2003年安徽省高中数学竞赛初赛

2003()(6,36)1.:A-B={x|xAx|B}.M={x|1x2002,xN+},N={y|2y2003,yN+},N-M().(A)M(B)N(C){1}(D){2003}2.f(x)=-(cosx)lg|x|().13.a+bm4a2+b2ab,m().(A)2(B)2(C)234(D)44.2x2-xy-y2-x-2y-1=03x2-4xy+y2-3x+y=0().(A)2(B)3(C)4(D)5.0a1.x1=a,x2=ax1,x3=ax2,x4=ax3,,xn=axn-1,,{xn}().(A)(B),(C)(D),6.1C,O1,C,O2,,.O2().(A)(7-43)(B)(7+43)(C)2-32(D)2+32(9,54)7.y=x+kx=1-y2,k.8.(4x2-2x-5)1+1x25,.9.n2003.(sin+icos)n=sinn+icosn,n.10.,,,504746.,.11.a=(cos,sin),b=(cos,sin),ab|ka+b|=3|a-kb|,k0.ab.12.xyz.x+y+z=15.(15,60)13.aR,f(x)=ax2+x-a(|x|1).(1)|a|1,:|f(x)|54;(2)f(x)178a.14.A(5,0)x24-y2=1(2x25,y0)P1,P2,,Pn.n,|P1A|,|P2A|,,|PnA|,d15,15?,n;,.(5=2124)15.A4(BCDE),2.5,,,?2316.3,ABCO,ABC,OO1O2O3.03©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(x),y,(B)(D).x,x(1,0),2,0.x=32,f32=-cos32lg320.321,2,(C).3.C.a+b2(a+b)2a2+b22.a=b.,m234,m234.4.D.2x2-xy-y2-x-2y-1=0,2x+y+1=0x-y-1=0.3x2-4xy+y2-3x+y=0,3x-y=0x-y-1=0.,,..5.B.(1)x1=a1,x2=ax1,1a,a1ax1,x1x2.(2)x1=a1,x3=ax2,1=a0,x2=ax1.0x1,1ax1.,a1ax2,x1x3.x1x3x5.(3)x2x4=ax1-x3a0=1,x2x4.x2x4x6.{xn},.46.A.4,O2r,O2D.O1O2BD.ADB=,sin=13.DEO2,DO2=rsin=3r,O1O2=r+12.,23r+r+12=BD=3.r=2-32,4r2=(7-43).7.k=-2k(-1,1].8.15.=(4x2-2x-5)1+5x2+10x4+.,(-5)1+45=15.9.501.(sin+icos)n=[i(cos-isin)]n=in[cos(-)+isin(-)]n=in(cosn-isinn)=in-1(sinn+icosn).in-1(sinn+icosn)=sinn+icosn,in-1=1.,n=4k+12003,k(k0).n2003-14+1=501.10.240.0,98,1,87,7,21,72+56+42+30+20+12+6+2=240.11.12.|ka+b|2=(3|a-kb|)28kab=(3-k2)a2+(3k2-1)b2.ab=(3-k2)a2+(3k2-1)b28k.a=(cos,sin),b=(cos,sin),,a2=1,b2=1,ab=k2+14k.k0,k2+12k,k2+14k2k4k=12.,ab12.12.91.15151,1+1++1=15,142,1,C214=91.13.(1)|f(x)|=|ax2+x-a|=|a(x2-1)+x||a(x2-1)|+|x||x2-1|+|x|=1-|x2|+|x|=-|x|-122+5454.(2)a=0,f(x)=x(|x|1)f(1)=1,.1320043©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(x).(i)a0,,x=-1x=1.f(-1)=a-1-a=-1,.f(1)=a+1-a=1.(ii)a0,x=1x=-1,f(x)=ax2+x-a(|x|1)178,-1-12a1,f-12a=178,a0]a-12,(a+2)a+18=0]a=-2.14.d0,,|P1A|,|PnA|.55,x24-y2=1(2x25,y0),A(5,0),lx=45,e=52.,|P1A|=5-2,n|PnA|=3.,3=(5-2)+(n-1)d.d=5-5n-1(n1).d15,15,155-5n-115.n(55-4,26-55).n,,n14.,n14,14n.,,n3.,n=3,4,5,,14.15.,,:(1),P55=120.(2).C45.AC14,4,,2(BDCE),C13,P22.C45C142C13P22=240.(3).C35.ABDCEP33,C35P35=60.,120+240+60=420.616.(1)6,OAOCO2AO2C,,OAO2C.,O2CAO.,OAO3B,AOO3B.,O3BO2C.,BCO2O3,BCO2O3.(2)ACO1O3,ABO1O2.(3)OO2AC,OO1BC,OO3AB.OO1O2O3,OO2O1O3,OO3O1O2.OO1O2O3.(4),:O1O2O3,O1O2O3,O.OO1OO2OO3,ABC.()200334,1993,,,,,,,23©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.