2013美赛题目

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2013年美赛MCM题目A评委点评中文翻译

介绍今年的焦点问题是如何实现质量和数量的平衡。

在质量方面，尽可能使热量均匀地分布。

目标是降低或避免矩形烤盘四个边角发生热量聚集的情况。

所以解决热量均匀分布这方面的问题，使用圆形烤盘是最佳的选择。

在数量方面，应该使烤盘充分的占据烤箱的空间。

所以我们的目的是使用尽可能多的烤盘来充分占据烤箱的空间，此时矩形烤盘是最佳选择。

对于这方面的问题的解决，就要考虑烤盘在烤箱水平截面上所占的比率。

在这个评论中，我们首先描述判断步骤，然后再讨论队伍对于三个问题的求解。

下一个话题就是论文的灵敏度和假设，紧随其后讨论确定一个给定方法的优势和劣势。

最后，我们简短的讨论一下参考和引用之间的区别。

过程第一轮的判别被称为“分流轮”。

这些初始轮的主要思想是确定论文应被给予更详细的考虑。

每篇论文应该至少阅读两次。

在阅读一篇论文的时候，评审的主要问题是论文是否包含所有必要的成分,使它成为一个候选人最详细的阅读。

在这些初始轮中，评审的时间是有限制的，所以我们要尽量让每一篇论文得到一个好的评判。

如果一篇论文解决了所有的问题，就会让评审觉得你的模型建立是合理的。

然后评审可能会认为你的论文是值得注意的。

有些论文在初轮评审中可能会得到不太理想的评论。

特别值得注意的是，一篇好的摘要应该要对问题进行简要概述，另外，论文的概述和方法，队员之间应该互相讨论，并且具体的结果应该在某种程度上被阐述或者表达出来。

在早期的几轮中，一些小细节能够有突出的表现，包括目录，它更便于评委看论文，同时在看论文的时候可能会有更高的期待。

问题求解也很重要。

最后，方法和结果要清晰简明的表达是至关重要的。

另外，在每个部分的开始，应该对那个部分进行一个概述。

在竞赛中，建模的过程是很重要的，同时也包括结论的表达。

如果结果没有确切和充分的表达，那么再好的模型和再大努力也是没有用的。

最后的回合最后一轮阅读的第一轮开始于评委会会议。

在这个会议中，评委将进行讨论，他们会分享他们各自认为的问题的关键方面。

2013美国数学竞赛AMC8题目及答案

2.is the value of ?friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $ to cover her portion of the total bill. What was the total bill?is in the grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds.Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?fair coin is tossed 3 times. What is the probability of at least two consecutive heads?Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?11.Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walkedless time on the treadmill. How many minutes less?12.At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pairof sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at halfthe regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?13.WhenClara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score.By which of the following might her incorrect sum have differed from the correct one?14.Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into .Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 is .15.If , , and , what is the product of , , and ?16.A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of -graders to -graders is , and the the ratio of -graders to -graders is . What is the smallest number of students that could be participating in the project?17.The sum of six consecutive positive integers is 2013. What is the largest of these six integers?18.Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?--Arpanliku 16:22, 27 November 2013 (EST) Courtesy of19.Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget andCassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowestscore in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?20.A rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?21.Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?22.Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?23.Angle of is a right angle. The sides of are the diameters of semicircles as shown. The area of the semicircle on equals , and the arc of the semicircle on has length . What is the radius of the semicircle on ?24.Squares , , and are equal in area. Points and are the midpoints of sides and , respectively. What is the ratio of the area of the shaded pentagon to the sum of the areas of the three squares?25.A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are inches, inches, and inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?1.50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is$6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars.that we can pair up every two numbers to make a sum of 1:Therefore, the answer is .of her seven friends paid to cover Judi's portion. Therefore, Judi's portion must be . Since Judi was supposed to pay of the total bill, the total bill must be .median here is obviously less than the mean, so option (A) and (B) are out.Lining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds.The average weight of the five kids is .Therefore, the average weight is bigger, by pounds, making the answer .1: Working BackwardsLet the value in the empty box in the middle row be , and the value in the empty box in the top row be . isthe answer we're looking for.It follows that , so .Solution 2: Jumping Back to the StartAnother way to do this problem is to realize what makes up the bottommost number. This method doesn't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Solution 1 would be faster since there's only two missing numbers.)We see that , making .Again, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for. We can write some equations:Now we can substitute into the first equation using the two others:Trey saw , then he saw .2 minutes and 45 seconds can also be expressed as seconds.Trey's rate of seeing cars, , can be multiplied by on the top and bottom (and preserve the same rate):.It follows that the most likely number of cars is .Solution 2minutes and seconds is equal to .Since Trey probably counts around cars every seconds, there are groups of cars that Trey most likely counts.Since , the closest answer choice is .,there are ways to flip the coins, in order.The ways to get two consecutive heads are HHT and THH.The way to get three consecutive heads is HHH.Therefore, the probability of flipping at least two consecutive heads is .is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters,However, because the first term is and not , the solution to the problem is10.To find either the LCM or the GCF of two numbers, always prime factorize first.The prime factorization of .The prime factorization of .Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940.For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. = 18.Thus the answer ==.we note that .We start off with a similar approach as the original solution. From the prime factorizations, the GCF is .It is a well known fact that . So we have,.Dividing by yields .Therefore,.11.We use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, let represent the time.On Monday, he was at a rate of . So,.For Wednesday, he walked at a rate of . Therefore,.On Friday, he walked at a rate of . So,.Adding up the hours yields + + =.We now find the amount of time Grandfather would have taken if he walked at per day. Set up the equation,.To find the amount of time saved, subtract the two amounts: - = . To convert this to minutes, we multiply by .Thus, the solution to this problem is12.First, find the amount of money one will pay for three sandals without the discount. We have .Then, find the amount of money using the discount:.Finding the percentage yields .To find the percent saved, we have13.Let the two digits be and .The correct score was . Clara misinterpreted it as . The difference between the two is which factors into .Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 is .14.The probability that both show a green bean is . The probability that both show a red bean is . Therefore the probability is15.Therefore,.Therefore,.To most people, it would not be immediately evident that , so we can multiply 6's until we get the desired number: ,so .Therefore the answer is .16.Solution 1: AlgebraWemultiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:Therefore, the ratio of 8th graders to 7th graders to 6th graders is . Since the ratio is in lowest terms, the smallest number of students participating in the project is .Solution 2: FakesolvingThe number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility).Then there are 6th graders and 7 th graders. The numbers of students is17.Solution 1The mean of these numbers is . Therefore the numbers are , so the answer isSolution 2Let the number be . Then our desired number is .Our integers are , so we have that .Solution 3Let the first term be . Our integers are . We have,18.Solution 1There are cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there are cubes. Hence, the answer is .Solution 2We can just calculate the volume of the prism that was cut out of the original box. Each interior side ofthe fort will be feet shorter than each side of the outside. Since the floor is foot, the height will be feet.So the volume of the interior box is .The volume of the original box is . Therefore, the number of blocks contained in the fort is .19.If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if Hannah did worse than Bridget, there is no way Bridget could have known that she didn't get the highest in the class. Therefore, Hannah did better than Bridget, so our order is .20.A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem, . The area is .21. The number of ways to get from Samantha's house to City Park is , and the number of ways to get from CityPark to school is . Since there's one way to go through City Park (just walking straight through), the number of different ways to go from Samantha's house to City Park to school .22.There are vertical columns with a length of toothpicks, and there are horizontal rows with a length of toothpicks. An effective way to verify this is to try a small case, . a grid of toothpicks. Thus, our answer is .23.Solution 1If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagorean theorem says that the other side has length 15, so the radius is .Solution 2We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircleson the left is equal to the biggest one, so the area of the largest is , and the middle one is , so the radius is .24.First let (where is the side length of the squares) for simplicity. We can extend until it hits the extensionof . Call this point . The area of triangle then is The area of rectangle is . Thus, our desired area is .Now, the ratio of the shaded area to the combined area of the three squares is .Solution 2Let the side length of each square be .Let the intersection of and be .Since , . Since and are vertical angles, they are congruent. We also have by definition.So we have by congruence. Therefore,.Since and are midpoints of sides, . This combined with yields .The area of trapezoid is .The area of triangle is .So the area of the pentagon is .The area of the squares is .Therefore,.Solution 3Let the intersection of and be .Now we have and .Because both triangles has a side on congruent squares therefore .Because and are vertical angles .Also both and are right angles so .Therefore by AAS(Angle, Angle, Side).Then translating/rotating the shaded into the position ofSo the shaded area now completely covers the squareSet the area of a square asTherefore,.25.Solution 1The radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses inches, and it gains inches on B. So, the departure from the length of the track means that the answer is .Solution 2The total length of all of the arcs is . Since we want the path from the center, the actual distance will be shorter. Therefore, the only answer choice less than is . This solution may be invalid because the actual distance can be longer if the path the center travels is on the outside of the curve, as it is in the middle bump.。

2013年美赛A题翻译题目

2013 MCM A 终极布朗尼锅
当在一个矩形锅里烘烤，热量被集中在四个角上，在角上的产品会煮过头（边缘部位程度会小一些）。

在一个圆形的锅里，热量会均匀分布在整个锅的外缘部位并且产品也不会被煮过头。

然而，因为大多数的烤箱的形状都是矩形的，用圆锅对于利用锅的内部空间是没有效率的。

建立一个模型，以显示热量穿过锅的外边缘上的分布，对于不同形状-矩形和圆形以及他们之间形状的锅。

假设：
1、矩形烤箱的宽长比是W/L.
2、每个锅的面积是A
3、烤箱中初始的两个隔板是均匀分布的
建立一个模型，可以用来选择最优的锅的形状，满足以下的条件：
1.烤箱中可以容纳锅的最大数量N。

2.使锅的热量H最大化均匀分布
3.优化组合（1）和（2）中的条件，他们的权重分配是p和（1-p），来说明在不同的W/L和p的值下，结果是怎么变化的。

除了你的MCM格式的解决方案，准备一到两页的广告片新布朗尼美食杂志突出自己的设计和结果。

2013年数学建模美赛题目中文翻译_共4页

Problem c:背景：社会正致力于运用和开发模型来预测地球的生物和环境情况。

很多科学研究总结了逐渐增长的地球环境和生物系统压力，但很少有人用全球范围的模型来检测这些观点。

联合国发表的千年生态系统评估综合报告发现：近三分之二的地球生命支持生态系统——包括净水，洁净的空气，稳定的气候——正在因非可持续性使用而逐渐衰减。

其中大部分破坏归咎于人类行为。

暴增的对于食物，淡水，燃料，木材的需求导致了剧烈的环境变化；从森林砍伐到空气，土壤和水污染。

尽管已存在大量关于局部习惯和地区因素的研究，目前的模型还不能告知决定人他们的局部策略是如何影响整个地球的健康的。

许多模型忽略了复杂的全球因素，这些模型无法判断重大政策的长期影响。

尽管科学家们意识到巨大环境和生物系统中存在的复杂关系和交叉作用，当前的模型通常忽略这些管理或限定了系统间的影响。

系统的复杂性体现在多元交互（多个元素的相关性），反馈，突发行为，即将发生的状态变化或触发点。

最近的自然杂志中一篇由22位国际知名科学家撰写的题为“迫近地球生物圈的状态变化”的文章讨论了许多有关科学模型对于预测行星健康系统潜在状态变化的重要性与必需性。

文章提供了两种具体定性的模型，并寻求更好的预测模型：1）通过在全球模型中加入相关系统的复杂性（包括局部情况对全球系统的影响，反之亦然）来优化生物状态预测。

2）辨别不同因素在产生非健康全球状态变化中的作用并展示如何运用有效的生态系统管理来预防或限制这些即将发生的状态变化。

研究最终归结于问题：我们是否能利用全球健康的局部或地区性组成部分预测潜在状态变化来帮助决策者制定基于对全球健康状况潜在影响的，有效的策略。

尽管有越来越多的警示信号出现，没人知道地球是否确实在接近全球性的转折点（极端状态），这种极端的状态是否是不可避免的。

自然杂志等研究指出了地球生态系统中的一些重要工作元素。

（例如：局部因素，全球变化，多维元素与关系，变化的时间与空间范围）。

2013美赛模拟题

2013年美赛模拟题PROBLEM A: Who Moved the Presidential Candidate's Cheese? New media is rapidly becoming an important part of a presidential candidate's media strategy. The use of the new media by most of the candidates had been unprecedentedly frequent and overwhelming in 2008 American president election cycle. It is most believed that U.S President Barack Obama, who had massively used new media as one of his campaign strategies, not only won the election, but also drawn the world's attention and gained himself the reputation as a "Internet President". Requirement 1: Model the effectiveness of the New media for president election.Requirement 2:Verify your model.Requirement 3: Suggestion for the presidential candidate.PROBLEM B: Air Politics in BeijingOne day last month, the reading was so high compared with the standards set by the U.S. Environmental Protection Agency that it was listed as "beyond index." In other words, it had soared right off the chart. But China's own assessment that day, Oct.9, was that Beijing's air was merely "slightly polluted."NASA:PM2.5 MAPNot even the most fervent propagandist would call the city's air clean, but the Chinese government made great efforts to improve air quality for the 2008 Olympic Summer Games. Beijing authorities moved huge steelworks out of the capital, switched city dwellers from coal to natural gas heating, raised emissions standards for trucks, and created new subway and bus lines. The cost of the cleanup was estimated at $10 billion, not including the investment in mass transit.Three years later, the difference between the Americans and the Chinese is at least in part about what they're measuring. And it highlights the rapid growth in the number of cars in Beijing.Chinese monitoring stations around the capital track large particulates of up to 10 micrometers. The number of those particles has dropped as a result of reforestation programs that lessen the dust storms that blew in from deserts. The Chinese have also been successful in reducing sulfur dioxide emissions by limiting coal heating and imposing stricter emissions standards.The U.S. monitor tracks tinier particles — less than 2.5 micrometers — that physicians say are capable of penetrating human lungs and other organs. Car and truck exhaust is a major source of fine particulate pollution, a particular problem in Beijing, where the number of registered cars has skyrocketed from to 5 million from 3.5 million in 2008.The embassy of US in Beijing installed its monitor in 2008 before the Olympics to advise its personnel about air quality,but then decided it should make readings public under diplomatic rules that require that information regarding health and security risks be made available.The measurements drew widespread attention last November, the first time a reading for fine particulate matter went above 500 micrograms per cubic meter, about seven times the U.S. standard for "acceptable" air quality.The expert suggests that its problem is political as well as technical. Politically, it is hard to revise the standard. After 10 years of saying things are getting better and better, if you reverse that, people will be justifiably angry. Although it is government data that are published in newspapers andbroadcast on television, other Chinese media are increasingly citing the U.S. figures.Please Survey data andwriting papers.。

2013美赛ABC试题及翻译

在一个圆形盘的热量被均匀地分布在整个外缘，在边缘处的食物烤的不会过头。

然而，因为大多数烤箱是方形的，使用原型的平底锅效率是不高的相对于烤箱中的空间。

Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.建立一个模型来说明不同形状的锅(圆形方形或其他形状)的外边缘热量的不同分布。

Assume1. A width to length ratio of W/L for the oven which is rectangular in shape.2. Each pan must have an area of A.3. Initially two racks in the oven, evenly spaced.假设1.矩形烤箱的的宽度与长度之比是W / L。

2013数学建模美赛题目(中文版)[1]

2013 ICM问题problem A:当用矩形平底锅高温加热物品时，热量一般集中于4个角落，因而在角落的物品会被焙烧过度（较小程度在角落的物品一部分会被焙烧过度）。

当用一个圆锅加热物品时，热量是均匀分布在整个外缘，因而物品不会在边缘被焙烧过度。

然而，大多数烤箱是长方形的，而圆型的锅被认为效率低的。

建立一个模型以显示不同形状如矩形圆形或者其他介于两者之间的形状的锅在整个外缘的热量分布。

假设1长方形烤箱的宽/长=W/L ；2 每个锅的面积是确定的常熟A；3 最初，烤箱里的烤架两两之间间隔均匀。

建立一个模型，该模型可用于在下列条件之下选择最佳形状的锅：1烤箱中，锅数量（N）最大；2均匀分布的热量（H）最大的锅；3 优化组合条件1和条件2，以比重p和（1-p）的不同分配来说明结果与W/L 和p的不同值的关系。

problem B :对世界来说，新鲜的水资源是限制发展的制约因素。

对2013年建立一个确实有效的，可行的和具有成本效益的水资源战略数学模型，以满足2025年[从下面的列表选择一个国家]预计的用水需求,并确定最佳水资源战略。

尤其是，你的数学模型必须解决水的存储，运动，盐碱化和保护等问题。

如果可能的话，用你的模型，探讨经济，物理和环境对于你的战略的影响。

提供一个非技术性的文件，向政府领导介绍你的方法，介绍其可行性和成本，以及为什么它是“最好的的水战略选择。

”国家有：美国，中国，俄罗斯，埃及，沙特阿拉伯3.网络建模的地球的健康背景:社会是感兴趣的发展和使用模型来预测生物和环境卫生条件我们的星球。

许多科学研究认为越来越多的压力在地球的环境和生物吗系统,但是有很少的全球模型来测试这些索赔。

由联合国支持的年生态系统评估综合报告》显示,近三分之二的地球的维持生命的生态系统——包括干净的水,纯净的空气,和稳定的气候-正在退化,被不可持续的使用。

人类是归咎于很多这次的损坏。

不断飙升的要求食品、新鲜水、燃料和木材有贡献到戏剧性的环境变化,从森林砍伐,空气,土地和水的污染。

2013美国大学生数学建模竞赛题目和翻译和A题图解

2013 Contest Problems MCM PROBLEMSPROBLEM A: The Ultimate Brownie PanWhen baking in a rectangular pan heat is concentrated in the 4 corne rs and the product gets overcooked at the corners(and to a lesser ext ent at the edges).In a round panthe heat is distributed evenly over t he entire outer edge andtheproduct is not overcooked at the edges.However,since mostovens are rectangular in shape using round pans isnot efficient with respect to using the space in an oven.Develop a model to show the distribution of heat across theouter edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.Assume1.A width to length ratio of W/L for the oven which isrectangular in shape.2.Each pan must have an area of A.3.Initially two racks in the oven, evenly spaced.Develop a model that can be used to select the best type of pan ( shape) under the following conditions:1. Maximize number of pans that can fit in the oven (N)2. Maximize even distribution of heat (H) for the pan3. Optimize a combination of conditions (1) and (2) where weights p a nd (1- p) are assigned to illustrate how the results vary with differ ent values of W/L and p.In addition to your MCM formatted solution, prepare a one to two pa ge advertising sheet for the new Brownie Gourmet Magazine highlightin g your当用方形的烤盘烤饼时，热量会集中在四角，食物就在四角（四条边的热量略小于四角）烤焦了。

2013年AMC8竞赛真题(英文版)

2013 AMC8 Problems1.Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?2.A sign at the fish market says, "50% off, today only: half-pound packages for just $3 perpackage." What is the regular price for a full pound of fish, in dollars?What is the value of?3.4.Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill? 5.Hammie is in thegrade and weighs 106 pounds. His quadruplet sisters are tiny babiesand weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?6.The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?7.Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?8.A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?9.The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?10.What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?11.Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?12.At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?13.When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?14.Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?15.If , , and , what is the product of , , and ?16.A number of students from Fibonacci Middle School are taking part in a community serviceproject. The ratio of -graders to -graders is , and the the ratio of -graders to-graders is . What is the smallest number of students that could be participating in the project?17.The sum of six consecutive positive integers is 2013. What is the largest of these six integers?18.Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?19.Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?20.A rectangle is inscribed in a semicircle with longer side on the diameter. What is thearea of the semicircle?21.Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?22.Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?23.Angle of is a right angle. The sides of are the diameters of semicirclesas shown. The area of the semicircle on equals , and the arc of the semicircle onhas length . What is the radius of the semicircle on ?24.Squares , , and are equal in area. Points and are the midpointsof sides and , respectively. What is the ratio of the area of the shaded pentagonto the sum of the areas of the three squares?25.A ball with diameter 4 inches starts at point A to roll along the track shown. The track iscomprised of 3 semicircular arcs whose radii are inches, inches, andinches, respectively. The ball always remains in contact with the track and does notslip. What is the distance the center of the ball travels over the course from A to B?。

2013年美赛数学建模b题的分析

Байду номын сангаас
美国2025年可提供的实际淡水量怎样预测？
（可以先预测不同地区不同领域供水量，再相加）
如果实际淡水量小于需求量，可通过那些方法提供不足的淡水？（例如：海水淡化、不同地区淡水转移）
这些方法那个更好？（比较的依据是什么？例如费用，环境）需要做出那些假设？
1. 从现在到2025年之前不会有战争，自然灾害等影响因素 2.同一个地区的居民年用水量相同 3.同一个地区的居民量的年增长率为常数 4.不同地区海水淡化的单位费用相同 5.不同地区淡水转移的单位费用相同
注：模型假设一般是在问题分析中根据需要提出的，所以可以先提出基本假设，以后再补充
Vx pre S
1.预测2025年供水量的模型（以一个州的降水量为例）
其中，表示州i 的可用降水量，表示该州的单位面积平均年降水量， Si 表示该州的淡水地表面积
Vx pre S
2.预测2025年需水量的模型（以一个州的居民用水量为例）
若 V大于0，则说明2025年美国的淡水量是足够的，然后再看每个州需要补充的淡水量，即 V V ，
si ri
若每个州需要补充的淡水量均大于0，则不需要采取任何措施，
否则，需要在不通州之间进行淡水调度。
若 V小于0，则说明2025年美国的淡水量是不足的，需要采取海水淡化等措施
注：为使表达更清晰，可以在论文中采用图、表等形式
其中，表示州i 在2025年人口数量，表示该州在2010年人口数量，表示该州的人口平均年增长率则该州在2025年的居民用水量为
i Vri w N 2 0 0 5
其中，w 表示该州居民的平均年用水量
3. 2025年美国需要补充的淡水量

2013年最新美国数学竞赛试题及答案(AMC2013A)

A1Square has side length . Point is on , and the area of is . What is ?2A softball team played ten games, scoring , and runs. They lost by one run inexactly five gam es. In each of the other gam es, they scored twice as m any runs as their opponent.How many total runs did their opponents score?3 A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of thepink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?4What is the value of5Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their tripTom paid , Dorothy paid , and Sammy paid . In order to share the costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What is ?6In a recent basketball game, Shenille attem pted only three-point shots and two-point shots. She wassuccessful on of her three-point shots and of her two-point shots. Shenille attem pted shots. How m any points did she score?7The sequence has the property that every term beginning with the third is thesum of the previous two. That is, Suppose that and. What is ?8Given that and are distinct nonzero real numbers such that , what is ?9In , and . Points and are on sides , , and ,respectively, such that and are parallel to and , respectively. What is the perimeter of parallelogram?(9th)（11 th）10Let be the set of positive integers for which has the repeating decimal representationwith and different digits. What is the sum of the elem ents of ?11Triangle is equilateral with . Points and are on and points and are onsuch that both and are parallel to . Furthermore, triangle and trapezoidsand all have the sam e perimeter. What is ?12he angles in a particular triangle are in arithmetic progression, and the side lengths are . Thesum of the possible values of equals where , and are positive integers. What is ?13Let points and . Quadrilateral is cut intoequal area pieces by a line passing through . This line intersects at point , where thesefractions are in lowest term s. What is ?14The sequence, , , ,is an arithm etic progression. What is ?15Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?16, , are three piles of rocks. The m ean weight of the rocks in is pounds, the m ean weightof the rocks in is pounds, the m ean weight of the rocks in the com bined piles and ispounds, and the m ean weight of the rocks in the combined piles and is pounds. What is thegreatest possible integer value for the mean in pounds of the rocks in the com bined piles and ?17A group of pirates agree to divide a treasure chest of gold coins am ong them selves as follows. Thepirate to take a share takes of the coins that rem ain in the chest. The number of coins initially in the chest is the sm allest number for which this arrangement will allow each pirate to receive apositive whole number of coins. How many coins doe the pirate receive?18Six spheres of radius are positioned so that their centers are at the vertices of a regular hexagon of side length . The six spheres are internally tangent to a larger sphere whose center is the center ofthe hexagon. An eighth sphere is externally tangent to the six sm aller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?19In , , and . A circle with center and radius intersects at pointsand . Moreover and have integer lengths. What is ?20Let be the set . For , define to m ean that either or. How m any ordered triples of elem ents of have the property that ,, and ?21Consider . Which of the following intervals contains ?22A palindrome is a nonnegatvie integer number that reads t he sam e forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome is chosen uniformly at random. Whatis the probability that is also a palindrome?23is a square of side length . Point is on such that . The square regionbounded by is rotated counterclockwise with center , sweeping out a region whosearea is , where , , and are positive integers and . What is ?24Three distinct segm ents are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segm ents are the three side lengths of a triangle with positive area?25Let be defined by . How m any complex numbers are there suchthat and both the real and the imaginary parts of are integers with absolute value atmost ?1. E2. C3. E4. C5. B6. B7. C8. D9. C10. D11. C12. A13. B14. B15. D16. E17. D18. B19. D20. B21. A22. E23. C24. E25. A。

美赛历年题目2005—2013

MCM 2013 A题：最佳巧克力蛋糕烤盘当你使用一个矩形的烤盘烘烤食物时，热量会集中在烤盘的四个角落，于是角落处的食物就会被烤糊（烤盘边缘处也有类似情形，但程度轻一些）。

当使用一个圆形烤盘时，热量会均匀地分布在整个边缘上，就不会再有边缘上烤糊的现象发生。

然而，由于大多数烤箱内部是矩形的，如果使用圆形烤盘，就不能充分利用烤箱的内部空间了。

建立一个模型，来描述热量在不同形状的烤盘表面的分布。

这些形状包括矩形、圆形以及两者之间的过渡形状。

假设，1、矩形烤箱的宽长比为 W/L。

2、每个烤盘的面积为A。

3、先考虑烤箱内有两个搁架且间隔均匀的情形。

建立一个模型用以选择满足下列条件的最佳烤盘的形状：(1)、使得烤箱中可以容纳的烤盘数量(N)最大。

(2)、使得烤盘上的热量分布(H)最均匀。

3、综合(1)、(2)两个条件，并且为(1)、(2)分别设置权值p和(1-p)，寻求最优。

然后描述结果随着 W/L 和 p 的值的变化是如何变化的。

除了撰写 MCM 论文之外，你还要为新的一期巧克力蛋糕美食杂志准备一个一至两页的广告，阐述你的设计和结果的亮点所在。

MCM 2013 B题：水，水，无处不在淡水资源是世界上许多地方持续发展的限制因素。

建立数学模型来确定一个有效的，可行的，低成本的2013年用水计划，来满足某国(从下方的列表中选择一个国家)未来(2025年)的用水需求，并确定最优的淡水分配计划。

特别的，你的数学模型必须包括储存、运输、淡化和节水等环节。

如果可能的话，用你的模型来讨论你的计划对经济，自然和环境的影响。

提供一个非技术性的意见书给政府领导概述你的方法，以及方法的可行性和成本，以及它为什么是“最好的用水计划的选择”。

国家：美国、中国、俄罗斯、埃及或者沙特阿拉伯。

ICM 2013 C题：地球健康的网络建模背景：全社会都在关注如何研究与应用模型来预测我们地球的生物和环境的健康状况。

许多科学研究表明地球的环境和生物系统所面对的压力正在增加,但是能够验证这一观点的全局性模型却很少。

2013美国数学竞赛12年级B卷试题

INSTRUCTIONS1. DO NOT OPEN THIS BOOKLET UNTIL YOUR PROCTOR TELLS YOU.2. This is a twenty-five question multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.3. Mark your answer to each problem on the AMC 12 Answer Form with a #2 pencil. Check the blackened circles for accuracy and erase errors and stray marks completely. Only answers properly marked on the answer form will be graded.4. SCORING: You will receive 6 points for each correct answer, 1.5 points for each problem left unanswered, and 0 points for each incorrect answer.5. No aids are permitted other than scratch paper, graph paper, rulers, compass, protractors, and erasers. No calculators are allowed. No problems on the test will require the use of a calculator.6. Figures are not necessarily drawn to scale.7. Before beginning the test, your proctor will ask you to record certain information on the answer form.8. When your proctor gives the signal, begin working on the problems. You will have 75 minutes to complete the test.9. When you finish the exam, sign your name in the space provided on the Answer Form.© 2013 Mathematical Association of AmericaThe Committee on the American Mathematics Competitions (CAMC) reserves the right to re-examine students before deciding whether to grant official status to their scores. The CAMC also reserves the right to disqualify all scores from a school if it is determined that the required security procedures were not followed.Students who score 100 or above or finish in the top 5% on this AMC 12 will be invited to take the 31st annual American Invitational Mathematics Examination (AIME) on Thursday, March 14, 2013 or Wednesday, April 3, 2013. More details about the AIME and other information are on the back page of this test booklet.The publication, reproduction or communication of the problems or solutions of the AMC 12 during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination via copier, telephone, e-mail, World Wide Web or media of any type during this period is a violation of the competition rules. After the contest period, permission to make copies of problems in paper or electronic form including posting on web-pages for educational use is granted without fee provided that copies are not made ordistributed for profit or commercial advantage and that copies bear the copyright notice.**Administration On An Earlier Date Will Disqualify Your School’s Results**1. All information (Rules and Instructions) needed to administer this exam is contained in the TEACHERS’ MANUAL, which is outside of this package. PLEASE READ THE MANUAL BEFORE FEBRUARY 20, 2013. Nothing is needed from inside this package until February 20.2. Your PRINCIPAL or VICE-PRINCIPAL must verify on the AMC 12 CERTIFICATION FORM (found in the T eachers’ Manual) that you followed all rules associated with the conduct of the exam.3. The Answer Forms must be mailed by trackable mail to the AMC office no later than 24 hours following the exam.4. The publication, reproduction or communication of the problems or solutions of this test during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination at any time via copier, telephone, e-mail, internet or media of any type is a violation of the competition rules.2013AMC 12 B DO NOT OPEN UNTIL WEDNEsDAy, fEbrUAry 20, 2013The American Mathematics Competitionsare Sponsored byThe Mathematical Association of America – MAA The Akamai Foundation ContributorsAcademy of Applied Sciences – AAs American Mathematical Association of Two-Year Colleges – AMATyC ...................................................... American Mathematical Society – AMs ........................................................................................................... American Statistical Association – AsA ...................................................................................................... Art of Problem Solving – Awesome Math Casualty Actuarial Society – CAs ................................................................................................................ D.E. Shaw & Co. ................................................................................................................................. Delta Airlines Jane Street Math For America Mu Alpha Theta – MAT ....................................................................................................................... National Council of Teachers of Mathematics – NCTM ................................................................................... Pi Mu Epsilon – PME ............................................................................................................................... Society for Industrial and Applied Math - SIAM ............................................................................................ 1.On a particular January day,the high temperature in Lincoln,Nebraska,was16degrees higher than the low temperature,and the average of the high and low temperatures was3◦.In degrees,what was the low temperature in Lincoln that day?(A)−13(B)−8(C)−5(D)−3(E)112.Mr.Green measures his rectangular garden by walking two of the sides andﬁnds that it is15steps by20steps.Each of Mr.Green’s steps is2feet long.Mr.Green expects a half a pound of potatoes per square foot from his garden.How many pounds of potatoes does Mr.Green expect from his garden?(A)600(B)800(C)1000(D)1200(E)14003.When counting from3to201,53is the51st number counted.When countingbackwards from201to3,53is the n th number counted.What is n?(A)146(B)147(C)148(D)149(E)1504.Ray’s car averages40miles per gallon of gasoline,and Tom’s car averages10miles per gallon of gasoline.Ray and Tom each drive the same number of miles.What is the cars’combined rate of miles per gallon of gasoline?(A)10(B)16(C)25(D)30(E)405.The average age of33ﬁfth-graders is11.The average age of55of their parentsis33.What is the average age of all of these parents andﬁfth-graders?(A)22(B)23.25(C)24.75(D)26.25(E)286.Real numbers x and y satisfy the equation x2+y2=10x−6y−34.What isx+y?(A)1(B)2(C)3(D)6(E)87.Jo and Blair take turns counting from1to one more than the last number saidby the other person.Jo starts by saying“1”,so Blair follows by saying“1,2”.Jo then says“1,2,3”,and so on.What is the53rd number said?(A)2(B)3(C)5(D)6(E)88.Line 1has equation3x−2y=1and goes through A=(−1,−2).Line 2hasequation y=1and meets line 1at point B.Line 3has positive slope,goes through point A,and meets 2at point C.The area of ABC is3.What is the slope of 3?(A)23(B)34(C)1(D)43(E)329.What is the sum of the exponents of the prime factors of the square root of thelargest perfect square that divides12!?(A)5(B)7(C)8(D)10(E)1210.Alex has75red tokens and75blue tokens.There is a booth where Alex cangive two red tokens and receive in return a silver token and a blue token,and another booth where Alex can give three blue tokens and receive in return a silver token and a red token.Alex continues to exchange tokens until no more exchanges are possible.How many silver tokens will Alex have at the end?(A)62(B)82(C)83(D)102(E)10311.Two bees start at the same spot andﬂy at the same rate in the followingdirections.Bee A travels1foot north,then1foot east,then1foot upwards, and then continues to repeat this pattern.Bee B travels1foot south,then1 foot west,and then continues to repeat this pattern.In what directions are the bees traveling when they are exactly10feet away from each other?(A)A east,B west(B)A north,B south(C)A north,B west(D)A up,B south(E)A up,B west12.Cities A,B,C,D,and E are connected by roadsAB,AD, AE,BC,BD,CD, andDE.How many diﬀerent routes are there from A to B that use each road exactly once?(Such a route will necessarily visit some cities more than once.)(A)7(B)9(C)12(D)16(E)1813.The internal angles of quadrilateral ABCD form an arithmetic progression.Tri-angles ABD and DCB are similar with∠DBA=∠DCB and∠ADB=∠CBD.Moreover,the angles in each of these two triangles also form an arithmetic pro-gression.In degrees,what is the largest possible sum of the two largest angles of ABCD?(A)210(B)220(C)230(D)240(E)25014.Two non-decreasing sequences of nonnegative integers have diﬀerentﬁrst terms.Each sequence has the property that each term beginning with the third is the sum of the previous two terms,and the seventh term of each sequence is N.What is the smallest possible value of N?(A)55(B)89(C)104(D)144(E)27315.The number2013is expressed in the form2013=a1!a2!···a m!b1!b2!···b n!,where a1≥a2≥···≥a m and b1≥b2≥···≥b n are positive integers and a1+b1is as small as possible.What is|a1−b1|?(A)1(B)2(C)3(D)4(E)516.Let ABCDE be an equiangular convex pentagon of perimeter 1.The pairwise intersections of the lines that extend the sides of the pentagon determine a ﬁve-pointed star polygon.Let s be the perimeter of this star.What is the diﬀerence between the maximum and the minimum possible values of s ?(A)0(B)12(C)√5−12(D)√5+12(E)√517.Let a ,b ,and c be real numbers such thata +b +c =2,anda 2+b 2+c 2=12.What is the diﬀerence between the maximum and minimum possible values of c ?(A)2(B)103(C)4(D)163(E)20318.Barbara and Jenna play the following game,in which they take turns.A number of coins lie on a table.When it is Barbara’s turn,she must remove 2or 4coins,unless only one coin remains,in which case she loses her turn.When it is Jenna’s turn,she must remove 1or 3coins.A coin ﬂip determines who goes ﬁrst.Whoever removes the last coin wins the game.Assume both players use their best strategy.Who will win when the game starts with 2013coins and when the game starts with 2014coins?(A)Barbara will win with 2013coins,and Jenna will win with 2014coins.(B)Jenna will win with 2013coins,and whoever goes ﬁrst will win with 2014coins.(C)Barbara will win with 2013coins,and whoever goes second will win with 2014coins.(D)Jenna will win with 2013coins,and Barbara will win with 2014coins.(E)Whoever goes ﬁrst will win with 2013coins,and whoever goes second will win with 2014coins.19.In triangle ABC ,AB =13,BC =14,and CA =15.Distinct points D ,E ,and F lie on segments BC ,CA ,and DE ,respectively,such that AD ⊥BC ,DE ⊥AC ,and AF ⊥BF .The length of segment DF can be written as m n ,where m and n are relatively prime positive integers.What is m +n ?(A)18(B)21(C)24(D)27(E)3020.For135◦<x<180◦,points P=(cos x,cos2x),Q=(cot x,cot2x),R=(sin x,sin2x),and S=(tan x,tan2x)are the vertices of a trapezoid.What is sin(2x)?(A)2−2√2(B)3√3−6(C)3√2−5(D)−34(E)1−√321.Consider the set of30parabolas deﬁned as follows:all parabolas have as focusthe point(0,0)and the directrix lines have the form y=ax+b with a and b integers such that a∈{−2,−1,0,1,2}and b∈{−3,−2,−1,1,2,3}.No three of these parabolas have a common point.How many points in the plane are on two of these parabolas?(A)720(B)760(C)810(D)840(E)87022.Let m>1and n>1be integers.Suppose that the product of the solutions forx of the equation8(lognx)(log m x)−7log n x−6log m x−2013=0 is the smallest possible integer.What is m+n?(A)12(B)20(C)24(D)48(E)27223.Bernardo chooses a three-digit positive integer N and writes both its base-5and base-6representations on a ter LeRoy sees the two numbers Bernardo has written.Treating the two numbers as base-10integers,he adds them to obtain an integer S.For example,if N=749,Bernardo writes the numbers10,444and3,245,and LeRoy obtains the sum S=13,689.For how many choices of N are the two rightmost digits of S,in order,the same as those of2N?(A)5(B)10(C)15(D)20(E)2524.Let ABC be a triangle where M is the midpoint of AC,and CN is the anglebisector of∠ACB with N on AB.Let X be the intersection of the median BM and the bisector CN.In addition BXN is equilateral and AC=2.What is BN2?(A)10−6√27(B)29(C)5√2−3√38(D)√26(E)3√3−4525.Let G be the set of polynomials of the formP(z)=z n+c n−1z n−1+···+c2z2+c1z+50, where c1,c2,...,c n−1are integers and P(z)has n distinct roots of the form a+ib with a and b integers.How many polynomials are in G?(A)288(B)528(C)576(D)992(E)1056WRITE TO US!Correspondence about the problems and solutions for this AMC 12and orders for publications should be addressed to:American Mathematics CompetitionsUniversity of Nebraska, P .O. Box 81606Lincoln, NE 68501-1606Phone 402-472-2257 | Fax 402-472-6087 | amcinfo@The problems and solutions for this AMC 12 were prepared by the MAA’s Committee on theAMC 10 and AMC 12 under the direction of AMC 12 Subcommittee Chair:Prof. Bernardo M. Abrego2013 AIMEThe 31st annual AIME will be held on Thursday, March 14, with the alternate on Wednesday, April 3. It is a 15-question, 3-hour, integer-answer exam. You will be invited to participate only if you score 120 or above or finish in the top 2.5% of the AMC 10, or if you score 100 or above or finish in the top 5% of the AMC 12. T op-scoring students on the AMC 10/12/AIME will be selected to take the 42nd Annual USA Mathematical Olympiad (USAMO) on April 30 - May 1, 2013. The best way to prepare for the AIME and USAMO is to study previous exams. Copies may be ordered as indicated below.PUBLICATIONSA complete listing of current publications, with ordering instructions, is at our web site: American Mathematics Competitions。

2013年美国数学建模B题

PROBLEM B: Water, Water, EverywhereFresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, and cost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your approach, its feasibility andcosts, and why it is the “best water strategy choice.”Countries: United States, China, Russia, Egypt, or Saudi Arabia水, 水, 无处不在(美国竞赛2013年B题)淡水资源逐渐成为这个世界大多数国家发展的极限约束。

美赛2013

For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________
Team Control Number
For office use only F1 ________________ F2 ________________
Linear Programming Analogy method Auto CAD 2010 Simulation
Dimensionless method
Team #20426
Page 1 of 21
Introduction
Selection of pan has been a important aspect for bakery, the good pan can not only make food tastier, but also improve the efficiency of the baking and save the power, and thus bring the greater benefits to the operators. The bakery is an urgent need to establish a model to analyze the distribution of the different shapes of the hotplate heat as well as the maximum number can be placed in an oven. By analyzing phenomenon of the point effect analogies the electrostatic field, we construct the heat field to analysis tip thermal effect, and take some properties of the electric field and equation analog-to-heat field, constructing their own equations, and mapping the heat field .By using matlab software find it can be used to reflect the actual situation. By analyzing and comparing the related data, we choose the most popular the length to width ratio of the oven. Based on this condition, we analysis the shape of pans under the ratio of the condition of the number of the oven, and get length to width ratio, the shape of the pan with maximum the number. In order to solve problem 2，we build two models. The first model, we analyze the edge of the heat distribution of the different shape of the pan, the second model, by analyzing the change of the uniform distribution on the number of the pans aபைடு நூலகம்d thermal the weight responding to the best shape of the pan

2013美赛A题

2013美赛A题PROBLEM A: The Ultimate Brownie PanWhen baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven.Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.Assume1. A width to length ratio of W/L for the oven which is rectangular in shape.2. Each pan must have an area of A.3. Initially two racks in the oven, evenly spaced. Develop a model that can be used to select the best type of pan (shape) under the following conditions:1. Maximize number of pans that can fit in the oven (N)2. Maximize even distribution of heat (H) for the pan3. Optimize a combination of conditions (1) and (2) where weights p and (1- p) are assigned to illustrate how the results vary with different values of W/L and p.In addition to your MCM formatted solution, prepare a one to two page advertising sheet for the new Brownie Gourmet Magazine highlighting your design and results.。

2013美赛A题

PROBLEM A: The Ultimate Brownie PanWhen baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven.Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.Assume1. A width to length ratio of W/L for the oven which is rectangular in shape.2. Each pan must have an area of A.3. Initially two racks in the oven, evenly spaced.Develop a model that can be used to select the best type of pan (shape) under the following conditions:1. Maximize number of pans that can fit in the oven (N)2. Maximize even distribution of heat (H) for the pan3. Optimize a combination of conditions (1) and (2) where weights p and (1- p) are assigned to illustrate how the results vary with different values of W/L and p.In addition to your MCM formatted solution, prepare a one to two page advertising sheet for the new Brownie Gourmet Magazine highlighting your design and results.。