Christopher Bradley's fascinating book, Challenges in Geometry: for Mathematical Olympians Past and Present, would make a wonderful addition to the personal library of any coaches of mathematical competitions, as well as anyone who has an interest in the intersection of geometry and number theory.
In this volume, Bradley explores the classes of triangles, circles, and other geometrical objects that are constrained to have various integer or rational properties such as side length, area, radius, etc. His typical approach is to define an interesting set of constraints, for example integer sided triangles with integer area and inscribed circle with integer radius, and then produce a parameterized system of variables that generates all (or some) of the solutions. As he indicates in his preface, these problems are not ones likely to be found in competition, but expose patterns of thinking and model techniques that competition questions commonly require.
Although not really set up as a textbook, the book does offer a number of exercises at the end of each section, with solutions at the back. I will certainly use it as a resource for problems for my mathematical problem solving course, making excerpts available to my students as appropriate.
While I enjoyed reading the book, pausing frequently to work out problems or proofs, I did find it to be fairly terse at times, requiring more effort than expected to connect the dots. When used with undergraduates (or even good high school students) it will most likely oblige the instructor or coach to provide a substantial amount of background and supplementation. As a book clearly targeted to this audience, I would have also liked more in the way of motivation and problem solving strategies, where instead the author presents solutions completely worked out with little hint as to how the solution was derived.
In any case, Challenges in Geometry offers a great treasure of interesting problems, potential avenues of exploration and research for students, and new insights into rational geometry.
David J. Stucki teaches computer science and mathematics at Otterbein College, in Westerville, Ohio. His most recent interests are in the history and philosophy of mathematics, computer science education, and algorithmic number theory, although he also maintains an interest in artificial intelligence, theory of programming languages, and foundations/theory of computation. He has participated in Otterbein's Mathematical Problem Solving seminar and has helped to coach the Otterbein teams participating in the annual ECC Undergraduate Mathematics Competition.