, its feasibility and costs, and why it is the ―best water strategy choice.‖ Countries: United States, China, Russia, Egypt, or Saudi Arabia 2014 MCM A:The Keep-Right-Except-To-Pass Rule 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. 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. 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? 2014 MCM B:College Coaching Legends Sports 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.

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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.

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