1995 MCM Problem A
The Single Helix

## Problem:

The problem proposed here consists of assisting a small biotechnological company in designing, proving, programming, and testing a mathematical algorithm to locate "in real time" all the intersections of a helix and a plane in general positions in space.

Similar programs for Computer Aided Geometric Design (CAGD) enable engineers to view a plane section of the object that they design, for example, an aircraft jet engine, an automobile suspension, or a medical device.  Moreover, engineers may also display on the plane section such quantities as air flow, stress, or temperature, coded by colors or level curves.  Furthermore, engineers may rapidly sweep such plane sections through the entire object, to gain a three-dimensional visualization of the object and its reactions to motion, forces, or heat.  To achieve such results, the computer programs must locate all the intersections of the viewed plane and every part of the designed object with sufficient speed and accuracy.  General "equation solvers" may in principle compute such intersections, but for specific problems specific methods may prove faster and more accurate than general methods.  In particular, general software for Computer Aided Geometric Design may prove too slow to complete computations in real time, or too large to fit in the finished medical devices being developed by the company.  The considerations just explained have led the company to the following problem.

Problem:

Design, justify, program, and test a method to compute all the intersections of a plane and a helix, both in general positions (at any locations and with any orientations) in space.

A segment of the helix may represent, for example, a helicoidal suspension spring or a piece of tubing in a chemical or medical apparatus.

The need for some theoretical justification of the proposed algorithm arises from the necessity of verifying the solution from several points of view.  For instance, through mathematical proofs of parts of the algorithm, and through tests of the final program with known examples.  Such documentation and tests will be required by government agencies for medical use.

1995 MCM Problem B
Aluacha Balaclava College

## Problem:

Background :

Aluacha Balaclava College, specializing in undergraduate education, has just hired a new Provost.  Problems with faculty compensation at the college caused the forced resignation of the former Provost so the new Provost needs to make the institution of a fair and reasonable compensation system her first priority.  As a first step in this process, she has hired your team as consultants to design a compensation system which reflects the following circumstances and principles:

Circumstances:

There are four possible faculty ranks:  Instructor, Assistant Professor, Associate Professor and Professor.  Faculty with Ph.D. degrees are hired at the rank of Assistant Professor.  Faculty who are working on a Ph.D. are hired at the rank of Instructor and promoted automatically to Assistant Professor upon completion of their degrees.  Faculty may apply for promotion from Associate Professor to Professor after serving at the rank of Associate for seven or more years.  The promotion decisions are made by the Provost with recommendations from a faculty committee and are not your concern.

Faculty  salaries are for the 10 month period September through June.  Raises are always effective beginning in September.  The total amount of money available for raises varies from year to year and is generally not known until March for the following year.

The starting salary this year for an Instructor with no prior teaching experience was \$27,000 and for an Assistant Professor was \$32,000.  Faculty can receive credit, upon hire, for as much as seven years of teaching experience at other institutions.

Principles:

• All faculty should get a raise any year that money is available.

• Faculty should get a substantial benefit from promotion.  If one is promoted in the minimum possible time, the benefit should be roughly equal to seven years of normal (non-promotion) raises.

• Faculty who get promoted on time (after 7 or 8 years in rank) and have careers of 25 or more years should make roughly twice as much at retirement as a new Ph.D. starting off.

• Faculty in the same rank with more experience should be paid more than others with less experience.

• But the effect of an additional year of experience should diminish over time.  In other words, if two faculty stay in the same rank, their salaries should tend to get closer over time.

The Project:

First design a new pay system without cost-of-living increases.  Then incorporate cost-of-living increases.  The final piece of this project is to design a transition process for existing faculty which will move all salaries towards your system without cutting anyone's salary.  The existing faculty salaries, ranks and year of service, are in Table 1.  Discuss any refinements you think would improve your system.

The Provost has asked for a detailed pay system plan that she can use for implementation as well as a short executive summary in clear language outlining the model, its assumptions, its strengths and weaknesses and the expected results which she can present to the Board and the faculty.