1996　美国大学生数模竞赛题

Problem　A

The world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large moving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, developa method for detecting the presence of a moving submarine, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed freqency and amplitude.

Problem　B

When determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge. Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest constains both the number of judges that can be obtained and amount of time that they can judge. For eample if P=100, then J=8 is typical.

Ideally, each judge would read paper and rank-order them, but there are too many papers for this. Instead, there will be a number of screening rounds in which each judge will read some number of papers and give them scores. Then some selection scheme is used to reduce the number of papers under consideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orders could be rejected. Alternatively, if the judges do not rank-order, but instead give them numerical score (say, from 1 to 100), then all papers below some cut-off level could be rejected.

The new pool of papers is then passed back to the judges, and the process is repeated. A concern is then the total number of papers that judge reads must be substantially less than P. The process is stopped when there are only W papers left. There are the winners. Typically for P=100, W=3.

Your task is to determine a selection scheme, using a combination of
rank-ordering, numerical scoring, and other methods, by which the final W papers will include only papers from among the "best" 2W papers. (By "best",we assume that there is an absolute rank-ordering to which all judges would agree.) For example, the top three papers. Among all such methods, the one that required each judge to read the least number of papers is desired.

Note the possibility of systematic bias in a numerical scoring scheme. For example, for a specific collection of papers, one judge could average 70 points, while another could average 80 points. How would you scale your scheme to accommodate for changes in the contest parameters (P, J, and W)?

1997 年美国大学生数模竞赛题

Problem　A: The Velociraptor Problem

The Velociraptor, Velociraptor mongoliensis, was a predatory dinosaur that lived during the late Cretaceous period, approximately 75 million years ago. Paleontologists think that it was a very tenacious hunter, and may have hunted in pairs or larger packs. Unfortunately, there is no way to observe its hunting behavior in the wild as can be done with modern mammalian predators. A group of paleontologists has approached your team and asked for help in modeling the hunting behavior of the velociraptor.
They hope to compare your results with field data reported by biologists studying the behaviors of lions, tigers, and similar predatory animals.

The average adult velociraptor was 3 meters long with a hip height of 0.5 meters and an approximate mass of 45 Kg. It is estimated that the animal could run extremely fast, at speeds of 60 km/hr., for about 15 seconds. After the initial burst of speed, the animal needed to stop and recover from a buildup of lactic acid in its muscles.

Suppose that Velociraptor prey on Thescelosaurus neglectus, a herbivorous biped approximately the same size as the Velociraptor. A biomechanical analysis of a fossilized thescelosaurus indicates that if could run at a speed of about 50km.hr. for long periods of time.

Part　1
Assuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thecelosaurus can always detect the velociraptor when in comes within 15 meters, but may detect the predator at even greater ranges (up to 50 meters) depending upon the habitat and weather conditions.
Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 meters.

Part　2
Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions and limitations given in Part 1.

Problem B: Mix Well For Fruitful Discussions

Small group meetings for the discussion of important issues, particularly long-rang planning, are gaining popularity. It is believed that large groups discourage productive discussion and that a dominant personality will usually control and direct the discussion. Thus, in corporate board meetings the board will meet in small groups to discuss issues before meeting as a whole. These smaller groups still run risk of control by a dominant personality. In an attempt to reduce this danger it is common to schedule several sessions with a different mix of people in each group.

A meeting of an Tostal Corporation will be attended by 29 Board Members of which nine are in-horse members(i.e., corporate employees). The meeting is to be an all-day affair with three sessions scheduled for the morning and four for the afternoon. Each session will take 45 minutes, beginning on the hour from 9:00 A.M. to 4:00 P.M., with lunch scheduled at noon. Each morning session will consist of six discussion group with each discussion group led by one of the corporation's six senior officers. None of these of officers are board members. Thus each senior officer will lead three
different discussion groups. The sessions will consist of only four
discussion groups.

The president of the corporation wants a list of board-member assignments to discussion group for each of seven sessions. The assignments should achieve as much of a mix of members as much as possible. The ideal assignment would have each board member with each other board member in a discussion group the same number of times while minimizing common membership of groups for the different sessions.

The assignments should also satisfy the following criteria:
　　1.For the morning sessions, no board member should be in the same senior officer's discussion group twice.
　　2.No discussion group should contain a disproportionate number of in-house members.
　　Give a list of assignments for members 1-9 and 10-29 and officers 1-6. Indicate how well the criteria in the precious paragraphs are met. Since it is possible that some board members will cancel at the last minute or that some not scheduled will show up, an algorithm that the secretary could use to adjust the assignments with an user to make assignments for future meetings involving different levels of participation for each type of attendee.

1998 年美国大学生数模竞赛题

Problem A:
Introduction:
Industrial and medical diagnostic machines known as Magnetic Resonance Imagers (MRI) scan a three-dimensional object such as a brain, and deliver their results in the form of a three-dimensional array of pixels. Each pixel consists of one number indicating a color or a shade of gray that encodes a measure of water concentration in a
small region of the scanned object at the location of the pixel. For instance, 0 can picture high water concentration in black (ventricles, blood vessels), 128 can picture a medium water concentration in gray (brain nuclei and gray matter), and 255 can picture a low water density in white (lipid-rich white matter consisting of myelinated axons). Such MRI scanners also include facilities to picture on a screen any horizontal or vertical slice through the three-dimensional array (slices are parallel to any of the three Cartesian coordinate axes). Algorithms for picturing slices through oblique planes, however, are proprietary. Current algorithms are limited in terms of the angles and parameter options available; are implemented only on heavily used dedicated workstations; lack input capabilities for marking points in the picture before slicing; and tend to blur and "feather out" sharp boundaries between the original pixels.

A more faithful, flexible algorithm implemented on a personal computer would be useful (1) for planning minimally invasive treatments, (2) for calibrating the MRI machines, (3) for investigating structures oriented obliquely in space, such as postmortem tissue sections in animal research, (4) for enabling cross-sections at any angle through a brain atlas consisting of black-and-white line drawings. To design such an algorithm, one can access the values and locations of the pixels, but not the initial data gathered by the scanner.

Problem:
Design and test an algorithm that produces sections of three-dimensional arrays by planes in any orientation in space, preserving the original gray-scale values as closely as possible.

Data Sets:
The typical data set consists of a three-dimensional array A of numbers A(i,j,k) which indicates the density A(i,j,k) of the object at the location (x,y,z)_{ijk} . Typically, A(i,j,k) can range from 0 through 255. In most applications; the data set is quite large. Teams should design data sets to test and demonstrate their algorithms. The data sets should reflect conditions likely to be of diagnostic interest. Teams should also characterize data sets that limit the effectiveness of their algorithms.

Summary:

The algorithm must produce a picture of the slice of the three-dimensional array by a plane in space. The plane can have any orientation and any location in space. (The
plane can miss some or all data points.) The result of the algorithm should be a model of the density of the scanned object over the selected plane.

Problem B:
Background:
Some college administrators are concerned about the grading at A Better Class (ABC) college. On average, the faculty at ABC have been giving out high grades (the average grade now given out is an A-), and it is impossible to distinguish between the good and mediocre students. The terms of a very generous scholarship only allow
the top 10% of the students to be funded, so a class ranking is required.

The dean had the thought of comparing each student to the other students in each class, and using this information to build up a ranking. For example, if a student obtains an A in a class in which all students obtain an A, then this student is only "average" in this class. On the other hand, if a student obtains the only A in a class, then that student is clearly "above average". Combining information from several classes might allow students to be placed in deciles (top 10%, next 10%, etc.) across the college.

Problem:
Assuming that the grades given out are (A+, A, A-, B+, ... ) can the dean's idea be made to work? Assuming that the grades given out are only (A, B, C, ... ) can the dean's idea be made to work? Can any other schemes produce a desired ranking? A concern is that the grade in a single class could change many student's deciles. Is this possible?

Data Sets:
Teams should design data sets to test and demonstrate their algorithms. Teams should characterize data sets that limit the effectiveness of their algorithms.