2007年新知杯上海市高中数学竞赛

(axii)110+282=166,(-byii)110,a1,a2,,am,b1,b2,,bn6(2ii2+1i2-1i2+1,i;2ii2+1,i2-1i2+1,i;,i2+1100,i10),,ax11,ax22,,axmm,by11,by22,,bynn.m=166,n=110100170,m-n50.,:a2i+1-b21,a2i+2-b22,,a2i+n-b2n(i=0,1,,m-n).,2008,.(,510006)2007:.(14,7,58,8,60)1.x1-1+2x2-4+3x3-9=12(x1+x2+x3)(x1,x2,x3)=.12.1,1EF,EF3ABCD.EF,EFM.3.{an}a1=0,an=a2n-1+i(n2).2007.4.-l-45,C,26,AB.ABC.5.f:P(m,n)P(m,n)(m0,n0).fx2a2+y2b2=1(x0,y0),.6.f(n)=cosn4.:f(1)f(3)f(2n-1)=.7.{xn}x1=0,x2=1,xn=xn-1+xn-22(n3).{xn}xn=.8.M:(x-1)2+(y-3)2=4,xP(a,0)MPBA,PB=PA.Pa.(60)9.(14)n,S(n)1x+1y=1n(x,y)(,1x+1y=12(6,3),(4,4),(3,6),S(2)=3).S(n)=2007n.10.(14)xx3sin-(sin+2)x2+6x-4=07320086©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=9sin2-4sin+3(1+cos)(2cos-6sin-3sin2+2).211.(16)2,y2=2px(p0),ABF.ABx(02),AOB.12.(16)n:OnA1,A2,,An,C2nAiOAj(1ijn),2007120.1.(2,8,18).2x1+x2+x3-2x1-1-4x2-4-6x3-9=0.(x1-1-1)2+(x2-4-2)2+(x3-9-3)2=0]x1-1-1=0,x2-4-2=0,x3-9-3=0.x1=2,x2=8,x3=18.2.8+.33,EF,M1214.,2.24+212=8+.3.-1003+2i.a1=0,a2=i,a3=-1+i,a4=-i,a5=-1+i,a6=-i,,n3,an=-1+i,n;-i,n.S2007=0+i+(-1+i)+1002[(-i)+(-1+i)]=-1003+2i.4.102.44,CPQ.ABPQ,ABC,|PQ|=(22)2+122-22212cos(180-45)=102.5.y=b21-xa2(0xa2).y=f(x)(sxt),P(x,f(x))P(x,f(x))x2a2+y2b2=1(x0,y0).xa2+f(x)b2=1(x0,f(x)0),f(x)=b21-xa2(0xa2).6.-12n2,n;12n2,n.kZ+,f(2k-1)f(2k+1)=cos(2k-1)4cos(2k+1)4=12cosk+cos2=(-1)k2.,f(1)f(3)=f(5)f(7)=83©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=f(4k+1)f(4k+3)=-12.n,f(1)f(3)f(2n-1)=-12n2;n,f(1)f(3)f(2n-1)=-12n-12cos(2n-1)4=-12n-12cosn-12+4=-12n-12(-1)n-1222=12n2.7.23-23-12n-1.xn=xn-1+xn-22xn+12xn-1=xn-1+12xn-2.x2+12x1=1,xn+12xn-1,1,xn+12xn-1=1,xn-23=-12xn-1-23.x1-23=-23,,xn-23-23-12.xn-23=-23-12n-1.,xn=23-23-12n-1.8.1-33a1+33.M(1,3),d=4.55,PM.PM+d2PM-d2=PBPA.PB=BAZPM2-d24=2AB22d2Z|PM|32dZ(a-1)2+3262Z1-33a1+33.9.1x+1y=1n(xynZ+)xn,yn.x=n+a,y=n+b(abZ+).1n+a+1n+b=1nZn2=ab.,S(n)(a,b).,S(n)n2.n=p11p22pkk,,p1,p2,,pk,iZ+(1ik).n2=p211p222p2kk.n2(21+1)(2k+1).(21+1)(2k+1)=2007=32223.k=1,1=1003k=2,1=1,2=334k=2,1=4,2=111k=3,1=2=1,3=111.n=p10031n=p1p3342n=p41p1112n=p1p2p1113.10.(x-1)(x2sin-2x+4)=0.3,,xx2sin-2x+4=02,=4-16sin0,sin0Z0sin14.9sin2-4sin+3=9sin-292+239239,0(1-cos)(2cos-6sin-3sin2+2)=2(1-cos)(1+cos)(1-3sin)=2sin2(1-3sin)=8932sin32sin(1-3sin)9320086©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=8927.u2398927=6218.sin=29,.umin=6218.11.02,ABy=tanx-p2,x=coty+p2.=2.y2-2pcoty-p2=0.yA=p(cos+1)sin,yB=p(cos-1)sin.xA=cotp(cos+1)sin+p2=p(1+cos)22sin2,xB=p(cos-1)22sin2.66,AByAPBQ.tanAOP=|AP||OP|=xAyA=1+cos2sin,tanBOQ=xB-yB=1-cos2sin.,tan(AOP+BOQ)=43sin.AOB=-(AOP+BOQ)=-arctan43sin.12.,n=90,7,ABO,AB45,,2C245=4544=19807120.,n=90.,n=91,:2007120.91A1,A2,,A91,AiOAj120,AiAj,G.Ge.AiOAj120,AjOAk120,AiOAk120,G.e=0,C291=40952007120,.e1,A1A2,,,Ai(3i91),A1A2.,d(A1)+d(A2)89+2=91,,d(A)A.d(A1)+d(A2)++d(A91)=2e,GAiAj,d(Ai)+d(Aj)91.,(d(A1))2+(d(A2))2++(d(A91))291e.91[(d(A1))2+(d(A2))2++(d(A91))2][d(A1)+d(A2)+d(A91)]2=4e2.4e291(d(A1))2+(d(A2))2++(d(A91))291e,e91242071.,91,C291-2071=20242007,.,120.,n91.()04©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.