1997 MCM Problem A
The Velociraptor Problem

Problem:

The velociraptor, Velociraptor mongoliensis, was a predator 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 even 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 biologist 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 that burst of speed, the animal needed to stop and recover from a buildup of lactate acid in its muscles.

Suppose that velociraptor preyed on Thescelosaurus neglectus, a bipedal herbivore approximately the same size as the velociraptor.  A biomechanical analysis of fossilized thescelosaurus indicates that it could run at a speed of about 50 km/hr almost indefinitely.

Part 1:  Assuming the velociraptor is a solitary hunter, design a mathematical model which describes a hunting strategy for a single velociraptor stalking and chasing a single prey as well as the evasive strategy of the prey.  Assume that the thescelosaurus can always detect velociraptor when it gets within 15 meters, but may detect this predator at even greater ranges (up to 50 meters) depending upon the nature of 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 the more realistic situtation that the velociraptor hunted in pairs, design a new model which describes a hunting strategy for two velociraptors stalking and chasing a single prey as well as the evasive strategy of the prey.  Use the same other assumptions and limitations in Part 1.

1997 MCM Problem B
Mix Well For Fruitful Discussions

Problem:

Small group meetings are gaining popularity for the discussion of important issues, particularly long range planning. It is believed that large groups stymie productive discussion and a dominant personality will usually control and direct the discussion.  Thus in Corporation Board meetings the Board will meet in small groups to discuss issues before meeting as a whole. These smaller groups still run the 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 the groups.

A meeting of An Tostal Corporation will be attended by 29 Board Members of which 9 are in-house members (ie. employees of the corporation).  The meeting is to be an all-day affair with 3 sessions scheduled for the morning and 4 for the afternoon.  The sessions will each be 45 minutes, beginning on the hour from 9:00 AM to 4:00 PM with lunch scheduled at noon Each morning session will consist of six discussion groups with each discussion group led by one of the Corporation's six senior officers. None of these officers are Board members.  Thus each senior officer will lead three different discussion groups.  The senior officers will not be involved in the afternoon sessions and each of these sessions will consist of only four different discussion groups.

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