2015_HiMCM_Problems
mathematica中findroot函数
mathematica中findroot函数FindRoot函数是Wolfram语言中的一个重要函数，它用于求解非线性方程组的数值解。
FindRoot函数的语法如下：FindRoot[f[x], {x, x0}]其中，f[x]是待求解的非线性方程或方程组，{x,x0}是初始解的范围，x0是初始解的初始值。
一般而言，FindRoot函数会尝试找到满足f[x]==0的解x，它采用了牛顿法或其变种的方法进行迭代求解。
牛顿法的基本思想是从初始解开始不断迭代，通过计算函数f[x]在当前解x附近的导数和函数值，来逼近方程f[x]==0的解。
下面举一个简单的例子来说明FindRoot函数的用法：```f[x_]:=x^3-2x-5;x0=2;sol = FindRoot[f[x], {x, x0}]```在使用FindRoot函数时，可以考虑以下几个因素：1.初始解的选择:初始解的选择会影响到迭代的速度和解的精度，一般情况下，可以先通过绘制函数图像来观察大致的解的位置，然后选择一个合适的初始解。
2.初始解的范围：可以通过设定{x,x0}的范围来限制解的区域，从而避免解的发散和无限迭代。
3. 可选参数：FindRoot函数还提供了多个可选参数，比如精度控制选项WorkingPrecision、迭代次数控制选项MaxIterations等。
可以根据具体需要进行调整。
4. 复杂方程组的求解：如果需要求解的是多元非线性方程组，可以将方程组写成向量形式，并定义一个向量函数，然后用向量作为参数调用FindRoot函数。
下面给出一个更复杂的例子来说明FindRoot函数的用法：```f[x_, y_] := {x^2 + y^2 - 1, Exp[x] + y - 2};{x0,y0}={-1,1};sol = FindRoot[f[x, y], {{x, x0}, {y, y0}}]```可以看到，FindRoot函数可以有效地求解非线性方程组的数值解，尤其是对于一些复杂的方程组，FindRoot函数提供了一个方便且高效的求解工具。
2016Himcm赛题A
2016HiMCM Problem A: Swim, Bike, and Run赛题A: 游泳、自行车、跑步A triathlon is a multiple-stage athletic endurance competition of three continuous and sequential events, usually long distance swimming, cycling, and running. Triathletes’overall course completion time includes their time for each event plus their time for transitions between the three events.铁人三项是多级别的运动耐力比赛，包括三个连续相继的项目，通常是长距离游泳、自行车和跑步。
运动员的总完成时间包括每项耗费时间及换项耗费时间。
Race organizers provide each participant a transition area where he/she can pre-position a bike, running shoes, performance gear and other equipment needed to transition from swimming to cycling and from cycling to running. An athlete’s time in the transition area (denoted as T1 for swimming to cycling and T2 for cycling to running) counts toward the total race time. The race begins with the swimming event and participants start the race in a sequence of waves of groups of swimmers at intervals of some number of minutes apart.主办方给每个参与者提供换项区域，他/她可以预先放置一辆自行车、跑鞋、户外装备和其他从游泳转换到骑车，从骑车转换到跑步的其他工具。
2017HiMCM竞赛AB题
2017 HiMCMProblem A: Drone Clusters as Sky Light DisplaysIntel®developed its Shooting Star™ drone and is using clusters of these drones for aerial light shows. In 2016, a cluster of 500 drones, controlled by a single laptop and one pilot, performed a beautifully choreographed light show (https:///watch?v=jNIAzeU8POQ).Your large city has an annual festival and is considering adding an outdoor aerial light show. The Mayor has asked your team to investigate the idea of using drones to create three possible sky displays.Part I– For each display:a) Determine the number of drones required and mathematically describe the initiallocation for each drone device that will result in the sky display (similar to a fireworksdisplay) of a static image.b) Determine the flight paths of each drone or set of drones that would animate yourimage and describe the animation. (Note that you do not have to actually write a program to animate the image, but you do need to mathematically describe the flight paths.)Display 1: Ferris wheelDisplay 2: DragonDisplay 3: Create your own imagePart II– Determine and discuss the requirements for your 3-display light show to include, but not limited to, the number of drones, required launch area, required air space, safety considerations, and duration of the aerial light show.Part III– Write a two-page memo to the Mayor to report the results of your investigation and make a recommendation as to whether or not to do the aerial light show.Your submission should consist of:∙One-page Summary Sheet,∙Two-page memo to the Mayor,∙Your solution of no more than 30 pages, for a maximum of 33 pages with your summary and memo.∙Note: Reference list and any appendices do not count toward the 33-page limit and should appear after your completed solution.2017 HiMCMProblem B: Ski SlopeWinter is coming! In February 2018, PyeongChang, South Korea will host the Winter Olympics. And, in 2022, Beijing, China will be the host city. The Winter Olympics have over fifty ski related events in the disciplines of Alpine, Nordic, Cross-Country, Ski Jumping, Snowboarding, and Freestyle.A group of wealthy winter sport fans are looking for a new mountain to develop into a ski resort that could perhaps host the Winter Olympics in the future. An agent, calling herself Ms. Mogul, represents them.Wasatch Peaks Ranch in Peterson, Utah, USA is for sale! This almost 13,000 acre ranch has an estimated 5,500 acres of potential ski slopes with an 11 mile ridgeline, a 4750 foot drop among its 24 peaks, and 15 bowls. Ms. Mogul wants your team to identify potential ski slopes and trails on the property in order to develop it as one of the top ski resorts in North America and a potential future Winter Olympics location.Part I– Given a brochure for Wasatch Peaks Ranch, a topographic map of this area, a partial list of North American ski resorts with comparison data, and other information available on the web, design the new Wasatch Peaks Ranch ski area to meet the following criteria:- Main slopes of varying lengths- Plenty of trails- A total of at least 160 km of slopes (main slopes and trails)- Distribution of slopes at approximately 20% rated beginner (● green circle), 40% rated intermediate (■ blue square), and 40% rated difficult (♦ black diamond).Part II– Rank your proposed ski area against existing ski areas/resorts in North American. Part III– Write a two-page memo to Ms. Mogul reporting the results of your design and the ranking of your proposed ski area.Your submission should consist of:∙One-page Summary Sheet,∙Two-page memo to Ms. Mogul,∙Your solution of no more than 30 pages, for a maximum of 33 pages with your summary and memo.∙Note: Reference list and any appendices do not count toward the 33-page limit and should appear after your completed solution.NOTE:Ski trail difficulty is measured by percent slope, not degree angle. A 100% slope is a 45-degree angle. In other words, when rise/run = 1, the slope is 100%. In general, beginner slopes (● green circle) are between 6% and 25%. Intermediate slopes (■ blue square) are between 25% and 40%. Difficult slopes (♦ black diamond) are 40% and higher. However, this is just a general "rule ofthumb." Although slope gradient is the primary consideration in assigning a trail difficulty rating, other factors come into play. A trail will be rated by its most difficult part, even if the rest of the trail is easy. Ski resorts often assign ratings to their own trails, rating a trail compared only with other trails at that resort. Also considered: width of the trail, sharpest turns, terrain roughness, and whether the resort regularly grooms the trail. Note that you may see differing symbols and colors in your research. Table 1 shows three examples of difficulty rating symbols.Attachments:Wasatch Peaks Ranch BrochureTopographic Map of Wasatch Peaks RanchSkiSlopeComparison.xlsReferences:/https:///ranches/wasatch-peaks-ranch/https:///ranches/wasatch-peaks-ranch/#prop-mapsWasatch Peaks RanchOgden, UT。
2015年AMC8试题及答案
Copyright © 2016 Art of Problem Solving How many square yards of carpet are required to cover a rectangular floor that is feet long and feetwide? (There are 3 feet in a yard.)First, we multiply to get that you need square feet of carpet you need to cover. Since thereare square feet in a square yard, you divide by to get square yards, so our answer is .Since there are feet in a yard, we divide by to get , andby to get . To find the area of the carpet, we then multiply these two values together to get .2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byFirst Problem Followed by Problem 21 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21• 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Placement:Easy GeometryRetrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_1&oldid=80323"SolutionSolution 2See AlsoPoint is the center of the regular octagon , and is the midpoint of the side What fraction of the area of the octagon is shaded?Since octagon is a regular octagon, it is split into equal parts, such as triangles, etc. These parts, since they are all equal, are of the octagon each. The shaded region consists of of these equal parts plus half of another, so the fraction of the octagon that is shaded isThe octagon has been divided up into identical triangles (and thus they each have equal area). Since the shaded region occupiesout of the total triangles, the answer is .For starters what I find helpful is to divide the whole octagon up into triangles as shown here:Now it is just a matter of counting the larger triangles remember that and are notfull triangles and are only half for these purposes. We count it up and we get a total ofof the shape shaded. We then simplify it to get our answer of .2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 1Followed by Problem 31 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsSolution 3See AlsoCopyright © 2016 Art of Problem Solving Jack and Jill are going swimming at a pool that is one mile from their house. They leave home simultaneously. Jill rides her bicycle to the pool at a constant speed of miles per hour. Jack walks tothe pool at a constant speed of miles per hour. How many minutes before Jack does Jill arrive?Using , we can set up an equation for when Jill arrives at the swimming pool:Solving for , we get that Jill gets to the pool inof an hour, which is minutes. Doing the same for Jack, we get that Jack arrives at the pool inof an hour, which in turn is minutes. Thus, Jill has to waitminutes for Jack to arrive at the pool.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 2Followed by Problem 41 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and Solutions The problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_3&oldid=81064"SolutionSee AlsoCopyright © 2016 Art of Problem SolvingThe Centerville Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy ateach end and the three girls in the middle. How many such arrangements are possible?There are ways to order the boys on the end, and there are ways to order the girls in the middle.We get the answer to be .2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 3Followed by Problem 51 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and Solutions The problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_4&oldid=73224"SolutionSee AlsoCopyright © 2016 Art of Problem SolvingBilly's basketball team scored the following points over the course of the first 11 games of the season:If his team scores 40 in the 12th game, which of the following statistics will show an increase?When they score a on the next game, the range increases from to . This means the increased.Because is less than the score of every game they've played so far, the measures of center will neverrise. Only measures of spread, such as the, may increase.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 4Followed by Problem 61 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22• 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_5&oldid=80825"SolutionSolution 2Copyright © 2016 Art of Problem Solving In , , and . What is the area of?We know the semi-perimeter of is . Next, we use Heron's Formula to find that the area of the triangle is just .Splitting the isosceles triangle in half, we get a right triangle with hypotenuseand leg . Using the Pythagorean Theorem , we know the height is. Now that we know the height, the area is.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 5Followed by Problem 71 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 •24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_6&oldid=80483"Solution 1Solution 2See AlsoEach of two boxes contains three chips numbered , , . A chip is drawn randomly from each box and thenumbers on the two chips are multiplied. What is the probability that their product is even?We can instead find the probability that their product is odd, and subtract this from . In order to get an odd product, we have to draw an odd number from each box. We have a probability of drawing an odd numberfrom one box, so there is a probability of having an odd product. Thus, there is a probability of having an even product.You can also make this problem into a spinner problem. You have the first spinner with equally divided sections, andYou make a second spinner that is identical to the first, with equal sections of ,, and . If the first spinner lands on , to be even, it must land on two. You write down the first combination of numbers . Next, if the spinner lands on , it can land on any number on the second spinner. We now have the combinations of . Finally, if the first spinner ends on , we have Since there arepossible combinations, and we have evens, the final answer is.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 6Followed by Problem 81 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_7&oldid=73737"SolutionSolution 2See AlsoCopyright © 2016 Art of Problem Solving What is the smallest whole number larger than the perimeter of any triangle with a side of length and aside of length ?We know from the triangle inequality that the last side, , fulfills . Adding to both sides of the inequality, we get , and becauseis the perimeter ofour triangle, is our answer.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 7Followed by Problem 91 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23• 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_8&oldid=78101"SolutionSee AlsoCopyright © 2016 Art of Problem Solving On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previousday. How many widgets in total had Janabel sold after working days?The sum of is The sum is just the sum of the first odd integers, which is2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 8Followed by Problem 101 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and Solutions The problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_9&oldid=79933"Solution 1Solution 2See AlsoCopyright © 2016 Art of Problem SolvingHow many integers betweenandhave four distinct digits?The question can be rephrased to "How many four-digit positive integers have four distinct digits?",since numbers between and are four-digit integers. There are choices for the first number, since it cannot be , there are only choices left for the second number since it must differ from the first, choices for the third number, since it must differ from the first two, and choices for the fourth number,since it must differ from all three. This means there are integersbetweenandwith four distinct digits.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 9Followed by Problem 111 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22• 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_10&oldid=81128"Solution 1See AlsoCopyright © 2016 Art of Problem SolvingIn the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and thefourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, whatis the probability that the plate will read "AMC8"?There is one favorable case, which is the license plate says "AMC8". We must now find how many total cases there are. There are choices for the first letter (since it must be a vowel), choices for the second letter (since it must be of consonants), choices for the third letter (since it must differ from the second letter), and choices for the number. This leads to total possible license plates. That means the probability of a license plate saying "AMC8" is.The probability of choosing A as the first letter is . The probability of choosing next is. Theprobability of choosing C as the third letter is(since there areother consonants to choose fromother then M). The probability of having as the last number is . We multiply all these to obtain2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 10Followed by Problem 121 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 •24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_11&oldid=73554"Solution 1Solution 2See AlsoHow many pairs of parallel edges, such asandorand, does a cube have?We first count the number of pairs of parallel lines that are in the same direction as. The pairs ofparallel lines are ,, , , ,and . These are pairs total. We can do the same for the lines in the same direction asand. This means there aretotal pairs of parallel lines.Pick a random edge. Given another edge, the probability that it is parallel to this edge is. Keep in mind we already used one edge. There are edges so pairs. So our answer is.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 11Followed by Problem 131 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsSolution 1Solution 2See AlsoCopyright © 2016 Art of Problem SolvingHow many subsets of two elements can be removed from the set so thatthe mean (average) of the remaining numbers is 6?Since there will be elements after removal, and their mean is , we know their sum is . We also knowthat the sum of the set pre-removal is . Thus, the sum of the elements removed is .There are onlysubsets of elements that sum to:.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 12Followed by Problem 141 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22• 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_13&oldid=73377"SolutionSee AlsoLet our numbers be , where is odd. Then our sum is . The only answer choice that cannot be written as , where is odd, is .If the four consecutive odd integers are and then the sum is . All the integers are divisible by except .If the four consecutive odd integers are and , the sum is , and divided by gives . This means that must be even. The only integer that does notgive an even integer when divided by is , so the answer is .From Solution 1, we have the sum of the numbers to be equal to . Taking mod 8 gives usfor some residue and for some odd integer . Since , we can express it as the equation for some integer . Multiplying 4 to each side of the equation yields , and taking mod 8 gets us , so . All the answer choicesexcept choice D is a multiple of 8, and since 100 satisfies all the conditions the answer is .The problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Copyright © 2016 Art of Problem SolvingAt Euler Middle School,students voted on two issues in a school referendum with the following results:voted in favor of the first issue and voted in favor of the second issue. If there were exactlystudents who voted against both issues, how many students voted in favor of both issues?We can see that this is a Venn Diagram Problem.First, we analyze the information given. There are students. Let's use A as the first issue and B asthe second issue.students were for the A, and students were for B. There were also students against both A andB.Solving this without a Venn Diagram, we subtract away from the total,. Out of the remaining,we havepeople for A andpeople for B. We add this up to get. Since that is more than what we need, we subtract fromto getThere are 198 people. We know that 29 people voted against both the first issue and the second issue. That leaves us with 169 people that voted for at least one of them. If 119 people voted for both of them, then that would leave 20 people out of the vote, because 149 is less than 198 people. 198-149 is 20, so to make it even, we have to take 20 away from the 119 people, which leaves us with2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 14Followed by Problem 161 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22• 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_15&oldid=80999"Solution 1Solution 2See AlsoCopyright © 2016 Art of Problem SolvingIn a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If of all the ninth graders are paired withof all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?Let the number of sixth graders be , and the number of ninth graders be . Thus, , whichsimplifies to. Since we are trying to find the value of, we can just substitute forinto the equation. We then get a value ofWe see that the minimum number of ninth graders is , because if there arethen there is ninth grader with a buddy, which would mean sixth graders with a buddy, and that's impossible. With ninth graders, of them are in the buddy program, so theresixth graders total, two of whom have a buddy. Thus,the desired fraction is .2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 15Followed by Problem 171 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22• 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_16&oldid=73512"Solution 1Solution 2See AlsoSo and .This gives , which gives , which then givesSolution 2, is obviously constantso , plug into the first one and it's miles to schoolWe set up an equation in terms of the distance and the speed In miles per hour. We have SoAn arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, is an arithmetic sequence with five terms, in which thefirst term is and the constant added is . Each row and each column in this array is an arithmetic sequence with five terms. What is the value of ?We begin filling in the table. The top row has a first term and a fifth term , so we have the common difference is . This means we can fill in the first row of the table:The fifth row has a first term of and a fifth term of , so the common difference is. We can fill in the fifth row of the table as shown:Copyright © 2016 Art of Problem SolvingWe must find the third term of the arithmetic sequence with a first term of and a fifth term of . The common difference of this sequence is, so the third term is.The middle term of the first row is, since the middle number is just the average in anarithmetic sequence. Similarly, the middle of the bottom row is . Applying this again forthe middle column, the answer is.The value ofis simply the average of the average values of both diagonals that contain. This is2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 17Followed by Problem 191 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24• 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_18&oldid=73460"Solution 2Solution 3See AlsoA triangle with vertices as , , and is plotted on a grid. What fraction of the grid is covered by the triangle?The area of is equal to half the product of its base and height. By the Pythagorean Theorem, we find its height is , and its base is . We multiply these and divideby to find the of the triangle is . Since the grid has an area of , the fraction of the grid covered by the triangle is .Note angle is right, thus the area is thus the fraction of the total isCopyright © 2016 Art of Problem SolvingBy the Shoelace theorem, the area of.This means the fraction of the total area isThe smallest rectangle that follows the grid lines and completely encloses has an area of,where splits the rectangle into four triangles. The area ofis therefore . That means thattakes upof the grid.Using Pick's Theorem, the area of the triangle is. Therefore, the triangle takes upof the grid.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 18Followed by Problem 201 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22• 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_19&oldid=75879"Solution 4Solution 5See AlsoRalph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If hebought at least one pair of each type, how many pairs of $1 socks did Ralph buy?So let there be pairs of socks, pairs ofsocks, pairs of socks.We have,, and.Now we subtract to find , and . It follows that is a multiple of and is amultiple of , so since , we must have .Therefore, , and it follows that. Now, asdesired.Since the total cost of the socks was and Ralph boughtpairs, the average cost of each pair ofsocks is.There are two ways to make packages of socks that average to . You can have:Twopairs and one pair (package adds up to )Onepair and onepair (package adds up to)So now we need to solvewhere is the number of packages and is the number of packages. We see our only solution (thathas at least one of each pair of sock) is , which yields the answer of.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 19Followed by Problem 211 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19• 20 • 21 • 22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?2015 AMC 8 Problems/Problem 20Solution 1Solution 2See AlsoIn the given figure hexagon is equiangular, andare squares with areasandrespectively, is equilateral and . What is the area of ?.Clearly, since is a side of a square with area , . Now, since , we have .Now, is a side of a square with area, so . Since is equilateral, . Lastly, is a right triangle. We see that, so is a right triangle with legs and . Now, its area is .2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded byProblem 20Followed byProblem 221 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 •21•22 • 23 • 24 • 25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'s American Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_21&oldid=73361"2015 AMC 8 Problems/Problem 21SolutionSee AlsoCopyright © 2016 Art of Problem SolvingCopyright © 2016 Art of Problem SolvingOn June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This processcontinues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in thegroup?As we read through this text, we find that the given information means that the number of students in thegroup has factors, since each arrangement is a factor. The smallest integer with factors is.2015 AMC 8 (Problems • Answer Key • Resources(/Forum/resources.php?c=182&cid=42&year=2015))Preceded by Problem 21Followed by Problem 231 •2 •3 •4 •5 •6 •7 •8 •9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 •19 • 20 • 21 • 22 • 23 • 24 •25All AJHSME/AMC 8 Problems and SolutionsThe problems on this page are copyrighted by the Mathematical Association of America ()'sAmerican Mathematics Competitions ().Retrieved from "/wiki/index.php?title=2015_AMC_8_Problems/Problem_22&oldid=73359"SolutionSee Also。
HiMCM真题
Adolescent PregnancyYou are working temporarily for the Department of Health and Environmental Control. The director is concerned about the issue of teenage pregnancy in their region. You have decided that your team will analyze the situation and determine if it is really a problem in this region. You gather the following 2000 data.County Age10-14PregnantAge15-17PregnantAge18- 19PregnantAge10-14birthsAge15-17birthsAge18-19birthsAge10-14births-unmarriedAge15-17births-unmarriedAge18-19births-unmarried1 29 350 571 17 281 437 16 164 1932 24 303 567 13 206 466 13 151 2333 40 422 691 29 307 546 28 251 3664 21 201 356 18 184 326 15 137 1805 16 156 357 11 109 254 10 99 1616 44 523 970 33 442 803 32 293 3967 17 263 434 9 201 345 7 113 1688 23 330 427 16 256 444 14 160 2109 13 123 221 10 113 199 9 78 10610 41 467 950 24 446 686 22 279 33111 28 421 713 18 343 615 15 219 32812 9 179 311 8 145 261 7 114 1621998Age Pregnancies Births10-14 320 23115-17 4041 322218-19 6387 51641999Age Pregnancies Births10-14 309 20815-17 3882 304818-19 6714 5391Build a mathematical model and use it to determine if there is a problem or not. Prepare an article for the newspaper that highlights your result in a novel mathematical relationship or comparison that will capture the attention of the youth.A South Sea Island ResortProblem:A south sea island chain has decided to transform one of their islands into a resort. This roughly circular island, about 5 kilometers across, contains a mountain that covers the entire island. The mountain is approximately conical, is about 1000 meters high at the center, appears to be sandy, and has little vegetation on it. It has been proposed to lease some fire-fighting ships and wash the mountain into the harbor. It is desired to accomplish this as quickly as possible.Build a mathematical model for washing away the mountain. Use your model to respond to the questions below.∙How should the stream of water be directed at the mountain, as a function of time?∙How long will it take using a single fire-fighting ship?∙Could the use of 2 (or 3, 4, etc.) fire-fighting ships decrease the time by more than a factor of 2 (or 3, 4, etc.)?∙Make a recommendation to the resort committee about what do.Bank RobbersThe First National Bank has just been robbed (the position of the bank on the map is marked). The clerk pressed the silent alarm to the police station. The police immediately sent out police cars to establish road blocks at the major street junctions leading out of town. Additionally, 2 police cars were dispatched to the bank.See the attached map.The Bank is located at the corner of 8th Ave. and Colorado Blvd. and is marked with the letter B. The main exits where the two road blocks are set up are at the intersection of Interstate 70 and Colorado Blvd, and Interstate 70 (past Riverside Drive). These are marked with a RB1 and RB2 symbol.∙Assume the robbers left the bank just before the police cars arrived. Develop an efficient algorithm for the police cars to sweep the area in order to force the bank robbers (who werefleeing by car) into one of the established road blocks.∙Assume that no cars break down during the chase or run out of gas.∙Further assume that the robbers do not decide to flee via other transportation means.Design of an Airline TerminalThe design of airline terminals varies widely. The sketches below show airline terminals from several cities. The designs are quite dissimilar. Some involve circular arcs; others are rectangular; some are quite irregular. Which is optimal for operations? Develop a mathematical model for airport design andoperation.Use your model to argue for the optimality of your specified design. Explain how it wouldoperate.Boston-Logan InternationalMunich InternationalCharlotte/Douglas InternationalRonald Reagan Washington National Pittsburgh InternationalForest ServiceYour team has been approached by the Forest Service to help allocate resources to fight wildfires.In particular, the Forest Service is concerned about wildfires in a wilderness area consisting of small trees and brush in a park shaped like a square with dimensions 80 km on a side. Several years ago, the Forest Service constructed a network of north-south and east-west firebreaks that form a rectangular grid across the interior of the entire wilderness area. The firebreaks were built at 5 km intervals.Wildfires are most likely to occur during the dry season, which extends from July through September in this particular region. During this season, there is a prevailing westerly wind throughout the day. There are frequent lightning bursts that cause wildfires.The Forest Service wants to deploy four fire-fighting units to control fires during the next dry season. Each unit consists of 10 firefighters, one pickup truck, one dump truck, one water truck (50,000 liters), and one bulldozer (w/ truck and trailer). The unit has chainsaws, hand tools, and other fire-fighting equipment. The people can be quickly moved by helicopter within the wilderness area, but all the equipment must be driven via the existing firebreaks. One helicopter is on standby at all times throughout the dry season.Your task is to determine the best distribution of fire-fighting units within the wilderness area. The Forest Service is able to set up base camps for those units at sites anywhere within the area. In addition, you are asked to prepare a damage assessment forecast. This forecast will be used to estimate the amount of wilderness likely to be burned by fire as well as acting as a mechanism for helping the Service determine when additional fire-fighting units need to be brought in from elsewhere.Gas Prices, Inventory, National Disasters, and the Mighty Dollar Information:It appears from the economic reports that the world uses gasoline on a very short supply and demand scale. The impact of any storm, let alone Hurricane Katrina, affects the costs at the pumps too quickly. Let’s re strict our study to the continental United States.Over the past six years, Canada has been the leading foreign supplier of oil to the United States, including both crude and refined oil products. (Petroleum Supply Monthly, Table S3 - Crude Oil and Petroleum Product Imports, 1988-Present. See page 5 for Canadian exports to the United States.)∙Canada was the largest foreign supplier of oil to the United States again in 2004, for the sixth year running (from1999, when the country displaced Venezuela, to 2004 inclusive).∙In 2002, Canada supplied the United States with 17 percent of its crude and refined oil imports — more than any other foreign supplier at over 1.9 million barrels per day.∙Western Canadian crude oil is imported principally by the U.S. Midwest and the Rocky Mountain states.∙Eastern Canada's offshore oil is imported principally by the U.S. East Coast states, and even by some Gulf Coast states.Many refiners are buying enough to serve motorists' current needs, but not enough to rebuild stocks. "They are looking to buy the oil when they need it,” according to The Washington Post. "When they are uncertain about the future, they hold back." (The Washington Post: Crude Oil Imports to U.S. Slow With War 3/31/03.)Build a better model for the oil industry for its use and consumption in the United States that is fair to both the business and the consumer. You can build your model based on a peak day.Write a letter to the President’s energy advisor summarizi ng your findinHow fair are major league baseball parks to the players?Consider the following major league baseball parks: Atlanta Braves, Colorado Rockies, New York Yankees, California Angles, Minnesota Twins, and Florida Marlins.Each field is in a different location and has different dimensions. Are all these parks “fair”? Determine how fair or unfair is each park. Determine the optimal baseball “setting” for major league baseball.Outfield DimensionsWall HeightFranchise LeftFieldLeftCenter Center FieldRightCenterRightFieldLeftFieldCenter FieldRightFieldArea of FairTerAngels 330 376 408 361 330 8 8 18 110,000 Braves 335 380 401 390 330 8 8 8 115,000 Rockies 347 390 415 375 350 8 8 14 117,000 Yankees 318 399 408 385 314 8 7 10 113,000 Twins 343 385 408 367 327 13 13 23 111,000 Marlins 330 385 404 385 345 8 8 8 115,000Modeling Ocean Bottom TopographyBackground Information:A marine survey ship maps ocean depth by using sonar to reflect a sound pulse off the ocean floor. Figure A shows the ship’s location atB on the surface of the ocean. The sonar apparatus aboard the ship is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. In two dimensions thisarc is shown withinFigure A triangle by , and the emanating sound pulses are displayed bythe dashed lines and the solid lines BA andBC.When a sonar sound pulse hits the bottom of the ocean, the pulse is reflected off the ocean bottom thesame way a billiard ball is reflected off a pool table; that is, the angle of incidenceequals the angle ofreflection as illustrated in Figure B. Since the ship is moving when the sound pulse is emitted, it will pick up a reflected sound pulse at location F in this picture. The actual depth of the water is the length of BD in Figure A.Figure AFigure BUseful Information:Oceanography vessels usually travel at a speed of 2m/s while Navy vessels travel at 20m/s. The sonar apparatus aboard these ships is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. The typical speed at which a sonar sound pulse is emitted is 1500m/s.Devise a model for mapping the topography of the ocean bottom. Write a letter to the science editor of your local paper summarizing your findings.Motel Cleaning ProblemMotels and hotels hire people to clean the rooms after each evening’s use. Develop a mathematical model for the cleaning schedule and use of cleaning resources. Your model should include consideration of such things as stay-overs, costs, number of rooms, number of rooms per floor, etc. Draft a letter to the manger of a major motel or hotel complex that recommends your model to help them in the management of their operation.School BusingConsider a school where most of the students are from rural areas so they must be bused. The buses might pick up all the students and go to the elementary school and then continue from that school to pick up more students for the high school.A clear alternative would be to have separate buses for each school even though they would need to trace over the same routes. There are, of course, restrictions on time (no student should be in the bus more than an hour), drivers, equipment, money and so forth.How can you set up school bus routes to optimize budget dollars while balancing the time on the bus for various school groups? Build a mathematical model that could be used by various rural and perhaps urban school districts. How would you test the model prior to implementation? Prepare a short article to the school board explaining your model, its assumptions, and its results.SkyscrapersSkyscrapers vary in height , size (square footage), occupancy rates, and usage. They adorn the skyline of our major cities. But as we have seen several times in history, the height of the building might preclude escape during a catastrophe either human or natural (earthquake, tornado, hurricane, etc). Let's consider the following scenario. A building (a skyscraper) needs to be evacuated. Power has been lost so the elevator banks are inoperative except for use by firefighters and rescue personnel with special keys.Build a mathematical model to clear the building within X minutes. Use this mathematical model to state the height of the building, maximum occupation, and type of evacuation methods used. Solve your model for X = 15 minutes, 30 minutes, and 60 minutes.The Art Gallery Security SystemAn art gallery is holding a special exhibition of small watercolors. The exhibition will be held in a rectangular room that is 22 meters long and 20 meters wide with entrance and exit doors each 2 meters wide as shown below. Two security cameras are fixed in corners of the room, with the resulting video being watched by an attendant from a remote control room. The security cameras give at any instant a “scan beam” of 30°. They rotate backwards and forwards over the field of vision, taking 20 seconds to complete one cycle.For the exhibition, 50 watercolors are to be shown. Each painting occupies approximately 1 meter of wall space, and must be separated from adjacent paintings by 1 meter of empty wall space and hang 2 meters away from connecting walls. For security reasons, paintings must be at least 2 meters from the entrances. The gallery also needs to add additional interior wall space in the form of portable walls. The portable walls are available in 5-meter sections. Watercolors are to be placed on both sides of these walls.To ensure adequate room for both patrons who are walking through and those stopped to view, parallel walls must be at least 5 meters apart throughout the gallery. To facilitate viewing, adjoining walls should not intersect in an acute angle.The diagrams below illustrate the configurations of the gallery room for the last two exhibits. The present exhibitor has expressed some concern over the security of his exhibit and has asked the management to analyze the security system and rearrange the portable walls to optimize the security of the exhibit.Define a way to measure (quantify) the security of the exhibit for different wall configurations. Use this measure to determine which of the two previous exhibitions was the more secure. Finally, determine an optimum portable wall configuration for the watercolor exhibit based on your measure of security.Falling LadderA ladder 5 meters long is leaning against a vertical wall with its foot on a rug on the floor. Initially, the foot of the ladder is 3 meters from the wall. The rug is pulled out, and the foot of the ladder moves away from the wall at a constant rate of 1 meter per second. Build a mathematical model or models for the motion of the ladder. Use your model (or models) to find the velocity at which the top of the ladder hits the floor and the distance the top of the ladder will be from the wall at the moment that it hits the ground.Traffic LightsMajor thoroughfares in big cities are usually highly congested. Traffic lights are used to allow cars to cross the highway or to make turns onto side streets. During commuting hours, when the traffic is much heavier than on any cross street, it is desirable to keep traffic flowing as smoothly as possible. Consider a two-mile stretch of a major thoroughfare with cross streets every city block. Build a mathematical model that satisfies both the commuters on the thoroughfare as well as those on the cross streets trying to enter the thoroughfare as a function of the traffic lights. Assume there is a light at every intersection along your two-mile stretch.First, you may assume the city blocks are of constant length. You may then wish to generalize to blocks of variable lengthWhat Is It Worth?In 1945, Noah Sentz died in a car accident and his estate was handled by the local courts. The state law stated that 1/3 of all assets and property go to the wife and 2/3 of all assets go to the children. There were four children. Over the next four years, three of the four children sold their shares of the assets back to the mother for a sum of $1300 each. The original total assets were mainly 75.43 acres of land. This week, the fourth child has sued the estate for his rightful inheritance from the original probate ruling. The judge has ruled in favor of the fourth son and has determined that he is rightfully due monetary compensation. The judge has picked your group as the jury to determine the amount of compensation.Use the principles of mathematical modeling to build a model that enables you to determine the compensation. Additionally, prepare a short one-page summary letter to the court that explains your results. Assume the date is November 10, 2003.。
2018HiMCM竞赛A题翻译
2017 HiMCMA题: 用作空中灯光显示屏的无人机集群
因特尔开发了自家Shooting Star™品牌的无人机，并把无人机集群用于空中灯光秀。
在2016年，一个500只无人机的集群，由一台笔记本和一个操作员控制，上演了一场精美的灯光秀(https:///watch?v=jNIAzeU8POQ).
你所在的大城市考虑在当地的年度活动中添加一场天空灯光秀。
市长要求你的团队评估一下使用无人机做三次空中灯光展示的想法。
第一部分：展示方案
a) 确定组成每次天空显示（类似烟花表演）的静态图案所需的无人机数量，以及用数学方法描述每架无人机的初始位置
b) 确定图案动画所需的无人机组或者单架无人机的飞行路线，并描述动画过程(你并不需要写出实际的动画程序，但是需要用数学方法描述飞行轨迹)
显示方案1: 摩天轮
显示方案2：龙形图案
显示方案3：自定义图案
第二部分：讨论决定三次灯光秀的要求，
包括但不限于：无人机数量，起降区域，所需空间，需要考虑的安全要求以及灯光秀时长
第三部分：给市长写一份两页的备忘报告你的评估结果并作出判断是否推荐此次灯光秀。
你提交的报告应包含：
你提交的内容总长度不超过33页，其中应该包括：
一页总结页
给市长的两页备忘
解决方案不超过30页，连同总结和备忘一共33页。
注意事项：参考列表和附录须附在文末，不计入33页限制。
项目10 常用的标准库和第三方库-习题答案
习题10一、填空题1.Python安装第三方库常用的是 pip 工具。
2.使用pip工具查看当前已安装的Python第三方库的完整命令是 pip list 。
3.random库中设置随机数种子的函数是 seed（）。
4.time库中使用函数是 sleep（）推迟调用线程的运行。
5.Numpy中的N维数组对象ndarray 不仅能方便地存取数组，而且拥有丰富的数组计算函数。
6.Matplotlib库具有丰富的绘图功能，是数据可视化的好帮手。
7. jieba 库是优秀的中文分词第三方库，能够将中文文本通过分词获得单个的词语。
8. wordcloud 库是非常优秀的词云可视化第三方库，词云是指对文本中出现频率较高的关键词汇通过彩色图形渲染，从而在视觉上予以突出。
9.PIL的全称是Python Image Library ，简称Python图像库,主要用于图像处理，在计算机视觉领域的研究中使用较多。
二、选择题1.表示海龟前进的方法是（C）A.turtle.penup()B.turtle.pendown()C.turtle.forword(d)D.turtle.backward(d)2.下列选项中，修改turtle画笔颜色的函数是（A）A.pencolor()B.speed()C.pensize()D.seth()3.设置下列( B )属性，可以使wordcloud支持中文。
A.font_stepB.font_pathC.modeD.font4.WordCloud类中能根据文本生成词云的方法是( D )。
A.fit_words()B.process_text()C.generate_from_frequencies()D.generate()三、编程题1、编写程序绘制平行四边形，并填充颜色为黄色，效果如图所示。
import turtleturtle.fillcolor("yellow")turtle.begin_fill()turtle.forward(100)turtle.left(60)turtle.forward(100)turtle.left(120)turtle.forward(100)turtle.left(60)turtle.forward(100)turtle.end_fill()turtle.done()2、利用Numpy模块和matplotlib.pyplot工具包编写程序绘制y=sin(2πx)及y=cos(2πx)的函数曲线图，效果如图所示。
HiMCM_B题初步思路及问题
Problem BProblem: City Crime and SafetyWhat can we make of the massive amount of crime statistics collected in major cities? Beyond just reporting numbers, how can we use these data to determine the safeness of a city?Assume that you and your modeling team live in My City, a large international hub of commerce, technology, finance and travel, with a current population of 2.8 million people impacted by a metropolitan area of an additional approximately 6 million people.The data set provided (My_City_Crime_Data.xlsx) shows two weeks from police reports in My City and includes crimes listed by case number, date of occurrence, primary and secondary crime descriptions, crime location, whether an arrest was made, whether or not this was domestic crime, and the beat number of the police route.Part I: Using mathematical modeling, analyze the data. Create a safety rating for My City. Use your safety rating to specify a measure of how safe My City is.Part II: In addition to the HiMCM contest format, prepare a 1-2 page non-technical report for the Mayor of My City to describe your findings.问题B问题：城市犯罪与安全我们怎么理解大量的在大城市里的犯罪统计数据？除了仅仅报告数字，我们如何利用这些数据来确定城市的安全性？假设你和你的建模团队住在MYCity，一个大型的国际商业，技术，金融和旅游中心，目前有280万人，且受都市群的约另外600万人的影响。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
2015 HiMCMProblem A: Preventing Road RageHave you ever been traveling down a highway when you see one or both of these signs?In some cases, the road simply loses a lane as it enters a less traveled section. In other cases, the road narrows because of construction or roadwork. When this happens on a fairly busy highway, it may be a recipe for road rage*.Assume we’re on a busy t wo-lane road reducing to a one-lane road. Once a driver sees a sign indicating the right lane is going to end, he or she has a choice: drive in the right lane or drive in the left lane. Often, many drivers move into the left lane fairly quickly. This causes the left lane to become congested and slow down. Some drivers remain in the right lane and are able to maintain their speed (and pass the vehicles in the left lane which are now moving slower). When a lane closure is approaching, there will be signs indicating the distance of the lane closure as it nears (e.g. 1 mile, ½ mile, 500 ft.). As the end of the right lane approaches, some of the right lane drivers merge into the left lane, while others continue heading toward the lane closure. It’s at this point that the probability of road rage rapidly increases. Horns blast and gestures are made. Some cars will pull half way into the right lane in an attempt to prevent any cars from passing only to have another vehicle swerve around them and continue toward the lane closure point. At times, a vehicle will stay in the right lane directly next to, and driving at the same speed as, a chosen vehicle in the left lane in an effort to make the travel toward the lane closure “fair.”Some states have done research on this phenomenon, but there is no consensus around best practices. Your team is tasked to provide a fresh analysis of this issue or add value to any existing analysis. Teams should not simply mimic previous research.Part I: Analyze the various driver actions and their implications in lane closure situations on a major highway.Part II: From your analysis in Part I, address and support fair and efficient driver actions:a) In the case of two lanes merging to one, what is the “fairest” way for drivers tobehave as they approach a lane closure? What is the most “efficient” way fordrivers to behave as they approach a lane closure? Is there a difference between“fair” and “efficient?”b) Use your analysis to address and support what drivers should do if a three-lanehighway is reducing to two lanes? How should driver behavior change, if at all, ifthe three-lane highway is reducing to one lane?c) Discuss and support any differences in fair and efficient driver behavior forlane closures on a secondary road with a 35 mile per hour speed limit versus ahighway with a 65 mile per hour speed limit.Part III: Policies and Practicesa) From your analysis in Parts I and II, develop guidelines for inclusion in theDepartment of Motor Vehicles driver education materials and signage used by theDepartment of Highway Safety to encourage appropriate driver actions whenapproaching a lane closure.b) In addition to the HiMCM contest format, write a one-page cover letter insupport of your guidelines and signage to the Director of the Department ofTransportation, urging adoption of your recommendations.*Road rage is aggressive or angry behavior by a driver of an automobile or other road vehicle. Such behavior might include rude gestures, verbal insults, deliberately driving in an unsafe or threatening manner, or making threats. Road rage can lead to altercations, assaults, and collisions that result in injuries and even deaths. It can be thought of as an extreme case of aggressive driving. (/wiki/Road_rage)2015 HiMCMProblem B: City Crime and SafetyWhat can we make of the massive amount of crime statistics collected in major cities? Beyond just reporting numbers, how can we use these data to determine the safeness of a city?Assume that you and your modeling team live in My City, a large international hub of commerce, technology, finance and travel, with a current population of 2.8 million people impacted by a metropolitan area of an additional approximately 6 million people.The data set provided (crime descriptions, crime location, whether an arrest was made, whether or not this was domestic crime, and the beat number of the police route.Part I:Using mathematical modeling, analyze the data. Create a safety rating for My City. Use your safety rating to specify a measure of how safe My City is.Part II: In addition to the HiMCM contest format, prepare a 1-2 page non-technical report for the Mayor of My City to describe your findings.。