Five Hundred Mathematical Challenges

Spectrum

by Edward J. Barbeau, Murray S. Klamkin, William O.J. Moser

This book contains 500 problems that range over a wide spectrum of mathematics and of levels of difficulty. Some are simple mathematical puzzlers while others are serious problems at the Olympiad level. Students of all levels of interest and ability will be entertained by the book. For many problems, more than one solution is supplied so that students can compare the elegance and efficiency of different mathematical approaches. A special mathematical toolchest summarizes the results and techniques needed by competition-level students. Teachers will find the book useful, both for encouraging their students and for their own pleasure. Some of the problems can be used to provide a little spice in the regular curriculum by demonstrating the power of very basic techniques. The problems were first published as a series of problem booklets almost twenty years ago. They have stood the test of time and the demand for them has been steady. Their publication in book form is long overdue.

Genres: Mathematics

236 pages, Paperback

First published January 1, 1995

Reviews

[Behrooz Parhami · Rating 5 out of 5]

This book is a wonderful source of mathematical/logical puzzles and diverse challenges of various difficulty levels. Problem statements are generally succinct and the fact that full solutions are provided adds to the book's educational value. The problems appear on pp. 1-46 and solutions on pp. 47-210. Beginning on page 211, six tool chests are briefly reviewed: Combinatorics; Arithmetic; Algebra; Inequalities; Geometry & trigonometry; Analysis.

I include in my review five samples to show the nature and level of the problems. I have chosen only problems that do not include diagrams.

Problem 1: The length of the sides of a right triangle are three consecutive terms of an arithmetic progression. Prove that the lengths are in the ratio 3:4:5.

Problem 101: Prove that each of the numbers 10201, 10101, and 100011 is composite in any base.

Problem 238: Show that, for all real values of x (radians), cos(sin x) > sin(cos x).

Problem 327: Let three concentric circles be given such that the radius of the largest is less than the sum of the radii of the two smaller ones. Construct an equilateral triangle whose vertices lie one on each circle.

Problem 445: Prove that if the top 26 cards of an ordinary shuffled deck contain more red cards than there are black cards in the bottom 26, then there are in the deck at least three consecutive cards of the same color.

The following humorous poem (from p. 20), purportedly written by a student in the fly leaf of an algebra textbook, does not apply to this book!

If there should be another flood
Hither for refuge fly
Were the whole world to be submerged
This book would still be dry