In Hungarian Problem Book III, we covered a large number of theorems in basic mathematics. Not much else is needed to tackle the problems in the current volume. We list them again without further discussion, and add two sections on theorems not covered in the earlier volume.

Theorems in Combinatorics

Principle of Mathematical InductionIf S is a set of positive integers such that 1 ∈ S, and n + 1 ∈ S whenever n ∈ S, then S is the set of all positive integers.

Well Ordering PrincipleAny non-empty set of positive integers has a minimum.

Extremal Value PrincipleEvery non-empty finite set of real numbers has a maximum and a minimum.

Mean Value PrincipleIn every non-empty finite set of real numbers, there is at least one which is not less than the arithmetic mean of the set, and at least one not greater.

Pigeonhole PrincipleSeveral pigeons are stuffed into several holes. If there are more pigeons than holes, then at least one hole contains at least two pigeons. If there are more holes than pigeons, then there is at least one empty hole.

Finite Union PrincipleThe union of finitely many finite sets is also finite.

Parity PrincipleThe sum of two even integers is even, the sum of two odd integers is also even, while the sum of an odd and an even integer is odd. Moreover, an odd integer can never be equal to an even integer.

Multiplication Principle |A × B| = |A| · |B|.

Problem Set: Combinatorics

Problem 1963.1

Let p and q be integers greater than 1. There are pq chairs, arranged in p rows and q columns. Each chair is occupied by a student of different height. A teacher chooses the shortest student in each row; among these the tallest one is of height a. The teacher then chooses the tallest student in each column; among these the shortest one is of height b. Determine which of a < b, a = b and a > b are possible by an appropriate rearrangement of the seating of the students.

First Solution Let A be the tallest among the shortest student in each row, and B be the shortest among the tallest student in each column. Let A be in row i and B be in column j, and let C be the student of height c who is in row i and column j. Note that C may coincide with A or B. We have a ≤ c ≤ b, so that a > b is impossible. Put the tallest student in each column in the last row. Then the shortest student in this row qualifies as both A and B, yielding a = b. Now let her change seats with any other student in the same column. This move does not disqualify her as B, and she is still taller than the shortest student in each row except possibly her new row. We can certainly put a shorter student there, and then a < b.

The present book is a continuation of Hungarian Problem Book III, #42 in the Anneli Lax New Mathematical Library Series. It is the translation of the second half of Volume Two of the original Hungarian work. It covers the years from 1947 to 1963, except that the competition was not held in 1956 because of political events, as was the case from 1944 to 1946. After World War II, the competition was renamed after the mathematician József Kürschák instead of after the physicist Loránd Eötvös.

The underlying philosophy of this book is the same as its predecessor. We assume that the reader is familiar with Book III, and will not duplicate the discussions conducted there. The two books should be used in conjunction.

The present book consists of four chapters. In Chapter 1, the contest problems are given in chronological order. There are forty-eight problems altogether. They are classified by subject into twelve sets. Within each set, the four problems are listed in ascending order of estimated difficulty. A Problem Index facilitates the location of solutions to individual problems.

In Chapter 2, the Theorems in Hungary Problem Book III are restated, and additional theorems are provided with proofs. In Chapter 3, the solutions to the problems are given set by set.

In Book III, we referred to Pólya's four-step method in problem-solving, focusing primarily on the first three steps, namely, understanding the problem, making a plan, and carrying out the plan. In this book, we will focus on the fourth step, looking back. This is carried out in Chapter 4.

The high school helpers I engaged when I worked on Book III have since moved on in various directions.

In this chapter, we look back at some of the problems and their solutions, in both this volume and in Hungarian Problem Book III. We look at various extensions and generalizations as well as related results. While this material is not needed for solving the problems at hand, they may provide valuable insight in solving other problems.

Discussion on Combinatorics

Most of our discussion in this section is on graph theory. We begin by looking back at the following problem which has become a classic.

Problem 1947.2

Prove that in any group of six people, either there are three people who know one another or three people who do not know one another. Assume that “knowing” is a symmetric relation.

This problem can be rephrased in graph theoretic terms in at least two ways other than that used in its solution. In the first version, which has a symmetric form, we have a graph on six vertices. We wish to conclude that either the graph or its complement contains a complete subgraph on 3 vertices.

This can be generalized as follows. What is the minimum number m(k) such that for any graph with m(k) vertices, either the graph or its complement contains a complete subgraph on k vertices?

Clearly, m(1) = 1 and m(2) = 2. The result of Problem 1947.2 shows that m(3) ≤ 6. In fact, m(3) ≤ 6 since neither a pentagon nor its complement contains a triangle.