## Hungarian Problem Book IV, Book 4The 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. |

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### Contents

Kürschák Mathematics Competition Problems | 1 |

Background | 9 |

Solutions to Problems | 21 |

Looking Back | 95 |

About the Editors | 131 |

### Common terms and phrases

Ä k Ä a1 C a2 a2 C C a2b2 C C anbn ABC and DEF ABCD American Mathematics Competitions assume basic lattice triangle bipartite graph circumcenter circumcircle collinear color Combinatorics common point complete subgraph congruent Consider convex ﬁgures convex hull convex quadrilateral countable cyclic quadrilateral deﬁne desired result diagonals digit Discussion on Algebra divisible divisor edge cover equal f f f Fermat numbers ﬁnite ﬁrst ﬁve follows fourth circle Geometry graph half-planes Hamiltonian cycle Hence hexagon Hungarian Problem Book hypothesis inﬁnite series International Mathematical Olympiads joined K¨ursch´ak Mathematical Competition lattice point inside lattice points Looking Back master median midpoint number of vertices parallel parallelogram Pigeonhole Principle plane point of intersection polynomials prime numbers Problem Set Prove Rearrangement Inequality segment side BC Solutions to Problems squares subset sufﬁcient symmetry tangent Theorem Let Titu Andreescu total number Zuming Feng