From erdos to kiev problems of olympiad caliber ross honsberger

Đây là cuốn sách tiếng anh trong bộ sưu tập "Mathematics Olympiads and Problem Solving Ebooks Collection",là loại sách giải các bài toán đố,các dạng toán học, logic,tư duy toán học.Rất thích hợp cho những người đam mê toán học và suy luận logic.

From Erdos to Kiev

Probkms of Olympiad Caliber

From Erdos to Kiev

Problems of Olympiad Caliber

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"""'d - , 10 " , " , " a o<l,_'o ""

Tht - ,'- Oftn ""<laOr " " ".-'''' ""k'" "

' ' IrunO«>I T' ' ' root" , o f - y~". _ _, 110<, "" - ,. t< , - , r - ,lot _ " "" , , _".,j ,,"'""II ' -

,." "'" bo , " N! oil< _

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I " t : _ .,_ _ "" ,_.t:oJ - , _ ,."" _

9 · e-.III.~_H_ '

'0 _ U ,,, U ~_

ll_' -" ll_' -"ll_' -"'-,ll_' -"'n t·- ",- -G.-",_ _ ,.V"' I(I« 5< w., 11 _, , _,,_ , CU ~ ll._ ') F.~.,_ t: "" """"., "" '_ _ ""D" 1l- "',

,-" "'" ' _ t:_, " ' _ 0 1 GoIb<" " I 1( ,

-<~ " '-"'" H_ _ MJ(),I".-U - · " " _ ,.•(''*'',_<1 _ " I' _5 ' ' _ , J, G« ,L"k "'''1l , )

'''' ' .~."'" ~ ~ ll_ 17 "-n'" £"1,,,""':,,,· _·_-I0h_._, ~

ll _ - ''''.1 " ,_of "

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, ,.m·I""", f"X '·"'''',*l~

Thi$ boot is dtda1ed to 1M Uk""', d 1_ "rdcss bentfaetond lhnna!iQ

-1M fOllndiug «blors d -1M probl< Ill.ioumoJ

en<.< }.{a''''-,i<o<wm

S<Y<" SoI",;o b)' Grorre E a<\<>POIIIOS I

t;; U " _ P r _ (""" ,I>< I9\lJ Inte< .,,,,,,,,1 OI)'mpod 29

" f'f<>bI<mr""" JolI,"" w.It" 119

F,om V.""", "o<><""tCom"",,''''"' '27

T Qu<>''''"' r""" ,I>< '966 N."",",IJ""'" H,l'hSdI<><>I

M '''" Com"",""'" or,I>< Po:oplo:", R<p<>l>I.: 0(0""" ,.,

From "'" 19*6Spo OI, poo<l

AOeometri<C""""",

I_bl '0,01""1 Lc>J>."'hmo

On,,-,,,Its R'JIo,-AnP:4 hdol Tn._&!<>

Two Pr-.ns f,om t"" 19U "'""rion 0Iymp d

From,II< I'IU <;.oudi> OI, " d

P.obl<rn on C1oI«l S<t>

from ,'" 1911 m.·l'<>lloII Tum Compe"'

T _p _r""" III< 19111 ti>n·Potioh totllt><moti«

C""""'",,""

An Enpp,,& !'Top<ny C""",",,"I <I Ir><ir<lc of T np

On Floots and l;cil>np

T"", _ " " from ,II< 19&1 [""""bonol OIpIIpood

On ,,, ,1< I'rop<""""

II "'opmy ofTrianp H -,"lIo A.'"nf)O"

from Ill< I'IIS 8<>1&0 So>t>nl Com """,

M U In '''''''''' OIympoad Ptobl<m r""" 1lo1

Th 1>10« S"IoO""" by 0.",., E ",lopO<lloo

TIl< _ lot " '''''luab'.

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.II , - 1I00h "p.>l>6<h<d by,II<C.<>a<!"n M"hrm.';"ol Soci<l~.••

II • ~ ,bo' ;, rpmod 01' ," kM,d or>d • pea' <m:I" '" On ,.", coni""",,, «, ," -It , All""'", nf 01' my ooI,,'''''' y p«>a<d

alone<Oft'In'N)oI lmes _ ••"''''1>1, "'""",h, hllf,he , in III< _ 0 '

collol:n"" OJ< or my O"'n ,n""'K><I. Whm • """'00ft ",he _ k of OIh<n

- , bI,~ '" ,II<" , oftt><<>PO<O'_ """""",,t ~1IC< 1 ho~

wn< I~ '0 tmy><lf , ,.", _ """"",bitfor ,heII>M

In p<a<fI100""'.

1 - Id lik to<>tmdmy"""""dunk> to [)on Alb«> fo< II;,""""sP

, , _, and _ ~"",".)'<an It" 01>0 pko ,.'hank Ill<

'''''''"<ha, Joe 8ohl<,and,hem<mbtnof"'"DoI<o> Ed"""", Boord

ro< ,II,,, _ , , , ofIII< _.pt. f1 lIy ",no ,honh d",

~, Ptd""o on<! ~1 R _ fOf '''''' ,«""""" «pen ,,.j cam.'

"''''''i ofn "••",,'" Ill"",,,, ,I>< p.b1"'r.,. pt<I«$>.

0.0'1' ",II""to ' ' My tId " n'I, ,he EdilO<·'n-O><fa{

,heG, ~ Ollk"", of,he _ tId,n "",mol QI-,

I Problem I

(n , fi ptObIm1 C<>fO<Of -,he 1911 "" , " OIl'""",d.It "" 1.- 1

byC C""""•.C to.1 M,,_mSr Uno"""y.•tIdbyJ'*'""'_.y 001

T 119lI) I'll

In , , "'" lG n of , ' ,he 71 ,.h,1< ballo.tId U" bbrko tId b<>od<<heum".bo.pol< ofl>lo<k boill.Now ,he f ,"1' "

"",,"to"" P"fonn<d _'<dl, Fino.'w<> ba'II ,hd••"n,,'O_

f""" <heum.tId II

(,) ,f, y."boIh I>Io<k "",of,,,,,,"~ pol< bocl.tIdI "' ;, 'h'., n '.(b) if "'" i,bla<~ •tid II" "' """".'M "' "'" ;, "'" bo<lc.tId 'Ml>Ia<~

""',' ,hrown ,

(,) ,f' boIh ""'''.'M' boIh,Now ,.tId.blad boll f,om ,hepol<" pII' ;'''0 ,he ",.

• - uD(Nf TO ,

n.:m_ ho , ''''<OI<,,, ,h '''''boll>"",- df -lh<

• ndonly """ pill bo<k.'hoo r.,h"';q "'" umber oftoolla in tI•• um b)' """.

E ,oaJIy lb<n II>< "'" ";11 _ 'Il< p<>IOl of <OII i~i- Ii.pboD Tho

q_ioftis "Wh.o,<010< is Illio boll?"

Solution

IIis , to _'hal,I><numt>ct""_ball>in wn olwoY'<1wIa<sbyOfl<"

<OdI {o i " (0) aCId (b~ and • po i cur «)~Si_ th< """

._ton0(10011> isi"'''''.l>Iy_"tilIn _ ••he""",lotfofbIKk bolll mUSl , l!» r.lIll1 all I 150.109 1 and perttal" O'

ll<l<I<I£b _ hkely ",,, 11 10m< ~ _ Si_ doll', koow hoi"" ,"'"

mWt1 bt IOmt ","",bolls lUIUnd _Iho um\><t orblo<k bolh , _

'hild,_., _ ~to"-"It.< " , _ , _ " lh< yin"tt>:h

11'1<.urn ofwhn loon n honF , ,lles ,," ;m"""j;"dr.n'.'onNd didn"' Ihlnk ofOOIIsi<leriIIa "Ii.in ,litt\no ~_

Tho"_lit,.h,'lti1o _ isWlC!la.Flb)' (ojand{h).ond il dmpsby

, ,und<f(,~ H-.:,10<""",0(,1>< numberol" '"«balb j'>,he10

Si_ " <>5<1 10 bqjn .,,11 md>' 11 it "' " ;"odd 'hlOU"''''''•

, "'" 1Ii«boll i.'bo ,.I1""""_n morc,h<""boll<>n~

_ bani ""'"

0 t'" 0Ih<r hond.if,,,, be, or h".bolk" , l><pnninl bocI-.

il would be """"•• H '0windupwi'"uiop tu«boll (.n _ "u'"\><Yl,

;",pl),,"1,IN: la>t boll , Id block

SlV"," SO",'TIONS V """OCO 0'ACf1.OPOUl.05

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•

(FfOIIl ,t>< '-too<ow OI""poa<I 191) 11990 l~»

n"p«>bI<rn «>D<tin>. l(I x l(Id,.<k«boa«l.,.j.",Ix t -I ,t><

um. ~«••'ll< , on ,he booed 8<&1""'"1'" ,he """" len <o,nt,.,he

"'be ~ _ " " I froIle.hl;ou, _ 0('" "",,,,,,,«Iv:- '" , rfOIIl '" '"

" ",,0<1'll< boa,d, ,,.ji.,."p~ 1Iy ,n,he _ , ""nt, "'_~

- , 'h< be;' 1 _ , roll only , ,he.,,1oJ ,,he 1>0>0«1 E",," ;,~

, ,h<rt ' " ,II ,","",-)'1 , >hull\<'ll<.9 _ '"th<

hI'" ""h 'h<~"pwo,d "CPSn«<>U1)'''' rn<tI 'll<owo>i" "" ,lin rKl

'h<" <:l "'j'<, • numb<r of 29<6&iUl.

N " ' _ ,"" "' 0(0 11y _h roc< _ _ of th<

""mix" I 2.)••, S.6,.,,,nFd I ,ho,,henumlx"onowo>i"fO<n

14<1.p',,? 1>I" -.• "_'IIa'.,, ,he"'be o<I,on.oq ,

" be, '" bot""" loa "p<>nl<d ,ho, so_,he "' "'".110

,I>< , ,<t.,n , ,ha, ",I""d,".mbe,ed hen 11,1>< nl;,

_ •• ' 10( I ~ 9 • 99 ua, ;U ""'1)' .n ,n ,. Who, ",he

p;e."",.11<l"0t<'>*~Il)o <lo:>rlylO_'h< hy 10,II< p<oblern he

r- ;"."",pi< - ,."

>i"",J'OUcan','urnback ",.h<Ief\ 0< """"<lown,,>< _td,W<>«.

1>0<""" ro« <on prin';'odf '"'ano<bet of,he_n! ;,

"" fino I<> Ih<"JPof '''''<lib<

Now, r""COOI ,.I" ,Il<,nlq<f~is "" ,10< lopof,he ",bo:••11<

OPPOS;" f_ ;.pr,n'inl'he«>rnplem<ola'l'i."1<' , _ x" ,he110"010of

<lib< 8<00 foa _ «>nor lOI> - pri""",.p _ _btt , ~'WO""""ren<><>of,II< ;n' " ~.,h< ""'" """" 0 7_>; UI<l

000 1, _ _ - y , , """', ofT _ ~ the • - •

C>«w Tho, is to , in , pUh , l", " / ,""

"""Ilk-"",."",, p'''' (l, 1_.) .~<m>l< 01001 ,II< IOQ'><IIC< rious in' 1s,

belin.inl with , ""mbet 0('110 pa; lheco ""'7 A' en<l "faU - po,,.(a 7 _ oj - "';Jh'Of"'JIIt "'" _., ,,, Of , - tho:

Tb f<>r <Urnpl<.to,II<np "loU'''' '"" lui " poi"(I 6) ,

ca """,.int"" - hcf ""'."her """' of I '" f>, 0< ~,I><

~'"mom""of ron.b<" • lor 6 Som,!>," for ooho:roompl<m<nt>.y

' fl.J).nd(l 4) So""', "m""boo _ 'hon ,h"" ,""""' pao", ,

_ Itt""01 ,""\19num""" <'n r.,1 to boo boll up ,n <dti pUn

Now,otMo",,>,,tu<polnO«'\lPl'.n<_numb«01II 99piau>In ,""

10'1"""",, pIy,nl ,ho, """" m boo'n _ boo, 01 ,""""' poln

Conaoq"""'~'"""" m boo ";l!><, '" ,h"" ,""""' pol", "nl

"",.,ninl In" """""a ,"""_hor'ho:po'" (I ,6) (2 I) (l () 001I14

_Sby_I'"'n~+S+."",byle>sdu,n I • 2+l.f"' "mum

or116+ Il • lSI m,n;"' ofl16 +6 _ lol2

" ;'.n y •.«1ciooto """firm',"" lb<o<h ." " n bl<('h

I, ldl,n'ho: , nll!>< _ ,ho,,ho:~oif<d "'~'" In fOtt

III J.Q.

II""'it ~",.bI<'hal ,be <norTT>OIl< no""". of """" of S.

( : ) _lS<716122_W1017lO<ll6OO,all r.lI in 0 mow _ of""I)' 10 ,n'<f"n'

Suppoo< 'N' a<h <0,')' in • 19C7" '9Ol1 "" M ;, • ",oJ n"",b<r of

"'"ud< "'" ,>««I,", I S _ " " , 'N''h<1< ,, I>a~b«n""f<r""~ ,,,-".n<!"".PI10 II"Itthe1"", "io 1)'2, 2IUbrna'", _ up'n

l<fO Pf<> ,•• h<n ,h ,I>< ololl til< <111' on" cO""'" "<"<O'd .917

Solulion

&0« 'Y Z" 2 '''''''In can b< d",<l<dIn odd,", "",II< , _in M.

on", ~'" ,m",,'" m,.h' b< ,. '''r.w , •• ma.y 2 ,,2 1Ubtr»"o<c<

~bI< ,nd tw>p< ,hOI "'" """",ho" 1911 ••, f< I<f\.;f~ul ,Ik

<!<>ired<0<1<_ 10 ,he. b<" _ 0"" <)' ""''J' , ,,*

S 1 II<J>nn"" In lllt boll"'" I<f\ com<l', ,hen. Oft< nil"" n "",,,., OIl'

""'" nd <o'wno.of 2 , 2 _.,." unhl " nti", I'IU " , _ subrnaU ,nII, cor""" d>sa_. Unr"" , Ita ,,;"1< on_,.tl< h<>k fi,,,

,ow.nd 1>"«>Illm of /If ,." ,",2 19017 _1 ' "ho<h r """"

''''" <!nom:! '.,,""'fIlI'h., Ihn opproad1 ,"', be ' y '" ,orr)' "'"

•• we hod hop«!

$uf<ly.t1,o••".<ho:<li " ;e",of.l"" many""","""" 1 " l mo" ,.," , 10 II< ( - ".1 on ' 001""' '",I><probl«n.no.d,lII<ull,

bm.ot -""",1cI._" ",b<,n fin,lIo,.de<om""",""' M ",h"h.•n d"",nj,"I'"

,I>< 1" l,.bma'oi<n Ir , n' ", , m'ria howe, , n' ,1><

m,p., ,.0 "" ,"odd"",n'" """" 'h.n1'1l!7.Geoo ·•001

,""'" "'UhrU' J<M!

<;"'_0"1'1>< v<T)' finl 'n'IY M _ 1,.•1>< M <an

PO""'''''''' '0'" Il'W1- ').9'JJ L ped poe«>. u,h of """'h 2.ndofIrol,h,J , J .1'111._'''-.t,.••,11"''''' no ,,,,,,,,,,,to< L· p«I poe«<<on b< run"" d,»<lrd '0'".b<I",oll xl""', ",h"h ,Ionl

_h.m> '" • """" po«< A "Iu<h ,., • 1 " J """'" "",h,to< """ I<fI

,

,

, ,

,

, ,

If, <q"_fh) • O ppcrK'0have ~ - ;,~ 00< mnark 'hal

'Il<,,1 of!!l) htn ht , Y"!

Pr <"" Iikd~ COIIoequcn<c; f(2) ~ )",

Solution

SitIO<all,heoodficimu ~ O U bI.ti,Wonofany"""n<&> " " 1Ief

f woold mak./(a) a'lt I.;",pI)'it\l.ha'01' lilt _ o f j ( ) • 0 "" ncp.~••numb< , -'lo -'" _., -'• UIi., ,II<>< - '" 1_/(4

-" /(.) _." ~,J'-'" .,.'-' " _1 I

- (x "XX ,)( ,,) ,(x '.J

_ x""(II" " <.j.'.' (_"1 'I', )J<_.'+ "" '."

TIl tach <odf ,,' _ b)o

'Il< ,omcf (.'} prool"'"of r-. Iaken t alo.im<

Aolm'Ut<lly,'hio _ ' , In lie V<f)' p<om;"nl H o _ 'hi ."k nl

p<odu< ,,, ,,'y"""'~<ly h lilt bril - ,"'"'IIa' ""'"Ib<

is no pr<r~ rm '. 0 _her." r _ ,ha•• odI ' """' In ,he

I"odu<l IIl< "",bt.of "'OI"'hef, 'h,he _ , 'hal

0,,'.-, ,.-(.,,, • S. r"'- pooi1,"' , ,.

- I'. I.

No '''' """,.ral_umben~, r, may\><"""'<oj,n,II<'Iot<du'"'

in«rvollO.1~_ 'h>', OtwOl" '"•• , 1 n_be•.' In'h"''''CfYaI ""h'ho' ,t>< u""&f\<d "'" '",be " <>.><111 ~

"" '''"' COI'lunu,,~ 01'ho: fu",""", ,m~/(,'l-l rot ""'" ~ 1><1 0

(1:_,. 1"1 do< '0 S v. K ~>&i"(1m tOlD

In'lOs ptobI<m mod '0 _ _ i' ;, paooi'ol< '" ,,,,,,;,"'",he

<n'i", o(; ( > _1.0 1,2• , )",'o'~_,j;oj<N"",_insuch ""1IN".fQt<ve<yn. II '01<&<"n (n _ 1O) • s(W I9Illa , .ineao:tI

dthe,h

Solution

LeI", pr<joOF , bu'II)'toCONlN<',he oXso'K(I",""i''''',I' be

"""",noon, , summa"",0IIf _ "'h Ill< f<>lloooi<l& """ ,.,:

1« _ t ,r ,1Ia' '",<>0:1 ~m'" bolon.to,ht at>d

1M.t, ,)<1<_,ha, t.&t>ol'btlo<I&0<1< toeao:tIof,he , ,j;1f< ,

t>s<u.'hoo(00, •• ,) pli<> ro<'unqIi<.'ho' , ""'OfIntho

, ,th ,,'Il<r or t.

LeI',bqjn "';,htho ol<d.RoMor''''''bW<: "hi<tIm,,'hol<\f",

,.~w or pon"ioni ,litinlqm:

(0' "_ (n+-19l1) (i f., ,., in, n ,he in a<MI (n 19J7) moo'

Th< r_ _ 'lIa'In 10Jl) can"," he in ,he sam< "' "'lII (n - ~) • , " 1>" " ,,,""., "';'10(• • I!ltTI ' B i , _

( _ 10••••• 1981).i'mo"be;" "h , ,h< ,hr n"",ben ( _ 50)••• ,

( + I\lllT) B~<timi"",;"" ,-. ~ os, be ""', (.+1917)iI ,'10 i".".

Tho' is ,,, lOY. r.,. """J ""'*'" •. ,II< "'''< .hl , I"~

•• (n 1931).(• • 2·1911)• mo" h.;" ,11< bo<!. 1 """",,,"".111<cnti - " " ' (0 19J1.1·19l7 ) , be.a'O&"lb<f;",1I< me _ ,

W U""'k' _ ,hil , h "",,(.),."'.1>Il$h' '1I<_

property•• _ (n_150) ""'10 , 11'<100 ' ,'hel"~ •• (. _150)

( _2 I~I , )' , 'i,<I~ in,he me bs« (i """"",I,,,.,II<_ " " " (64/; I~ -50).(64H~-~).(6U 110 - ~l• ,ll.

1<>110 ,too, In - lSOI mOO' lwa,.11< "1>,,

Now''''n )'0"mu<lII<_ n o J ,.• ,<p« I _,Ion< numb< 19J1ood 150 Th<r n',",O"''''''' 'ho'

T!><f< mol'b< 'or""", way.oflOl""lI 'h pm" t ,_ " , l Oeo<,.',

001.'"",do,pi.ol' t", ,., ,hhl< ,,, -" ,)'1W."""",_ U,,, h"'c«>110<

'''''' "h.he 'olklwl"I"'l' problem.

IK, t9l!7 11'l\lO.10Sjf

SurPO>< 'wo p(.ol""" fi A ,I>.fl 8 (.k< '""" ,," * - ,-01''''

In"""••ub!«( '0 ,11< (wo '.'"

i'l no ,nO<&<1 "'".f<0««1an ov-I upon 10m" /.,

l"l no ,.0<&<1 '""f be di,,~#or n_b<,.1",,,,,._.

n , ~'" """ ~l>o""",,1>1< to play"(lit loot,

FOf mpl< fOf limll ofL. 10.• P"" ""$h' l"O«<da, fol , ,;

A " w"h to ,<Iom,,,,,,,.,.n of I. 2 I III from play

""-'.1 J, •.~.1.1 9."n,,-.;ul>l<k

~ pl., lnol <Iomln ,nl.n.'h,nl bu' ", "><If

" ,nlJ 6 1 I •nd 9 , bIo~

" _ _ UJ>lJsTO ,'"

At<pI"" ~·"h7 (lu"n,J 6.1 ond 9~

.nd Bplay>I (1<"',.,jw J. 6 <><1 9~

Now, If ~ <me ~ ~y"""""'" ploy6.<10m;""" ''''' l Bwould";n "';,h 9: bu<A

can n,n ",,",he J.rot'h<n boOh 6.0<19<anb<pIay«!,«,h<f""" by8.nd,1'1<

"'hrr by A

Thrt<l"ot< II """,,bit for ';,hn pia)'<\' '0win.Pr., ~" ,No.r",

. yhm;' L ,""" ;" . ;n ""'tI,for

We nnp/>o>o",'ho' we 'n: "'" '«l\lit<d'0fuo<i. "';on, .""'<tYro< tM./' SImpi)' ""how'ho' ,.,.m'" "," Sine<oom<bo<I,m'" WIn ('boyan" boIh be

lll<fi"',.I\o bIt , p1ay~if can """"'ho'th< no ni,,& bot of

pia, r", B. 'h<n , mn" to< lin<or ploy'ho' an odo<>t h",h Ih b&< '"""noll<""""""'ully.<1.&1 i> '1'I<f<IS10m<w'no,,, "'01." fo< A.We

pro««! ""'''O<Ily.

Suppoo< '0 ,lit """,to" •t>rn.'ho''lorn: ;" ,.; nn;n l<lY 10< /I I th;"

~B_Id obi< <0 p"IT)'Oft,~'" mo>Y< "'""" by A. Now ,I>< "ombtr I.

beI.,.di of <>'<f)''."'p.can OIIlybeplay'" 011A",wry fi", ,.,.'f " OS

_ tol>tplaya! at 11 "'.""'I<$)' m'"b<.bI< In "",.jl< 'h,>, say by f'fplyinl ,;'h ',,,,,n,,.,,,,f,,,,,, ,"' p,J, ,n _ I ,n+1• ll.t<doe<d

by ••> """">011&) d, - 'ho' n m'£hl ""

OnIll< 0Ih<r hand n rtt< '" tqon""h.nyIn ,., "P'nL,.nd miJh'

,boos< '0 " ,h ,tiroPO""""" ''''rc<''~, u.r"""na'dy tI." fi", """'"

"""ld_I< B"';,h , ""' ,0( 1 ""''''''' 110"If'" 1<.,.fOO:

"1'0" A , , ""I<w AtqI.,wllh , Th", "'""'''I''''''IYr", 11 _ _

Il'y -_." ,"""nti.1 '0 _ , , " "" «>11"'" 'll< _ 0(i,. ow dow.r.II.• r""'" 0( , nd;' ' '';'~ro a ,,", 0(. w'••'OSbIe

tqy;, lO'h<rcold"""r"" r",A."1'-"''' il""""'cn ,,',ima """"" ,.

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A Decomposilion of a Triangle

I An Easy Warm-up

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4., _ I (~ It' lo(l II, 10

A Problem from the

Solution

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n

Nine Unused Problems from

the 1987 Inlernational Olympiad

I From the USA

lC M.,_~.1917 2711; ~mila """boll;' ' ; " ,. 1989 IJH

N<>I """'" ,.«>Mid<r alo"l"ri"l" ,W(> 'bo"'"1 on<>.

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