2002年安徽省高中数学竞赛

2002(6,36)1.P={x|x2=1}Q={x|mx=1}.QP,m().(A)0(B)1(C)2(D)32.ab,ab,().(A)a2b2(B)ba1(C)lg(a-b)0(D)12a12b3.x2+y2=k2f(x)=3sinxk,k().(A)|k|3(B)|k|2(C)|k|1(D)1|k|24.OP=(2,1),OA=(1,7),OB=(5,1).XOP(O).,XAXB,AXB().(A)90(B)arccos41717(C)arccos-41717(D)+arccos-417175.,1.,().(A)(B)2(C)3(D)36.2AB,4ABC1C2C3C4.|sinACiB-sinACjB|13(ij,ij=1,2,3,4),(Ci,Cj).,().(A)(B)(C)(D)(9,54)7.f(x)=x2+2bx+1g(x)=2a(x+b),xab.y=f(x)y=g(x)xOy(a,b)A.,AaObS.8.{an},an,n3,nN,an=an-1-an-2.19851000,19954000,,2002.9.10,,3.,.10.mABx2a2-y2b2=1m2b2a.,ABM(abm).11.ABCDABEF120,MNACBF,AM=FN.AB=1,MN.12.(a+b+c)10,.(20,60)13.f(x)f(x+2)=f(-x+2),y(0,1),x22.f(x).14.aN+,a2,A={y|y=ax,xN+},B={y|y=(a+1)x+b,xN+}.[1,a]b,ABÁ?,bAB;,.115.1,OABC,I.AIBICIDEF,ADBECFOABC.:AAID=BBIE=CCIF.1.(D).P={-1,1},QP,QÁ,{-1},{1}.5320032©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=Á,m=0;Q={-1},m=-1;Q={1},m=1.m0,-1,13.2.(D).3.(B).f(x)=3sinxk,,x2+y2=k2f(x).xk=2,f(x)Pk2,3.k2k22+(3)2|k|2.4.(C).OX=(x0,y0),OP=(2,1),x02=y01.x0=2y0,XA=(1-2y0,7-y0),XB=(5-2y0,1-y0).XAXB=(5-2y0)(1-2y0)+(1-y0)(7-y0)=5y20-20y0+12.,y0=2,XAXB,,OX=(4,2),XA=(-3,5),XB=(1,-1),|XA|=34,|XB|=2,,cosAXB=XAXB|XA||XB|=(-3,5)(1,-1)342=-868=-41717.5.(C).(1)2,SA=SB=SC=BC=1,SASB,SASC,VS2ABC=VA2SBC=312;(2)2,SB=SC=BC=1,SA=22,SAAB,SAAC,VS2ABC=224;(3)2,SA=SB=AB=AC=BC=1,SC=2,SAAC,SBBC,VS2ABC=VA2SBC=212.26.(B).ACiB[0,],sinACiB[0,1](i=1,2,3,4).[0,1]0,13,13,23,23,1,sinAC1B,sinAC2B,sinAC3B,sinAC4B().|sinACiB-sinACjB|13(ij)(Ci,Cj).7..(a,b),x2+2bx+1=2a(x+b).=[2(b-a)]2-4(1-2ab)0,a2+b21.aOb,(a,b)A().,S12=.8.3000.a1=a,a2=b,6a,b,b-a,-a,-b,a-b.a7=a,a8=b,60.S1995=S3326+3=0+a+b+(b-a)=2b=4000.S1985=S3306+5=0+a+b+(b-a)+(-a)+(-b)=b-a=1000.b=2000,a=1000.S2002=S3336+4=0+a+b+(b-a)+(-a)=2b-a=3000.9.51100.3xy,0x10,0y10,|x-y|3,(x,y)3OABC,S1x-y=3.P=100-(10-3)2100=51100.10.a(m+2a)2a2+b2.63©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.|MM|=|AA|+|BB|2=12|AF|e+|BF|e=12e(|AF|+|BF|)|AB|2e=m2e.M=|MN|=|MM|+|MN|m2e+a2c=am2c+a2c=a(m+2a)2a2+b2.11.32,1.55,MPABP,PN,PNAB.MPND2AB2E,MPN=120.AM=FN=x,MP=22x,PN=2-x2.MN2=x22+2-x22-222x2-x2cos120=12(x2-2x+2)=12x-222+34(0x2).,x=22,MNmin=32;x=02,MNmax=1.12.66.ak1bk2ck3,k1+k2+k3=10(k1k2k3N),k1k2,k3.k1=10,k2=0,1;k1=9,k2=01,2;k1=8,k2=012,3;k1=1,k2=0129,10;k1=0,k2=01210,11.1+2+3++10+11=66,,66.13.f(x)x=2,f(x)=a(x-2)2+b.4a+b=1.f(x)xx1x2,|x1-x2|=22,2-ba=22.a=12,b=-1.f(x)=12(x-2)2-1.14.b[1,a]ABÁ,y0Ay0B,y0=am(mN+)y0=(a+1)n+b(nN+).mnN+,am=(a+1)n+b(1ba),n=am-ba+1(mnN+,1ba).am-b=[(a+1)-1]m-b=(a+1)m-C1m(a+1)m-1++Cm-1m(-1)m-1(a+1)+(-1)m-b,nN+,(-1)m-ba+1.(1)m,1-ba+1.a2,1ba,b=1.(2)m,-1-ba+1.2b+1a+1,b=a.,b,b=1b=a.b=1,AB={y|y=a2k,kN+};b=a,AB={y|y=a2k+1,kN+}.615.6,OAG,IKBCK.OR,ABCr,AG=2R,IK=r.IABC,BAA=CAA]AB=AC]AGBC.IKBC,IKAG.,RtIDKRtAGA]IDAG=IKAA]AAID=AGIK=2Rr.BBIE=2Rr,CCIF=2Rr.AAID=BBIE=CCIF.()7320032©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.