From Erdos to KievProbkms of Olympiad Caliber

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From Erdos to KievProblems of Olympiad Caliber

Ross HonsbergerUn;".rsily"/II'"",/",,

"""""'"M.J Do#",""",.,TilE MATIlUMT1CAl ASllOXIATION Of' AMERICA

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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""" ,.,

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Seven Solutions byGeorge Evagelopoulos

A' ,he n' ,iM< (19'H)~ E ~1oo p

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

2. The Cube and the Ch~kerboard

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"6. Problem MI043 from Kvant

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On Ill< 0Ih

A Decomposilion of a Triangle

I. An Easy Warm-up

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A Problem from the1991 AIME Examination

.jl-:-~-,----~~--~-'., ., ., 11

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•Solution

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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

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9\1 l) 61.1-).100. 1 -100.100'

·od ,ho: o~.. """,bt, c( ,.'" ,ft 0 100l4,1P' '""I'" """Id btI ~ I~ (10001 ~ 611.

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"4. From Russia

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Solution

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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

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Siocc A and 0

Solulion

50"", I" 'ho io:rn,.. of "ABC. 'ho 1o>e

From the1988 International Olympiad

(6...,1< ;noq...h,)

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• 10 71.(1+2+ +7OHj:-,-

.(1 .. 2+ -+101+2111

.(1.1+ .. , +70) + .9U.a._reel.

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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 ;, .,_,. "

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A Kie" Olympiad Problem

1__ ~h'oo_'IIo,..-.,.ltin_.. p '10",",,",_01lop, _,. North II I••~

•Solution

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Some Student Favorites

Tho pr-.n. io lhi>....-bon 'r'" lOf"h< III< 10m<.umt>

",."'" ,..._ TO ....

S.N«:I''''' 0',• .-pI • '< 'l

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Four Unused Problems from the1988 International Olympiad

I. From Bulgaria

(C..... MOI"'-I""'.... I'IQ. n~)

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Solution

I 'm no< ...11 "'.. ,hat ,tit """""''' p««

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th.. " Ii.... '" b< 'NO rot ~ • 1. 00

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..... ,II<

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_inl'" .... mel"" hyPO'h".... -..or. 1'" d,vK:I,. Ind"",,,,,,.

1" d,_... IrandMllyl( 1" d,_•.

•

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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

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IIi

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".'''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

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3. Proposer Unspecifiro

(C"'. 1n',,_,.... tIIo, .... d,....... bel A, .... A, «>ulol _.>«l .yni«ly by ~".I'lIeoo-l\o

•

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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

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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 ~

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". 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\ "

"'£.====""

-,

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"~'" .....

..... 1'" (,;;; .. ,/Ii .. I)'

A Japanese TempleGeometry Problem

Wi,~ 'h< J«'''' I'''' 01' ,I>< Edo p< ',"''''''' U.S- """"'.. o.n 1'«10<IM'n",.poI", M,.........'. Th< follo~,., probI..'ho', ",ho:"",. X ""lh' be ,,,,,,," "" L ,ho: "'.." ......... 01' A .tId 6.", ''''''l'''ho: ..m

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ncllO£ ..

Solutions

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Ou, fin' ....uI... is, 0011.., I'i«< of"""" brO.. So*;_~y(Wd"'m>bo'J.vi",...).

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Two Problems from theSecond Balkan Olympiad, 1985

ION.' M,.,_'"....... 1911. 71_11'Prgbk,m 1

Th< ""'"., al r_1" ,.",..,.. ;, ,'~~,.. • pIor! ol) "',"'- -..bOed. ..ll1.".,.... «>1'1""'" 1>"...,.. "..., _ 1>"...........pok.. by at...., 200 of ,II< propk at ,II< ,fI&.

Solulion

,A IIm,1>, ..."',,"" """(1","",, .,....!,.

1< ..

The Analytical Approach

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l.« .ho: c........_rd or lh< wno«< b< 8(0. 0),q .... 0),.01 A(4b. k).1bt or OC.~ -'.,-.nd CD • ..t DE Ort p

no"", ••_ 'fO 0""

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nOM .._ TO .,'"

F;""lIy. ftom It>< ..... oI"ooso........ !III........00109 ... +~ -~.II:i~

Ol oc."+

A Property of Pedal Triangles

(P ,016, C..... ",~,_,_. 19S7. 62)

Th< ' ....n,.. DEF .st"'m".... by dfOl'POn, potp

,-,,_TOOJEY

Thio io ,he dio

Bu' f""" 6FLT,,,,, '" fT. isinA.Th. , ,

fT. i Fl. i~ ..nA,

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nOM ,.._ TO .,....

"(l>"8(") _I(£LENDM).

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:

(LENDN _l>MfD+l>DNE+ (l>DEF _ 6£L£).

Th", ,he _'i I...,.. ,he._ "JUII100' '" ,too "'.. 01 01>1...., ...,...., .nd

Three More Solutionsby George Evagelopoulos

Tb< 'h... probIe, S"ll ,h

w..... It< Q"",kly '1\0' 'ho: ;""l",I;'y ho'd' rOt ...nil< ."".,b

_ t< pea«,,hail ". and ... 11>... a1t....1o

1 ... ~,+ ... + .... ~~..,+~••,+. _+x,., (... 1 .......).. (l .. I)x•• ,.

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(I .. x," , x.r-'.... ,s (l .. Ilx,.oj'.,.,,

·.+ll(l+I~."r.

~·;..lIy. ,_. «1m ",,01.. (8).0

"'Solution

Clea~y ,he _ .... _ divno/:'1. lh< "''1'" "'" or b/2". Si... we at<'."'-001,... 'bU", ony "'''it

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raoMUoOSlOX'EV

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• nd 00'" •• wwoo< 'hO! ,I>< .'.,

AI"" r""" "" ." ;0''11'' >. ol ",,,,mum .., ,Of «>Uf>

b ..... +~.> ..... +~" .. ~.• """ " ,

...k'"1 k•• l.nd K•• K•• , +K,.

_inllY. ,... ,nl'''' .', ,n

"....,+., ,. x•.,. x,

,. (.'''',nl

.....,., .._ TO .J....

••So., ••• *,.*,.···.l._,.' '" J(~_I),

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So .l, .l,.· .. +l•. , .l•• (1")+*,+", • .t.,_,. (I ..)+*•• I.(..l,+---.l•.,+~+l+l {.-Ul•• I}.1·So_,+I.1·So.,+)"'J(~_I).J

_lo. ..

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,

(..,.; +1'>"; + +....;)'" < (..,aj +-i + ... +....;)"'•

..,+,,+. '+P. - P,+P>+' '+P.,,1>

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io ..riel!)' ,nct

[","'" ''''.j~ IIot!j'II"~!: /''"

l~ho .1It Iop,,,hm 01 f(".

S;oe< ,he p• • ncI.,.".U pW,i", n....b

f ..... 1'1') • "0_".""':(1< ,be~..ti", 01' f( 'I"" ........ ".&1 is bas ,""10m< iiI" os I I", , .. 0,

""

ftCUR[ '..

F'('I ., (1:,...;)(1:,...:101' ... )- (1:,...:101 ••)(1:'''': 101"')(1:,...;)1

+(1) 1:, ;101 .. _ f'. 1:....;......t : r,..; ,.• , .11:,.•;)(I:,...;101' )- II:,...;101"')'

II: , ;j'Th .nd ••

Wb

Subject Index

8la;" ball> ;" an .m . ,A

00 2-«>Iori.... of> -

'". __ ~11

. , III

., liem

'", ••• 2t l .tMl " .." ... or «1...1'mo ,•• "" ."p"-,', of. , ho,",,,&.n .n'" 01)0'.On ,ho: lox of> "">t" ",.,O'(od "' ." Iopo '...... ",_m on ,,,.olio". ".....On ,he pe!Jl

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.,,",,, ,-,....~,.....',.'.~, .,-*--

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•

--,..._......'--_0·

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, ~IIIII