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From Erdos to KievProbkms of Olympiad Caliber
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-- ..... ---."lICE ~ ."U
From Erdos to KievProblems of Olympiad Caliber
Ross HonsbergerUn;".rsily"/II'"",/",,
"""""'"M.J Do#",""",.,TilE MATIlUMT1CAl ASllOXIATION Of' AMERICA
.-..... t, '_~_f ~
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_..--,_ ... _--_....._-,,-.-_--.....,.",
Tht OOt.C'''''' "'''lH[.."nC'''. ExPOSI1IO.,~ n.. """""",,,,.,..-.,.'_of n.","_,..""",. ,....._pn , I,.."'.., P r-...."' ,"' .. H, c"",,,, .. , c,,, IJ ~" ...... Y , ,.. ' ."..""""""Dob.o..,_ _ .. " «d.N!...,..,,, _ ..,..,.......''''' ... ''.......-ofl..''''''.. '''''''' _.............,.,., .._-
1 r"',,·""".·.. _ .._,..........- ...,..""'..,"1'lI< "'1 r"", "" oM In.. _ , _ ' ... """"'_ ."h.." t4••• _""of'..."""'d -., 10 " ., ".., ".. a..o
Thi$ boot. is dtda1ed to 1M Uk""', d1_ "rdcss bentfaeton d lhnna!iQ
-UOs-w .. FM"" -1MfOllndiug «blorsd 1Mprobl
Contents
S< '911 USA ~'"poad IIf"'m ,I>< 1981lnte< '966 N."",",I J""'" H,l'h SdII
M.........'''" Com"",""'" or ,I>< Po:oplo:", RI.: 0(0""" ,.,
•
From "'" 19*6 Spo...... OI,...poo
~
~;:!I"qH1'qHi!jn!jP
l~
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8~
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iii~~;r~~;~~~~~!~f;jl:51
l/,l'fj!,liiild
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t!i,!l!l,lil,i!j!lj·tttil~l~]i~
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In
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,!;;.;....
;,..!~-[-o'~g;:;g!"~':!i~j!~!l1!
• '
••.II.,-.............1I00h "p.>l>6
Seven Solutions byGeorge Evagelopoulos
A' ,he n' ,iM< (19'H)~ E ~1oo p
• .-uD(Nf TO .,...
n.:m_...ho....., ''''
SlV"," SO",'TIONS .V """OCO 0'ACf1.OPOUl.05
2. The Cube and the Ch~kerboard
,
•
(FfOIIl ,t>< '-too<um. ~« •• 'll< ......,.. on ,he booed 8
•p;e."",.11< l"0t«w. Tho, is to ..,.. in .....,. pUh ,. l", ..." /. ,"" """Ilk-"",."",, p'''' (l, 1_.) .~l< 01001 ,II< IOQ'>
,II""' it ~",.bI< 'hal ,be OIl< no""". of """" of S.
(:) _lSp< ,hOI "'" """" ,ho" 1911 ••, f< I< v. u,h of """'h 2.nd ofIrol,h, J. ,. J. . 1'111. _'''-.t,.•• ,11"'''''.... no. ,,,,,,,,,,,to
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nGURE J
C '" M ;, Ill< ,..1..of". fi'" mIrYpl.. ,lie ...... In ",< Io9l.".,....po«eI A,_
B., " ;, _ di!fiolllO _ U.I' 'M """of"""",n.. In"'1 _ -r":« A,CO""" ..««I 1, ,
a b" e d c
f g h
he ..... of I!I< m,nn 'n A,• {4~ ~ ..
,If,...
• '_"_~O~I.VN_,ko"ly "'.. r""" 110" _ 1. and .. if_............. 1.1....... at>ol
... """Id _""'1 .....1(;) .".l (:J....k,",... ~G) r",.n k"O,U. __ ,n.
•{(·j-E ..x"-·,.-.
• ..l ,II< ..10.. 01 fll) is " .... b)' L:;' _. ",2'-', S".,. ... ~ (;). 'hn1
.. (I +2)" by Ill< bo""",",' 'heo
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"6. Problem MI043 from Kvant
(1:_,. 1"1. do< '0 S. v. K...~>&i" (1m tOlDIn 'lOs ptobId ...1M. t, ,)"...." ...,,,""., .. "';'10(•• I!ltTI ..' ..... B.... i , ... _(. _ 10••••• 1981).i' mo" be;" .."h .,... .... ,h< ,hr.. n"",ben (. _ 50)••• .,.(. + I\lllT). B~
"(bl~~(~_l~1
W.ho"'.I.....y""'td'Iu' (~_ 100.• - \(I•• ~ 19)11 ,""" hold. Sonh«! In _ '00.• _ ~").•,1110 ,too, In - lSOI mOO' .lwa,.11< .."1>,,
Now ''''n. )'0" mu110.fl 8. (.k< '""" ...,,".. *- ,-01''''In"""••ub!«( '0 ,11< (wo '.'"
i'l no ,nO
" __ UJ>lJs TO .,'"A t I (1, say by f'fplyinl.,;'h ',,,,,n,,.,,,,f,,,,,, ,"' p, J, ,n _ I ,n + 1•... ll.t011&) d,...-. 'ho' n m'£hl "" .
On Ill< 0Ih
A Decomposilion of a Triangle
I. An Easy Warm-up
,r yOU ...... ""'... '0 ",bd,_. 11''''''' oq..... ;",o. "1 yOU wuI
" .-........ roKItVor'*"'> l~. " + J. " +6. " + ~. __ 1- e-q..n~1.;roo>
"TIm __.,ed .. po"dP,_lOO In C..... JoID'_oro,,,,,,,,,, of. 6«.,.,..,c>cil"", .. IUbd;.O\ ••,""",',,;011, c.._ ",'h It 01....h"'
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"_'" lid< IK"'o • ..." .. o.•1lIf.....,'"'tr«pI. A£ On(! AI'. ...k,n, aACf »00«1« Ion roaOQ.ilo",..n n... EF,,..,~ DB, DC, DE. ond OF. pi""'" ,he JI'1OOI of
,
AIME-I987
N"", 1l
•
fl,).l'•... )" ,I,I·},I-)', __ ,2-3" 1_
_ I< ~ hardl)' -.no "",'h...h,1t .....'ion,&. it ;'",".;nl, ''''" tI,., ,liefi", "umb« on "", "nnl m", b.kn'I'''' '011I""'"
1.2 +l_ll(*+ I).
"""", ,.,.. .....t""', n', """'h III'" ""'" '''' '011I)" ~ !l(* .. Il. 11""'&l(l t I) _.t' ... !> 1-3" .. JSoIl9oI.
A' ........ +"). /.)' • r .719,2.>_ 2+1, • ..d • _ Ill. • ".nll" ">qn.
0. tria;.
m .. 12H --- .. 601 .\ ..'36(2...... OSI}-IlH')(12l1) • )',J' _ )".
"Problem 2
1".., ..."'_.. tho: proI>I01 un'"~ " .. 4»p1o«d b)' On .... bluer nYmb«. Th on ,''' finop"" II...".,... numb« " po< ~'" t>
" ........oOS....,.,...11>"'''''' ''''' ..... n"",ba of' ........ 01 w,1I < 'J< ,,0< pn>bo.biti" 01.f,,,,,,,,bl
is Ii.... by
Ii ((10 .. 12k)' .. II .. 6(10 .. 12J
A Problem from the1991 AIME Examination
.jl-:-~-,----~~--~-'., ., ., 11
Cl
•Solution
In .....orlhf_"".- 'NI /(11< 1)' + oj ;,'tw: .....lh .,,1>< hl'P¢t<or. ';&h'·'""o&l< ..,11I-. (a _ IJ and ... ...._ma S.('"! ,1Ie ..mo£o UI,""'k,n, ("I, 17, I) • Pytho..,..o. tripl
eas, '0 ..nf, 'ha~ if .. is _ (like 17). , .....(... j("" -1)·Il"" +m
io. 1')1""_. lripk. _ (17,1~.1'}) .. n.. dair«l .npl
Nine Unused Problems fromthe 1987 Inlernational Olympiad
I. From the USA
lC..... M.,_~. 1917. 2711;. ~mila. """boll;' ';". ,. 1989. IJHc_. ,he ._he,N m 02 Sl6,. I 1919712.
As i' " """..... from ,he lo
• ........008 TO .r",Tho q...."'" is "Wl>o' is til< ....__0/..... ;" """ an .....,,;,;nl
"
'"'"'"'"
'" cnl (I)
n,;, q...ti,y ,,. P
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;",p1y On ."..-., ""'" ...."" ~,.ot
: .. 1-4S ~"pu ""'_
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• 1_2. ('rl .1_ 1101-100·9\1" 'iiii"iiiil )-1101.100.100
9\1 l) 61.1-).100. 1 -100.100'
·od ,ho: o~.. """,bt, c( ,.'" ,ft 0 100l4,1P' '""I'" """Id btI ~ I~ (10001 ~ 611.
Th.. ,... ''''''1'' ....,'h c( 0 tu" if> 'l>l"",,,', 1 +1(0 - 2). 11"01 UI o~.. I'
• "000 ,;,,,>OS TO .,EV.. • ",nolh~ , ... po$OU" '"",.'". For a bi".ry _./. can ....iDokIim",ly II.IC. for ...mplI1""'..i"iu~,.ch1< IMI ,to< 11""" b
S_{I.I.2,4.I.16. ... 1", __J("","III'l.i'.l'.- .1"·'_1"_1).
For .hn >
'N" ...f•• I+I+""+~" +,/FOl
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• ",'C«
•• nd ,....'"""'" woulll appel"o be I ......,...bl< ...y 10 oPl'fO'l"'!o It>< problom.A«cnl~Y.,._
'"and let '" It)' 10 """" .hal
/•• "",""'+6"'70 s I .. ...,-. flO'" (A~ ... _ "'. bound«h.b«i,.,'" r", ""h;;:; or ., .. a, .. · ,,,'•• ,.
"~ho oppel"'" ,. ""PI' .nd"''''''•. '?: 1+J1 .
f,om whi
l"I.T.f,fJ, •...•~) :!i 12 • 2.n) ...."". ..• ~:!ill • .n I"I.T..
• _T•• , •.~.'"Now. f""" ,... illd"""", "''''',...... V,~ ... hal'<
""•...• ,fiN S (1 • .n)"!,.,,'.C.C.C.""'.'0.,..R
•'em>..... old. m'" _
•Now. when ,lie " ...ben d".... ~bk.,,u'ca fot&lQf.lof4 11 4ft otd
• .-U_10.'..'-'( '('-'J ) .,>- .. ">_&.+16 ~-.'" -,- ,- -,- -, --,-- , --,-
;::--;;'1- ..
"4. From Russia
(C"'"M.,_,,-. 1'll!7. JI09; ...'«moll .. ..,."',""",....,," 191'1. 1'(l)_2.•ll) a 1, '("J_3..
Th< "'b;I«1 of ,m, pn>l>I. r...oly of ooq"'nc«
n, «0), 1'(>1"1). *(>'(.1))•... ,
in .hoctl .""h l,.
"odd noml>cf. o..M 2:
.(nJJ.If ~ is • _ of n. ~ dlrid ;"0> of~_ k• ........ ho...
• _n.The" io _.di_oI....... (",k}'.. poitofCltmon. _.dolo...,._ both 4."" k...,.=., on'ho: ",herha..... ,hey...,l boIII be .... 'IIIn ,.r,; ';'IIrnpItrne1lt&ry -. bu' t••_1< ..1". Of~II>«< i,.I"" Ih •. clrly ... ,," 'n ..... 2';';" n, am", >'fn)" n, , ..>in)" .oJ"" t>oIds r.,." a lood. (,(lJ a 2 < la"" .(aJ a) < al..........ho .. flnall}' ""',
,,_) n _,CO" 'ho n1. But"" tum ,n a o'posi" inlqa,.;!h •••$10
".".,.;, I .•nd';_ .(.) iouo ''''''' ... pI'>"",. Of < ••..rtf"",,,,,, 11.... lit< r,"1 1. 2.2.1.S;"'" pnme ..,.... of III< only.,... '0 1'" ,I>< 2', ".,,«1
• ...,.. ••_ TO OJ""
From 0 poin' 0 on U.. hypo'..,... of IJC .... nil" ,no.p< "fe.pt~"'""" DE ..01 DF.t< d",,,,... ,je anov>ly. AD ho. miNm"m .....'h ......... () .. '"Ill< fOOl or ,he .It,'odt from ..t. -til _ ,,;,,., ,he problem run, iI.
In ,lOs ...mp!o.., ...t>m .. " .. """,.." II"""",•. ,he""""""'''''' btl...... £F and AD .."OO '1",1< 0 ••_ a.""..... In lhe ...mpl
•If 0;' _ .11owed '" mo.. "" BC hclo.......or'$hcHI1d "",_,«
...,.. ..005 TO "ttY
Solution
C1."'ly ,be ollma _fi&un'ioo may be a".ph in ...hi ....' of,be
•1_.jfthi> is. "".""urncaso~ ,hu("; ') 0II";. I
• ,_ ....... 'O~I~V
•Cb,ly'........ ,ym""''Y'''''~I''I'''~.I) • 0(0. S)..... "'•. 21 • ",•. 41. W;,~a' • I. 'J • lHnd., • U. I ..., ol..ody ""'1'>'.10""1 m)'$< ."h"" obl.l'n 0, • S. ond. lor' « "'...,II< ..nllt f,,",,~"
.... I )10 '''''' t (I - (- W) l"'~_
whn. ,II< "i...... t>..."l
• ._..005 TO m;"
(il Choo< ,h,,,,1 ell will ,.b«nd... UIfI< of IlIO' - (x + y) at """"";"t of """-,,,,,,. 8y 'h< chool Be. ,..«>n
"S,,,,,larIy.' ~;"IO A oft O,o, .•nd we bo..
tI. '0,0,." .00,0,.
4"0,c.240,0,o,eu•. ,n 11:, •,he .np .00,Cen nd'n'..... '. ,,_..milo, ,...O&It.«n.m. 1'Il!1.)(I9, .....I.. I
"
/lk{h,ninlo, 1"''«'"'''' _term .......-..h>ch.1I ,hel
fiN"""" ll.l.. . 1917).•no! 10 """"
Two Problems from the 1988USA Olympiad
Prob...... I
I.< ZO labob lo" -.I 1m " .
Pt~•• ,h., no ....n'" how'..... "bol. m'I'" bo ....If'Od. 'ho:......., •..m•• IO-blo:m. I f~.'" m""lf ..-.JII,nl ba,......,.1It d' ........ ,...,..." "Pl'I'oocllt., r..I" '" ",.....,"'" T 1Ii .....1y by wI»l""""1 fo<;I'l
• __ ........ 'O.,.V~'\I ,ndOK
Siocc A and 0
Solulion
50"", I" 'ho io:rn,.. of "ABC. 'ho 1o>e
From the1988 International Olympiad
(6...,1< ;noq...h,)
" ,LX.l ~i..,
Q • union or d......' ,",,"'.... ,I>< >11m or..- I
•
0.,- ••,·t .......
11\... "-, • • •
\ .. ".,..u~[ II
tpea'..... " Ih< 'nks
•nd ,11< ...... or ,he .-...911.2 .... 101 •I _(1.1+ +10),-;:(1+2+_ +701
• 10 71.(1+2+ +7OHj:-,-
.(1 .. 2+ -+101+2111
.(1.1+ .. , +70) + .9U.a._reel.
..
A Geometric Gem ofDuane DeTemple
T1I< 1>."r.1 ....1' of ,.... 1«
• ._ uoOlS 10 ~II;V."",n:l,,,.""'h ",..I> A.6 ."'.....1.,.. ond .1,.,...1>« K,. TO" K" Ie, 'n: ,he r"",...""''''.. of ABCD. """" 'IIot K,K,K,Ko ;, .,_,. "
_ ,_« K,K,K,I:,"
A Kie" Olympiad Problem
1__ ~h'oo_'IIo,..-.,.ltin_.. p '10",",,",_01lop, _,. North II I••~
•Solution
I. J
•
. ,nGUOE 7l'
..I>
•
••
ncuu ..
Clearly.'hen,_hof''''' .......'.... d•• d,.d, ;.""' """*"110
Some Student Favorites
Tho pr-.n. io lhi>....-bon 'r'" lOf"h< III< 10m<.umt>
",."'" ,..._ TO ....
S.N«:I''''' 0',• .-pI • '< 'l
"SoIl1lion
!klln.,nl ",'h eo. any of the G, con b< obu.>ncd by oddonl ,he d,rr"'n«<
""~" ,0(,."
.....,.",.", •...•
1.,.. "... "I .
1.
"+1">" oj + I. ond '''''' .Iw:rqua..(ul ., ;, '" "~. '" ."", to< On , .... """,btt. Il~..r, .11....
•SoIulion
The ~ rooIS of It' • 1 ... 1......' •.. , ..... ' .........
1. _, 1•....
"n.: '",oel< of _ in ,I>< ,,-pl<
_on I aocl I - .... ~ "'-" In Fit"'" XI. By .1>< ..... or """"... 'M_, of 1 - .... IS .....,. by
11_.11'_1' +1'_1< po.. (i.11Add,nl ..... on -.. pai". ,I>< 1'''.0 ,,,,.1 r: s .. _" by
Es .. L(-- 1)'(1 _Jl' ,","
(Of .o
• n""' .._TO ......
.. ~ [21
Four Unused Problems from the1988 International Olympiad
I. From Bulgaria
(C..... MOI"'-I""'.... I'IQ. n~)
~ Ih< .....ol, ,f w , If " d;""bIo: "" l'_
Solution
I 'm no< ...11 "'.. ,hat ,tit """""''' p««
•~I. ~ ... ...,."ioo "'~ '" ""'" ,II< poobI ....,.
th.. " Ii.... '" b< 'NO rot ~ • 1. 00
("I • 2, 2,6.'-,)f..ll.n....1_ .... _ ................. _Jorlo.:
..-.. -,., .-' -r··... - .-11 ••-,
- -'-1(..- ..... ') -(r"" r-·•.-.-. , .
O"_D .. + __ .. _I. .., .
''OJ; - II .. /l.IJ~ .... -(0 -I!J.
111...... o;mplr
.,. .... -tl~.~(o.• r)... -II 0-"
..... ,II<
•F""" .hi/>, - 2b..., + ~, _ '" + U _ 2(2< + dJ,which .. ...., ,..... on odd """''''''' 1M""""......... foll_ ,10o
_inl'" .... mel"" hyPO'h".... -..or. 1'" d,vK:I,. Ind"",,,,,,.
1" d,_... IrandMllyl( 1" d,_•.
•
:f#R. illi:ja1 .. ·1 ..·-C- .. ·121 .. ·1
2. Proposed by East Germany
(e"", M.,--, 19U.1171Tholock on ...Itn>isl.of,h......- A,II. and C._h ofwh,,~mo~ "" ...m~' ,j;1r",n' posi......., 011
If'h< corm" combo..,,,,,, isla, ~.4 fo, .._ .,11. and C. rIlIbi till.. num...... ,he poaeon_ prJ"""" ,mpI;os ,ha, ri'''''.til oom< ,_ of ,he "umb
•It ''''''' " ... 1No' 'lid .. Quite cuy '" .n...... folio....
Sine< w< _nl ,ho """,bm (1.2.3,41 .horouJIII, ""..,.. up. 11_,.Th
•
II!
IIi
. " q"'~1!lJ!"'fq!~ g'~!liif~~i~ ~] j.o~ ~l~"'~~l!1!~~ti .Iii E j~£ i~I"L~,;;.::!>,.H U j_0 -~ l;~'S l,- ~~ .~~ f~;~~H~:~l !;:::j" ~~I "!"il!~'! E.!,:i>. !"'~~f.ll e~~.l! ;'ili."f:i; ill ~~,~ ~!:H :j ... ~ '"~s ~;jl,:h;j~l :i~ ~~ ~-.l! c8'i1c_~e~. ..;_~z. _~~ "&> •]j l;"~l!:jl;d.l!l!'I," ''', 'h", 'j'jl~= ~8.l!~f~ -.:~"'h l ct"1 !i~lH!;;l~l' I, I:: I:'I 'il"I",- 1j;!;l
".'''l;'''~~, - ~,E .... ,.:-, 1~7i Jl~.Ai:' ]t ~l'j.l'! :~1i,:~"",8;;'.l! ,.l'! ~ ~~" -!"'- -,,,,,0=;1 "j"~ JlE E-.~..:,.:g .. :l!", '=8.n.. SLI~Hrn~J~ I ~ I 1.5lli H;i!; i ha1
i·1 -11·1·1
•
•,•
~
• __ .._ro.,...Thi. 0f&U"'C'1' 1'>
3. Proposer Unspecifiro
(C"'. 1n',,_,.... tIIo, .... d,....... bel A, .... A, «>ulol _.>«l .yni«ly by ~".I'lIeoo-l\o
•
, • ,ncnE JJ
nom"'.. 1.l.).'.~.6,1,.nd .11""noe«. of T. "".,." "" ,.... """'plemrn" of'll
•lb" )'O$OO'l'lpml., .11 11.... ..-""__«I by ,II<..r,,~ ,~I
•Solulloll
.-.._>0.,....
Tho __ or , ".Ino < < _,......"he...""n~,_ur In Ill< po
00UIt IJI'l ;, lah. "". ; La ...... on< _ ... o(,...k. It ,,
"the "mo-u....o
From the 1988 AIME Examination
y ....fIt< .... I .'" .."".... " ,tit ;"Jf\
•Probl
"Soluli«l ., U..."
Eot4 0, if. _ 6 .. '""t
•Sol"... r..., .... ptoot
An Unused Bulgarian Problemon the Medial Triangleand the Gergonne Triangle
The ,....",.. ..K. "'~'''I ,I>< If "''''''01.'''-
4 ... 6. ..·, 6.".4". 4e_4e'
], ...n ,mp",..'" P'op
,~
.he bite
'""hholo&h IA /JC. AO
'" ._..OOsTO.'..
1M ... lOW .bove,ho"b< .."",0< of a'" In.1I< """-"'I , ....0ll< .. "...lIdlO'h< bos«,oro( 4 ....ncI'ht..r.... ,""110_"..0( t. "·m '" [",'be"" ..1)'p
Two Solutions by John Morvayfrom the 1982 uningradHigh School Olympiad
Ie..... M,,_"-. 1911. 101)Ihn M......y II (,.... Dalla>, T.,o~ olld ... ,""~ hool ma"y ""'1lOf'ldOn& """lKO'lllpobIl>!I« .. ,,",, , .... ""1"""" io complel
o.and rOIl' o< It""" 10k." ""l"'hlolor "" _ fo<
'Il< ",Ilh.omn>"«P' roo lOy II of'hem whl< ""'"1'",,,1(""" 01U" _hoI< 01> """u" ,10< lOU' «,,,n, could noI''''''''.''0"''''' 'han k••• "",,"ado«1ioI\o'n"' ....a t
i,~~~ II n \(I - li'-ll:lf - ..;I • ~, _,in~
d- ! ~"91~ !:(~ lit;:;, i;i~:' I in! tg>
~ 2.':"1'".!::< ~-~~~ C~. ,..,~!. ~il"l'!!:.
p~- I go:. e: r"a 0 !J!.I ~ ;I"i .... . ~;::l ~!;'!!. ~ · i'l· ='" '- ,,&.- ~ 2. ~~~ · - - •" ~! H ~ ~! e ~~~ ~ i~~i~ c ~3,. 1·'· ~" • i i ~ ~ ~ l:!~.. ~ .. '- ~ ~ = c" ~~.it ~l~ ~I~ii • · ~ ~ ~ - '"~ 15 ~il.l" i,-=. ~i,"i, g• ..~ w T.~. ;;-1 8 a;~ ~.~~ f;j~ .-0 s:· •i- -I. I" .. ~'!l " ! ..;: 1iH2. ~ i"" o' ~ t-i! .. !;I!-i1~ ~~; .~ "i~~ !r-e-i, :~;. ~J~ =: ~;!.- '~I !Ii . I l- ~ ~ i ~ '-1
~ ~;< _2.51 i2._
,.Solution
l>ooIi,,"" _.,.,;, _hI"' _ or l>:h1"Now. ;,',_ «.l11)'! 'ha, ....... in' led in_ it". I -:l!- Sin« 'h",,1es<
thon ~
,.Thtrtf""'.odd,n&( I _ {r,/l'll,fl,ol"" don_."" ,...,."...1'1...bI
". fll
'""""" ....bi
1...11Io......' "'1 ",~,,,,' ......J" to
'" ..""' .......m.''''',10< l, l . .u>VTtI .. • 110. nJhI.np.
n- po..IIeIs_',........" ... ",;"l , I ,opo,"""'" h..Su•• folloo¢Sf SI' ln·v,·i"
,L'XS-,~,
ncuu .,
'"CoN
'"...,.. ....... ro.,...
beaI".. tho 1>0...nll " 1/ 'M E.t< equal (,.... o..~ To... 1M.'"n< ""Il< A (. qua".,.,,,," .bo., II _101 matt 'lI< onns.... d........ at /I "'f'«',,,,'y parallel '" til< IHla Ae.nd A.6).Th.. 0 ,,,""_"",..orIIEo.nd ..."" 0 .. ~ po ,...lto 011 •In £>011#: ,I><
..... DE" < I"Io< ABC IS mpktfd oS .....0«1 ,boot,
From the 1987 Spanish Olympiad
fOf
.··'-b+ 1_('_1)('''' + ,.+---+~~ I)
(..... 'lI< _1 .. ,...
'" 'OOM IJtDOs TO .,nII .. ou'.•nd
,·" ..y" .. ---.y a. w."mnotd'olhow.t>o-..... '..." _ Iii.1>0 "",....bIe. 'ho' ... ,"", .• _< _ equo,;o" we pol
l_(" .. 2)< fin, "1"';0" .. jus< ,
"+I- le.+ I - O,
'(' ) ..,
'"
( "')"" I11-+1) -;j";"'i'
10+1)("")'" _r'.". ,
(,. _,_)M'. l"""+1 .+rNow. II d .....' k""""" 'ha' ,lit I< tro.. of ,I><...,,,..., I¢prn"""'- It>ll 'ht.er_
-"•• 1< J,""o>Ipn, 2'"
'"no. ,,,,,,1>1< ";,h ,hi. is tho' n
A Problem from Johann Walter
w.... ;ndob«d ,o"""'n" w.ll... (I"""y, for "'.._,'1;......"""Germany)rot ,,,. (01 dII' ...tUntrod""",",
11w Siml*t:l Solulion
In ,... ...,." ,..., 10m< odd ~,;.. ;".......
•• 2.0+1. ,.l/I,I. ,.20' .. 1...... '0 ""P, ,,,. ....n l'yt.....,...n _'inn...., uId ha ..
,.,. Most Eltganl SolulMm
(d... to O. Knq>a&O
'"Fo< (P.f.') • Ix .. ~.X .. .y••J>d:.re an odd. ,hr.... : _ld be C"al, mallna
l("" -"» an '&0""" 'pin." _'ac< lhr ".'hot I..."., rUl M.._"k.T
From the 1987 Balkan Olympiad
Proposed by Bulgaria
Two
'" ..""" ....... ro.,...OWOSO', , .1>AO£. ond,~-
AM.~_00,
11Ibl
,_,HO ,,., 0AU
From VariousKurschlik Competitions
J. From lhe 1982 Competition
$ul'PO"'.""h 01'.... ,nl
,.Solution
.."., ••1>00; TO "ltV
Pa.. I
let. pOi, or ;n"'..... Iuol
'"it"l""""y " ....._ of ,II< d,"'"", " m"" -.r Infi..'lot> ;0 """"II, u>«l. lei an '."p of n
OM ._..DOli TO "'"
1- 'I"~- I
" . . . ',.,. ," " ,• I I , I.-. _. ...
,
u, ... ,
•" " "
,hcf< ",jo< t"" _h""""ricpoi.. (I, ,> and (., .),.oclI wi, h""""",h""""tio:4isuncc ~,II,. on< "" ean .Iu, -..' In 1~.J1 lon. Th.. 11m... _.-;••n u..«>lon,.od._ III l
'"2. From the 1983 Competition
Tho • fM'O"" 1',.,.". •P• ........... poIn' Q.... ,.~n In ,"" pi,,,,,,. If ,I>n<""","Of. ~1 l«I.. ,hat fo,
(i) t>ct_n 'w-.mdi", bl.. ,art<(ii) t>ct-. a bl.. ray on. 11...."~- """ is 6
on
(An oll', '" "
Si""" tl1< ""ul< ,lid"",,,,, hypol _ '""'" ...,01'«1""'".Ih< .....,(n' l/-p .. '0 • ."......._ C/. Il ldb< .... ir""""d..JONI...... '0
.fOUnd Q. 10 ",lab
,.tqood tbe cl
..~. -.~-...~. -.~.\L_~·h.- ~ - ~-,~ ~"-,', -,-, -,-, -,.,
• '-'--__ r ,r
flGUR[ "
8>' ;ndUCl_. tire....... «10>0I.... ,ha, ,Ire '"'lOlled "_w'", io 01",*)'$pOIOibk.
4. Probltm 2
l.
,.Solution
._ 010$ 1'0 .,...
0... ...- ~"""" _ d,l6-w_l'''+2_1(2'+1).
_ ....IIdII •• odd rO 2'" +-1,
and It>< """,_ foilowu'fCady.
I,"._ .....y to ... ,hat w" 2p. ,.n odd 1"''''. ,""'-tl>< J't"'...._, 1'..._d.oa _",oood n...... be_I«,""",.
2'" ,_
Si""',...._"'",...."....1, " ,., .. ".. implyin.''''''''''........_or, ..1I>dl
1'.-, "
",-">q, -P< ,.11«,_. "'PI' ,nan&!<AK ,n ,"" ,......" ...P XB
,. nOM uo(oJ TO lin'0'.xrz • l:L46C ... 6XBC -+ l>. YAC -+ 6ZAB± 6A yz ± t>.,ZX ± t>.CXY
• 4.l> "''''An ±l>BZK±l>CXY•
........_hol'h< .... th_'.............,_10 be.- or,ulltfO 01 tho:_ ",,
~~l ..n A _ ~oin' A) .1«1) 11- • sin' 81• i""'ll""C - 'q > _6.
~ - '/l,;"' A • lll. _ '6 .."'8. U. _~ ..n'C > -6.106 > ~(..n' A+ ..n' B. Nn' CI
~>""''' ...n'II ....,C..n ,nll'nl ,IIoII0' B_ II,___',-_'..1-_'•• 1• (I __, n -
,.To fi..... n.. \all< we _ ,,, "- .11>, ,
1+l, ..... up'" IIIO'.S _,_
Two Questions from the 1986National Junior High SchoolMathematics Competition of thePeople's Republic of China
CC"'-' 1>1","'-'"",__ 19U. 1)(11
Qunloono 0' ' ... j\o..... "'ch ",hool ""'.fKD".-....
k (;);, '..... (;;);, f.ll
,.• ,. ,., - '-• , , ,
• , , ,, , • ,• , , ,
flCU.[ ..
.- nDOs TO 't'"
F.- ,to. ..bit it os dea, eq...' ......001 ,fl;.)_.. '...., to "OC"«0
....THEMATICS COMnnr.". Of 1H[""'OIX Of CH_ ,.s..,......." (.) ....... II.., f>BCD .. is< under .11 oo"d"KIftL Th«O«Gfo
,.i
from the 1986 Spanish Olympiad
4""'-' ....__• 1931. 61)
Suppos< II>. ......'" or n.. ..,., or ....,•.,"""l!< AIIC ... "'''pm. l.<...,...,.. ,,10..10 p" ,he CCol, ;0.., .......-
•
SoIulion
," ,.. , ""","'"" liven '" e>. _,_. '990. 17 IThe fi,...h,"'lto ""' ",ho, ,~ hul< , ..p. 011.1>. .._ size...._I~ \l>.48(' [0'a.",pit. """ion C'(" to l>.4 tIC'••nd .._ ,he «'11' rood "'...... """'ltl.""_.
1 2 I I•• Jl>C.4r • J 'jl>AtIC' _jl>AtIC'
Sinu""~ fo'}'.nd ,. Hmcc "" ..... ""'r_ 'ho' .• ;,."....., '"«II
,.
F1Cl.'lE'1
__ l;Jloo)s TO .....
N-. ~ is f.my .....r>tty k..,..~ ••old I p"".'" ""II kf>OWll 10 IrmpiadC
A Geometric Construction
(PII'" C,.... M"'_,~. 19.... l1JHn. ;, • kwdtho..,....,..,'~ ... '0 G
,. ..."'" u_ TO .,..Clt""",F "ny
In "&h
,.•
,I----f---'.,-----
An InequalityInvolving Logarithms
(PIIlJ. C.... M",Iw:....",
'"em".
(Ioc"b
On Isosceles Right-angledPedal Triangles
lrI2l~. C,..., "'",Iw""JI~_. 1911, ,:Ill)
If """",nd"",i>" .... d..wo '0 ,he »
'"Solution
w. "",n "'" '''empl to ..,.'" .h., probltm o:ompietrdio",: ""-"'"01 to FiEll" 11.;r P;, tilt inl
,.
;~ .• in Fi&u
,.•
nCuRE 70(0}
•
•L-,!- •
lb..i' .. really. oimpl< ....tlef to Ioco« I' at .he in_ion of the ala K•...:1 L, ....... ;,c..... tha' P .. the ......Y"'"n'i_.....8(" .._ pod.I,"".'"D£f "on _I«n&l>' 'no.V< wtuAIIC ..Nlnn.
Thd;' .. rat .........111>"".. It>e ....".,.
'"If P os .... poOn' iMide ,he .riaft&I< ..hoA Be. aDd ,11"_",0
Two Problems from the 1987Austrian Olympiad
""""" ,
(,_I)(x+2).0,
,;.;". ,II< ,,,.,.. 1 _2 E..,je""y. ,ben. >' + .• _ 2 ,.. "...."opoly"","",'.."h '""g'(x.l)_(.,.')_O.
(>'_1)(.
,.Solurlon
,-,"-ro.,...
Thu ..... on< o(,~"" p«>bItm$ t'" «10'1'''''.'' in ,1m oIyropIod ...,'" p.m •""",0( '!lIoun '"_. While" _0', ..'1'"'< ... '0 "om ",," many of'h< , .11 ,he _~'" 1, .,.•,. . m", b< _0_ Tho, .. '" ..y.....(.) ......._ of -... of. " """''' pOlynOm.., 011 o( ..boo< ,_~"'f
'"Ik ,be .", ... in« 'lid i...... Ihot _ c( ,II< ""ft'n"..• ,•., , "'Ol' b< .., 1 or _ I •• 'ho , ,»d.., ..lt odd,
In 'Iu> """. " 1.' ,I>< IW"l of ,lit ".,,, ;s - 0,:
.,+., ... _- ..... -.,-L
"' .. _I +~_b_1 _~-li-1.
••'b) _ (~- 1]'(.• + 1)""'[. -1* - li- 111
Sin«;, ,..I••» alOod ".. 10"" 0
,. ....,.. .0_ 'fO .,....l .. ); A..i"..... ddin...... r"""""I, b< 1.•0l
'"AUhoc r""' .Tho,.{""_,,,,,,,hodt ,"'. P.(") ..."",, J •.r + ., _ I, ,J + ,r _.• _l.ot On)-. 0( • " ..... """ 0( ,I>«
'".......... ,
__ •• _ TO OJf;>'
(An .1",,,,,,,;,,, """'"'" "CO""" j" C"", Mo,__ 1919,260.)
n". "'obIn« x,x,··· .. In .._ U< "'" m.'-'otal ,...."..,... 1ow ""l"""'"oIlrnCOh" nuJhl be ""'''Cd f""" 'ho.1
,.4ropp«1. TIM'''' """,ton ol"l ISIh< I«i"""'" f'1m l pI"""".~"" I
,. .-........ mu..E>p.ood;,,* ohio "",""""," ;"10 ~. ",""..If_ .... Qbu;n
I_I -l 8.--.--(I-bl(l ...) I-to I ... 'I _J. A(l .. x) .. 11(1 - hj.
for ,. _I .... ~ _,¥
From the 1988 CanadianOlympiad (slightly revised)
(e_ M"' • 1m. 163)
L< ..... '" pooiU.. ;11" _ ..,'"nJ ....... ,Wmeml>n< I""OUKO '" l(I d>< ..I .a9;
...(SI • ...(... .., •. , ) _ ll.,."'! .., ) • '9_
$olUrioJI
f""" ~",h
L,o(A).(.,+..,+ ...••• )+(.,..,+··· ....-, •• )+- +(.,.., - ...)• (2' _ Ijno{S).
Nw.;" "'''rm''';OI 'ho r"" 'ho ad (.,. ",,, ......,,1 oh< ....L ,0(,0) would be 'ho old E ,o(A) , >
,.Inl
,~
....p bill "'"~< .. ,tunk of,he >lan< Ib«>ty of"l.""""to .. 4, H' .. e,)(... ..,)
.. ...(1 ••,)(1 • ..,Hl +.,1"'" 9' .. 91
Sin« ,I>< ,,""'" _poOl""" of 92" I 2 2), ,II< OfIly 001.'"", on _"..,nits
A Problem on Closed Sets
SuWOO< _10 pol.' "" II.. al'< at
on
m""pn 'W b1.."""'t>."""";01 'ho' .8
from the 1987 Austrian-PolishTeam Competition
Polo"'" "'"P''' ,,""h ..... "....... bKh...fd... forward collrdpoI~ fo.....mplo IS.II9lI.•nd )(l103. I~ Ihi$ poblI, N Oft ,nfi,,"y 01 m
'"FOf
11. .. IS'JlISI IEN~=-~ ~-~
92' 915
'"
(p .. 9 5' .. 5611.'od S" 1'16)
Oft II.. I"'" lw>cI. """b,ofQ. 0>01 ........ny ,.J,.."1 ha'" ""y ....nywPt>. If. m
Two Problems from the 1987Austrian-Polish MathematicsCompetition
K ..·• M.'_'_....,1987. lJ. A••h""",,,,,, ""..,""' .. ,;..... ,n C"""M.,'..·.,.."......-. 19l1'J. 269: nIKl X."""""Id fN! ,_ ,m ;., "';,ho.., _,""., .be>.., dn..1>1",.. up ""••,01"""".00 ,"" ....... "'nd ='..nly""n~ .... _,.... ,m'.I
.. .- ..DOsro ......, •, , ,-- ,.. ,,., -II Il' ,
I ". I.-
= • ~_11,~MulE 1'1 .- -- -• ,,
.m I.m
= • Its_II,''''•• •• •0' I I , ,, •• I••= ·)'_,.....
II "., •- II I I ,, ..
I••= • "",,1,111, , • • • "" , , , , , ," I'
"= • l_l'.:zl,__'Ior_....."........
nC::UlE .(\-00)
.1.5""."'·f'OUSH "'.,".....ncs COM,,","nO'" '"H"""'.... Ih11, IAI a" b< ."~ n..mbOf; .. jIXI· AwI\''''& "'" ....,_ '0 X ..{'.2•...• .lOOj.IIo ""'" 'ho' A NIl< ho.........f .. 200 .....b
,. " __'lOa..,.!Nfl« .... """Id lil< ,,, ...._ (;;~ ..
"l&.'-..oLISH MATHr.",n
An Engaging PropertyCon~rning the Incirc1eof a Triangle
•
••L_-'~.,.._"-.i-------",<
I.< ,"" "".._ d 'ha' of6AIK
n.. prohlrm
,.
•
r"" llIo' ",.pfM>Ii.......,H lKeq..I.It .......pk .......Jl!n,.;__'hoI 'h" io."",by~... holf·'um .bo<""""".. ood< h..... S....Ior!)' ' ..carried,,, S. and ...,110",{>1'.1'$,
If ,'" ......,.. of It.< ""F"" OF< /'(l ••'. RS • b'. and TU. ''' I)~ ,hen. bmI... "1' iIpo",,1tt '" //C. ....np A"'" """I", to l>AB
·M
~'~l
On Floors and Ceilings
1'b>.in'.....1,..I~ombcn "'cropopin """ki...hic~ ...... io..'« '"'." '" fond .... de.h,.. ,~ ,'" """"n..,' ,he ".1...m..... 10 .ny I hop
,.Solution
.""'" U_TO ~,••
""'hap> we d 10
,.
n... o1~). k (Of.U ~ 1hI' ..",ry ,he 'n< >1
,
tlGuU ...
/.
' I I II • •
Two Problems from theInlernational Olympiad 1987
(.) ,he d;..._ bd...... _~ ,w"of'he poo~.. " ;',Q_.,'"(bJ ...,~ ,_of'he poo.." ............. , , "'...~".... ''''nal< """"1_'-.-
ldon', """''') Odea """',he ......;..... l
,. ._..I)()s TO ....
+------.
"6_ J(~' -I'l' Ii - Il'.. (t - Il';(i ,t' ... I,
..1IiclI;' ......)'1 i"''''''''1 for """'l"'" posi'i.. ''''~''*.nd '(ll .. ,'1".""rf"'" "l""". ,0
'"
F1Gf,J.' "'Solution
l'uhaps " .. fir.. 'hln."'."'\.A 3C"
_AN·KQ.
,.
_.,1>< riv- ......... K.nd" moh AK!-M...,..h
,.
&.,·CII.AlI.
• fIo! ,f,h~;, ,.1>< ,ho:...". .. rA + AS. ,..", CT _101 ho,.. ,. «l"'" tiS....."""......,y. If ... can _'ho' CT • U. ,he "'~""«>nO" ~"""O ,..,1>0, IT • liS. for IlI
On Arithmetic Progressions
(PII66,,,,,,.I«III
,.CIeo,Iy ,.........
k' •• ("'O
A Property of TrianglesHaving an Angle of 30"
IPll00. C,... 1.1",,,,,,",,,,,__.1917. 1601
.........1. lei I.nel 0 II< ,he ,n«>I",.nel "'1 Tnyk c;.-"7 "'_."nU", ,n coIl.obot••"'" ~,th Do 0_IM ,"",.poi;,. M, onno
Solution
L,.EFA. 1>,.BFA••od ""fill'....4EFB_l 1>,.AFB. But 4AFB .. 4C on,.CAF_ 4CIIFoo->".0«ICF• ....l~ 6,CIIF_jA,
.nd ..... lllt>ooo:u 1>,. B•... 110.. 4/I1F.!B + lA' bo'. from 6AIB..._
.Uf!< Fill ,110
• """""'" or ..,""""'".,
N_ ......... 'his '0 provo 'n••PA°B ...",""..,1. oDd ..._od< , , II"",,,,...rad,..;, ....... ",Iocr ,h•• A6 "< Am< d'ra.1 .... l_..... ,n ,II<tIa."l '0 lhr ",toe, .rm. IO.nd F:D _ ... ,I>< ptrp
From the 1985 Bulgarian SpringCompetition (Grade II)
K"'" .11.,__, '9117. l.lIIt
"'-1; (~) -(~). (~). (~).. (~)
_'ha'
Solution
a"uly s. is ,he ...m of ....,~ 'Ill'" to"""",,1 O
FIlOM .._ TO .,""
•• ,0, "OJ + 0, ...... i [/(1) -/(-1)).
Bu'''" "'" -..- ho 01"","'y.11 ,"'.. onycho_ ,'" ,hie< ",,,, ....... 01" .....y. I......ncI";'. ",h
S.."" I ~ ... + ...' • 0. ... ha... I -.,' I .. .,' .. -", ,mpl)'l"ll
S. • j 12" .. I-II (- I)....."!.. jI2 I-II.. 2]
.jI2 2] ... 0 .. ""'0 _.
In .~ co ,........ haY<
l(2" - 2) S S. S ja"" 2)•
••(')""( ')""'" (')""( ')""'i 1- 2" S-,-:5 'i 1+ 2" -
NO'>'. fOfO < (xl < I.... ha.., Iu'n._,..." I, ..... fOf~ > I.....100haw 10..._-,,"".' Tho:mo«."" _ 00. ,.. hmS:'''/2. 1.I'"nl'h< dnir
An Unused InternationalOlympiad Problem from Britain
iC...., M.,_It_. 1981. n,JOl"'"", ~ ...... -.... RO)~I 0.0''''s.:_ al E,h",,""'" Sow.... C""""h.......
Pro>< 'ha' ,II< "",,,,,,,, '" h", «>n>«>o"", _,;'" ;n......... nn..-,pm«' "l"""'.
Solution
iii"", _ ........... ;ntqn .... "",I""" '" ,ny pnm< 'han'
Now. nt'1' _,,'" on,_. N" "n..", pr'm< """""_''''''.-:!'",.,')'.
Si"", only.,.., al'lI< h.. ;n'..." ,n ,II< """I" di""blo~ '.' 'f... "I....._lei "' , or'ho q....,"'" foo- ,hoo. prod,... "".... 'ho l'''''' 1_ '. ""',,' '"'n po",n' ,n ,,,.do",. pr'm< do
.l!-- 2,-g.-.l!- ;It I,H _; .. i. Ii! ~.il1~;1 ,,;y ~Bi5 a2,!' "i' Si 5 Z'llP3~ I'~· ~i[i ~~3:' l' i ~ i: ~ p.l:if;_~-~~- ~ .. ~.l,··~'i·!-
",,, ". - ~ - ~ ~-Ba~rl.£l ~l'~ ::~~~':di.
l:}~ i};~H !.l i ill:! ~ {~~e!.diu~~q i"~1 .r r "" ~If;,j.a ~~ i-!~.- "I: i I Ii:i>"'i""112,." , .......... i--_,!'-.~- ~~'" ~::':' "i "i 'ol ;.~~i.-;~::;i~ - ~il'", i~ ~.ifi '&'" ~.:; i'~ i'll, 'ol '1 '1 '1 rI 1 .,~ !5:~'ll ~ I"' 't 't 't ---;.2, .'-.t. lf~,=2,: ! -1 : : ~ =ifJi ~.11i~' ;r : ;; ::~ .:..~ dJl2,; ~ ~5 h.~~· i .... 2:f'!!f~ ;'~~j"lf d ..., ~ - '1 f~
i~_'5~ ~:'lI.~ u~ - ~ !l. 5~l!l"'~ ,'.,;. . 'll ~ ~1• '" "I
A Rumanian OlympiadProposal
511"""", A. " .•nd C'r< ,h... di_ "",n" Oft ,ho: .."" >id< "fa plane •.Tho: poon" A'. "'••nd C .rt _ .,bo"."ly ,n ••nd ,he """""n" 01'AA'. "If' . ..... CC .rt 1-. U, .nd II. S is ,he "'",_ of '''''0$1< UJII.
A.A'. tr. C "'1,n r, ,he"",n'S """""' ,,,,o"hnll,. Who,,, ,ho: 1001'S..A'. If' •C usu"".11 _bit J-poon, """~""Oft..n o?(t, " ..........bIt rot• pu' Of .11 ,h... 01' A', "'.C ,,, dt.)
•
•
"
Th. probIttn, rtom Ru........ is ,...... lr)' Danotl hd< (I) ...n~....".,..,., of 'ho: k;",j 01' _"y .....bItm ,he ",'em Eu",peon "",""n.f
".Solulloro
If. panicl
From the 1984 BulgarianOlympiad
(0..., "'0'__, 19C1, lUJ~.
~'1)"'._J'
r", ,n ".Op.6. 10.1•. It. I·
f", : • l. ,lIt ",."oon isS< 1' ••• 9.
'"
'" .-........ ro.,...
S"·1'. S,
.,., "'" ob< oolu,,,,,, (~., .•)• (I,O.ll-H_... "'" ""'" .... ,"'" is,he only ,oMx.n.
For •• 1.1,. 1. ... ho",
••S··,,_,"_6.(J'_1)(J' .. 1). ,.1I1.
!lOll« II>< radon Oft ,I>< "&h' ddf.nd " .....1'0 'h< Qlbe I).
implyi"l~
Two Erdos Problems
....mt.' ~ """""" '" P.ul E
". fOpoood.od IOIrtd ~ Pa.1 EnIO mlJl>' I
'"of,.tim.. Itnlth. ,.; 'M"" prop< b"......oddo .., '0~ ..Solulioll
w, _k" I«lU ha~ "••, Ii<HI..... tll,•.. , ,b.} .."" bt add, ...... b, < ....Tb"" ..l
From the 1985 BulgarianOlympiad
l« P,."'. _, P, bo: .fty ........ pol." ,. opo«. "" I"", ,. ,"" ...". pl.o.....od.._ < d",ribulal. "."_~,om.""'tioO&l« ",,'1 t>< fonn....,,«1."&1... ,"'" ''''"l''''''ho" ,,,"of
'"A, po;nl 1'.. , ..... Of< .....b< ....y Cb'onlI.IICGraph>,b,. H..old [loo" ...nd 0.. Finkbt,,,.,.. In /11•.-,.",1_.Vol.0, DoIi:Io.. ..,.........'''''' E.posi';"'.. MAA, '979, pop:s J__~
C"",""",. ,1>
._TH'.''''M;u:••,.~oo.'M''''' '"
".,
-, ---)-~-~-,..---;-1\ "
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-,
If ,I>< foo, red ""nl'" .1< ,I>< fo«< ofa '«1,.,., P,P,P,P. "".1<9~;;)) ......,...' ......., roI~ If any''''' of ,I>< foo, «II'" f""" P, '0 ,I><'nto«> of '1M: I< < drwred ,,-..'''''''''_101 101\0»-. "Th< ..".h«I ,on I""" • ny of P,. P•.•nd p,. ,I>< ,_,""""follo~.
~'....ny. '1>
From a Chinese Contest
II Fa" E"-
ill
10 < 210(0 .. 1) < 1.0 .. 1.
2';""10+ I). {';;; .. .r.+1l' _ (210+ I).Thrnf_(MIJi'"
2n < \,;;; .. ..r. .. II' - (20 +1) .. 2ond,-. :z" .. I 'hfOOl&ho"~'" .....
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A Japanese TempleGeometry Problem
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Two Problems from theSecond Balkan Olympiad, 1985
ION.' M,.,_'"....... 1911. 71_11'Prgbk,m 1
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Solulion
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The Analytical Approach
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A Property of Pedal Triangles
(P ,016, C..... ",~,_,_. 19S7. 62)
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f,om ..h;ch til< d1 A.F'P'" ,GS 'ho' ,he .... of l>(U n«d. '0 b< JUbt'OC1«l f''''''' ,he p«Iol'rio ..... DEF;" _ ro _ ,... riIll' omouo, '0 'rion&k> MfD.nd DNE roobu,o ,he ......... (LENDN:
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Three More Solutionsby George Evagelopoulos
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w..... It< Q"",kly '1\0' 'ho: ;""l",I;'y ho'd' rOt ...nil< ."".,b
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"'Solution
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The Power Mean Inequality
s..- ."", _ 10k. '0 PfO"< """ "I< val... of "'" f,,,,,;,,,,(.+lb+xJ'0'+16'+);1
...._ J"."'" ,tun 61.,. an. _by< ..,,1 numb<.......Ii>ed "'",.,,, .....n ,noq ...h'y. II a.....J, f«, < •. ,tu,
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f ..... 1'1') • "0_".""':(1< ,be~..ti", 01' f( 'I"" ........ ".&1 is bas ,""10m< iiI" os I I", , .. 0,
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ftCUR[ '..
F'('I ., (1:,...;)(1:,...:101' ... )- (1:,...:101 ••)(1:'''': 101"')(1:,...;)1
+(1) 1:, ;101 .. _ f'. 1:....;......t : r,..; ,.• , .11:,.•;)(I:,...;101' )- II:,...;101"')'
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Subject Index
8la;" ball> ;" an .m . ,A
00 2-«>Iori.... of> -
'". __ ~11
. , III
., liem
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