2013美赛题目
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
2013 Contest ProblemsMCM PROBLEMSPROBLEM 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.当在一个矩形的锅中烘烤时热量集中在烤箱的4个角落中，并在角落处的食物烤的过了（在某种程度上边缘也是这样）。
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
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
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Problem A:
最终布朗尼托盘当我们在一个矩形盘中烘焙东西的时候热量主要集中在平底锅的4个角，所以东西的四个角容易糊（4个边上也容易糊但是比4个角稍微好点）。
而在圆形的盘中，热量是平均分配在外边缘上，因此东西不容易糊。
不过，因为大多数烤箱是矩形状的，所以使用圆形的托盘相比矩形的托盘比较浪费空间。
请建立一个模型来讨论在不同形状托盘的外边缘上热量的分配----矩形到圆形以及之间所有的形状。
条件：
1.矩形烤箱宽度和长度的比值（W/L）
2.任何托盘的面积都是A
3.有两个放置托盘的烤架，同样的面积
请建立一个模型并得到一个最终的托盘形状，并符合下列条件
1.使烤箱能容下的托盘数量（N）最多
2.使托盘的热量（H）分配最平均
3.请结合起来优化1.2两个条件（给你权重p（1的权重）和1-p（2的权重）描述最后的结果与W/L以及p的函数关系）
Problem B:
淡水资源是很多地区发展的限制条件。
请建立一个数学模型来得到一个有效的，可用的并且花费最少的2013年水资源策略以实现以下国家（任选其一）期望在2025年的水资源需求，并且找到最好的策略。
特别地，你的数学模型必须有水的储量和转移；水的去盐渍化；和保护。
如果可能，用你的模型来讨论你的策略涉及的经济，物理和环境方面。
提供一个非技术的现状分析报告给你的政府领导，其中描述你的策略的方法，可行性和花费，以及它为什么是最好的选择。
国家：美国，中国，俄罗斯，埃及，沙特阿拉伯。