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2004IMO1.XOY=90,PXOY,OP=1,XOP=30,POXOYMN.OM+ON-MN.2.u.:n!=ua-ub(n,a,b).3.n1,n2,,nkk(k2),1n1n2nk,ab1-1n11-1n21-1nkab1-1n11-1n21-1nk-1.:n1n2nk(4a)2k-1.4.DEFABCBCCAAB(),EFBC,D1BC(BDC),D1D1E1DE,D1F1DF,ACABE1F1,E1F1,BC(A)PBC,PBCDEF,PD1.:EFE1F1PD1.5.p1,p2,,p25200425,T,T,(p1p2p25)2004(1,p1,1+p21+p1p2+p3(p1p2p25)2004).6.abc2.:sinasinbsinc.1.O1POX,OY(AB),PAB.1OXOYxy,1.P(32,12).O1(a,a),(32-a)2+(12-a)2=a2,a2-(3+1)a+1=0.,=(3+1)2-4=23.a=3+1-232=3+1-4122().3045,12a=3+1-4122,,PO1OXOY.22,MNPO1,MNOXOY,M1N1P,O1,M1N1OXOY.M1N1O,O1M2N2(Q),M2N2OXOY.OM1+ON1-M1N1OM2+ON2-M2N2=(OB+BN2)+(OA+AM2)-(N2Q+QM2)=2OA.,2OA=OM+ON-MN.,MNPO1,OM+ON-MN,2OA=2a=3+1-412.2..p,p8u,dup,ud-1=pvk,v1,p8k.m,p8m.t(t0),udmpt=1+pt+vkt,p8kt.:t.t=0,ud=1+pvk(p8k)(p8m)umd=(1+pvk)m=1+pvkm+p2vk2C2m+=1+pv(km+pvk2C2m+)=1+pvk1,p8k1.t,udmpt+1=(1+pt+vkt)p=1+pt+1+v(kt+C2ppv+t-1k2t+)=1+pt+1+vkt+1,p8kt+1.(p,p|C2p.)..,n!=ur(us-1),rs.p,np().pn!,1.p8up(us-1).p|(us-1).dup,d|s.s=dmpt,t0,p8m.us-1=pM,p8M=t+v,t=-v.us-1=udmp-v-1.=i=1npinpan,ap.b=udp-v.dpuv,b1.,us-1udp-v-1bpan-1.n,bpan-1nn-1.(bpannn,pannlogbn.),,n,us-1n!,ur(us-1)n!.,n,.,.:pap|a,p+18a,p,N+.3..n1,n2,,nkab,r(1rk),n1n2nr(2r+1a)r.:,ni(1ik),ni2i+1a.abk-1i=1(1-1ni)1ab,ba+1.nini2i+1a,aa+1abki=1(1-1ni)1-ki=11ni1-1aki=112i+11-12a,1-1a+11-12a.2aa+1...ri,ni2i+1a.n1n2nr,n1n2nrnrr(2r+1a)r..k.k=1,1-1n1ab1n1(4a)21-1..ba,ba+1.1-1n1aa+1=1-1a+1.n1a+14a.k,k.r,1rk-1.ki=r+1(1-1ni)ABk-1i=r+1(1-1ni),A=ari=1ni,B=bri=1(ni-1).ki=r+1ni(4A)2k-r-1=(4a)2k-r-1(ri=1ni)2k-r-1.ki=1ni(4a)2k-r-1(ri=1ni)2k-r(4a)2k-r-1(2r+1a)r2k-r.,,r=k().,,42k-r-12r(r+1)2k-r42k-1a2k-r-1ar2k-ra2k-1.2+r(r+1)2r+11+r2r(r1),..34.3,PD1D1E1D1F1EFD2E2F2,E1D2F1.E1D1CE1E2E,,D1E1E1E2=D1CEE2.F1FF2F1BD1,,F2F1F1D1=FF2BD1.PBCD1E2F2DEF,,,PBCD1E2F2,.PD1D1D2,,E2D2D2F2=BD1D1C.,EE2=DD1=FF2(DD1E2EDD1F2F),D1E1E1E2E2D2D2F2F2F1F1D1=D1CEE2BD1D1CFF2BD1=FF2EE2=1.D1E2F2,E1D2F1.5.p12,2(p1p2p25)2004,T=1.p1=2,:p1,p2,,pkk,pipi+1p2005i(i=1,2,,k),p1=2,(p1p2pk)2004{1,2,3,,Tk},Tk=p20051-1p1-1p20052-1p2-1p2005k-1pk-1.(p1p2pk)2004Tk.,1nTk,n(p1p2pk)2004.k=1,1nT1=1+2+22++22004.nn22004.k,1nTk+1.Tk+1=Tk(1+pk+1++P2004k+1)0i2004,Tk(pi+1k+1+pi+2k+1++p2004k+1)nTk(pik+1+pi+1k+1++p2004k+1).i=2004,0.,1n-Tk(pi+1k+1+pi+2k+1++p2004k+1)Tkpik+1.mi,0n-Tk(pi+1k+1+pi+2k+1++p2004k+1)-mipik+1pik+1.,0miTk.n-Tk(pi+1k+1+pi+2k+1++p2004k+1)-mipik+1pk+1,n-Tk(pi+1k+1+pi+2k+1++p2004k+1)-mipik+1=m0+m1pk+1++mi-1pi-1k+1.ji-1,0mjpk+1-1p2005k-1p20051-1p1-1p20052-1p2-1p2005k-1pk-1=Tk.(pu+1-1p2005u-1,p1-1=1.)mi+1=mi+2==m2004=Tk,n=m0+m1pk+1++m2004p2004k+1(0miTk,0i2004).,mi(p1p2pk)2004..,p12,T=1;p1=2,T=p20051-1p1-1p20052-1p2-1p200525-1p25-1.6.0abc.sina0,sinb0,sinc0,|cosa|1,|cosb|1,|cosc|1.sinasinbsinc.a=2,b=c=2.sina=sinb=sinc=1..a2.(1)a+b+c=2,sinc=sin(2-a-b)=-sin(a+b)sina|cosb|+sinb|cosa|sina+sinb.4(2)a+b+c2,abc,abc.4.OROPOQ,OQ=OP=OR=1,QOR=a,QOP=b,POR=c.QPOQ,H.HOR,G.QOH=,HOR=,02,02.sina=QG=QH2+GH2=sin2+cos2sin2=sin2+sin2cos2|sin|.sinb=sin2(c-)+sin2cos2(-c)|sin(c-)|.,.,sin20cos=cos(c-)=0.=2,32,c-=2,32.0c.,sina+sinb|sin|+|sin(c-)||sin(+c-)|=sinc.(2004IMO)