数学奥林匹克初中训练题8

(8)(7,42)1.abc,,a+b2b+c2c+a2().(A)(B)(C)(D)2.1,EFABCD1ADBC,ABGDCH1520.1().(A)15(B)20(C)35(D)403.ABCD().(A)ACBD,ACBD(B)ACBD,A=C(C)ACBD,AC(D)ACAC,A=C4.x-y=1,x4-xy3-x3y-3x2y+3xy2+y4().(A)-1(B)0(C)1(D)25.b111,c.x2-bx+c=0().(A)50(C)55(C)57(D)586.,,,4.().(A)727(B)527(C)78(D)2164(7,28)1.a+b+c=0,abc.ca.2.:1212-100+5000+3232-300+5000+5252-500+5000++992992-9900+5000=.3.2,RtABCS1S2S3,S1=S2+S3.2RtABCS1S2S3,S1=S2+S3.,.4.5(),3,,,9220071©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(20)tx2-x-1=0.a,bc(at+1)(bt+c)=1.(1)bc(a);(2)a,bc12008?,a;,.(25),1:11505110010013119,,1314;,1008.?3(25)3,ABCADBECF.BPFCPDAPE1,APFDPBEPC.1.C.abc,ab,a+b2.2.C.EF.ABEF,SEGF=SABG,SEGF=15.,SEFH=20.,35.43.D.4,ACBD,ACBD,ACAC,(A)(C).(B)ABCD,(B).4.C.x-y=1,x3-y3=(x-y)(x2+xy+y2)=x2+xy+y2=(x-y)2+3xy=1+3xy,x4-xy3-x3y-3x2y+3xy2+y4=x(x3-y3)+y(y3-x3)+3xy(y-x)=(x-y)(x3-y3)-3xy(x-y)=1+3xy-3xy=1.5.A.b=2,b2-4c=4-4c0,c1,;b=4,b2-4c=16-4c0,c4,,c=1,2,3;b=6,b2-4c=36-4c0,c9,,c=1,2,,8;b=8,b2-4c=64-4c0,c16,,c=1,2,,15;b=10,b2-4c=100-4c0,c25,,c=1,2,,24.3+8+15+24=50().6.A.5.55,.6,,27-6=21.434,2134=727.1.-2ca-12.03©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=0,abc,a0,c0.1+ba+ca=0,ba=-1-ca.abc,1baca.1-1-caca.-2ca-12.2.50.k2k2-100k+5000+(100-k2)(100-k)2-100(100-k)+5000=k2k2-100k+5000+10000-200k+k2k2-100k+5000=2(k2-100k+5000)k2-100k+5000=2.k=1,3,5,,49,,=225=50.3..,S1S2S3=AB2BC2CA2.RtABC,AB2=BC2+CA2.,S1=S2+S3.4.6.4.1,,,2.,(3+4+1+2+2)2=6.2,x,3+4+2+x+x=9+2x,.(1)tx2-x-1=0,,t2=t+1.(at+1)(bt+c)=1,abt2+act+bt+c-1=0,ab(t+1)+act+bt+c-1=0.,(ab+ac+b)t+ab+c-1=0.t,abc,,ab+ac+b=0,ab+c-1=0.b=aa2-a-1,c=-a+1a2-a-1.(2).b=12008,aa2-a-1=12008,a2-2009a-1=0,.c=12008,-a+1a2-a-1=12008,a2+2007a+2007=0,.a.100,1300,,100.10089=112,112.xy.x+y=112,xy50.11211=12321314,xy50.100,12,900+1312=10561314.,50,50100.1x50,51y100,x+y=112,13x+11y=1314.x=41,y=71.4171.APFDPBEPCxyz.SDPBSDPC=BDDC,SDABSDAC=BDDC,SDPBSDPC=SDABSDAC,y1=x+y+1z+2.,x-y=yz-1.,y-z=zx-1,z-x=xy-1.xyz,yz-1=x-y0,xy-1=z-x0.,xyyz,xz.,x=y=z.0=x-y=yz-1=z2-1,z=1.,x=y=z=1.1.(,210008)1320071©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.