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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
THE GREEN BOOK,OF MATHEMATICAL PROBLEMS,Kenneth Hardy and Kenneth S Williams. Carleton University Ottawa,DOVER PUBLICATIONS INC,Mineola New York. Copyright 1985 by Kenneth Hardy and Kenneth S Williams. All rights reserved under Pan American and International Copyright. Conventions, Published in Canada by General Publishing Company Ltd 30 Les. mill Road Don Mills Toronto Ontario, Published in the United Kingdom by Constable and Company Ltd. 3 The Lanchesters 162 164 Fulham Palace Road London W6 9ER. Bibliographical Note, This Dover edition first published in 1997 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 n Publication Data,Hardy Kenneth. Green book, The green book of mathematical problems Kenneth Hardy and. Kenneth S Williams, Originally published The green book Ottawa Ont Canada. Integer Press 1985,Includes bibliographical references. ISBN 0 486 69573 5 pbk,1 Mathematics Problems exercises etc I Williams. Kenneth S II Title,QA43 H268 1997,5IO 76 dc21 96 47817. Manufactured in the United States of America, Dover Publications Inc 31 East 2nd Street Mineola N Y 11501. 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 lOO 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 Oust 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. Dedicated to the contestants of the,William LoweD Putnam Mathematical Competition. To Carole with love,Nota tion IX,The Problems 1,The Hints 25. The Solutions 41,Abbreviations 169,References 171,xl denotes the greatest integer i x where. x is a real number,x denotes the fractional part of the real. number x that is x x x,ln x denotes the natural logarithm of x. exp x denotes the exponential function of x,p n l denotes Euler s totient function defined. for any natural number n,GCD a I b denotes the greatest common divisor of. the integers a and b,denotes the binomial coefficient nl kl n kll. where nand k are non negative integers,the symbol having value zero when n k l. denotes a matrix with a ij as the i j th, det A denotes the determinant of a square matrix A. THE PROBLEMS,Problems problems,problems all day long. Will my problems work out right or wrong,The Everly Brothers. 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. o b d a a b a 2b a n l b,11 a c c c d c 2d c n l d. Evaluate the limit L lim Q a b c d,3 Prove the following inequality. tit x x 0 x 1,2 PROBLEMS 4 12, 4 Do there exist non constant polynomials p z ln the complex. variable z such that Ip z 1 R on Izl R where R and. p z is monic and of degree n, 5 Let f x be a continuous function on O al where a 0. such that f x f a x does not vanish on O a Evaluate the. f x f a x dx,6 For E 0 evaluate the limit,lim x sinCt dt. 7 Prove that the equation,7 0 x4 y4 z4 2y2Z2 2z 2x2 2x 2y2 24. has no solution in integers x y Z,8 Let a and k be positive numbers such that a 2k. Set xO a and define xn recursively by,8 0 xn xn 1 k. Prove that,exists and determine its value,PROBLEMS 4 12 3. o denote a fixed non negative number and let a and. b be positive numbers satisfying,Define x recursively by. Prove that lim x exists and determine its value,10 Let a b c be real numbers satisfying. a 0 c 0 b2 ac,max ax 2bxy cy,11 Evaluate the sum,n n l n 2r 1. for n a positive integer,12 Prove that for m 0 1 2. 12 0 S n 12m l 22m l n 2m l,is a polynomial in n n l. 4 PROBLEMS 13 21,13 Let a b c be positive integers such that. GCD a b GCD b c GCD c a 1, Show that t 2abc bc ca ab is the largest integer such that. bc x ca y ab z t,is insolvable in non negative integers x y z. 14 Determine a function fen such that the nth term of the. 14 0 1 2 2 3 3 3 4 4 4 4 5,is given by fen, 15 Let a 1 a 2 an be given real numbers which are not all. zero Determine the least value of,where xl xn are real numbers satisfying. 16 Evaluate the infinite series,S 1 rr rr 3T, 17 F x is a differentiable function such that F a x F x. for all x satisfying 0 x a Evaluate fa F x dx and give an.