MathematDiicaamlo nds © 200b3y TheM athematiAcsasolc iatioofAn m eric(aI ncorporated) Libraoryf C ongresCsa talCoagr dN umbe2r 002114177 ISBN0 -88853-332-9 Printiendt heU niteSdt atoefsA merica Current (plraidsnittg iintg) : 109 78 6 5 4 3 2 1 TeDh olMhcaeitmaEpanxotisi ictailo ns NUMBER TWE NTY-SIX MathematDiicaamlo nds RosHso nsberger UniveorfWs aittye rloo Publaindsd hiesdtt erbydi bu TheM athematiAcsaslo ciaotfiA omne rica THE DOLCIANIM ATHEMATICAL EXPOSTIIONS Publbiys hed THEM ATHEMATICALA SSOCIATIOONF AMERICA CommiotnPt uebaelt iisco n Chair GERALDL .A LEXANDERSON, TheD OLCIANI MATHEMATICEAXLP OSITOINS serioefst heM athematical AssociatoifAo mne ricwaa se stablitshhreodu agg he neroguisftt ot heA ssociafrtoimo n MaryP .D olciaPnrio,f esosfoM ra themataitcH su nteCro lleogfet heC itUyn iversity ofN ew YorkI.nm akintgh eg ifPtr,o fesDsoolrc iahneir,s ealnef x ceptiotnaallelnyt ed ands uccessefuxlp osiotfom ra themathiacdst ,h ep urpoosfef urthertiheni gd eaolf excelleinnmc aet hemateixcpaols ition. TheA ssociatfioorin t,ps a rtw,as d elighttoea dc ceptth eg racioguess tuirnei ti atintgh er evolvfiunngdf ort hisse riferso mo new hoh ass ervtehde A ssociawtiitohn distincbtoitoahns ,a m embero ft heC ommittoeneP ublicatainodan ssa m embero f thBeo arodf G overnorIstw. a sw itghe nuipnlee asturheat th Beo arcdh osteon amet he seriienhs e rh onor. Theb ookisn t hes eriaerses elecftoertd h eliurc iedx positsotryyla ned s timulat ingm athematciocnatle Tnytp.i claylt,h ecyo ntaainna mplseu pployfe xercimsaensy, witahc companysionlgu tiTohnesy.a rei ntendteobd e s ufficienetlleym entfaorrty h e undergradaunadet vee tnh em athematiicnacllliyhn iegdh -scshtouodle tnout n derstand ande njobyu,ta lstoo b ei nteresatnidsn ogm etimcehsa llentgoit nhgem ore advanced mathematician. 1M.a themaGteimcsa,l Ross Honsberger 2.M athemaGteimIcsI,a l Ross Honsberger 3.Ma themaMtoircsaell s, Ross Honsberger 4.Ma htemaPtliucmasl, RossH onsber(geedr). 5.Gr eMaotm einMnta st hiemc(asBt ef165o0rH)eo,w arEdv es 6.Ma xiamndMa i niwmiat hCaoluctu lus, Ivan Niven 7.G reMaotm einMnta sht etmia(cAsft1 6e5r0H )o,w arEdv es 8.Ma pC olnogPr,oyih ledarndat ,hF eo ur-PCrboolleoDamrv, iBd arn ette 9.Ma themaGteimIcsIa Il, Ross Honsberger 10M.o rMeat hemaMtoircsaell s, RossH onsberger 11O.la dnd N eUwn solPvrbeolde imPnsl eaG neomaendtN ryu mbTehre ory, Victor Kleea ndS taWna gon 12P.ro blfeoMmras t hieamcnsaiY,to nugan dO ld, PauRl. H almos 1.3E xcusri snCia locnuAlnIu tnse:ary op tflh Ceo nntouiuasnd t hDesi crReobteert, M. Young 1.4T hWeo hasCconuutymP robBloeomk , GeorgTe. G ilbeMratr,k K rusemeyaenrd, LoreCn. L arson 1.5L ioHnu ngta indnO thMeatrh etmiaPcualr sAu Ciotlstl:ie oocMfna htetmiacs, VerasndeS ,tr oibeyRs a lP.p Bho aJsr.e,,d i tbeydG eralLd.A lexanderasnodn DalHe. M ugler 16L.i nAeglaerbP rar obBloeomk , PauRl. H amlos 17F.ro mE rdtOsKo i ePvro:b loeOfmyl sm piCaaldi bRoessr, Honsberger 18W.hi cWhayD itdhB eiy cclGeo .?, a.ndO thIetnrr inggMua tihemaMytsirtceiaels , Joseph DK.o nEha.u sDearn,V ellemaann,dS taWna gon 19I.Pn 6 yal'sF ootMsiteslpcnlseea:oP urosb laendmE ssy ssa, RossH onsberger 20.D ioapnthuasnd D oipahnteEi qunatsiI,o.G .nB ashmako(vUap datbeydJ oseph Silvennaann dt ranslbaytA ebde S henitzer) LogaisAcg l ebra, 21. PauHla lmoasn dS teveGni vant EulTehrMe:a setroU fs A ll, 22. WilliDaumn ham ThBee nginignasnd E votliouoAfngl ebra, 23. I.G .B ashmakoavnadG .S .S mirnova (Tranbsy.A beS henitzer) MathemaCthiecsaftlrnA ourmto sdut nhWeo rld, 24. RossH onsberger Coungot nFi ranmweorMkasht:e miacttsAod i t hDee snoi Rfgi gSitudrc tures, 25. Jack E.G raver MathemaDtiiancmsdao,l 26. RossH onsberger MAA ServiCceen ter P. BOo.x 9 1112 WashingtDoCn 2,0 090-1112 1-800-331-1MAAf ax3:0 1-206-9789 Preafce Tmhiisscc oeolelflll eegacmecntemoebinsnorot tiuinaansl ris lynii sga hntts fmraofmnmiayn t emh iTenmhmodaeoarjsi fti.tt it shcyt iaa omcnlpos r e 75 cofErmuoegcm eel ctoiormdmyegb,aei anonl maagnenmteutdbobr erryria, a l thamenooodtfrs h yte, c adfnioc sloblcmeboufc ywsoo esrladitl oaenbgsle y fershman. Thineasort t etoi e nimgsioptitnnthvr ; eeuw chpfltereirewme i places nariiedtsoo nnipalesrry efea gpo i tearfpnomr or aletlsioeownn.t ed Tehsews riailnyte sstiata,es iry nunela rtm eseea, lnte ydhn eedm atical tienrWhmtiameal enoeactf so. un itrcspheeo ree njify ntoc sireona mgte i on otfhe xespsii,lehts a ot nphwaeacitdtoli t mleothb mxnheiaece ninnyetdt e rioecflh enmeeswnsrit etaahrd yii nintmgso a ttp hleemaastuirces. Although netctoerp syrssb o ategbhrfoolteyoitneos rnh mi oeegioly fnauoa rstube, l e tgoti havle etim htf otIi'mulsr ygeuswoh rtituet,l ah lelm mlof ritenh de enxgc.i ti Tsheeaci rtneida oenbnpdreseie a mnnoanda rdyeyd n etr . Af eexsw(ee wrssic ootainlshrsupoet)tru h iittrn bhokaoeulno gedkhd o ut a fsimefitts cecoheef(anl swl lioilaioltgensnuhiante st vetghof) eneeuo nds sr yfouuejrrnot yhmeern t. Tbhcoeoo wnkaic ntlo phufiu di nbaeosdl snueii sbxnac,j dna eedtcx ta geinnedreaxl. Iw oluitlktadoehk p ieptts ooo r ettsDuhVsalnaenolin retk my aP nr of ando ttfDhh oeEe1 d cBimioftetaaoomhnrrrbreid ieei aarcrnlse d w p atrimo n gecnrottifmlht aeei.nT c uhiisemssc uim brcm boiphueopr ostcktofeha v eeidr deadInid dceg aerttptaiol.htIoi y aeetns gamf p a ultileeonn ax dts eure mwya tram htmEaoelPn saektaidsB nnr edeev fietorruhraline ynfi agrRi uledi geantniedeac xlshipsienneet i mreycati atninlhugp rs uioct.bourh lngieihp c ta t I Godm uslto vmea thematicians He'gsi vuesns om uctoh e njoy! J 1 I I .� ,;, Contents Pr.e.fa ce vii SecTRtheiemWo nainerP -krR.aoa bpcleker ty 1. 1 SecTRtheieLso .itn ol ne ss 2. 5 SecAtpaipSnolt.dnie csk s 3. 9 SecASt uiRrpoeoinsfsE u irlPndtago u sl 4. 13 SecMtiisGoclnee. la.lna.in.neg osu s 5. 19 SecAAtncpi apooTtlfnu'sTii r hoaenno rem 6. 35 SecFPtoriufoorPrbn uoPl tmae.n pm aesmr s 7. 43 SecTptoBiiaocPosonsrneb f dlQrue oammns t um 8. 49 SecTDtwiioso Itnni tneggueirssh ed 9. . . . . . .7 9 . SecAPt riooofBpn ieCt nrohoteemyffi i caile nts 10. 83 SecNMtiiinsoePocn rbe .ll elmasneous 11. 87 SecAPt riiCoonobon ils. nes -miT ng 12. . 130 SecS-terimLeoiaPgntou tlliyacgreo ns 13. 190 SecSPtiorixbfo ltrCnehoa memns a dian Open 14. MatChheamlalteincgse . . . . . .1 71. . SecTthPirrToeehneiteG ntoe yro emmest ry 15. 152 SecTGtwefrioEmoo usmnGc el.oi mdeetarny 16. 153 SecTTthhieouo rensS- eeM-.qH ueednlcuend 17. 134 SecTtwioo PMbnri losecmesl laneous 18. . . . . . 151 SecASt uiPrporronoRif pesP egiorunltlgyya g ro ns 19. 175 SecTthSirhSoeotineron trT iNheuesmo breyr 20. . . 136 SecTthGirePoeornemso e btlrey.m 21. . . ..... 196 SecTthPoirrbFoelrTneeoh mBmesa O llkyamnp iad 22. 1990 175 SecAJt ai"poFPanarn noe"bs lee m 23. . .. .. . . .1 8.1 . . . SecStlaDii oocunig nhgn ut 24. . .. .. .. .. .. 1.78 . . . . SecAt PfirtrohoonTuebm o rnl oaefTmmw oe ntnsht e 25. 1980 159 MatheimacDtailil monds x MoCrhea. l .l. e .n. g .e. s 203 SoTlToCuh hteai lolnesn ges . 209 In.d.i.c.e. s. . . . . . . 237
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