Mathematical Contest in Modeling & Interdisciplinary Contest in Modeling for Undergraduates
The Mathematical Contest in Modeling (MCM) is a unique international contest for undergraduate students. The MCM is designed to stimulate and improve problem-solving and writing skills. Students participate as team members rather than as individuals, creating an environment for sharing knowledge and skills. Your institution may take part in the MCM effort by encouraging a member of your department to serve as a faculty advisor. Advisors help organize the teams, distribute contest materials, and return solution papers to COMAP.
MCM is a contest where teams of undergraduates use mathematical modeling to present their solutions to real world problems.
MCM will take place February 9 - 13, 2012..
The Interdisciplinary Contest in Modeling (ICM) is an international contest for undergraduate students. ICM is designed to develop and advance interdisciplinary problem-solving skills as well as competence in written communication. The interdisciplinary problem has changed its emphasis from strictly environmental issues to reflect a quantitative situation in mathematics; operations research; systems engineering; information, industrial or physical security; or resource and environment protection and management. Your institution may be part of the ICM effort by encouraging a member of your department to serve as a faculty advisor and by promoting the participation of faculty and students from associated departments. Advisors help organize the teams, distribute contest materials, and return solution papers to COMAP.
Course Math 142 on Mathematical Modeling: Lecture, three hours; discussion, one hour. Prerequisites: courses 32B, 33B. Introduction to fundamental principles and spirit of applied mathematics. Emphasis on manner in which mathematical models are constructed for physical problems. Illustrations from many fields of endeavor, such as the physical sciences, biology, economics, and traffic dynamics.
Fall 2006 Math 142
FACULTY ADVISORS: Braxton Osting (firstname.lastname@example.org),
Jinsun Sohn (email@example.com), and Luminita Vese (firstname.lastname@example.org).
Interested students: please contact one of the coaches and attend a meeting about the competition on Dec. 12 at 10am-12pm, room MS 6627.
Our graduate students
Pascal Getreuer, and
have participated and won several "outstanding paper" awards as undergraduate students at Washington University and Univ. of Boulder, Colorado. Below you can find some of their outstanding papers.
IMPORTANT LINKS AND RESOURCES:
* COMAP: The Consortium for Mathematics and its Applications
* COMAP Contests
A guide to the Mathematical Contest in Modeling by Pascal Getreuer (graduate student at UCLA) and colleagues
An official guide to the Mathematical Contest in Modeling, with sample problems and solutions.
SAMPLE PROBLEMS WITH SOLUTIONS OF OUTSTANDING WINNERS:
2005 MCM PROBLEM A: Flood planning
Lake Murray in central South Carolina is formed by a large earthen dam, which was completed in 1930 for power production. Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam.
Two particular questions:
Rawls Creek is a year-round stream that flows into the Saluda River a short distance downriver from the dam. How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend?
Could the flood be so massive downstream that water would reach up to the S.C. State Capitol Building, which is on a hill overlooking the Congaree River?
by Ryan Bressler, Braxton Osting, and Christina Polwarth from Washington University.
2004 MCM PROBLEM B: A Faster QuickPass System
"QuickPass" systems are increasingly appearing to reduce people's time waiting in line, whether it is at tollbooths, amusement parks, or elsewhere. Consider the design of a QuickPass system for an amusement park. The amusement park has experimented by offering QuickPasses for several popular rides as a test. The idea is that for certain popular rides you can go to a kiosk near that ride and insert your daily park entrance ticket, and out will come a slip that states that you can return to that ride at a specific time later. For example, you insert your daily park entrance ticket at 1:15 pm, and the QuickPass states that you can come back between 3:30 and 4:30 pm when you can use your slip to enter a second, and presumably much shorter, line that will get you to the ride faster. To prevent people from obtaining QuickPasses for several rides at once, the QuickPass machines allow you to have only one active QuickPass at a time.
You have been hired as one of several competing consultants to improve the operation of QuickPass. Customers have been complaining about some anomalies in the test system. For example, customers observed that in one instance QuickPasses were being offered for a return time as long as 4 hours later. A short time later on the same ride, the QuickPasses were given for times only an hour or so later. In some instances, the lines for people with Quickpasses are nearly as long and slow as the regular lines.
The problem then is to propose and test schemes for issuing QuickPasses in order to increase people's enjoyment of the amusement park. Part of the problem is to determine what criteria to use in evaluating alternative schemes. Include in your report a non-technical summary for amusement park executives who must choose between alternatives from competing consultants.
by Sasha Aravkin, Tracy Lovejoy, and Casey Schneider-Mizell from Washington University.
by Moorea Brega Alejandro Cantarero Corry Lee, from University of Boulder, Colorado.
2003 MCM PROBLEM A: The Stunt Person
An exciting action scene in a movie is going to be filmed, and you are the stunt coordinator! A stunt person on a motorcycle will jump over an elephant and land in a pile of cardboard boxes to cushion their fall. You need to protect the stunt person, and also use relatively few cardboard boxes (lower cost, not seen by camera, etc.).
Your job is to:
determine what size boxes to use
determine how many boxes to use
determine how the boxes will be stacked
determine if any modifications to the boxes would help
generalize to different combined weights (stunt person & motorcycle) and different jump heights
Note that, in "Tomorrow Never Dies", the James Bond character on a motorcycle jumps over a helicopter.
by Ernie Esser, Jeff Giansiracusa, and Sheng-Fong Pai from Washington University.
2003 MCM PROBLEM B: Gamma Knife Treatment Planning
Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are the gamma knife unit, heavy charged particle beams, and external high-energy photon beams from linear accelerators.
The gamma knife unit delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy helmet. All 201 beams simultaneously intersect at the isocenter, resulting in a spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed a “shot.” Shots can be represented as different spheres. Four interchangeable outer collimator helmets with beam channel diameters of 4, 8, 14, and 18 mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.
The goal of radiosurgery is to deplete tumor cells while preserving normal structures. Since there are physical limitations and biological uncertainties involved in this therapy process, a treatment plan needs to account for all those limitations and uncertainties. In general, an optimal treatment plan is designed to meet the following requirements.
1. Minimize the dose gradient across the target volume.
2. Match specified isodose contours to the target volumes.
3. Match specified dose-volume constraints of the target and critical organ.
4. Minimize the integral dose to the entire volume of normal tissues or organs.
5. Constrain dose to specified normal tissue points below tolerance doses.
6. Minimize the maximum dose to critical volumes.
In gamma unit treatment planning, we have the following constraints:
1. Prohibit shots from protruding outside the target.
2. Prohibit shots from overlapping (to avoid hot spots).
3. Cover the target volume with effective dosage as much as possible. But at least 90% of the target volume must be covered by shots.
4. Use as few shots as possible.
Your tasks are to formulate the optimal treatment planning for a gamma knife unit as a sphere-packing problem, and propose an algorithm to find a solution. While designing your algorithm, you must keep in mind that your algorithm must be reasonably efficient.
by Mark Blunk, Sam Coskey, and Luke Winstrom
from Washington University.
2002 MCM PROBLEM A: Wind and Waterspray
An ornamental fountain in a large open plaza surrounded by buildings squirts water high into the air. On gusty days, the wind blows spray from the fountain onto passersby. The water-flow from the fountain is controlled by a mechanism linked to an anemometer (which measures wind speed and direction) located on top of an adjacent building. The objective of this control is to provide passersby with an acceptable balance between an attractive spectacle and a soaking: The harder the wind blows, the lower the water volume and height to which the water is squirted, hence the less spray falls outside the pool area.
Your task is to devise an algorithm which uses data provided by the anemometer to adjust the water-flow from the fountain as the wind conditions change.
by Ryan Card, Ernie Esser, and Jeff Giansiracusa from Washington University.