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Hungarian Problem Book IV

Robert Barrington Leigh and Andy Liu, editors
Mathematical Association of America
Publication Date: 
Number of Pages: 
MAA Problem Books
Problem Book
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Ana Momidic-Reyna
, on

Anticipated to widen the originality of elementary mathematics problems and deepen the creativity and diversity of their solutions, Hungarian Problem Book IV proves to be a valuable tool for students interested in preparing for mathematics competitions and for all those involved in organizing them. The book is a precious collection of problems from the Kürschák Mathematics Competition, which is the oldest high school mathematics competition in the world.

Robert Barrington Leigh and Andy Liu have worked diligently in the translation of the original forty-eight problems from the Hungarian Kürschák Competition of 1947 to 1963, editing and organizing them by subject: combinatorics, graph theory, number theory, divisibility, sums and differences, algebra, geometry, tangent lines and circles, geometric inequalities, combinatorial geometry, trigonometry and solid geometry. The experienced reader will find some new and intriguing problems here.

Chapter 2 provides theorems (with proofs) that enable the reader to understand the problems and their solutions. In the following chapter, various alternate solutions are presented, including several new solutions. Among the most unique and ingenious solutions is the second alternate solution given to problem 1949.3, which relies on a model that appears to be unrelated to the problem. Only a few problems, such as 1958.2 and 1948.3, have only one solution; for most two to three solutions are given. There are even some problems (1949.2, 1951.1, 1953.3, 1963.2 etc.) that get five or six solutions. This diversity of solutions enhances the process of teaching and learning creative mathematical problem solving.

Hungarian Problem Book IV is of course a sequel to Hungarian Problem Book III. The latter discusses Pólya’s four-step method for problem-solving, focusing mostly on the first three steps (understanding the problem, making a plan, and carrying out the plan). The final chapter of Hungarian Problem Book IV, "Looking Back", illustrates the usage of the fourth step in Pólya’s problem-solving process, which is looking back and eliciting further insights into the problems. An example is the discussion of problems in number theory. The authors begin by proving that an integer can be expressed as the sum of two squares if and only if twice that number can be so expressed. Then he deviates from this problem to another one, as he tries to determine all positive integers m such that (m–1)! is divisible by m. He draws the solution from Wilson’s Theorem, and proves it using geometric intuition. Next, he digresses once again to consider the Fermat’s Little Theorem (which is a result very close to Wilson’s theorem) and proceeds to prove it. Finally, he explores Waring’s Problem and looks back at yet another related problem which involves the Fermat Numbers. Thus this discussion exhibits an astonishing interplay between results of different problems.

The problems and their solutions draw on numerous famous theorems and concepts; to name a few: Ramsey’s Theorem, Hamiltonian cycles, Farey fractions, Chebyshev’s Inequality, Vieta’s Formulae, Cantor’s Diagonalization Method, Hall’s Theorem, Euler’s formula, etc., all of which are introduced and explained.

Hungarian Problem Book IV enriches its readers’ problem-solving technique and challenges their creative thinking.

A native of Macedonia, Ana Momidic-Reyna has an M.S. in Mathematics and has also worked for the high energy physicists at Fermilab. While waiting for the opportunity to work on her Ph.D. in mathematics, she keeps up with the field by reading as many mathematics books as she can.

Foreword by George Berzsenyi
List of Winners
1. Kürschák Mathematics Competition
2. Background
2.1 Theorems in Combinatorics
2.2 Additional Theorems in Combinatorics
2.3 Theorems in Number Theory
2.4 Theorems in Algebra
2.5 Additional Theorems in Algebra
2.6 Theorems in Geometry
3. Solutions to Problems
3.1 Problem Set: Combinatorics
3.2 Problem Set: Graph Theory
3.3 Problem Set: Number Theory
3.4 Problem Set: Divisibility
3.5 Problem Set: Sums and Differences
3.6 Problem Set: Algebra
3.7 Problem Set: Geometry
3.8 Problem Set: Tangent Lines and Circles
3.9 Problem Set: Geometric Inequalities
3.10 Problem Set: Combinatorial Geometry
3.11 Problem Set: trigonometry
3.12 Problem Set: Solid Geometry
4. Looking Back
4.1 Discussion on Combinatorics
4.2 Discussion on Number Theory
4.3 Discussion on Algebra
4.4 Discussion on Geometry