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Challenges in Geometry: for Mathematical Olympians Past and Present

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Over the past couple of years, many books have appeared that concern mathematical problem solving, in particular, in the vein of mathematical olympiads. Challenges in Geometry is one of them, and it is in a class by itself. I would qualify it as 'a bit excentric'. Let me remind you that olympiads are for (talented) high school pupils.

The title mentions geometry, but the book almost exclusively concerns combinatorial and number theoretic problems inspired by geometric configurations. For example: characterize all integer-sided triangles with an angle of 60 degrees. Using the cosine rule, this boils down to solving the diophantine equation c^2 = a^2 - ab + b^2.

The formula density is truly amazing, in many places exceeding that of the accompanying prose. Only 10 pages are without formulae, and these include the preface (2 p.), references (2 p.), and index (3 p.). Fortunately, there are also 63 figures for the visually inclined. There is even some attention for the historic context of some problems.

The reader does need to have a strong background in Euclidean geometry. Theorems by Apollonius, Ceva, de Moivre, Menelaus, and Ptolemy are applied without further explanation. But also modular arithmetic, Gaussian integers, unimodular matrices, determinants, partial derivatives, complex numbers, 2-variable Taylor series, and more pop up. This cannot be considered typical high school knowledge nowadays.

A few of the problems treated by Bradley are more widely known, such as counting the number of lattice points in a lattice polygon (Pick's Theorem), and characterizing Euler bricks (rectangular blocks whose edges and face diagonals all have integer lengths) and the related -but as yet undiscovered- perfect cuboid (which in addition has an integer main diagonal).

The proof style is quite terse. This makes for quick reading and helps maintain a good overview. But the proofs contain 'rabbits' pulled from the hat, without any guidance, thereby hindering the understanding.

The book offers many exercises, all with solutions (15 p.). The appendix treats areal co-ordinates, also known as Barycentric co-ordinates; a useful topic hardly treated in textbooks. This makes the book a good resource for olympiad coaches, but prospective olympians might well be scared off.

The title of the book is a misnomer. This book hardly deals with geometry, it is rather a number theory book. If you are preparing for the International Mathematics Olympiad (IMO) and hope to learn geometry, this is not the book to study it from. Anything but this book. This is a number theroy book I can say. I finished the first two chapters and now I gave up as I want to solve geometry problems.