Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory

357 Pages·2000·7.11 MB·English

by  Jiri Herman

JiHreir maRna daKnu cera
v
JaroSmiimrs a
Equatiaonnds
Inequalities
ElemenPtraorbyl aenmds
TheorienmA sl geabnrda
Number Theory
TranslabtyeK da rDli lcher
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Canad1Maant hematSioccaile ty
� :7 Sociemtaet hematidquuC ea nada
CMBSo oikMnsa thematics
Ouvramgaetsh edmea dteIiS aqM uCe s
EquatiaonndsI nequalities
1 HERMAN /KutERA/SIMSA
AbeliGanr oupansd RepresentoaftiF oinnsiP taerti allOyr derSeedt s
2 ARNOLD
ConveAxn alyasnidsN onlinOepatri mization
3 BORWEIN/LEWIS
OrthogoPnoallyn omiaflosExr p onentiWaeli ghts
4 LEVIN/LUBINSKY
ReflectGiroonu pasn dI nvariTahneto ry
5 KANE
JifiH ermanR adaKnu cera
v
JaromSiirm sa
Equatainodn s
Inequalities
ElententParroyb leantnsd
Theoreinnt sAl gebraan d
NUinberT heory
TranslbaytK eadrD lil cher
Springer
Herman RadanK ucera
Jiff
Brno Gr ar School DepartmoefnM ta thematics
amm
KptJ.ar os1e4 FaculotfyS cienMcaes,ar ykU niversity
tf.
658 Brno Janackonvaom .2 a
70
CzecRhe public 6629 5B rno
CzecRhe public
Jaromfr imsa
S Translator
MathematiIcnaslti tute Karl Dilcher
Academoyf S ciencoefths e Czech DepartmoefnM ta thematiancds S tatistics
Republic DalhouUsniiev ersity
iZk.ova2 2 HalifNaoxv,a S cotiaB 3H 3J5
Z
6166 2B rno Canada
CzecRhe public
Editors-in-Chief
Redacteurs-en-chef
JonathaBno rwein
PeteBro rwein
Centref orE xperimenatanld C onstructMiavteh ematics
DepartmoefnM ta thematicasn dS tatistics
SimoFnr asUenri versity
Burnaby,B ritisCho lumbiVaS A 1S6
Canada
MathematSiucbsj eCclta ssific(a2t0i0on01 )6:G 202,0Kx x
Libraoryf C ongresCsa taloging-in-PDuabtlai cation
Herman, Jiff.
[Metodyr esemnaft ematiculcylcohIh . E nglish]
Equationasn di nequal:ie tlieemse ntparroyb leamnsd t heoreimnsa lgebarnadn umber
theoryJ iffH erman,R adaKnu eerJaar,o mirS ims;a translabtyeK dar lD ilcher.
I
p.e m.- (CMSb ookisn m athematics)
Includbesi bliograrpehfiecraelna cnedis n dex.
ISBN0 -387-98942-p0a per)
(alk.
1.E quations-Numeriscoallu tio2n.sIn .e qualiti3e.Prs o.b lems olvinIg.K. u cera
Radan. SimSaJ,ar omir. Title. Series.
n. m. IV.
QA218.H471230 00
512.4-9-d'c21 99-047384
ona cid-frpeaep er.
Printed
2000 SpringeVre-rlaNge wY orkIn,c .
©
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10010U,S A),e xcepfto rb rieefx cerpts connectwiiotnrh e vieowrss cholaralnya lysUisse.
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softwareo,r b ys imiloarrd issimilmare thodolnoogwykn own orh ereafter developiesfd o rbidden.
The useo fg enerdaels cripntaimveestr ,a den ames,t rademarektsc,. ,t hipsu blicaetvieoinn, ft he
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Preface
Thisb ooki si ntendase da t ext fora problem-soclovuirnsgae tt hefi rst­
ors econyde-aurn iverslietvyeas l ,a t extf ore nrichmcelnasts esf otral ented
high-soclhs otudenotrsf ,o rm athematciocsm pteititornai ninIgtc. an also
be as as ourcoefs upplementmaartye rifalo ran y coursdee aling with
used
alegbraeiqcu atioorni sn equaliotrit eoss u,p plemean stt andaerlde mentary
numbetrh eoryc ourse.
Theraree alreamdanyy excellbeonotk osn t hem arketth actan beu sed
foar problem-soclovuirnsgHe o.w evesro,m ea rem erelcyo llectoifpo rnosb ­
lemsfr om a varieotfyfi elds alacnkdc ohesiOotnh.e rpsr esepnrto blems
accorditnotg o pibcu,t p rovildiet tolren ot heoretbiaccalk rgoundM.o st
problebmo oks haal viem itneudm beorf r athcehral lengipnrgo blemWhsil.e
these probletmesn dt ob eq uitbee autifutlh,e cyan appeafro rbiddanidn g
discoguirnatgo a beginnisntgu denetv,e nw itwehl l-motivaatnedcd a refully
writtseonl untsio. a conseqnucees,t udenmtasy decidteh atp roblem
As
solviinsg onfloyrt hef ew hipgerhf ormeirnts h eir calnasdsa ,b andotnh is
impotrantp arto ft heimra thematicaaln,d i needd overaleld,u cation.
Oneo ft her easonwsh yp roblesmo lviinsog ft enf ountdo b ed ifficulits
thef actth ait nr ecendte cadest here beenl esse mphasiosn t echnical
has
skillisnN ortAhm ericanh igh-schmoaotlh ematiFucrst.h ermorseu,c shk ills
arer areltya ughatt u nivresitwyh,e rmeo stc oursaerseq uitteh eoretoirc al
structure-oriean rteseudl.ta , l acokf ' 'mathematfliucealn cyi"so ften
As
evideenvte inn u ppery earastu nivseirttyh;i rse ductesh ee njoymeonftt h e
subjeacntd i mpaiprrso gresansd success.
A seconrde asoni st hamto sts tudtesna ren otu sedt om orec omplexo r
multiy-elraepdr oblemsI.tw ouldc ertainbleyw rongt og ivteh ei mpres-
Preacfe
vi
siotnh aat lmla thematipcralo blesmhso ulsdu ccumtbo a straightforward
approacIhn.d eemdu,c ho ft hea ttractivoefnm eastsh ematilcisesi nt he
satisfacdteiroinv ferdo ms olvidnigffi culptr obleaftmesr muche fforatn d
severfualt ilaet temptOsn. t heo thehra ndb,e in"gs utck"t ooo ften and
fort ool oncg an bvee ryd isocuragianngd c ounterproduancdt iivsue l,t i­
matelay w astoef t imet hact oulbde b ettesrp enlte arning panrdac ticing
new techniques.
Thisb ooka ttempttosa ddretshses ei ssuaensd o fferas p artirale med.y
Thisi sd oneb y emphasizinbgas ica lgebroaipce ratianodn so thetre chni­
caslki lltsh aarte reinforicnne udm erouesx amplaesn dex ercisHeosw.e ver,
evetnh ee asiesto nesr equiar sem altlwis t.W e thereforet hhaottph ei psr o­
cesosf p racticidnoge sn otb ecompeu relryo teb,u tw ilrle tatihne r eader's
interesrta,i shei osr h erl evoeflc onfidenacned,e ncoureaa gttempttoss olve
someo ft hem orec hallenpgrionbgl ems.
Anotheari mo ft hibso oki st of amiliartizhee r eadewri thm ethodfso r
solvinpgr obleimnse lementmaartyh ematiaccsc,ei sbslteo b eginningu ni­
versiantdy advancedh igh-schosotlu denTthsi.s c an bed onei nd ifferent
ways;f ori nstantchee,a uthorosfs omeb ooksi ntrodusceev ergale neral
methods( e.,gi .nductianoanl,og y,o rt hep igoenholper incipanlde )i llus­
trateea cho new ithc oncreptreo blefrmosm differeanrte aso fm athematics
andw ithva ryindgeg rees ofd ifficuyl.t
Oura pproachho,w eveirsd, i ffereWnet p.r esenat r elatisveellfy- condt aine
overviewo vers omep artosf e lementamrayth ematitchsa td on ot receive
mucha ttentiionhn igh-schoanodl universietdyu catiWoen .g iveo nly
enought heoretibcaalc gkrountdo m aket hesteo piscesl f-contanainde dr ig­
orousan,d concnetratoen s olvipnagr ticuplraorb leimnst het hreem ain
areasc orrespondtiont gh efi rstth reec hapteorfts h ibso ok.
Thesec haptearres f airly indepoefne daecnhot t hewri,t ho nlay limited
numbeor fc ross-refeWreitnhciesen.a ch chatperc,l ustoerfss e ctioannsd
subscetioanrse t ietdo getheeri thbeyrt opiocr,b yt he methnoeeddesd to
solvteh ee xamplanesd exerciTsheesp .r oblem-bcohoarka ctoefrt hist ext
isu nderlinbeydt hel argen umbeorf e xercitsheescy;an bes olvebdy using
a methoodr m ethodpsr eviouisnltyr uocdedW.e sugesgt thatt her eader
firsctar efullsyt udayn yr elevanexta mplesb,e foraet tempttionsg o lvaen y
330
oft heex ercises.
760
Thei ndividpuralo ble(mdsi vidiendta op prxoimately examplaensd
exercisaesr) e voafr yidnegg reeso fd iffictuylfr, om completsetlrya ight­
forwarwdh,e rteh eu seo fa methoudn decro nsirdaetiowni lilm mediately
leatdo a solutitoonm ,u chm ored ifficulptr oblewmhso seso lutiownisl l
sometimreesq uicroens ideraebffloer tT.h em ored emandinegx erciasrees
(.* )
markedw itha na steriskA nswertso exercsiesc an bef ounidn t he
all
finalch atperw,h eraed ditiohnianltan sd instructtioto hnesm ored iffictu l
oneasr eg iveasn well.
Thep roblemwesr et akenfr om a varieotfym ainly EastEeurrno pean
sources, as sMuacthh ematiOclaly mpiaadnsd o ther competMiatniyo ns.
Preacfe vii
otfha terhmene oortteeh fasaeesoicrwsrtlcitieoyEshb neelegl eai ksihn-gs p
re.Aai ndm ecprrofirotstreean orltptfie rh ocoetntb th ilhaoeetnmi sr
soslhu,iot ncui iblopapednrcil sitcbeh noesl,i se ghs -t.sWu ced heonotls
was
bletiehewvavtietelhtn iih oms cni sanetu acstcticesiorsmnefuu alltlaiytv ee
woimrnak t anhidel mildatuitssanvi rtdercir .ssca i httneye ss
Oonofeou bvrhjes base tee csontta eirlrestanspa pttsroioo vlaevc ihnesg
ag ipvrSeoonbm wlpeteeri msom.edve viitsesffd roeeialor nlune teni ons
plaaocttet h,iwre meerwet t hspoui r irlaolnne bap ltaoteerfhmrt e s
book.
Tbhoiiatoss rak ianto otsfsnhl e eCc zeoedoncifdht ion
I" Mfesotorhmtl oahvdMeipestorimrd ryesIgao ne-t" tbilceamls
pmaitueimsicba1k'gthcnulha9l Moe ht9 asd 6Ua nriiyBnvkrA e npfroarrsc.toio mtr yrt ehcet ion
( )
osfom mimenit otssrhp,m er a ihtmbnaseailee runenin ft acT lhh arneeg ed.
shsoewrc"ittct iohoym npppsreea o"ttteb hi onlettfdce hip shmeot as ne -rts
wdeterbelesude oot,tmf he es weie nrpcerio otnrmbhtpaleooie rnma st ed
bootdfbhy oe ao lkp.ih nwaadasbde edxte idc.a l
TChzeee docitfhth iiosun bsbstyeo ah douei kts n hp hoe­arvse been
An
ceinanrclasteil sf scso hermc eosntadunarydMd-ea sntfcthohsero m oalt ical
OltyrmitaipnCnhi zReieaepn dIcug hthab asb ll uesioseocarnen . le d arg u
basaits UM nasiicavnore ufuyrurtkmsrs aieett tshye ea afmt.Coa hrte ircss
sescco.hpn ordAoeaE llrnvsiygem oltrififihsn twsreiacohs orh tnya pters
wsasu ctcroebiutsyetths sd reful ia alannn lte cyowr prler yao tbelde m­
socloivtuniFhrl noaesl1gf e 9 D 9a8Ul vnhaeioitrHun sals iiitfeyax ,,
CaInwtfaas ted thlca ato.ap u abrraosttsneblhe owyidoo s qkrt uke eid
weelivlacne wl naias w tsrih ado menfa g tebh aencmadkansgtd ri ocual
ab.Ti bhl'osesio tutktrrihueeseexc, at wanmtodprhr laieklesnn ee,g dv e e l
odfil offittfecyh xu eme akeirp tca irwstseesiulas cisl utoa fleu oadrrr c ley
asisgrneaneaddd o iehrncxokgiem ssre esw.
Tthreaa flnlteesssthflol ,oaa l ttoCheowzexecria h ntm gbhp oilosek ,
myba seu fiaoct roiaa utnbc re lhesaeede rpu srcf'oraho gt irigamoh n­
scmhaottoahelc.eF h miweanhertato stitlhpr chlaeeswey afi i ,tdnl hedlir s
boaro sikocou uhfsir tedc,ifueee tlqn ni ausiesna,ndt ie ieqosu.a lit
Brannod Halifax JHiefrim an
J1u9l9y9 RaKduacne ra
Ja rSoimmi8ra
KDairllc her

Contents
Preface
v
Symbols
xi
Alg1eb FrraiIocd mefntuPanoiwdtE oliqre.euass.a r t i so.ns . . 1.
1 1
2 FiSn.i .t e. .. . . . 5.
3 Poal.ly. sn. . .o . . m. i .. . . 23
ums
4 SmyemPtoorlmiynica. .l s.. . . .
. . . . . .
5 SyosEfttq eiu.moa .ns. s . . .4368 . .
. . . . . . . . . . . .
6 IrErtqaiu.to.a in. .os . n . a l. 6.0 .
.
7 SoAmptepio Clofoins Nmcm upableerxs 7 5
. . . . . .
. .
Al1ge DbfirInaieecnq uaaliitPniteersdi ro otp.n.i s es . . ..9 0.. .. .
2 89
2 BiasMcte h..o . d. .s. . . . . . . 9.4
3 TUhseAe l Fgrooeasmfb u. r.l ai ..c ..1 0..6
. .
4 TMhhetedooS fq u.ar. e .s.. . .. . 1 3.1
. . . . . . .
5 TDhrieais anmcCntiuayd ncI' hnste. yq. u a1l2i4
. . .
6 TIhnePd ru.ic nt.ci. .io .pn. l e15 3
. . . . .
7 Ch'esIeb nqy.us ah.lev.. i ...t y. 15 4 . .
IenqBueeetwaMnles ia t.ni es. .15 0. .
. .
89 ApopIner nNrmduabi.tex ir osna1l6 9
.
Contents
x
NumbeBrT chasCeo iory. n c.e.. p .t s.. .
3 117473
1 PrNium.m.e b . e.. r .s.
2 Cocn.eg s.r. . u. . .e . . n .. 138
. . . . . .
3 ConiOgrnVnu iareea.. n b .c. .l ..e. e s 189
. . . . . . . .
4 DoipEhqaun.at ti.in o..e. n s2 10
. . . .
5 SoloDvafib oEip.aql ht.ui.. a it noytn is2n 17e
. . .
6 InPtareFrtga ePcartan ird. ot .. n ..a l23. 8
. . .
7 BReasper esen.t .a tio25n1 s
.
8 DritiP'crhs.il n.e..c. . i. ..p le 258
. . . .
9 Pol.y .n o.m ials 268
.
10 . . . 274
. . .
.
HintsanH dAn iaswAnenrsntd tssC ow h earpst er
4 283
1 HiAnnttssCoaw h p1ae. trn. e.sd.r 238
2 HiaAnnntdswsC ae hp2r tse rt o 294
3 3 . 312
.
Bibliography
337
Index
341