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Page 1: Math Problem Book I

M

ath

Problem

Book

I

compiledby

K

in

Y�Li

DepartmentofMathematics

HongKongUniversityofScienceandTechnology

Copyrightc �����HongKongMathematicalSocietyIMO�HK�Committ

PrintedinHongKong

Page 2: Math Problem Book I

Preface

Thereareover�ftycountriesintheworldnowadaysthatholdmath

ematicalolympiadsatthesecondaryschoollevelannually�InHungary

RussiaandRomaniamathematicalcompetitionshavealonghistorydat

ingbacktothelate�����sinHungary�scase�Manyprofessionalorama

teurmathematiciansdevelopedtheirinterestinmathbyworkingonthese

olympiadproblemsintheiryouthsandsomeintheiradulthoodsaswell�

Theproblemsinthisbookcamefrommanysources�Forthoseinvolved

ininternationalmathcompetitionstheynodoubtwillrecognizemanyof

theseproblems�Wetriedtoidentifythesourceswheneverpossiblebut

therearestillsomethatescapeusatthemoment�Hopefullyinfuture

editionsofthebookwecan�llinthesemissingsourceswiththehelpofthe

knowledgeablereaders�

Thisbookisforstudentswhohavecreativemindsandareinterestedin

mathematics�Throughproblemsolvingtheywilllearnagreatdealmore

thanschoolcurriculacano erandwillsharpentheiranalyticalskills�We

hopetheproblemscollectedinthisbookwillstimulatethemandseduce

themtodeeperunderstandingofwhatmathematicsisallabout�Wehope

theinternationalmathcommunitiessupportoure ortsforusingthesebril

liantproblemsandsolutionstoattractouryoungstudentstomathematics�

Mostoftheproblemshavebeenusedinpracticesessionsforstudents

participatedintheHongKongIMOtrainingprogram�Weareespecially

pleasedwiththee ortsofthesestudents�Infacttheoriginalmotivation

forwritingthebookwastorewardtheminsomewaysespeciallythosewho

workedsohardtobecomereserveorteammembers�Itisonly�ttingto

listtheirnamesalongwiththeirsolutions�Againthereareunsungheros

iii

whocontributedsolutionsbutwhosenameswecanonlyhopetoident i

infutureeditions�

Asthetitleofthebooksuggestthisisaproblembook�Soverylitt

introductionmaterialscanbefound�Wedopromisetowriteanotherbo

presentingthematerialscoveredintheHongKongIMOtrainingprogram

Thisforcertainwillinvolvethededicationofmorethanoneperson�Als

thisisthe�rstofaseriesofproblembookswehope�Fromtheresults

theHongKongIMOpreliminarycontestswecanseewavesofnewcreati

mindsappearinthetrainingprogramcontinuouslyandtheyareyoung

andyounger�Maybethenextproblembookintheserieswillbewritten

ourstudents�

FinallywewouldliketoexpressdeepgratitudetotheHongKo

QualityEducationFundwhichprovidedthesupportthatmadethisbo

possible�

KinY�

HongKo

April���

iv

Page 3: Math Problem Book I

Advicesto

the

Readers

Theonlywaytolearnmathematicsistodomathematics�Inthis

bookyouwill�ndmanymathproblemsrangingfromsimpletochallenging

problems�Youmaynotsucceedinsolvingalltheproblems�Veryfew

peoplecansolvethemall�Thepurposesofthebookaretoexposeyouto

manyinterestingandusefulmathematicalideastodevelopyourskillsin

analyzingproblemsandmostimportantofalltounleashyourpotential

ofcreativity�Whilethinkingabouttheproblemsyoumaydiscoverthings

youneverknowbeforeandputtinginyourideasyoucancreatesomething

youcanbeproudof�

Tostartthinkingaboutaproblemveryoftenitishelpfultolookat

theinitialcasessuchaswhenn���������Thesecasesaresimpleenough

toletyougetafeelingofthesituations�Sometimestheideasinthese

casesallowyoutoseeapatternwhichcansolvethewholeproblem�For

geometryproblemsalwaysdrawapictureasaccurateaspossible�rst�

Haveprotractorrulerandcompassreadytomeasureanglesandlengths�

Otherthingsyoucantryintacklingaproblemincludechangingthe

givenconditionsalittleorexperimentingwithsomespecialcases�rst�

Sometimesmaybeyoucanevenguesstheanswersfromsomecasesthen

youcanstudytheformoftheanswersandtracebackward�

Finallywhenyou�gureoutthesolutionsdon�tjuststopthere�You

shouldtrytogeneralizetheproblemseehowthegivenfactsarenecessary

forsolvingtheproblem�Thismayhelpyoutosolverelatedproblemslater

on�Alwaystrytowriteoutyoursolutioninaclearandconcisemanner�

Alongthewayyouwillpolishtheargumentandseethestepsoftheso

lutionsmoreclearly�Thishelpsyoutodevelopstrategiesfordealingwith

otherproblems�

v

Thesolutionspresentedinthebookarebynomeanstheonlywa

todotheproblems�Ifyouhaveaniceelegantsolutiontoaprobleman

wouldliketosharewithothers�infutureeditionsofthisbook�pleasesen

ittousbyemailatmakyli�ust�hk�Alsoifyouhavesomethingyoucann

understandpleasefeelfreetocontactusbyemail�Wehopethisbookw

increaseyourinterestinmath�

Finallywewillo eronelastadvice�Don�tstartwithproblem��Re

thestatementsoftheproblemsandstartwiththeonesthatinterestyout

most�Werecommendinspectingthelistofmiscellaneousproblems�rst�

Haveafuntime�

vi

Page 4: Math Problem Book I

Table

ofContents

Preface����������������������������������������������������������������iii

AdvicestotheReaders�������������������������������������������������v

Contributors����������������������������������������������������������ix

AlgebraProblems�������������������������������������������������������

GeometryProblems�����������������������������������������������������

NumberTheoryProblems�����������������������������������������������

CombinatoricsProblems�������������������������������������������������

MiscellaneousProblems�������������������������������������������������

SolutionstoAlgebraProblems�������������������������������������������

SolutionstoGeometryProblems�����������������������������������������

SolutionstoNumberTheoryProblems����������������������������������

SolutionstoCombinatoricsProblems������������������������������������

SolutionstoMiscellaneousProblems�������������������������������������

Page 5: Math Problem Book I

Contributors

ChanKinHang����������������HongKongteammember

ChanMingChiu����HongKongteamreservemember

ChaoKhekLun����HongKongteammember

ChengKeiTsi����HongKongteammember

CheungPokMan��������HongKongteammember

FanWaiTong����HongKongteammember

FungHoYin����HongKongteamreservemember

HoWingYip������������HongKongteammember

KeeWingTao����HongKongteamreservemember

LamPoLeung����HongKongteamreservemember

LamPeiFung����HongKongteammember

LauLapMing��������HongKongteammember

LawKaHo������������HongKongteammember

LawSiuLung����HongKongteammember

LeeTakWing����HongKongteamreservemember

LeungWaiYing����HongKongteammember

LeungWingChung��������HongKongteammember

MokTzeTao������������HongKongteammember

NgKaMan����HongKongteamreservemember

NgKaWing��������HongKongteammember

PoonWaiHoi������������HongKongteammember

PoonWingChi����HongKongteamreservemember

TamSiuLung����HongKongteamreservemember

ToKarKeung��������HongKongteammember

WongChunWai��������HongKongteammember

WongHimTing��������HongKongteammember

YuKaChun����HongKongteammember

YungFai����HongKongteammember

ix

Page 6: Math Problem Book I

Problem

s

Page 7: Math Problem Book I

AlgebraProblems

Polynomials

���CruxMathematicorumProblem

��Find�withoutcalculus�a�fth

degreepolynomialp�x�suchthatp�x���isdivisibleby�x����and

p�x���isdivisibleby�x�����

��ApolynomialP�x�ofthenthdegreesatis�esP�k��

�kfork�

����������n�FindthevalueofP�n����

�������PutnamExam�LetP�x�beapolynomialwithrealcoe�cients

suchthatP�x���foreveryrealx�Provethat

P�x��f ��x���f ��x�������f n�x��

forsomepolynomialsf ��x��f ��x������f n�x�withrealcoe�cients�

�������RussianMathOlympiad�Isitpossibleto�ndthreequadratic

polynomialsf�x��g�x��h�x�suchthattheequationf�g�h�x�����has

theeightroots����������������

�������PutnamExam�Determineallpolynomialswhosecoe�cientsare

all��thathaveonlyrealroots�

�������Putnam

Exam�Isthereanin�nitesequencea��a��a�����of

nonzerorealnumberssuchthatforn�

����������thepolynomial

Pn�x��a��a�x�a�x������anxnhasexactlyndistinctrealroots�

�������AustrianPolishMathCompetition�LetP�x�beapolynomial

withrealcoe�cientssuchthatP�x���for��x���Showthat

therearepolynomialsA�x��B�x��C�x�withrealcoe�cientssuchthat

�a�A�x����B�x����C�x���forallrealxand

�b�P�x��A�x��xB�x�����x�C�x�forallrealx�

�ForexampleifP�x��x���x��thenP�x����x���x������x�x���

�������IMO�Letf�x��xn��xn�����wheren��isaninteg e

Provethatf�x�cannotbeexpressedasaproductoftwopolynomia

eachhasintegercoe�cientsanddegreeatleast��

��Provethatiftheintegeraisnotdivisibleby�thenf�x��x��x�

cannotbefactoredastheproductoftwononconstantpolynomialswi

integercoe�cients�

��������SovietMathOlympiad�Given�ndistinctnumbersa��a������a

b ��b������bn�ann�ntableis�lledasfollows�intothecellinthei

rowandjthcolumniswrittenthenumberai�b j�Provethatift

productofeachcolumnisthesamethenalsotheproductofeachro

isthesame�

���Leta��a������anandb ��b������bnbetwodistinctcollectionsofnpo

itiveintegerswhereeachcollectionmaycontainrepetitions�Ifthetw

collectionsofintegersai�aj���i�j�n�andb i�b j���i�j�

arethesamethenshowthatnisapowerof��

RecurrenceRelations

���Thesequencexn

isde�nedby

x����

xn���

��xn

���xn

n�����������

Provethatxn��

� �

or�forallnandthetermsofthesequenceare

distinct�

��������NanchangCityMathCompetition�De�nea����a���an

an���

a� n����

an

forpositiveintegern�Provethat�anan����i s

perfectsquareforeverypositiveintegern�

����ProposedbyBulgariafor����IMO�De�nea����a���andan

�an���an��forn���Showthatforpositiveintegerk�anisdivisib

by�kifandonlyifnisdivisibleby�k�

Page 8: Math Problem Book I

����AmericanMathematicalMonthlyProblem

E�����Letxandybe

distinctcomplexnumberssuchthat

xn�yn

x�y

isanintegerforsome

fourconsecutivepositiveintegersn�Showthat

xn�yn

x�y

isaninteger

forallpositiveintegersn�

Inequalities

���Forrealnumbersa��a��a������ifan���an����an

forn���������

thenprovethat

An���An����An

forn���������

whereAn

istheaverageofa��a������an�

���Leta�b�c��andabc���Provethat

a c�

b a�

c b�a�b�c�

��������MoscowMathOlympiad�Usetheidentity����������n��

n��n����

toprovethatfordistinctpositiveintegersa��a������an�

�a� ��a

� ������a

� n���a

� ��a

� ������a

� n����a

� ��a

� ������a

� n���

Canequalityoccur�

��������IMOshortlistedproblem�Leta������an

�an����bea

sequenceofrealnumbers�Provethat

v u u tn X k��

ak�

n X k��

p k�pak�p ak����

��������ChineseTeamSelectionTest�For��a�b�c�d�ea n

a�b�c�d�e���showthat

ad�dc�cb�be�ea�

� ��

��������WuhuCityMathCompetition�Letx�y�zberealnumberssu

thatx�y�z���Showthat

��x

��y

��z

�����x

��y

��z

����

��������IMO�Letnbea�xedintegerwithn���

�a�DeterminetheleastconstantCsuchthattheinequality

X��i�j�n

xixj�x

� i�x

� j��C

� X ��i�n

xi�

holdsforallnonnegativerealnumbersx��x������xn�

�b�ForthisconstantC�determinewhenequalityholds�

��������BulgarianMathCompetition�Letn��and��xi��f

i���������n�Provethat

�x��x������xn���x�x��x�x������xn��xn�xnx���

h n �i

where�x�isthegreatestintegerlessthanorequaltox�

���Foreverytripletoffunctionsf�g�h������R�provethattherea

numbersx�y�zin�����suchthat

jf�x��g�y��h�z��xyzj�

� ��

����ProposedbyGreatBritainfor����IMO�Ifx�y�zarerealnumbe

suchthatx��y��z����thenshowthatx�y�z�xyz���

Page 9: Math Problem Book I

����ProposedbyUSAfor����IMO�Provethatforpositiverealnumbers

a�b�c�d�

a

b��c��d

b

c��d��a

c

d��a��b

d

a��b��c

�� ��

���Leta��a������an

andb ��b������bn

be�npositiverealnumberssuch

that

�a�a��a������an

and

�b�b �b ����bk�a�a����akforallk���k�n�

Showthatb ��b ������b n�a��a������an�

����ProposedbyGreecefor����IMO�Leta�b�c��andmbeapositive

integerprovethat

am

b�c

bmc�a

cma�b

�� �� a�b�c

� m���

���Leta��a������an

bedistinctpositiveintegersshowthat

a� ��

a� ������

ann�n

���

� �n�

��������WestGermanMathOlympiad�Ifa��a������an

��anda�

a��a������an�thenshowthat

n X i��

ai

�a�ai

n�n��

���Provethatifa�b�c���then

a�

b�c

b�c�a

c�a�b

�a��b��c�

���Leta�b�c�d��and �

��a�

���b

���c

���d���

Provethatabcd���

����DuetoPaulErd�os�Eachofthepositiveintegersa������anislessth

�����Theleastcommonmultipleofanytwooftheseisgreaterth

�����Showthat

� a�

�����

� an

���

n����

���Asequence�Pn�ofpolynomialsisde�nedrecursivelyasfollows�

P��x���

andforn���

Pn���x��Pn�x��

x�Pn�x��

Provethat

��

p x�Pn�x��

�n��

foreverynonnegativeintegernandallxin������

��������IMOshortlistedproblem�LetP�x�betherealpolynomialfun

tionP�x��ax��bx��cx�d�ProvethatifjP�x�j��forallxsu

thatjxj���then

jaj�jbj�jcj�jdj���

����AmericanMathematicalMonthlyProblem�����LetP�z��az�

bz��cz�d�wherea�b�c�darecomplexnumberswithjaj�jbj�jcj

jdj���ShowthatjP�z�j�

p �foratleastonecomplexnumber

satisfyingjzj���

��������HungarianIsraeliMathCompetition�Findallrealnumbers

withthefollowingproperty�foranypositiveintegern�thereexists

integermsuchthat

� � ���m n� � ��� �n�

��������BritishMathOlympiad�Ifnisapositiveintegerdenotebyp�

thenumberofwaysofexpressingnasthesumofoneormorepositi

integers�Thusp������asthereare�vedi erentwaysofexpressi

�intermsofpositiveintegers�namely

����������������������and��

Page 10: Math Problem Book I

Provethatp�n�����p�n��p�n�����foreachn���

FunctionalEquations

���Findallpolynomialsfsatisfyingf�x���f�x�f�x������

��������GreekMathOlympiad�Letf�����Rbeafunctionsuch

that

�a�fisstrictlyincreasing

�b�f�x���

� x

forallx��and

�c�f�x�f�f�x��� x���forallx���

Findf����

��������E�otv�osK�ursch�akMathCompetition�Thefunctionfisde�ned

forallrealnumbersandsatis�esf�x��xandf�x�y��f�x��f�y�

forallrealx�y�Provethatf�x��xforeveryrealnumberx�

����ProposedbyIrelandfor����IMO�Supposef�R

Rsatis�es

f������f�a�b��f�a��f�b�foralla�b�Randf�x�f�

� x���for

x����Showthatf�x��xforallx�

��������PolishMathOlympiad�LetQ�

bethepositiverationalnumbers�

Determineallfunctionsf�Q�

Q�

suchthatf�x����f�x���

andf�x���f�x��foreveryx�Q��

��������IMOshortlistedproblem�LetRdenotetherealnumbersand

f�R������satisfy

f� x

��� ��� �

f�x��f

� x�

� �� �

f� x

�� ��

foreveryx�R�Showthatfisaperiodicfunctioni�e�thereisa

nonzerorealnumberTsuchthatf�x�T��f�x�foreveryx�R�

���LetNdenotethepositiveintegers�Supposes�NNisanincreasing

functionsuchthats�s�n����nforalln�N�Findallpossiblevalues

ofs�������

���LetNbethepositiveintegers�Isthereafunctionf�NNsucht h

f���� �n���nforalln�N�wheref� �x��f�x�andfk�� �x�

f�fk �x���

����AmericanMathematicalMonthlyProblemE����LetRdenotet

realnumbers�Findallfunctionsf�RRsuchthatf�f�x���x��

orshownosuchfunctioncanexist�

���LetRbetherealnumbers�Findallfunctionsf�RRsuchthatf

allrealnumbersxandy�

f� xf�y��x

� �xy�f�x��

��������IMO�Determineallfunctionsf�RRsuchthat

f�x�f�y���f�f�y���xf�y��f�x���

forallx�yinR�

��������ByelorussianMathOlympiad�LetRbetherealnumbers�Fin

allfunctionsf�RRsuchthat

f�f�x�y���f�x�y��f�x�f�y��xy

forallx�y�R�

��������CzechoslovakMathOlympiad�LetZbetheintegers�Find

functionsf�ZZsuchthat

f�����f���andf�x��f�y��f�x��xy��f�y��xy�

forallintegersx�y�

��������SouthKoreanMathOlympiad�LetAbethesetofnonnegati

integers�Findallfunctionsf�AAsatisfyingthefollowingtw

conditions�

�a�Foranym�n�A��f�m��n����f�m�����f�n����

Page 11: Math Problem Book I

�b�Foranym�n�Awithm�n�f�m���f�n���

����AmericanMathematicalMonthlyProblemE�����LetQdenotethe

rationalnumbers�Findallfunctionsf�QQsuchthat

f�����

and

f� x�y

x�y

� �f�x��f�y�

f�x��f�y�

forx��y�

����MathematicsMagazineProblem�����Findallfunctionsf�RR

suchthat

f�x�yf�x���f�x��xf�y�

forallx�yinR�

Maximum�Minimum

��������AustrianMathOlympiad�Forpositiveintegersn�de�ne

f�n���n��n����n��������n���

���n���

��n�

Whatistheminimumoff�n����f�n��

��������PutnamExam�Giventhatfx��x������xng�f��������ng��nd

thelargestpossiblevalueofx�x��x�x������xn��xn�xnx�interms

ofn�withn����

GeometryProblems

��������BritishMathOlympiad�TriangleABChasarightangleat

TheinternalbisectorsofanglesBACandABCmeetBCandC

atPandQrespectively�ThepointsM

andN

arethefeetoft

perpendicularsfromPandQtoAB�FindangleMCN�

��������LeningradMathOlympiad�SquaresABDE

andBCFG

a

drawnoutsideoftriangleABC�ProvethattriangleABCisisosceles

DGisparalleltoAC�

���ABisachordofacirclewhichisnotadiameter�ChordsA�B�an

A�B�intersectatthemidpointPofAB�Letthetangentstothecirc

atA�andB�intersectatC��Similarlyletthetangentstothecirc

atA�andB�intersectatC��ProvethatC�C�isparalleltoAB�

��������HunanProvinceMathCompetition�TwocircleswithcentersO

andO�intersectatpointsAandB�AlinethroughAintersectst

circleswithcentersO�

andO�

atpointsY�Z�respectively�Lett

tangentsatYandZintersectatX

andlinesYO�andZO�interse

atP�Letthecircumcircleof�O�O�BhavecenteratOandinterse

lineXBatBandQ�ProvethatPQisadiameterofthecircumcirc

of�O�O�B�

��������BeijingCityMathCompetition�InadiskwithcenterO�the

arefourpointssuchthatthedistancebetweeneverypairofthem

greaterthantheradiusofthedisk�Provethatthereisapairofpe

pendiculardiameterssuchthatexactlyoneofthefourpointsliesinsi

eachofthefourquarterdisksformedbythediameters�

���Thelengthsofthesidesofaquadrilateralarepositiveintegers�T

lengthofeachsidedividesthesumoftheotherthreelengths�Pro

thattwoofthesideshavethesamelength�

��������SichuanProvinceMathCompetition�Supposethelengthsoft

threesidesof�ABCareintegersandtheinradiusofthetriangleis

Provethatthetriangleisarighttriangle�

��

Page 12: Math Problem Book I

GeometricEquations

��������IMO�AcirclehascenteronthesideABofthecyclicquadri

lateralABCD�Theotherthreesidesaretangenttothecircle�Prove

thatAD�BC�AB�

��������RussianMathOlympiad�CirclesS�andS�withcentersO��O�

respectivelyintersecteachotheratpointsAandB�RayO�Bintersects

S�atpointFandrayO�BintersectsS�atpointE�Thelineparallel

toEFandpassingthroughBintersectsS�andS�atpointsM

and

N�respectively�Provethat�Bistheincenterof�EAFand�MN�

AE�AF�

���PointCliesontheminorarcABofthecirclecenteredatO�Suppose

thetangentlineatCcutstheperpendicularstochordABthroughA

atEandthroughBatF�LetDbetheintersectionofchordABand

radiusOC�ProvethatCE�CF�AD�BDandCD��AE�BF�

���QuadrilateralsABCPandA� B� C� P�areinscribedintwoconcentric

circles�IftrianglesABCandA� B� C�areequilateralprovethat

P� A��P� B��P� C��PA���PB���PC���

���LettheinscribedcircleoftriangleABCtouchssideBCatDsideCA

atEandsideABatF�LetGbethefootofperpendicularfromDto

EF�Showthat

FG

EG

�BF

CE

��������IMOshortlistedproblem�LetABCDEFbeaconvexhexagon

suchthat

�B��D��F�����and

AB

BC

�CD

DE

�EF

FA

���

Provethat

BC

CA

�AE

EF

�FD

DB

���

��

SimilarTriangles

��������BritishMathOlympiad�P�Q�andRarearbitrarypointsont

sidesBC�CA�andABrespectivelyoftriangleABC�Provethatt

threecircumcentresoftrianglesAQR�BRP�andCPQformatriang

similartotriangleABC�

���HexagonABCDEFisinscribedinacirclesothatAB�CD�E

LetP�Q�RbethepointsofintersectionofACandBD�CEandD

EAandFB

respectively�ProvethattrianglesPQRandBDFa

similar�

��������IMOshortlistedproblem�LetABCDbeacyclicquadrilater

LetEandFbevariablepointsonthesidesABandCD�respective

suchthatAE�EB�CF�FD�LetPbethepointonthesegme

EFsuchthatPE�PF�AB�CD�Provethattheratiobetweent

areasoftrianglesAPDandBPCdoesnotdependonthechoiceof

andF�

TangentLines

���TwocirclesintersectatpointsAandB�Anarbitrarylinethrough

intersectsthe�rstcircleagainatCandthesecondcircleagainatD

Thetangentstothe�rstcircleatCandtothesecondcircleat

intersectatM�TheparalleltoCM

whichpassesthroughthepoi

ofintersectionofAM

andCDintersectsACatK�ProvethatBK

tangenttothesecondcircle�

��������IMO�Twocircles��and��arecontainedinsidethecircle

andaretangentto�atthedistinctpointsM

andN�respective

��passesthroughthecenterof���Thelinepassingthroughthetw

pointsofintersectionof��and��meets�atAandB�respective

ThelinesMAandMBmeets��atCandD�respectively�Proveth

CDistangentto���

����ProposedbyIndiafor����IMO�CirclesG�andG�toucheachoth

externallyatapointW

andareinscribedinacircleG�A�B�Ca

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Page 13: Math Problem Book I

pointsonGsuchthatA�G�andG�areonthesamesideofchordBC�

whichisalsotangenttoG�andG��SupposeAW

isalsotangentto

G�andG��ProvethatW

istheincenteroftriangleABC�

Locus

���PerpendicularsfromapointPonthecircumcircleof�ABCaredrawn

tolinesAB�BCwithfeetatD�E�respectively�Findthelocusofthe

circumcenterof�PDEasPmovesaroundthecircle�

���SupposeAisapointinsideagivencircleandisdi erentfrom

the

center�Considerallchords�excludingthediameter�passingthrough

A�Whatisthelocusoftheintersectionofthetangentlinesatthe

endpointsofthesechords�

���Given�ABC�LetlineEFbisects�BACandAE�AF�AB�AC�

FindthelocusoftheintersectionPoflinesBEandCF�

��������PutnamExam�LetC�andC�becircleswhosecentersare��

unitsapartandwhoseradiiare�and��Findthelocusofallpoints

M

forwhichthereexistspointsXonC�andYonC�suchthatM

is

themidpointofthelinesegmentXY�

CollinearorConcyclicPoints

��������IMO�DiagonalsACandCEoftheregularhexagonABCDEF

aredividedbytheinnerpointsM

andN�respectivelysothat

AM

AC

�CN

CE

�r�

DeterminerifB�M

andNarecollinear�

��������PutnamExam�IfA�B�C�Darefourdistinctpointssuchthat

everycirclethroughAandBintersectsorcoincideswitheverycircle

throughCandD�provethatthefourpointsareeithercollinearor

concyclic�

��

��������PutnamExam�Givenanin�nitenumberofpointsinaplan

provethatifallthedistancesbetweeneverypairareintegersthent

pointsarecollinear�

��������IMOshortlistedproblem�TheincircleoftriangleABCtouch

BC�CAandAB

atD�E

andF

respectively�X

isapointinsi

triangleABCsuchthattheincircleoftriangleXBCtouchesBC

DalsoandtouchesCXandXBatYandZrespectively�Proveth

EFZYisacyclicquadrilateral�

��������IMO�IntheconvexquadrilateralABCD�thediagonalsACan

BD

areperpendicularandtheoppositesidesAB

andDC

aren

parallel�SupposethepointP�wheretheperpendicularbisectors

AB

andDC

meetisinsideABCD�ProvethatABCD

isacyc

quadrilateralifandonlyifthetrianglesABPandCDPhaveequ

areas�

��������PutnamExam�Showthatifaconvexquadrilateralwithsid

lengthsa�b�c�dandarea

p abcdhasaninscribedcirclethenitis

cyclicquadrilateral�

ConcurrentLines

���In�ABC�supposeAB�AC�LetPandQbethefeetofthepe

pendicularsfromBandCtotheanglebisectorof�BAC�respective

LetDbeonlineBCsuchthatDA AP�ProvethatlinesBQ�P

andADareconcurrent�

��������ChineseNationalMathCompetition�DiagonalsACandB

ofacyclicquadrilateralABCDmeetsatP�Letthecircumcenters

ABCD�ABP�BCP�CDPandDAPbeO�O��O��O�andO�respe

tively�ProvethatOP�O�O��O�Oareconcurrent�

��������IMO�LetA�B�CandDbefourdistinctpointsonalineinth

order�ThecircleswithdiametersACandBDintersectatthepoin

XandY�ThelineXYmeetsBCatthepointZ�LetPbeapoint

thelineXYdi erentfromZ�ThelineCPintersectsthecirclewi

��

Page 14: Math Problem Book I

diameterACatthepointsCandM�andthelineBPintersectsthe

circlewithdiameterBDatthepointsBandN�Provethatthelines

AM�DNandXYareconcurrent�

���AD�BE�CFarethealtitudesof�ABC�IfP�Q�Rarethemidpoints

ofDE�EF�FD�respectivelythenshowthattheperpendicularfrom

P�Q�RtoAB�BC�CA�respectivelyareconcurrent�

��������ChineseMathOlympiadTrainingTest�ABCDEFisahexagon

inscribedinacircle�ShowthatthediagonalsAD�BE�CFareconcur

rentifandonlyifAB�CD�EF�BC�DE�FA�

���AcircleintersectsatriangleABCatsixpointsA��A��B��B��C��C��

wheretheorderofappearancealongthetriangleisA�C��C��B�A��A��

C�B��B��A�SupposeB�C��B�C�meetsatXC�A��C�A�meetsat

YandA�B��A�B�meetsatZ�ShowthatAX�BY�CZareconcurrent�

��������IMOshortlistedproblem�AcirclepassingthroughverticesB

andCoftriangleABCintersectssidesAB

andACatC�andB�

respectively�ProvethatBB� �CC�andHH�areconcurrentwhereH

andH� aretheorthocentersoftrianglesABCandAB� C� �respectively�

PerpendicularLines

��������APMO�LetABCbeatriangleandDthefootofthealtitude

fromA�LetEandFbeonalinepassingthroughDsuchthatAE

isperpendiculartoBE�AFisperpendiculartoCF�andEandFare

di erentfromD�LetM

andNbethemidpointsofthelinesegments

BCandEF�respectively�ProvethatANisperpendiculartoNM�

��������APMO�LetABCbeatriangle�LetM

andN

bethepoints

inwhichthemedianandtheanglebisectorrespectivelyatAmeet

thesideBC�LetQandPbethepointsinwhichtheperpendicularat

NtoNAmeetsMAandBA�respectivelyandOthepointinwhich

theperpendicularatPtoBAmeetsANproduced�ProvethatQOis

perpendiculartoBC�

��

���LetBB�andCC�bealtitudesoftriangleABC�AssumethatAB

AC�LetM

bethemidpointofBC�HtheorthocenterofABCand

theintersectionofB� C�andBC�ProvethatDH AM�

��������ChineseTeamSelectionTest�ThesemicirclewithsideBC

�ABCasdiameterintersectssidesAB�ACatpointsD�E�respe

tively�LetF�GbethefeetoftheperpendicularsfromD�Etosi

BCrespectively�LetM

betheintersectionofDGandEF�Proveth

AM

BC�

��������IMO�AcirclewithcenterOpassesthroughtheverticesAan

CoftriangleABCandintersectsthesegmentsABandACagain

distinctpointsK

andN�respectively�Thecircumcirclesoftriangl

ABCandKBN

intersectatexactlytwodistinctpointsB

andM

ProvethatOM

MB�

��������ChineseSenoirHighMathCompetition�Acirclewithcenter

isinternallytangenttotwocirclesinsideitatpointsSandT�Suppo

thetwocirclesinsideintersectatM

andNwithNclosertoST�Sho

thatOM

MNifandonlyifS�N�Tarecollinear�

���AD�BE�CFarethealtitudesof�ABC�LinesEF�FD�DEmeetlin

BC�CA�ABinpointsL�M�N�respectively�ShowthatL�M�N

a

collinearandthelinethroughthemisperpendiculartothelinejoini

theorthocenterHandcircumcenterOof�ABC�

GeometricInequalities�Maximum�Minimum

���������IMO�LetP��P������P�n��bedistinctpointsonsomehalf

theunitcirclecenteredattheoriginO�Showthat

j��OP����OP�����������

OP�n��j���

����Lettheanglebisectorsof�A��B��CoftriangleABCintersect

circumcircleatP�Q�R�respectively�Provethat

AP�BQ�CR�BC�CA�AB�

��

Page 15: Math Problem Book I

���������APMO�LetABCbeatriangleinscribedinacircleandletl a�

ma�Ma�lb�mb�Mb�lc�mc�Mc�wherema�mb�mcarethelengths

oftheanglebisectors�internaltothetriangle�andMa�Mb�Mcare

thelengthsoftheanglebisectorsextendeduntiltheymeetthecircle�

Provethat

l asin

�A

l b

sin

�B

l c

sin

�C

���

andthatequalityholdsi ABCisequilateral�

�����MathematicsMagazineProblem

�����LetIandO

betheincen

terandcircumcenterof�ABC�respectively�Assume�ABCisnot

equilateral�soI��O��Provethat

�AIO����ifandonlyif

�BC�AB�CA�

����SquaresABDEandACFGaredrawnoutside�ABC�LetP�Qbe

pointsonEGsuchthatBPandCQareperpendiculartoBC�Prove

thatBP�CQ�BC�EG�Whendoesequalityhold�

����PointPisinside�ABC�DeterminepointsDonsideABandEon

sideACsuchthatBD�CEandPD�PEisminimum�

SolidorSpaceGeometry

�����ProposedbyItalyfor����IMO�Whichregularpolygonscanbeob

tained�andhow�bycuttingacubewithaplane�

���������IsraeliMathOlympiad�Fourpointsaregiveninspaceingeneral

position�i�e�theyarenotcoplanarandanythreearenotcollinear��

Aplane�iscalledanequalizingplaneifallfourpointshavethesame

distancefrom��Findthenumberofequalizingplanes�

��

NumberTheoryProblems

Digits

���������PutnamExam�Provethateverypositiveintegerhasamultip

whosedecimalrepresentationinvolvesalltendigits�

����Doesthereexistapositiveintegerasuchthatthesumofthedig

�inbase���ofais����andthesumofthedigits�inbase���ofa�

������

�����ProposedbyUSSRfor����IMO�Letan

bethelastnonzerodig

inthedecimalrepresentationofthenumbern��Doesthesequen

a��a������an����becomeperiodicaftera�nitenumberofterms�

ModuloArithmetic

���������PutnamExam�Provethatthenumberofoddbinomialcoe�

cientsinanyrowofthePascaltriangleisapowerof��

����Leta��a��a������a��andb ��b��b������b��betwopermutationsoft

naturalnumbers�������������Showthatifeachofthenumbersa�b

a�b ��a�b ������a��b ��isdividedby��thenatleasttwoofthemw

havethesameremainder�

���������CzechSlovakMatch�Leta��a�����beasequencesatisfyinga�

��a���and

an������n

��an������n

��an

foralln���Dothereexistindicesp�qandrsuchthatapaq�ar�

PrimeFactorization

�����AmericanMathematicalMonthlyProblemE�����LetAn

bethes

ofpositiveintegerswhicharelessthannandarerelativelyprimeto

Forwhichn���dotheintegersinAnformanarithmeticprogressio

��

Page 16: Math Problem Book I

���������IMO�Provethatthesetofintegersoftheform

�k���k�

��������containsanin�nitesubsetinwhicheverytwomembersare

relativelyprime�

���������ChineseMathOlympiadTrainingTest�Determinethesmallest

valueofthenaturalnumbern��withthepropertythatwhenever

thesetSn

�f��������ngispartitionedintotheunionoftwosub

setsatleastoneofthesubsetscontainsthreenumbersa�bandc�not

necessarilydistinct�suchthatab�c�

BasenRepresentations

���������IMO�Canyouchoose����pairwisedistinctnonnegativeintegers

lessthan���suchthatnothreeareinarithmeticprogression�

�����AmericanMathematicalMonthlyProblem

�����Letpbeanodd

primenumberandrbeapositiveintegernotdivisiblebyp�Forany

positiveintegerkshowthatthereexistsapositiveintegermsuchthat

therightmostkdigitsofmr�whenexpressedinthebasep�areall��s�

�����ProposedbyRomaniafor����IMO�Showthatthesequencefang

de�nedbyan

��n

p ��forn�����������wherethebracketsdenote

thegreatestintegerfunction�containsanin�nitenumberofintegral

powersof��

Representations

����Findall�even�naturalnumbersnwhichcanbewrittenasasumof

twooddcompositenumbers�

����Findallpositiveintegerswhichcannotbewrittenasthesumoftwo

ormoreconsecutivepositiveintegers�

�����ProposedbyAustraliafor����IMO�Observethat������������

IsthereanintegerNwhichcanbewrittenasasumof����consecutive

positiveintegersandwhichcanbewrittenasasumof�morethanone�

consecutiveintegersinexactly����ways�

��

����Showthatifp��isprimethenpncannotbethesumoftwopositi

cubesforanyn���Whataboutp��or��

�����DuetoPaulErd�osandM�Sur�anyi�Provethateveryintegerkcan

representedinin�nitelymanywaysintheformk������������m

forsomepositiveintegermandsomechoiceofsigns�or��

���������IMOshortlistedproblem�A�nitesequenceofintegersa��a����

an

iscalledquadraticifforeachi�f��������ng�ja i�ai��j�i��

�a�Provethatforanytwointegersbandc�thereexistsanatur

numbernandaquadraticsequencewitha��bandan�c�

�b�Findtheleastnaturalnumbernforwhichthereexistsaquadrat

sequencewitha���andan������

����Provethateveryintegergreaterthan��canberepresentedasasum

threeintegers��whicharepairwiserelativelyprimeandshowth

��doesnothavethisproperty�

ChineseRemainderTheorem

���������ChineseTeam

SelectionTest�De�nexn

�xn����for

positiveintegersn�Provethatanintegervaluecanbechosenforx�

thatx���isdivisibleby�����

�����ProposedbyNorthKoreafor����IMO�Doesthereexistaset

withthefollowingproperties�

�a�ThesetM

consistsof����naturalnumbers�

�b�EveryelementinM

andthesumofanynumberofelementsin

havetheformmk�wherem�karepositiveintegersandk���

Divisibility

����Findallpositiveintegersa�bsuchthatb��and�a��isdivisible

�b���

��

Page 17: Math Problem Book I

����Showthattherearein�nitelymanycompositensuchthat�n����n��

isdivisiblebyn�

����Provethattherearein�nitelymanypositiveintegersnsuchthat�n��

isdivisiblebyn�Findallsuchn�sthatareprimenumbers�

���������RomanianMathOlympiad�Findallpositiveintegers�x�n�such

thatxn��n��isadivisorofxn����n�����

���������BulgarianMathCompetition�Findallpairsofpositiveintegers

�x�y�forwhich

x��y�

x�y

isanintegeranddivides�����

���������RussianMathOlympiad�Isthereasequenceofnaturalnumbers

inwhicheverynaturalnumberoccursjustonceandmoreoverforany

k����������thesumofthe�rstktermsisdivisiblebyk�

���������PutnamExam�LetA���andA����Forn���thenumber

An

isde�nedbyconcatenatingthedecimalexpansionsofAn��

and

An��fromlefttoright�ForexampleA��A�A�����A�A�A��

����A��AA��������andsoforth�DetermineallnsuchthatAn

isdivisibleby���

���������BulgarianMathCompetition�Ifk���showthatkdoesnot

divide�k�����Usethisto�ndallprimenumberspandqsuchthat

�p��qisdivisiblebypq�

����Showthatforanypositiveintegern�thereisanumberwhosedecimal

representationcontainsndigitseachofwhichis�or�andwhichis

divisibleby�n�

����Forapositiveintegern�letf�n�bethelargestintegerksuchthat�k

dividesnandg�n�bethesumofthedigitsinthebinaryrepresentation

ofn�Provethatforanypositiveintegern�

�a�f�n���n�g�n��

�b��divides

� �n n� ���n��

n�n�

ifandonlyifnisnotapowerof��

��

�����ProposedbyAustraliafor����IMO�Provethatforanypositivei

tegerm�thereexistanin�nitenumberofpairsofintegers�x�y�su

that

�a�xandyarerelativelyprime�

�b�ydividesx��m�

�c�xdividesy��m�

����Findallintegersn��suchthat�n��n������n���n

isdivisib

byn�

���������PutnamExam�Showthatifnisanintegergreaterthan�th

ndoesnotdivide�n���

�����ProposedbyRomaniafor����IMO�Fork���letn��n������nk

positiveintegerssuchthat

n�

� � ��n�

����n�

� � ��n�

��������nk

� � ��nk�

����n�

� � ��nk

���

Provethatn��n������nk���

���������APMO�Determinethelargestofallintegernwiththeproper

thatnisdivisiblebyallpositiveintegersthatarelessthan�p n�

���������UkrainianMathOlympiad�Findthesmallestintegernsuchth

amonganynintegers�withpossiblerepetitions�thereexist��intege

whosesumisdivisibleby���

PerfectSquares�PerfectCubes

����Leta�b�cbepositiveintegerssuchthat

� a�

� b�

� c�Ifthegreat e

commondivisorofa�b�cis�thenprovethata�bmustbeaperfe

square�

���������E�otv�osK�ursch�akMathCompetition�Letnbeapositiveintege

Showthatif���p��n���isanintegerthenitisasquare�

��

Page 18: Math Problem Book I

���������PutnamExam�Provethatforanyintegersa�b�c�thereexistsa

positiveintegernsuchthat

p n��an��bn�cisnotaninteger�

���������IMOshortlistedproblem�Letkbeapositiveinteger�Provethat

therearein�nitelymanyperfectsquaresoftheformn�k���wheren

isapositiveinteger�

����Leta�b�cbeintegerssuchthat

a b�

b c�

c a���Provethatabcisthe

cubeofaninteger�

DiophantineEquations

����Findallsetsofpositiveintegersx�yandzsuchthatx�y�zand

xy�yz�zx�

�����DuetoW�Sierpinskiin�����Findallpositiveintegralsolutionsof

�x��y��z�

�����DuetoEuleralso����MoscowMathOlympiad�Ifn���thenprove

that�ncanberepresentedintheform�n��x��y�withx�yodd

positiveintegers�

���������IMOshortlistedproblem�Findallpositiveintegersxandysuch

thatx�y��z��xyz�wherezisthegreatestcommondivisorofx

andy�

����Findallpositiveintegralsolutionstotheequationxy�yz�zx�

xyz���

����Showthatiftheequationx��y����

xyzhaspositiveintegral

solutionsx�y�z�thenz���

���������CzechSlovakMatch�Findallpairsofnonnegativeintegersxand

ywhichsolvetheequationpx�yp���wherepisagivenoddprime�

����Findallintegersolutionsofthesystemofequations

x�y�z��

and

x��y

��z

����

��

CombinatoricsProblems

CountingMethods

���������ItalianMathematicalOlympiad�Givenanalphabetwiththr

lettersa�b�c��ndthenumberofwordsofnletterswhichcontain

evennumberofa�s�

����FindthenumberofnwordsfromthealphabetA�f�����g�ifa

twoneighborscandi erbyatmost��

���������RomanianMathOlympiad�LetA��A������An

bepointson

circle�Findthenumberofpossiblecoloringsofthesepointswith

colorsp���suchthatanytwoneighboringpointshavedistinctcolo

PigeonholePrinciple

���������AustrianPolishMathCompetition�Doesthesetf�����������

containasubsetAconsistingof����numberssuchthatx�Aimpl

�x��A�

���������PolishMathOlympiad�Supposeatrianglecanbeplacedinsi

asquareofunitareainsuchawaythatthecenterofthesquareisn

insidethetriangle�Showthatonesideofthetrianglehaslengthle

than��

����Thecellsofa���squarearecoloredwithtwocolors�Proveth

thereexistatleast��rectangleswithverticesofthesamecoloran

withsidesparalleltothesidesofthesquare�

����Forn���let�nchesspiecesbeplacedatthecentersof�nsquares

ann�nchessboard�Showthattherearefourpiecesamongthemth

formedtheverticesofaparallelogram�If�nisreplacedby�n���

thestatementstilltrueingeneral�

����Thesetf����������gispartitionedintothreesubsets�Showthat

leastoneofthesubsetscontainsthreedi erentnumbersa�b�csu

thata�b�c�

��

Page 19: Math Problem Book I

InclusionExclusionPrinciple

����Letm�n���FindthenumberofsurjectivefunctionsfromBm

f��������mgtoBn�f��������ng�

����LetAbeasetwith�elements�Findthemaximalnumberof�element

subsetsofA�suchthattheintersectionofanytwoofthemisnota�

elementset�

�����a������HongKongChinaMathOlympiad�Studentshavetakena

testpaperineachofn�n���subjects�Itisknownthatforany

subjectexactlythreestudentsgetthebestscoreinthesubjectand

foranytwosubjectsexcatlyonestudentgetsthebestscoreinevery

oneofthesetwosubjects�Determinethesmallestnsothattheabove

conditionsimplythatexactlyonestudentgetsthebestscoreinevery

oneofthensubjects�

�b������AustrianPolishMathCompetition�Thereare����clubs�

Eachhas��members�Ifeverytwoclubshaveexactlyonecommon

memberthenprovethatall����clubshaveacommonmember�

CombinatorialDesigns

���������ByelorussianMathOlympiad�Inthebegining��beetlesare

placedatdi erentsquaresofa���squareboard�Ineachmoveevery

beetlecreepstoahorizontalorverticaladjacentsquare�Ifnobeetle

makeseithertwohorizontalmovesortwoverticalmovesinsuccession

showthataftersomemovestherewillbeatleasttwobeetlesinthe

samesquare�

���������GreekMathOlympiad�Linesl ��l������lk

areonaplanesuch

thatnotwoareparallelandnothreeareconcurrent�Showthatwe

canlabeltheCk �

intersectionpointsoftheselinesbythenumbers

��������k��sothatineachofthelinesl ��l������lk

thenumbers

��������k��appearexactlyonceifandonlyifkiseven�

���������TournamentsoftheTowns�Inalotterygameapersonmust

selectsixdistinctnumbersfrom������������toputonaticket�The

��

lotterycommiteewillthendrawsixdistinctnumbersrandomlyfro

�������������Anyticketwithnumbersnotcontaininganyoftheses

numbersisawinningticket�Showthatthereisaschemeofbuyi

�ticketsguaranteeingatleastawinningticketbut�ticketsisn

enoughtoguaranteeawinningticketingeneral�

���������ByelorussianMathOlympiad�Bydividingeachsideofanequ

lateraltriangleinto�equalpartsthetrianglecanbedividedinto

smallerequilateraltriangles�Abeetleisplacedoneachvertexofthe

trianglesatthesametime�Thenthebeetlesmovealongdi erentedg

withthesamespeed�Whentheygettoavertextheymustmake

���or����turn�Provethatatsomemomenttwobeetlesmustme

atsomevertex�Isthestatementtrueif�isreplacedby��

Covering�ConvexHull

���������AustralianMathOlympiad�Therearenpointsgivenonapla

suchthattheareaofthetriangleformedbyevery�ofthemisatmo

��Showthatthenpointslieonorinsidesometriangleofareaatmo

��

���������PutnamExam�Showthatanycontinuouscurveofunitleng

canbecoveredbyaclosedrectanglesofarea����

���������PutnamExam�LetFbea�nitecollectionofopendiscsint

planewhoseunioncoversasetE�Showthatthereisapairwisedisjoi

subcollectionD������Dn

inFsuchthattheunionof�D�������D

coversE�where�DisthediscwiththesamecenterasDbuthavi

threetimestheradius�

���������IMO�Determineallintegersn��forwhichthereexistnpoin

A��A������An

intheplaneandrealnumbersr ��r������rn

satisfyi

thefollowingtwoconditions�

�a�nothreeofthepointsA��A������An

lieonaline�

�b�foreachtriplei�j�k���i�j�k�n�thetriangleAiAjAkh

areaequaltor i�r j�r k�

��

Page 20: Math Problem Book I

���������IMO�Determineall�nitesetsSofatleastthreepointsinthe

planewhichsatisfythefollowingcondition�foranytwodistinctpoints

AandBinS�theperpendicularbisectorofthelinesegmentABisan

axisofsymmetryofS�

��

MiscellaneousProblems

���������RussianMathOlympiad�Therearenseatsatamerrygoaroun

Aboytakesnrides�Betweeneachridehemovesclockwiseacerta

number�lessthann�ofplacestoanewhorse�Eachtimehemoves

di erentnumberofplaces�Findallnforwhichtheboyendsupridi

eachhorse�

���������IsraeliMathOlympiad�Twoplayersplayagameonanin�ni

boardthatconsistsof���squares�PlayerIchoosesasquarean

marksitwithanO�ThenplayerIIchoosesanothersquareandmar

itwithX�Theyplayuntiloneoftheplayersmarksaroworacolum

of�consecutivesquaresandthisplayerwinsthegame�Ifnoplay

canachievethisthegameisatie�ShowthatplayerIIcanpreve

playerIfromwinning�

���������USAMO�Acalculatorisbrokensothattheonlykeysthatst

workarethesincostansin���cos���andtan��buttons�Thed

playinitiallyshows��Givenanypositiverationalnumberq�showth

pressingsome�nitesequenceofbuttonswillyieldq�Assumethatt

calculatordoesrealnumbercalculationswithin�niteprecision�A

functionsareintermsofradians�

���������E�otv�osK�ursch�akMathCompetition�Eachofthreeschools

attendedbyexactlynstudents�Eachstudenthasexactlyn��a

quaintancesintheothertwoschools�Provethatonecanpickthr

studentsonefromeachschoolwhoknowoneanother�Itisassum

thatacquaintanceismutual�

����Isthereawaytopack��������bricksintoa��������box�

����Isitpossibletowriteapositiveintegerintoeachsquareofthe�r

quadrantsuchthateachcolumnandeachrowcontainseverypositi

integerexactlyonce�

����Therearenidenticalcarsonacirculartrack�Amongallofthemth

havejustenoughgasforonecartocompletealap�Showthatthere

��

Page 21: Math Problem Book I

acarwhichcancompletealapbycollectinggasfromtheothercars

onitswayaroundthetrackintheclockwisedirection�

���������RussianMathOlympiad�Attheverticesofacubearewritten

eightpairwisedistinctnaturalnumbersandoneachofitsedgesis

writtenthegreatestcommondivisorofthenumbersattheendpoints

oftheedge�Canthesumofthenumberswrittenattheverticesbethe

sameasthesumofthenumberswrittenattheedges�

����Canthepositiveintegersbepartitionedintoin�nitelymanysubsets

suchthateachsubsetisobtainedfromanyothersubsetbyaddingthe

sameintegertoeachelementoftheothersubset�

���������RussianMathOlympiad�Isitpossibleto�llinthecellsofa

���tablewithpositiveintegersrangingfrom�to��insuchaway

thatthesumoftheelementsofevery���squareisthesame�

���������GermanMathematicalOlympiad�Showthatforeverypositive

integern���thereexistsapermutationp ��p������pn

of��������n

suchthatp k��dividesp ��p ������p kfork���������n���

����Eachlatticepointoftheplaneislabeledbyapositiveinteger�Each

ofthesenumbersisthearithmeticmeanofitsfourneighbors�above

belowleftright��Showthatallthenumbersareequal�

���������TournamentoftheTowns�Inapartynboysandngirlsare

paired�Itisobservedthatineachpairthedi erenceinheightisless

than��cm�Showthatthedi erenceinheightofthekthtallestboy

andthekthtallestgirlisalsolessthan��cmfork���������n�

���������LeningradMathOlympiad�Onemayperformthefollowingtwo

operationsonapositiveinteger�

�a�multiplyitbyanypositiveintegerand

�b�deletezerosinitsdecimalrepresentation�

ProvethatforeverypositiveintegerX�onecanperformasequenceof

theseoperationsthatwilltransformXtoaonedigitnumber�

��

���������IMOshortlistedproblem�Fourintegersaremarkedonacir c

Oneachstepwesimultaneouslyreplaceeachnumberbythedi eren

betweenthisnumberandnextnumberonthecircleinagivendirecti

�thatisthenumbersa�b�c�darereplacedbya�b�b�c�c�d�d�a

Isitpossibleafter����suchstepstohavenumbersa�b�c�dsuchth

thenumbersjbc�adj�jac�bdj�jab�cdjareprimes�

���������NanchangCityMathCompetition�Thereare����coinson

table�Someareplacedwiththeheadsidesupandsomethetailsid

up�Agroupof����personswillperform

thefollowingoperation

the�rstpersonisallowedturnoveranyonecointhesecondperson

allowedturnoveranytwocoins����thekthpersonisallowedtu

overanykcoins����the����thpersonisallowedtoturnovereve

coin�Provethat

���nomatterwhichsidesofthecoinsareupinitiallythe����perso

cancomeupwithaprocedureturningallcoinsthesamesidesu

attheendoftheoperations

���intheaboveprocedurewhethertheheadorthetailsidesturn

upattheendwilldependontheinitialplacementofthecoins�

�����ProposedbyIndiafor����IMO�Showthatthereexistsaconv

polygonof����sidessatisfyingthefollowingconditions�

�a�itssidesare��������������insomeorder�

�b�thepolygoniscircumscribableaboutacircle�

����Thereare��white��black��redchipsonatable�Inonestepy

maychoose�chipsofdi erentcolorsandreplaceeachonebyachip

thethirdcolor�Canallchipsbecomethesamecoloraftersomestep

����Thefollowingoperationsarepermittedwiththequadraticpolynom

ax��bx�c�

�a�switchaandc

�b�replacexbyx�t�wheretisarealnumber�

Byrepeatingtheseoperationscanyoutransformx��x��intox�

x���

��

Page 22: Math Problem Book I

����Fivenumbers���������arewrittenonablackboard�Astudentmay

eraseanytwoofthenumbersaandbontheboardandwritethe

numbersa�bandabreplacingthem�Ifthisoperationisperformedre

peatedlycanthenumbers����������������everappearontheboard�

����Nine���cellsofa�����squareareinfected�Inoneunittimethe

cellswithatleast�infectedneighbors�havingacommonside�become

infected�Cantheinfectionspreadtothewholesquare�Whatifnine

isreplacedbyten�

���������ColombianMathOlympiad�Weplaythefollowinggamewith

anequilateraltriangleofn�n�����dollarcoins�withncoinsoneach

side��Initiallyallofthecoinsareturnedheadsup�Oneachturnwe

mayturnoverthreecoinswhicharemutuallyadjacent�thegoalisto

makeallofthecoinsturnedtailsup�Forwhichvaluesofncanthisbe

done�

���������ChineseTeamSelectionTest�Everyintegeriscoloredwithone

of���colorsandall���colorsareused�Forintervals�a�b���c�d�having

integersendpointsandsamelengthsifa�chavethesamecolorand

b�dhavethesamecolorthentheintervalsarecoloredthesameway

whichmeansa�xandc�xhavethesamecolorforx���������b�a�

Provethat�����and����havedi erentcolors�

��

Page 23: Math Problem Book I

Solutions

Page 24: Math Problem Book I

SolutionstoAlgebraProblems

Polynomials

���CruxMathematicorumProblem

��Find�withoutcalculus�a�fth

degreepolynomialp�x�suchthatp�x���isdivisibleby�x����and

p�x���isdivisibleby�x�����

Solution��DuetoLawKaHoNgKaWingTamSiuLung�Note

�x����dividesp�x���andp��x����so�x����dividestheirsum

p�x��p��x��Also�x����dividesp�x���andp��x����so�x����

dividesp�x��p��x��Then�x�����x����dividesp�x��p��x��whichis

ofdegreeatmost��Sop�x��p��x���forallx�Thentheevendegree

termcoe�cientsofp�x�arezero�Nowp�x�����x�����Ax��Bx����

Comparingthedegree�and�coe�cientswegetB��A��and

���B�A���whichimpliesA�����andB������Thisyields

p�x����x�����x������x���

��ApolynomialP�x�ofthenthdegreesatis�esP�k��

�kfork�

����������n�FindthevalueofP�n����

Solution�For��r�n�thepolynomial� x r

� �x�x�������x�r���

r�

isofdegreer�Considerthedegreenpolynomial

Q�x��

� x �� �� x �� �

����

� x n� �

BythebinomialtheoremQ�k�������k��kfork�����������n�

SoP�x��Q�x�forallx�Then

P�n����Q�n����

� n��

� �� n��

� �����

� n��

n

� ��n

�����

�������PutnamExam�LetP�x�beapolynomialwithrealcoe�cients

suchthatP�x���foreveryrealx�Provethat

P�x��f ��x���f ��x�������f n�x��

��

forsomepolynomialsf ��x��f ��x������f n�x�withrealcoe�cients�

Solution��DuetoCheungPokMan�WriteP�x��aR�x�C�x��whe

aisthecoe�cientofthehighestdegreetermR�x�istheproductof

realrootfactors�x�r�repeatedaccordingtomultiplicitiesandC�x�

theproductofallconjugatepairsofnonrealrootfactors�x�zk��x�zk

Thena���SinceP�x���foreveryrealxandafactor�x�r��n

wouldchangesignneararealrootrofoddmultiplicityeachrealro

ofPmusthaveevenmultiplicity�SoR�x��f�x��forsomepolynom

f�x�withrealcoe�cients�

Nextpickonefactorfromeachconjugatepairofnonrealfacto

andlettheproductofthesefactors�x�z k�beequaltoU�x��iV�x

whereU�x��V�x�arepolynomialswithrealcoe�cients�Wehave

P�x��af�x���U�x��iV�x���U�x��iV�x��

��paf�x�U�x��

���paf�x�V�x��

��

�������RussianMathOlympiad�Isitpossibleto�ndthreequadrat

polynomialsf�x��g�x��h�x�suchthattheequationf�g�h�x�����h

theeightroots����������������

Solution�Supposetherearesuchf�g�h�Thenh����h��������h���w

betherootsofthe�thdegreepolynomialf�g�x���Sinceh�a�

h�b��a��bifandonlyifa�baresymmetricwithrespecttotheax

oftheparabolaitfollowsthath����

h����h����

h����h���

h����h����h���andtheparabolay�h�x�issymmetricwithr

specttox�����Alsowehaveeitherh����h����h����h���

h����h����h����h����

Nowg�h�����g�h�����g�h�����g�h����aretherootsofthequadra

polynomialf�x��sog�h�����g�h����andg�h�����g�h�����whi

impliesh����h����h����h����Forh�x��Ax��Bx�C�thiswou

forceA���acontradiction�

�������PutnamExam�Determineallpolynomialswhosecoe�cientsa

all��thathaveonlyrealroots� ��

Page 25: Math Problem Book I

Solution�Ifapolynomiala�xn�a�xn�������anissuchapolynomial

thensoisitsnegative�Hencewemayassumea����Letr ������rnbe

theroots�Thenr� ������r� n�a� ���a�andr� ����r� n�a� n�Iftheroots

areallrealthenbytheAMGMinequalityweget�a� ���a���n�a

��nn

Sincea��a�

���wemusthavea�

��andn���Bysimple

checkingwegetthelist

��x����

��x����

��x

��x����

��x

��x����

��x

��x

��x����

��x

��x

��x����

�������Putnam

Exam�Isthereanin�nitesequencea��a��a�����of

nonzerorealnumberssuchthatforn�

����������thepolynomial

Pn�x��a��a�x�a�x������anxnhasexactlyndistinctrealroots�

Solution�Yes�Takea�

��a�

��andproceedbyinduction�

Supposea������anhavebeenchosensothatPn�x�hasndistinctreal

rootsandPn�x�or�asxdependinguponwhethern

isevenorodd�SupposetherootsofPn�x�isintheinterval��T�T��

Letan�������n���MwhereM

ischosentobeverylargesothat

Tn���M

isverysmall�ThenPn���x��Pn�x����x�n���M

isvery

closetoPn�x�on��T�T�becausejPn���x��Pn�x�j�Tn���M

for

everyxon��T�T��SoPn���x�hasasignchangeveryclosetoevery

rootofPn�x�andhasthesamesignasPn�x�atT�SincePn�x�and

Pn���x�takeondi erentsignwhenxtheremustbeanother

signchangebeyondT�SoPn���x�musthaven��realroots�

�������AustrianPolishMathCompetition�LetP�x�beapolynomial

withrealcoe�cientssuchthatP�x���for��x���Showthat

therearepolynomialsA�x��B�x��C�x�withrealcoe�cientssuchthat

�a�A�x����B�x����C�x���forallrealxand

�b�P�x��A�x��xB�x�����x�C�x�forallrealx�

�ForexampleifP�x��x���x��thenP�x����x���x������x�x���

Solution��Belowallpolynomialshaverealcoe�cients��Weinduct

onthedegreeofP�x��IfP�x�isaconstantpolynomialc�thenc��

��

andwecantakeA�x��c�B�x��C�x����Nextsupposethedegr

ncaseistrue�ForthecaseP�x�isofdegreen���IfP�x���f

allrealx�thensimplyletA�x��P�x��B�x��C�x����Otherwis

P�x�hasarootx�in�����or������

Casex�in������ThenP�x���x�x��Q�x�andQ�x�isofdegree

withQ�x���forallxn������SoQ�x��A��x��xB��x�����x�C��x

whereA��x��B��x��C��x���forallxin������Usingx���x�

x���x������x�x��wehave

P�x���x�x���A��x��xB��x�����x�C��x��

���x�A��x��x

�B��x��

z

Ax

�x�A��x��x�B��x�����x��C��x

zBx

����x���x�C��x��x

�B��x��

z

Cx

wherethepolynomialsA�x��B�x��C�x���forallxin������

Casex�in������ConsiderQ�x��P���x��Thisreducestot

previouscase�WehaveQ�x��A��x��xB��x�����x�C��x��whe

thepolynomialsA��x��B��x��C��x���forallxin������Then

P�x��Q���x��A����x�

z�

Ax

�xC����x�

z�

Bx

����x�B����x�

z�

Cx

wherethepolynomialsA�x��B�x��C�x���forallxin������

�������IMO�Letf�x��xn��xn�����wheren��isanintege

Provethatf�x�cannotbeexpressedasaproductoftwopolynomia

eachhasintegercoe�cientsanddegreeatleast��

Solution�Supposef�x��b�x�c�x�fornonconstantpolynomialsb�

andc�x�withintegercoe�cients�Sincef������wemayassum

b����

��andb�x��

xr�������Sincef��������r�

��L

z ������zrbetherootsofb�x��Thenjz ����zrj�jb���j��and

jb����j�j����z ���������z r�j�

r Y i��

jzn��

i

�zi���j��r���

��

Page 26: Math Problem Book I

Howeverb����alsodividesf�������acontradiction�

��Provethatiftheintegeraisnotdivisibleby�thenf�x��x��x�a

cannotbefactoredastheproductoftwononconstantpolynomialswith

integercoe�cients�

Solution�Supposefcanbefactoredthenf�x��

�x�b�g�x�or

f�x���x��bx�c�g�x��Inthefomercaseb��b�a�f�b����Now

b��b�mod��byFermat�slittletheoremorsimplycheckingthecases

b�����������mod���Then�dividesb�b��a�acontradiction�In

thelattercasedivdingf�x��x��x�abyx��bx�c�wegetthe

remainder�b��b�c�c����x��b�c��bc��a��Sincex��bx�cis

afactoroff�x��bothcoe�cientsequal��Finally

��b�b��b

�c�c�������b�c��bc��a��b��b��bc���a

implies�a�b��b��bc�isdivisibleby��Thenawouldbedivisible

by�acontradiction�

��������SovietMathOlympiad�Given�ndistinctnumbersa��a������an�

b ��b������bn�ann�ntableis�lledasfollows�intothecellintheith

rowandjthcolumniswrittenthenumberai�b j�Provethatifthe

productofeachcolumnisthesamethenalsotheproductofeachrow

isthesame�

Solution�Let

P�x���x�a���x�a������x�an���x�b ���x�b ������x�b n��

thendegP�n�NowP�bj���bj�a���b j�a������bj�an��c�some

constantforj���������n�SoP�x��chasdistinctrootsb ��b������bn�

ThereforeP�x��cforallxandso

c�P��ai������n���ai�b ���ai�b ������ai�b n�

fori���������n�Thentheproductofeachrowis����n��c�

���Leta��a������anandb ��b������bnbetwodistinctcollectionsofnpos

itiveintegerswhereeachcollectionmaycontainrepetitions�Ifthetwo

��

collectionsofintegersai�aj���i�j�n�andb i�b j���i�j�

arethesamethenshowthatnisapowerof��

Solution��DuetoLawSiuLung�Considerthefunctionsf�x�

n X i��

xai

andg�x��

n X i��

xbi�Sincetheai�sandb i�saredistinctfand

aredistinctpolynomials�Now

f�x���

n X i��

x�ai

��

X��i�j�n

xai�aj

�f�x

����

X��i�j�n

xai�aj�

Sincetheai�aj�sandtheb i�b j�sarethesamesof�x���f�x��

g�x���g�x���Sincef����g����n�n���sof�x��g�x���x

��kQ�x�forsomek��andpolynomialQsuchthatQ�������Then

f�x��g�x��

f�x���g�x��

f�x��g�x�

��x����kQ�x��

�x���kQ�x�

��x���kQ�x��

Q�x�

Settingx���wehaven��k���

RecurrenceRelations

���Thesequencexn

isde�nedby

x����

xn���

��xn

���xn

n�����������

Provethatxn��

� �

or�forallnandthetermsofthesequenceare

distinct�

Solution��DuetoWongChunWai�Thetermsxn�sareclearlyration

byinduction�Writexn�pn�q n�wherepn�qnarerelativelyprimeint

gersandq n���Thenq ���andpn���q n�����q n�pn���qn��pn

Soq n��dividesq n��pn�whichimplieseveryq nisoddbyinductio

Henceeveryxn��

� ��

Nexttoshoweveryxn

����let��arctan��thenxn

�tann

byinduction�Supposexn

��andnistheleastsuchindex�Ifn

��

Page 27: Math Problem Book I

�miseventhen��x�m

�tan�m���xm����x� m�wouldimply

xm

���acontradictiontonbeingleast�Ifn��m��isoddthen

��x�m��

�tan����m������x�m������x�m�wouldimply

x�m

����Then����xm����x� m�wouldimplyxm

����

p ����is

irrationalacontradiction�Finallyifxm

�xn

forsomem�n�then

xm�n�tan�m��n����xm

�xn�����xmxn����acontradiction�

Thereforethetermsarenonzeroanddistinct�

��������NanchangCityMathCompetition�De�nea����a���and

an���

a� n����

an

forpositiveintegern�Provethat�anan����isa

perfectsquareforeverypositiveintegern�

Solution��DuetoChanKinHang��Sincean��dependsonan��and

an�itisplausiblethatthesequencesatis�esalinearrecurrencerelation

an���can���c�an�Ifthisissothenusingthe�rst�termswe�nd

c���c������De�neb ��a��b��a��bn����bn���b nforn���

Thenb �����a��Supposeak�b kfork�n���then

an���

b� n����

b n

���b n�b n������

b n

���b n���b n���b n��

��bn���b n�b n���

Soak�b kforallk�

Nextwritingoutthe�rstfewtermsof�anan����willsuggest

that�anan������an�an�����Thecasen��istrueas������

�������Supposethisistrueforn�k�Usingtherecurrencerelations

and� ��a� k������akak�����akak����a� k�wegetthecasen�k��

asfollow�

�ak��ak������ak����ak���ak���

���a

� k����ak�ak������

���a

� k����akak���a

� k��

���a

� k�����ak��ak�a

� k

by� �

���ak���ak����ak���ak�����

��

����ProposedbyBulgariafor����IMO�De�nea����a���andan

�an���an��forn���Showthatforpositiveintegerk�anisdivisib

by�kifandonlyifnisdivisibleby�k�

Solution�Bythebinomialtheoremif���

p ��n�An�Bn

p ��th

���

p ��n�An�Bn

p ��Multiplyingthese�equationswegetA� n

�B� n

����n�ThisimpliesAn

isalwaysodd�Usingcharacterist

equationmethodtosolvethegivenrecurrencerelationsonan�we�n

thatan

�Bn�Nowwriten��km�wheremisodd�Wehavek�

�i�e�nisodd�ifandonlyif�B� n

�A� n�����mod����i�e�Bn

odd��Nextsupposecasekistrue�Since���

p ���n��An�Bn

p ���

A�n�B�n

p ��soB�n

��AnBn�Thenitfollowscasekimpliesca

k���

����AmericanMathematicalMonthlyProblem

E�����Letxandy

distinctcomplexnumberssuchthat

xn�yn

x�y

isanintegerforsom

fourconsecutivepositiveintegersn�Showthat

xn�yn

x�y

isaninteg

forallpositiveintegersn�

Solution�Fornonnegativeintegern�lett n��xn�yn���x�y��

t ����t ���andwehavearecurrencerelation

t n���btn���ctn���

whereb���x�y��c�xy�

Supposet nisanintegerform�m���m���m���Sincecn��xy�n

t� n���t nt n��isanintegerforn�m�m���socisrational�Sin

cm��isintegercmustinfactbeaninteger�Next

b�

t mt m���t m��t m��

cm

Sobisrational�Fromtherecurrencerelationitfollowsbyinducti

thatt n�f n���b�forsomepolynomialfn��ofdegreen��withinteg

coe�cients�Notethecoe�cientofxn��inf n��is�i�e�f n��ismon

Sincebisarootoftheintegercoe�cientpolynomialfm�z��t m���

bmustbeaninteger�Sotherecurrencerelationimpliesallt n�sa

integers�

��

Page 28: Math Problem Book I

Inequalities

���Forrealnumbersa��a��a������ifan���an����an

forn���������

thenprovethat

An���An����An

forn���������

whereAn

istheaverageofa��a������an�

Solution�Expressinginak�therequiredinequalityisequivalentto

a������an���

n��n��

an�

n�n���

an�����

�Fromthecasesn�����weeasilyseethepattern��Wehave

a������an���

n��n��

an�

n�n���

an��

n X k��

k�k���

�ak����ak�ak������

���Leta�b�c��andabc���Provethat

a c�

b a�

c b�a�b�c�

Solution��DuetoLeungWaiYing�Sinceabc���weget���bc��a�

���ac��band���ab��c�BytheAMGMinequality

�a c�

c b�

a c�

a c�

c b��

�r a� bc��a�

Similarly�b�a�a�c��band�c�b�b�a��c�Addingtheseand

dividingby�wegetthedesiredinequality�

��

Alternativelyletx�

�p ab�c��y�

�p ca�b�andz�

�p bc�a

Wehavea�x�y�b�z�x�c�y�zandxyz�

�p abc���Usingth

andtherearrangementinequalityweget

a c�

b a�

c b�

x�

yz

�z�

xy

�y�

zx

�xyz

� x� yz

�z�

xy

�y�

zx

� �x

��y

��z

�x�y�y

�z�z

�x�a�b�c�

��������MoscowMathOlympiad�Usetheidentity����������n�

n��n����

toprovethatfordistinctpositiveintegersa��a������an

�a� ��a

� ������a

� n���a

� ��a

� ������a

� n����a

� ��a

� ������a

� n��

Canequalityoccur�

Solution�Forn���a� ��a� ����a� ����a� ��a�������andsoca

n��istrue�Supposethecasen�kistrue�Forthecasen�k�

withoutlossofgeneralitywemayassumea��a������ak���No

��a

� ������a

� k�������a

� ������a

� k��

��a

� k����a

� k���a

� ������a

� k�

��a

� k����a

� k����

����������ak�����

��

��a

� k����a

� k��

�ak������a� k

�a

� k���a

� k���

So�a� ������a� k�����a� ������a� k������a� ������a� k����follow

Equalityoccursifandonlyifa��a������anare��������n�

��������IMOshortlistedproblem�Leta������an

�an����be

sequenceofrealnumbers�Provethat

v u u tn X k��

ak�

n X k��

p k�pak�p ak����

��

Page 29: Math Problem Book I

Solution��DuetoLeeTakWing�Letxk�p ak�p ak���Thenak�

�xk�xk�������xn���So

n X k��

ak�

n X k��

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Page 36: Math Problem Book I

inthisnewequationweget�equations�Addingtheseandcancelling

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�f�n��Soifn��eq�wheree�qarenonnegativeintegersandqodd

thenf�n���ef�q��Tode�nesuchafunctionweneedtode�neitat

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andlettinga�f�y����weget

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Page 37: Math Problem Book I

Puttingy�aandlettingb�f����weget

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f� f�y���� �y�f����

Foreveryrealnumbera�lety�a�f����thenf

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easilycheckedtobesolutions�

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getf��c��f�c��c���Soc����Forx�f�y��A�f�x��c���

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allfunctionsf�RRsuchthat

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Solution��DuetoYungFai�Clearlyfromtheequationf�x�isn

constant�Puttingy���wegetf�f�x������f����f�x��Replaci

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Page 38: Math Problem Book I

xbyx��in� �wegetf���f�x��f�x���f�����x���Eliminating

f�x���inthelasttwoequationsweget

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wegetf����

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xy�Thenf�x�f����

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allx�R�Sof�����f���f��������resultinginacontradiction�

Thereforef�����f���f�������andf�x�isadegree�polynomial�

Finallysubstitutingf�x��ax�bintotheoriginalequationwe

�nda��andb���i�e�f�x��xforallx�R�

��������CzechoslovakMathOlympiad�LetZbetheintegers�Findall

functionsf�ZZsuchthat

f�����f���andf�x��f�y��f�x��xy��f�y��xy�

forallintegersx�y�

Solution�Wehave� �f����f�n��f����n��f��n�andf�n��

f�����f��n��f�����n��Sincef�����f����thisgivesf����n��

f�����n�foreveryintegern�Sof�k�hasthesamevalueforevery

oddk�Thenequation� �impliesf�n��f��n�foreveryintegern�So

weneedto�ndf�n�fornonnegativeintegersnonly�

Ifweletx����k����y�n�thenxandx��xyareodd�The

functionalequationgivesf�n��f�y��f�y��xy��f�n��k�����

Ifweletx�n�y����k����thensimilarlywegetf�n��f�x��

f�x��xy��f�n���k�����f�n��k�����Sof�n��f�nm�forevery

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Forapositiveintegern�wecanfactorn��em�wheree�m

are

nonnegativeintegersandmodd�Thenf�n��f��e��Soanysuchfunc

tionfisdterminedbythevaluesf����f����f����f����f����f��������

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above�Finallywechecksuchfunctionssatisfytheequations�Clearly

f�����f����Ifxory���thenthefunctionalequationisclearly

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f�x��f�y��f��e��f��d��f�x����y���f�y����x���

��

��������SouthKoreanMathOlympiad�LetAbethesetofnonnegati

integers�Findallfunctionsf�AAsatisfyingthefollowingtw

conditions�

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f�m���f�n�����f�m���f�n������Sof�m��f�n��Thismea

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rationalnumbers�Findallfunctionsf�QQsuchthat

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and

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Solution�Wewillshowf�x��xistheonlysolutionbyaseriesof

observations�

���Settingy���wegetf�����f�x��f������f�x��f�����which

yields�f������f�x��f������f������Nowfisnotconstant

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Addingthesewegetf�a��f�b���f�a

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� �f�yf�x���xf�y��Settingy���wegetf�f�x��x�Thenfis

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Incasen�

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permutationsis��n���n����n������becauseoftheequalityca

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Page 41: Math Problem Book I

SolutionstoGeometryProblems

��������BritishMathOlympiad�TriangleABChasarightangleatC�

TheinternalbisectorsofanglesBACandABC

meetBCandCA

atPandQ

respectively�ThepointsM

andN

arethefeetofthe

perpendicularsfromPandQtoAB�FindangleMCN�

Solution��DuetoPoonWaiHoi�Usingprotractortheangleshould

be��� �ToprovethisobservethatsincePisonthebisectorof�BAC�

wehavePC�PM�LetLbethefootoftheperpendicularfromC

toAB�ThenPM

kCL�So�PCM

��PMC��MCL�Similarly

�QCN��NCL�So�MCN�� ��PCQ���� �

��������LeningradMathOlympiad�SquaresABDE

andBCFG

are

drawnoutsideoftriangleABC�ProvethattriangleABCisisoscelesif

DGisparalleltoAC�

Solution��DuetoNgKaManNgKaWingYungFai�FromBdraw

aperpendiculartoAC�andhencealsoperpendiculartoDG��Letit

intersectACatXandDGatY�Since�ABX���� ��DBY��BDY

andAB�BD�therighttrianglesABXandBDYarecongruentand

AX�BY�SimilarlytherighttrianglesCBXandBGYarecongruent

andBY�CX�SoAX�CX�whichimpliesAB�CB�

���ABisachordofacirclewhichisnotadiameter�ChordsA�B�and

A�B�intersectatthemidpointPofAB�Letthetangentstothecircle

atA�andB�intersectatC��Similarlyletthetangentstothecircle

atA�andB�intersectatC��ProvethatC�C�isparalleltoAB�

Solution��DuetoPoonWaiHoi�LetOC�

intersectsA�B�

atM�

OC�intersectsA�B�atN�andOC�intersectABatK�SinceOC�is

aperpendicularbisectorofA�B��soOM

A�B��SimilarlyON

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areconcyclic�So�ONM

��OPM�Since

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therighttrianglesOA�C�

andOB�C��wegetOM

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OB� ��ON�OC��Bytheconverseoftheintersectingchordtheorem

��

wegetM�N�C��C�areconcyclic�So�OC�C���ONM

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ThenC�C�kKP�thatisC�C�isparalleltoAB�

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andO�intersectatpointsAandB�AlinethroughAintersectst

circleswithcentersO�

andO�

atpointsY�Z�respectively�Lett

tangentsatYandZintersectatX

andlinesYO�andZO�interse

atP�Letthecircumcircleof�O�O�BhavecenteratOandinterse

lineXBatBandQ�ProvethatPQisadiameterofthecircumcirc

of�O�O�B�

Solution��FirstweneedtoshowP�O��O��Bareconcyclic�Thenw

willshow�����QBP��XBP�Since�XYP��PZX

areboth��

itsu�cestoshowX�Y�B�P�Zareconcyclic��ConnectO�AandO�

In�YPZ�

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SoY�Z�P�Bareconcyclic�Since�XYP��XZP���� �sothepoin

Y�X�Z�P�Bareconcyclic�Then�QBP��XBP�������XZP

��� �ThereforePQisadiameterofthecircumcircleof�O�O�B�

��������BeijingCityMathCompetition�InadiskwithcenterO�the

arefourpointssuchthatthedistancebetweeneverypairofthem

greaterthantheradiusofthedisk�Provethatthereisapairofpe

pendiculardiameterssuchthatexactlyoneofthefourpointsliesinsi

eachofthefourquarterdisksformedbythediameters�

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Page 42: Math Problem Book I

Solution��DuetoLeeTakWing�Bythedistanceconditiononthe

fourpointsnoneofthemequalsOandnopairofthemareonthesame

radius�LetusnamethepointsA�B�C�Dintheorderarotatingradius

encounteredthem�SinceAB�OA�OB�so�AOB��OBA��BAO�

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�����������������SoA�B�Ceachisonadi erentquarterdisk�

Similarly�����EOD��EOA��AOD������ThereforeDwill

lieontheremainingquarterdisk�

���Thelengthsofthesidesofaquadrilateralarepositiveintegers�The

lengthofeachsidedividesthesumoftheotherthreelengths�Prove

thattwoofthesideshavethesamelength�

Solution��DuetoChaoKhekLunandLeungWaiYing�Supposethe

sidesarea�b�c�dwitha�b�c�d�Sinced�a�b�c��dand

ddividesa�b�c�wehavea�b�c��d�Noweachofa�b�cdvides

a�b�c�d�

�d�Letx�

�d�a�y�

�d�bandz�

�d�c�Then

a�b�c�dimpliesx�y�z���Soz���y���x���Then

�d�a�b�c�

�d ��

�d ��

�d ���d�

acontradiction�Thereforetwoofthesidesareequal�

��������SichuanProvinceMathCompetition�Supposethelengthsofthe

threesidesof�ABCareintegersandtheinradiusofthetriangleis��

Provethatthetriangleisarighttriangle�

Solution��DuetoChanKinHang�Leta�BC�b�CA�c�ABbe

thesidelengthsrbetheinradiusands��a�b�c����Sincethearea

��

ofthetriangleisrs�weget

p s�s�a��s�b��s�c����s�s�The

�s�a��s�b��s�c��s��s�a���s�b���s�c��

Now��a�b�c���s���s��a���s��b���s��c���b�c�a��c

a�b��a�b�c��In�mod��eachofb�c�a�c�a�b�a�b�ca

thesame�Soeithertheyarealloddoralleven�Sincetheirproduct

eventheyarealleven�Thena�b�cisevenandsisaninteger�

Thepositiveintegersx�s�a�y�s�b�z�s�csatisfyxyz

x�y�z�Supposex�y�z�Thenyz��forotherwisexyz��x

x�y�z�Thisimpliesx���y���z���s���a���b���c�

Thereforethetriangleisarighttriangle�

GeometricEquations

��������IMO�AcirclehascenteronthesideABofthecyclicquad

lateralABCD�Theotherthreesidesaretangenttothecircle�Pro

thatAD�BC�AB�

Solution�LetM

beonABsuchthatMB�BC�Then

�CMB�

������ABC

��CDA

��CDO�

ThisimpliesC�D�M�Oareconcyclic�Then

�AMD��OCD�

�DCB

�������DAM

��AMD��ADM

So�AMD��ADM�ThereforeAM

�ADandAB�AM

�MB

AD�BC�

��������RussianMathOlympiad�CirclesS�andS�withcentersO��O

respectivelyintersecteachotheratpointsAandB�RayO�Bintersec

S�atpointFandrayO�BintersectsS�atpointE�Thelineparal

toEFandpassingthroughBintersectsS�andS�atpointsM

an

N�respectively�Provethat�Bistheincenterof�EAFand�MN

AE�AF�

��

Page 43: Math Problem Book I

Solution�Since

�EAB�

� ��EO�B������O�BE������FBO���BAF�

ABbisects�EAFand�O�BE������EAB�����

� ��EAF�Now

�EBA��FBA��EBA������ ��O�BA��������O�BE������

� ��EAF�Then�EBF������EAF�whichimpliesBistheincenter

of�EAF�becausetheincenteristheuniquepointPonthebisectorof

�EAFsuchthat�EPF�����

� ��EAF��Then�AEB��BEF�

�EBM

sinceEFkMN�SoEBAM

isanisoscelestrapezoid�Hence

EA�MB�SimilarlyFA�NB�ThereforeMN

�MB�NB

AE�AF�

���PointCliesontheminorarcABofthecirclecenteredatO�Suppose

thetangentlineatCcutstheperpendicularstochordABthroughA

atEandthroughBatF�LetDbetheintersectionofchordABand

radiusOC�ProvethatCE�CF�AD�BDandCD��AE�BF�

Solution��DuetoWongChunWai�Notethat�EAD��ECD��FCD�

�FBDarerightangles�SoA�D�C�EareconcyclicandB�D�C�Fare

concyclic�Then�ADE��ACE��ABC��DFC�saythemeasure

oftheseanglesis��Also�BDF��BCF��BAC��DEC�say

themeasureoftheseangleis�Then

CE�CF��DEcos��DFcos����DEcos���DFcos��AD�BD�

CD

���DEsin��DFsin����DEsin���DFsin��AE�BF�

���QuadrilateralsABCPandA� B� C� P�areinscribedintwoconcentric

circles�IftrianglesABCandA� B� C�areequilateralprovethat

P� A��P� B��P� C��PA���PB���PC���

Solution�LetObethecenterofbothcirclesandEbethemidpoint

ofA� B� �From�PA� B� withmedianPE�bycosinelawwegetPA���

PB�����PE��EB����From�PC� EwithcevianPO�noteC� O�

��

�OE�bycosinelawagainwegetPC����PE����PO���OE

Puttingthesetogetherweget

PA���PB���PC�����EB���OE

����PO

���OE

��B� O���PO

��C� O�

���PO

��P� O���

SimilarlyP� A��P� B��PC�����PO��P� O���

Alternativelytheproblemcanbesolvedusingcomplexnumbe

WithoutlossofgeneralityletthecenterbeattheoriginA�be

rei��r�cos �isin �andPbeatRei��Let��e��i���Wehave

PA���PB���PC��

�jRei��rei�j��jRei��rei��j��jRei��rei��

�j�

��R

���Re

� Rrei

��� ������

��� ��r

��R

���r

��

SimilarlyP� A��P� B��P� C���R���r��

���LettheinscribedcircleoftriangleABCtouchssideBCatDsideC

atEandsideABatF�LetGbethefootofperpendicularfromD

EF�Showthat

FG

EG

�BF

CE

Solution��DuetoWongChunWai�LetIbetheincenterof�AB

Then�BDI�

����

�EGD�Also�DEG

� ��DIF

�DI

So�BDI��EGDaresimilar�ThenBD�ID

�DG�EG�Likewis

�CDI��FGDaresimilarandCD�ID�DG�FG�Therefore

FG

EG

�DG�EG

DG�FG

�BD�ID

CD�ID

�BD

CD

�BF

CE

��������IMOshortlistedproblem�LetABCDEFbeaconvexhexag

suchthat

�B��D��F�����and

AB

BC

�CD

DE

�EF

FA

���

��

Page 44: Math Problem Book I

Provethat

BC

CA

�AE

EF

�FD

DB

���

Solution�LetPbesuchthat�FEA��DEPand�EFA��EDP�

whereP

isontheoppositesideoflinesDE

andCD

asA�Then

�FEA��DEParesimilar�So

FA

EF

�DP

PE

and

���

EF

ED

�EA

EP

Since�B��D��F������weget�ABC��PDC�Also

AB

BC

�DE�FA

CD�EF

�DP

CD

Then�ABC��PDC

aresimilar�Consequentlyweget�BCA

�DCPand� �CB�CD�CA�CP�Since�FED��AEP�by� �

�FED��AEParesimilar�Alsosince�BCD

��ACP�by� �

�BCD��ACParesimilar�SoAE�EF�PA�FD

andBC�CA�

DB�PA�Multiplyingtheseandmovingallfactorstotheleftsidewe

getthedesiredequation�

Usingcomplexnumberswecangetanalgebraicsolution�Let

a�b�c�d�e�fdenotethecomplexnumberscorrespondingtoA�B�C�D�

E�F�respectively��Theoriginmaybetakenanywhereontheplane��

SinceABCDEFisconvex�B��Dand�Faretheargumentsofthe

complexnumbers�a�b���c�b���c�d���e�d�and�e�f���a�f��

respectively�Thenthecondition�B��D��F�����impliesthat

theproductofthesethreecomplexnumbersisapositiverealnumber�

ItisequaltotheproductoftheirabsolutevaluesAB�BC�CD�DE

andEF�FA�Since�AB�BC��CD�DE��EF�FA����wehave

a�b

c�b

�c�d

e�d

�e�f

a�f

���

So

���a�b��c�d��e�f���c�b��e�d��a�f�

��b�c��a�e��f�d���a�c��f�e��b�d��

��

Then

BC

CA

�AE

EF

�FD

DB

�� � � �b�c

a�c

�a�e

f�e

�f�d

b�d

� � � ����

SimilarTriangles

��������BritishMathOlympiad�P�Q�andRarearbitrarypointsont

sidesBC�CA�andABrespectivelyoftriangleABC�Provethatt

threecircumcentresoftrianglesAQR�BRP�andCPQformatriang

similartotriangleABC�

Solution�LetthecircumcentersoftrianglesAQR�BRPandCPQ

A� �B�andC� �respectively�Agooddrawingsuggeststhecirclespa

throughacommonpoint�Toprovethisletcircumcirclesoftriangl

AQRandBRPintersectatRandX�Then�QXR�������CAB

�ABC��BCAand�RXP�������ABC��CAB��BCA�

�PXQ�������QXR��RXP�������BCA�whichimplies

isonthecircumcircleoftriangleCPQ�Now

�C� A� B� ��C� A� X��XA� B�

�� ��QA� X�

� ��RA� X

�� ��QA� R��CAB�

Similarly�A� B� C���ABCand�B� C� A���BCA�Sotriangl

A� B� C�andABCaresimilar�

���HexagonABCDEFisinscribedinacirclesothatAB�CD�E

LetP�Q�RbethepointsofintersectionofACandBD�CEandD

EAandFB

respectively�ProvethattrianglesPQRandBDFa

similar�

Solution��DuetoNgKaWing�LetObethecenterofthecirc

andletL�M�NbetheprojectionsofOonBD�DF�FB�respective

ThenL�M�NaremidpointsofBD�DF�FB�respectively�LetSbet

projectionofOonAE�SinceAB�EF�wegetFB�AEandhen

��

Page 45: Math Problem Book I

ON

�OS�Let�AOB

��COD

��EOF����Then�RON

� ��SON

� ��ARB

� ��AOB��EOF����HenceON�OR�

cos��Similarly�POL��QOM

��andOL�OP�OM�OQ�

cos�� N

extrotate�PQRaroundOatangle�sothattheimageQ�

ofQ

liesonthelineOM�theimageR�ofRliesonthelineON

andtheimageP�ofPliesonlineOL�ThenON�OR� �OL�OP��

OM�OQ��cos��So�P� Q� R� ��LMN

aresimilar�SinceL�M�N

aremidpointsofBD�DF�FB�respectivelywehave�LMN��BDF

aresimilar�Therefore�PQR��BDFaresimilar�

��������IMOshortlistedproblem�LetABCDbeacyclicquadrilateral�

LetEandFbevariablepointsonthesidesABandCD�respectively

suchthatAE�EB�CF�FD�LetPbethepointonthesegment

EFsuchthatPE�PF�AB�CD�Provethattheratiobetweenthe

areasoftrianglesAPDandBPCdoesnotdependonthechoiceofE

andF�

Solution�Let�UVW�denotetheareaof�UVW

andletd�X�YZ�

denotethedistancefromX

tolineYZ�WehaveAE�EB�CF�

FD�a�b�wherea�b���SincePE�PF�AB�CD�wehave

d�P�AD��

CD

AB�CD

d�E�AD��

AB

AB�CD

d�F�AD��

�APD��

CD

AB�CD

�AED��

AB

AB�CD

�AFD�

a�CD

AB�CD

�ABD��

b�AB

AB�CD

�ACD��

d�P�BC��

CD

AB�CD

d�E�BC��

AB

AB�CD

d�F�BC��

�BPC��

CD

AB�CD

�BEC��

AB

AB�CD

�BFC�

b�CD

AB�CD

�BAC��

a�AB

AB�CD

�BDC��

��

SinceA�B�C�Dareconcyclicsin�BAD�sin�BCDandsin�ABC

sin�ADC�So

�APD�

�BPC�

�a�CD��ABD��b�AB��ACD�

b�CD��BAC��a�AB��BDC�

�a�CD�AB�AD�sin�BAD�b�AB�CD�AD�sin�ADC

b�CD�AB�BC�sin�ABC�a�AB�CD�BC�sin�BCD

�AD

BC

�a�sin�BAD�b�sin�ADC

b�sin�ABC�a�sin�BCD

�AD

BC

TangentLines

���TwocirclesintersectatpointsAandB�Anarbitrarylinethrough

intersectsthe�rstcircleagainatCandthesecondcircleagainatD

Thetangentstothe�rstcircleatCandtothesecondcircleat

intersectatM�TheparalleltoCM

whichpassesthroughthepoi

ofintersectionofAM

andCDintersectsACatK�ProvethatBK

tangenttothesecondcircle�

Solution�LetLbetheintersectionofAM

andCD�Since

�CMD��CAD��CMD��CAB��DAB

��CMD��BCM

��BDM

������

soA�C�M�Dareconcyclic�SinceLKkMC�

�LKC�������KCM

�������KCL��LCM

�������ACB��CAB��CBA��LBA�

SoA�B�L�Kareconcyclic�Then

�KBA��KLA��CMA��CDA��BDA�

ThereforeBKistangenttothecirclepassingthroughA�B�D�

��������IMO�Twocircles��and��arecontainedinsidethecircle

andaretangentto�atthedistinctpointsM

andN�respective

��

Page 46: Math Problem Book I

��passesthroughthecenterof���Thelinepassingthroughthetwo

pointsofintersectionof��and��meets�atAandB�respectively�

ThelinesMAandMBmeets��atCandD�respectively�Provethat

CDistangentto���

Solution��DuetoWongChunWai�LetX�Ybethecentersof������

respectively�Extend�� YXtomeet��atQ�JoinANtomeet��atE�

SinceABistheradicalaxisof������soAC�AM

�AE�AN�This

impliesC�M�N�Eareconcyclic�LetUbetheintersectionoflineCE

withthetangentto��atM�Then�UCM

��ENM

��ANM

�UMC�SoCEistangentto���SimilarlyCEistangentto���Now

YE�YQand �

CYE������ECY�����

� ��CXY

�����������CYQ���CYQ�

Theseimply�CYE��CYQarecongruent�Hence�CQY��CEY�

��� �Similarly�DQY���� �ThereforeCDistangentto���

����ProposedbyIndiafor����IMO�CirclesG�andG�toucheachother

externallyatapointW

andareinscribedinacircleG�A�B�Care

pointsonGsuchthatA�G�andG�areonthesamesideofchordBC�

whichisalsotangenttoG�andG��SupposeAW

isalsotangentto

G�andG��ProvethatW

istheincenteroftriangleABC�

Solution�LetPandQbethepointsoftangencyofG�withBCand

arcBAC�respectively�LetDbethemidpointofthecomplementary

arcBCofG�notcontainingA�andLbeapointonG�sothatDLis

tangenttoG�andintersectssegmentPC�Consideringthehomothety

withcenterQthatmapsG�ontoG�weseethatQ�P�Darecollinear

becausethetangentatP

�namelyBC�andthetangentatD

are

parallel�Since�BQD��CBD

subtendequalarcs�BQD��PBD

aresimilar�HenceDB�DP

DQ�DB�Bytheintersectingchord

theoremDB��DP�DQ�DL��SoDL�DB�DC�ThenDhas

thesamepowerDB��DC�withrespecttoG�andG��HenceDis

ontheradicalaxisAW

ofG�andG��SoL�W

andDW

istangent

toG�andG��

��

SinceDisthemidpointofarcBC�soAW

bisects�BAC�Also

�ABW

��BWD��BAD��WBD��CBD��CBW

andBW

bisects�ABC�ThereforeW

istheincenterof�ABC�

Comments�The�rstpartoftheproofcanalsobedonebyinversi

withrespecttothecirclecenteredatD

andofradiusDB

D

ItmapsarcBContothechordBC�BothG�

andG�

areinvaria

becausethepowerofDwithrespecttothemisDB��DC��Hen

W

is�xedandsoDW

istangenttobothG�andG��

Locus

���PerpendicularsfromapointPonthecircumcircleof�ABCaredraw

tolinesAB�BCwithfeetatD�E�respectively�Findthelocusoft

circumcenterof�PDEasPmovesaroundthecircle�

Solution�Since�PDB��PEB

�����P�D�B�E

areconcycl

Hencethecircumcircleof�PDEpassesthroughBalways�ThenP

isadiameterandthecircumcenterof�PDEisatthemidpointM

PB�LetObethecircumcenterof�ABC�thenOM

PB�Itfollow

thatthelocusofM

isthecirclewithOBasadiameter�

���SupposeAisapointinsideagivencircleandisdi erentfrom

t

center�Considerallchords�excludingthediameter�passingthrou

A�Whatisthelocusoftheintersectionofthetangentlinesatt

endpointsofthesechords�

Solution��DuetoWongHimTing�LetObethecenterandand

betheradius�LetA�bethepointonOAextendedbeyondAsu

thatOA�OA� �r��SupposeBCisonesuchchordpassingthrou

AandthetangentsatBandCintersectatD� �BysymmetryD�

onthelineODwhereDisthemidpointofBC�Since�OBD� ���

OD�OD��OB���OA�OA� ��So�OADissimilarto�OD� A

Since�ODA���� D�isonthelineLperpendiculartoOAatA� �

ConverselyforD�onLletthechordthroughAperpendicul

toOD�intersectthecircleatBandC�LetDbetheintersection

��

Page 47: Math Problem Book I

thechordwithOD� �Now�OAD��OD� A� aresimilarrighttriangles�

SoOD�OD� �OA�OA� �OB��OC��whichimplies�OBD��

�OCD� ���� �ThereforeD�isonthelocus�Thisshowsthelocusis

thelineL�

���Given�ABC�LetlineEFbisects�BACandAE�AF�AB�AC�

FindthelocusoftheintersectionPoflinesBEandCF�

Solution�ForsuchapointP�sinceAB�AE�AF�ACand�BAE�

�FAC�so�BAE��FACaresimilar�Then�AEP

�PCA�So

A�E�C�Pareconcyclic�Hence�BPC��CAE��BAC���There

forePisonthecircleCwhosepointsX

satisfy�BXC��BAC��

andwhosecenterisonthesamesideoflineBCasA�

ConverselyforPonC�LetBP�CPintersecttheanglebisector

of�BACatE�F�respectively�Since�BPC��BCA���so�EPF�

�EAC�HenceA�E�C�Pareconcyclic�So�BEA

�FCA�Also

�BAE��FAC�So�BAE��FACaresimilar�ThenAB�AC�

AE�AF�ThereforethelocusofPisthecircleC�

��������PutnamExam�LetC�andC�becircleswhosecentersare��

unitsapartandwhoseradiiare�and��Findthelocusofallpoints

M

forwhichthereexistspointsXonC�andYonC�suchthatM

is

themidpointofthelinesegmentXY�

Solution��DuetoPoonWaiHoi�LetO��O�bethecentersofC��C��

respectively�Ifwe�xYonC��thenasX

movesaroundC��M

will

traceacircle�Y

withradius

� �

centeredatthemidpointmY

ofO�Y�

AsYmovesaroundC��mY

willtraceacircleofradius

� �

centeredat

themidpontPofO�O��Sothelocusisthesolidannuluscenteredat

Pwithinnerradius

� ��

� ���andouterradius

� ��

� ����

CollinearorConcyclicPoints

��������IMO�DiagonalsACandCEoftheregularhexagonABCDEF

aredividedbytheinnerpointsM

andN�respectivelysothat

AM

AC

�CN

CE

�r�

��

DeterminerifB�M

andNarecollinear�

Solution��DuetoLeeTakWing�LetAC�x�thenBC�x�p

CN

�xr�CM

�x���r��Let�XYZ�denotetheareaof�XY

Since�NCM

���� ��BCM

����and�BCM���CMN���BCN

so

x����r�sin���

�p�

�x�r���r�sin���

x�r

�p�

Cancellingx�andsolvingforr�wegetr�

� p ��

��������PutnamExam�IfA�B�C�Darefourdistinctpointssuchth

everycirclethroughAandBintersectsorcoincideswitheverycirc

throughCandD�provethatthefourpointsareeithercollinear

concyclic�

Solution�SupposeA�B�C�Dareneitherconcyclicnorcollinear�Th

theperpendicularbisectorpofABcannotcoincidewiththeperpe

dicularbisectorqofCD�Iflinespandqintersecttheircommonpoi

isthecenteroftwoconcentriccirclesonethroughAandB�theoth

throughCandD�acontradiction�Iflinespandqareparallelth

linesABandCDarealsoparallel�ConsiderpointsPandQonpan

q�respectivelymidwaybetweentheparallellinesABandCD�Clear

thecirclesthroughA�B�PandC�D�Qhavenocommonpointaga

acontradiction�

��������PutnamExam�Givenanin�nitenumberofpointsinaplan

provethatifallthedistancesbetweeneverypairareintegersthent

pointsarecollinear�

Solution�SupposetherearethreenoncollinearpointsA�B�Csu

thatAB�randAC�s�ObservethatifPisoneoftheotherpoin

thenbtthetriangleinequalityjPA�PBj�����������r�Hence

wouldbeonthelineH�

joiningA�B

orononeofthehyperbol

Hi�fX�jXA�XBj�igfori���������r��orontheperpendicul

bisectorHrofAB�SimilarlyjPA�PCj�����������s�SoPis

oneofthesetsKj�fX�jXA�XCj�jgforj���������s�Sin

��

Page 48: Math Problem Book I

linesAB

andACaredistincteveryintersectionHi�Kj

isonlya

�niteset�Sotherecanonlybe�nitelymanypointsthatareintegral

distancesfrom

A�B�C�acontradiction�Thereforethegivenpoints

mustbecollinear�

��������IMOshortlistedproblem�TheincircleoftriangleABCtouches

BC�CAandAB

atD�E

andF

respectively�X

isapointinside

triangleABCsuchthattheincircleoftriangleXBCtouchesBCat

DalsoandtouchesCXandXBatYandZrespectively�Provethat

EFZYisacyclicquadrilateral�

Solution�IfEFkBC�thenAB�ACandADisanaxisofsymmetry

ofEFZY�HenceEFZYisacyclicquadrilateral�IflinesEFandBC

intersectatP�thenbyMenelaus�theorem�AF�BP�CE���FB�PC�

EA����SinceBZ�BD�BF�CY�CD�CEandAF�EA���

XZ�YX�weget�XZ�BP�CY���ZB�PC�YX����Bytheconverse

oftheMenelaus�theoremZ�Y�Parecollinear�Bytheintersecting

chordtheoremPE�PF�PD��PY�PZ�HenceEFZYisacyclic

quadrilateralbytheconverseoftheintersectingchordtheorem�

��������IMO�IntheconvexquadrilateralABCD�thediagonalsACand

BD

areperpendicularandtheoppositesidesAB

andDC

arenot

parallel�SupposethepointP�wheretheperpendicularbisectorsof

AB

andDCmeetisinsideABCD�ProvethatABCD

isacyclic

quadrilateralifandonlyifthetrianglesABPandCDPhaveequal

areas�

Solution��DuetoLeungWingChung�SettheoriginatP�Suppose

AandCareontheliney�pandBandDareonthelinex�q�Let

AP�BP�r�CP�DP�s�ThenthecoordinatesofA�B�C�Dare

��p r��p��p���q�p r��q����p s��p��p���q��

p s��q���

respectively�Now�ABP��CDPhaveequalareasifandonlyif

� �� � � ��pr��p�

p

q

p r��q�

� � � ��� �

� � � �p s��p�

p

q

�ps��q�

� � � ��

��

i�e���

p r��p�

p r��q��pq������

p s��p�p s��q��pq����Sin

f�x����

p x��p�p x��q��pq���isstrictlydecreasingwhenx�

andjqj�thedeterminantsareequalifandonlyifr�

s�which

equivalenttoABCDcyclic�

��������PutnamExam�Showthatifaconvexquadrilateralwithsid

lengthsa�b�c�dandarea

p abcdhasaninscribedcirclethenitis

cyclicquadrilateral�

Solution�Sincethequadrilateralhasaninscribedcirclewehavea

c�b�d�Letkbethelengthofadiagonalandangles�andselect

sothat

k��a

��b���abcos��c��d

���cdcos�

Ifwesubtract�a�b����c�d��anddivideby�wegettheequati

� �ab���cos���cd���cos��Fromthearea�absin��cdsin���

p abcd�weget

�abcd�a

�b����cos����c�d

����cos����abcdsin�sin�

Using� �wecancancelabcdtoobtaintheequation

�����cos�����cos�����cos����cos����sin�sin

����cos�����

whichimplies��������Thereforethequadrilateraliscyclic�

ConcurrentLines

���In�ABC�supposeAB�AC�LetPandQbethefeetofthepe

pendicularsfromBandCtotheanglebisectorof�BAC�respective

LetDbeonlineBCsuchthatDA AP�ProvethatlinesBQ�P

andADareconcurrent�

Solution�LetM

betheintersectionofPCandAD�LetB�bet

mirrorimageofB

withrespecttolineAP�SinceBB� AP

an

��

Page 49: Math Problem Book I

AD AP�soBB�kAD�Then�BCB� ��DACaresimilar�SinceP

isthemidpointofBB� �soPCintersectsADatitsmidpointM�Now

AQ

PQ

�MC

PC

�AM

B� P�

AM

BP

�BPQ��MAQaresimilar�Thisimplies�BQP��MQA�Soline

BQpassesthroughM�too�

��������ChineseNationalMathCompetition�DiagonalsAC

andBD

ofacyclicquadrilateralABCDmeetsatP�Letthecircumcentersof

ABCD�ABP�BCP�CDPandDAPbeO�O��O��O�andO�respec

tively�ProvethatOP�O�O��O�Oareconcurrent�

Solution�LetlinePO�intersectthecircumcircleof�BCPandseg

mentADatpointsQandR�respectively�Now�PDR��BCA�

�PQBand�DPR��QPB�So�DRP��QBP����andPO�

AD�NextcircumcirclesofABCDandDAPsharethecommonchord

AD�soOO

AD�HencePO�

andOO

areparallel�Similarly

POandOO�areparallel�SoPO�OOisaparallelogramanddiag

onalO�OpassesthroughthemidpointGofOP�SimilarlyPO�OO�

isaparallelogram

anddiagonalO�O�

passesthroughG�Therefore

OP�O�O��O�OconcuratG�

��������IMO�LetA�B�CandDbefourdistinctpointsonalineinthat

order�ThecircleswithdiametersACandBDintersectatthepoints

XandY�ThelineXYmeetsBCatthepointZ�LetPbeapointon

thelineXYdi erentfromZ�ThelineCPintersectsthecirclewith

diameterACatthepointsCandM�andthelineBPintersectsthe

circlewithdiameterBDatthepointsBandN�Provethatthelines

AM�DNandXYareconcurrent�

Solution���DuetoYuChunLing�LetARbeparalleltoBPand

DR�beparalleltoCP�whereRandR�arepointsonlineXY�Since

BZ�ZD

�XZ�ZY�CZ�ZA�wegetBZ�AZ�CZ�DZ�Since

�CZPissimilarto�DZR�and�BZPissimilarto�AZR�so

ZP

ZR

�BZ

AZ

�CZ

DZ

ZP

ZR��

��

HenceRandR� mustcoincide�Therefore�BPCissimilarto�ARD

SinceXY

AD�AM

CM�CM

kDR�DN

BN

an

BN

kAR�thelinesAM�DN�XYaretheextensionsofthealtitud

of�ARD�hencetheymustbeconcurrent�

Solution���DuetoMokTzeTao�SettheoriginatZandthe

axisonlineAD�Letthecoordinatesofthecircumcentersoftriangl

AMCandBNDbe�x����and�x�����andthecircumradiiber �an

r ��respectively�ThenthecoordinatesofAandCare�x��r ����an

�x��r �����respectively�LetthecoordinatesofPbe���y���Sin

AM

CPandtheslopeofCPis�

y �

x��r �

�theequationofA

worksouttobe�x��r ��x�y �y�x� ��r� ��LetQbetheintersection

AM

withXY�thenQhascoordinates���

r� ��x� �

y �

��SimilarlyletQ�

theintersectionofDNwithXY�thenQ� hascoordinates���

r� ��x� �

y �

Sincer� ��x� ��ZX��r� ��x� ��soQ�Q� �

Solution��LetAM

intersectXYatQandDN

intersectXY

Q� �ObservethattherighttrianglesAZQ�AMC�PZCaresimilar

AZ�QZ�PZ�CZ�ThenQZ�AZ�CZ�PZ�XZ�YZ�PZ�Sim

larlyQ� Z�XZ�YZ�PZ�ThereforeQ�Q� �

���AD�BE�CFarethealtitudesof�ABC�IfP�Q�Rarethemidpoin

ofDE�EF�FD�respectivelythenshowthattheperpendicularfro

P�Q�RtoAB�BC�CA�respectivelyareconcurrent�

Solution�LetYbethefootoftheperpendicularfromQtoBCan

H

betheorthocenterof�ABC�Notethat�ACF������CAB

�ABE�Since�CEH

������CDH�C�D�H�Eareconcyclic�

�ACF��EDH�NowAH

kQY�ED

kQR�so�EDH

��RQ

Hence�RQY

��EDH

��ACF�Similarly�ABE

��FDH

�PQY�Nextsince�ACF��ABE�QYbisects�PQR�Fromthes

itfollowstheperpendicularsfromP�Q�RtoAB�BC�CAconcur

theincenterof�PQR�

��������ChineseMathOlympiadTrainingTest�ABCDEFisahexag

��

Page 50: Math Problem Book I

inscribedinacircle�ShowthatthediagonalsAD�BE�CFareconcur

rentifandonlyifAB�CD�EF�BC�DE�FA�

Solution��DuetoYuKaChun�SupposeAD�BE�CFconcursatX�

FromsimilartrianglesABX

andEDX�wegetAB�DE�BX�DX�

SimilarlyCD�FA�DX�FX

andEF�BC�FX�BX�Multiplying

thseweget�AB�CD�EF���DE�FA�BC����soAB�CD�EF�

BC�DE�FA�

Fortheconverseweusethesocalledmethodoffalseposition�

Suppose� �AB�CD�EF�BC�DE�FAandAD

intersectBE

atX�NowletCX

meetthecircleagainatF� �Bythe�rstpartwe

getAB�CD�EF��BC�DE�F� A�Dividingthisby� �wehave

EF� �EF�F� A�FA�IfF�isonopenarcAF�thenF� A�FAand

EF

EF�yieldingF� A�FA

��

EF� �EF�acontradiction�If

F�isontheopenarcEF�thenFA�F� AandEF��EFyielding

EF� �EF���F� A�FA�acontradiction�SoF� �F�

AlternativelywecanuseCeva�stheoremanditsconverse�Let

ACandBEmeetatG�CEandADmeetatH�EAandCFmeetat

I�Leth�kbethedistancesfromA�CtoBE�respectively�Then

AG

CG

�h k�

ABsin�ABG

BCsin�CBG

Similarly C

HEH

�CDsin�CDH

DEsin�EDH

and

EI

AI

�EFsin�EFI

FAsin�AFI

Now�ABG��EDH��CBG��EFI��CDH��AFI�ByCeva�s

theoremanditsconverseAD�BE�CFareconcurrentifandonlyif

��

AG�CH�EI

CG�EH�AI

�AB�CD�EF

BC�DE�FA

���AcircleintersectsatriangleABCatsixpointsA��A��B��B��C��C��

wheretheorderofappearancealongthetriangleisA�C��C��B�A��A��

��

C�B��B��A�SupposeB�C��B�C�meetsatXC�A��C�A�meets

YandA�B��A�B�meetsatZ�ShowthatAX�BY�CZareconcurre n

Solution�LetDbetheintersectionofAXandB�C��SinceAX�B�C

B�C�areconcurrentby�thetrigonometricformof�Ceva�stheorem

��

DC��B�B��AC�

DB��AB��C�C�

�sinC�AD�sinB�B�C��sinB�C�B�

sinDAB��sinC�B�C��sinB�C�C�

Then

sinBAX

sinXAC

�sinC�AD

sinDAB�

�sinC�B�C��sinB�C�C�

sinB�B�C��sinB�C�B�

�Similarly

sinCBY

sinYBA

�sinA�C�A��sinC�A�A�

sinC�C�A��sinC�A�C�

sinACZ

sinZCB

�sinB�A�B��sinA�B�A�

sinA�A�B��sinA�B�A�

Using�C�B�C�

��C�A�C�

andsimilarangleequalityweseeth

theproductofthethreeequationsinvolvingX�Y�Zaboveisequal

��BytheconverseofthetrigonometricformofCeva�stheoremwes

thatAX�BY�CZareconcurrent�

��������IMOshortlistedproblem�Acirclepassingthroughvertices

andCoftriangleABCintersectssidesAB

andACatC�andB

respectively�ProvethatBB� �CC�andHH�areconcurrentwhere

andH� aretheorthocentersoftrianglesABCandAB� C� �respective

Solution��DuetoLamPoLeung�Letd�X�L�denotethedistan

fromapointXtoalineL�Fortheproblemwewillusethefollowi

lemma�

Lemma�LetlinesL��L�intersectatP�formingfourangleswithve

texP��SupposeH�H�lieonanoppositepairoftheseangles�

d�H�L���d�H� �L���d�H�L���d�H� �L���thenH�P�H� arecollinea

Proof�LetHH�intersectL��L�atX�Y�respectively�Then

HH�

H� X�

HX

H� X���

d�H�L��

d�H� �L��

��

d�H�L��

d�H� �L��

���

HY

H� Y���

HH�

H� Y�

��

Page 51: Math Problem Book I

SoX�YisonbothL�andL��henceitisP�ThereforeH�P�H� are

collinear�

FortheproblemletBB� �CC�intersectatP�Since�ABH

�����A

�AC� H� �soBH

kC� H� �SimilarlyCH

kB� H� �Let

BH�CC� intersectatLandCH�BB�intersectatK�Now

�PBH��ABH��C� BP�������A���B� CP

��ACH��B� CP��PCH�

SoK�B�C�Lareconcyclic�Then�LHK��BHCaresimilar�Also

�BHC��B� H� C�aresimilarbecause

�CBH������ACB������AC� B� ��C� B� H�

andsimilarly�BCH

�B� C� H� �Therefore�LHK��B� H� C�are

similar�SoKH�B� H� �LH�C� H� �SinceBHkC� H� andCHkB� H� �

sod�H�BB� ��d�H� �BB� ��

d�H�CC� ��d�H� �CC� ��Bythelemma

HH� alsopassesthroughP�

PerpendicularLines

��������APMO�LetABCbeatriangleandDthefootofthealtitude

fromA�LetEandFbeonalinepassingthroughDsuchthatAE

isperpendiculartoBE�AFisperpendiculartoCF�andEandFare

di erentfromD�LetM

andNbethemidpointsofthelinesegments

BCandEF�respectively�ProvethatANisperpendiculartoNM�

Solution��DuetoCheungPokMan�Therearemanydi erentpic

turessoitisbettertousecoordinategeometrytocoverallcases�

SetAattheoriginandlety�b���betheequationoftheline

throughD�E�F��Notethecaseb��impliesD

�E

�F�which

isnotallowed��LetthecoordinatesofD�E�Fbe�d�b���e�b���f�b��

respectively�SinceBE AEandslopeofAEisb�e�sotheequa

tionoflineAEisex�by��b��e�����Similarlytheequationof

lineCFisfx�by��b��f����andtheequationoflineBCis

dx�by��b��d�����

��

FromthesewefoundthecoordinatesofB�Care�d�e�b�de b��� d

f�b�df b��respectively�ThenthecoordinatesofM�Nare�d�e�f

� b

de�df

�b

���e

�f�

�b��respectively�SotheslopeofANis�b��e�f�andt

slopeofMN

is��de�df

�b

��d��e�f�b�Theproductoftheseslopes

���ThereforeAN MN�

��������APMO�LetABCbeatriangle�LetM

andN

bethepoin

inwhichthemedianandtheanglebisectorrespectivelyatAme

thesideBC�LetQandPbethepointsinwhichtheperpendicular

N

toNAmeetsMAandBA�respectivelyandOthepointinwhi

theperpendicularatPtoBAmeetsANproduced�ProvethatQO

perpendiculartoBC�

Solution���DuetoWongChunWai�SettheoriginatN

andt

xaxisonlineNO�LettheequationoflineABbey�ax�b�th

theequationoflinesACandPOarey��ax�bandy��

� ax�

respectively�LettheequationofBC

bey�

cx�ThenB

hasc

ordinates�

bc�a

�bc

c�a

��Chascoordinates��

bc�a

��bc

c�a

��M

h

coordinates�

ab

c��a��

abc

c��a���Ahascoordinates��

b a����Ohasc

ordinates�ab���andQhascoordinates���

ab c��ThenBChasslope

andQOhasslope�

� c�ThereforeQO BC�

Solution���DuetoPoonWaiHoi�ThecaseAB

�ACisclea

WithoutlossofgeneralitywemayassumeAB�AC�LetANinterse

thecircumcircleof�ABCatD�Then

�DBC��DAC�

� ��BAC��DAB��DCB�

SoDB�DCandMD BC�

WithAasthecenterofhomothetyshrinkDtoO�BtoB� and

toC� �Then�OB� C� �� ��BAC��OC� B�andBCkB� C� �LetB�

cutPN

atK�Then�OB� K��DAB��OPK�SoP�B� �O�K

a

concyclic�Hence�B� KO��B� PO����andsoB� K�C� K�Sin

��

Page 52: Math Problem Book I

BCkB� C� �thisimpliesA�K�M

arecollinear�ThereforeK

�Q�

Since�B� KO����andBCkB� C� �wegetQO BC�

���LetBB�andCC�bealtitudesoftriangleABC�AssumethatAB��

AC�LetM

bethemidpointofBC�HtheorthocenterofABCandD

theintersectionofB� C�andBC�ProvethatDH AM�

Solution�LetA�bethefootofthealtitudefrom

AtoBC�Since

A� �B� �C� Mlieontheninepointcircleof�ABC�sobytheintersecting

chordtheoremDB� �DC� �DA� �DM�Since�AC� H������AB� H�

pointsA�C� �H�B�lieonacircle��

withthemidpointX

ofAH

as

center�Since�HA� M

���� �sothecircle��

throughH�A� �M

has

themidpointYofHM

ascenter�SinceDB� �DC��DA� �DM�the

powersofDwithrespectto��and��arethesame�SoD�andH�

areontheradicalaxisof������ThenDH

XY�Bythemidpoint

theoremXYkAM�ThereforeDH AM�

��������ChineseTeamSelectionTest�ThesemicirclewithsideBCof

�ABCasdiameterintersectssidesAB�ACatpointsD�E�respec

tively�LetF�GbethefeetoftheperpendicularsfromD�Etoside

BCrespectively�LetM

betheintersectionofDGandEF�Provethat

AM

BC�

Solution�LetHbethefootoftheperpendicularfromAtoBC�Now

�BDC������BEC�SoDF�BDsinB

�BCcosBsinB

and

similarlyEG�BCcosCsinC�Now

GM

MD

�EG

FD

�cosCsinC

cosBsinB

�cosC

cosB

AB

AC

SinceBH�ABcosB�HG�AEcosC�weget

BH

HG

�ABcosB

AEcosC

�ACcosB

ADcosC

and

BH

HG

GM

MD

DA

AB

���

BytheconverseofMenelaus�theoremon�BDG�pointsA�M�Hare

collinear�ThereforeAM

BC�

��������IMO�AcirclewithcenterOpassesthroughtheverticesAand

CoftriangleABCandintersectsthesegmentsABandACagainat

��

distinctpointsK

andN�respectively�Thecircumcirclesoftriang l

ABCandKBN

intersectatexactlytwodistinctpointsB

andM

ProvethatOM

MB�

Solution�LetCM

intersectthecirclewithcenterOatapoint

Since�BMC�������BAC�������KAC��KLC�soBM

paralleltoKL�Now

�LKM

��LKN��NKM

��LCN��NBM

�������BMC��BAC��KLM�

ThenKM

�LM�SinceKO�LO�soOM

KL�HenceOM

BM

��������ChineseSenoirHighMathCompetition�Acirclewithcenter

isinternallytangenttotwocirclesinsideitatpointsSandT�Suppo

thetwocirclesinsideintersectatM

andNwithNclosertoST�Sho

thatOM

MNifandonlyifS�N�Tarecollinear�

Solution��DuetoLeungWaiYing�Considerthetangentlinesat

andatT��SupposetheyareparallelthenS�O�Twillbecollinear

thatM

andN

willbeequidistantfromST�contradictingN

isclos

toST��LetthetangentlinesmeetatK�then�OSK������OT

impliesO�S�K�TlieonacirclewithdiameterOK�AlsoKS��KT

impliesK

isontheradicalaxisMN

ofthetwoinsidecircles�

M�N�Karecollinear�

IfS�N�Tarecollinearthen

�SMT��SMN��TMN��NSK��KTN�������SKT

SoM�S�K�T�Oareconcyclic�Then�OMN

��OMK��OSK

��� �

ConverselyifOM

MN�then�OMK

������OSK

impl

M�S�K�T�Oareconcyclic�Then

�SKT�������SMT

�������SMN��TMN

�������NSK��KTN�

��

Page 53: Math Problem Book I

Thus�TNS�������NSK��SKT��KTN������Therefore

S�N�Tarecollinear�

���AD�BE�CFarethealtitudesof�ABC�LinesEF�FD�DEmeetlines

BC�CA�ABinpointsL�M�N�respectively�ShowthatL�M�N

are

collinearandthelinethroughthemisperpendiculartothelinejoining

theorthocenterHandcircumcenterOof�ABC�

Solution�Since�ADB������AEB�A�B�D�Eareconcyclic�By

theintersectingchordtheoremNA�NB�ND�NE�Sothepowerof

N

withrespecttothecircumcirclesof�ABC��DEFarethesame�

HenceNisontheradicalaxisofthesecircles�SimilarlyL�M

arealso

onthisradicalaxis�SoL�M�Narecollinear�

Sincethecircumcircleof�DEFistheninepointcircleof�ABC�

thecenterNoftheninepointcircleisthemidpointofHandO�Since

theradicalaxisisperpendiculartothelineofcentersOandN�sothe

linethroughL�M�NisperpendiculartothelineHO�

GeometricInequalities�Maximum�Minimum

���������IMO�LetP��P������P�n��bedistinctpointsonsomehalfof

theunitcirclecenteredattheoriginO�Showthat

j��OP����OP�����������

OP�n��j���

Solution�Whenn���thenj��OP�j���Supposethecasen�kistrue�

Forthecasen�k���wemayassumeP��P������P�k��arearranged

clockwise�Let�� OR���OP�����������

OP�k��

and� OS���OP�������

OP�k���

Bythecasen�k�j�� ORj���Also�� ORliesinside�P�OP�k���Since

j��OP�j���j�����

OP�k��j�OSbisects�P�OP�k���Hence�ROS���� �

Thenj��OP�����������

OP�k��j�j�� OR�� OSj�j�� ORj���

����Lettheanglebisectorsof�A��B��CoftriangleABCintersectits

circumcircleatP�Q�R�respectively�Provethat

AP�BQ�CR�BC�CA�AB�

��

Solution��DuetoLauLapMing�Since�ABQ��CBQ�weha

AQ�CQ�Bycosinelaw

AQ

��AB

��BQ

���AB�BQcos�ABQ

CQ

��CB

��BQ

���CB�BQcos�CBQ�

IfAB

��CB�thensubtractingtheseandsimplifyingwegetAB

CB��BQcos�ABQ��BQ�IfAB�CB�thenBQisadiamet

andweagaingetAB�CB��AB��BQ�SimilarlyBC�AC��C

andCA�BA��AP�Addingtheseinequalitiesanddividingby�w

getthedesiredinequality�

���������APMO�LetABCbeatriangleinscribedinacircleandletl a

ma�Ma�lb�mb�Mb�lc�mc�Mc�wherema�mb�mcarethelengt

oftheanglebisectors�internaltothetriangle�andMa�Mb�Mca

thelengthsoftheanglebisectorsextendeduntiltheymeetthecirc

Provethat

l asin

�A

l b

sin

�B

l c

sin

�C

���

andthatequalityholdsi ABCisequilateral�

Solution��DuetoFungHoYin�LetA� bethepointtheanglebisect

of�Aextendedtomeetthecircle�Applyingsinelawto�ABA� �w

getAB�sinC�Ma�sin�B�A ���Applyingsinelawto�ABD�weg

AB�sin�C�A ���ma�sinB�So

l a�

ma

Ma

sinBsinC

sin�B�A ��sin�C�A ��

�sinBsinC�

BytheAMGMinequality

l asin

�A

l b

sin

�B

l c

sin

�C

�sinBsinC

sin

�A

�sinCsinA

sin

�B

�sinAsinB

sin

�C

withequalityifandonlyifsinA�sinB�sinCandC�A �

�B�A �

������� �whichisequivalentto�A��B��C�

��

Page 54: Math Problem Book I

�����MathematicsMagazineProblem

�����LetIandO

betheincen

terandcircumcenterof�ABC�respectively�Assume�ABCisnot

equilateral�soI��O��Provethat

�AIO����ifandonlyif

�BC�AB�CA�

Solution��DuetoWongChunWai�LetDbetheintersectionofrayAI

andthecircumcircleof�ABC�ItiswellknownthatDC�DB�DI�

�DC�DBbecause�CAD��BADandDB�DIbecause�BID�

�BAD��ABI��CAD��CBI��DBC��CBI��DBI��Since

ABDCisacyclicquadrilateralbyPtolemy�stheoremAD�BC�

AB�DC�AC�DB��AB�AC��DI�ThenDI�AD�BC��AB�AC��

Since�AODisisosceles�AIO����ifandonlyifDI�AD���which

isequivalentto�BC�AB�AC�

Comments�Inthesolutionabovewesee�AIO����ifandonlyif

�BC�AB�AC�Alsotheconverseofthewellknownfactistrue

i�e�thepointIonADsuchthatDC�DB�DIistheincenterof

�ABC�Thisisbecause�BID��DBIifandonlyif�CBI��ABI�

since�DBC��BADalways��

����SquaresABDEandACFGaredrawnoutside�ABC�LetP�Qbe

pointsonEGsuchthatBPandCQareperpendiculartoBC�Prove

thatBP�CQ�BC�EG�Whendoesequalityhold�

Solution�LetM�N�ObemidpointsofBC�PQ�EG�respectively�Let

HbethepointsothatHEAGisaparallelogram�Translatingby� GA�

thenrotatingby���aboutA��GHAwillcoincidewith�ABCandO

willmovetoM�SoHA�BC�HA BC�OE�OG�MA�EG

MA�LetLbeonMN

suchthatALkEG�SinceNLkPB�PB

BC�BC HA�soLNOAisaparallelogram�ThenAO�LN�Since

MA EG�soMA AL�whichimpliesML�MA�Therefore

BP�CQ��MN���LN�ML�

���AO�MA����BM

�OE��BC�EG�

EqualityholdsifandonlyifLcoincideswithA�i�e�AB�AC�

��

����PointPisinside�ABC�DeterminepointsDonsideABandE

sideACsuchthatBD�CEandPD�PEisminimum�

Solution�TheminimumisattainedwhenADPEisacyclicqua d

lateral�ToseethisconsiderthepointG

suchthatG

liesont

oppositesideoflineAC

asB��ABP

�ACG

andCG

B

LetE

betheintersectionoflinesAC

andPG�LetD

bethei

tersectionofAB

withthecircumcircleofAPE�SinceADPE

is

cyclicquadrilateral�BDP

�AEP

�CEG�Usingthede�n

tionofG�wehave�BDP��CEGarecongruent�SoBD�CEan

PD�PE�GE�PE�GP�

ForotherD� �E�onsidesAB�AC�respectivelysuchthatBD�

CE� �bythede�nitionofG�wehave�BPD� ��CGE�arecongruen

ThenPD� �GE�andPD� �PE� �GE� �PE� �GP�

SolidorSpaceGeometry

�����ProposedbyItalyfor����IMO�Whichregularpolygonscanbeo

tained�andhow�bycuttingacubewithaplane�

Solution��DuetoFanWaiTongKeeWingTaoandTamSiuLun

Observethatiftwosidesofapolygonisonafaceofthecubeth

thewholepolygonliesontheface�Sinceacubehas�faceson

regularpolygonwith���or�sidesarepossible�Letthevertic

ofthebottomfaceofthecubebeA�B�C�Dandtheverticesont

topfacebeA� �B� �C� �D�withA�ontopofA�B�ontopofBand

on�ThentheplanethroughA�B� �D� cutsanequilateraltriangle�T

perpendicularbisectingplanetoedgeAA�cutsasquare�Thepla

throughthemidpointsofedgesAB�BC�CC� �C� D� �D� A� �A� Acuts

regularhexagon�Finallyaregularpentagonisimpossibleotherwi

the�vesideswillbeon�vefacesofthecubeimplyingtwooft

sidesareonparallelplanesbutnotwosidesofaregularpentagona

parallel�

���������IsraeliMathOlympiad�Fourpointsaregiveninspaceingener

position�i�e�theyarenotcoplanarandanythreearenotcollinea

��

Page 55: Math Problem Book I

Aplane�iscalledanequalizingplaneifallfourpointshavethesame

distancefrom��Findthenumberofequalizingplanes�

Solution�Thefourpointscannotalllieononesideofanequalizing

planeotherwisetheywouldlieinaplaneparalleltotheequalizing

plane�Henceeitherthreelieononesideandoneontheotherortwo

lieoneachside�Intheformercasethereisexactlyoneequalizing

planewhichisparalleltotheplanePcontainingthethreepointsand

passingthroughthemidpointofthesegmentjoiningthefourthpointx

andthefootoftheperpendicularfromxtoP�Inthelattercaseagain

thereisexactlyoneequalizingplane�Thetwopairofpointsdetermine

twoskewlinesinspace�Considerthetwoplaneseachcontainingone

ofthelineandisparalleltotheotherline�Theequalizingplaneis

theplanemidwaybetweenthesetwoplane�Sincethereare�����

waysofdividingthefourpointsintothesetwocasesthereareexactly

�equalizingplanes�

��

SolutionstoNumberTheoryProblems

Digits

���������PutnamExam�Provethateverypositiveintegerhasamultip

whosedecimalrepresentationinvolvesalltendigits�

Solution�Letnbeapositiveintegerandp��������������k�whe

kissolargethat��k�n�Thenthenconsecutiveintegersp���p

������p�nhavedecimalrepresentationsbeginningwith�����������

andoneofthemisamultipleofn�

����Doesthereexistapositiveintegerasuchthatthesumofthedig

�inbase���ofais����andthesumofthedigits�inbase���ofa�

������

Solution�Yes�Infactthereissuchanumberwhosedigitsconsist

��sand��s�Letk������Considera�����

�����

��������k�Th

thesumofthedigitsofaisk�Now

a����

��

���

��

�������

�k��

��

X��i�j�k

���i��j�

Observethattheexponentarealldi erentbytheuniquenessofba

�representation�Thereforethesumofthedigitsofa�inbase��

k��Ck ��k��

�����ProposedbyUSSRfor����IMO�Letan

bethelastnonzerodig

inthedecimalrepresentationofthenumbern��Doesthesequen

a��a������an����becomeperiodicaftera�nitenumberofterms�

Solution�SupposeafterNtermsthesequencebecomesperiodicwi

periodT�Thenai�jT

�aifori�N�j����������Bythepigeonho

principletherearetwonumbersamong��N�����N�����N������th

havethesameremainderwhendividedbyT�say��m

���k�modT

withN�m�k�Then��k���m

�jTforsomeintegerj�

��

Page 56: Math Problem Book I

Observethat��k��

��k���k����impliesa��k

a��k���Let

n���k���jT�Since��k���N�an���a��k�jT

�a��k���jT

�an�

Since�n����������k���m��n����������������n���so�an

�an��

�an

�mod����Thisimpliesan

���Howeverintheprime

factorizationofn��theexponentof�isgreaterthantheexponentof

�whichimpliesan

isevenacontradiction�

ModuloArithmetic

���������PutnamExam�Provethatthenumberofoddbinomialcoe�

cientsinanyrowofthePascaltriangleisapowerof��

Solution�Byinduction���x��

m

���x

�m

�mod���Ifwewriten

inbase�sayn��a�

��a�

������ak�wheretheai�saredistinct

nonnegativeintegersthen

���x�n����x��

a�

������x��

ak

����x

�a��������x

�ak��mod���

Inexpandingtheexpressioninfrontof�mod��wegetthesumofxnS�

whereforeachsubsetSoff��������kg�nS�

X i�S

�ai�Sincethereare�k

subsetsoff��������kg�thereareexactly�ktermseachwithcoe�cient

��Thisimpliesthereareexactly�koddbinomialcoe�cientsinthe

nthrowofthePascaltriangle�

����Leta��a��a������a��andb ��b��b������b��betwopermutationsofthe

naturalnumbers�������������Showthatifeachofthenumbersa�b ��

a�b ��a�b ������a��b ��isdividedby��thenatleasttwoofthemwill

havethesameremainder�

Solution�Supposea�b ��a�b ������a��b ��havedistinctremainderswhen

dividedby���Bysymmetrywemayassumea�b ����mod����Let

x��a�b ������a��b ����Ononehandx��������mod����Onthe

otherhandsinceaibi����mod����fori����������wegeta�

���b ��Sox��a����a����b ����b�����������������mod����a

contradiction�

��

���������CzechSlovakMatc h�Leta��a�����beasequencesatisfyinga�

��a���and

an������n

��an������n

��an

foralln���Dothereexistindicesp�qandrsuchthatapaq�ar�

Solution��DuetoLauLapMing�The�rstfewtermsare����������

���������Sincethedi erencesofconsecutivetermsaremultiplesof

wesuspectan

���mod��foralln�Clearlya��a����mod���

an�an�����mod���then

an������n

�������n

��������mod���

Sobyinductionallan

���mod���Thenapaq��arforallp�q�r

�����mod���

PrimeFactorization

�����AmericanMathematicalMonthlyProblemE�����LetAn

bethes

ofpositiveintegerswhicharelessthannandarerelativelyprimeto

Forwhichn���dotheintegersinAnformanarithmeticprogressio

Solution�SupposeAn

isanarithmeticprogression�Ifnisoddan

n���then����AnimpliesAn�f��������n��g�whichimpliesn

prime�Ifnisevenandnotdivisibleby�then����An����Animp

An

�f����������n��g�whichimpliesnisapowerof��Finally

nisevenanddivisibleby�thenletpbethesmallestprimen

dividingn�Eitherp���mod��orp���mod���Intheform

casesince��parethe�rsttwoelementsofAn

andn���An�

��k�p����

n��forsomek�Thisimpliesn���mod���

contradiction�Sop���mod���Then�p��isdivisibleby�an

so�p����An�ConsequentlyAn

�f��pg�whichimpliesn��

consideringtheprimefactorizationofn�ThereforeAnisanarithmet

progressionifandonlyifnisaprimeapowerof�orn���

���������IMO�Provethatthesetofintegersoftheform

�k���k

��������containsanin�nitesubsetinwhicheverytwomembersa

relativelyprime�

���

Page 57: Math Problem Book I

Solution�Weshallgivearecipeforactuallyconstructinganin�nite

setofintegersoftheformai��ki

���i���������eachrelatively

primetoalltheothers�Leta���������Supposewehavenpairwise

relativelyprimenumbersa���k����

a���k��������an��kn

���

Weformtheproducts�a�a����an�whichisodd�Nowconsiderthe

s��numbers��������������s�Atleasttwoofthesewillbecongruent

�mods�say������mods��orequivalently�����������ms

forsomeintegerm�Theoddnumbersdoesnotdivide���soitmust

divide�������hence�������lsforsomeintegerl�Since������

isdivisiblebysandsisodd������isrelativelyprimetos�This

implies���������ki��fori���������n�Sowemayde�nean���

�������Thisinductiveconstructioncanberepeatedtoformanin�nite

sequence�

Comments�ByEuler�stheoremwemaytaketheexponent��tobe

��s��theEuler�functionofs�whichequalsthenumberofpositivein

tegerslessthansthatarerelativelyprimetos�then��s ���mods��

���������ChineseMathOlympiadTrainingTest�Determinethesmallest

valueofthenaturalnumbern��withthepropertythatwhenever

thesetSn

�f��������ngispartitionedintotheunionoftwosub

setsatleastoneofthesubsetscontainsthreenumbersa�bandc�not

necessarilydistinct�suchthatab�c�

Solution��DuetoLamPeiFung�We�rstshowthat������hasthe

propertythenwewillshowitistheleastsolution�

SupposeS��ispartitionedintotwosubsetsX��X��Withoutloss

ofgeneralitylet�beinX��If����isinX��thenwearedone�

Otherwise�isinX��If�����isinX��thenwearedone�Otherwise

��isinX��If�������isinX��thenwearedone�Otherwise��is

inX��Finallyeither��������isinX�or��������isinX��

Ineithercasewearedone�

Toshow���isthesmallestwewillshowthatS��canbepar

titionedintotwosubsetseachofwhichdoesnotcontainproductsof

itselements�De�neCtobe!prime"inS��ifCisnottheproduct

ofelementsofS���The!primes"inS��consistof����p��pwhere

p����isausualprimenumber�SincethesmallestprimeinS��is

���

�nonumberinS��istheproductofmorethanfour!primes"�P

allthe!primes"andnumbersthatcanbewrittenasproductsoffo

!primes"inonesubsetX��andletX��S��nX��

NoproductsinX�areinX�becausenumbersinX�haveatlea

two!prime"factorssotheirproductscanbewrittenwithatleastfo

!prime"factors�Nextlookingattheproductof����p��p�poddprim

�����weseethataproductoftwo!primes"cannotbefactoredin

aproductoffour!primes"�SonoproductsinX�areinX��

BasenRepresentations

���������IMO�Canyouchoose����pairwisedistinctnonnegativeintege

lessthan���suchthatnothreeareinarithmeticprogression�

Solution�Weconsiderthegreedyalgorithmforconstructingsuch

sequence�startwith��andateachstepaddthesmallestinteg

whichisnotinarithmeticprogressionwithanytwoprecedingterm

Weget�����������������������������Inbase�thissequenceis

����������������������������������������

�Notethissequenceisthenonnegativeintegersinbase���Since����

base�is�����������soswitchingthisfrombase�tobase��weg

the����thtermofthesequenceis����������Toseethissequen

workssupposex�y�zwithx�y�zarethreetermsofthesequen

inarithmeticprogression�Considertherightmostdigitinbase�whe

xdi ersfromy�thenthatdigitforzisa�acontradiction�

�����AmericanMathematicalMonthlyProblem

�����Letpbeanod

primenumberandrbeapositiveintegernotdivisiblebyp�Fora

positiveintegerkshowthatthereexistsapositiveintegermsuchth

therightmostkdigitsofmr�whenexpressedinthebasep�areall�

Solution�Weprovebyinductiononk�Fork���takem���Nex

supposemr�inbasep�endsink��si�e�

mr���p�����pk����apk�higherterms��

���

Page 58: Math Problem Book I

Clearlygcd�m�p����Then

�m�cpk�r�mr�rmr��cpk�����crpkri

���p�����pk����a�rmr��c�pk�higherterms�

Sincegcd�mr�p����thecongruencea�rmr��c���modp�issolv

ableforc�Ifc �isasolutionthen�m�c �pk�rwillendin�k�����s

asrequired�

�����ProposedbyRomaniafor����IMO�Showthatthesequencefang

de�nedbyan

��n

p ��forn�����������wherethebracketsdenote

thegreatestintegerfunction�containsanin�nitenumberofintegral

powersof��

Solution�Write

p �inbase�asb ��b�b �b �����whereeachb i��or��

Since

p �isirrationaltherearein�nitelymanyb k���Ifb k���then

inbase��k��p ��b ����bk���bk���Letm���k��p ���then

�k��p ������k��p ���m��k��p ��

� ��

Multiplyingby

p �andadding

p ��weget�k��m���

p ���k�

p� ��

Then��m���

p ����k�

Representations

����Findall�even�naturalnumbersnwhichcanbewrittenasasumof

twooddcompositenumbers�

Solution�Letn���anddbeitsunitsdigit�Ifd���thenn�

����n����willdo�Ifd���thenn�����n����willdo�Ifd���

thenn����n���willdo�Ifd���thenn�����n����willdo�If

d���thenn�����n����willdo�Forn����directcheckingshows

only��������������������������������������

canbesoexpressed�

����Findallpositiveintegerswhichcannotbewrittenasthesumoftwo

ormoreconsecutivepositiveintegers�

���

Solution��DuetoCheungPokMan�Foroddintegern��k���

n�k��k����Forevenintegern���supposen�m��m���

�����m�r����m�r��r�����withm�r���Then�m�r�r���

andoneof�m�r�r��isodd�Sonmusthaveanodddivisorgreat

than��Inparticularn��j�j�����������cannotbewritten

thesumofconsecutivepositiveintegers�Fortheotherevenintege

n��j��k���withj�k���If�j�k�thenn���j�k����j�k���

������j�k��If�j�k�thenn��k��j�����k��j���������k��j

�����ProposedbyAustraliafor����IMO�Observethat����������

IsthereanintegerNwhichcanbewrittenasasumof����consecuti

positiveintegersandwhichcanbewrittenasasumof�morethanon

consecutiveintegersinexactly����ways�

Solution�ForsuchNwehaveN�

���� X i�

��m�i�������m�������

Nisoddandisdivisibleby����������Alsothereareexactly��

positiveintegerpairs�n�k�suchthatN�

k X i��

�n�i��

�k����n�� k

Hence�Ncanbefactoredas�k�����n�k�inexactly����ways��No

if�N�abwith��a�b�thenn����b�a����k�a����Th

means�Nhasexactly���������������positivedivisors�No

write�Ninprimefactorizationas���e�

����e�

�����Thenweg

�����������e �����e��������Sofe��e�g�f������g�Therefor

N�����������or�����������Asallthestepscanbereversedthe

aretheonlyanswers�

����Showthatifp��isprimethenpncannotbethesumoftwopositi

cubesforanyn���Whataboutp��or��

Solution�Supposenisthesmallestpositiveintegersuchthatpnist

sumoftwopositivecubessaypn�a��b���a�b��a��ab�b���Th

a�b�pkanda��ab�b��pn�k�Sincea�b���sok���Sin

a��ab�b��ab���son�k�Now�ab��a�b����a��ab�b��

p�k�pn�k

and��k�n�sopj�ab�Sincep���sopjaorpj

Sincea�b�pk�sopjaandpjb�saya�pAandb�pB�Th

A��B��pn���contradictingthesmallestpropertyofn�

���

Page 59: Math Problem Book I

Forp���supposea��b���n�Ifa�b���then�ja��jband

�a�����b �����n���Soa�b��kandn��k���

Forp���supposea��b���n�Ifa�b��kanda��ab�b��

�n�k����then�j�abimplies�ja��jband�a�����b �����n���

Otherwisewehave��a��ab�b��abandthena�b���Sointhis

casea�bare���k��kandn��k���

�����DuetoPaulErd�osandM�Sur�anyi�Provethateveryintegerkcanbe

representedinin�nitelymanywaysintheformk������������m�

forsomepositiveintegermandsomechoiceofsigns�or��

Solution�We�rstshoweveryintegercanbesorepresentedinatleast

oneway�Ifkcanberepresentedthenchangingallthesignswesee

�kalsocanberepresented�Soitsu�cestodothenonnegativecases�

Thekeyobservationistheidentity

�m���

���m���

���m���

���m���

����

Now�������������������������������������

����������������������Bytheidentityifkcanberepresented

thenk��canberepresented�Sobyinductioneverynonnegative

integer�andhenceeveryinteger�canberepresented�Toseethere

arein�nitelymanysuchrepresentationsweusetheidentityagain�

Observe�������m������m������m������m������m�

�����m������m������m�����Soforeveryrepresentationwe

canadd�moretermstogetanotherrepresentation�

���������IMOshortlistedproblem�A�nitesequenceofintegersa��a������

an

iscalledquadraticifforeachi�f��������ng�ja i�ai��j�i��

�a�Provethatforanytwointegersbandc�thereexistsanatural

numbernandaquadraticsequencewitha��bandan�c�

�b�Findtheleastnaturalnumbernforwhichthereexistsaquadratic

sequencewitha���andan������

Solution�Part�a�followsfromthelastproblembylettingk�c�

b�Forpart�b�considerak

insuchaquadraticsequence�Wehave

���

ak

�����������k�

�k�k�����k������Soa��

������Al

ak

�����������k�

�mod���Since�������������

isod

n����Toconstructsuchaquadraticsequencewithn�����rstno

�������������������Nowwewrite�����������������

����������Then

�����������������������������

����

����

��������

����Provethateveryintegergreaterthan��canberepresentedasasum

threeintegers��whicharepairwiserelativelyprimeandshowth

��doesnothavethisproperty�

Solution��DuetoChanKinHangandNgKaWing�Letk���Fro

����������wesee������k���worksfor�k�From��������

wesee������k���worksfor�k���From����������wes

������k���worksfor�k���

For�k���wesplitintocases��k� ��and��k� ���Weha

��k� �������k� ������k� ���and��k� �������k� ������k� ��

For�k���wesplitintocases��k� ��and��k� ���Weha

��k� �������k� ������k� ���and��k� �������k� ������k� ��

For�k���wesplitintocases��k� ��and��k� ����Weha

��k� �������k� ������k� ���and��k� ��������k� ������k� ��

Finally��doesnothavetheproperty�Otherwise���a�b�

wherea�b�carerelativelyprimeanda�b�c�Thena�b�careodd�

a���then������a�b�c�������showsthisisimpossib

Ifa���thenb���c��anda�b�c�������againimpossible

ChineseRemainderTheorem

���������ChineseTeam

SelectionTest�De�nexn

�xn����for

positiveintegersn�Provethatanintegervaluecanbechosenforx�

thatx���isdivisibleby�����

���

Page 60: Math Problem Book I

Solution�Lety n�

xn

�n�theny n�y n���

� �n�whichimplies

y n�y ��

� ��

� �������

� �n�

Thisgivesxn��x�����n���Wewantx�����x����������tobe

divisibleby������������whichmeans

x����mod���x����mod���x�����mod����

Since������arepairwiserelativelyprimebytheChineseremainder

theoremsuchx�exists�

�����ProposedbyNorthKoreafor����IMO�DoesthereexistasetM

withthefollowingproperties�

�a�ThesetM

consistsof����naturalnumbers�

�b�EveryelementinM

andthesumofanynumberofelementsinM

havetheformmk�wherem�karepositiveintegersandk���

Solution��DuetoCheungPokMan�Letn��������������Choose

ndistinctprimenumbersp ��p������pn�Letd��e��e��e�

���nen�where

e iisasolutionofthenequationsx����modp i�andx���modp j�

forevery��j�n�j��i��Sincethep i�sarepairwiserelatively

primesuchasolutionexistsbytheChineseremaindertheorem��Since

e ��e������en���modp ���disap ithpower�Sincee ����e ������en�

��modp ����disap �thpowerandsoon�Itfollowsd��d���������d

areallperfectpowersandanysumofthemisamultipleofd�lessthan

orequaltond�henceisalsoaperfectpower�

Divisibility

����Findallpositiveintegersa�bsuchthatb��and�a��isdivisibleby

�b���

Solution�Sinceb�

��so�b���

�a���henceb�

a�Leta�

qb�r���r�b�thenbydivisionweget

�a��

�b��

��a�b��a��b������a�qb�

�r��

�b��

���

Since��

�r��

�b��

���therearenosolutions�

����Showthattherearein�nitelymanycompositensuchthat�n����n

isdivisiblebyn�

Solution�Weusethefactx�yjxk�ykforpositiveintegerk�Consid

n���

t

���

t

fort���������Byinductionwecanshow�tj��t

��n�����

t��AlternativelybyEuler�stheorem��

t

�����t ��

��mod�t���Thenn����tk�Son���

t

���

t

j���

t�k����t�k

�n����n���

����Provethattherearein�nitelymanypositiveintegersnsuchthat�n�

isdivisiblebyn�Findallsuchn�sthatareprimenumbers�

Solution�Lookingatthecasesn��to��suggestforn��k�k

����������weshouldhavenj�n���Thecasek��isclear�Suppo

casekistrue�Now��

k��

������k

������k����

k

����Bycase

��k

���mod���so��

k����k������������������mod���

��k��

��isdivisibleby�k���completingtheinduction�

Ifaprimendivides�n���thenbyFermat�slittletheoremnj�n

��too�Thennj��n������n������son���

���������RomanianMathOlympiad�Findallpositiveintegers�x�n�su

thatxn��n��isadivisorofxn����n�����

Solution��DuetoChengKeiTsiandLeungWaiYing�Forx�

���n��n�����n����n������n��n���Forx������n��n���

�n����n������n��n���Forx������n��n�����n��

�n��������n��n����Sotherearenosolutionswithx������

Forx���ifn���thenwegetx�xn��n����xn����n���

Now

xn����n

����

��x����xn��n���

�xn���n���x����n��

��x����xn��n���

���

Page 61: Math Problem Book I

becauseforn���xn���n���x��n���x���x����andfor

n���xn���n���x�x��n����n������Henceonlyn��

andx��arepossible�Nowxn

��n���x��isadivisorof

xn����n�����x�����x����x������ifandonlyifx��

isadivisorof���Sincex�����x��or���Sothesolutionsare

�x�y�������and�������

���������BulgarianMathCompetition�Findallpairsofpositiveintegers

�x�y�forwhich

x��y�

x�y

isanintegeranddivides�����

Solution�Suppose�x�y�issuchapair�Wemayassumex�y�oth

erwiseconsider�y�x��Thenx��y�

�k�x�y��wherekj�����

���������Ifp��or�or��dividesk�thenbythefactthat

primep���mod��dividingx��y�

impliespdividesxory�we

maycancelp�togetanequationx� ��y� �

k��x��y ��withk�

notdivisibleby�����

Sincex� ��y� �

x� �

x�

x��y ��

wemusthavex� ��y� �

��x��y ���Completingsquaresweget

��x��������y �����

����whichgives�x��y��������or������

Itfollows�x�y����c�c����c�c���c��c���c��c��wherecisapositive

divisorof�������

���������RussianMathOlympiad�Isthereasequenceofnaturalnumbers

inwhicheverynaturalnumberoccursjustonceandmoreoverforany

k����������thesumofthe�rstktermsisdivisiblebyk�

Solution�Leta�

��Supposea������ak

hasbeenchosentohave

theproperty�Letnbethesmallestnaturalnumbernotyetappeared�

BytheChineseremaindertheoremthereisanintegerm

suchthat

m��a������ak�modk���andm��a������ak�n�modk����

Wecanincreasem

byalargemultipleof�k����k���toensureit

ispositiveandnotequaltoanyoneofa������ak�Letak���m

and

ak���n�Thesequenceconstructedthiswayhavetheproperty�

���������PutnamExam�LetA���andA����Forn���thenumber

An

isde�nedbyconcatenatingthedecimalexpansionsofAn��

and

An��fromlefttoright�ForexampleA��A�A�����A�A�A��

���

����A��AA��������andsoforth�DetermineallnsuchthatA

isdivisibleby���

Solution�TheFibonaccinumbersFn

isde�nedbyF����F���an

Fn

�Fn���Fn��forn���NoteAn

hasFn

digits�Sowehavet

recursionAn���Fn�

�An���An�������Fn�

�An���An���mod��

ByinductionthesequenceFn

�mod��is����������������The�r

eighttermsofAn�mod���are������������������Notethenumbe

starttorepeatafterthesixthterm��Infacttherecursionimpl

An���An�mod���byinduction�SoAnisdivisibleby��ifandon

ifn��k��forsomepositiveintegerk�

���������BulgarianMathCompetition�Ifk���showthatkdoesn

divide�k�����Usethisto�ndallprimenumberspandqsuchth

�p��qisdivisiblebypq�

Solution�Supposekj�k����forsomek���Thenkisodd�Wri

k�pe��

���perr�wherep i�saredistinctprimes�Letp i����mi

witheq iodd�Letmj�minfm������mrg�Sincemi���modmj��w

getpeii

���modmj�andsok��mjq��forsomepositiveinteg

q�Sincepj

jkandkj�k�����so��

m

jq

��k��

����modp j

Then�pj�� q����m

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����modpj�becauseq jisodd�Howeve

�pj�����modpj�byFermat�slittletheoremsincegcd���pj����

�pj�� q���modpj��acontradiction�

Supposep�qareprimeand�p��qisdivisiblebypq�Then�p

��q�modp��Ifp�qareoddthen�p���modp�byFermat�slitt

theoremand�q���p����modp��So�pq�����p����modp

Similarly�pq����modq��Then�pq������modpq��contradic

ingthe�rstpartoftheproblem�Ifp���thenq��orq���

q���then�����q����modq�byFermat�slittletheoremwhi

impliesq���Thereforethesolutionsare�p�q�������������������

����Showthatforanypositiveintegern�thereisanumberwhosedecim

representationcontainsndigitseachofwhichis�or�andwhich

divisibleby�n�

Solution�Wewillprovethatthe�nnumberswithndigitsof��s

��shavedi erentremainderswhendividedby�n�Henceoneofthem

���

Page 62: Math Problem Book I

divisibleby�n�Forn���thisisclear�Supposethisistrueforn�k�

Nowifa�bare�k���digitnumberswhereeachdigitequals�or�

anda�b�mod�k����thentheunitsdigitsofa�barethesame�If

a���a� �i�b���b��i�whereiistheunitsdigitthen�k��divides

a�b����a� �b��isequivalentto�kdividesa� �b��Sincea� �b�are

kdigitnumbers�withdigitsequal�or��wehavea��b��Soa�b�

completingtheinduction�

����Forapositiveintegern�letf�n�bethelargestintegerksuchthat�k

dividesnandg�n�bethesumofthedigitsinthebinaryrepresentation

ofn�Provethatforanypositiveintegern�

�a�f�n���n�g�n��

�b��divides

� �n n� ���n��

n�n�

ifandonlyifnisnotapowerof��

Solution��DuetoNgKaManandPoonWingChi��a�Writenin

base�as�arar�����a�� ��Then

ai��ar���ai��ai����ar���ai������

n �i� �� n �i

��

� �

So

g�n��

r X i��

ai�

r X i��

� n �i� �� n �i

��

�� �n�

r X i��

n �i� �n�f�n���

�b�LetMn���n����n����Sinceg��n��g�n��using�a�weget

f�Mn��f���n�����f�n����g�n��g��n��g�n��

Sothelargestksuchthat�kdividesMn

isk�g�n��Now�divides

Mnifandonlyifg�n����whichisequivalenttonnotbeingapower

of��

�����ProposedbyAustraliafor����IMO�Provethatforanypositivein

tegerm�thereexistanin�nitenumberofpairsofintegers�x�y�such

that

���

�a�xandyarerelativelyprime�

�b�ydividesx��m�

�c�xdividesy��m�

Solution�Note�x�y�������issuchapair�Nowif�x�y�iss u

apairwithx�y�thenconsider�y�z��wherey��m

�xz�Th

everycommondivisorofzandyisadivisorofm�andhenceofx�

gcd�z�y����Now

x��z

��m���y

��m��x

�m�y

��my

��m�x

��m�

isdivisiblebyy�Sincegcd�x�y����yjz��m�so�y�z�isanoth

suchpairwithy�y��x�z�Thiscanberepeatedin�nitelyma

times�

����Findallintegersn��suchthat�n��n������n���n

isdivisib

byn�

Solution�Foroddn��j�����since�n�k�n�kn

�modn�f

��k�j�so�n��n������n���n

isdivisiblebyn�Forevenn�wri

n��st�wheretisodd�Then�sj�n��n������n���n�Nowif

isevenandlessthann�then�sjkn�Ifkisoddandlessthann�th

byEuler�stheoremk�s��

���mod�s��sokn

���mod�s��Th

���n��n������n���n

�n �

�mod�s��whichimplies�s��jn

contradiction�Soonlyoddn��hastheproperty�

���������PutnamExam�Showthatifnisanintegergreaterthan�th

ndoesnotdivide�n���

Solution�Supposenj�n��forsomen���Since�n��isodd

nisodd�Letpbethesmallestprimedivisorofn�Thenpj�n���

�n���modp��ByFermat�slittletheorem�p�����modp��Let

bethesmallestpositiveintegersuchthat�k���modp��Thenkj

�becauseotherwisen�kq�rwith��r�kand���n���k�q�r

�r�modp��contradictingkbeingsmallest��Similarlykjp���

kjgcd�n�p����Nowd�gcd�n�p���mustbe�sincedjn�d�p�

andpisthesmallestprimedivisorofn�Sok�

�and��

�k

��modp��acontradiction�

���

Page 63: Math Problem Book I

�����ProposedbyRomaniafor����IMO�Fork���letn��n������nkbe

positiveintegerssuchthat

n�

� � ��n�

����n�

� � ��n�

��������nk

� � ��nk�

����n�

� � ��nk

����

Provethatn��n������nk���

Solution�Observethatifni��forsomei�thenni��willequal�

andthechaine ectcausesallofthemtobe��Soassumenoniis��

Letp kbethesmallestprimenumberdividingnk�Thenp kj�nk�

���

So�nk�

���modp k��Letmk

bethesmallestpositiveintegerm

suchthat�m���modp k��Thenmk

jnk��

andmk

jpk��by

Fermat�slittletheorem�Inparticular��mk

�p k���p kandso

thesmallestprimedivisorp k��ofnk��islessthanp k�Thenweget

thecontradictionthatp k�p k�������p ��p k�

���������APMO�Determinethelargestofallintegernwiththeproperty

thatnisdivisiblebyallpositiveintegersthatarelessthan�p n�

Solution��DuetoLauLapMing�Thelargestnis����Since����

�������and��

�p ����

�����hastheproperty�Nextifn

hasthepropertyandn�

����then�������dividen�Hencen�

�����������Then�������dividen�son��������������

���������Then�������������dividen�son������������

������������Letkbetheintegersuchthat��k�

�p n���k���

Then�k��k��k���k���k���k���k���kdividen�andwegetthefollowing

contradiction

n��k�k�k��k��k��k��k��k

���k�����k�����k������k������k������k���

�k���n�

���������UkrainianMathOlympiad�Findthesmallestintegernsuchthat

amonganynintegers�withpossiblerepetitions�thereexist��integers

whosesumisdivisibleby���

Solution�Takingseventeen��sandseventeen��sweseethatthe

smallestsuchintegerncannotbe��orless�Wewillshow��isthe

���

answer�Considerthestatement!amongany�k��integersthereex

kofthemwhosesumisdivisiblebyk�"Wewill�rstshowthatift

statementistruefork�k�andk��thenitistruefork�k�k��

Supposeitistruefork�k�andk��Sincethecasek�k�istru

for�k�k���integerswecantakeout�k���ofthemandpickk�

themwithsumdivisiblebyk�toformagroup�Thenreturntheoth

k���integerstotheremainingintegersandrepeatthetakingan

picking�Totallywewewillget�k���groups�Sincethecasek�

istruefromthe�k���sumss ������s�k���ofthegroupsconsideri

thenumbersdi�s i�gcd�k��k���wecangetk�ofthemwhosesum

divisiblebyk��Theunionofthek�groupswithsums i�sconsists

k�k�numberswhosesumisthendivisiblebyk�k��

To�nishtheproblemsince���

�����wehavetoshowt

statementistruefork��and��Among�������numbe

therearetwooddortwoevennumberstheirsumiseven�Amo

�������integersconsider�mod��oftheintegers�If���ea

appearsthenthesumofthosethreewillbe��mod��otherwisethe

aretwochoicesfor�integersandthreeofthemwillbecongruent�m

��theirsumis��mod���

Comments�Thestatementistrueforeverypositiveintegerk�Allw

havetoconsideristhecasek�pisprime�Suppose�p��intege

aregiven�Thereare

m�

� �p��

p

� ���p�����p�������p���

�p����

waysinpickingpofthem�Ifnopofthemhaveasumdivisibleby

thenconsider

S�

X �a������ap�p���

wherethesum

isoverallm

pickingsa������ap�ByFermat�slitt

theorem

S���������m����modp��

Ontheotherhandinexpansionthetermsae��

���a

epp

haveexpone

sume ������e p�p���Hencethenumbersofnonzeroexponents

���

Page 64: Math Problem Book I

inthetermswillbepositiveintegersj�p���Sincep�jofthee iis

�thecoe�cientoftheterminthefullexpansionofSis

� �p���j

p�j

��p��

e ������ep

� ���p���j����p����p�j���

�p�j��

�p��

e ������ep

� �

whichisdivisiblebyp�Soallcoe�cientsaredivisiblebyp�hence

S���modp��acontradiction�

PerfectSquares�PerfectCubes

����Leta�b�cbepositiveintegerssuchthat

� a�

� b�

� c�Ifthegreatest

commondivisorofa�b�cis�thenprovethata�bmustbeaperfect

square�

Solution�Byalgebra

� a�

� b�

� cisequivalentto

a�c

c

cb�c

�Sup

pose

a�c

c

cb�c

�p q�wherep�qarepositiveintegersandgcd�p�q��

��Then

ap�q

�c qand

bp�q

�c pbysimplealgebra�So

a

p�p�q�

b

q�p�q�

c pq�

Nowgcd�p�q���impliesgcd�p�p�q��q�p�q��pq����Sincegcd�a�b�c�

��wehavea�

p�p�q��b�

q�p�q�andc�

pq�Therefore

a�b��p�q���

���������E�otv�osK�ursch�akMathCompetition�Letnbeapositiveinteger�

Showthatif���p��n���isanintegerthenitisasquare�

Solution�If���p��n����m�anintegerthen����n������m�

����Thisimpliesmisevensaym��k�So��n��k���k�Thisimplies

kisevensayk��j�Then�n��j�j����Sincegcd�j�j������

eitherj�

�x��j���

y�

orj�

x��j���

�y��Intheformer

caseweget���y��mod���whichisimpossible�Inthelattercase

m��k��j��x�isasquare�

���

���������PutnamExam�Prov ethatforanyintegersa�b�c�thereexists

positiveintegernsuchthat

p n��an��bn�cisnotaninteger�

Solution�LetP�x��x��ax��bx�candn�jbj���Obser

thatP�n��P�n����mod���SupposebothP�n�andP�n���a

perfectsquares�Sinceperfectsquaresarecongruentto�or��mod�

soP�n��P�n����mod���HoweverP�n����P�n���n��

isnotdivisibleby�acontradiction�SoeitherP�n�orP�n���

notaperfectsquare�Thereforeeither

p P�n�or

p P�n���isnot

integer�

���������IMOshortlistedproblem�Letkbeapositiveinteger�Proveth

therearein�nitelymanyperfectsquaresoftheformn�k���where

isapositiveinteger�

Solution�Itsu�cestoshowthereisasequenceofpositiveintege

aksuchthata� k����mod�k�andtheak�shavenomaximum�L

a��a��a�����Fork���supposea� k

����mod�k��Th

eithera� k����mod�k���ora� k��k���mod�k����Intheform

caseletak���ak�Inthelattercaseletak���ak��k���Thensin

k��andakisodd

a� k���a

� k��kak���k���a

� k��kak�a

� k��k����mod�k

���

Sincea� k��k��forallk�thesequencehasnomaximum�

����Leta�b�cbeintegerssuchthat

a b�

b c�

c a���Provethatabcist

cubeofaninteger�

Solution�Withoutlossofgeneralitywemayassumegcd�a�b�c��

�Otherwiseifd�gcd�a�b�c��thenfora� �a�d�b� �b�d�c� �c�d�t

equationstillholdsfora� �b��c� anda� b� c� isacubeifandonlyifabcis

cube��Multiplyingbyabc�wegetanewequationa�c�b�a�c�b��ab

Ifabc����thenwearedone�Otherwiseletpbeaprimed

visorofabc�Sincegcd�a�b�c����thenewequationimpliesthat

dividesexactlytwoofa�b�c�Bysymmetrywemayassumepdivid

���

Page 65: Math Problem Book I

a�b�butnotc�Supposethelargestpowersofpdividinga�barem�n�

respectively�

Ifn��m�thenn����m

andpn��

ja�c�b�c��abc�Hence

pn��jc�b�forcingpjc�acontradiction�Ifn��m�thenn��m��and

p�m��jc�b�b�a��abc�Hencep�m��ja�c�forcingpjc�acontradiction�

Thereforen��mandabc�

Y pjabc

p�m

isacube�

DiophantineEquations

����Findallsetsofpositiveintegersx�yandzsuchthatx�y�zand

xy�yz�zx�

Solution��DuetoCheungPokMan�Since������������������

wehaveyz�zyify���Hencetheequationhasnosolutionify���

Since��x�y�theonlypossiblevaluesfor�x�y�are�����������and

������Theseleadtotheequations����z����z�zand���z�z��

Thethirdequationhasnosolutionsince�z�z�forz��and�������

isnotasolutiontoxy�yz�zx�Thesecondequationhasnosolution

eithersince�z�

z�The�rstequationleadstotheuniquesolution

��������

�����DuetoW�Sierpinskiin�����Findallpositiveintegralsolutionsof

�x��y��z�

Solution�Wewillshowthereisexactlyonesetofsolutionnamely

x�y�z���Tosimplifytheequationweconsidermodulo��We

have�����y��x��y��z�����z�mod���Itfollowsthatz

mustbeevensayz��w�Then�x��z��y���w

��y���w��y��

Now�w��yand�w��yarenotbothdivisibleby�sincetheirsum

isnotdivisibleby��So�w��y�

�xand�w��y�

��Then

����w

�����y

���mod��and����w

�����y

���mod���From

thesewegetwisoddandyiseven�Ify��then���w��y��x��

or��mod��acontradiction�Soy���Then�w��y��implies

w��andz���Finallywegetx���

���

�����DuetoEuleralso����MoscowMathOlympiad�Ifn���thenpr o

that�ncanberepresentedintheform�n��x��y�withx�yod

positiveintegers�

Solution�Afterworkingoutsolutionsforthe�rstfewcasesapatte

beginstoemerge�If�x�y�isasolutiontocasen�thenthepatte

suggeststhefollowing�If�x�y���isoddthen��x�y����j�x�yj�

shouldbeasolutionforthecasen���If�x�y���iseventh

�jx�yj�����x�y����shouldbeasolutionforthecasen���Beforew

con�rmthisweobservethatsince�x�y����jx�yj���max�x�y�

oddexactlyoneof�x�y����jx�yj��isodd�Similarlyexactlyo

of��x�y����j�x�yj��isodd�Alsoif�x�y�isasolutionandone

x�yisoddthentheotherisalsoodd�

Nowwecon�rmthepatternbyinduction�Forthecasen�

�x�y�������with���������leadstoasolution�����forca

n���Supposeincasen�wehaveasolution�x�y��If�x�y���

oddthen�� x

�y

� � �� j�x�yj

� � ���x

���y

���n

���If�x�y�

iseventhen�� jx

�yj

� � �� �x�y

� � ���x

���y

���n

���Therefo r

thepatternistrueforallcasesbyinduction�

���������IMOshortlistedproblem�Findallpositiveintegersxandysu

thatx�y��z��xyz�wherezisthegreatestcommondivisorof

andy�

Solution�Suppose�x�y�isapairofsolution�Letx�az�y�b

wherea�barepositiveintegers�andgcd�a�b��

���Theequati

impliesa�b�z�z�

�abz��Hencea�czforsomeintegercan

wehavec�b��z�cbz��whichgivesc�

b��z

bz���

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c�

b���

b��

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or������Ifz���then��c�

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thatb��or�so�x�y�������or������

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Page 66: Math Problem Book I

Ingeneralcz

��

b�z��z�

bz���

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b�z�

bz���

�Nowintegercz

��b�

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bz���

��impliesb�

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sob�z�Itfollowsthatc�

b��z

bz���

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���soc���Nowbisan

integersolutionofw��z�w�z�����Sothediscriminantz��z��

isasquare�Howeveritisbetween�z�����and�z����acontradiction�

Thereforetheonlysolutionsare�x�y�������������������and������

����Findallpositiveintegralsolutionstotheequationxy�yz�zx�

xyz���

Solution�Bysymmetrywemayassumex�y�z�Dividingboth

sidesbyxyz�weget

� z�

� y�

� x���

�xyz

�So

����

�xyz

�� z�

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� x�

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Thenx��or��Ifx���thentheequationimpliesy�z���Ifx���

then

� z�

� y�

� ��

� yz

�So

� ��

� ��

� yz

�� z�

� y�

� y�Theny���Sim

plecheckingsyieldy���z���Thereforetherequiredsolutionsare

�x�y�z����������������������������������������������������������

����Showthatiftheequationx��y����

xyzhaspositiveintegral

solutionsx�y�z�thenz���

Solution��DuetoChanKinHang�Supposetheequationhaspositive

integralsolutionsx�y�zwithz����Thenx��y�forotherwise�x����

x�zwouldgivex��z�����andsox���z����Astheequationis

symmetricinx�y�wemayassumex�y�Amongthepositiveintegral

solutions�x�y�z�withx�yandz����let�x��y��z��beasolution

withx�

leastpossible�Nowx��y �z �x��y� �����

�hasx�

as

aroot�Theotherrootisx�

�y �z ��x�

��y� �����x��Wehave

��x���y� �����x���y� ������y �����y ��Now�y��x��z��isalso

apositiveintegralsolutionwithy ��x�andz �����Howevery ��x�

contradictsx�beingleastpossible�

���

���������CzechSlovakMatch�Findallpairsofnonnegativeintegersxa n

ywhichsolvetheequationpx�yp���wherepisagivenoddprim

Solution�If�x�y�isasolutionthen

px�yp����y����yp�������y

��y���

andsoy���pn

forsomen�Ifn���then�x�y�������andpm

bearbitrary�Otherwise

px��pn���p��

�pnp�p�pnp�� �

� p �� pnp�� �����

� p p��� p

�n�p�pn�

Sincepisprimeallofthebinomialcoe�cientsaredivisiblebyp�Hen

alltermsaredivisiblebypn���andallbutthelastbypn���Therefo

thehighestpowerofpdividingtherightsideispn��andsox�n�

Wealsohave

��pnp�p�pnp�� �

� p �� pnp�� �����

� p p��� p

�n�

Forp�

��thisgives��

��n

�����n�whichimpliesn�

�an

�x�y��������Forp���

� p p��

� isnotdivisiblebyp��soeveryter

butthelatontherightisdivisiblebyp�n���whilethelasttermisno

Since�isdivisiblebyp�n���thisisacontradiction�

Thereforetheonlysolutionsare�x�y�������foralloddprim

pand�x�y�������forp���

����Findallintegersolutionsofthesystemofequations

x�y�z��

and

x��y

��z

����

Solution�Suppose�x�y�z�isasolution�Fromtheidentity

�x�y�z����x

��y

��z

�����x�y��y�z��z�x��

weget��

���z����x����y��Since��

���z�����x�

���y��Checkingthefactorizationof�weseethatthesolutionsa

�����������������������������������

���

Page 67: Math Problem Book I

SolutionstoCombinatoricsProblems

CountingMethods

���������ItalianMathematicalOlympiad�Givenanalphabetwiththree

lettersa�b�c��ndthenumberofwordsofnletterswhichcontainan

evennumberofa�s�

Solution���DuetoChaoKhekLunandNgKaWing�Foranon

negativeeveninteger�k�n�thenumberofnletterwordswith�ka�s

isCn �k�n��k�Theansweristhesumofthesenumberswhichcanbe

simpli�edto������n������n���usingbinomialexpansion�

Solution���DuetoTamSiuLung�LetSnbethenumberofnletter

wordswithevennumberofa�sandTnbethenumberofnletterwords

withoddnumberofa�s�ThenSn�Tn

��n�AmongtheSn

words

thereareTn��wordsendedinaand�Sn��wordsendedinborc�So

wegetSn�Tn����Sn���SimilarlyTn�Sn����Tn���Subtracting

thesewegetSn�Tn�Sn���Tn���SoSn�Tn�S��T��������

ThereforeSn���n������

����FindthenumberofnwordsfromthealphabetA�f�����g�ifany

twoneighborscandi erbyatmost��

Solution�Letxn

bethenumberofnwordssatisfyingthecondition�

Sox�

���x�

���Lety nbethenumberofnwordssatisfyingthe

conditionandbeginningwith���Byinterchanging�and�y nisalso

thenumberofnwordssatisfyingtheconditionandbeginningwith

���Consideringa��or�infrontofannwordwegetxn��

�xn��ynandy n���xn�y n�Solvingfory ninthe�rstequationthen

substitutingintothesecondequationwegetxn����xn���xn���

Forconveniencesetx��x���x����Sincer���r����hasroots

��

p �andx����x����wegetxn

�����

p ��n����

p ��n�

where�����

p���������

p �����Thereforexn�����

p��n���

���

p ��n������

���������RomanianMathOlympiad�LetA��A������An

bepointsona

circle�Findthenumberofpossiblecoloringsofthesepointswithp

colorsp���suchthatanytwoneighboringpointshavedistinctcolors�

���

Solution�LetCn

betheanswerfornpoints�WehaveC��p�C�

p�p���andC�

�p�p����p����Forn��pointsifA�

andA

havedi erentcolorsthenA������AncanbecoloredinCnwayswh

An��

canbecoloredinp��ways�IfA�

andAn

havethesam

colorthenA������An

canbecoloredinCn��

waysandAn��

c

becoloredinp��ways�SoCn��

��p���Cn��p���Cn��

f

n���whichcanbewrittenasCn���Cn

��p����Cn�Cn�

ThisimpliesCn���Cn

��p���n���C��C���p�p���n�Th

Cn��p���n�����n�p���forn��byinduction�

PigeonholePrinciple

���������AustrianPolishMathCompetition�Doesthesetf�����������

containasubsetAconsistingof����numberssuchthatx�Aimpl

�x��A�

Solution�LetA�bethesubsetofS�f������������gcontaining

numbersoftheform�nk�wherenisanonnegativeintegerandk

anoddpositiveinteger�ThennotwoelementsofA�haveratio��

simplecountshowsA�has����elements�Nowforeachx�A��for

asetSx�fx��xg�S�NotetheunionofallSx�scontainsS�Sobyt

pigeonholeprincipleanysubsetofShavingmorethan����elemen

mustcontainapairinsomeSx�henceofratio��Sonosubsetof��

numbersinShastheproperty�

���������PolishMathOlympiad�Supposeatrianglecanbeplacedinsi

asquareofunitareainsuchawaythatthecenterofthesquareisn

insidethetriangle�Showthatonesideofthetrianglehaslengthle

than��

Solution��DuetoToKarKeung�Throughthecentercofthesquar

drawalineL�paralleltotheclosestsideofthetriangleandaseco

lineL�perpendiculartoL�atc�ThelinesL�andL�dividethesqua

intofourcongruentquadrilaterals�Sincecisnotinsidethetriang

thetrianglecanlieinatmosttwo�adjacent�quadrilaterals�Byt

pigeonholeprincipletwooftheverticesofthetrianglemustbelong

thesamequadrilateral�Nowthefurthestdistancebetweentwopoin

���

Page 68: Math Problem Book I

inthequadrilateralisthedistancebetweentwooppositeverticeswhich

isatmost��Sothesideofthetrianglewithtwoverticeslyinginthe

samequadrilateralmusthavelengthlessthan��

����Thecellsofa���squarearecoloredwithtwocolors�Provethat

thereexistatleast��rectangleswithverticesofthesamecolorand

withsidesparalleltothesidesofthesquare�

Solution��DuetoWongChunWai�Letthecolorsbeblackandwhite�

Forarowsupposetherearekblackcellsand��kwhitecells�Then

thereareCk ��C

��k

�k���k�����pairsofcellswiththesame

color�Sothereareatleast������pairsofcellsonthesamerowwith

thesamecolor�NextthereareC� ����pairsofcolumns�Sothereare

�������combinationsofcolorandpairofcolumns�Forcombination

i��to��iftherearej ipairsinthesamecombinationthenthere

areatleastj i��rectanglesforthatcombination�Sincethesumof

thej i�sisatleast��sothereareatleast

� X i�

��ji������������

suchrectangles�

����Forn���let�nchesspiecesbeplacedatthecentersof�nsquaresof

ann�nchessboard�Showthattherearefourpiecesamongthemthat

formedtheverticesofaparallelogram�If�nisreplacedby�n���is

thestatementstilltrueingeneral�

Solution��DuetoHoWingYip�Letm

bethenumberofrowsthat

haveatleast�pieces��Theneachoftheremainingn�mrowscontains

atmost�piece��Foreachofthesemrowslocatetheleftmostsquare

thatcontainsapiece�Recordthedistances�i�e�numberofsquares�

betweenthispieceandtheotherpiecesonthesamerow�Thedistances

canonlybe��������n��becausetherearencolumns�

Sincethenumberofpiecesinthesemrowsaltogetherisatleast

�n��n�m��n�m�thereareatleast�n�m��m�ndistances

recordedaltogetherforthesemrows�Bythepigeonholeprincipleat

leasttwoofthesedistancesarethesame�Thisimpliesthereareat

leasttworowseachcontaining�piecesthatareofthesamedistance

apart�These�piecesyieldaparallelogram�

���

Forthesecondquestionplacing�n��piecesonthesquares

the�rstrowand�rstcolumnshowstherearenoparallelograms�

����Thesetf����������gispartitionedintothreesubsets�Showthat

leastoneofthesubsetscontainsthreedi erentnumbersa�b�cs u

thata�b�c�

Solution�BythepigeonholeprincipleoneofthesubsetssayX�mu

containatleast����elementssayx�

�x�

�����x���Formt

di erencesx��x��x��x������x���x�andremovex��becausea�b

aretobedi erent�ifitappearsonthelist�Ifoneoftheremaini

di erencesbelongstoX�thenwearedone�

Otherwisebythepigeonholeprincipleagainoneofthesubse

sayY���X��mustcontainatleast����elementsfromthesedi erenc

y j�xi j�x��sayy ��y ������y ��Considerthedi erencesy �

y ��y��y ������y��y �andremovey �andxi �

iftheyappearont

list�Ifoneofthesedi erencesbelongtoY�thenwearedone�Ifo

ofthemsayy j�y ��xi j�xi ����xi ��xi j��belongtoX�thenl

xi ��xi j�xi j�xi �aredi erentelementsofXand�xi j�xi ���xi ��x

andwearedone�

Thuswemayassume�ofthesedi erencesz k�y jk

�y ��belo

totheremaingingsubsetZandsayz ��z ������z ��Form

t

di erencez ��z ��z��z ��z�z ��z��z �andremovez ��yj ��xi j�

ifth

appearonthelist�Theremainingdi erencez k�z ��y jk

�y j�

xi jk

�y ij�

mustbelongtooneofX�YorZ�Asabovewegetthr

distinctelementsa�b�cinoneofX�YorZsuchthata�b�c�

InclusionExclusionPrinciple

����Letm

�n���FindthenumberofsurjectivefunctionsfromBm

f��������mgtoBn�f��������ng�

Solution�Fori���������n�letAibethesetoffunctionsf�Bm

���

Page 69: Math Problem Book I

Bn

suchthati��f��������f�m��Bytheinclusionexclusionprinciple

jA������Anj

X��i�n

jAij�

X��i�j�n

jAi�Ajj�

X��i�j�k�n

jAi�A��A�j����

�� n �� �n

���m

�� n �� �n

���m

�� n �� �n

���m

�����

ThenumberofsurjectionsfromBm

toBn

is

nm

�jA������Anj�

n X i��

����i

� n i� �n�i�m�

����LetAbeasetwith�elements�Findthemaximalnumberof�element

subsetsofA�suchthattheintersectionofanytwoofthemisnota�

elementset�

Solution�LetjSjdenotethenumberofelementsinasetS�Let

B������Bn�AbesuchthatjBij���jBi�Bjj���fori�j�������n�

Ifa�AbelongstoB������Bk�thenjBi�Bjj��fori�j�������k�

Since��jAj�jB������Bkj����k�wegetk���Fromthiswe

seethateveryelementofAisinatmost�Bi�s�Then�n�����so

n���Toshow�ispossiblejustconsider

B��f�����g�B��f�����g�B��f�����g�B�f�����g�

B��f�����g�B��f�����g�B��f�����g�B��f�����g�

�����a������HongKongChinaMathOlympiad�Studentshavetakena

testpaperineachofn�n���subjects�Itisknownthatforany

subjectexactlythreestudentsgetthebestscoreinthesubjectand

foranytwosubjectsexcatlyonestudentgetsthebestscoreinevery

oneofthesetwosubjects�Determinethesmallestnsothattheabove

conditionsimplythatexactlyonestudentgetsthebestscoreinevery

oneofthensubjects�

���

�b������AustrianPolishMathCompetition�Thereare����club

Eachhas��members�Ifeverytwoclubshaveexactlyonecomm

memberthenprovethatall����clubshaveacommonmember�

Solution��a��DuetoFanWaiTong�Fori���������n�letSi

thesetofstudentswhogetthebestscoreintheithsubject�Suppo

nobodygetsthebestscoreineveryoneofthensubjects�Letx

onestudentwhoisbestinmostnumberofsubjectssaym�m�

subjects�WithoutlossofgeneralitysupposexisinS��S������S

Fori���������m�letS� i�Sinfxg�ThenthemsetsS� iarepairwi

disjointandsoeachsharesa�distinct�commonmemberwithSm�

SinceSm��hasthreememberssom���Thismeanseachstudent

bestinatmostthreesubjects�Bytheinclusionexclusionprinciple

jS��S������Snj

X��i�n

jSij�

X��i�j�n

jSi�Sjj�

X��i�j�k�n

jSi�Sj�Skj

��n�

� n �� �

jS��S������Snj�

whichimpliesn���Thereforeifn���thenthereisatleasto

studentwhogetthebestscoreineveryoneofthensubjects�There

exactlyonesuchstudentsbecauseonlyonestudentgetsthebestsco

inapairofsubjects�

Finallywegiveanexampleofthecasen��withnobodybest

allsubjects�

S��fx��x��x�g�S��fx��x�x�g�S��fx��x�x�g�

S�fx��x��x�g�S��fx��x��x�g�S��fx��x��x�g�

S��fx��x�x�g�

�b�Letn�����andk����LetC��C������Cn

bethenclubs�F

eachmemberofC��formalistoftheindicesoftheotherclubsth

thismemberalsobelongsto�SinceC�

andanyotherclubCiha

exactlyonecommonmembertheklistsofthekmembersofC�a

���

Page 70: Math Problem Book I

disjointandtogethercontainallintegersfrom�ton�Bythepigeonhole

principleoneofthelistssayx�slistwillcontainatleastm�dn��

k

e

numbers��Thenotationmeansmistheleastintegergreaterthanor

equalto

n��

k

��

Nextwewillshowthisxisamemberofallnclubs�Supposexis

notamemberofsomeclubCi�Theneachofthem��clubsthatx

belongtowillshareadi erentmemberwithCi�otherwisetwoofthe

m��clubswillshareamemberyinCiandalsox�acontradiction��

SinceCihaskmemberssok�m���n��k

���whichimplies

k��k���n�Sincek��k���

�����

n�

�����thisisa

contradiction�Soxmustbeamemberofallnclubs�

Comments�Itisclearthatthetwoproblemsareessentiallythesame�

Asthenumberofmembersinthesetsgetslargetheinclusionexclusion

principlein�a�willbelesse ective�Theargumentinpart�b�ismore

convenientandshowsthatfornsetseachhavingkmembersandeach

pairhavingexactlyonecommonmemberifn�k��k���thenalln

setshaveacommonmember�

CombinatorialDesigns

���������ByelorussianMathOlympiad�Inthebegining��beetlesare

placedatdi erentsquaresofa���squareboard�Ineachmoveevery

beetlecreepstoahorizontalorverticaladjacentsquare�Ifnobeetle

makeseithertwohorizontalmovesortwoverticalmovesinsuccession

showthataftersomemovestherewillbeatleasttwobeetlesinthe

samesquare�

Solution��DuetoCheungPokManandYungFai�Assignanordered

pair�a�b�toeachsquarewitha�b�����������Dividethe��squares

into�types�TypeAconsistsofsquareswithbothaandboddtype

BconsistsofsquareswithbothaandbevenandtypeCconsistsof

theremainingsquares�ThenumbersofsquaresofthetypesA�Band

Care����and��respectively�

Assumenocollisionoccurs�Aftertwosuccessivemovesbeetlesin

typeAsquareswillbeintypeBsquares�Sothenumberofbeetlesin

���

typeAsquaresareatmost��atanytime�Thenthereareatmost

beetlesintypeAortypeBsquaresatanytime�Alsoafteronemov

beetlesintypeCsquareswillgototypeAortypeBsquares�Sothe

areatmost��beetlesintypeCsquaresatanytime�Hencetherea

atmost��beetlesontheboardacontradiction�

���������GreekMathOlympiad�Linesl ��l������lk

areonaplanesu

thatnotwoareparallelandnothreeareconcurrent�Showthatw

canlabeltheCk �

intersectionpointsoftheselinesbythenumbe

��������k��sothatineachofthelinesl ��l������lk

thenumbe

��������k��appearexactlyonceifandonlyifkiseven�

Solution��DuetoNgKaWing�Ifsuchlabelingexistsforaninteg

k�thenthelabel�mustoccuronceoneachlineandeachpointlabel

�liesonexactly�lines�Hencetherearek����si�e�kiseven�

Converselyifkiseventhenthefollowinglabelingworks�f

��i�j�k���givetheintersectionoflinesl iandl jthelab

i�j��wheni�j�k�thelabeli�j�kwheni�j�k�Fort

intersectionoflinesl kandl i�i���������k���givethelabel�i�

when�i�k�thelabel�i�kwhen�i�k�

Alternativelywecanmakeuseofthesymmetryofanoddnum

bersidedregularpolygontoconstructthelabelingasfollows�for

evenconsiderthek��sidedregularpolygonwiththeverticeslabel

��������k���For��i�j�k���theperpendicularbisector

thesegmentjoiningverticesiandjpassesthroughauniqueverte

givetheintersectionoflinesl iandl jthelabelofthatvertex�Fort

intersectionoflinesl kandl i�i���������k���givethelabeli�

���������TournamentsoftheTowns�Inalotterygameapersonmu

selectsixdistinctnumbersfrom������������toputonaticket�T

lotterycommiteewillthendrawsixdistinctnumbersrandomlyfro

�������������Anyticketwithnumbersnotcontaininganyoftheses

numbersisawinningticket�Showthatthereisaschemeofbuyi

�ticketsguaranteeingatleastawinningticketbut�ticketsisn

enoughtoguaranteeawinningticketingeneral�

���

Page 71: Math Problem Book I

Solution�Considerthenineticketswithnumbers

��������������

��������������

��������������

��������������������

��������������������

��������������������

��������������������

��������������������

��������������������

Forthe�rstthreeticketsiftheyarenotwinningthentwoofthe

sixnumbersdrawnmustbeamong����������Forthenextthreetick

etsiftheyarenotwinningthentwoofthesixsumbersmustbe

�������������Forthelastthreeticketsiftheyarenotwinningthen

threeofthesixnumbersmustbeamong�������������Sinceonlysix

numbersaredrawnatleastoneofthenineticketsisawinningticket�

Foranyeightticketsifonenumberappearsinthreeticketsthen

thisnumberandonenumberfromeachofthe�veremainingtickets

maybethesixnumbersdrawnresultinginnowinningtickets�

Soofthe��numbersontheeightticketswemayassume�atleast�

��appearedexactly�timessaytheyare�����������Considerthetwo

ticketswith�onthem�Theremaining��numbersonthemwillmiss

�atleast�oneofthenumbers�����������say���Now��appearsin

twoothertickets�Then����andonenumberfromeachofthefour

remainingticketsmaybethesixnumbersdrawnbythecommittee

resultinginnowinningtickets�

���������ByelorussianMathOlympiad�Bydividingeachsideofanequi

lateraltriangleinto�equalpartsthetrianglecanbedividedinto��

smallerequilateraltriangles�Abeetleisplacedoneachvertexofthese

trianglesatthesametime�Thenthebeetlesmovealongdi erentedges

withthesamespeed�Whentheygettoavertextheymustmakea

���or����turn�Provethatatsomemomenttwobeetlesmustmeet

atsomevertex�Isthestatementtrueif�isreplacedby��

Solution�Weputcoordinatesattheverticessothat�a�b��for��b�

a���correspondstothepositionof

� a b� inthePascaltriangle�First

markthevertices

������������������������������������������������������������

���

Afteronemoveifnobeetlesmeetthenthe��beetlesatthemark

verticeswillmoveto��unmarkedverticesand��otherbeetlesw

movetothemarkedvertices�Afteranothermovethese��beetlesw

beatunmarkedvertices�Sincethereareonly��unmarkedvertice

twoofthemwillmeet�

If�isreplacedby�thendividetheverticesintogroupsasfollow

f�����������������g�f�����������������g�

f�����������������������g�f�����������������g�

f�����������������������g�f�����������������������g�

Letthebeetlesineachgroupmoveinthecounterclockwisedirecti

alongtheverticesinthegroup�Thenthebeetleswillnotmeetata

moment�

Covering�ConvexHull

���������AustralianMathOlympiad�Therearenpointsgivenonapla

suchthattheareaofthetriangleformedbyevery�ofthemisatmo

��Showthatthenpointslieonorinsidesometriangleofareaatmo

�� Solution��DuetoLeeTakWing�LetthenpointsbeP��P������P

Suppose�PiPjPk

havethemaximumareaamongalltriangleswi

verticesfrom

thesenpoints�NoPlcanlieontheoppositeside

thelinethroughPiparalleltoPjPkasPjPk�otherwise�PjPkPlh

largerareathan�PiPjPk�SimilarlynoPlcanlieontheoppositesi

ofthelinethroughPjparalleltoPiPkasPiPkorontheoppositesi

ofthelinethroughPkparalleltoPiPjasPiPj�Thereforeeachoft

npointslieintheinteriororontheboundaryofthetrianglehavi

Pi�Pj�Pk

asmidpointsofitssides�Sincetheareaof�PiPjPk

is

most�sotheareaofthistriangleisatmost��

���������PutnamExam�Showthatanycontinuouscurveofunitleng

canbecoveredbyaclosedrectanglesofarea����

���

Page 72: Math Problem Book I

Solution�Placethecurvesothatitsendpointsliesonthexaxis�

Thentakethesmallestrectanglewithsidesparalleltotheaxeswhich

coversthecurve�Letitshorizontalandverticaldimensionsbeaand

b�respectively�LetP�andP�beitsendpoints�LetP��P��P��Pbe

thepointsonthecurveintheordernamedwhichlieoneoneachof

thefoursidesoftherectangle�ThepolygonallineP�P�P�P�PP�has

lengthatmostone�

Thehorizontalprojectionsofthesegmentsofthispolygonalline

adduptoatleasta�sincethelinehaspointsontheleftandright

sidesoftherectangle�Theverticalprojectionsofthesegmentsofthis

polygonallineadduptoatleast�b�sincetheendpointsareonthe

xaxisandthelinealsohaspointsonthetopandbottomsideofthe

rectangle�

Sothepolygonallinehaslengthatleast

p a���b����Bythe

AMGMinequality�ab�a���b���andsotheareaisatmost����

���������PutnamExam�LetFbea�nitecollectionofopendiscsinthe

planewhoseunioncoversasetE�Showthatthereisapairwisedisjoint

subcollectionD������Dn

inFsuchthattheunionof�D�������Dn

coversE�where�DisthediscwiththesamecenterasDbuthaving

threetimestheradius�

Solution�WeconstructsuchDi�sbythegreedyalgorithm�LetD�

beadiscoflargestradiusinF�SupposeD������Djhasbeenpicked�

ThenwepickadiscDj��

disjointfrom

eachofD������Djandhas

thelargestpossibleradius�SinceFisa�nitecollectionthealgorithm

willstopata�naldiscDn�ForxinE�supposexisnotintheunion

ofD������Dn�ThenxisinsomediscDofradiusrinF�NowD

is

notoneoftheDj�simpliesitintersectssomediscDjofradiusr j�r�

Bythetriangleinequalitythecentersisatmostr�r junitsapart�

ThenDiscontainedin�Dj�Inparticularxisin�Dj�ThereforeEis

containedintheunionof�D�������Dn�

���������IMO�Determineallintegersn��forwhichthereexistnpoints

A��A������An

intheplaneandrealnumbersr ��r������rn

satisfying

thefollowingtwoconditions�

���

�a�nothreeofthepointsA��A������An

lieonaline�

�b�foreachtriplei�j�k���i�j�k�n�thetriangleAiAjAkh

areaequaltor i�r j�r k�

Solution��DuetoHoWingYip�Forn���noteA��������A�

������A�

�������A

�������r ��r ��r ��r ����satisfyt

conditions�Nextwewillshowtherearenosolutionsforn���Suppo

thecontraryconsidertheconvexhullofA��A��A��A�A���Thisist

smallestconvexsetcontainingthe�vepoints��Therearethreecase

TriangularCase�WemayassumethepointsarenamedsoA��A��A

aretheverticesoftheconvexhullwithA�A�

insidesuchthatA

isoutside�A�A�AandAisoutside�A�A�A��Denotethearea

�XYZby�XYZ��Wegetacontradictionasfollows�

�A�AA����A�A�A����r��r �r ����r��r ��r ��

��r��r ��r ���r��r ��r ��

��A�A�A���A�A�A����A�A�A���

PentagonalCase�Wemayassumer ��

minfr��r��r��r�r�g�Dra

lineLthroughA�paralleltoA�A�Since�A�A�A��r ��r ��r

r ��r ��r ��A�A�A��A�

isonlineLoronthehalfplaneof

oppositeA��AandsimilarlyforA��SinceA��A��A�cannotallbe

L�weget�A�A�A������contradictingconvexity�

QuadrilateralCase�WemayassumeA�isinsidetheconvexhull�Fir

obsevethatr ��r ��R��r �Thisisbecause

�r��r ��r ����r��r �r ����r��r ��r ���r��r �r ��

istheareaSoftheconvexhull�So�S���r ��r ��r ��r ��Also

S��A�A�A����A�A�A����A�AA����AA�A��

���r ��r �r ��r ��r ��

Fromthelastequationwegetr ����r ��r ��r ��r �����S����

NextobservethatA��A��A�

notcollinearimpliesoneside

�A�A�A�islessthan�����ThenoneofthequadrilateralsA�A�A�A

���

Page 73: Math Problem Book I

orA�A�A�A�isconvex�Bythe�rstobservationofthiscaser ��r ��

r ��r i�wherer i�r orr ��Sincer ��r ��r ��r �wegetr ��r �or

r �SimilarlyconsideringA��A��Anotcollinearwealsogetr ��r �

orr ��Thereforethreeofthenumbersr ��r��r��r�r�arenegativebut

theareaofthecorrespondingtriangleispositiveacontradiction�

���������IMO�Determineall�nitesetsSofatleastthreepointsinthe

planewhichsatisfythefollowingcondition�foranytwodistinctpoints

AandBinS�theperpendicularbisectorofthelinesegmentABisan

axisofsymmetryofS�

Solution�Clearlynothreepointsofsuchasetiscollinear�otherwise

consideringtheperpendicularbisectorofthetwofurthestpointsofS

onthatlinewewillgetacontradiction��LetH

betheconvexhull

ofsuchasetwhichisthesmallestconvexsetcontainingS�SinceSis

�nitetheboundaryofHisapolygonwiththeverticesP��P������Pn

belongingtoS�LetPi�

Pj

ifi�j�modn��Fori���������n�

theconditiononthesetimpliesPiisontheperpendicularbisectorof

Pi��Pi���SoPi��Pi�PiPi���Consideringtheperpendicularbisector

ofPi��Pi���weseethat�Pi��PiPi����PiPi��Pi���Sotheboundary

ofHisaregularpolygon�

NexttherecannotbeanypointPofSinsidetheregularpolygon�

�ToseethisassumesuchaPexists�Placeitattheoriginandthe

furthestpointQofSfromPonthepositiverealaxis�Sincetheorigin

Pisintheinterioroftheconvexpolygonnotalltheverticescanlie

onortotherightoftheyaxis�SothereexistsavertexPj

tothe

leftoftheyaxis�SincetheperpendicularbisectorofPQisanaxisof

symmetrythemirrorimageofPjwillbeapointinSfurtherthanQ

fromP�acontradiction��SoSisthesetofverticesofsomeregular

polygon�Converselysuchasetclearlyhastherequiredproperty�

Comments�Theo�cialsolutionisshorterandgoesasfollows�Suppose

S�fX������Xngissuchaset�ConsiderthebarycenterofS�whichis

thepointGsuchthat

�� OG�

��OX���������OXn

n

���

�Thecasen��yieldsthemidpointofsegmentX�X�andthec a

n��yieldsthecentroidoftriangleX�X�X���Notethebarycent

doesnotdependontheorigin�Toseethissupposewegetapoint

usinganotheroriginO� �i�e�

���O� G�istheaverageof

���O� Xifori�������

Subtractingthetwoaveragesweget�� OG�

���O� G��

�� OO� �Adding

��� O�G

tobothsidesweget�� OG�

��OG� �soG�G� �

BytheconditiononS�afterre#ectionwithrespecttotheperpe

dicularbisectorofeverysegmentXiXj�thepointsofSarepermut

only�SoGisunchangedwhichimpliesGisoneverysuchperpe

dicularbisector�HenceGisequidistantfromallXi�s�Thereforet

Xi�sareconcyclic�ForthreeconsecutivepointsofS�sayXi�Xj�X

onthecircleconsideringtheperpendicularbisectorofsegmentXiX

wehaveXiXj�XjXk�ItfollowsthatthepointsofSarethevertic

ofaregularpolygonandtheconverseisclear�

���

Page 74: Math Problem Book I

SolutionstoMiscellaneousProblems

���������RussianMathOlympiad�Therearenseatsatamerrygoaround�

Aboytakesnrides�Betweeneachridehemovesclockwiseacertain

number�lessthann�ofplacestoanewhorse�Eachtimehemovesa

di erentnumberofplaces�Findallnforwhichtheboyendsupriding

eachhorse�

Solution�Thecasen��works�Ifn��isoddtheboy�stravel

���������n����n�n�����placesbetweenthe�rstandthelast

rides�Sincen�n�����isdivisiblebyn�hislastridewillrepeatthe�rst

horse�Ifniseventhisispossiblebymovingforward��n�����n�

������n��placescorrespondingtohorses����n���n�������n ����

���������IsraeliMathOlympiad�Twoplayersplayagameonanin�nite

boardthatconsistsof���squares�PlayerIchoosesasquareand

marksitwithanO�ThenplayerIIchoosesanothersquareandmarks

itwithX�Theyplayuntiloneoftheplayersmarksaroworacolumn

of�consecutivesquaresandthisplayerwinsthegame�Ifnoplayer

canachievethisthegameisatie�ShowthatplayerIIcanprevent

playerIfromwinning�

Solution��DuetoChaoKhekLun�Dividetheboardinto���blocks�

Thenbisecteach���blockintotwo���tilessothatforeverypair

ofblockssharingacommonedgethebisectingsegmentinonewillbe

horizontalandtheothervertical�Sinceevery�veconsecutivesquares

ontheboardcontainsatileafterplayerIchoseasquareplayerII

couldpreventplayerIfromwinningbychoosingtheothersquarein

thetile�

���������USAMO�Acalculatorisbrokensothattheonlykeysthatstill

workarethesincostansin���cos���andtan��buttons�Thedis

playinitiallyshows��Givenanypositiverationalnumberq�showthat

pressingsome�nitesequenceofbuttonswillyieldq�Assumethatthe

calculatordoesrealnumbercalculationswithin�niteprecision�All

functionsareintermsofradians� ��

Solution�Wewillshowthatallnumbersoftheform

p m�n�whe

m�narepositiveintegerscanbedisplayedbyinductiononk�m�

�Sincer�s�

p r��s��theseincludeallpositiverationals��

Fork���pressingcoswilldisplay��Supposethestateme

istrueforintegerlessthank�Observethatifxisdisplayedth

usingthefacts �

tan��ximpliescos���sin ��

������ an

tan������� ����x�Sowecandisplay��x�Thereforetodispl

p m�nwithk�

m�n�wemayassumem

n�Bytheindu

tionstepn�kimplies

p �n�m��m

canbedisplayed�Thenusi

��tan��p �n�m��m

andcos��

p m�n�wecandisplay

p m�

Thiscompletestheinduction�

���������E�otv�osK�ursch�akMathCompetition�Eachofthreeschools

attendedbyexactlynstudents�Eachstudenthasexactlyn��a

quaintancesintheothertwoschools�Provethatonecanpickthr

studentsonefromeachschoolwhoknowoneanother�Itisassum

thatacquaintanceismutual�

Solution��DuetoChanKinHang�Considerastudentwhohast

highestnumbersayk�ofacquaintancesinanotherschool�Callth

studentx�hisschoolX

andthekacquaintancesinschoolY�Sin

n���n�k�xmusthaveatleastoneacquaintancesayz�inthethi

schoolZ�NowzhasatmostkacquaintancesinschoolXandhence

hasatleast�n����kacquaintancesinschoolY�Addingthenumber

acquaintancesofxandzinschoolY�wegetk��n����k�n���

andsoxandzmusthaveacommonacquaintanceyinschoolY�

����Isthereawaytopack��������bricksintoa��������box�

Solution�Assigncoordinate�x�y�z�toeachofthecellswherex�y�z

����������Letthecell�x�y�z�begivencolorx�y�z�mod���No

each�����brickcontainsall�colorsexactlyonce�Ifthepacki

ispossiblethenthereareexactly���cellsofeachcolor�However

directcountingshowsthereare���cellsofcolor�acontradiction�

suchapackingisimpossible�

���

Page 75: Math Problem Book I

����Isitpossibletowriteapositiveintegerintoeachsquareofthe�rst

quadrantsuchthateachcolumnandeachrowcontainseverypositive

integerexactlyonce�

Solution�Yesitispossible�De�neA�����andAn���

� Bn

An

An

Bn

� �

wheretheentriesofBn

arethoseofAn

plus�n���So

A������

A��

� ��

�� �

A��

� B ��

�� C A�

����

NotethatifeverycolumnandeveryrowofAn

contain���������n��

exactlyoncetheneverycolumnandeveryrowofBn

willcontain

�n����������nexactlyonce�SoeverycolumnandeveryrowofAn��

willcontain���������nexactlyonce�Now�llthe�rstquadrantusing

theAn�s�

����Therearenidenticalcarsonacirculartrack�Amongallofthemthey

havejustenoughgasforonecartocompletealap�Showthatthereis

acarwhichcancompletealapbycollectinggasfromtheothercars

onitswayaroundthetrackintheclockwisedirection�

Solution��DuetoChanKinHang�Thecasen��isclear�Suppose

thecasen�kistrue�Forthecasen�k����rstobservethatthere

isacarAwhichcanreachthenextcarB��Ifnocarcanreachthe

nextcarthenthegasforallcarswouldnotbeenoughforcompleting

alap��LetusemptythegasofBintoAandremoveB�Thenthek

carsleftsatisfythecondition�Sothereisacarthatcancompletea

lap�Thissamecarwillalsobeabletocompletethelapcollectinggas

fromothercarswhenBisincludedbecausewhenthiscargetstocar

A�thegascollectedfromcarAwillbeenoughtogetittocarB�

���������RussianMathOlympiad�Attheverticesofacubearewritten

eightpairwisedistinctnaturalnumbersandoneachofitsedgesis

writtenthegreatestcommondivisorofthenumbersattheendpoints

���

oftheedge�Canthesumofthenumberswrittenattheverticesbet

sameasthesumofthenumberswrittenattheedges�

Solution�Observethatifa�b�thengcd�a�b��bandgcd�a�b��a�

So�gcd�a�b��a�b�Ifthesumofthevertexnumbersequalsthesu

oftheedgenumbersthenwewillhavegcd�a�b���a�b���foreve

pairofadjacentvertexnumberswhichimpliesa��borb��aatt

twoendsofeveryedge�Ateveryvertexthereare�adjacentvertice

Thea��borb��aconditionimpliestwooftheseadjacentvert

numbersmustbethesameacontradiction�

����Canthepositiveintegersbepartitionedintoin�nitelymanysubse

suchthateachsubsetisobtainedfromanyothersubsetbyaddingt

sameintegertoeachelementoftheothersubset�

Solution�Yes�LetAbethesetofpositiveintegerswhoseodddig

positions�fromtheright�arezeros�LetBbethesetofpositiveintege

whoseevendigitpositions�fromtheright�arezeros�ThenAand

arein�nitesetandthesetofpositiveintegersistheunionofa�B

fa�b�b�BgasarangesovertheelementsofA��Forexamp

����������������������B��

���������RussianMathOlympiad�Isitpossibleto�llinthecellsof

���tablewithpositiveintegersrangingfrom�to��insuchaw

thatthesumoftheelementsofevery���squareisthesame�

Solution�Place�����������������onthe�rstfourthandseven

rows�Place�����������������onthesecond�fthandeigthrow

Place�����������������onthethirdsixthandninthrows�Th

every���squarehassum���Considerthistableandits���rotatio

Foreachcell�llitwiththenumber�a�b���whereaisthenumb

inthecelloriginallyandbisthenumberinthecellafterthetable

rotatedby��� �Byinspection�to��appearsexactlyonceeachan

every���squarehassum��������������

���������GermanMathematicalOlympiad�Showthatforeverypositi

integern���thereexistsapermutationp��p������pn

of�������

suchthatp k��dividesp��p������pkfork���������n���

���

Page 76: Math Problem Book I

Solution��Thecasesn��������suggestthefollowingpermutations��

Forevenn��m�considerthepermutation

m�����m���������m�m�m�

Foroddn��m���considerthepermutation

m�����m���������m�m�m��m���

Ifk��j������j�m�then�m�����������m�j��j�m�j��If

k��j����j�m�then�m�����������m�j��j�j�m�j����

����Eachlatticepointoftheplaneislabeledbyapositiveinteger�Each

ofthesenumbersisthearithmeticmeanofitsfourneighbors�above

belowleftright��Showthatallthenumbersareequal�

Solution�Considerthesmallestnumbermlabelledatalatticepoint�

Ifthefourneighboringnumbersarea�b�c�d�then�a�b�c�d����m

anda�b�c�d�m

implya�b�c�d�m�Sinceanytwolattice

pointscanbeconnectedbyhorizontalandverticalsegmentsifone

endislabelledm�thenalongthispathallnumberswillequaltom�

Thereforeeverynumberequalsm�

���������TournamentoftheTowns�Inapartynboysandngirlsare

paired�Itisobservedthatineachpairthedi erenceinheightisless

than��cm�Showthatthedi erenceinheightofthekthtallestboy

andthekthtallestgirlisalsolessthan��cmfork���������n�

Solution��DuetoAndyLiuUniversityofAlbertaCanada�Letb ��

b ������b nbetheheightsoftheboysandg ��g ������g n

bethoseofthegirls�Supposeforsomek�jb k�g kj����Inthecase

g k�b k����wehaveg i�b j�g k�b k���for��i�kand

k�j�n�Considerthegirlsofheightg i�where��i�kandthe

boysofheightb j�wherek�j�n�Bythepigeonholeprincipletwo

ofthesen��peoplemustbepairedoriginally�Howeverforthatpair

g i�b j���contradictsthehypothesis��Thecaseb k�g k���is

handledsimilarly��Sojb k�g kj���forallk�

���

���������LeningradMathOlympiad�Onemayperformthefollowingt w

operationsonapositiveinteger�

�a�multiplyitbyanypositiveintegerand

�b�deletezerosinitsdecimalrepresentation�

ProvethatforeverypositiveintegerX�onecanperformasequence

theseoperationsthatwilltransformXtoaonedigitnumber�

Solution�BythepigeonholeprincipleatleasttwooftheX��num

bers

��������������������

z�

X��digits

havethesameremainderwhendividedbyX�Sotakingthedi eren

oftwoofthesenumberswegetanumberoftheform�����������

whichisamultipleofX�Performoperation�a�onX

togetsuch

multiple�Thenperformoperation�b�todeletethezeros�ifany��

thenewnumberhasmorethanonedigitswedothefollowingstep

���multiplyby��togetanumber�������������deletethezeroan

multiplyby�togetanumber�������������deletethezerostog

������now�������������������������������anddele

zerowegetthesingledigit��

���������IMOshortlistedproblem�Fourintegersaremarkedonacirc

Oneachstepwesimultaneouslyreplaceeachnumberbythedi eren

betweenthisnumberandnextnumberonthecircleinagivendirecti

�thatisthenumbersa�b�c�darereplacedbya�b�b�c�c�d�d�a

Isitpossibleafter����suchstepstohavenumbersa�b�c�dsuchth

thenumbersjbc�adj�jac�bdj�jab�cdjareprimes�

Solution��DuetoNgKaManandNgKaWing�Iftheinitialnumbe

area�w�b�x�c�y�d�z�thenafter�stepsthenumberswillb

a���w��x��y��z��

b���x��y��z��w��

c���y��z��w��x��

d���z��w��y��z��

Fromthatpointona�b�c�dwillalwaysbeevensojbc�adj�jac

bdj�jab�cdjwillalwaysbedivisibleby��

���

Page 77: Math Problem Book I

���������NanchangCityMathCompetition�Thereare����coinsona

table�Someareplacedwiththeheadsidesupandsomethetailsides

up�Agroupof����personswillperform

thefollowingoperations�

the�rstpersonisallowedturnoveranyonecointhesecondpersonis

allowedturnoveranytwocoins����thekthpersonisallowedturn

overanykcoins����the����thpersonisallowedtoturnoverevery

coin�Provethat

���nomatterwhichsidesofthecoinsareupinitiallythe����persons

cancomeupwithaprocedureturningallcoinsthesamesidesup

attheendoftheoperations

���intheaboveprocedurewhethertheheadorthetailsidesturned

upattheendwilldependontheinitialplacementofthecoins�

Solution��DuetoChanKinHang����Thenumber����maynot

bespecial�Soletusreplaceitbyavariablen�Thecasesn��and

�aretruebutthecasen��isfalse�whenbothcoinsareheads

upinitially��Sowesuspectthestatementistrueforoddnanddo

inductiononk�wheren��k���Thecasesk����aretrue�Suppose

thecasekistrue�Forthecasek���wehaven��k��coins�

Firstsupposeallcoinsarethesamesideupinitially�Fori�

��������k�lettheithperson#ipanyicoinsandletthe��k���i�th

person#ipstheremaining�k���icoins�Theneachcoinis#ipped

k��timesandattheendallcoinswillbethesamesideup�

Nextsupposenotallcoinsarethesamesidesupinitially�Then

thereisonecoinheadupandanothertailup�Markthesetwocoins�

Letthe�rst�k��persons#iptheother�k��coinsthesameside

upbythecasek�Thenthereareexactly�kcoinsthesamesideup

andonecoinoppositesideup�The�kthperson#ipsthe�kcoinsthe

samesideupandthe�k��stperson#ipsallcoinsandthissubcase

issolved�

Sothek��caseistrueineitherwayandtheinductionstepis

completeinparticularcasen�����istrue�

���Iftheproceduredoesnotdependontheinitialplacementthen

insomeinitialplacementsofthecoinsthecoinsmaybe#ippedwith

���

allheadsupandmayalsobe#ippedwithalltailsup�Reversingt

#ippingsontheheadsupcasewecanthengofromallcoinsheadsu

toalltailsupin���������������#ippings�Howeverforeachco

togofromheaduptotailupeachmustbe#ippedanoddnumber

timesandthe����coinsmusttotaltoanoddnumberof#ippings

contradiction�

�����ProposedbyIndiafor����IMO�Showthatthereexistsaconv

polygonof����sidessatisfyingthefollowingconditions�

�a�itssidesare��������������insomeorder�

�b�thepolygoniscircumscribableaboutacircle�

Solution�Forn��������������de�ne

xn�

� � �n��

ifn�����mod��

���

ifn���mod��

���

ifn���mod��

andan�xn�xn��withx�����x��Thesequencean

is

����������������������������������������

ConsideracirclecenteredatOwithlargeradiusrandwindapolygon

lineA�A����A����A����withlengthAiAi���aiaroundthecircle

thatthesegmentsAiAi��aretangenttothecircleatsomepoint

withAiPi�xiandPiAi���xi���ThenOA��

p x� ��r��OA���

De�ne

f�r���tan��x� r��tan��x� r������tan��x����

r

���A�OP���P����OA�������P�OP���P�OP�

������P����OP�����

Nowfiscontinuouslimr���f�r�������and

limr���f�r����Byt

intermediatevaluetheoremthereexistsrsuchthatf�r�����F

suchr�A����willcoincidewithA��resultinginthedesiredpolygon

���

Page 78: Math Problem Book I

Comments�Thekeyfactthatmakesthepolygonexistsisthatthere

isapermutationa��a������a����of������������suchthatthesystem

ofequations

x��x��a��x��x��a������x�����x��a����

havepositiverealsolutions�

����Thereare��white��black��redchipsonatable�Inonestepyou

maychoose�chipsofdi erentcolorsandreplaceeachonebyachipof

thethirdcolor�Canallchipsbecomethesamecoloraftersomesteps�

Solution�Write�a�b�c�forawhitebblackcredchips�Sofrom

�a�b�c��inonestepwecanget�a���b���c���or�a���b���c���

or�a���b���c����Observethatinall�casesthedi erence

�a�����b�����a�����b�����a�����b����a�b�mod���

Soa�b�mod��isaninvariant�Ifallchipsbecomethesamecolor

thenattheendwehave��������or��������or���������Soa�b�

��mod��attheend�Howevera�b����������mod��inthe

beginning�Sotheanswerisno�

����Thefollowingoperationsarepermittedwiththequadraticpolynomial

ax��bx�c�

�a�switchaandc

�b�replacexbyx�t�wheretisarealnumber�

Byrepeatingtheseoperationscanyoutransformx��x��intox��

x���

Solution�Considerthediscriminant$�b���ac�Afteroperation�a�

$�b���ca�b���ac�Afteroperation�b�a�x�t���b�x�t��c�

ax����at�b�x��at��bt�c�and$���at�b����a�at��bt�c��

b���ac�So$isaninvariant�Forx��x���$���Forx��x���

$���Sotheanswerisno�

����Fivenumbers���������arewrittenonablackboard�Astudentmay

eraseanytwoofthenumbersaandbontheboardandwritethe

���

numbersa�bandabreplacingthem�Ifthisoperationisperformedr

peatedlycanthenumbers����������������everappearontheboar

Solution�Observethatthenumberofmultiplesof�amongthe�

numbersontheblackboardcannotdecreaseaftereachoperation�

a�baremultiplesof�thena�b�abwillalsobemultiplesof��Ifo

ofthemisamultipleof�thenabwillalsobeamultipleof���T

numberofmultiplesof�canincreaseinonlyonewaynamelywheno

ofaorbis��mod��andtheotheris��mod���thena�b���mod

andab���mod���Nownotethereisonemultipleof�inf��������

andfourmultiplesof�inf����������������g�Sowhenthenumber

multiplesof�increasestofourthe�fthnumbermustbe��mod�

Since�����mod���so����������������canneverappearont

board�

����Nine���cellsofa�����squareareinfected�Inoneunittimet

cellswithatleast�infectedneighbors�havingacommonside�becom

infected�Cantheinfectionspreadtothewholesquare�Whatifni

isreplacedbyten�

Solution��DuetoCheungPokMan�Colortheinfectedcellsblackan

recordtheperimeteroftheblackregionateveryunittime�Ifac

hasfourthreetwoinfectedneighborsthenitwillbecomeinfect

andtheperimeterwilldecreaseby���respectivelywhenthatc

iscoloredblack�Ifacellhasoneornoinfectedneighborsthenitw

notbeinfected�Observethattheperimeteroftheblackregioncann

increase�Sinceinthebeginningtheperimeteroftheblackregion

atmost�������anda�����blackregionhasperimeter��t

infectioncannotspreadtothewholesquare�

Ifnineisreplacedbytenthenitispossibleasthetendiagon

cellswheninfectedcanspreadtothewholesquare�

���������ColombianMathOlympiad�Weplaythefollowinggamewi

anequilateraltriangleofn�n�����dollarcoins�withncoinsonea

side��Initiallyallofthecoinsareturnedheadsup�Oneachturnw

mayturnoverthreecoinswhicharemutuallyadjacent�thegoalis

���

Page 79: Math Problem Book I

makeallofthecoinsturnedtailsup�Forwhichvaluesofncanthisbe

done�

Solution�Thiscanbedoneonlyforalln�����mod���Belowby

atrianglewewillmeanthreecoinswhicharemutuallyadjacent�For

n���clearlyitcanbedoneandforn���#ipeachofthefour

triangles�Forn�����mod��andn���#ipeverytriangle�Then

thecoinsatthecornersare#ippedonce�Thecoinsonthesides�not

corners�are#ippedthreetimeseach�Soallthesecoinswillhavetails

up�Theinteriorcoinsare#ippedsixtimeseachandhaveheadsup�

Sincetheinteriorcoinshavesidelengthn���bytheinductionstep

allofthemcanbe#ippedsotohavetailsup�

Nextsupposen���mod���Colortheheadsofeachcoinred

whiteandbluesothatadjacentcoinshavedi erentcolorsandany

threecoinsinarowhavedi erentcolors�Thenthecoinsinthecorner

havethesamecolorsayred�Asimplecountshowsthatthereare

onemoreredcoinsthanwhiteorbluecoins�Sothe�oddoreven�

paritiesoftheredandwhitecoinsaredi erentinthebeginning�As

we#ipthetrianglesateachturneither�a�bothredandwhitecoins

increaseby�or�b�bothdecreaseby�or�c�oneincreasesby�and

theotherdecreasesby��Sotheparitiesoftheredandwhitecoins

staydi erent�Inthecaseallcoinsaretailsupthenumberofredand

whitecoinswouldbezeroandtheparitieswouldbethesame�Sothis

cannothappen�

���������ChineseTeamSelectionTest�Everyintegeriscoloredwithone

of���colorsandall���colorsareused�Forintervals�a�b���c�d�having

integersendpointsandsamelengthsifa�chavethesamecolorand

b�dhavethesamecolorthentheintervalsarecoloredthesameway

whichmeansa�xandc�xhavethesamecolorforx���������b�a�

Provethat�����and����havedi erentcolors�

Solution�Wewillshowthatx�yhavethesamecolorifandonlyif

x�y�mod�����whichimplies�����and����havedi erentcolors�

Letthecolorsbe�����������andletf�x�bethecolor�number�

ofx�Sinceall���colorswereusedthereisanintegermisuchthat

f�mi��ifori�������������LetM

�min�m��m������m����������

���

Considera�xedintegera�M

andanarbitrarypositiveinteg

n�Sincethereare����waysofcoloringapairofintegersatleasttw

ofthepairsa�i�a�i�n�i����������������arecoloredthesam

waywhichmeansf�a�i ���f�a�i ��andf�a�i ��n��f�a�i ��

forsomeintegersi ��i�suchthat��i ��i �������Letd�i ��

Sincethereare�nitelymanycombinationsoforderedpairs�i��d�an

nisarbitrarytherearein�nitelymanyn�ssayn��n������havingt

samei ��sandd�s�

Sincethesenk�smaybearbitrarilylargetheunionoftheinterva

�a�i ��a�i ��nk�willcontaineveryintegerx�a������Soforeve

suchx�thereisaninterval�a�i ��a�i ��nk�containingx�Sin

f�a�i ���f�a�i ��d�andf�a�i ��nk��f�a�i ��d�nk��

intervals�a�i ��a�i ��nk���a�i ��d�a�i ��d�nk�arecolor

thesameway�Inparticularf�x��f�x�d��Sof�x�hasperiod

whenx�a������Sincea�M����colorsareusedfortheintege

x�a�����andsod�����Considertheleastpossiblesuchperi

d�

Nextbythepigeonholeprincipletwooff�a�������f�a�����

�������f�a����������arethesamesayf�b��f�c�witha�����

b�c�a����������Foreveryx�a����������choosealar

integermsothatxisin�b�b�md��Sincef�b�md��f�b��f�c�

f�c�md��intervals�b�b�md���c�c�md�arecoloredthesameway�

particularf�x��f�x�c�b��Sof�x�hasperiodc�b����wh

x�a����������Sotheleastperiodoff�x�forx�a�������

mustbe����Finallysinceacanbeascloseto�aswelikefmu

haveperiod���onthesetofintegers�Sinceall���colorsareused

twoof���consecutiveintgerscanhavethesamecolor�

���