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--- 1998_年美国大学生数学建模竞赛_mcm_试题 [2008/02/08 16:10]
amao 创建
+++ 1998_年美国大学生数学建模竞赛_mcm_试题 [2014/12/30 22:36] (当前版本)
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 ====== 1998 年美国大学生数学建模竞赛 MCM 试题 ======
 ===== 1998 MCM A: MRI Scanners =====
 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 low water density in white (lipid-right white matter consisting of myelinated axons). Such MRI scanners also include facilities to pictures on a screen any horizontal or vertical slide 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.
+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
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 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.
 ===== 1998 MCM B: Grade Inflation =====
+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 is 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.

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