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OlympiadInequalitiesThomasJ.MildorfAugust4,2006Itisthepurposeofthisdocumenttofamiliarizethereaderwithawiderangeoftheoremsandtechniquesthatcanbeusedtosolveinequalitiesofthevarietytypicallyappearingonmathematicalolympiadsorotherelementaryproofcontests.TheStandardDozenisanexhibitionoftwelvefamousinequalitieswhichcanbecitedandappliedwithoutproofinasolution.Itisexpectedthatmostproblemswillfallentirelywithinthespanoftheseinequalities.TheExamplessectionprovidesnumerouscompletesolutionsaswellasremarksoninequality-solvingintuition,allintendedtoincreasethereader'saptitudeforthematerialcoveredhere.Itisorganizedinroughorderofdi±culty.Finally,theProblemssectioncontainsexerciseswithoutsolutions,rangingfromeasyandstraightforwardtoquitedi±cult,forthepurposeofpracticingtechniquescontainedinthisdocument.Ihavecompiledmuchofthisfrompostsbymypeersinanumberofmathematicalcommunities,particularlytheMathlinks-ArtofProblemSolvingforums,1aswellasfromvariousMOPlectures,2KiranKedlaya'sinequalitiespacket,3andJohnScholes'site.4Ihavetriedtotakenoteoforiginalsourceswherepossible.Thisworkinprogressisdistributedforpersonaleducationaluseonly.Inparticular,anypublicationofallorpartofthismanuscriptwithoutexplicitpriorconsentoftheauthor,aswellasanyoriginalsourcesnotedherein,isstrictlyprohibited.Pleasesendcomments-suggestions,corrections,missinginformation,orotherinterestingproblems-totheauthorattmildorfATmitDOTedu.Withoutfurtherdelay...1://://¶olya,Inequalities,CambridgeUniversityPress.(Theformeriselementaryandgearedtowardscontests,thelatterismoretechnical.)4[a;b]and(a;b)respectively.AfunctionfissaidtobeconvexonIifandonlyif¸f(x)+(1¡¸)f(y)¸f(¸x+(1¡¸)y)forallx;y2Iand0·¸·1.Conversely,iftheinequalityalwaysholdsintheoppositedirection,thefunctionissaidtobeconcaveontheinterval.AfunctionfthatiscontinuousonIandtwicedi®erentiableonI0isconvexonIifandonlyiff00(x)¸0forallx2I(Concaveiftheinequalityis°ipped.)Letx1¸x2¸¢¢¢¸xn;y1¸y2¸¢¢¢¸ynbetwosequencesofrealnumbers.Ifx1+¢¢¢+xk¸y1+¢¢¢+ykfork=1;2;:::;nwithequalitywherek=n,thenthesequencefxigissaidtomajorizethesequencefyig.Anequivalentcriterionisthatforallrealnumberst,jt¡x1j+jt¡x2j+¢¢¢+jt¡xnj¸jt¡y1j+jt¡y2j+¢¢¢+jt¡ynjWeusethesede¯nitionstointroducesomefamousinequalities.Theorem1(Jensen)Letf:I!Rbeaconvexfunction.Thenforanyx1;:::;xn2Iandanynonnegativereals!1;:::;!nwithpositivesum,!1f(x1)+¢¢¢+!nf(xn)¸(!1+¢¢¢+!n)fµ!1x1+¢¢¢+!nxn!1+¢¢¢+!n¶Iffisconcave,thentheinequalityis°ipped.Theorem2(WeightedPowerMean)Ifx1;:::;xnarenonnegativerealsand!1;:::;!narenonnegativerealswithapostivesum,thenf(r):=µ!1xr1+¢¢¢+!nxrn!1+¢¢¢+!n¶1risanon-decreasingfunctionofr,withtheconventionthatr=0istheweightedgeometricmean.fisstrictlyincreasingunlessallthexiareequalexceptpossiblyforr2(¡1;0],whereifsomexiiszerofisidentically0.Inparticular,f(1)¸f(0)¸f(¡1)givestheAM-GM-HMinequality.Theorem3(HÄolder)Leta1;:::;an;b1;:::;bn;¢¢¢;z1;:::;znbesequencesofnonnegativerealnumbers,andlet¸a;¸b;:::;¸zpositiverealswhichsumto1.Then(a1+¢¢¢+an)¸a(b1+¢¢¢+bn)¸b¢¢¢(z1+¢¢¢+zn)¸z¸a¸a1b¸b1¢¢¢z¸z1+¢¢¢+a¸znb¸bn¢¢¢z¸znThistheoremiscustomarilyidenti¯edasCauchywhentherearejusttwosequences.Theorem4(Rearrangement)Leta1·a2·¢¢¢·anandb1·b2·¢¢¢·bnbetwonondecreasingsequencesofrealnumbers.Then,foranypermutation¼off1;2;:::;ng,wehavea1b1+a2b2+¢¢¢+anbn¸a1b¼(1)+a2b¼(2)+¢¢¢+anb¼(n)¸a1bn+a2bn¡1+¢¢¢+anb1withequalityontheleftandrightholdingifandonlyifthesequence¼(1);:::;¼(n)isde-creasingandincreasingrespectively.2Theorem5(Chebyshev)Leta1·a2·¢¢¢·an;b1·b2·¢¢¢·bnbetwonondecreas-ingsequencesofrealnumbers.Thena1b1+a2b2+¢¢¢+anbnn¸a1+a2+¢¢¢+ann¢b1+b2+¢¢¢+bnn¸a1bn+a2bn¡1+¢¢¢+anb1nTheorem6(Schur)Leta;b;cbenonnegativerealsandr0.Thenar(a¡b)(a¡c)+br(b¡c)(b¡a)+cr(c¡a)(c¡b)¸0withequalityifandonlyifa=b=corsometwoofa;b;careequalandtheotheris0.Remark-Thiscanbeimprovedconsiderably.(Seetheproblemssection.)However,theyarenotaswellknown(asofnow)asthisformofSchur,andsoshouldbeprovenwheneverusedonacontest.Theorem7(Newton)Letx1;:::;xnbenonnegativerealnumbers.De¯nethesymmetricpolynomialss0;s1;:::;snby(x+x1)(x+x2)¢¢¢(x+xn)=snxn+¢¢¢+s1x+s0,andde¯nethesymmetricaveragesbydi=si=¡ni¢.Thend2i¸di+1di¡1Theorem8(Maclaurin)Letdib