2007年北京市中学生数学竞赛初二

7.23,2-1,1-22.x=2y,62y3+42y2-172y+18y-6+52=0.y=13.x1=23.,3x3+22x2-(17-92)x-(6-52)=(x-23)(3x2+32x+92-15).=3(x-23)(x-2+1)(x+22-1).,x1=23,x2=2-1,x3=1-22.8.k-88+18k(k-16).ab(ab).4(a+b)2ab=k,4a2+(8-k)ab+4b2=0.t=ab,4t2+(8-k)t+4=0.t=18[(k-8)+k(k-16)].9.6223.a=2007,DDC=.BDE=2,CD=atan,BD=a(1-tan).,BDEa(1-tan)(1+tan2+sec2)=acos-sincoscos2+sin2+1cos2=acos-sincos2cos2+2sincoscos2-sin2=2a=6223.10.20119.x2+(x+1)2=v2,(2x+1)2=2v2-1.u=2x+1,u2-2v2=-1.,(u0,v0)=(1,1).un+vn2=(u0+v02)2n+1.,(u1,v1)=(7,5),(u2,v2)=(41,29),(u3,v3)=(239,169),u4400.x=20119.()2007()(5,25)1.abc3,abc=16,ab-bc+ca().(A)249(B)253(C)263(D)2642.191504.().(A)27(B)24(C)21(D)183.2007().(A)675(B)1326(C)1329(D)13324.x,[x]x,,[3114]=3,[-7159]=-8.x3x+77=4().(A)4(B)3(C)2(D)15.1,8,4,.()().104©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(A)10(B)12(C)14(D)16(7,35)21.12,,21,32,53,74,.,2007.2.331,2,3,4,5,6,7,8,9,.,.M,m.M-m.33.3,ABC,AB=AC,PQACAB,AP=PQ=QB=BC.PCQ=.4.1a+1b1+1a+1c1+1a1+1b+1d1+1a1+1b1+1c-1+1a1+1b1+1c1+1d.45.4,ABCD,EFGABBCCD.ABCD40cm2.MFNPcm2.(15)abc.b2+c2-a22bcc2+a2-b22aca2+b2-c22ab1,:1,-1.5(10)5,ABC,ABC=46,DBC,DC=AB,DAB=21.CAD.(15)n,8.n..1.C.{a,b,c}={1,2,8}.b=1,c=2,a=8,ab-bc+ca8-1+28=263.2.B.n-1,n,n+1.191504=1n-1+1n+1n+1=3n2-1n(n2-1),191n3-1512n2-191n+504=0.(n-8)(191n2+16n-63)=0.n1,191n2+16n-630.n=8.3.D.2007=32223,,200720072007-[20073]-[2007223]+[20073223]=1332().4.B.[3x+77]=4,43x+775,7x283.x=7,8,9.5.A.66,EEFBDF.FBD.ABFE,DEDA=DFDB,DE8=2545]DE=5.1420081©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=12DEAB=10.1.1008017.iai.,a1=2,a2i=a2i-1+i,a2i+1=a2i+(i+1).2007=21003+1,,a2007=1+2(1+2++1003)+1004=10042+1=1008017.218.,3mM7,mM.M-m{-4,-3,-2,-1,1,2,3,4}.7,873.30.A=2,ABC=ACB=90-,BCQ=BQC=45+2,PCQ=ACB-BCQ=45-32.(AP2cos2+BQ)2sin=BC,2sin(2cos2+1)=1]2sin(3-4sin2)=1]sin3=3sin-4sin3=12]=1050().PCQ=45-32=30.4.-1.1a+1b(1+1a)+1c(1+1a)(1+1b)+1d(1+1a)(1+1b)(1+1c)=-1+(1+1a)+1b(1+1a)+1c(1+1a)(1+1b)+1d(1+1a)(1+1b)(1+1c)=-1+(1+1a)(1+1b)(1+1c)(1+1d).519.FPADQ,,QAD.PQ=12AE=14AB,FP=34AB.,MN=35BC.SMFNP=12FPMN=123435SABCD=9.b2+c2-a22bc+c2+a2-b22ac+a2+b2-c22ab=1,b2+c2-a22bc-1+a2+c2-b22ac-1+a2+b2-c22ab+1=0]b2+c2-a2-2bc2bc+a2+c2-b2-2ac2ac+a2+b2-c2+2ab2ab=0](b-c)2-a22bc+(a-c)2-b22ac+(a+b)2-c22ab=0]a(b-c+a)(b-c-a)+b(a-c+b)(a-c-b)2abc+c(a+b+c)(a+b-c)2abc=0](a+b-c)(ab-ac-a2+ab-bc-b2+ac+bc+c2)2abc=0](a+b-c)(c2-(a-b)2]2abc=0](a+b-c)(c+a-b)(c-a+b)2abc=0.,a+b-c=0c+a-b=0b+c-a=0.24©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(1)a+b-c=0,b2+c2-a22bc=b2+c2-(b-c)22bc=1;c2+a2-b22ac=c2+a2-(c-a)22ac=1;a2+b2-c22ab=a2+b2-(a+b)22ab=-1.(2)c+a-b=0,b2+c2-a22bc=1,c2+a2-b22ac=-1,a2+b2-c22ab=1.(3)b+c-a=0,b2+c2-a22bc=-1,c2+a2-b22ac=1,a2+b2-c22ab=1.(1)(2)(3),b2+c2-a22bcc2+a2-b22aca2+b2-c22ab1,-1.88,ABDADAED,EAD=21,AE=AB.,DE=BD.ADC=21+46=67.ADE=ADB=180-67=113,CDE=113-67=46.CE.DC=AB,,CDEABDAED.OAEDC.ODE=OED=46,,OD=OE.DC=AE,AO=CO]OCA=OAC]COE=2ACO.COE=246=92.,2ACO=COE=92.,ACO=46=OAC.,DAC=DAE+EAC=67.n14,.n=14,9999993,9999994,,1000000614,8.n14,.,,n15.n=15,.a1,a2,,a1515,15,0,,:(1)a1,a2,,a150,aiaj,aiaj0,ai,ai+1,,ai+9,aj11,,ai,ai+1,,ai+9.niai,10ni,ni+1,,ni+9.108.(2)a1,a2,,a150(ai):(i)i1i8,ai7.,ai,ai+1,,ai+788.,8.(ii)i9i15,ai8,ai-8,ai-7,,ai-1,88.,8.,15,,8.15.,n15.()3420081©1994-2008ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.