2015年美国大学生数学建模竞赛A题
2015年数学建模国赛A题
二、 问题分析
问题一要建立直杆影子长度变化的数学模型， 首先需知道太阳影子长度计算 公式，故引入太阳高度角[1]这个概念。即若已知某时刻太阳高度角的大小和直 杆高度，根据其满足的三角函数关系便可得到此时太阳影子长度。太阳高度角与 观测地地理纬度、地方时角和太阳的赤纬[2]相关。其中太阳赤纬是太阳直射点 所在纬度，与日期有关；时角由当地经度及其所用时区时间决定，故根据影长、 太阳赤纬、时角计算公式可求得直杆影子长度变化模型，并根据模型分析影子长 度关于各参数的变化规律。将附件一中直杆的有关数据直杆影长变化模型中，可 求出该直杆的具体影长变化公式。根据所建立的模型，运用 MATLAB 软件便可得 到影子长度随时间的变化曲线。 问题二需根据某固定直杆在水平地面上的太阳影子顶点坐标数据， 建立数学 模型确定直杆所处的地点。首先由问题一可推测影子长度与时间的关系，故可将 太阳影子长度与对应时间进行拟合，得到影长与时间关系模型。当某个时刻影长 得到极小值时，该时刻为太阳与直杆距离最近，即地方时正午 12 时，结合当地 所使用的标准时间便可得到当地经度。 最后利用太阳高度角与直杆长度以及影长 满足的三角关系式，便可得到影长关于直杆高度、直杆所在地点的纬度的函数关 系式，即得到了有关太阳影子顶点坐标与直杆地点经纬度的模型。将附件一中影 子顶点坐标数据应用于该直杆位置模型，可得到直杆所在位置。用相对误差分析 法分析误差[3](168-169 页)，若所得的相对误差小于 2.5%，认为得到的模型合 理。 问题三可根据光照成影原理和太阳高度角计算公式建立影长与时间变 化模型，根据相关数据，运用 MATLAB 软件拟合可得到直杆所在位置的经纬 度。令年份均为 2015 年，根据太阳赤纬角计算公式，可求解具体的日期。 将附件 2 和附件 3 时间和对应直杆影长数据分别代入模型中，通过拟合计
(最新整理)数学建模美赛试题
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地球资源的消耗速度快,越来越多的人关注人类社会的未来.自1960年以来，已经有许多专家研究可持续发展.然而大多数人的研究对象是整个世界，一个国家或一个地区。
几乎没有人选择48个最不发达国家(LDC)在联合国为研究对象列表。
然而,LDC国家集团共享许多相同的点。
他们的发展道路也有法律的内涵。
本文选择这些国家为研究对象针对发现常规的可持续发展道路。
本文组织如下.第二部分介绍研究的背景和本研究的意义。
第三节描述了我们对可持续发展的理解细节和显示我们的评估系统的建立过程和原理，那么我们估计每一个国家的LDC和获得可持续发展的能力和等级。
第四节提供了一个最糟糕的国家毛里塔尼亚计划指数在第三节。
第五节演示了在第四节的合理性和可用性计划。
最后在第六节总结本文的主要结论和讨论的力量和潜在的弱点。
地球上的资源是有限的。
三大能源石油、天然气和煤炭可再生。
如何避免人类的发展了资源枯竭和实现可持续发展目标是现在的一个热门话题.在过去的两个世纪,发达国家已经路上,先污染，再控制和达到高水平的可持续发展。
发展中国家希望发展和丰富。
然而,因为他们的技术力量和低水平的经济基础薄弱,浪费和低效率的发展在这些国家是正常的.所以本文主要关注如何帮助发展中国家特别是48在联合国最不发达国家实现可持续发展是列表可持续发展的理解是解决问题的关键.可持续发展的定义经历了一个长期发展的过程.在这里，布伦特兰可持续发展委员会的简短定义的"能力发展可持续- — - — - -以确保它既满足现代人的需求又不损害未来的能力代来满足自己的需求"［1]无疑是最被广泛接受的一个在各种内吗定义.这个定义方面发挥了重要作用在很多国家的政策制定的过程.然而，为了证明一个国家的现状是否可持续不可持续的,更具体的定义是必要的更具体的概念,我们认为,如果一个国家的发展是可持续的,它应该有一个基本的目前的发展水平，一个平衡的国家结构和一个光明的未来。
2015年美赛O奖论文A题Problem_A_32150
Team Control Number
For office use only F1 ________________ F2 ________________
32150
Problem Chosen
A
F3 ________________ F4 ________________
2015 Mathematical Contest iow to Eradicate Ebola? The breakout of Ebola in 2014 triggered global panic. How to control and eradicate Ebola has become a universal concern ever since. Firstly, we build up an epidemic model SEIHCR (CT) which takes the special features of Ebola into consideration. These are treatment from hospital, infectious corpses and intensified contact tracing. This model is developed from the traditional SEIR model. The model’s results (Fig.4,5,6), whose parameters are decided using computer simulation, match perfectly with the data reported by WHO, suggesting the validity of our improved model. Secondly, pharmaceutical intervention is studied thoroughly. The total quantity of the medicine needed is based on the cumulative number of individuals CUM (Fig.7). Results calculated from the WHO statistics and from the SEIHCR (CT) model show only minor discrepancy, further indicating the feasibility of our model. In designing the delivery system, we apply the weighted Fuzzy c- Means Clustering Algorithm and select 6 locations (Fig.10, Table.2) that should serve as the delivery centers for other cities. We optimize the delivery locations by each city’s location and needed medicine. The percentage each location shares is also figured out to facilitate future allocation (Table.3,4). The average speed of manufacturing should be no less than 106.2 unit dose per day and an increase in the manufacturing speed and the efficacy of medicine will reinforce the intervention effect. Thirdly, other critical factors besides those discussed early in the model, safer treatment of corpses, and earlier identification/isolation also prove to be relevant. By varying the value of parameters, we can project the future CUM . Results (Fig.12,13) show that these interventions will help reduce CUM to a lower plateau at a faster speed. We then analyze the factors for controlling and the time of eradication of Ebola. For example, when the rate of the infectious being isolated is 33% - 40%, the disease can be successfully controlled (Table.5). When the introduction time for treatment decreases from 210 to 145 days, the eradication of Ebola arrives over 200 days earlier. Finally, we select three parameters: the transmission rate, the incubation period and the fatality rate for sensitivity analysis. Key words: Ebola, epidemic model, cumulative cases, Clustering Algorithm
2015年美国(国际)大学生数学建模竞赛
比赛时间：美国东部时间：2015年2月5日（星期四）下午8点-2月9日下午8点（共4天）北京时间：2015年2月6日（星期五）上午9点-2月10日上午9点农历：十二月十八～十二月廿二重要说明：●COMAP是所有的规则和政策的最后仲裁者，对不遵循竞赛规则和程序的任何队伍，拥有唯一的自由裁量权，取消参赛资格或拒绝登记。
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2015数模美赛A题翻译
PROBLEM A: Eradicating EbolaThe world medical association has announced that their new medication could stop Ebola and cure patients whose diseases not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement.A消除埃博拉病毒世界医学协会已经宣布他们的新疗法可以阻止埃博拉疫情和治愈非晚期患者。
构建一个现实的、合理的和有用的模型,不仅要考虑疾病的传播,所需药物的数量,可能且可行的给药系统,给药地点,生产疫苗或药物的速度,而且还要考虑其他关键因素（你的团队认为有必要要考虑的）作为模型的一部分以优化消除埃博拉病毒，或至少是现行毒株。
2015年美赛数学建模题目
2015 Contest ProblemsMCM PROBLEMSPROBLEM A: Eradicating EbolaThe world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement.PROBLEM B: Searching for a lost planeRecall the lost Malaysian flight MH370. Build a generic mathematical model that could assist "searchers" in planning a useful search for a lost plane feared to have crashed in open water such as the Atlantic, Pacific, Indian, Southern, or Arctic Ocean while flying from Point A to Point B. Assume that there are no signals from the downed plane. Your model should recognize that there are many different types of planes for which we might be searching and that there are many different types of search planes, often using different electronics or sensors. Additionally, prepare a 1-2 page non-technical paper for the airlines to use in their press conferences concerning their plan for future searches.ICM PROBLEMSPROBLEM C: Managing Human Capital in OrganizationsClick the title below to download a PDF of the 2015 ICM Problem C.Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.Managing Human Capital in OrganizationsPROBLEM D: Is it sustainable?Click the title below to download a PDF of the 2015 ICM Problem D.Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.Is it sustainable?。
2015美国数学竞赛AMC12试题及答案
Problem1What is the value ofProblem2Two of the three sides of a triangle are20and15.Which of the following numbers is not a possible perimeter of the triangle?Problem3Mr.Patrick teaches math to15students.He was grading tests and found that when he graded everyone's test except Payton's,the average grade for the class was80.after he graded Payton's test,the class average became81.What was Payton's score on the test?Problem4The sum of two positive numbers is5times their difference.What is the ratio of the larger number to the smaller?Problem5Amelia needs to estimate the quantity,where and are large positiveintegers.She rounds each of the integers so that the calculation will be easier to do mentally.In which of these situations will her answer necessarily be greater than the exact value of?Problem6Two years ago Pete was three times as old as his cousin Claire.Two years before that, Pete was four times as old as Claire.In how many years will the ratio of their ages be ?Problem7Two right circular cylinders have the same volume.The radius of the second cylinderis more than the radius of the first.What is the relationship between the heights of the two cylinders?Problem8The ratio of the length to the width of a rectangle is:.If the rectangle hasdiagonal of length,then the area may be expressed as for some constant.What is?Problem9A box contains2red marbles,2green marbles,and2yellow marbles.Carol takes2 marbles from the box at random;then Claudia takes2of the remaining marbles at random;and then Cheryl takes the last2marbles.What is the probability that Cheryl gets2marbles of the same color?Problem10Integers and with satisfy.What is?Problem11On a sheet of paper,Isabella draws a circle of radius,a circle of radius,and all possible lines simultaneously tangent to both circles.Isabella notices that she has drawn exactly lines.How many different values of are possible?Problem12The parabolas and intersect the coordinate axes in exactly four points,and these four points are the vertices of a kite of area.What is?Problem13A league with12teams holds a round-robin tournament,with each team playing every other team exactly once.Games either end with one team victorious or else end in a draw.A team scores2points for every game it wins and1point for every game it draws.Which of the following is NOT a true statement about the list of12scores?Problem14What is the value of for which?Problem15What is the minimum number of digits to the right of the decimal point needed toexpress the fraction as a decimal?Problem16Tetrahedron has and .What is the volume of the tetrahedron?Problem17Eight people are sitting around a circular table,each holding a fair coin.All eight people flip their coins and those who flip heads stand while those who flip tails remain seated.What is the probability that no two adjacent people will stand?Problem18The zeros of the function are integers.What is the sum of the possible values of?Problem19For some positive integers,there is a quadrilateral with positive integerside lengths,perimeter,right angles at and,,and.Howmany different values of are possible?Problem20Isosceles triangles and are not congruent but have the same area and the sameperimeter.The sides of have lengths of and,while those of have lengthsof and.Which of the following numbers is closest to?Problem21A circle of radius passes through both foci of,and exactly four points on,the ellipse with equation.The set of all possible values of is an interval. What is?Problem22For each positive integer n,let be the number of sequences of length n consisting solely of the letters and,with no more than three s in a row and nomore than three s in a row.What is the remainder when is divided by12?Problem23Let be a square of side length1.Two points are chosen independently at random onthe sides of.The probability that the straight-line distance between the points is atleast is,where and are positive integers and.Whatis?Problem24Rational numbers and are chosen at random among all rational numbers in the interval that can be written as fractions where and are integers with .What is the probability that is a real number?Problem25A collection of circles in the upper half-plane,all tangent to the-axis,is constructedin layers as yer consists of two circles of radii and that areexternally tangent.For,the circles in are ordered according to their points of tangency with the-axis.For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair.Layer consists of the circles constructed in this way.Let,andfor every circle denote by its radius.What isDIAGRAM NEEDED。
2015 AMC10A美国数学竞赛试题
What is the value ofProblem2A box contains a collection of triangular and square tiles.There are tiles in the box, containing edges total.How many square tiles are there in the box?Problem3Ann made a3-step staircase using18toothpicks.How many toothpicks does she need to add to complete a5-step staircase?Problem4Pablo,Sofia,and Mia got some candy eggs at a party.Pablo had three times as many eggs as Sofia,and Sofia had twice as many eggs as Mia.Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs.What fraction of his eggs should Pablo give to Sofia?Problem5Mr.Patrick teaches math to students.He was grading tests and found that when hegraded everyone's test except Payton's,the average grade for the class was.After he graded Payton's test,the test average became.What was Payton's score on thetest?The sum of two positive numbers is times their difference.What is the ratio of thelarger number to the smaller number?Problem7How many terms are there in the arithmetic sequence,,,...,,?Problem8Two years ago Pete was three times as old as his cousin Claire.Two years before that, Pete was four times as old as Claire.In how many years will the ratio of their ages be :?Problem9Two right circular cylinders have the same volume.The radius of the second cylinderis more than the radius of the first.What is the relationship between the heights of the two cylinders?Problem10How many rearrangements of are there in which no two adjacent letters are alsoadjacent letters in the alphabet?For example,no such rearrangements could include either or.The ratio of the length to the width of a rectangle is:.If the rectangle hasdiagonal of length,then the area may be expressed as for some constant.What is?Problem12Points and are distinct points on the graph of.What is?Problem13Claudia has12coins,each of which is a5-cent coin or a10-cent coin.There are exactly17different values that can be obtained as combinations of one or more of her coins.How many10-cent coins does Claudia have?Problem14Problem15Consider the set of all fractions where and are relatively prime positive integers. How many of these fractions have the property that if both numerator anddenominator are increased by,the value of the fraction is increased by?Problem16If,and,what is the value of?Problem17A line that passes through the origin intersects both the line and the line.The three lines create an equilateral triangle.What is the perimeter of the triangle?Problem18Hexadecimal(base-16)numbers are written using numeric digits through as well as the letters through to represent through.Among the firstpositive integers,there are whose hexadecimal representation contains onlynumeric digits.What is the sum of the digits of?Problem19The isosceles right triangle has right angle at and area.The raystrisecting intersect at and.What is the area of?Problem20A rectangle has area and perimeter,where and are positive integers.Which of the following numbers cannot equal?Problem21Tetrahedron has,,,,,and.What is the volume of the tetrahedron?Problem22Eight people are sitting around a circular table,each holding a fair coin.All eight people flip their coins and those who flip heads stand while those who flip tails remain seated.What is the probability that no two adjacent people will stand?Problem23The zeros of the function are integers.What is the sum of the possible values of?Problem24For some positive integers,there is a quadrilateral with positive integerside lengths,perimeter,right angles at and,,and.Howmany different values of are possible?Problem25Let be a square of side length.Two points are chosen independently at random onthe sides of.The probability that the straight-line distance between the points is atleast is,where,,and are positive integers with.Whatis?。
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PROBLEM A: Eradicating Ebola
The world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement.
问题一：根除病毒
世界医疗联盟声称，他们的新药物可以制止埃博拉病毒，并且治愈病情没有恶化的患者。
建立一个现实的，明智的，并且有用的模型，需要考虑的不仅是疾病的扩散、所需要的药物、可能并且可行的发放系统、发放的地点、生产疫苗或药物的速度，还有你们队伍认为在优化消灭埃博拉（或至少它的现在的同类血亲）模型之中重要的因素。
除了建立模型以外，还需写一封1-2页非技术性信给世界医疗联盟，让他们在声明中阐述。
SIR。