2012年美国大学生数学建模竞赛题目B题英语达人全配套翻译
2012年美国数学奥林匹克(USAMO)试题及其解答
2012 年美国数学奥林匹克试题（USAMO）及其解答
田开斌 解答
1、求所有整数 n≥3，使得对于任意 n 个正实数a1 、a2 、a3 、 … … 、an ，如果满足 Max�a1 、a2 、a3 、 … … 、an � ≤ n · Min�a1 、a2 、a3 、 … … 、an �，则其中可以取出三个数， 它们能够构成一个锐角三角形的三条边的长度。 解：对于任意 n≤12，在序列 1、1、√2、√3、√5、√8、√13、√21、√34、√55、√89、12中取前 n 个数，都有 Max�a1 、a2 、a3 、 … … 、an � ≤ n · Min�a1 、a2 、a3 、 … … 、an �，但任意三个数都不能构成 锐角三角形的三条边。所以满足条件的 n≥13。 下面我们用反证法证明 n≥13 时，都满足条件。 我们给 n 个正实数从小到大排序为a1 ≤ a2 ≤ a3 ≤ ⋯ ≤ an ，若其中任意三个数，都不 能构成一个锐角三角形的三条边，则有a1 2 ≤ a2 2 ，ak 2 + ak+1 2 ≤ ak+2 2 ，其中 1≤k≤n-2。 于是知an 2 ≥ fn · a1 2 ，即an ≥ �fn · a1 ，其中fn 为斐波那契数列的第 n 项。又当 n≥13 时， 根据数学归纳法易知都有�fn ＞n，此时则有an ≥ �fn · a1 ＞na1 ，与 Max�a1 、a2 、a3 、 … … 、an � ≤ n · Min�a1 、a2 、a3 、 … … 、an �矛盾。所以当 n≥13 时，都 满足条件。 综上所述知，满足条件的 n 为所有不小于 13 的自然数。 2、一个圆被 432 个点等分为 432 段弧，将其中 108 个点染成红色，108 个点染成绿色， 108 个点染成蓝色，108 个点染成黄色。求证：可以在每种颜色的点中各选 3 个点，使得由 同色点构成的四个三角形都全等。 解：我们记 f(m)表示 m 除以 432 的余数，其中 0≤f(m)≤431。 我们从某点开始，按顺时针方向依次给 432 个点排序为 0、1、2、3、……431。设 108 个 红点所在位置依次为a1 、a2 、a3 、 … … 、a108 ，108 个绿点依次为b1 、b2 、b3 、 … … 、b108 ， 108 个蓝点依次为c1 、c2 、c3 、 … … 、c108 ，108 个黄点依次为d1 、d2 、d3 、 … … 、d108 。 记Xi = �f(a1 + i)、f(a2 + i)、f(a3 + i)、 … … 、f(a108 + i)� ∩ �b1 、b2 、b3 、 … … 、b108 � b1、b2、b3、……、b108=108（j=1、2、3、……108），所以
2012 mcm 美国大学生数学建模竞赛
2012 MCM/ICM Photograph & Video Release FormI hereby grant permission to the rights of my image, likeness and sound of my voice as recorded on audio or video tape. I understand that my image may be edited, copied, exhibited, published or distributed and waive the right to inspect or approve the finished product wherein my likeness appears. Additionally, I waive any right to royalties or other compensation arising or related to the use of my image or recording. I also understand that this material may be used in diverse outreach efforts within an unrestricted geographic area. Photographic, audio or video recordings may be used for the following purposes:•conference presentations•educational presentations or courses•informational presentations•on-line educational courses•educational videos and DVDs•reporting on commercial and not for profit outletsBy signing this release I understand this permission signifies that photographic or video recordings of me may be electronically displayed via the Internet or in the public educational setting.There is no time limit on the validity of this release nor is there any geographic limitation on where these materials may be distributed.This release applies to photographic, audio or video recordings collected as part of the sessions listed on this document only.By signing this form I acknowledge that I have completely read and fully understand the above release and agree to be bound thereby. I hereby release any and all claims against any person or organization utilizing this material for educational purposes.School ___________________________________________ Team Control Number __________________ Student Full Name_______________________________________________________________________ Street Address/P.O. Box___________________________________________________________________ City ___________________________________ State __________________ Zip Code________________ Country_____________________________________Phone_______________________________________Email Address_________________________________________________________________Signature_____________________________________ Date____________________________If this release is obtained from a student/presenter under the age of 19, then the signature of thatstudents/presenter’s parent or legal guardian is also required.Parent’s Signature______________________________ Date____________________________。
美国数学建模题目2001至2012翻译
2001年A题（一）Choosing a Bicycle Wheel选择自行车车轮有不同类型的车轮可以让自行车手们用在自己的自行车上。
两种基本的车轮类型是分别用金属辐条和实体圆盘组装而成（见图1）。
辐条车轮较轻，但实体车轮更符合空气动力学原理。
对于一场公路竞赛，实体车轮从来不会用作自行车的前轮但可以用作后轮。
职业自行车手们审视竞赛路线，并且请一位识文断字的人推断应该使用哪种车轮。
选择决定是根据沿途山丘的数量和陡度，天气，风速，竞赛本身以及其他考虑作出的。
你所喜爱的参赛队的教练希望准备妥当一个较好的系统，并且对于给定的竞赛路线已经向你的参赛队索取有助于确定宜用哪种车轮的信息。
这位教练需要明确的信息来帮助作出决定，而且已经要求你的参赛队完成下面列出的各项任务。
对于每项任务都假定，同样的辐条车轮将总是装在前面，而装在后面的车轮是可以选择的。
任务1. 提供一个给出风速的表格，在这种速度下实体后轮所需要的体能少于辐条后轮。
这个表格应当包括相应于从百分之零到百分之十增量为百分之一的不同公路陡度的风速。
（公路陡度定义为一座山丘的总升高除以公路长度。
如果把山丘看作一个三角形，它的陡度是指山脚处倾角的正弦。
）一位骑手以初始速度45kph从山脚出发，他的减速度与公路陡度成正比。
对于百分之五的陡度，骑上100米车速要下降8kph左右。
任务2. 提供一个例证，说明这个表格怎样用于一条时间试验路线。
任务3. 请判明这个表格是不是一件决定车轮配置的适当工具，并且关于如何作出这个决定提出其他建议。
MCM2001B题Escaping a Hurricane's Wrath逃避飓风怒吼(一场恶风…)1999年，在Floyd飓风预报登陆之前，撤离南卡罗来纳州沿海地区的行动导致一场永垂青史的交通拥塞。
车水马龙停滞在州际公路I-26上，那是内陆上从Charleston通往该州中心Columbia相对安全处所的主要干线。
正常时轻松的两个小时驱车路要用上18个小时才能开到头。
2012年美国大学生数学建模竞赛B题特等奖文章翻译
We develop a model to schedule trips down the Big Long River. The goalComputing Along the Big Long RiverChip JacksonLucas BourneTravis PetersWesternWashington UniversityBellingham,WAAdvisor: Edoh Y. AmiranAbstractis to optimally plan boat trips of varying duration and propulsion so as tomaximize the number of trips over the six-month season.We model the process by which groups travel from campsite to campsite.Subject to the given constraints, our algorithm outputs the optimal dailyschedule for each group on the river. By studying the algorithm’s long-termbehavior, we can compute a maximum number of trips, which we define asthe river’s carrying capacity.We apply our algorithm to a case study of the Grand Canyon, which hasmany attributes in common with the Big Long River.Finally, we examine the carrying capacity’s sensitivity to changes in thedistribution of propulsion methods, distribution of trip duration, and thenumber of campsites on the river.IntroductionWe address scheduling recreational trips down the Big Long River so asto maximize the number of trips. From First Launch to Final Exit (225 mi),participants take either an oar-powered rubber raft or a motorized boat.Trips last between 6 and 18 nights, with participants camping at designatedcampsites along the river. To ensure an authentic wilderness experience,at most one group at a time may occupy a campsite. This constraint limitsthe number of possible trips during the park’s six-month season.We model the situation and then compare our results to rivers withsimilar attributes, thus verifying that our approach yields desirable results.Our model is easily adaptable to find optimal trip schedules for riversof varying length, numbers of campsites, trip durations, and boat speeds.No two groups can occupy the same campsite at the same time.Campsites are distributed uniformly along the river.Trips are scheduled during a six-month period of the year.Group trips range from 6 to 18 nights.Motorized boats travel 8 mph on average.Oar-powered rubber rafts travel 4 mph on average.There are only two types of boats: oar-powered rubber rafts and motorizedTrips begin at First Launch and end at Final Exit, 225 miles downstream.*simulates river-trip scheduling as a function of a distribution of trip*can be applied to real-world rivers with similar attributes (i.e., the Grand*is flexible enough to simulate a wide range of feasible inputs; andWhat is the carrying capacity of the riverÿhe maximum number ofHow many new groups can start a river trip on any given day?How should trips of varying length and propulsion be scheduled toDefining the Problemmaximize the number of trips possible over a six-month season?groups that can be sent down the river during its six-month season?Model OverviewWe design a model thatCanyon);lengths (either 6, 12, or 18 days), a varying distribution of propulsionspeeds, and a varying number of campsites.The model predicts the number of trips over a six-month season. It alsoanswers questions about the carrying capacity of the river, advantageousdistributions of propulsion speeds and trip lengths, how many groups canstart a river trip each day, and how to schedule trips.ConstraintsThe problem specifies the following constraints:boats.AssumptionsWe can prescribe the ratio of oar-powered river rafts to motorized boats that go onto the river each day.There can be problems if too many oar-powered boats are launched with short trip lengths.The duration of a trip is either 12 days or 18 days for oar-powered rafts, and either 6 days or 12 days for motorized boats.This simplification still allows our model to produce meaningful results while letting us compare the effect of varying trip lengths.There can only be one group per campsite per night.This agrees with the desires of the river manager.Each day, a group can only move downstream or remain in its current campsiteÿt cannot move back upstream.This restricts the flow of groups to a single direction, greatly simplifying how we can move groups from campsite to campsite.Groups can travel only between 8 a.m. and 6 p.m., a maximum of 9hours of travel per day (one hour is subtracted for breaks/lunch/etc.).This implies that per day, oar-powered rafts can travel at most 36 miles, and motorized boats at most 72 miles. This assumption allows us to determine which groups can reasonably reach a given campsite.Groups never travel farther than the distance that they can feasibly travelin a single day: 36 miles per day for oar-powered rafts and 72 miles per day for motorized boats.We ignore variables that could influence maximum daily travel distance, such as weather and river conditions.There is no way of accurately including these in the model.Campsites are distributed uniformly so that the distance between campsites is the length of the river divided by the number of campsites.We can thus represent the river as an array of equally-spaced campsites.A group must reach the end of the river on the final day of its trip:A group will not leave the river early even if able to.A group will not have a finish date past the desired trip length.This assumption fits what we believe is an important standard for theriver manager and for the quality of the trips.MethodsWe define some terms and phrases:Open campsite: Acampsite is open if there is no groupcurrently occupying it: Campsite cn is open if no group gi is assigned to cn.Moving to an open campsite: For a group gi, its campsite cn, moving to some other open campsite cm ÿ= cn is equivalent to assigning gi to the new campsite. Since a group can move only downstream, or remain at their current campsite, we must have m ÿ n.Waitlist: The waitlist for a given day is composed of the groups that are not yet on the river but will start their trip on the day when their ranking onthe waitlist and their ability to reach a campsite c includes them in theset Gc of groups that can reach campsite c, and the groups are deemed “the highest priority.” Waitlisted groups are initialized with a current campsite value of c0 (the zeroth campsite), and are assumed to have priority P = 1 until they are moved from the waitlist onto the river.Off the River: We consider the first space off of the river to be the “final campsite” cfinal, and it is always an open campsite (so that any number of groups can be assigned to it. This is consistent with the understanding that any number of groups can move off of the river in a single day.The Farthest Empty CampsiteOurscheduling algorithm uses an array as the data structure to represent the river, with each element of the array being a campsite. The algorithm begins each day by finding the open campsite c that is farthest down the river, then generates a set Gc of all groups that could potentially reach c that night. Thus,Gc = {gi | li +mi . c},where li is the groupÿs current location and mi is the maximum distance that the group can travel in one day.. The requirement that mi + li . c specifies that group gi must be able to reach campsite c in one day.. Gc can consist of groups on the river and groups on the waitlist.. If Gc = ., then we move to the next farthest empty campsite.located upstream, closer to the start of the river. The algorithm always runs from the end of the river up towards the start of the river.. IfGc ÿ= ., then the algorithm attempts tomovethe groupwith the highest priority to campsite c.The scheduling algorithm continues in this fashion until the farthestempty campsite is the zeroth campsite c0. At this point, every group that was able to move on the river that day has been moved to a campsite, and we start the algorithm again to simulate the next day.PriorityOnce a set Gc has been formed for a specific campsite c, the algorithm must decide which group to move to that campsite. The priority Pi is a measure of how far ahead or behind schedule group gi is:. Pi > 1: group gi is behind schedule;. Pi < 1: group gi is ahead of schedule;. Pi = 1: group gi is precisely on schedule.We attempt to move the group with the highest priority into c.Some examples of situations that arise, and how priority is used to resolve them, are outlined in Figures 1 and 2.Priorities and Other ConsiderationsOur algorithm always tries to move the group that is the most behind schedule, to try to ensure that each group is camped on the river for aFigure 1. The scheduling algorithm has found that the farthest open campsite is Campsite 6 and Groups A, B, and C can feasibly reach it. Group B has the highest priority, so we move Group B to Campsite 6.Figure 2. As the scheduling algorithm progresses past Campsite 6, it finds that the next farthest open campsite is Campsite 5. The algorithm has calculated that Groups A and C can feasibly reach it; since PA > PC, Group A is moved to Campsite 5.number of nights equal to its predetermined trip length. However, in someinstances it may not be ideal to move the group with highest priority tothe farthest feasible open campsite. Such is the case if the group with thehighest priority is ahead of schedule (P <1).We provide the following rules for handling group priorities:?If gi is behind schedule, i.e. Pi > 1, then move gi to c, its farthest reachableopen campsite.?If gi is ahead of schedule, i.e. Pi < 1, then calculate diai, the number ofnights that the group has already been on the river times the averagedistance per day that the group should travel to be on schedule. If theresult is greater than or equal (in miles) to the location of campsite c, thenmove gi to c. Doing so amounts to moving gi only in such a way that itis no longer ahead of schedule.?Regardless of Pi, if the chosen c = cfinal, then do not move gi unless ti =di. This feature ensures that giÿ trip will not end before its designatedend date.Theonecasewhere a groupÿ priority is disregardedisshownin Figure 3.Scheduling SimulationWe now demonstrate how our model could be used to schedule rivertrips.In the following example, we assume 50 campsites along the 225-mileriver, and we introduce 4 groups to the river each day. We project the tripFigure 3. The farthest open campsite is the campsite off the river. The algorithm finds that GroupD could move there, but GroupD has tD > dD.that is, GroupD is supposed to be on the river for12 nights but so far has spent only 11.so Group D remains on the river, at some campsite between 171 and 224 inclusive.schedules of the four specific groups that we introduce to the river on day25. We choose a midseason day to demonstrate our modelÿs stability overtime. The characteristics of the four groups are:. g1: motorized, t1 = 6;. g2: oar-powered, t2 = 18;. g3: motorized, t3 = 12;. g4: oar-powered, t4 = 12.Figure 5 shows each groupÿs campsite number and priority value foreach night spent on the river. For instance, the column labeled g2 givescampsite numbers for each of the nights of g2ÿs trip. We find that each giis off the river after spending exactly ti nights camping, and that P ÿ 1as di ÿ ti, showing that as time passes our algorithm attempts to get (andkeep) groups on schedule. Figures 6 and 7 display our results graphically.These findings are consistent with the intention of our method; we see inthis small-scale simulation that our algorithm produces desirable results.Case StudyThe Grand CanyonThe Grand Canyon is an ideal case study for our model, since it sharesmany characteristics with the Big Long River. The Canyonÿs primary riverrafting stretch is 226 miles, it has 235 campsites, and it is open approximatelysix months of the year. It allows tourists to travel by motorized boat or byoar-powered river raft for a maximum of 12 or 18 days, respectively [Jalbertet al. 2006].Using the parameters of the Grand Canyon, we test our model by runninga number of simulations. We alter the number of groups placed on thewater each day, attempting to find the carrying capacity for the river.theFigure 7. Priority values of groups over the course of each trip. Values converge to P = 1 due to the algorithm’s attempt to keep groups on schedule.maximumnumber of possible trips over a six-month season. The main constraintis that each trip must last the group’s planned trip duration. Duringits summer season, the Grand Canyon typically places six new groups onthe water each day [Jalbert et al. 2006], so we use this value for our first simulation.In each simulation, we use an equal number of motorized boatsand oar-powered rafts, along with an equal distribution of trip lengths.Our model predicts the number of groups that make it off the river(completed trips), how many trips arrive past their desired end date (latetrips), and the number of groups that did not make it off the waitlist (totalleft on waitlist). These values change as we vary the number of new groupsplaced on the water each day (groups/day).Table 1 indicates that a maximum of 18 groups can be sent down theriver each day. Over the course of the six-month season, this amounts to nearly 3,000 trips. Increasing groups/day above 18 is likely to cause latetrips (some groups are still on the river when our simulation ends) and long waitlists. In Simulation 1, we send 1,080 groups down river (6 groups/day?80 days) but only 996 groups make it off; the other groups began near the end of the six-month period and did not reach the end of their trip beforethe end of the season. These groups have negligible impact on our results and we ignore them.Sensitivity Analysis of Carrying CapacityManagers of the Big Long River are faced with a similar task to that of the managers of the Grand Canyon. Therefore, by finding an optimal solutionfor the Grand Canyon, we may also have found an optimal solution forthe Big Long River. However, this optimal solution is based on two key assumptions:?Each day, we put approximately the same number of groups onto theriver; and?the river has about one campsite per mile.We can make these assumptions for the Grand Canyon because they are true for the Grand Canyon, but we do not know if they are true for the Big Long River.To deal with these unknowns,wecreate Table 3. Its values are generatedby fixing the number Y of campsites on the river and the ratio R of oarpowered rafts to motorized boats launched each day, and then increasingthe number of trips added to the river each day until the river reaches peak carrying capacity.The peak carrying capacities in Table 3 can be visualized as points ina three-dimensional space, and we can find a best-fit surface that passes (nearly) through the data points. This best-fit surface allows us to estimatethe peak carrying capacity M of the river for interpolated values. Essentially, it givesM as a function of Y and R and shows how sensitiveM is tochanges in Y and/or R. Figure 7 is a contour diagram of this surface.The ridge along the vertical line R = 1 : 1 predicts that for any givenvalue of Y between 100 and 300, the river will have an optimal value ofM when R = 1 : 1. Unfortunately, the formula for this best-fit surface is rather complex, and it doesn’t do an accurate job of extrapolating beyond the data of Table 3; so it is not a particularly useful tool for the peak carrying capacity for other values ofR. The best method to predict the peak carrying capacity is just to use our scheduling algorithm.Sensitivity Analysis of Carrying Capacity re R and DWe have treatedM as a function ofR and Y , but it is still unknown to us how M is affected by the mix of trip durations of groups on the river (D).For example, if we scheduled trips of either 6 or 12 days, how would this affect M? The river managers want to know what mix of trips of varying duration and speed will utilize the river in the best way possible.We use our scheduling algorithm to attempt to answer this question.We fix the number of campsites at 200 and determine the peak carrying capacity for values of R andD. The results of this simulation are displayed in Table 4.Table 4 is intended to address the question of what mix of trip durations and speeds will yield a maximum carrying capacity. For example: If the river managers are currently scheduling trips of length?6, 12, or 18: Capacity could be increased either by increasing R to be closer to 1:1 or by decreasing D to be closer to ? or 12.?12 or 18: Decrease D to be closer to ? or 12.?6 or 12: Increase R to be closer to 4:1.ConclusionThe river managers have asked how many more trips can be added tothe Big Long Riverÿ season. Without knowing the specifics ofhowthe river is currently being managed, we cannot give an exact answer. However, by applying our modelto a study of the GrandCanyon,wefound results which could be extrapolated to the context of the Big Long River. Specifically, the managers of the Big Long River could add approximately (3,000 - X) groups to the rafting season, where X is the current number of trips and 3,000 is the capacity predicted by our scheduling algorithm. Additionally, we modeled how certain variables are related to each other; M, D, R, and Y . River managers could refer to our figures and tables to see how they could change their current values of D, R, and Y to achieve a greater carrying capacity for the Big Long River.We also addressed scheduling campsite placement for groups moving down the Big Long River through an algorithm which uses priority values to move groups downstream in an orderly manner.Limitations and Error AnalysisCarrying Capacity OverestimationOur model has several limitations. It assumes that the capacity of theriver is constrained only by the number of campsites, the trip durations,and the transportation methods. We maximize the river’s carrying capacity, even if this means that nearly every campsite is occupied each night.This may not be ideal, potentially leading to congestion or environmental degradation of the river. Because of this, our model may overestimate the maximum number of trips possible over long periods of time. Environmental ConcernsOur case study of the Grand Canyon is evidence that our model omits variables. We are confident that the Grand Canyon could provide enough campsites for 3,000 trips over a six-month period, as predicted by our algorithm. However, since the actual figure is around 1,000 trips [Jalbert et al.2006], the error is likely due to factors outside of campsite capacity, perhaps environmental concerns.Neglect of River SpeedAnother variable that our model ignores is the speed of the river. Riverspeed increases with the depth and slope of the river channel, makingour assumption of constant maximum daily travel distance impossible [Wikipedia 2012]. When a river experiences high flow, river speeds can double, and entire campsites can end up under water [National Park Service 2008]. Again, the results of our model don’t reflect these issues. ReferencesC.U. Boulder Dept. of Applied Mathematics. n.d. Fitting a surface to scatteredx-y-z data points. /computing/Mathematica/Fit/ .Jalbert, Linda, Lenore Grover-Bullington, and Lori Crystal, et al. 2006. Colorado River management plan. 2006./grca/parkmgmt/upload/CRMPIF_s.pdf .National Park Service. 2008. Grand Canyon National Park. High flowriver permit information. /grca/naturescience/high_flow2008-permit.htm .Sullivan, Steve. 2011. Grand Canyon River Statistics Calendar Year 2010./grca/planyourvisit/upload/Calendar_Year_2010_River_Statistics.pdf .Wikipedia. 2012. River. /wiki/River .Memo to Managers of the Big Long RiverIn response to your questions regarding trip scheduling and river capacity,we are writing to inform you of our findings.Our primary accomplishment is the development of a scheduling algorithm.If implemented at Big Long River, it could advise park rangerson how to optimally schedule trips of varying length and propulsion. Theoptimal schedule will maximize the number of trips possible over the sixmonth season.Our algorithm is flexible, taking a variety of different inputs. Theseinclude the number and availability of campsites, and parameters associatedwith each tour group. Given the necessary inputs, we can output adaily schedule. In essence, our algorithm does this by using the state of theriver from the previous day. Schedules consist of campsite assignments foreach group on the river, as well those waiting to begin their trip. Given knowledge of future waitlists, our algorithm can output schedules monthsin advance, allowing managementto schedule the precise campsite locationof any group on any future date.Sparing you the mathematical details, allow us to say simply that ouralgorithm uses a priority system. It prioritizes groups who are behindschedule by allowing them to move to further campsites, and holds backgroups who are ahead of schedule. In this way, it ensures that all trips willbe completed in precisely the length of time the passenger had planned for.But scheduling is only part of what our algorithm can do. It can alsocompute a maximum number of possible trips over the six-month season.We call this the carrying capacity of the river. If we find we are below ourcarrying capacity, our algorithm can tell us how many more groups wecould be adding to the water each day. Conversely, if we are experiencingriver congestion, we can determine how many fewer groups we should beadding each day to get things running smoothly again.An interesting finding of our algorithm is how the ratio of oar-poweredriver rafts to motorized boats affects the number of trips we can send downstream. When dealing with an even distribution of trip durations (from 6 to18 days), we recommend a 1:1 ratio to maximize the river’s carrying capacity.If the distribution is skewed towards shorter trip durations, then ourmodel predicts that increasing towards a 4:1 ratio will cause the carryingcapacity to increase. If the distribution is skewed the opposite way, towards longer trip durations, then the carrying capacity of the river will always beless than in the previous two cases—so this is not recommended.Our algorithm has been thoroughly tested, and we believe that it isa powerful tool for determining the river’s carrying capacity, optimizing daily schedules, and ensuring that people will be able to complete their trip as planned while enjoying a true wilderness experience.Sincerely yours,Team 13955。
2012美国大学生数学建模题目(英文原版加中文翻译)
2012 MCM ProblemsPROBLEM 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 mode l 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.“多少钱树的叶子有多重？”怎么可能估计的叶子（或树为此事的任何其他部分）的实际重量？会如何分类的叶子吗？建立了一个数学模型来描述和分类的叶子。
2012年美国数学建模题目中文版3篇
2012年美国数学建模题目中文版第一篇：2012年美国数学建模题目解析2012年美国数学建模竞赛题目分为3个部分：A、B、C 部分，其中A、B两部分每个题目都设计成了开放式问题，而C部分则是两道严谨的数学证明题目。
A部分共有四个问题，分别为：1、搜索引擎的自动补充功能对于使用者的输入进行了什么样的预测和补全？如果这种功能可以被改变，在搜索引擎中进行必要的优化，会对搜索引擎的使用产生什么影响？2、在一个公共交通的网络中，如何合理地分配车辆保证所有的车辆在一定时间内都能够按时到达各自的终点站？3、如何在餐馆排队时，给不同的桌子和不同的人分配最佳位置，以便让顾客在餐厅等待的时间最短？4、针对特定的树木，如何编写算法来找到该树生长的变化，在叶片的数量和大小、气孔的数量和大小等方面的特征？对于这四个问题，考生需要通过分析问题，理清思路，构思模型，进行数据分析，最后得出自己的结论。
需要注意的是，每个问题都是非常开放式的，没有标准答案，最终得分并不会仅仅取决于观点是否正确，具体的解题过程、数据展示和准确度也是非常关键的。
B部分共有三个问题，分别为：1、如何通过旅游者在社交网络上的信息，帮助旅游者更好地定制旅游计划？2、如何在残缺不全的传媒报道中，找到事实并从中解读该事件？3、针对滑雪者在滑雪过程中的各种情况，如何预测他们的滑雪技巧以及未来的滑雪表现？对于B部分的三个问题，其实也都是很自由的问题，可以根据自己所擅长领域进行分析，构思自己的模型和算法，注重细节和数据展示。
C部分共有两个题目：1、已知一个最小二乘问题，其正则化后的解为稀疏的，试设计一个迭代算法在有效的处理机制下对其进行数值求解。
2、已知一个对象向一条线段上匀速运动，在线段的中途，运动的对象突然重力下落，如果目标是在最短的时间内捕捉该运动的对象，该怎样运动才是最优策略？对于C部分两个题目，需要在数学基础扎实的基础上进行思考，深入分析，构建出严谨的证明过程，注重逻辑和方法。
2012年全国大学生数学建模竞赛B题
太阳能小屋的设计摘要本文提出了一种光伏电池“反馈式铺设”的方案，把每个问题下的铺设任务分离为几个有序的步骤来进行，步骤之间联系紧密，后一步骤的进行甚至可以反过来改变前一步骤的结果，经过多次反馈后，可以得到较优的解。
在此方案的基础上，对太阳能小屋铺设问题进入了深入的研究。
针对问题一，我们首先处理了大同市典型气象年气象数据，得到了各个表面的年辐射总量，由于各个表面的年辐射总量不一样，所以针对于各个表面的各个光伏电池的发电量、盈利额度、单位面积效益等都要分别进行计算，尤其是屋顶的情况，公示推导繁琐，数据处理起来颇为复杂，数据处理完毕后，选取候选光伏电池，建立矩形排样的模型，将各表面的非矩形约束条件转化为矩形约束来求解，铺设完成后，根据铺设的结果选取适当的光伏电池进行二次再铺设，而后将光伏电池组件进行分组，选取逆变器，调整二次再铺设后的铺设结果，使得逆变器的选取以及光伏电池组件的分组达到较佳的状态。
在问题二中，我们首先利用matlab算取了架空方式安装下的最佳倾斜角度和旋转角度，然后将架空方式安装的光伏电池进行相应的变换投影到太阳能小屋表面，根据变换后的图形利用类似问题一中提出的算法进行铺设。
在问题三中，根据题目给出的小屋建筑要求，将约束条件转化为lingo中的语言，目标函数是使得问题一中的盈利表面的面积尽可能大，房屋屋顶的倾斜角尽可能接近问题二中的最佳倾斜角。
lingo求解出的房屋尺寸在solidworks中进行三维建模，建模后的房屋进行光伏电池的铺设，根据光伏电池的铺设情况对房屋的尺寸进行小范围的调整，使得房屋的设计达到较优，即使小屋的全年发电总量尽可能大，单位发电量的费用尽可能小。
关键词：太阳能小屋光伏电池逆变器反馈式铺设最小劣度矩形排样一、问题重述在设计太阳能小屋时，需在建筑物外表面（屋顶及外墙）铺设光伏电池，光伏电池组件所产生的直流电需要经过逆变器转换成220V交流电才能供家庭使用，并将剩余电量输入电网。
美国大学生数学建模竞赛试题AB题中文
A 题热水澡一个人进入浴缸洗澡放松。
浴缸的热水由一个水龙头放出。
然而浴缸不是一个可以水疗泡澡的缸，没有辅助加热系统和循环喷头，仅仅就是一个简单的盛水容器。
过一会，水温就会显著下降。
因此必须从热水龙头里面反复放水以加热水温。
浴缸的设计就是当水达到浴缸的最大容量，多余的水就会通过一个溢流口流出。
做一个有关浴缸水温的模型，从时间和地点两个方面来确定在浴缸中泡澡的人能采用的最佳策略，从而泡澡过程中能保持水温并在不浪费太多水的情况下使水温尽量接近最初的水温。
用你的模型来确定你的策略多大程度上依赖于浴缸的形状和容量，浴缸中的人的体型/体重/体温，以及这个人在浴缸中做出的动作。
如果这个人在最开始放水的时候加入了泡泡浴添加剂，这将会对你的模型结果有什么影响？要求提交一页MCM的总结，此外你的报告必须包括一页给浴缸用户看的非技术性的解释，其中描述了你的策略并解释了在泡澡过程中为什么保持平均的水温会非常困难。
B题太空垃圾地球轨道周围的小碎片的数量受到越来越多的关注。
据估计，目前大约有超过50万片太空碎片被视为是宇宙飞行器的潜在威胁并受到跟踪，这些碎片也叫轨道碎片。
2009年2月10号俄罗斯卫星科斯莫斯-2251与美国卫星iridium-33相撞的时候，这个问题在新闻媒体上就愈发受到广泛讨论。
已经提出了一些方法来清除这些碎片。
这些方法包括小型太空水流喷射器和高能量激光来瞄准具体的碎片，还有大型卫星来清扫碎片等等。
这些碎片数量和大小不一，有油漆脱离的碎片，也有废弃的卫星。
碎片高速转动使得定位清除变得困难。
建一个随时间变化的模型来确定一个最佳选择或组合的选择提供给一家私人公司让它以此为商业机遇来解决太空碎片问题。
你的模型应该包括对成本、风险、收益的定量和/或定性分析以及其他重要因素的分析。
你的模型应该既能够评估单个的选择也能够评估组合的选择，且能够探讨一些重要的”what if ”情景。
用你的模型来确定是否存在这样的机会，在经济上很有吸引力；或是根本不可能有这样的机会。
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PROBLEM B: Camping along the Big Long River
问题B：沿着大长河露营
Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. 游客到大长河（225英里）可以享受风景和令人兴奋的白色水急流。
The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip that requires
河是徒步旅行者无法进入的，所以唯一享受它的方法是采取河之旅，需要几天的露营several days of camping. River trips all start at First Launch and exit the river at Final Exit, 225
河旅行都开始在第一次启动和退出河在最后退出，225英里下游。
miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4
乘客乘橡皮艇桨驱动平均每小时4英里，或摩托艇，平均每小时8英里的速度行驶。
mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 nights of
旅程从开始到完成6到18个晚上，
camping on the river, start to finish.. The government agency responsible for managing this river
负责管理这条河的政府机关想要，以最少的接触与其他群体在河的小船，享受荒野之旅。
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
×旅行每年在一六个月期间旅行下来，大长江（一年的其余太冷对于河旅行）。
period (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long
大长河上有Y个露营地，在整个河流廊分布相当均匀。
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 鉴于流行起来的漂流，公园管理人员被要求允许更多的旅行来到河上来旅行。
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 in the best way 他们想确定他们如何可能安排的最佳组合，不同的时间（以夜河）和推进（电机或桨）将利用营地以可能的最佳方式。
possible. In other words, how many more boat trips could be added to the Big Long River’s
换句话说，如何使更多的游船可以被添加到大长河泛舟季节？
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 河流管理者雇你来指导他们如何在发展最好的安排和方法，确定承载力的河流，
that no two sets of campers can occupy the same site at the same time. In addition to your one
记住，没有两支露营队可以在同一时间占据相同的露营地。
page summary sheet, prepare a one page memo to the managers of the river describing your key findings.除了你的一页表，准备了一一页的备忘录的管理人员的描述你的调查结果。
问题：露营沿大长河
游客到大长河（225英里）可以享受风景和令人兴奋的白色水急流。
河是徒步旅行者无法进入的，所以唯一享受它的方法是采取河之旅，需要几天的露营。
河旅行都开始在第一次启动和退出河在最后退出，225英里下游。
乘客乘橡皮艇桨驱动，而旅行的平均每小时4英里或摩托艇，平均每小时8英里的速度行驶。
车次范围从6到18个晚上露营的河流，开始完成。
政府机构负责管理这条河都要享受之旅荒野经验，以最少的接触与其他群体的小船在河。
目前，×旅行旅行下来，大长江每年在一六个月期间（一年的其余太冷河旅行）。
有你的营地在大长河，分布相当均匀在整个河流廊。
鉴于流行起来的漂流，公园管理人员被要求允许更多的去旅行下来河。
他们想确定他们如何可能安排的最佳组合，不同的时间（以夜河）和推进（电机或桨）将利用营地以可能的最佳方式。
换句话说，如何更多的游船可以被添加到大长河泛舟季节？河流管理者雇你来指导他们如何在发展最好的安排和方法，确定承载力的河流，记住，没有设置营可以占据相同的网站在同一时间。
除了你的一页表，准备了一一页的备忘录的管理人员的描述你的调查结果。