The Green Book of Mathematical Problems

THE

Green Book

OF

Mathematical

Problems

Kenneth Hardy and

Kenneth S. Williams

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 1985 by Kenneth Hardy and Kenneth S. Williams.

All rights reserved

Bibliographical Note

This Dover edition, first published in 1997 and reissued in 2013, is an unabridged and slightly corrected republication of the work first published by Integer Press, Ottawa, Ontario, Canada in 1985, under the title The Green Book: 100 Practice Problems for Undergraduate Mathematics Competitions.

Library of Congress Cataloging-in-Publication Data

Hardy, Kenneth.

[Green book]

The green book of mathematical problems / Kenneth Hardy and Kenneth S. Williams.

        p. cm.

Originally published: The green book. Ottawa, Ont., Canada: Integer Press, 1985.

Includes bibliographical references.

eISBN-13: 978-0-486-16945-3

1. Mathematics—Problems, exercises, etc. I. Williams, Kenneth S. II. Title.

QA43.H268 1997

Manufactured in the United States by Courier Corporation

69573504

www.doverpublications.com

PREFACE

There is a famous set of fairy tale books, each volume of which is designated by the colour of its cover: The Red Book. The Blue Book, The Yellow Book, etc. We are not presenting you with The Green Book of fairy stories, but rather a book of mathematical problems. However, the conceptual idea of all fairy stories, that of mystery, search, and discovery is also found in our Green Book. It got its title simply because in its infancy it was contained and grew between two ordinary green file covers.

The book contains 100 problems for undergraduate students training for mathematics competitions, particularly the William Lowell Putnam Mathematical Competition. Along with the problems come useful hints, and in the end (just like in the fairy tales) the solutions to the problems. Although the book is written especially for students training for competitions, it will also be useful to anyone interested in the posing and solving of challenging mathematical problems at the undergraduate level.

Many of the problems were suggested by ideas originating in articles and problems in mathematical journals such as Crux Mathematicorum, Mathematics Magazine, and the American Mathematical Monthly, as well as problems from the Putnam competition itself. Where possible, acknowledgement to known sources is given at the end of the book.

We would, of course, be interested in your reaction to The Green Book, and invite comments, alternate solutions, and even corrections. We make no claims that our solutions are the best possible solutions, but we trust you will find them elegant enough, and that The Green Book will be a practical tool in the training of young competitors.

We wish to thank our publisher. Integer Press; our literary adviser; and our typist, David Conibear, for their invaluable assistance in this project.

Kenneth Hardy and Kenneth S. Williams

Ottawa, Canada

May, 1985

Dedicated to the contestants of the

William Lowell Putnam Mathematical Competition

To Carole with love

KSW

CONTENTS

Notation

The Problems

The Hints

The Solutions

Abbreviations

References

NOTATION

THE PROBLEMS

Problems, problems,

problems all day long.

Will my problems work out right or wrong?

The Everly Brothers

1.   If {bn: n = 0,1,2, . . . } is a sequence of non-negative real numbers, prove that the series

converges for every positive real number a.

2.   Let a,b,c,d be positive real numbers, and let

Evaluate the limit .

3.   Prove the following inequality:

4.   Do there exist non-constant polynomials p(z) in the complex variable z such that |p(z)| < Rn on |z| = R, where R > 0 and p(z) is monic and of degree n ?

5.   Let f(x) be a continuous function on [0,a], where a > 0, such that f(x) + f(a–x) does not vanish on [0,a]. Evaluate the integral

6.   For ε > 0 evaluate the limit

7.   Prove that the equation

has no solution in integers x,y,z.

8.   Let a and k be positive numbers such that a² > 2k. Set x0 = a and define xn recursively by

Prove that

exists and determine its value.

9.   Let x0 denote a fixed non-negative number, and let a and b be positive numbers satisfying

Define xn recursively by

Prove that exists and determine its value.

10.   Let a,b,c be real numbers satisfying

Evaluate

11. Evaluate the sum

for n a positive integer.

12.   Prove that for m = 0,1,2, . . .

is a polynomial in n(n+1).

13.   Let a,b,c be positive integers such that

Show that ℓ = 2abc − (bc+ca+ab) is the largest integer such that

is insolvable in non-negative integers x,y,z.

14.   Determine a function f(n) such that the nth term of the sequence

is given by [f(n)].

15.   Let a1, a2, . . . , an be given real numbers, which are not all zero. Determine the least value of

where x1, . . . , xn are real numbers satisfying

16.   Evaluate the infinite series

17.   F(x) is a differentiable function such that F′(a−x) = F′(x) for all x satisfying 0 ≤ x ≤ a. Evaluate and give an example of such a function F(x).

18.   (a) Let r,s,t,u be the roots of the quartic equation

Prove that if rs = tu then A²D = C².

(b) Let a,b,c,d be the roots of the quartic equation

Use (a) to determine the cubic equation (in terms of p,q,r) whose roots are

19.   Let p(x) be a monic polynomial of degree m ≥ 1, and set

where n is a non-negative integer and denotes differentiation with respect to x.

Prove that fn(x) is a polynomial in x of degree (mn – n). Determine the ratio of the coefficient of xmn–n in fn(x)