2003年上海市高中数学竞赛CASIO杯

,1d2+1e2+1f2+1g2=121d+1e+1f+1g2.1g=1d+1e+1f+21d+1e+1f2-21d2+1e2+1f2=1d+1e+1f+2d+e+fdef.SDEF=14SABC=pr4,SDEF=(d+e+f)def,r2=2p(d+e+f)def=defd+e+f.tp2+1-32rr2g12-3=2+3.r2g=defd+e+f1d+1e+1f+2d+e+fdef=x+y+zxy+yz+zx+2,x=1d,y=1e,z=1f,(x+y+z)2xy+yz+zx3.(x+y+z)2-3(xy+yz+zx)=12[(x-y)2+(y-z)2+(z-x)2]0,,.2003(CASIO)(7,70)1.ax2+bx+c=0(a0)1,4.(b,c).2.ABC,ABC=60,AC=3.ST,SA=SB=SC=2,TA=TB=TC=3.ST.3.1+2x3x.4.f(x)=|x2-a|[-1,1]M(a)=.5.ABC5,P,PA=4,PB=3.PC.6.{an}nSn=cnan,n,c,a1a2.a100a99.7.[x]x.x,x+99x=[x]+99[x].,x.8.xx2-6x+a=0x2+26x+b=0,1,ab.9.x,f(x+4)+f(x-4)=f(x)f(x)..10.xy,xy(x2+y2)(3x2+y2)1k.k.(16)ab,i,z4z2+(2a+i)z-8b(9a+4)-2(a+2b)i=0..(16)abca+b+c=2,abc=4.(1)abc;(2)|a|+|b|+|c|.(18)abc,x3-ax2+bx-c=0.abc,.1.x2-y2=1(y0)2.3+223.x14.1-a,a12,a,a125.25-12323©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(9a+4)+[x-2(a+2b)]i=0Z4x2+2ax-8b(9a+4)=0,x-2(a+2b)=0Z5a2+16b-142=1,x=2(a+2b).,a=15cos,b=14sin+14,x=215cos+12sin+12=355sin(+)+1,,sin=23.,xmax=355+1.(1)a=max{a,b,c}.a0,b+c=2-a,bc=4a.,bcx2-(2-a)x+4a=0,,=(2-a)2-44a0Za3-4a2+4a-160Z(a2+4)(a-4)0Za4.a=4,b=c=-1,4.(2)abc=40abc0.abc0,abc(0,2),(1).abc.,a0,b0,c0.,|a|+|b|+|c|=a-(b+c)=a-(2-a)=2a-224-2=6.a=4,b=c=-1,.,|a|+|b|+|c|6.x1x2x3.x1+x2+x3=a,x1x2+x2x3+x3x1=b,x1x2x3=c.x1x2x3Zx1x2x3R+x1+x2x3,x2+x3x1,x3+x1x2.x1x2x3R+]abcR+.:abcR+x1x2x3R+.,x1x2x3=c0x1x2x3R+x1x2x3.x1x2x3,,x10,x20,x30.x1+x2+x3=a0,,x1-x2-x3.,x1(x2+x3)-(x2+x3)2.,b=x1(x2+x3)+x2x3-(x2+x3)2+x2x3=-x22-x2x3-x23=-x2+x322-34x230..x1x2x3R+.x1+x2x3,x2+x3x1,x3+x1x2Za-2x30,a-2x10,a-2x20](a-2x1)(a-2x2)(a-2x3)0.(a-2x1)(a-2x2)(a-2x3)0]a-2x30,a-2x10,a-2x20.a-2x1a-2x2a-2x3.,,a-2x10,a-2x20,a-2x30.,2a-2(x2+x3)0,2x10..,.(a-2x1)(a-2x2)(a-2x3)=a3-2(x1+x2+x3)a2+4(x1x2+x2x3+x3x1)a-8x1x2x3=-a3+4ab-8c,,,,abcR+,-a3+4ab-8c0.()3320045©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.