2007年南昌市高中数学竞赛

2007:()(),.(6,36)1.(i)()abc,A={x|x=a2-4a},B={y|y=b2+2b-3},C={z|z=c2+6c+10-5}().(A)A=B=C(B)ABC=Á(C)ABC=R(D)(ii)()x22a2+y2b2=1x2a2-y22b2=1(ab0),().(A)32(B)153(C)64(D)3062.(i)()y=2sinxcosx-3sinx+cosx-2().(A)12,3(B)[2-2,2+2](C)[2-1,2+1](D)1-22,1+22(ii)()abcd,a+b=1,x=ac+bd,y=ac+bd,z=ac+bd.,().(A)xyz(B)zxy(C)yzx(D)zyx3.(i)()ABC,sin2A2+sin2B2+sin2C2=cos2B2().(A)ac=b2(B)a2+c2=b2(C)a+c=2b(D)1a+1c=2b(ii)()ABCD2,EAB,EADEBCEDEC,EAEB,.().(A)13(B)23(C)3(D)434.,3sin2+2sin2=1,3sin2-2sin2=0,sin(+2)=().(A)12(B)22(C)32(D)15.7,.,.().(A)3(B)7(C)21(D)356.(1+2)10()(,[x]x).(A)1(B)3(C)5(D)7(9,54)7.(i)()A={2,4,7,8,13,15}.MM4A,M4A,M=.(ii)()1,2,,933()..8.(i)()f(x)=2cosk4x+32.k1320085.(ii)()y=kx-2y2=8xAB.AB2,|AB|=.9.(i)()cos10cos401-sin10=.(ii)()1,2,3,4,5,6.,.,.10.12+32++(2n-1)222+42++(2n)2=1314,n=.11.{an}a1=8,a2=26,an=an-1+an-2+an-1an-2.a10=.12.f(x):RR,xyR,f(x)f(y)=f(2xy+3)+3f(x+y)-3f(x)+6x.f(x)=.(20,60)13.a,b,x[0,1],1-x+1+x2-bxa.a,b.114.1,ABCDBCFE,,DCAB,CFBE,BCAB,EFBE,AEDF,P,AP=AB,DP=DC.:APE=DPF.15.{an}a2=15,2+4an+1anan-4n+2+anan+1-4n-22+4an-1.:Sn=1a1+1a2++1an.1.(i)A.A=B=C={t|t-4}.(ii)D.2a2-b2=a2+2b2,a2=3b2.e2=2a2-b22a2=5b26b2=56,e=306.2.(i)B.2sinxcosx-3=(sinx+cosx)2-4,,y=(sinx+cosx)+2.-2sinx+cosx2,2-2y2+2.(ii)B.z2-x2=ac+bd+2abcd-(ac+bd)=2abcd0,x2-y2=(ac+bd)-(a2c+b2d+2abcd)=ac(1-a)+bd(1-b)-2bdcd=abc+abd-2abcd=ab(c+d-2cd)0.,zxy.3.(i)C.sin2A2+sin2B2+sin2C2=cos2B2Z1-cosA2+1-cosC2=cosBZ1-cosB=12(cosA+cosC)Z2sin2B2=cosA+C2cosA-C2Z2sinB2=cosA-C2Z2sinB2cosB2=cosB2cosA-C2ZsinB=12(sinA+sinC)Z2b=a+c.23(ii)A.2,ABP,PCD.EPPC,EPPD,EPPCD.,V=13PESPCD=1313422=13.24.D.cos(+2)=coscos2-sinsin2=cos3sin2-sin3sincos=0,0+232,+2=2.sin(+2)=1.5.B.,77(7-1)2=21,,2137;,1,2,,7,7.,3,(1,2,4),(2,3,5),(3,4,6),(4,5,7),(5,6,1),(6,7,2),(7,1,3).6.C.(1+2)10=(3+22)5.a=3+22,b=3-22,Sk=ak+bk.abx2-6x+1=0,an=6an-1-an-2,bn=6bn-1-bn-2,Sn=6Sn-1-Sn-2.S0=2,S1=6,…SS,…Sn=6…Sn-1-…Sn-2.…S0=2,…S1=6,…S2=4,…S3=8,…S4=4,…S5=6.0b1,0b51.[a5]=5.7.(i){11}.A1={x|x=a-4,aA}={-2,0,3,4,9,11},A2={x|x=a+4,aA}={6,10,11,12,17,19},MjA1A2={11}.M,,M={11}.(ii)114.1,2,,9,5,.9,,P=95!4!9!=114.8.(i)13.T=8k2,k412156.,k13.(ii)215.A(x1,y1),B(x2,y2).y=ky28-2,ky2-8y-16=0,y1+y2=8k,y1y2=-16k.8k=y1+y2=k(x1+x2)-4=4k-4,k2-k-2=0.k=1,y=kx-2y2=8x,,k=2.y=2x-2,y2=8x,A(2-3,2-23),B(2+3,2+23).,|AB|=215.9.(i)2.cos401-sin10=cos401-cos80=cos402sin40=22sin80=22cos10,,cos10cos401-sin10=2.(ii)30.33200856,65,,,3!.,53!=30.10.20.2nk=1k2=2714nk=1(2k)2,2n(2n+1)(4n+1)6=27144n(n+1)(2n+1)6,4n+1=277(n+1).,n=20.11.3144-1.an+1=(an-1+1)(an-2+1),{an}.log3(an+1)=log3(an-1+1)+log3(an-2+1).bn=log3(an+1).b1=2,b2=3,bn=bn-1+bn-2(n3).,b10=144.a10=3b10-1=3144-1.12.f(x)=2x+3.xy-3f(x)+6x=-3f(y)+6y.f(x)-2x.f(x)=2x+a.(2x+a)(2y+a)=2(2xy+3)+a+3(2x+2y+a)-3(2x+a)+6x,2(x+y)(3-a)=a2-a-6=(a-3)(a+2),(a-3)x+y+a2+1=0.xy.a=3.,f(x)=2x+3.13.1-x+1+x-2=(1-x+1+x-2)(1-x+1+x+2)1-x+1+x+2=2(1-x2-1)1-x+1+x+2=-2x2(1-x+1+x+2)(1-x2+1).1-x+1+x2-bxa,-2x2(1-x+1+x+2)(1-x2+1)-bxa,bxa-22(1-x+1+x+2)(1-x2+1).f(x)=(1-x+1+x+2)(1-x2+1)(x[0,1]),x=cos20,4,t=sin+cos,1t2.,(1-x+1+x+2)(1-x2+1)=(2sin+2cos+2)(2sincos+1)=(2+2t)t2.t[1,2]t,2+2(2+2t)t28,2+2f(x)8.1-x+1+x-2=-2x2f(x)-x24,2x2f(x)x24.a2,b14.,a2,b14.,a2,b,x[0,1],1-x+1+x-2=-2x2f(x)-bxa,x2-ab2f(x).x=0,04b,.a=2.1-x+1+x-2-bx2(x[0,1]),-2x2f(x)-bx2,b2f(x)(x[0,1]).,bmin2f(x)=28=14.14.4,AABA.AP=AB,PA.DDCD.DP=DC,43PD.Q,ADPQ.PQADM,BCT.4BCABBCCD,BC.,TB2=TPTQ=TC2.TMBCFE.,ME=MF.MEK,MK=MD,PD=PK,DF=EK,EPK=DPF.ABEDCF,DCAB,CFBE,,ABEDCF,AEDF=ABDC=APDP=APKP,AEEK=APKP.,PEAPK.APE=EPK=DPF.15.an.n=1,2+4a1+1a1a1-2+a192+4a1-1.a11,2,a13.2a1-2+a19-4a1+11,2a1-2+a19-4a1-11.a31-10a21-11a1+1080,(a1-10)(a21-11)2.,a13a111.a31-12a21+11a1+360,(12-a1)(a21+11)168.,3a110.a1=3.n=2,2+14159+15a3-102+27,71215a3-101321.31513a3-101807,34+313a335+57.,a3=35.a1+1=4=22,a2+1=16=42,a3+1=36=62.:an=(2n)2-1.n=1,2,3.n(n3),n+1.2+1n22n+12n-1+4n2-1an+1-4n-22+22n2-1,1+1n2-22n-14n2-1an+1-4n-21+22n2-1-22n-1,2n3-3n2+2n-1n2(2n-1)(2n+1)(2n-1)an+1-2(2n+1)4n3-6n2+2n+1(2n-1)(2n2-1).(2n-1)2(2n2-1)4n3-6n2+2n+1an+12n+1-2n2(2n-1)22n3-3n2+2n-1,(2n+1)(4n3-6n2+2n+1)-24n3-6n2+2n+1an+12n+1-2(2n+1)(2n3-3n2+2n-1)+12n3-3n2+2n-1,(2n+1)-24n3-6n2+2n+1an+12n+1-25320085(2n+1)+12n3-3n2+2n-1.,(2n+1)2+2(2n+1)-2(2n+1)4n3-6n2+2n+1an+1(2n+1)2+2(2n+1)+2n+12n3-3n2+2n-1.,(2n+1)2+2(2n+1)-1an+1(2n+1)2+2(2n+1)+1.an+1,,an+1=(2n+1)2+2(2n+1)=[2(n+1)]2-1.,n,an=(2n)2-1.Sn.1an=1(2n-1)(2n+1)=1212n-1-12n+1,Sn=nk=11ak=12nk=112k-1-12k+1=121-12n+1=n2n+1.()(108)(7,42)1.xOy,l:y=-43x+4xyAB,lO90,yA,lC.ABC().(A)325(B)625(C)1225(D)24252.ABC,AB=AC=a,BC=b,A=36.m=a+ba-b,n=(a+b)2ab,p=a3b3.mnp().(A)mnp(B)npm(C)pmn(D)m=n=p3.ABC,BDABCACD,CEACBABE.BE+CD=BC,A().(A)30(B)45(C)60(D)904.ABCD:.ABCD().(A)(B)(C)(D)5.ab(a,b)A=a+b2G=ab,AG,.,().(A)1(B)2(C)3(D)416.1,PABCAB.BCPC().(A)BC32PC(B)BC33PC63