2012美赛题目翻译
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美赛B题参考资料
Typical Day on the Rogue RiverIf you've never taken a river trip before, it may seem daunting, but read through this description and we think you'll be ready to climb aboard!MorningImagine waking to the smell of coffee brewing or bacon cooking. For the early birds we have coffee, fruit and juice ready by 7 or 7:30 and full breakfast by 8. Breakfast normally consists of something hearty and traditional like pancakes and bacon, french toast and sausage, or scrambled eggs and Canadian bacon. We also have juice, fruit, yogurt and cereal for those who want to eat lighter.Breakfast is a good time to talk about what the day will be like, so people can pack accordingly. For example, if there is a long hike planned, you might put your hiking shoes in your day bag so they will available for the hike.We will have everything packed and ready to hit the river by 9:30 or 10. We don't usually care to be on the river earlier, because it is too cool before then to be getting wet. You will have a chance to read, chat, hike, fish or otherwise relax while the rafts are being loaded.On The WaterRogue River rafting is done in three types of boats.An oar raft is a large raft that is rowed by the guide. Your job here is to relax, watch the scenery go by, and hang on in the rapids.A paddle raft is a bit smaller. Everyone gets a paddle, and works as a team to maneuver the raft. The guide sits in back, steers and give commands.An inflatable kayak, or ducky is a one- or two- person boat. With no guide, it is the most challenging -and fun-way to run the river.Unless you know what kind of boat you want to be in, we recommend that you rotate between boats a few times each day, in order to get the whole river experience.Once on the river we will float for two to three hours before stopping for lunch. We may also stop to take a short hike to a waterfall or other spot of interest.Lunch normally takes about an hour, and gives people another chance to stretch their legs with a short hike or relax.After lunch we float another two to three hours, again perhaps with an off-river activity. We arrive in camp between 3 and 5, depending on the miles we cover and the stops we make.CampWhen we hit camp you will grab your bag and find a nice place to camp. Again, there is time to hike, swim, read, fish, relax or play games. Often a guide will organize a game or a hike to a place of special interest.We serve hors d'oeuvre around 7 and dinner around 8. It is too warm to eat earlier. Dinner is usually a social time when everyone gathers to eat and share their day's experiences. The lead guide usually takes advantage of the gathering to tell everyone what tomorrow will bring.Some folks tend to stay up late, talking, perhaps singing. Instruments are welcome if they are small enough to pack (no pianos, please) and add a great deal to any camp scene. Sometimes, especially on the last night, people put together skits, make up songs, and otherwise celebrate the completion of a wonderful trip together.Dates and PricesTrip ItineraryBefore You Go....Trip Details。
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2012年美国赛的公费队选拔赛题目Problem A: Choosing a Bicycle Wheel (2001 MCM)Cyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike.Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course.Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on theright.The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the windspeeds for different road grades starting from zero percent to ten percent in onepercent increments. (Road grade is defined to be the ratio of the total rise of a hilldivided 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 riderwill lose about 8 kph for a five percent grade over 100 meters.∙Task 2. Provide an example of how the table could be used for a specific time trial course.∙Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.Problem B: Escaping a Hurricane's Wrath (An Ill Wind...)(2001 MCM)Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed:1.Under what conditions does the plan for turning the two coastal-bound lanes of I-26into two lanes of Columbia-bound traffic, essentially turning the entire I-26 intoone-way traffic, significantly improve evacuation traffic flow?2.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?3.Several smaller highways besides I-26 extend inland from the coast. Under whatconditions would it improve evacuation flow to turn around traffic on these?4.What effect would it have on evacuation flow to establish more temporary shelters inColumbia, to reduce the traffic leaving Columbia?5.In 1999, many families leaving the coast brought along their boats, campers, andmotor homes. Many drove all of their cars. Under what conditions should there berestrictions on vehicle types or numbers of vehicles brought in order to guaranteetimely evacuation?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 tocompare strategies. Required: Prepare a short newspaper article, not to exceed twopages, 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.PROBLEM C: Creating Food Systems: Re-Balancing Human-Influenced Ecosystems (2009 ICM, Please see the PDF file “ICM_2009.pdf”)。
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。
美赛C题翻译
Using Networks to Measure Influence and Impact（使用网络系统测定影响因子（因素））One of the techniques to determine influence of academic research is to build and measure properties of citation or co-author networks. 衡量学术研究的方法之一是建立和衡量检索频率或共同作者。
Co-authoring a manuscript usually connotes a strong influential connection between researchers. 合著论文通常表明研究者之间有很强的联系。
One of the most famous academic co-authors was the 20th-century mathematician Paul Erdös who had over 500 co-authors and published over 1400 technical research papers. 合著论文最有名的例子是20世纪的数学家Paul Erdös ，他有500多个合著论文者，且发表了1400篇研究论文。
It is ironic, or perhaps not, that Erdös is also one of the influencers in building the foundation for the emerging interdisciplinary science of networks, particularly, through his publication with Alfred Rényi of the paper “On Random Graphs” in 1959. 或许有些沨刺意味，或许没有，Erdös 还是建立刚刚出现的跨学科网络的最有影响力的人之一。
2012年河南省翻译竞赛真题
2012年河南省翻译竞赛翻译竞赛英译中参赛原文Over-regulated AmericaThe home of laissez-faire is being suffocated by excessive and badly written regulationAmericans love to laugh at ridiculous regulations. A Florida law requires vending-machine labels to urge the public to file a report if the label is not there. The Federal Railroad Administration insists that all trains must be painted with an “F” at the front, so you can tell which end is which. Bureaucratic busybodies in Bethesda, Maryland, have shut down children’s lemon ade stands because the enterprising young moppets did not have trading licences. The list goes hilariously on.But red tape in America is no laughing matter. The problem is not the rules that are self-evidently absurd. It is the ones that sound reasonable on their own but impose a huge burden collectively. America is meant to be the home of laissez-faire. Unlike Europeans, whose lives have long been circumscribed by meddling governments and diktats from Brussels, Americans are supposed to be free to choose, for better or for worse. Yet for some time America has been straying from this ideal.Consider the Dodd-Frank law of 2010. Its aim was noble: to prevent another financial crisis. Its strategy was sensible, too: improve transparency, stop banks from taking excessive risks, prevent abusive financial practices and end “too big to fail” by authorising regulators to seize any big, tottering financial firm and wind it down. This newspaper supported these goals at the time, and we still do. But Dodd-Frank is far too complex, and becoming more so. At 848 pages, it is 23 times longer than Glass-Steagall, the reform that followed the Wall Street crash of 1929. Worse, every other page demands that regulators fill in further detail. Some of these clarifications are hundreds of pages long. Just one bit, the “Volcker rule”, which aims to curb risky proprietary trading by banks, includes 383 questions that break down into 1,420 subquestions.Hardly anyone has actually read Dodd-Frank. Those who have struggle to make sense of it, not because so much detail has yet to be filled in: of the 400 rules it mandates, only 93 have been finalised. So financial firms in America must prepare to comply with a law that is partly unintelligible and partly unknowable. Flaming water-skisDodd-Frank is part of a wider trend. Governments of both parties keep adding stacks of rules, few of which are ever rescinded. Republicans write rules to thwart terrorists, which make flying in America an ordeal and prompt legions of brainy migrants to move to Canada instead. Democrats write rules to expand the welfare state. Barack Obama’s health-care reform of 2010 had many virtues, especially its attempt to make health insurance universal. But it does little to reduce the system’s staggering and increasing complexity. Every hour spent treating a patient in America creates at least 30 minutes of paperwork, and often a whole hour. Next year the number of federally mandated categories of illness and injury for which hospitals may claim reimbursement will rise from 18,000 to 140,000. There are nine codes relating to injuries caused by parrots, and three relating to burns from flaming water-skis.Two forces make American laws too complex. One is hubris. Many lawmakers seem to believe that they can lay down rules to govern every eventuality. Examples range from the merely annoying (eg, a proposed code for nurseries in Colorado that specifies how many crayons each box must contain) to the delusional (eg, the conceit of Dodd-Frank that you can anticipate and banevery nasty trick financiers will dream up in the future). Far from preventing abuses, complexity creates loopholes that the shrewd can abuse with impunity.The other force that makes American laws complex is lobbying. The government’s drive to micromanage so many activities creates a huge incentive for interest groups to push for special favours. When a bill is hundreds of pages long, it is not hard for congressmen to slip in clauses that benefit their chums and campaign donors. The health-care bill included tons of favours for the pushy. Congress’s last, failed attempt to regulate greenhouse gases was even worse.Complexity costs money. Sarbanes-Oxley, a law aimed at preventing Enron-style frauds, has made it so difficult to list shares on an American stockmarket that firms increasingly look elsewhere or stay private. America’s share of initial public offerings fell from 67% in 2002 (when Sarbox passed) to 16% last year, despite some benign tweaks to the law. A study for the Small Business Administration, a government body, found that regulations in general add $10,585 in costs per employee. It’s a wonder the jobless rate isn’t even higher than it is.A plea for simplicityDemocrats pay lip service to the need to slim the rulebook –Mr Obama’s regulations tsar i s supposed to ensure that new rules are cost-effective. But the administration has a bias towards overstating benefits and underestimating costs (see article). Republicans bluster that they will repeal Obamacare and Dodd-Frank and abolish whole government agencies, but give only a sketchy idea of what should replace them.America needs a smarter approach to regulation. First, all important rules should be subjected to cost-benefit analysis by an independent watchdog. The results should be made public before the rule is enacted. All big regulations should also come with sunset clauses, so that they expire after, say, ten years unless Congress explicitly re-authorises them.More important, rules need to be much simpler. When regulators try to write an all-purpose instruction manual, the truly important dos and don’ts are lost in an ocean of verbiage. Far better to lay down broad goals and prescribe only what is strictly necessary to achieve them. Legislators should pass simple rules, and leave regulators to enforce them.Would this hand too much power to unelected bureaucrats? Not if they are made more accountable. Unreasonable judgments should be subject to swift appeal. Regulators who make bad decisions should be easily sackable. None of this will resolve the inevitable difficulties of regulating a complex modern society. But it would mitigate a real danger: that regulation may crush the life out of America’s economy.选自The Economist, Feb 18th-24th, p8翻译竞赛中译英参赛原文“悦读”的“姿势”从一定意义说，一个民族的发展史就是它的阅读史，一个人亦如此。
2012年美赛C题论文
1.2 Methods of the Analysis
As investigators, we now know well about the 83nodes, 400 links over 21,000 words of message traffic, 15 topics (three have been deemed to be suspicious), 7 known conspirators, and 8known non-conspirators. We now call the known information about the company INTELLIGENCE. From the practice, the method which is always used in the intelligence work is a combination of qualitative reasoning and quantified reasoning, that is called Analytic Hierarchy Process (called AHP in the later part). At the same time, we also use Structural Model Analysis (SMA).
1.3 Assumptions
There is only linguistic communication between people. Body language never exists. People can talk to each other freely, without the liof known conspiractors and non-conspiractors is correct. The talk happens only between two people. Talks between 3 or over 3 members never exists.
美赛题目翻译
美赛题目翻译————————————————————————————————作者：————————————————————————————————日期：2012美赛A题：一棵树的叶子（数学中国翻译）“一棵树的叶子有多重？”怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。
模型要考虑和回答下面的问题：•为什么叶子具有各种形状？•叶子之间是要将相互重叠的部分最小化，以便可以最大限度的接触到阳光吗？树叶的分布以及树干和枝杈的体积影响叶子的形状吗？•就轮廓来讲，叶形（一般特征）是和树的轮廓以及分枝结构有关吗？•你将如何估计一棵树的叶子质量？叶子的质量和树的尺寸特征（包括和外形轮廓有关的高度、质量、体积）有联系吗？除了你的一页摘要以外，给科学杂志的编辑写一封信，阐述你的主要发现B：沿着Big Long River野营【数学中国翻译】0 i" k1 T3 h' B# u" ]游客在“大长河”(225英里)可以享受到秀丽的风光和令人兴奋的白色湍流。
这条河对于背包客来说是进不去的，因此畅游这条长河的唯一办法就是在这条河上露营上几天。
这次旅行从开始的下水点到最终结束点，共225英里，且是顺流而下的。
乘客可以选择平均4英里/小时的以浆作为动力的橡胶筏或者平均8英里/小时的机动帆船旅行。
整个旅行从开始到结束会经历6至18个夜晚。
负责管理这条河的政府机构希望到这里的每一次旅行都能够享受到野外经历，以最少的接触到在河上其它的船只。
目前，每年在六个月期间(一年的其余部分的天气对于河流旅行来说太冷)，共有X次旅行，有Y 处露营地，露营地均匀的分布整个河道。
由于漂流的受欢迎程度的上升，公园管理者已经被要求允许更多的旅行次数。
所以他们想确定怎样可能安排一个最优的混合的旅行方案，不同的时间(单位为夜)和推动方式(马达或浆)，最大限度的利用露营地。
2012年美国数学建模MCM题目(中英对照版)
2012 Contest ProblemsMCM 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 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.“一棵树的叶子有多重？”怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。
数模1998-2016年历年美赛题目(中文)
2016 年美赛题目翻译Program A一个人用热水从一个水龙头里灌满一个浴缸，然后安顿在浴缸中，清洗和放松。
不幸的是，浴缸不是一个温泉式浴缸，一个二次加热系统和循环射流，而是一个简单的水容器。
过了一会儿，洗澡就明显地凉快，所以人增加了一个恒定滴热水从水龙头加热洗浴用水。
该浴缸的设计是在这样一种方式，当浴缸达到容量，多余的水通过溢流泄流。
在空间和时间上开发一个浴缸的水的温度模型，以确定最佳的策略，在浴缸的人可以采取保持温度，即使在整个浴缸和尽可能接近的初始温度，没有浪费太多的水。
使用你的模型来确定你的策略取决于浴缸的形状和体积，浴缸的形状/体积/温度，浴缸中的人的运动。
如果这个人用了一个泡泡浴剂，而最初填充浴缸，以协助清洗，这会影响你的模型的结果？除了要求的一页摘要MCM提交，你的报告必须包括一一页的非技术性解释的浴缸，描述你的策略，解释为什么它是如此难以在洗澡水温度得到均匀地保持用户Program B小碎片在轨道上绕地球金额已日益受到关注。
据估计，超过50万件的空间碎片，也被称为轨道碎片，目前都正在跟踪的潜在危害飞船。
这个问题本身在新闻媒体上变得更广泛的讨论时，俄罗斯卫星的Kosmos-2251和美国铱卫星-33 2009年2月10日，上相撞。
已经提出许多方法以除去碎屑。
这些方法包括小的，基于空间的水射流，并用于针对碎片的特定部分高能激光器和大型卫星，旨在清扫杂物，等等。
碎片的大小和质量范围从漆片的废弃卫星。
碎片“高速轨道捕获做出困难。
开发时间依赖模型来确定一个私人公司可以采取作为一个商业机会，以解决空间碎片问题的替代品的最佳替代品或组合。
您的模型应该包括成本，风险，收益定量和/或定性的估计，以及其他的重要因素。
您的模型应该能够评估独立的替代方案以及替代品的组合，并能够探索各种重要的“如果什么？”的情景。
使用你的模型，确定经济上有吸引力的机会是否存在没有这样的机会是可能的。
如果可行的商业机会的存在作为替代的解决方案，提供了用于去除碎屑的不同选项的比较，并包括特定建议作为对碎片应如何除去。
年美赛C题翻译
年美赛C题翻译One of the techniques to determine influence of academic research is to build and measure properties of citation or co-author networks.学术研究的技术来确定影响之一是构建和引文或合著网络的度量属性。
Co-authoring a manuscript usually connotes a strong influential connection between researchers.与人合写一手稿通常意味着一个强大的影响力的研究人员之间的联系。
One of the most famous academic co-authors was the 20th-century mathematician Paul Erd?s who had over 500 co-authors and published over 1400 technical research papers.最著名的学术合作者是20世纪的数学家保罗鄂尔多斯曾超过500的合作者和超过1400个技术研究论文发表。
It is ironic, or perhaps not, that Erd?s is also one of the influencers in buildingthe foundation for the emerging interdisciplinary science of networks,particularly, through his publication with Alfred Rényi of the paper “On Random Graphs”in 1959.讽刺的是,或者不是,鄂尔多斯也是影响者在构建网络的新兴交叉学科的基础科学,特别是通过与阿尔弗雷德Renyi的出版论文的“随机”在1959年。
全国大学生英语竞赛C类(2012真题以及答案)剖析
2012 National English Contest forCollege Students(Level C – Preliminary)(总分：150分时间：120分钟)Part I listening Comprehension (30 marks)Section A (5 marks)In this section, you will hear five short conversations. Each conversation will be read only once .After each conversation, there will be a twenty-second pause. During the pause, read the question and the three choices marked A, B and C, and decide which is the best answer .Then mark the corresponding letter on the answer sheet with a single line through the centre.1.What does the man say we can do to deal with oil crisis?A. To make full use of oil.B. To use aslittle oil as possibleC. To find alternative energy.2. Where does this conversation most probably take place?A. In an insurance company.B. In a bank.C. In a supermarket.3. According to the man, who is going to take over the position they are talking about?A. Janice.B. Someone else.C. Meryl.4. What does the woman say about the man’s report for the meeting?A. He has to get it ready before tomorrow noon.B. He has done well enough.C. He has enough time to prepare it.5. How many people will be on the earth by the year 2020 according to the professor?A. Some 5.8 billion.B. Nearly 7 billion.C. Over 8.5 billion.Section B (10 marks)In this section, you will hear two long conversations. Each conversation will be readonly once. At the end of each conversation, there will be a one-minute pause. During the pause, read the questions and the three choices marked A, Band C, and decide which is the best answer. Then mark the corresponding letter on the answer sheet with a single line through the centreConversation One6. What’s the main job of Simon’s organization?A. They send out radio signals to communicate with other planets.B. They look for life and intelligence on other planets.C. They study stars that have planets orbiting around them.7. Why does the organization search for radio signals from space?A. Their presence may prove the existence of aliens.B. They may help scientists find out how the universe started.C. They convey messages about life on the earth.8. Does Simon believe those stories about aliens visiting our planet?A. No, he doesn’t believe them at all.B. Yes, he does believe those stories because of his study.C. Yes, he believes them although he has no evidence.9. Which of the following factor is considered to be important in forming life according to the passage?A. Gravity.B. Minerals.C. Water.10. When did the “big bang” occur?A. No one knows.B. 12 billion years ago.C.20 billion year ago.Conversation Two11. Why did Bob make the news last month?A. Because he criticized traditional jobsB. Because he earned a lot of money.C. Because he tried new strategies in finding a job.12. In which way do the bestsellers like “The Brand Called You” and “The PersonalBranding Phenomenon” help people?A. They advise people to promote themselves as brands.B. They teach people how to be a perfect partner.C. They give people tips for job interview.13. When did Bob come up with the idea of using the internet to find a job?A. Ten years ago when he was a college student.B. When he began to do research between jobs.C. After he launched the campaign “Give Bob a Job”.14. What was the purpose of Bob’s making the video?A. To fulfill his ambition as a director.B. To sell his products like Teddy bears and T-shirts.C. To advertise his skills and talents.15. How can the internet help Bob in his job search exactly?A. By helping him make friends.B. By passing on the video he made.C. By providing job information for him.Section C (5 marks)In this section, you will hear five short news items. After each item, which will be read only once, there will be a pause. During the pause, read the question and the three choices marked A, B and C, and decide which is the best answer. Then mark the corresponding letter on the answer sheet with a single line through the centre.16. How many troops are going to withdraw from Afghanistan by the end of next year?A. 43,000.B. 10,000.C. 33,000.17. What is the finding of the study?A. Low-calorie foods may lead to more weight.B. High-calorie foods may lead to more weight.C. High-calorie foods may lead to less weight.18. What do the conflicting reports show about radiation?A. The radiation level given by Tokyo Electric is much smaller than that given byother tests.B. The radiation level given by other tests is smaller than that given by TokyoElectric.C. The radiation level is close to 10,000 times the normal level.19. Why are Caribbean Service and Europe Today shut down by BBC?A. Because of their outdated services.B. Because of cuts in government funding.C. Because of lack of target audience.20. What’s the cause of demonstrations in cities and towns across Syria?A. They are calling for more freedom.B. Workers demand higher pay.C. Some thirty people were killed by the police.Section D (10 marks)In this section, you will hear a short passage. There are 10 missing words or phrases. Fill in the blanks with the exact words or phrases you hear. The passage will be read twice. Remember to write the answers on the answer sheet.Recovery from jetlag can take as long as a day for every time zone crossed. So if you’re flying east-west for your holiday or on business, it is likely to mean a few days of feeling tired or even unwell.Jetlag is the (21)__________ of the body’s natural cycle and some factors make it worse. The artificial atmosphere inside a plane can affect the body in a number of ways and add to the effects of jetlag.Exercising before flights helps to offset the effects on the body of reduced (22) _________, and aerobic exercise afterwards helps to reoxygenate it. Drink plenty of water. Children may need more. Drinking carrot juice before flying overcomes oxygen deficiency.Eat light, frequent meals. Heavy meals (23) __________ the blood circulation, which can lead to dizziness and fainting. For two weeks before you fly, eat plenty of foodcontaining vitamins A and E; they will build up your (24) __________ and help to keep you fit. Sleep as much as you can before the flight and on it. On board, wear earplugs and eye pads because darkness (25) __________ secretion of the hormones that enhance sleep.Avoid alcohol, which restricts the brain’s oxygen intake. Like tea and coffee, alcohol increases the dehydration effect of flying. If you need to drink to relax, (26) __________ that the cabin environment intensifies the effect of alcohol.Wear loose, comfortable, warm clothes and limber up during flight. Sitting down for several hours slows down the (27) _________, leading to local stiffness, cramps and dizziness. Re-adjust to local time as soon as you can. Bright light helps the body stay alert, so if you are going somewhere sunny, stay outside.Do not smoke before or during the flight since smoking (28) ___________ the blood oxygen level. If you need to calm your nerves, try aromatherapy oils which have a sedative effect. (29) _________ your doctor before flying if you are on medication. The effects of some drugs are strengthened at high altitudes and some may produce (30) _________.Part II Vocabulary and Structure (15 marks)There are 15 incomplete sentences in this section. For each blank there are four choices marked A, B, C and D. Choose the one that best completes the sentence. Then mark the corresponding letter on the answer sheet with a single line through the centre.31. What we all work for is to free ________ time for the things we really want to do.A. offB. upC. asideD. in32. You _________ his words seriously. He was talking nonsense.A. won’t takeB. may not takeC. mustn’t have takenD. needn’t have taken33. Never hesitate to _________ the first opportunity that comes along.A. seizeB. drawC. who; thatD. obtain34. His mother ________ hated city life longed to return to the village in _______shegrew up.A. that; whereB. who; whichC. who; thatD. who; where35. All things _______, I think I’d better take your advice.A. consideringB. to be consideredC. consideredD. have been considered36. Twelve European countries ________ over to the Euro on January 1st, 2002.A. transformedB. switchedC. reversedD. altered37. She isn’t anything ________ unpleasant as people say she is.A. as likeB. likeC. asD. like as38. Stop complaining. You really ________ my nerves.A. get downB. get alongC. get offD. get on39. There wasn’t _______ truth in what he said.A. a grain ofB. a ray ofC. a point ofD. a drop of40. ________ for my savings, I wouldn’t be able to survive these miserable days.A. Was it notB. Were it notC. Had it been notD. Hadn’t it been41. There was a power cut this morning. ________, I couldn’t do anything with mycomputer.A. SupposedlyB. PresumablyC. ConsequentlyD. Essentially42. Alarm clocks needlessly wake ______ households. I want to design somethingtargeted at the individual sleeper.A. wholeB. totalC. fullD. high43. _______ both parties agree on these issues will a contract be signed soon.A. If onlyB. UnlessC. ShouldD. Only if44.—I’ve run out of cash. Could you lend me a few pounds this evening?—_____ I’ll just have to find time to get to the bank and make a withdrawal.A.I can lend you some now.B. I’m not sure I’ve got any either.C. Sorry, I haven’t got a penny in my account.D. My credit card must be left home.45. —We’re all going to the New Skyline Restaurant for our end-of-year get-together.Can we count you in?—Oh, thanks, Sara, but _____A. I didn’t find you then.B. I’ve been to the restaurant once.C. I’ll pass this time.D. I’ll meet you there at 5:00.Part III Cloze (10 marks)Read the following passage and fill in each blank with one word. Choose the correct word in one of the following three ways: according to the context, by using the correct form of the given word, or by using the given letters of the word. Remember to write the answers on the answer sheet.Did you know that for every 20,000 novels written, only one gets published? So the (46) ____ (likely) that I’ll ever fulfill my ambition of becoming a professional mystery writer doesn’t seem very high. But the prospect of turning my lifelong passion into my livelihood and achieving fame and (47) for ______ at the same time is just too exciting for me to be put off by dull statistics.So what does it (48) t_______ to become a writer? Reading is important—all writers need to research their genre thoroughly to familiarize themselves (49) _______ its codes and conventions. My bookshelves at home are stacked with the novels of all the great mystery writers, which I’ve read and in many cases, re-read, despite (50) _______ (know) all the time “who’d done it”.Of course, being a writer requires imagination. You have to develop your own personal style rather than simply copy the work of “the greats”. I’ve turned out (51) d________ of short crime stories for my university student magazine—some havebeen published, some not, but I’ve always aimed to produce original and imaginative material.Last but not (52) ________, successful writers possess enormous self-discipline. I’ve often sacrificed my social life in order to devote the necessary time and effort to producing a good quality story. And more than once that has meant (53) _______ the midnight oil.We’ve got a novel inside us. Getting it out in anywhere near publishable form is no (54) ______ task, but with imagination and determination, and the help of an expert on “master class”, who could possibly fail to (55) re_______ their ambition?Part IV Reading Comprehension (40 marks)Read the following passages. Each passage is followed by several questions. Respond to the questions using information from the passage. Remember to write the answers on the answer sheet.Section A (10 marks)Questions 56—60 are based on the following passage.The term culture shock was introduced forthe first time in 1958 to describe the anxietyproduced when a person moves to a completelynew environment. This term expresses thefeeling of not knowing what to do or how to dothings in a new environment. This termexpresses the feeling of not knowing what to do or how to do things in a new environment. We can describe culture shock as the physical and emotional discomfort a person suffers when coming to live in a place different from the place of origin.Often, the way that we live before is not accepted or considered as normal in the new place. Everything is different, for example, not speaking the language, not knowing how to use banking machines, not knowing how to use telephones, and so forth.The symptoms of culture shock can appear at different times. Although a person can experience real pain from culture shock, it is also an opportunity for growing andlearning new perspectives. Culture shock can help people develop a better understanding of themselves and stimulate personal creativity.Culture shock has many stages. Each stage can be ongoing, or appear only at certain times. In the first stage, the new arrival may feel very happy about all of the new experience. This time is called the honeymoon stage. Afterward, the second stage presents itself. A person may encounter some difficulties in daily life. For example, communication difficulties may occur, such as not being understood. In this stage, there may be feelings of impatience, anger and sadness. Transition between the old methods and those of the new country is a difficult process and takes time to complete. The third stage is characterized by gaining some understanding of the new culture. A new feeling of pleasure and sense of humor may be experienced. A person may start to feel a psychological balance. The individual is more familiar with the environment and wants to belong. This starts an evaluation of the old ways versus those of the new. In the fourth stage, the person realizes that the new culture has good and bad things to offer. This stage can be one of double integration or triple integration, depending on the number of cultures that the person has to process. This integration is characterized by a more solid feeling of belonging. The person starts to define himself or herself and establish goals for living. The fifth stage is the stage that is called the reentry shock. This occurs when a person returns to the newly acquired customs are considered improper in the old culture.Questions 56—60Complete the summary with words from the passage, changing the form where necessary, with only one word for each blank.Introduced in 1958, culture shock is a term used to (56) __________ physical and emotional discomfort people experience when they come to or live in a new place. Although many people suffer a lot from culture shock, it is an (57) _________ for personal improvement. The five stages of culture shock may last for a long time or show up once in a while. In the first stage, the (58) __________ stage, the new arrivalmight be OK with everything. Afterward, (59) ________ from old methods to the new one calls for time and energy. During the third and fourth stage, people may gain more understanding of the new culture and possess objective views. The double or triple integration is (60) __________ by a feeling of belonging. The last stage, called reentry shock, refers to the feeling people have when returning to their home country.Section B (10 marks)Questions 61-65 are based on the following passage.For some people, it would be unthinkable. But Gabe Henderson is finding freedom in a recent decision; he canceled his MySpace account. The 26 years old graduate student stopped his account after realizing that a lot of his online friends were really just acquaintances. “The superficial emptiness clouded the excitement I had once felt,”Henderson wrote in an article. “It seems we have lost, to some degree, the special depth that true friendship is about.”Journalism professor Michael Bugeja, who is a strong supporter of face-to-face communication, read Henderson’s column and saw it was a sign of hope. Though he’s not anti-technology, Bugeja often lectures students about “interpersonal intelligence”—knowing when, where, and for what purpose technology is most appropriate. He points out the students he’s seen walking across campus, holding hands with each other while talking on cell phones to someone else He’s also seen them in coffee shops, surrounded by people, but staring instead at a computer screen. “True friends,”he says, “need to learn when to stop blogging and go across campus to help a friend.”These days, young people are more wired than ever—but they’re also getting more worried. Increasingly, they’ve had to deal with online bullies, who are posting anything from embarrassing photos to online threats. And increasingly, young people also are realizing that things they post on their profiles can come back to hurt them when applying for schools or jobs.Social networking can be an “extremely effective”way to publicize events tolarge groups. It can even help build a sense of community on campus. People joined Facebook as a way to meet others. However, it has limitations. A good Internet profile could make even the most boring person seem some what interesting. People are also not always happy with text messages on the cell phones. Cell phones can be a quick way to say “have a good day”. But friends can also cancel a night out with a text message to avoid having to explain. “Our generation needs to get over this fear of confrontation and rejection.”Henderson says. “The focus needs to be on quality communication, in all ways.” Back in his life, Henderson is enjoying spending more face-to-face time with his friends and less with his computer. He says his decision to quit his social-networking Internet accounts was a good one. “I’m not sacrificing friends,” he says, “because if a picture, some basic information about their life and a Web page is all my friendship has become, then there was nothing to sacrifice to begin with.”Questions 61—63Decide whether the following statements are true (T) or false (F) according to the passage.61. Although Henderson quit MySpace, most of his online friends are actually close friends.62. According to Bugeja, young people should know when to use technology and when to stop using it.63. Social networking helps build a sense of community but has a negative effect on young people’s life if it is used inappropriately.Questions 64—65Answer the following questions briefly according to the passage.64. What does Henderson lose by using social networking websites like Myspace?65. What do young people need to overcome in order to focus on quality communication?Section C (10 marks)Questions 66—70 are based on the following passage.“Five …Four …Three …Two …One …See ya! ”andChance McGuire, twenty-five, isairborne off a 600 –foot concretedam in Northern California. In onesecond he falls 15 feet, in twoseconds 60 feet, and after threeseconds and 130 feet, he is flying at66 miles an hour.McGuire is a practitioner ofwhat he calls the king of all extreme sports. BASE—an acronym for building, antenna, span (bridge) and earth (cliffs)—jumping has one of the sporting world’s highest fatality rates: in its 18-year history, forty –six participants have been killed. Yet the sport has never been more popular, with more than a thousand jumpers in the United States, and more seeking to get into it every day. It is an activity without margin for error. If your chute malfunctions, don’t bother researching for a reserve—there isn’t time. There are no second chances.Still, the sport may be a perfect fit with the times. Americans may have more in common with McGuire than they know or care to admit. America has embarked on a national orgy of thrill seeking and risk taking. The rise of adventure and extreme sports such as BASE jumping, snowboarding, ice climbing, skateboarding, and paragliding is merely the most vivid manifestation of this new national behavior.The rising popularity of extreme sports speaks of an eagerness on the part of millions of Americans to participate in activities closer to the edge, where danger, skill, and fear combine to give weekend warriors and professional athletes alike a sense of pushing out personal boundaries. According to American Sports Data Inc., a consulting firm, participation in so-called extreme sports is way up. Snowboarding has grown 113 percent in five years and now boasts nearly 5.5 million participants. Mountain hiking, skateboarding, scuba diving—their growth curves reveal a nation that loves to play with danger. Contrast that with activities such as baseball, touch football, and aerobics, all of which were in steady decline throughout the 1990s.The pursuits that are becoming more popular have one thing in common: the perception that they are somehow more challenging than a game of touch football.“Every human being with two legs and two arms is going to wonder how fast, how strong, how enduring he or she is,” says Eric Perlman, a mountaineer and film maker specializing in extreme sports. “We are designed to experience or die.”Questions 66—68Complete the following sentences with information given in the passage in a maximum of 4 words for each blank.66. _______ rates doesn’t stop people from getting into BASE jumping every day.67. The rise of extreme sports manifests the national behavior of _______ and ________.68. The combination of fear, skill and danger gives both amateurs and professionals asense of ______.Questions 69—70Choose the best answer according to the passage.69. Which of the following activities reveals a nation that loves to play activities closer to the edge?A. Touch football.B. Baseball.C. Scuba diving.D. Aerobics.70. What does Eric mean by saying “We are designed to experience or die”?A. Life with great challenges is a meaningful one.B. Life without great experiences is very common.C. People may die while doing extreme sports.D. Extreme sports are essential parts of human life.Section D (10 marks)Questions 71—75 are based on the following passage.Albert Einstein was one of the greatest thinkers the world has ever known. Heformulated theories of relativity, successfully described the nature of the universe and came up with the most famous equation in the world. David Beckham is the footballer whose skills and precision have made him one of the most gifted sportsmen of his generation. Who is more intelligent?How Gardner’s theory of multiple intelligences (MI) dares us to put these two men on neighboring pedestals. Instead of regarding intelligence as a single quantity measurable by pen-and-paper tests, Gardner, an education professor at Harvard University, divides human intelligence into no fewer than eight separate categories ranging from mathematical to musical competence. (74)His ideas have provoked vigorous debate about how one defines intelligence. Gardner’s point is that quantity measures only one capacity, the sort of mental agility that is valued in academic achievement, and that this single number does not do justice to human potential. So he has created his own spheres of achievement. Some categories are easily reconcilable with general perceptions about IQ. For example, “linguistic”intelligence confers a mastery of language, and is the preserve of such people as poets, writers and linguists. “Logical mathematical” intelligence marks out people who take a reasoning approach to physical things, and seek underlying principles. Einstein is the standard –bearer of this group, which also includes philosophers. These two categories are the main components of what we generally think of as “intelligence”.“Musical”intelligence characterizes musicians, composers and conductors. “Spatial” intelligence is about being able to picture perspective, to visualize a world in one’s head with great accuracy. Chess players, artists and architects would rate highly in this category. Dancers, athletes and actors are lumped under the “bodily-kinesthetic” heading; these individual, like Beckham, are able to control their bodies and movements very carefully.Then come two types of “personal”intelligence-intrapersonal, the ability to gauge one’s own mood, feelings and mental states, and interpersonal, being able to gauge it in others and use the information. These two categories could be interpreted as emotional intelligence. Psychiatrists are particularly adept at the former, whilereligious leaders and politicians are seen as people who can exploit the latter.Charles Darwin is perhaps the perfect embodiment of the eighth intelligence –“naturalist”. This label describes people with a deep understanding of the natural world and its objects. Zoologists and botanists can count themselves among this group.(75)These eight categories certainly reflect the fact that, in these areas, there is a spectrum of human ability ranging from the hopeless to the brilliant. But are these really intelligences, or could these competences be more accurately described as gifts or talents?Questions 71—73Answer the following questions briefly according to the passage.71. What are the main components of “intelligence” we generally think of?72. What kind of abilities does emotional intelligence involve?73. Which intelligences are represented by Einstein and Beckham respectively? Questions 74—75Translate the underlined sentences in the passage into Chinese.74. His ideas have provoked vigorous debate about how one defines intelligence.75. These eight categories certainly reflect the fact that, in these areas, there is a spectrum of human ability ranging from the hopeless to the brilliant.Part V Translation (10 marks)Translate the following sentences into English, using the hints given in brackets. Remember to write the answer on the answer sheet.76. 既然没有退路了，我们不妨试试他的方法。
2012美赛A题分析
1.单纯的称量叶子的质量是没有研究价值的，因为不同的植株，不同的叶子的其质量不会相同，这样不同植被的数据无任何联系，与后面的分类要求矛盾。
2.考虑到植物的呼吸作用与光合作用，可以从此处下手：
利用有关实验仪器（空气中氧气，二氧化碳的浓度测量以及湿度的测量），建立数学模型，分析植株一昼夜的呼吸量与光合作用量（生物书上有详细介绍），这是完全可以实现的。其呼吸作用量，为植株的不同部分提供能量（可以考虑不同的部分建立不同的权重），在这里不但有叶子自身的需要，还有通过树干的韧皮部运输到植被其他地方以作用，用作诸如树干的形成层分生细胞以及其他作用（好好研究树干的作用）。而光合作用于呼吸作用的差值便是植株一昼夜的糖分积累，则积累到植株的各个部分（叶子、树干、根等）。
5.关于叶子的分类，也是建立在地理环境对植株生理作用的影响，虽然是建立在叶子质量上的分类，我的理解是对有机物积累能力的分类：可以考虑在主体上按热带、亚热带、温带、寒带等划分（叶的形状），在同一纬度上，通过其积累有机物能力划分（质量的不同）。
6.关于“最小面积的”的遮蔽以及“最大范围的扩展”，可以从不同的地理环境对植物的光合作用、呼吸作用以及蒸腾作用的角度分析。
注:之所以落叶是因为在秋末时，树停止吸收水分无机物等，以减少呼吸、蒸腾作用，3.叶子的输送量与树干的韧皮部等密切相关，这就跟我们的网速一样，另一个方面，也就是说叶子的质量与树干密切相关，可以找一下有关文献。
4.据此，想一下，如果把一天内植株的能量积累看做A，叶子积累b,树干积累c，其他组织积累d，公式为A=b+c+d，在一年的不同时期有不同的权重，然后 ，就可以得到叶子以及其他组织在不同时期重量的范围值。（还可以更忽悠一些）
7.植株的叶的形状与树干的结构也是从最大光合作用的角度考虑。
2012数学建模美国赛B题参考答案
PROBLEM B: Camping along the Big Long RiverVisitors 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 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.Nowadays the heavy metal pollution is so common that people pay more and more attention to it. The aim of this paper is to calculate the maximum of methylmercury in human body during their lifetime and the maximum number of fish the average adult can safely eat per month. From City Officials research[1], we get information that the mean value of methylmercury in bass samples of the Neversink Reservoir is 1300 ug/kg and the average weight of bass people consume per month is 0.7 kg. According to the different consuming time in every month, we construct a discrete dynamical system model for the amount of methylmercury that will be bioaccumulated in the average adult body. In ideal conditions, we assume people consume bass at fixed term per month. Based on it, we construct fixed-ingestion model and we reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3505 ug. As methylmercury ingested is not only coming from bass but also from other food, hence, we make further revise to our model so that the model is closer to the actual situation.As a result, we figure out the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3679 ug. As a matter of fact, although we assume people consume one fish per month, the consuming time has great randomness. Taking the randomness into consideration, we construct a random-ingestion model at the basis of the first model. Through computer simulations, we obtain the maximum of methylmercury in human body is 4261 ug. We also calculate the maximum amount is 4420 ug after random-ingestion model is revised. As it is known to us, different countries and districts have different criterions for mercury toxicity. In our case, we adopt LD50 as the toxic criterions(LD50 is the dosage at which 50% of the humans exposed to a particular chemical will die. The LD50 for methylmercury is 50 mg/kg.). We speculate mercury toxicity has effect on the ability of eliminating mercury, therefore, we set up variable-elimination model at the basis of the first model. According to the first model, the amount of methylmercury in human body is 50 ug/kg, far less than 50 mg/kg, so we reach the conclusion that the fish consumption restrictions put forward by the reservoir advisories can protect the average adult. If the amount of methylmercury ingested increases, the amount of bioaccumulation will go up correspondingly. If 50 mg/kg is the maximum amount of methylmercury in human body, we can obtain the maximum number of fish that people consume safely per month is 997.Keywords: methylmercury discrete dynamical system model variable-elimination modeldiscrete uniform random distribution model random-ingestion modelIntroductionWith the development of industry, the degree of environmental pollution is also increasing. Human activities are responsible for most of the mercury emitted into the environment. Mercury, a byproduct of coal, comes from acid rain from the smokestack emissions of old, coal-fired power plants in the Midwest and South. Its particles rise on the smokestack plumes and hitch a ride on prevailing winds, whichoften blow northeast. After colliding with the Catskill mountain range, the particles drop to the earth. Once in the ecosystem, micro-organisms in the soil and reservoir sediment break down the mercury and produce a very toxic chemical form known as methylmercury . It has great effect on human health.Public officials are worried about the elevated levels of toxic mercury pollution in reservoirs providing drinking water to the New Y ork City . They have asked for our assistance in analyzing the severity of the problem. As a result of the bioaccumulation of methylmercury , if the reservoir is polluted, we can make sure that the amount of methylmercury in fish is also increasing. If each person adheres to the fish consumption restrictions as published in the Neversink Reservoir advisory and consumes no more than one fish per month, through analyzing, we construct a discrete dynamical system model of time for the amount of methylmercury that will bioaccumulate in the average adult person. Then we can obtain the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime. At the same time, we can also get the time that people have taken to achieve the maximum amount of methylmercury . As we know, different countries and districts have different criterions for the mercury toxicity . In our case, we adopt the criterion of Keller Army Community Hospital. If the maximum amount of methylmercury in human body is far less than the safe criterion, we can reach the conclusion that the reservoir is not polluted by mercury or the polluted degree is very low, otherwise we can say the reservoir is great polluted by mercury . Finally, the degree of pollution is determined by the amount of methylmercury in human body .Problem Onediscrete dynamical system modelThe mean value of methylmercury in bass samples of the Neversink Reservoir is 1300 ug/kg and the average weight of bass is 0.7 kg. According to the subject, people consume no more than one fish per month. For the safety of people, we must consider the bioaccumulation of methylmercury under the worst condition that people absorb the maximum amount of methylmercury . Therefore, we assume that people consume one fish per month.Assumptions● The amount of methylmercury in fish is absorbed completely and instantly bypeople.● The elimination of mercury is proportional to the amount remaining.● People absorb fixed amount of methylmercury at fixed term per month. ● We assume the half-life of methylmercury in human body is 69.3 days. SolutionsLet 1α denote the proportion of eliminating methylmercury per month, 1β denote the accumulation proportion. As we know, methylmercury decays about 50 percent every 65 to 75 days, if no further methylmercury is ingested during that time. Consequently ,111,βα=-69.3/3010.5.β=Through calculating, we get10.7408.β=L et’s define the following variables ：ω denotes the amount of methylmercury at initial time, n denotes the number of month,n ω denotes the amount of methylmercury in human body at the moment people have just ingested the methylmercury in the month n ,1x denotes the amount of methylmercury that people ingest per month and113000.7910x ug ug=⨯=. Moreover, we assume0=0.ωThough,111,n n x ωωβ-=⋅+we get1011x ωωβ=⋅+2201111x x ωωββ=⋅+⋅+ ⋅⋅⋅10111111n n n x x x ωωβββ-=⋅+⋅+⋅⋅⋅+⋅+121111(1)n n n x ωβββ--=++⋅⋅⋅++⋅11111.1n n x βωβ--=-With the remaining amount of methylmercury increasing, the elimination of methylmercury is also going up. We know the amount of ingested methylmercury per mouth is a constant. Therefore, with time going by, there will be a balance between absorption and elimination. We can obtain the steady-state value of remaining methylmercury as n approaches infinity.1*1111111lim 3505.11n n n x x ug βωββ-→∞-===--The value of n ω is shown by figure 1.Figure 1. merthylmercury completely coming from fish and ingested at fixed term per monthIf the difference of the remaining methylmercury between the month n and 1n - is less than five percent of the amount of methylmercury that people ingest per month, that is,115%.n n x ωω--<⋅Then we can get11=3380ug.ωAt the same time, we can work out the time that people have taken to achieve 3380 ug is 11 months.From our model, we reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3505 ug.If people ingest methylmercury every half of a month, however, the sum of methylmercury ingested per month is constant, consequently,11910405,0.86.2x ug β===As a result, we obtain the maximun amount of methylmercury in human body is 3270ug. When the difference is within 5 %, we get the time people have taken to achieve it is 11 months.Similarly, if people ingest methylmercury per day , we get the maximum amount is 3050ug, and the time is 10 months.Revising ModelAs a matter of fact, the amount of methylmercury in human body is not completely coming from fish. According to the research of Hong Kong SAR Food andEnvironmental Hygiene Department [1], under normal condition, about 76 percent of methylmercury comes from fish and 24 percent comes from other seafood. In order to make our model more and more in line with the actual situation, it is necessary for us to revise it. The U.S. environmental Protection Agency (USEPA) set the safe monthly dose for methylmercury at 3 microgram per kilogram (ug/kg) of body weight. If we adopt USEPA criterion, we can calculate the amount of methylmercury that the average adult ingest from seafood is 50.4 ug per month.Assumptions● The amount of methylmercury in the seafood is absorbed completely andinstantly by people.● The elimination of methylmercury is proportional to the amount remaining. ● People ingest fixed amount of methylmercury from other seafood every day . ● We assume the half-life of methylmercury in human body is 69.3 days. SolutionsLet 0ωdenote the amount of methylmercury at initial time, t denote the number of days, t ω denote the remaining amount on the day t , and 2x denote theamount of methylmercury that people ingest per day . Moreover, we assume0=0.ωIn addition, we work out2x =50.4/30=1.68 ug.The proportion of remaining methylmercury each day is 2β, then69.320.5.β=Through calculating, we get20.99.β=Because of12221,1t t x βωβ--=- we obtain steady-state value of methylmercury1*2222211lim 168.11t t t x x ug βωββ-→∞-===--If the difference of remaining methylmercury between the day t and 1t - is less than five percent of the amount of methylmercury that people ingest every day, that is,125%.t t x ωω--<⋅We have301= 160 ug.ωSo we can reach the conclusion that the maximum amount of methylmercury the average adult human will bioaccumulate from seafood is 160 ug and the time that people take to achieve the maximum is 301 days.Let 1x denote the amount of methylmercury people ingest through bass at fixedterm per month, so the amount of methylmercury an average adult accumulate on the day t is1221221if t is a positive integer and not divisible by 30if t is a positive integer and divisible by 30.t t t t x x x ωωβωωβ--=⋅+⎧⎨=⋅++⎩ The value of t ωis shown by figure 2.Figure 2. merthylmercury coming from fish and other seafood and ingested at fixed term per dayThe change of t ω reflects the change of the amount of methylmercury inhuman body . Through revising model, we can figure out the maximum amount of methylmercury the average adult human will bioaccumulate in their lifetime is 3679 ug.Problem TwoRandom-ingestion modelAlthough people consume one fish per month, the consuming time has great randomness. We speculate the randomness has effect on the bioaccumulation of methylmercury , therefore, we construct a new model.AssumptionsThe amount of methylmercury in fish is absorbed completely and instantly bypeople.● The elimination of methylmercury is proportional to the amount remaining. ● People consume one fish per month, but the consuming time has randomness. ● We assume the half-life of methylmercury in human body is 69.3 days.Let 0L denote the amount of methylmercury at initial time, n L denote theamount of methylmercury at the moment people have just ingested methylmercury in the month n , and x denote the amount of methylmercury that people absorb each time.We assume0=0.LWe have910.x ug =We define 1β the proportion of remaining methylmercury every day .Through69.310.5,β=we can get10.99.β=Let i obey discrete uniform random distribution with maximum 30 and minimum 1 and n t denote the number of days between the day1n i - of the month 1n - and the day n i of the month n , then we have-130-,n n n t i i =+(1)1.n tn n L L x β-=⋅+ The value of n L is shown by figure 3.Figure 3. merthylmercury completely coming from fish and ingested at random per monthFigure 3 shows the amount of methylmercury in human body has a great change due to the randomness of consuming time. Through the computer simulation, if we have numberless samples, n L will achieve the maximum value.That is,4261.n L ug =Revising modelIn order to make our model more accurate, we need to make further revise. We take methylmercury coming from other seafood into consideration. We know the amount of methylmercury that people ingest from other seafood every day is 1.68 ug.In that situation, we have1212.30(-1)30(-1)n n n n n n L L x if n n i L L x x if n n i ββ=⋅+≠⨯+⎧⎨=⋅++=⨯+⎩ Through the computer simulation, we can get a set of data about n L shown by figure4.Figure 4. remaining merthylmercury coming from fish consumed at random per month and otherfood consumed at fixed term per dayThough the revised model, we reach the conclusion that if we have numberless samples, n L will achieve the maximum value.That is,4420.n L ug =Variable-eliminateion modelAs a matter of fact, the state of human health can affect metabolice rate so that the ability of eliminating methylmercury is not constant. We have koown the amount of methylmercury in human body will affect human health. So we can draw the conclusion that the amount of methylmercury in human body will affect the abilitity of eliminating methylmercury .Assumptions● The amount of methylmercury in fish is absorbed completely and instantly bypeople.● the elimination of methylmercury is not only proportional to the amountremaining, but also affected by the change of human health which are caused by the amount of methylmercury .● People absorb fixed amount of methylmercury at fixed term per month throughconsuming bass.● We assume the half-life of methylmercury in human body is 69.3 days.● In condition that no further methylmercury is ingested during a period of time, welet χ denote the eliminating proportion per month. We have known methylmercury decays about 50 percent every other day 5 to a turn 5 days, so we determine the half-life of methylmercury in human body is 69.3 days. Then we have69.3/301(1)0.5χ⋅-=. By calculating, we getχ=0.2592.We adopt LD50 as the toxic criterions, then we get the maximum amount ofmethylmercury in human body is 63.510⨯ ug. L et’s define the following variables ：ω denotes the amount of methylmercury at initial time,ndenotes the number of month,nω denotes the amount of methylmercury in human body at the moment people have just ingested the methylmercury in the monthn,n χ denotes the ability of eliminating methylmercury in the month n . γ denotes the effect on human health caused by methylmercury toxicity .1161 3.510r n n ωχχ-⎛⎫⎡⎤=⋅- ⎪⎢⎥ ⎪⨯⎣⎦⎝⎭1(1)n n n ωωχϕ-=⋅-+Hence, we have101(1)ωωχϕ=⋅-+20212(1)(1)(1)ωωχχϕχϕ=⋅-⋅-+⋅-+[]01233(1)...(1)(1)(1)...(1)(1)...(1)...(1)1n n n n n ωωχχϕχχχχχχ=⋅--+⋅-⋅--+--++-+We define the value of γ is 0.5, then we get the maximum amount of maximum in human body is 3567 ug, that is,*=3567 ug n ωNot taking the effect on the ability of eliminating maximum caused by methylmercury toxicity into account in model one,we obtain the maximum amount is 3510 ug. The difffference proves methylmercury toxicity has effect on eliminating methylmercury . We find out through calculating when r increases, the amount of methylmercury go up correspondingly. The reason for it is that methylmercurytoxicity rises as a result of r increasing. Correspondingly, the effect on human health will increase, which is in accordance with fact.Problem ThreeAccording to the first model revised, we can get the maximum amount of bioaccumulation methylmercury is 3679 ug. We assume the average weight of an adult is 70 kg and the amount of methylmercury in human body is 53 ug/kg, far less than 50 mg/kg. Therefore, according to our model, the fish consumption restrictions put forward by the reservoir advisories can protect the average adult from reaching the LD50(LD50 is the dosage at which 50% of the humans exposed to a particular chemical will die. The LD50 for methylmercury is 50 mg/kg).We assume the lethal dosage of methylmercury is not gradually increasing. If the amount of methylmercury people ingests goes up rapidly , the bioaccumulation amount will reach to a higher value. Moreover, the value probably endangers human safety . Let LD50 be the maximum amount of methylmercury in human body , that is,*n =50 m g/kg 70 kg=3500 m g.ω⨯Let 1x denote the amount of methylmercury people ingest per month. According to the first model,1*1111111lim.11n n n x x βωββ-→∞-==--We can figure out1 x =907.2 m g.We know the mean value of methylmercury in bass samples is 1.3 mg/kg, hence, we can obtain the maximum amount of fish that people consume safely per month is1m ax 698.1.3x M kg =≈The maximum number of fish is 698/0.7=997.ConclusionIn problem one, the paper calculates the final steady-state value at the same time interval per month, per half a month and per day . Through comparing the results, we get the final bioaccumulation amount of methylmercury is less, when discrete time unit is smaller. It shows when the interval of consuming fish is smaller and the sum of methylmercury ingested is constant for a period of time, the possibility of poisoning is lower.In problem two, we analyze the change of the amount of methylmercury under the condition that consuming time is random. We find out the amount o f methylmercury in human body is changing constantly in fixed range, when people have just consumed fish. Moreover, the maximum is 4261 ug, which is far bigger than3505 ug. So we can reach the conclusion that people are more endangered when the consuming time is irregular.In order to closer to the actual situation, we construct a model in which the half-life of methylmercury in human body is not constant. Through analyzing the data of computer simulation, the maximum amount of methylmercury will increase, that is, the risk of poisoning will be higher.References[1] Dr.D.N.Rahni, PHD. Airborne Mercury Contamination and the NeversinkReservoir./dnabirahni/rahnidocs/Envsc/Airborne%20Mercury%20C ontamination%20and%20the%20Neversink%20Reservoir.doc[2] Hu Dong Bai Ke. Bass. /wiki%E9%B2%88%E9%B1%BC.[3] Centre for Food Safety Food and Environmental Hygiene Department TheGovernment of the Hong Kong Special Administrative Region. Mercury in Fish and Food Safety..hk/english/Programmme/programme_rafs/Programme_rafs_fc _01_19_mercury_in_fish.html.。
2012年大学英语竞赛答案
2012 National English Contest for College Students(Level C - Preliminary)参考答案及作文评分标准Part I Listening Comprehension (30 marks)Section A (5 marks)1—5 CBBAC Section B (10 marks)6—10 BAACB 11—15 CABCBSection C (5 marks)16—20 ACABA Section D (10 marks)21. disruption 22. air pressure 23. slow down 24.immune system 25.stimulates 26. bear in mind 27. circulation 28. decreases 29. Consult 30. side effects Part II Vocabulary and Structure (15 marks)31—35 BDABC 36—40 BDDAB 41—45 CADBCPart III Cloze (10 marks)46. likelihood 47. fortune 48. take 49. with 50. knowing51. dozens 52. least 53. burning 54.easy/simple 55. realizePart IV Reading Comprehension (40 marks) Section A (10 marks)56. describe 57. opportunity 58. honeymoon 59. transition 60. characterized Section B (10 marks)61. F 62. T 63. T64. The special depth that true friendship is about.65. They need to get over fear of confrontation and rejection. Section C (10 marks)66. High fatality 67. thrill seeking; risk taking 68. pushing out personal boundaries 69. C 70. A Section D (10 marks)71.“Linguistic”intelligence and“Logical mathematical”intelligence.72. The ability to gauge one's own mood, feelings and mental states, and the ability to gauge it in others and use the information.73.“Logical mathematical”intelligence and the“bodily-kinesthetic”intelligence.74. 他的观点引发了激烈的讨论，应如何定义智力。
美赛历年赛题及其翻译-推荐下载
2015年：A题一个国际性组织声称他们研发出了一种能够阻止埃博拉，并治愈隐性病毒携带者的新药。
建立一个实际、敏捷、有效的模型，不仅考虑到疾病的传播、药物的需求量、可能的给药措施、给药地点、疫苗或药物的生产速度，而且考虑你们队伍认为重要的、作为模型一部分的其他因素，用于优化埃博拉的根除，或至少缓解目前（治疗）的紧张压力。
除了竞赛需要的建模方案以外，为世界医学协会撰写一封1-2页的非技术性的发言稿，以便其公告使用。
B题回顾马航MH370失事事件。
建立一个通用的数学模型，用以帮助失联飞机的搜救者们规划一个有效的搜索方案。
失联飞机从A地飞往B地，可能坠毁在了大片水域（如大西洋、太平洋、印度洋、南印度洋、北冰洋）中。
假设被淹没的飞机无法发出信号。
你们的模型需要考虑到，有很多种不同型号的可选的飞机，并且有很多种搜救飞机，这些搜救飞机通常使用不同的电子设备和传感器。
此外，为航空公司撰写一份1-2页的文件，以便在其公布未来搜救进展的新闻发布会上发表。
2014美赛A题翻译问题一:通勤列车的负载问题在中央车站,经常有许多的联系从大城市到郊区的通勤列车“通勤”线到达。
大多数火车很长(也许10个或更多的汽车长)。
乘客走到出口的距离也很长，有整个火车区域。
每个火车车厢只有两个出口,一个靠近终端, 因此可以携带尽可能多的人。
每个火车车厢有一个中心过道和过道两边的座椅，一边每排有两个座椅，另一边每排有三个座椅。
走出这样一个典型车站,乘客必须先出火车车厢,然后走入楼梯再到下一个级别的出站口。
通常情况下这些列车都非常拥挤,有大量的火车上的乘客试图挤向楼梯，而楼梯可以容纳两列人退出。
大多数通勤列车站台有两个相邻的轨道平台。
在最坏的情况下,如果两个满载的列车同时到达,所有的乘客可能需要很长时间才能到达主站台。
建立一个数学模型来估计旅客退出这种复杂的状况到达出站口路上的时间。
假设一列火车有n个汽车那么长,每个汽车的长度为d。
站台的长度是p,每个楼梯间的楼梯数量是q。
2012年美国国际大学生数学建模竞赛(MCM+ICM)题目+翻译
2012 Contest 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 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美赛A题：一棵树的叶子（数学中国翻译）“一棵树的叶子有多重？”怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。
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2012美赛A题：一棵树的叶子
（数学中国翻译）
“一棵树的叶子有多重？”怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。
模型要考虑和回答下面的问题：
•为什么叶子具有各种形状？
•叶子之间是要将相互重叠的部分最小化，以便可以最大限度的接触到阳光吗？树叶的分布以及树干和枝杈的体积影响叶子的形状吗？
•就轮廓来讲，叶形（一般特征）是和树的轮廓以及分枝结构有关吗？
•你将如何估计一棵树的叶子质量？叶子的质量和树的尺寸特征（包括和外形轮廓有关的高度、质量、体积）有联系吗？
除了你的一页摘要以外，给科学杂志的编辑写一封信，阐述你的主要发现
B：沿着Big Long River野营
【数学中国翻译】
0 i" k1 T3 h' B# u" ]
游客在“大长河”(225英里)可以享受到秀丽的风光和令人兴奋的白色湍流。
这条河对于背包客来说是进不去的，因此畅游这条长河的唯一办法就是在这条河上露营上几天。
这次旅行从开始的下水点到最终结束点，共225英里，且是顺流而下的。
乘客可以选择平均4英里/小时的以浆作为动力的橡胶筏或者平均8英里/小时的机动帆船
旅行。
整个旅行从开始到结束会经历6至18个夜晚。
负责管理这条河的政府机构希望到这里的每一次旅行都能够享受到野外经历，以最少的接触到在河上其它的船只。
目前，每年在六个月期间(一年的其余部分的天气对于河流旅行来说太冷)，共有X次旅行，有Y 处露营地，露营地均匀的分布整个河道。
由于漂流的受欢迎程度的上升，公园管理者已经被要求允许更多的旅行次数。
所以他们想确定怎样可能安排一个最优的混合的旅行方案，不同的时间(单位为夜)和推动方式(马达或浆)，最大限度的利用露营地。
换句话说，在长河的漂流季，将会有多少更多的乘船旅行可以加进来?河流的管理者现在雇佣你，为他们提出最佳排程方式和河流承载能力的建议，记住两个露营者不能在同一时间内占据同一个露营地。
除了你的一页摘要，准备一页备忘录，对河流的管理者描述你的主要发现。