The Kürschák Mathematics Competition is the oldest high school mathematics competition in the world, dating back to 1894. This book is a continuation of Hungarian Problem Book III and takes the contest through 1963. Forty-eight problems in all are presented in this volume. Problems are classified under combinatorics, graph theory, number theory, divisibility, sums and differences, algebra, geometry, tangent lines and circles, geometric inequalities, combinatorial geometry, trigonometry and solid geometry. Multiple solutions to the problems are presented along with background material. There is a substantial section entitled 'Looking Back', which provides additional insights into the problems. Hungarian Problem Book IV is intended for beginners, although the experienced student will find much here. Beginners 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.