2004年北京市中学生数学竞赛高一

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2004()(6,36)1.f(x2)=[f(x)]2().(A)f(x)=x2(B)f(x)=ax2+5(C)f(x)=x2+x(D)f(x)=-x2+20042.Ry=sinx,y=sin|x|,y=sin2004,y=sin2-x,().(A)0(B)1(C)2(D)33.||x|-1|=a.a().(A)0(B)015(C)1(D)24.abca+b0,b+c0,c+a0,f(x)R,,x1x2,f(x1)f(x2).().(A)2f(a)+f(b)+f(c)=0(B)f(a)+f(b)+f(c)0(C)f(a)+f(b)+f(c)0(D)f(a)+2f(b)+f(c)=20045.abcd,a91,b93,c95,d97.().(A)ab(B)bc(C)cd(D)da6.a1a2a3a4a5,a1a2a3,a2a3a4,1.a3a4a5.().(A)a1a3a5(B)a1a3a5(C)1a11a31a5(D)16a113a312a5(8,64)7.f(x)=2x,x0,3,0x1,log13x,x1.f(f(f(-2004))).8.a=1+2++2004.a17.9.f(x)=x2+x-1.ab21,f(a-1)=f(b2)=0,a1+ab2=.10.1,RtABCCRtDEFDF,EABC1AB.ADCE5cm2,,ABFDcm2.11.abRa2+b2=10,a-b.12.abxx4+m=9x2,a+b=4.m.13.cos20cos40cos60cos80=.14.2004n.n.23(8,40)1.f(x)=x21+x2.,f12004+f(1)+f(2004)=.2.(log34+log29)2-(log34-log29)2=.3.{an},a1=1,a2=2,anan+1an+2=an+an+1+an+2,an+1an+21.a1+a2++a2004=.4.2,O1O224,O1O2=10.MNP.235.3,ABMNBCKLACPQ.SABMN-SBCKL=m(m).NQ2-PK2=.(15)abc0.:a44a4+b4+c4+b4a4+4b4+c4+c4a4+b4+4c412.(15)2+2k(kZ).:xx2-(1-cos3)x+cos=0,x2-(1-sin3)x+sin=0,x2-1+cos1-sinx+14=0.4(15)4,OIABC,OIB=30.:BAC=60.(15){an}:an=n-n-(n=1,2,),x2-x-1=0.:(1)n,an+2=an+1+an;(2){an},.1.A.f(x)=ax2+bx+c.a=1,b=0,c=0.2.D.,y=sin|x|,y=sin2004,y=sin2-x.53.C.:y=||x|-1|(5),Oxy=ay=||x|-1|,a=1.4.B.xR,f(x),f(0)=0.f(x),,x0,f(x)0.a+b0,a-b]f(a)f(-b)=-f(b)]f(a)+f(b)0.,f(b)+f(c)0,f(c)+f(a)0.2[f(a)+f(b)+f(c)]0.5.B.x90,1,2,3,4,x290,1,4,7.,92,3,5,6,8,bc3320055.6.A.2a2=a1+a3,a23=a2a4,a-13+a-15=2a-14.a3+a5a3a5=2a4.a4=a23a2,,a3+a5a3a5=2a2a23=a1+a3a23]a23+a3a5=a1a5+a3a5]a23=a1a5.7.-12.,f(-2004)=2-2004=122004.01220041,f(f(-2004))=f122004=3.,f(f(f(-2004)))=f(3)=log133=-12.8.1.a=1+2++2004=200420052=2009010.200901017118177,1.9.-1.f(x)=x2+x-1,f(a-1)=f(b2)=0,ab21a-1b2f(x)=x2+x-1.1a+b2=-1,b2a=-1.1a+b2=b2a=-1.,1+ab2=-a.10.10.ECFBADCE,BC=AC,EF=DE.OCP=PEO=90,EOC+EPC=180.,sinEOC=sinEPC.,SECFB=12EFBCsinEPC=12DEACsinEOC=SADCE=5cm2.SABFD=SADCE+SECFB=5+5=10(cm2).11.[-25,25].abRa2+b2=10,(a-b)2=2(a2+b2)-(a+b)22(a2+b2)=20.|a-b|25.12.12125.x4-9x2+m=0.a2b2y2-9y+m=0,y=x2.,a2+b2=9,a2b2=m.a+b=4,(a+b)2=16,ab=72.13.116.cos20cos40cos60cos80=2sin20cos20cos40cos60cos802sin20=2sin40cos40cos60cos804sin20=2sin80cos80cos608sin20=sin16016sin20=116.14.62.2004n.n,1.1+2++n2004,n(n+1)22004.n2+n-40080.n62.1.32.ab=1,f(a)+f(b)=a21+a2+b21+b2=1+a2+1+b21+a2+b2+a2b2=1.,f(1)=12,f(2004)+f12004=1.,f12004+f(1)+f(2004)=32.2.16.(log34+log29)2-(log34-log29)2=4log34log29=4lg4lg3lg9lg2=42lg2lg32lg3lg2=16.3.4008.a1=1,a2=2anan+1an+2=an+an+1+43an+2,a3=3.anan+1an+2=an+an+1+an+2,an+1an+2an+3=an+1+an+2+an+3,(an+3-an)(an+1an+2-1)=0.an+1an+21,,an+3=an.,{an}.a1+a2++a2004=(a1+a2+a3)+(a4+a5+a6)++(a2002+a2003+a2004)=(1+2+3)668=4008.64.323.6,AB=46,CD=EF=8,O1P=103,O2P=203.PC=PF=83,PD=PE=163.MA=MF=x,ND=NB=y.MB=ME.,AB-x=8+x.,x=26-4.y=26-4.,SAO1O2B=(2+4)462=126.SAO1FM=2(26-4),SBO2DN=4(26-4),SO1FP=83,SO2DP=323.SMPN=126-2(26-4)-4(26-4)-83-323=323.5.3m.NQ2=AN2+AQ2-2ANAQcosNAQ=AB2+AC2-2ABACcosNAQ,BC2=AB2+AC2-2ABACcosBAC.NAQ+BAC=180,cosNAQ=-cosBAC,,NQ2+BC2=2AB2+2AC2.PK2+AB2=2BC2+2AC2.NQ2-PK2=3AB2-3BC2=3m.4a4+b4+c4=2a4+a4+b4+a4+c42a4+2a2b2+2a2c2,,a44a4+b4+c4a42a4+2a2b2+2a2c2=a22(a2+b2+c2).b4a4+4b4+c4b22(a2+b2+c2),c4a4+b4+4c4c22(a2+b2+c2).a44a4+b4+c4+b4a4+4b4+c4+c4a4+b4+4c4a22(a2+b2+c2)+b22(a2+b2+c2)+c22(a2+b2+c2)=a2+b2+c22(a2+b2+c2)=12.a=b=c0,.123,1=(1-cos3)2-4cos,2=(1-sin3)2-4sin,3=1+cos1-sin2-1.,cos0,10.,x2-(1-cos3)x+cos=0.sin0,20.,x2-(1-sin3)x+sin=0.cos0sin0,2+2k(kZ),1-sin0,1+cos1-sin1.,3=1+cos1-sin2-10,,x2-1+cos1-sinx+14=0.,,.77,AIOD,BD.DBC.,IBD=IBC+CBD=ABC2+CAD=ABC2+BAC2,BID5320055=IBA+IAB=ABC2+BAC2.,IBD=BID]ID=BD.BOIA1,A1O,IA1=IB,OIA1=OIB=30.,BIA1=OIB+OIA1=60.,A1BI,A1B=A1I.ID=BD,A1DBI,BA1I.BAD=BA1D=12BA1I=30.,BAC=2BAD=60.(1)x2-x-1=0,,+=1,=-1.n+2-n+2=(+)(n+1-n+1)-(n-n)=(n+1-n+1)+(n-n).,n+2-n+2-=n+1-n+1-+n-n-,an+2=an+1+an.(2){an},an=n-n-(n=1,2,),a1=--=1,a2=2-2-=+=1.n,an+2=an+1+an,,{an}.,.,(an+2,an+1)=d1.n,an+2=an+1+an,d=(an+2,an+1)=(an+1,an)=(an,an-1)==(a4,a3)=(a3,a2)=(1,1)=1,d1.,{an}.()2004()(6,36)1.ab0.,().(A)1a1b(B)1a-b1a(C)|a||b|(D)a2b22.y=log12(2x-x2)().(A)(0,2)(B)[1,+)(C)[1,2)(D)(0,1]3.SA{1,2,,26},S25.S().(A)10(B)11(C)12(D)134.0,2.sin+cos1().(A)sin+cos1(B)sin+cos1(C)sin+cos=1(D)15.1,ABCD-ABCDAABBMABBC.,M().(A)(B)(C)(D)6.A(a,b),B(c,d),63

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