Table Of Contentgoo{s o Aa xasonJJas
larusglalul
rv,rv,rv,'er1o ¡d ;oq 'c our
@ 2007,2012 AoPS Incorporated. A1l Rights Reserved.
Reproduction of any portion of this book without the written permission of AoPS Incorporated is strictly
prohibited, except for "fair use" or other noncommercial uses as defined in Sections 107 and 108 of the
U.S. Copyright Act.
Publishedby: AoPSlncorporated
PO. Bor 2185
Alpine, CA 91903-2185
(619) 659-1.612
booksoartofproblemsolvÍng. com
ISBN-13: 97 8-1-934124-06-2
Visit the Art of Problem Solving website at http: //www, artofproblemsolving . com
ELr;fllE Scan this code with your mobile device to visit the Art of Problem Solving website, to
ffii+ view our other books, our free videos and interactive resources, our ontnJcommunity,
ffiffi a¡d ou¡ online school'
Cover image designed by Vanessa Rusczyk using KaleidoTile software.
Printed in the United States of America.
Printed in 2012.
HoM 10 nsf rHrs 8ooy
goo{
Hol,l/ lo nsa lqls
lauru q(z 5o1r,¡u8 droqlalus
Ma qara^a llrel lqa qasl rvzel 1o 1ae:u ruelqauruqJs rs ql so¡t,nr8 droqlaus' 1o¡s eup lols o¡ droqlaus'
Iu rt,a qalra^a }qe} +qa qasl *re{. 1o IaEJu ue+qaxrpuJs rs lo 1rl ¡o so¡r,a dr.oq¡arus +qe+ don pou,1
JeJ+,
IDLorl, qor\. lo po 4¡qau don prsco^ar soura}{rJuS ou don¡ o',¡u' fonlI mlpaJs+eup rl um)q qa$aJ +{ruu IJ
soulaoua {rep fnsl ¡o1p {,on'
rt ]osl oJ lqa sa)lIous oJ lqrs qooI qa8ru .u,rlq sa.rrare¡ droqlaurs' ¡qa solnFous lo lqasa d:oqlaurs tar¡
qa Jo^aJap ru lqa qnrt rt:f +o soI^a lqa dtoqlaus qalota raepruS ]qa sa;lrou 1¡ fon J¿u,l so1la soura
1ax¡,
oJ lqa droqlarus, lIret,s O)/ qaJensa lqaf mrg e¡ qa ¡p1{. so¡aap es lon raep +qa sa)qou gr,au r¡ don ceu
so1t-a e¡ o¡ ¡qa droqlaxrs, rt,s SE iudor,¡eu1 1o raep lqa saJlrou'qolq lo ruela snJa ¡qel donr" soln+rous
p¡a JoJra)+, eup pso qacensa (.on urul ¡rup 1qe1 ¡ra qoo{,s solnuous eJa srurdlar or aesraJ +o nupaJsleup
lqeu lonr. olau'
gxdlEuPlrou oJ lrous
¡q:on8qon¡ lqa qool' don rl.rll saa ^eJrons sqepap qoxas eup lrous'
¡ourad¡: lqrs rvlr qa e 8auar.el droqlaur-solaruS ¡arqurbna o:s}¡e}a3{. Jqasa era
lqa ,,1>als,, 1o qarounuS e qp1ar. dr.oqlaru so1r,at¡
¡rudor,¡uu1: IqIs MrII qa so¡ua+{n.¡.r8 rudorl¿ul lqel lon sqonlp laeru' Il u¡8q¡ qa e
A
JorrunlÉ'¿ solnqou laoqutbna'oJ p )pnqou'
MVUNIND¡¡ gaMJ€ a rJ fon saa lqrs qoxi lqrs MrII doru} on} e Jonrn1ou rursle{a oJ
t dItjuII'
t
r
HOW TO USE THIS BOOK
Sidenote: This box will contain material which, although interesting, is not part of
t. ,N the main material of the text. It's OK to skip over these báxes, buiif you
read them, you might learn something interestingl
Bogusf Sfoilution: Just like the impossible cube shown to the left, there's something
wrong with any "solution" that appears in this box.
Exercises, Review Problems, and Challenge Problems
Most sections end with several Exercises. These will test your understanding of the material that was
covered in the section. You should try to solve all of the exercises. Exercises marked with a * are more
difficult.
Most chapters conclude with several Review Problems. These are problems that test your basic
understanding of the material covered in the chapter. Your goal should be to solve most or all of the
Review Problems for every chapter-if you're unable to do this, it means that you haven't yet mastered
the material, and you should probably go back and read the chapter again.
All of the chapters (except for Chapter 1) end with Challenge Problems. These problems are
generally more difficult than the other problems in the book, and will really test your mastery of the
material. Some of them are very, very hard-the hardest ones are marked with a *. Don't expect to
be able to solve all of the Challenge Problems on your first try-these are difficult problems even for
experienced problem solvers. If you are able to solve a large number of Challenge Problems, then
congratulations, you are on your way to becoming an expert problem solver!
Hints
Many problems come with one or more hints. You can look up the hints in the Hints section in the
back of the book. The hints are numbered in random order, so that when you're looking up a hint to a
problem you don't accidentally glance at the hint to the next problem at the same time.
It is very important that you first try to solve each problem without resorting to the hints. Only after
vou'r'e seriously thought about a problem and are stuck should you seek a hint. Also, for problems
rrhich have multiple hints, use the hints one at a time; don't go to the second hint until you've thought
about the first one.
Solutions
I:,e solutions to all of the Exercises, Review Problems, and Challenge Problems are in the separate
:-'-:tions book. If you are using this textbook in a regular school class, then your teacher may decide
: --: t,-, make this solutionsbook available to you, and instead present the solutions him,/herse1f. However,
HOM IO NS:IHIS 8OO)
r¡ fon ur.a nsruS lqrs qoo¡ 1o 1ae¡.u ou (,,on¡. or¡u' lqau fon droqeqlf qet.a e )odd oJ lqa solnqous q,!rLI'
ru r,a{nlq )esa +qaJa e¡a so¡ua t.a¡d rudoueul ¡¡rq8s 1o 4aad ru rurup:
I. {¡le{a e sarrons B¡aurd} lo sol^a aeJ{r d;oqlaur qa¡o:a looquS pl lqa solnqou l6ou,l nsa lua
solnuous qoo{ Es E Jrn}lq lo E^orp ra¿lld lqrulDuS eqonl lqa d;oqlauL }rrs+' ^on sqonlp lqrq{ li¡r1l
eqonl e droqlaru qa¡or.a panpruS 1o looI e+ lqa soinqou' Ou lqa otqar qeup' eJ+ar fon,^a }{ron8ql
qer,p eqonl u droqlaur' pou,+ JaaI qep eqonl 1oo¡ru8 e1 ¡qa solnuou r¡ (,on,r.a rae¡Á. s1np>'
ygar don so1r,a e droq¡aru' r1,s nsnu¡(,. e 8oop rpae 1o JaEp +qa solnqou' ar,au r¡ fon lqrul> i,on
Z. ^
lD.rort, qo +o sol^a lqa dr,oqlaru Jqa solnqous qoo¡ rur3{rl sqom (-on e bnrr¡ar or. uloJa Jou)rsa
nef 1o so1.r.a 1qa droqlau' or rrt ur8ql ger,a u rou.rd1a1a1d pr¡¡araut soln+lou urartqop lquu (,onr.s'
'€ 1¡ fon qer,a ¡o ¡oo{ et lqa solnpou ru orpaJ lo solr,a e dr.oqlaru' urpla e uo+a oJ lqe¡ d:oqlaur' ¡oua
qpJ{ }o rl ru e maal oJ +{^o lo ruela snJa ¡r¿l fon pJa EqIa 1o soi^a r1 ou fonr ol',,u' l'vlqon1 raso:¡ru8
+o +qa
solnlrou.
uosonrJas
Hara era sorua o1qar. 8oop rason:ras ¡o: (,.on 1o ¡nrlqar. dnrsna (,onr s1np(,. oJ ülElqaurE+r)s:
¡4a yt4 ol ¿toq1aw go1aruS qoons' ql geupo:laqorz¡f eup ¿lrqprp ¿nsrzf¡' M{raraes ]qa qoo¡
lqe¡ lon,ra raepruS :r8ql uon,, r,r,r¡ Bo rulo 8:ae¡ pa1u¡ o¡ oua sdar¡rc snq[ap erae-ronu¡ru3 eup
droqeq111¡l-14a yy ol ¿toq¡aw go¡ma8 qoo¡s ro.r,ar e .u,rpa reu8a o¡ dr.oqlaru so1r,u8 lodrcs er:r.oss
rueuf pr¡¡ar.aul uraes oJ rue+qaueE)s'
lqa r¡r¡r¡ ' pJ}oJdroqlausoI^fuD ' ,fo¡u maqsr}a' ¡qa dnqqsqars o¡ lqs qoo{ EJa EIso +{ra ^.aqupslals
oJ lqa Vu oJ lcJoqlaut SoIr.ruS rv.aqsrla' .u.lnJ{r Joulerus rueuÁ. r.asonr.cas ¡or slnpau+s:
-
€ prS)nSSTOU JOJnü1
-
oulua pPssas
- lasonr)a {sls oJ qool$' Joulas}s' eup ol{raJ ^aqsrlas
- e ¿-¡f¡ 1n¡orrp
-
e urelq eup droqlaur so¡.r,ru8 4¡po
- eup ulnJlr uora¡
. on ,feu qoua fon:droqlaru so¡r,ru8 s1>r¡s )eup darqeds u,.nr d:rzas¡( qi, dp¡lr¡rdeqrrS ru \-pllons
ulE+lr Joulas+s' CoJ unppIá s]qooI slnpauls lu lqa nur+ap +qa rue[o:foulasls pra {IOEI IS'
S+elas?
I IyIH)O¡NJS' eup }qa V II g ¡or. ¡'g q¡3q srlrool s+npau}s' soula o, }qa qasl-{rlol\,u rou+asts
p¡a +{ra V{'rIJ/VII I,fVfS/\^IO sarras oJ rou¡asls )t.r.qrcq era nsap lo rqoosa tqá )'- S taEu.r Ior +qa
lu}aruelrouel I lE+{raruEtr)e1 6lfrudrup(' lira yruar.rruu ¿ar€ ous 1r,¡e1q lae8na )VIt\rl(' +qé weupaI-
q:o1 ¡ourdalrlrou' lqa Her^etp-rt IIJ l4le}qaule+r)s Jonrueulau}' eup pra ¡S¡1 {lpl{raruelrrel JEIaul
saerJq' i lora palurls eqonl lqasa Jou+asls eJa ou de8a r,rr' eup q.u{s to lqasa pup uleuf o+qaJ
Jou}asls eJa e^ErleqIa ou lqa vrl oJ ¿roqlaru so1^u3 {^aqsr+a'
HOW TO USE THIS BOOK
A Note to Teachers
We believe students leam best when they are challenged with hard problems that at first they may not
know how to do. This is the motivating philosophy behind this book.
Rather tha¡ fust introducing new material and then giving students exercises, we present problems
at the start of each section that students should try to solve before new material is presented. The goal
is to get students to discover the new material on their own. Often, complicated problems are broken
into smaller parts, so that students can discover new techniques one piece at a time. After the problems,
new material is formally presented in the text, and fuIl solutions to each problem are explained, along
with problem-solving strategies.
We hope that teachers will find that their stronger students will discover most of the material in this
book on their own by working through the problems. Other students may leam better from a more
traditional approach of first seeing the new material, then working the problems. Teachers have the
flexibility to use either approach when teaching ftom this book.
The book is linear in coverage. Generally, students and teachers should progress straight through
the book in order, without skipping chapters. Sections marked with a * contain supplementary material
that may be safely skipped. In general, chapters are not equal in length, so different chapters may take
different amounts of classroom time.
Links
The Art of Problem Solving website has a page containing links to websites with content relating to
material i¡r this book, as well as an errata list for the book. This page can be found at:
http artofproblemsolving com/Booklinks/Intermcounting/links php
: //rülüw. . .
v3)NoMtScc=¡i=\_s
VJ)uo^l1lapSaurauls
Joulasls
Ma monlp IIa +o }qeu{ lILa JollomruS roulas¡s ¡or. e¡1or,r,ru8 ns 1o nsa e saIa)qou oJ lqarJ dr.oq¡arus ru lqrs
qOOI:
JIra ¿¡urarrruu rt rtelqauEprs 3oruda¡¡¡¡ous, e sarras oJ rou+asts Jor lI S' urppla ¿up qr8q srqoo¡
slnpau+s. Jqa y¡ I) g, Vl rt,f I0, eup VI I) tZ routas+s ¿ra rupqdla-rqorra tasls +{ret er.a lE{au
qf oaar p6g,ggg slnpau¡s aL,ard daer. ¡od srorars ou ]qa ¿¡hl I0 eup vl\ll IZ er.a qr^llap lo
lela tqa vurarr)uu q^llEllouul rt lEllraulúllrs ÍxEurluEllou )vllha(' M{tlrq $ e ruora prJJr)nlt'
sllor.}-eusmar rou¡as1. ydd:oxute¡a¡{, 1g'ggg slnpauts ar,a¡{,.(,,ae¡ der+r)rdela ru +qa y¡hI:',lqau'
qesap ou +lra rasn1}s oJ +qa V{rtJ sup yIrt¡IA Joulasls, eqonl g00 slnpau+s ¿Ja lu^llap 1o der4:rdela
Iu lqa nSV rt lplqaruullrúI g¡lurd¡ep ¡¡svt{to( eup 1qa ¡gy {nu¡or, I IElqaurPtrrPI g¡furdrep
)nsvfl^rtO(, e ¿-pel, 6-qonr axeünueEou nr MILTJq apJq s+npaul unsl sqoM BII oJ qrs or ILaJ r\-orI'
¿asnlls üoru +ILa nsvl lO eup ¡SyINIO era nsap ]o u^rla s+npauls +o lqa IAIplq ol(,rudrep 5murua:
droSreru'eup +o rqoosa tqa lI S'+au€ r Jor tqa lulaluElloupI I^lEtqarue¡¡ae¡ g¡ÁudrEp )ll\lO(' \lora
ruJorur€qou pqonl l{ra yNI) Jou+as+s Jeu qa Jonup ou }{ra V{r) r^aqsr+a el qttd : /lpurt 'uee ' 016'
lqa l rteupalqr,o¡ 3oruda¡r¡rou, rl,.{.r,fq {^ps Jonupap nr 166g qf geupor. 1aqo:21>d r»rqerp ¡ns:zi-¡'
eup S¿ur ¡pupaJ^alpa. Jqa erru oJ +qa NleupalqJo+ lorudaqlrou rs 1o dror,rpa e :qe¡au8n.r8'
auBBBruS uelqaue+rJeI axda¡rau¡a lqel rs qolq routdalr¡rr,a eup apnJelrouel S+npauts rourdala
qopr es rupr^rpnels eup ru +aeurs' lqa rt JpupalqJol 3ourdaq4ou rs oJJarap el llra upqoupI la\.aI
Jor ruora upleuJap slnpauls uup e+ +ILa :a8roue¡ ¡arral ¡o:lass axdauaurap d:oqlau sol\.arsylora
ruJoJutpqou )eu qa Jonup el 1,^r,^4¡'ulEupaTqJol ' oJ6'
Iqa nSV l rtel{raruullreI le¡au+ sau¡rq )nSñ IIS('mq,rrlr mes Jonupap 4l 1OSO qi-drot.assor
Saor3a ga:zsauú. ¿qa nSVIA]JS rs E JJaa ur+€ qaueqJs rorudalr¡rou odau lo pI ¡ur]ap slPlas
rurppla pup qr3g srqoo¡ slnpau}s Vs oddosap }o uros} ruplqaul¿trrs roudaqqous' }qa nSY{l"JS
elloms slnpau+s e JnII uoulq +o morl onl lqarJ solngous' ¡er.a¡n¡{. n,:r¡au fnsqqreuous EJa
rabnrr.ap ¡or. aecq dr.oqlaut. 1,¡oJa lfrJoJul¿+rou if p^pileqla e+ u.uu1'nseru.ls ' oJ6'
IILa vularl¡Eu ¿aSrous ¡4¡e1q 1au8na )vuJ^rtI.('Mlnrq ^es Jomrpap ru I619 lya EurrneI
YUrt\1lI-
rourdalrlrou quu8s rto3a¡qar- uaeFl Z'000 oJ +qa uplrou,s Jruasl slnpauls 1qai, u.raal' )outdala
ACKNOWLEDGEMENTS
against, arrd socialize with one another, forming friendships and sharpening their mathematical
skitls. The contest is written for high school students, although some exceptional junior high
students attend each year. The competition consists of several events, which include a team
round, a power question (in which a team solves proof-oriented questions), an individual round,
and two relay rounds. More information ls available at ¡.nru. arml . com.
¡ The Harvard-MlT Mathematics Tournament, which is an annual math tournament for high
school students, held at MIT and at Harvard in alternating years. It is run exclusively by MIT and
Harvard students, most of whom themselves participated in math contests in high school. More
information is available at web .mit . edu/hmmt/.
Some other sources for problems in the book are listed below.
MATHCOUNTS MATHCOUNTS, the premier national math contest for U.S. middle school students.
Putnam The William Lowell Putnam Competition, an annual math competition for undergraduate
sLudents in North A meric¿.
PSS Problem-solaing Strategies by Arthur Engel. (See the references on page 366 for the complete
citation.)
ToT The International Mathematics Tournament of Towns, a mathematical problem solving competition
for students throughout the worid.
We have also included problems from various countries' national math Olympiad contests. These
problems are cited by country name in the text.
How We Wrote This Book
This book was written using the tr+TEX document processing system. We thank the authors of the various
trAT¡[ packages that we used while preparing this book, and also the authors of The Wl¿X Companion for
writing a reference book that is not only thorough but also very readable. The diagrams were prepared
using METFPO5T.
About Us
Thls book is a collaboraüve effort of the staff of the Art of Problem Solving. Richard Rusczyk wrote
the original lecture notes for the online course that was the inspiration for this book; he also read
several drafts of this book and made many, many helpful suggestions. Extensive proofreading of the
manuscript was done by Greg Brockman, Chris Chang, Larry Evans, Sean Markan, Jeff Nanney, Adrian
Sanborn, Nathan Savir, and Valentin Vornicu. Special thanks to Hyun Soo Kim for his assistance with
selecting many of the problems. Vanessa Rusczyk designed the cover.
CON.J3NIS
)oulauls
Ho^t1 lo nsa I{rIs goo{ III
ypuoar,¡apBaurau¡s AII
T
^
Ua^Ia oI 3onulru8 yr ¿r,oqeqr¡r¡f lfesI)s I
I'I Ju4opnrlrou 1
f'z gesrr 3onuquS ¡arqurbnas ' ' ' Z
It Í::,:'l::r?L?rr"r*;*".' ""' ' ' II
f't I8
I9 deSJEI,s {reu81a eup 1qa gruouneI lqaoJaur ¿0
I'9 snülürEllou No+epou ' 7c
I'L gnuluer(. ' Z9
Z
SaE
Pup loSrr
z'f lulJopnJlrou ' ZL
z'z sals ' Z8
Z'€ 6dar.eqous ou 5ps .2€
ZV ¡:nlq eup 1o3rr '
,2'g óneulrJrars ' ,a
Z'g gnruruerf ' '9
IX
