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### Hungarian Problem Book IV Translated and edited by Robert Barrington Leigh and Andy Liu

Catalog Code: HP4
Print Edition ISBN: 978-0-88385-831-8
Electronic Edition ISBN: 978-0-88385-913-0
115 pp., Paperbound, 2011
List Price: \$40.95
Member Price: \$33.95
Series: Problem Books

This book, which is a continuation of Hungarian Problem Book III, contains the problems from the Kürschák Mathematics Competition in the period 1947-1963. In all, 48 problems are presented.

Hungarian Problem Book IV is intended for beginners, who are encouraged to work the problems in each section and then to compare their results against the solutions presented in the book. They will find ample material in each section to help them improve their problem-solving techniques.

The volume’s problems are classified under 12 subject headings: combinatorics, graph theory, number theory, divisibility, sums and differences, algebra, geometry, tangent lines and circles, geometric inequalities, combinatorial geometry, trigonometry, and solid geometry.
Solutions to the problems are presented along with background material. Furthermore, the last section, titled “Looking Back,” provides additional insights into the problems.

• Foreword by George Berzsenyi
• Preface
• List of Winners

1. Kürschák Mathematics Competition Problems: 1947; 1948; 1949; 1950; 1951; 1952; 1953; 1954; 1955; 1957; 1958; 1959; 1960; 1961; 1962; 1963.

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. 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
•

### Excerpt

Problem 1950.1 (p. 23):

On a certain day, a number of readers visited a library. Each went only once. Among any three readers, two of them met at the library on that day. Prove that there were two particular instants such that each reader was in the library at one of the two instants.

First Solution. Consider the moment a when the first reader A leaves and the moment b when the last reader B arrives. We may assume that
a < b as otherwise every two readers meet. Suppose a reader C is not present at either moment. If C arrives before a, C must be present at a since C leaves after A. Hence C arrives after a. If C leaves after b, C must be present at b since C arrives before B. Hence, C leaves before b. It follows that among A, B, and C, no two have met. This is a contradiction.

Second Solution. Suppose the librarian makes an announcement twice, trying to catch all readers, Clearly, the first moment a should be when the first reader A leaves. If every reader has heard the first announcement, there is nothing else to prove. Otherwise, the second announcement would be made at a later moment b when the first reader B who has not heard the first announcement leaves. Suppose some reader C has not heard either announcement. Then C must have arrived after b, so among A. B, and C, no two have met. This is a contradiction.