Comap Competition

As an undergraduate student, I participated in the Consortium for Mathematics and Its Applications' Mathematical Contest in Modeling contest for three consecutive years. For each year below, I have listed the problem, my team's ranking, and our submitted paper. Competition Description

The contest is held each year, usually at the beginning of February. Teams of up to three undergraduate students all attending the same institution have four uninterrupted days, from Thursday evening to Monday evening, to research and present a solution in paper form to one of two open ended questions, which are revealed Thursday evening. Any source of information may be utilized, with the limitation that team members cannot discuss the problem or solution with any other person outside of their team. Each institution may have more than one team, and a faculty member serves as an adviser leading up to the contest and a liaison during the contest with COMAP.

All three years I competed with the same team: Sarah Kaiser, Nathan Youngblood, and myself. Dr. Nathan Gossett was our adviser, and we operated from the Electronics lab in the Physics department at Bethel. 2011 Competition

Team rank / number: Meritorious / 11202 (official results) Team paper: CQ: The Call Heard ’Round The Network (PDF; 1.1MB) Problem statement:

The VHF radio spectrum involves line-of-sight transmission and reception. This limitation can be overcome by "repeaters," which pick up weak signals, amplify them, and retransmit them on a different frequency. Thus, using a repeater, low-power users (such as mobile stations) can communicate with one another in situations where direct user-to-user contact would not be possible. However, repeaters can interfere with one another unless they are far enough apart or transmit on sufficiently separated frequencies.

In addition to geographical separation, the "continuous tone-coded squelch system" (CTCSS), sometimes nicknamed "private line" (PL), technology can be used to mitigate interference problems. This system associates to each repeater a separate subaudible tone that is transmitted by all users who wish to communicate through that repeater. The repeater responds only to received signals with its specific PL tone. With this system, two nearby repeaters can share the same frequency pair (for receive and transmit); so more repeaters (and hence more users) can be accommodated in a particular area.

For a circular flat area of radius 40 miles radius, determine the minimum number of repeaters necessary to accommodate 1,000 simultaneous users. Assume that the spectrum available is 145 to 148 MHz, the transmitter frequency in a repeater is either 600 kHz above or 600 kHz below the receiver frequency, and there are 54 different PL tones available.

How does your solution change if there are 10,000 users?

Discuss the case where there might be defects in line-of-sight propagation caused by mountainous areas.
2011 MCM Problem B 

2010 Competition

Team rank / number: Meritorious / 7820 (official results) Team paper: Who's on First (PDF; 0.7MB) Problem statement:

Explain the “sweet spot” on a baseball bat.

Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.

Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?

Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?
2010 MCM Problem A 

2009 Competition

Team rank / number: Meritorious / 5171 (official results) Team paper: Roundabouts: The Next Generation (PDF; 0.7MB) Problem statement:

Many cities and communities have traffic circles—from large ones with many lanes in the circle (such as at the Arc de Triomphe in Paris and the Victory Monument in Bangkok) to small ones with one or two lanes in the circle. Some of these traffic circles position a stop sign or a yield sign on every incoming road that gives priority to traffic already in the circle; some position a yield sign in the circle at each incoming road to give priority to incoming traffic; and some position a traffic light on each incoming road (with no right turn allowed on a red light). Other designs may also be possible.

The goal of this problem is to use a model to determine how best to control traffic flow in, around, and out of a circle. State clearly the objective(s) you use in your model for making the optimal choice as well as the factors that affect this choice. Include a Technical Summary of not more than two double-spaced pages that explains to a Traffic Engineer how to use your model to help choose the appropriate flow-control method for any specific traffic circle. That is, summarize the conditions under which each type of traffic-control method should be used. When traffic lights are recommended, explain a method for determining how many seconds each light should remain green (which may vary according to the time of day and other factors). Illustrate how your model works with specific examples.
2009 MCM Problem A