The problems from the 2015 International Mathematical Olympiad are in this issue, with a new twist. The solutions are written by the USA team members. They all won medals at the IMO, so their insights might be worth studying! We also have our report from the USAMO/USAMJO, and articles about sorting hats, mind switchers, and unpaintable paint cans. —Walter Stromquist, editor
JOURNAL SUBSCRIBERS AND MAA MEMBERS:
To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.
Vol. 87, No. 4, pp. 241-320
Letter from Editor
The Sorting Hat Goes to College
By Andrew Beveridge and Stan Wagon
We describe the solution to a combinatorial optimization problem that arises in higher education: the assignment of first-year students to introductory seminars. We trace the evolution of our implementation at Macalester College, a small liberal arts school. We first describe how the classic Assignment Problem for bipartite graphs can be used. Then we show how Integer Linear Programming leads to more flexibility, as it allows the optimization to be fine-tuned to handle more challenging data sets.
Mind Switches in Futurama and Stargate
By Ron Evans and Lihua Huang
We generalize two mind-switching problems that arise in connection with the popular sci-fi television series Futurama and Stargate SG-1. Optimal solutions to these problems are found by answering the following question about a permutation δ expressed as a product of nontrivial disjoint cycles. “When writing δ as a product of distinct transpositions, none occurring as factors in the disjoint cycle representation of δ, what is the smallest number of transpositions that can be used?”
By Vincent Coll and Michael Harrison
We generalize hypersurfaces of revolution by allowing the profile curve to be dependent on more than one variable. We call these more general objects spherical arrays and we use them to introduce the reader to volume calculations, which are normally reserved for a more advanced course in differential geometry. The rich symmetry of the spherical arrays allow us to build objects with properties reminiscent of those for which Gabriel’s Horn became a cause célèbre.
Proof Without Words: An Electrical Proof of the AM-HM Inequality
By Alfred Witkowski
The AM-HM inequality arises from comparing the resistances in two circuits.
Proving the Reflective Property of an Ellipse
By Stephan Berendonk
The tangent to an ellipse at a point P can be constructed as the exterior angle bisector of the angle that the point P makes with the two foci of the ellipse. Typically this fact is proved by showing that the exterior angle bisector cannot meet the ellipse in a second point, and therefore must be the tangent. This note gives an alternative proof that is more in line with the notion of tangent as a limit of secants.
Viviani à la Kawasaki: Take Two
By Burkard Polster
We consider a natural twist to a beautiful one-glance proof of Viviani’s theorem, and its implications for general triangles.
Bisections and Reflections
By Jorgen Berglund and Ron Taylor
We consider the process of reflecting a point on the side of a polygon across successive angle bisectors. As we iterate this process, we find an interesting and sometimes periodic pattern. In particular, we show that for any polygon with an odd number of sides, a point on any side is returned to its original position by a finite number of reflections that is either one or two times the number of sides. Additionally, for a polygon with an even number of sides, a point is guaranteed to return to its original position as soon as possible or not at all.
Proof Without Words: Ptolemy’s Inequality
By Claudi Alsina and Roger B. Nelsen
Ptolemy: In a convex quadrilateral with sides of length a, b, c, d (in that order) and diagonals of length p and q, we have pq ≤ ac + bd.
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World
NEWS AND LETTERS
43rd USA Mathematical Olympiad, 5th USA Junior Mathematical Olympiad
By Jacek Fabrykowski and Steven R. Dunbar
55th International Mathematical Olympiad
By Po-Shen Loh
2014 Carl B. Allendoerfer Awards