对称式和轮换对称式的性质及其应用

(,642450):2005-08-03().,,,..11.1,,,..1.2,,,,,.,,.1.3,,.1.4,:1.2.3.4.5mn,m+n.22.11(x+y+z)(xy+yz+zx).:xyz,5,xyz,,,,.:x(xy+yz+zx)=x2y+xyz+zx2,,=x2y+xyz+zx2+y2z+yzx+xy2+z2x+zxy+yz2=x2y+zx2+y2z+xy2+z2x+yz2+3xyz.2.2,,.,,.2:(xy-1)2+(x+y-2)(x+y-2xy).(1996,)2©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.:x+y=u,xy=v,=(v-1)2+(u-2)(u-2v)=v2-2v+1+u2-2u-2uv+4v=(u-v)2-2(u-v)+1=(u-v-1)2=(x+y-xy-1)2=(x-1)2(y-1)2.3ABCabc,a-b1+ab+b-c1+bc+c-a1+ca=0.ABC.(1989,):abc,,abc.:,f,f=(a-b)(1+bc)(1+ca)+(b-c)(1+ab)(1+ca)+(c-a)(1+bc)(1+ab)=a(b2-c2)+b(c2-a2)+c(a2-b2)=0.a=b,f=0,fa-b.fabc,,fb-cc-a.,fg=(a-b)(b-c)(c-a).fg,fg,f=k(a-b)(b-c)(c-a)=0.,a-bb-cc-a0,abc,.,.2.34xy,xy+x+y=71,x2y+xy2=880.x2+y2.(14):,,.:x+y=u,xy=v.u+v=71,uv=880.uvt2-71t+880=0.t=16t=55.,u=16,v=55u=55,v=16,x+y=16,xy=55x+y=55,xy=16.x2-16x+55=0,=62,;x2-55x+16=0,=2961,.x+y=16,xy=55.x2+y2=(x+y)2-2xy=146.5xyz=1,x+y+z=2,x2+y2+z2=16.1xy+2z+1yz+2x+1zx+2y=.(2003,()):xyz,1,.:x+y+z=2x2+y2+z2+2(xy+yz+zx)=4.x2+y2+z2=16xy+yz+zx=-6.x+y+z=2,z=2-x-y.,1xy+2z=1xy-2x-2y+4=1(x-2)(y-2).,1yz+2x=1(y-2)(z-2),1zx+2y=1(z-2)(x-2).1xy+2z+1yz+2x+1zx+2y320064©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=1(x-2)(y-2)+1(y-2)(z-2)+1(z-2)(x-2)=z-2+x-2+y-2(x-2)(y-2)(z-2)=x+y+z-6xyz-2(xy+yz+zx)+4(x+y+z)-8=2-61-2(-6)+42-8=-413.2.46abc.:a4(a-b)(a-c)+b4(b-c)(b-a)+c4(c-a)(c-b)0.(2000,):f,f=-(b-c)a4-(c-a)b4-(a-b)c4(a-b)(b-c)(c-a).5,3,5,2.f=-(b-c)a4-(c-a)b4-(a-b)c4(a-b)(b-c)(c-a)=k(a2+b2+c2)+p(ab+bc+ca).a=0,b=1,c=2,5k+2p=7;a=1,b=2,c=3,14k+11p=25.p=1,k=1.f=(a2+b2+c2)+(ab+bc+ca)=12[(a+b)2+(b+c)2+(c+a)2].abc,,f0.,.7abx2-3x+1=0,cdx2-4x+2=0.ab+c+d+bc+d+a+cd+a+b+da+b+c=B.:a2b+c+d+b2c+d+a+c2d+a+b+d2a+b+c=7B-7.(19911992):a+b=3,ab=1;c+d=4,cd=2.a+b+c+d=3+4=7.a2+b2=(a+b)2-2ab=7,c2+d2=(c+d)2-2cd=12,,a2+b2+c2+d2=19.a2b+c+d=a2+7a-7ab+c+d=7a-a(7-a)b+c+d=7ab+c+d-a.abcd,b2c+d+a=7bc+d+a-b,c2d+a+b=7cd+a+b-c,d2a+b+c=7da+b+c-d.a2b+c+d+b2c+d+a+c2d+a+b+d2a+b+c=7ab+c+d+bc+d+a+cd+a+b+da+b+c-(a+b+c+d)=7B-7.:a3b+c+d+b3c+d+a+c3d+a+b+d3a+b+c=49B-68.2.5,.8x3+x3y3+y3=17,x+xy+y=5.(1990,):x+y=u,xy=v,4©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=17,u+v=5.3-uv=6.u=2,v=3u=3,v=2,x+y=2,xy=3x+y=3,xy=2.,xyt2-2t+3=0t2-3t+2=0.,t=1t=2.x1=1,y1=2;x2=2,y2=1.1.x+y=3,x2+y2-xy=4.x4+y4+x3y+xy3.(13)(:x+y=u,xy=v.:36.)2.:a2(b+c)+b2(c+a)+c2(a+b)-a3-b3-c3-2abc.(:a+b=c,=0.:(a+b-c)(b+c-a)(c+a-b).)3.a21b-1c+b21c-1a+c21a-1ba1b-1c+b1c-1a+c1a-1b=.(1989,)(:,k(a+b+c).:a+b+c.)4.0abc1a+1b+1c=1a+b+c.:abc.(1999,())(:,f=(ab+bc+ca)(a+b+c)-abc.a+b=0,f=0.(a+b)(b+c)(c+a)=0.)5.x+xy+y=1,x2+x2y2+y2=17(x,y)=.(1996,)(:x+y=u,xy=v.:x1=3+172,y1=3-172;x2=3-172,y2=3+172.)(),20068,:11::(1)();(2)()21,():40(),5,10,,,314150:(430062)512006431,520064©1994-2009ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.