Murray Klamkin Problems

68 Pages·2005·0.375 MB·English

by  Andy Liu, Bruce Shawyer

A T O M
ASTE F ATHEMATICS
A T O M
(cid:19)
IME{ { N LES ATHEMATIQUES
Volume / Tome VII
THE MURRAY KLAMKIN PROBLEMS
CANADIAN COLLECTION | Part 1.
Edited by
Andy Liu
University of Alberta
and
Bruce Shawyer
Memorial University of Newfoundland
Published by the Canadian Mathematical Society, Ottawa, Ontario
and produced by the CMS ATOM O(cid:14)ce, St. John’s, NL
Publi(cid:19)e par la Soci(cid:19)et(cid:19)emath(cid:19)ematique du Canada, Ottawa (Ontario)
et produit par le Bureau ATOM de la SMC, St. John’s, NL
Printed in Canada by / imprim(cid:19)e au Canada par
The University of Toronto Press Incorporated
ISBN 0-919558-16-X
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, including
photocopying, recording, or any information or retrieval system now known or
to be invented, without permission in writing from the publisher: The Canadian
MathematicalSociety,577KingEdwardAvenue,P.O.Box450,StationA,Ottawa,
Ontario K1N 6N5, Canada, except so far as may be allowed by law.
Tous droits r(cid:19)eserv(cid:19)es. Aucune partie de cet ouvrage ne peut ^etre
reproduiteouutilis(cid:19)eeparquelque proc(cid:19)ed(cid:19)eouquelquefac(cid:24)onquece soit,ycompris
les m(cid:19)ethodes(cid:19)electroniques ou m(cid:19)ecaniques, les enregistrements ou les syst(cid:18)emes de
mise en m(cid:19)emoire et d’information, sans l’accord pr(cid:19)ealable (cid:19)ecrit de l’(cid:19)editeur, la
Soci(cid:19)et(cid:19)emath(cid:19)ematiquedu Canada, 577,rue King-Edward,C.P. 450,succursaleA,
Ottawa (Ontario) K1N 6N5, Canada, sauf dans les limites prescrites par la loi.
c 2005
(cid:13)
Canadian Mathematical Society / Soci(cid:19)et(cid:19)e math(cid:19)ematique du Canada
A T O M
ASTE F ATHEMATICS
A T O M
(cid:19)
IME{ { N LES ATHEMATIQUES
Volume / Tome VII
THE MURRAY KLAMKIN PROBLEMS
CANADIAN COLLECTION | Part 1.
Edited by
Andy Liu
University of Alberta
and
Bruce Shawyer
Memorial University of Newfoundland
The ATOM series
The booklets in the series, A Taste of Mathematics, are published by
the Canadian Mathematical Society (CMS). They are designed as enrichment
materials for high school students with an interest in and aptitude for
mathematics. Some booklets in the series will also cover the materials useful
for mathematical competitions at national and international levels.
La collection ATOM
Publi(cid:19)es par la Soci(cid:19)et(cid:19)e math(cid:19)ematique du Canada (SMC), les livrets
de la collection Aime-t-on les math(cid:19)ematiques (ATOM) sont destin(cid:19)es au
perfectionnement des (cid:19)etudiants du cycle secondaire qui manifestent un int(cid:19)er^et
et des aptitudes pour les math(cid:19)ematiques. Certains livrets de la collection
ATOM servent (cid:19)egalement de mat(cid:19)eriel de pr(cid:19)eparation aux concours de
math(cid:19)ematiques sur l’(cid:19)echiquier national et international.
Editorial Board / Conseil de r(cid:19)edaction
Editor-in-Chief / R(cid:19)edacteur-en-chef
Bruce Shawyer
MemorialUniversityofNewfoundland /Universit(cid:19)eMemorialdeTerre-Neuve
Associate Editors / R(cid:19)edacteurs associ(cid:19)es
Edward J. Barbeau
UniversityofToronto/Universit(cid:19)edeToronto
Malgorzata Dubiel
SimonFraserUniversity/Universit(cid:19)eSimonFraser
Joseph Khoury
UniversityofOttawa/Universit(cid:19)ed’Ottawa
Antony Thompson
DalhousieUniversity/Universit(cid:19)eDalhousie
Managing Editor / R(cid:19)edacteur-g(cid:19)erant
Graham P. Wright
UniversityofOttawa/Universit(cid:19)ed’Ottawa
iv
Table of Contents
Preface 1
1 Quickies 2
2 Combinatorics and Number Theory 27
3 Functions and Polynomials 39
4 Expressions and Identities 51
v
QUICKIES
Murray Klamkin was famous for his Quickies, problems that had
quick and neat solutions. We present all the Quickies published in CRUX
MATHEMATICORUM, with some slight editing by Bruce Shawyer.
PROBLEMS AND SOLUTIONS
The problems have been selected by Andy Liu and arranged into sets
according to topic.
ThesolutionsareaspublishedinCRUXMATHEMATICORUM,withsome
slight editing by Bruce Shawyer. Solutions from pre-LATEX editions were coded
by students at Memorial University of Newfoundland, funded by the Canadian
Mathematical Society. Special thanks are due to Karelyn Davis, Alyson Ford,
Don Hender, Shawna Gammon, Paul Marshall, Shannon Sullivan and Rebecca
White.
These problems exhibit the special talents of Murray Klamkin. They cover
a very wide range of topics, and show a great deal of insight into what is possible
in there areas. They demonstrate that Murray Klamin was a problem setter par
excellence. We are very greatful to have known him and to have been inspired by
him.
Problemnumbersandreferencesareto[year:pagenumber]areasinCRUX
?
MATHEMATICORUM. When a problem numberis followedbyastar , this
means that the problem was proposed without a solution.
Andy Liu Bruce Shawyer
Department of Mathematics Department of Mathematics
University of Alberta Memorial University of Newfoundland
Edmonton St. John’s
Alberta Newfoundland and Labrador
1
Problems posed by Murray Klamkin.
Unlessotherwisestated, theseproblemswereproposedbyMurrayKlamkinalone.
Quickies Part 1
1|54
Combinatorics and Number Theory Part 1
429, 1456, 2054, 1863, 1027, 969, 1561, 2034, 1752, 1434.
Vectors and Matrices Part 4
1200, 1721, 1482, 1693, 2005, 398, 1314, 3024, 1207, 1242.
Functions and Polynomials Part 1
299, 254, 1423, 2014, 1110, 1283.
Expressionsand Identities. Part 1
1304, 287, 1594, 1522, 830, 1996, 1362.
Numerical Approximations Part 4
1003, 1213, 1371.
Algebraic Inequalities. Part 2
347, 1642, 2615, 1703, 1734, 1445, 2064, 2095, 2044, 1652,
1742, 1674, 1774, 323, 805, 1394, 2734, 2839, 1662.
Trigonometric Inequalities. Part 3
1414, 908, 1352, 1712, 1542, 1613, 1503, 2084, 1332, 1801,
1271, 1060, 958, 1962.
Geometric Inequalities. Part 4
1165, 1473, 1574, 1764, 1296, 506, 1131, 1985, 1945.
The Triangle. Part 3
1872, 1605, 1385, 2848, 210, 1076, 1532, 2618.
Cevian Lines. Part 4
2614, 548, 485, 2613, 1621, 1631.
Central Symmetry. Part 3
1062, 1348, 1513.
Conic Sections. Part 2
2616, 1975, 1935, 1405, 520.
Solid Geometry. Part 2
375, 1784, 1553, 1581, 330, 478, 2617, 1261.
Higher Dimensions. Part 2
2651, 224, 1086, 1465, 2733, 2024, 1793.
Calculus. Part 4
1178, 1494, 1322, 1147, 273.
Problems dedicated to Murray Klamkin Part 4
1241, 2619, 2620, 2621
Klamkin Problems of September 2005 Part 4
K{01 through K{15
2
1 Quickies
1
. Determine the extreme values of r =h +r =h +r =h +r =h where h , h ,
1 1 2 2 3 3 4 4 1 2
h , h are the four altitudes of a given tetrahedron T and r , r , r , r are the
3 4 1 2 3 4
correspondingsigned perpendicular distances from any point in the space of T to
the faces.
2
. Determine the minimum value of the product
P =(1+x +y )(1+x +y ):::(1+x +y )
1 1 2 2 n n
where x ;y 0, and x x :::x =y y :::y =an.
i i 1 2 n 1 2 n
(cid:21)
3. Prove that if F(x;y;z) is a concave function of x, y, z, then F(x;y;z) 2 is
(cid:0)
f g
a convex function of x, y, z.
4
. If a, b, c are sides of a given triangle of perimeter p, determine the maximum
values of
(i) (a b)2+(b c)2+(c a)2;
(cid:0) (cid:0) (cid:0)
(ii) a b + b c + c a;
j (cid:0) j j (cid:0) j j (cid:0) j
(iii) a b b c + b c c a + c a a b:
j (cid:0) jj (cid:0) j j (cid:0) jj (cid:0) j j (cid:0) jj (cid:0) j
5
. IfA, B,C arethreedihedralanglesofatrihedralangle,showthatsinA,sinB,
sinC satisfy the triangle inequality.
6
. Are there any integral solutions (x;y;z) of the Diophantine equation
(x y z)3 =27xyz
(cid:0) (cid:0)
other than ( a;a;a) or such that xyz =0?
(cid:0)
7
. Does the Diophantine equation
(x y z)(x y+z)(x+y z)=8xyz
(cid:0) (cid:0) (cid:0) (cid:0)
have an in(cid:12)nite number of relatively prime solutions?
8
. It is an easy result using calculus that if a polynomial P(x) is divisible by
its derivative P (x), then P(x) must be of the form a(x r)n. Starting from the
0
(cid:0)
known result that
P (x) 1
0
=
P(x) x r
i
(cid:0)
X
where the sum is over all the zeros r of P(x) counting multiplicities, give a
i
non-calculus proof of the above result.
9
. Solve the simultaneous equations
x2(y+z)=1; y2(z+x)=8; z2(x+y)=13:
3
10
. Determine the area of a triangle of sides a, b, c and semiperimeter s if
a b c
(s b)(s c)= ; (s c)(s a)= ; (s a)(s b)= ;
(cid:0) (cid:0) h (cid:0) (cid:0) k (cid:0) (cid:0) l
where h, k, l are consistent given constants.
11
. Provethat
3(x2y+y2z+z2x)(xy2+yz2+zx2) xyz(x+y+z)2
(cid:21)
where x, y, z 0.
(cid:21)
12
. Determine all integral solutions of the Diophantine equation
(x8+y8+z8)=2(x16+y16+z16):
13
. Determine all the roots of the quintic equation
31x5+165x4+310x3+330x2+155x+33=0:
14
. If F(x) and G(x) are polynomials with integer coe(cid:14)cients such that
F(k)=G(k) is an integer for k =1, 2, 3, :::, prove that G(x) divides F(x).
15
. GiventhatABCDEF isaskewhexagonsuchthateachpairofoppositesides
are equal and parallel. Provethat the midpoints of the six sides are coplanar.
16
. If a, b, c, d are the lengths of sides of a quadrilateral, show that
pa pb pc pd
; ; ; ;
(4+pa) (4+pb) (4+pc) (4+pd)
are possible lengths of sides of another quadrilateral.
17
. Determine the maximum value of the sum of the cosines of the six dihedral
angles of a tetrahedron.
18
. Which is larger
(p32 1)1=3 or 3 1=9 3 2=9+ 3 4=9?
(cid:0) (cid:0)
p p p
19
. Provethat
a b c b c a 1 1 1
3 min + + ; + + (a+b+c) + +
(cid:2) b c a a b c (cid:21) a b c
(cid:26) (cid:27) (cid:18) (cid:19)
where a, b, c are sides of a triangle.
20. Let ! =ei(cid:25)=13. Express 1 as a polynomial in ! with integral coe(cid:14)cients.
1 !
(cid:0)
21
. Determine all integral solutions of the simultaneous Diophantine equations
x2+y2+z2 =2w2 and x4+y4+z4 =2w4.
4
22
. Prove that if the line joining the incentre to the centroid of a triangle
is parallel to one of the sides of the triangle, then the sides are in arithmetic
progressionand, conversely, if the sides of a triangle are in arithmetic progression
then the line joining the incentre to the centroid is parallel to one of the sides of
the triangle.
23
. Determine integral solutions of the Diophantine equation
x y y z z w w x
(cid:0) + (cid:0) + (cid:0) + (cid:0) =0
x+y y+z z+w w+x
(joint problem with Emeric Deutsch, Polytechnic University of Brooklyn).
24
. For x, y, z >0, prove that
1 1 x
(i) 1+ 1+ ,
(x+1) (cid:21) x(x+2)
(cid:26) (cid:27)
(ii) [(x+y)(x+z)]x[(y+z)(y+x)]y[(z+x)(z+y)]z [4xy]x[4yz]y[4zx]z.
(cid:21)
25
. If ABCD is a quadrilateral inscribed in a circle, prove that the four lines
joining each vertex to the nine point centre of the triangle formed by the other
three vertices are concurrent.
26
. How many six digit perfect squares are there each having the property that
if each digit is increased by one, the resulting number is also a perfect square?
27
. LetV W ,i=1,2,3,4,denotefourceviansofatetrahedronV V V V which
i i 1 2 3 4
are concurrent at an interior point P of the tetrahedron. Prove that
PW +PW +PW +PW maxV W longest edge:
1 2 3 4 i i
(cid:20) (cid:20)
28
. Determine the radius r of a circle inscribed in a given quadrilateral if the
lengths of successive tangents from the vertices of the quadrilateral to the circle
are a, a, b, b, c, c, d, d, respectively.
29. Determine the four roots of the equation x4+16x 12=0.
(cid:0)
30
. Prove that the smallest regular n{gon which can be inscribed in a given
regular n{gon is one whose vertices are the mid-points of the sides of the given
regular n{gon.
31. If 311995 divides a2+b2, prove that 311996 divides ab.
32
. Determine the minimum value of
S = (a+1)2+2(b 2)2+(c+3)2+ (b+1)2+2(c 2)2+(d+3)2) +
(cid:0) (cid:0)
p (c+1)2+2(d 2)2+(a+3)2+p (d+1)2+2(a 2)2+(b+3)2
(cid:0) (cid:0)
p p
where a, b, c, d are any real numbers.
33
. A set of 500 real numbers is such that any number in the set is greaterthan
one-(cid:12)fth the sum of all the other numbers in the set. Determine the least number
of negative numbers in the set.