2009我爱数学初中生夏令营数学竞赛

©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=NAP=DAUA=BAC.PKAB,NKAC.NKBD.,LMBD.AKNABDA,OAUA,CMLCDBC,OCUC.UAUCOAOCAC,,.=ANAD=AKAB=AOAAUA=COCCUC=CMCD=CLCB.,DNDA=DMDC=BKBA=BLBC=1-.,KLACMN,KLMN.7.IMO6.2009:50,150;15,150.1.a2+b2+c2+d2+(1-a)2+(a-b)2+(b-c)2+(c-d)2+(d-1)2,a+b+c+d?.2.ABC1O,ABAC=AB-AC=1.ABC,.3.:101?.1.nn+2009-111005n.2.310+72+310-72.3.AB60,120m.A72m,AB24m.,AB.4.112+21+123+32++1nn+1+(n+1)n1920,2021.n.5.xx4-6x3+ax2+6x+b=0x-1,x+272.().6.21,1463,,.7.ABCDO,EOB,OE=1nBE,CEOF,AFDOG.DGGC=.32200910©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=70,ACB=60,DE=DA,DEI.9.,.,,.6,,,;4,,,;2,,,.200911,,,,2,2009.10.n.20092n+1,n+1n,n.1.a,=3a2-(2+2b)a+.,,a=2+2b23=1+b3.,,d=1+c3,b=a+c3,c=d+b3.b=c=15,a=d=25.,,a+b+c+d25+15+15+25=65.2.AB+AC=(AB-AC)2+4ABAC=5,AB=5+12,AC=5-12.OAD,OD=AC=5-12,AD=1-OD=3-52.OACCAD,OAC=CAD,ADAC=5-12=ACOA.,OACCAD.,CD=AC=OD,ACD=AOC=OCD,OAC=OCA=2AOC.5AOC=180]AOC=36]ABC=18.ABE,AE=1,EB=AB-1=5-12.EBOOBA,EBO=OBA,OBBE=5+12=ABOB.EBOOBA.,BE=OE,OE=5-12.,OAE=36,OBA=36,AOB=180-36-36=108.ACB=54126,BAC=10836.,ABC,ABC=18,ACB=54,BAC=108ABC=18,ACB=126,BAC=36.3.x,x3+101=y2(yN+).x,y.x=2n,y=2m+1(nmN).2n3+25=m2+m,,,.42©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=2n+1,y=2m(nmN).(2n+1)3+101=(2m)2]8n3+12n2+6n+102=4m2]4n3+6n2+3n+51=2m2.3n+51,n,n=2n1-1.x=4n1-1(n1N+).x30,y232,.x31,y230,y30.x=3n+1,y=3m(nmN).33n3+33n2+32n+102=9m2.1029,.,x32.x=3n-1(nN+).x=12n-1(nN+).y2014965,2378,,x31267,x1368.,x11,23,35,47,59,71,83,95,.,x11,23,71,83.,x35,47,59,95,.,x=35,47,59,.x=95,953+101=857375+101=857476=9262,.,95.1.1008015.1-nn+200911005]nn+200910041005]nn+20091004210052]n10042=1008016.,nn+2009-111005n1008015.2.234.=x0.x3=20+33100-492x]x3-332x-20=0.x=34y0.4y3-6y-20=0](y-2)(2y2+4y+5)=0.y=2.=234.3.72+323m,643m.C,A72mD.BCA=60,AB=120,AD=72,BD=120-24=96.7296120=345,ADB=90.RtBCD,CD=x.BC=2x.(2x)2-x2=962.x=323,2x=643.AC=72+323m,BC=643m.4.39.1kk+1+(k+1)k=(k+1)k-kk+1(k+1)2k-k2(k+1)=(k+1)k-kk+1k2+k=1k-1k+1.=1-1n+1.19201-1n+12021,20n+121]202-1n212-1.n212-252200910©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=41-2=39.5.-1,1,2,4.1-6+a+6+b=0,(-2)4-6(-2)3+a(-2)2+6(-2)+b=72]a=7,b=-8.x4-6x3+7x2+6x-8=0,(x+1)(x-1)(x-2)(x-4)=0.()-1,1,2,4.6.11291.xy(xy),d,x=dx1,y=dy1.dx1y1,d(x1-y1)=21,d(1+x1y1)=1463.(1463,21)=7,,d=17.d=1,x1-y1=21,x1y1=1462.x1+y1=6289,.d=7,x1-y1=3,x1y1=208.x1+y1=841=29.,x1=16,y1=13]x=112,y=91.,11291.7.1n+1.n+1.OE=1,CE=12+(n+1)2.DF.CFD=90.EOCDFC]ECDC=OCCF.CF=2(n+1)2(n+1)2+1,EF=CF-CE=(n+1)2-1(n+1)2+1.FHBO,H.COEFHE]FHOC=EHOE=EFCE=(n+1)2-1(n+1)2+1.FH=(n+1)2-1(n+1)2+1(n+1),EH=(n+1)2-1(n+1)2+1.AGOAFH,OGAO=FHAH=(n+1)2-1(n+1)2+1(n+1)(n+1)+1+(n+1)2-1(n+1)2+1=nn+2,OG=n(n+1)n+2.DGGC=n+1-n(n+1)n+2n+1+n(n+1)n+2=1n+1.8.25.BAC=50,EDA=ABC=70,DEA=DAE=55,CIA=90+12ABC=180-CEA.,CIAE.,DEI=CAI=1250=25.9.59.6,x,x6.y,4.x+2y=4x6+2y,x=18y.z,2.x+2y+4z=2x6+2y+4z,62©1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.=2y+4z.5y=2z.y2.y=2n(nN+).x+y+z=18y+y+52y=43n.120=31+28+31+3043n31+28+31+30+31=151.,x+y+z=43n=129.,129-31-28-31-30=9.,2,200959.10131.m,m+1,,m+2n2n+1.m2+(m+1)2++(m+n)2=(m+n+1)2+(m+n+2)2++(m+2n)2.Sk=12+22++k2.Sm+n-Sm-1=Sm+2n-Sm+n.2mn2+2n3-m2+n2=0,m=2n2+n.m2009m+2n,2n2+n20092n2+3n.,n=31.()1.x2-2xy+126y2=2009(x,y).()2.ABCDE,AB=DE,BC=EA,ABEA,BCDE.:ABCDAC=AD.()3.xyzR+,a=x(y-z)2,b=y(z-x)2,c=z(x-y)2.:a2+b2+c22(ab+bc+ca).()4..n,n,,.()5.1,2,,9X=(x1,x2,,x9)A.XA,f(X)=x1+2x2++9x9,M={f(X)XA}.M(MM).()16.1,OIABC.:OD,DEF,OIDEF.()72200910