数学奥林匹克的不等式

OlympiadInequalitiesxxxxxxggg———999KKK„„„IIIEvanChens¼÷April30,2014ThegoalofthisdocumentistoprovideaeasierintroductiontoolympiadinequalitiesthanthestandardexpositionOlympiadInequalities,byThomasMildorf.Iwasmotivatedtowriteitbyfeelingguiltyforgettingfree7'sonproblemsbysimplyregurgitatingafewtricksIhappenedtoknow,whileotherstudentswereunabletosolvetheproblem.–H›š©(0ª°=ŒPcyc(cyclicsum)Œ1=ŒPsym(sym-metricsum). ‹ Šx„BXcyca2=a2+b2+c2Xcyca2b=a2b+b2c+c2aXsyma2=a2+a2+b2+b2+c2+c2Xsyma2b=a2b+a2c+b2c+b2a+c2a+c2b:1„„„šššBBB1.1sssGGGIIIŒŒŒMuirheadIIInå„šššš1(sGI/AM-GM).äa1,a2,...,anºcæxGa1+a2+


+annnpa1:::an:I_˄Eöºa1=a2=


=an ‹P1dïIa2+b22ab;a3+b3+c33abc:Š.I w†1ú,„}L ‹‹‹‹PPP2.fIa2+b2+c2ab+bc+caŒa4+b4+c4a2bc+b2ca+c2ab.1s¼÷1„šBIII)(AM-GMï—a2+b22abŒ2a4+b2+c24a2bc:^ b2+c22bcŒ2b4+c2+a24b2ca:c2+a22caŒ2c4+a2+b24c2ab:ð w†1—a2+b2+c2ab+bc+caŒa4+b4+c4a2bc+b2ca+c2ab:ôôôÒÒÒLLL3.fIa3+b3+c3a2b+b2c+c2a.ôôôÒÒÒLLL4.fIa5+b5+c5a3bc+b3ca+c3ababc(ab+bc+ca).;/—ú†'‚'‹‚a3+b3+c3/'„abc/„'‚†Ô\mixed„/Ô„1dïoú†e.g.(a+b+c)3a3+b3+c3+24abcàºiŠŠa3+b3+c3ˆ‰åŒóŠêi24abc@å(AM-GM1ãz†}(„š/Muirheadš‚œfšixx1x2


xnŒy1y2


yn—x1+x2+


+xn=y1+y2+


+yn;¼Ïk=1;2;:::;n1 x1+x2+


+xky1+y2+


+yk;1ª(xn)Ë(majorizes)(yn)ë(xn)(yn)9Úð1 ššš5(MuirheadI).‚œa1;a2;:::;anºcæx(xn)Ë(yn)å„IºXsymax11ax22:::axnnXsymay11ay22:::aynn:‹‹‹PPP6.àº(5;0;0)(3;1;1)(2;2;1)Ea5+a5+b5+b5+c5+c5a3bc+a3bc+b3ca+b3ca+c3ab+c3aba2b2c+a2b2c+b2c2a+b2c2a+c2a2b+c2a2b:1dï—a5+b5+c5a3bc+b3ca+c3ababc(ab+bc+ca).èMuirhead/1„/ª°„ ‹Ö6(3;0;0)(2;1;0)F/(Muirheadï—ú2(a3+b3+c3)a2b+a2c+b2c+b2a+c2a+c2b@åï(†Ia3+b3+c3a2b+b2c+c2aB„/(AM-GM†ãz2s¼÷2ûýx„I1.2JJJ!!!„„„IIInå„Lî‹‹‹PPP7.‚œabc=1,fIa2+b2+c2a+b+c.IIIsGIiŠ„!xý#@åI(AM-GM抄!xºŒF/óŠ„!xº)(abc=1ŸIï9ëa2+b2+c2a1=3b1=3c1=3(a+b+c):àºIþ(/J!‚œŠa,b,cýXk0IùIiŠXk2,qÿ0Ÿ†„I@åB1Ł(0abc=1„öàº(2;0;0)(43;13;13))(Muirhead1Œ†¹Õ„ÍÞ/ïåŠLö։ˆÍ¹Õ_ïåÍN†(‚œI/J!_ï J!„ ö 1.3ôôôÒÒÒLLL1.a7+b7+c7a4b3+b4c3+c4a3.2.åa+b+c=1,G1a+1b+1c3+2
(a3+b3+c3)abc.3.a3bc+b3ca+c3aba+b+c.4.å1a+1b+1c=1,G(a+1)(b+1)(c+1)64.5.(USA2011)åa2+b2+c2+(a+b+c)24,Gab+1(a+b)2+bc+1(b+c)2+ca+1(c+a)23:6.åabcd=1,Ga4b+b4c+c4d+d4aa+b+c+d.2ûûûýýýxxx„„„IIIäf:(u;v)!Rºýx-a1;a2;:::;an2(u;v)G-úša1+a2+


+ann=a‚œI/J!„8êñ ö óIf(a1)+f(a2)+


+f(an)'¼¼ nf(a)å ¹Õš©ýxf/øýx‚œ¼ûx f00(x)0åÏx f00(x)01š©f/ùýxè‚œfºøýxf1ºùýx2.1Jensen/Karamataššš8(JensenI).‚œfºøýxGf(a1)+


+f(an)nfa1+


+ann:åfºùýxIøÍ3s¼÷2ûýx„Iššš9(KaramataI).‚œfºøýx(xn)Ë(yn)Gf(x1)+


+f(xn)f(y1)+


+f(yn):åfºùýxIøÍ‹‹‹PPP10(Shortlist2009).åa+b+c=1a+1b+1cfI1(2a+b+c)2+1(a+2b+c)2+1(a+b+2c)2316:IIIHŠö(‰ŸLIù1(2a+b+c)2+1(a+2b+c)2+1(a+b+2c)2316
1a+1b+1ca+b+c:þ(I/J!†@åï¹G-a+b+c=3I19ŠëXcyc116a1(a+3)20:å-f(x)=116x1(x+3)2,ïIf(0;3)/øýxE(Jensen1㌆‹‹‹PPP11.fI1a+1b+1c21a+b+1b+c+1c+a9a+b+c:IIIŸLIù1a+1b+1c1a+b2+1b+c2+1c+a21a+b+c3+1a+b+c3+1a+b+c3:¹G-abc-f(x)=1=xàº(a;b;c)a+b2;a+c2;b+c2a+b+c3;a+b+c3;a+b+c3)(Karamata1ãz†J!f(x)=1x‹‹‹PPP12(APMO1996).åa,b,c,/ Òb„ŠfIpa+bc+pb+ca+pc+abpa+pb+pc:III¹G-abcn(a;b;c)(b+ca;c+ab;a+bc))(Karamataf(x)=px2.2TangentLineTrick#úša=a1+


+ann‚œf/øýx BŒ„/ïåIf(x)f(a)+f0(a)(xa):‚œïIð_1ïåä¹Õëtangentlinetrick4s¼÷2ûýx„I‹‹‹PPP13(DavidStoner).åa+b+c=3fI18Xcyc1(3c)(4c)+2(ab+bc+ca)15:IIIL9ŠëXcyc18(3c)(4c)c26:àº18(3c)(4c)c2c+32,c(c1)2(2c9)0@å w†1ãz†‹‹‹PPP14(Japan).fIPcyc(b+ca)2a2+(b+c)235IIIŸL/J!@åï¹G-a+b+c=3@åI„/Xcyc(32a)2a2+(3a)235:)(tangentlinemethodïå~ú(32a)2(3a)2+a2151825(a1),1825(a1)22a+12a26a+90:2.3n1EVŒå¹Õ/n1EV—/´›„¹Õï/ˆ (ššš15(n1EV).äa1,a2,...,anºæxúša1+a2+


+anäf:R!Rºýx—f c}ÐÞåf(a1)+f(a2)+


+f(an)T0'Gaig n1ŠxøIIIIOlympiadInequalities,byThomasMildorf,page15.I„óÕ/)(Kara-mataIŠai¨(w‹‹‹PPP16(IMO2001/APMOC2014).äa,b,cºcæxfI1Pcycapa2+8bc2III-ex=bca2ey=cab2ez=abc2úš x+y+z=0óI1f(x)+f(y)+f(z)2dUf(x)=1p1+8exï—úf00(x)=4ex(4ex1)(8ex+1)52@å)(n1EVï¹G-x=yät=ex@åŸL1Š12p1+8t+1p1+8=t2:êiŠxàdïå(®Mô¥ãz5s¼÷3ˆd͌9_‹‹‹PPP17(Vietnam1998).äx1,x2,...,xnºcæxÿ³Pni=111998+xi=11998fInpx1x2:::xnn11998:IIIš©yi=19981998+xiàdy1+y2+


+yn=1„/nYi=11yi1(n1)n:äf(x)=ln1x1
@åŸLŠf(y1)+


+f(yn)nf1n
ï—f00(y)=12y(y2y)2:àdfê ÐÞ@åý¹G-y1=y2=


=yn1ädºtêI(n1)ln1t1+ln11(n1)t1nln(n1):_ïåô¥(®M2.4ôôôÒÒÒLLL1.)(JensenIAM-GM2.åa2+b2+c2=1,fI1a2+2+1b2+2+1c2+216ab+c2+16bc+a2+16ca+b23.åa+b+c=3,fIXcyca2a2+a+134:4.(MOP2012)åa+b+c+d=4,fI1a2+1b2+1c2+1d2a2+b2+c2+d23ˆˆˆdddÍÍ͌ŒŒ999___3.1WeightedPowerMeanAM-GMïå9Úågeneralizeššš18(WeightedPowerMean).äa1;a2;:::;anºcæxw1,w2,...,wnºcæxÿ³w1+w2+


+wn=1¼Ïæxrš©P(r)=8:(w1ar1+w2ar2+


+wnarn)1=rr6=0aw11aw22:::awnnr=0:års,GP(r)P(s)I_˄Eö/a1=a2=


=any%/åw1=w2=


=wn=1nå„P(r)IùP(r)=8