2009年北京市中学生数学竞赛高一年级决赛试题及参考解答

20098375():zxss.chinajournal.net.cn:[e-mail]2009(40,8)1.ab,c=a+2bd=5a-4b,aba,b=.:3.aba,b=,:0=cd=(a+2b)(5a-4b)=5(a)2+10ab-4ab-8(b)2=5+6cos-8=6cos-3,cos=12,=3.2.1cos290+13sin250.:433.1cos290+13sin250=1cos70-13sin70=3sin70-cos703sin70cos70=32sin70-12cos7034sin140=sin70cos30-cos70sin3034sin140=4sin(70-30)3sin40=43sin403sin40=433.13.1,OMB,MOA,MA=4,MB=43.NAB.().:4(6-)3.22,,MOC,r,MC=4+2r,MB2=MAMC,48=4(4+2r),r=4cm,OC=OA=AM=4cm.OB,OBM,sinMOB=MBOM=438=32,MOB=60,AB60,NAB,AN=30,ON,MON=30,SMON=12OMONsin30=128412=8(cm2).AON=4230360=43(cm2),=8-43=24-43=4(6-)3(cm2).4.320+142+320-142.:4.320+142+320-142=x0320098375():zxss.chinajournal.net.cn:[e-mail]3=20+142+33(20+142)(20-142)(320+142+320-142)+20-142=40+6x,x3-6x-40=0,,4,(x-4)(x2+4x+10)=0.=42-410=-240,x2+4x+10=0.,x3-6x-40=0x=4,320+142+320-142=x=4.5.,m,y=mx2+(2m+1)x-(3m+2)y=-x+1.():(1,0),(-3,4),(32,-12).3y=mx2+(2m+1)x-(3m+2)=m(x+3)(x-1)+(x-2),A(1,-1),B(-3,-5).AByy=-x+1C(1,0),D(-3,4).ABy=-x+1E(32,-12).CDE.(15)ABC4r,t.:ABCabx(t-22r)x2+22r2x-2tr2=0.ABC,C=90,AB=c,AC=b,CB=a,CCD=t,O,r.OA,OB,OC,SABC=12abSABC=SADC+SBDC=12btsin45+12atsin45=24t(a+b),ab=12t(a+b)SABC=SOBC+SOAC+SOAB=12r(a+b+c)=12r(a+b+a+b-2r)=r(a+b-r)=(a+b)r-r2,ab=2(a+b)r-2r2a+b,aba+b=22r222r-t,ab=2tr222r-t,,abx2-22r222r-tx+2tr222r-t=0,(t-22r)x2+22r2x-2tr2=0.(15)f:Z+Z+,(1)f(1)=1;(2)x,yZ+,f(x+y)=f(x)+f(y)+xy.f:Z+Z+,(1)f(1)=1;(2)x,yZ+,f(x+y)=f(x)+f(y)+xy,n,m,f(2n)=2f(n)+n2,f(3n)=f(n)+f(2n)+2n2,f(4n)=f(n)+f(3n)+3n2,f(mn)=f(n)+f((m-1)n)+(m-1)n2.,f(mn)=mf(n)+(1+2+3++(m-1))n2.1320098375():zxss.chinajournal.net.cn:[e-mail]1+2+3++(m-1)=m(m-1)2,f(mn)=mf(n)+m(m-1)2n2,mn.,n=1f(m)=m(m+1)2(3)(3)f.f(1)=1(1+1)2=1,f(x+y)=(x+y)(x+y+1)2=(x2+x)+(y2+y)+2xy2=x(x+1)2+y(y+1)2+xy=f(x)+f(y)+xy,f(m)=m(m+1)2.5(15)5,ABCD,BADBCM,DCN,CMNO,CMNCBDK.:(1)OCBD;(2)AKC=90.(1)ABCD,BADBCM,DCN,BMA=MAD=BAM,BA=BM,MC=CN.OC,OCNCM.OB,OM,OD,BAD=,OCD=BCD+OCM=+12(180-)=90+2,BMO=180-OMC=180-OCM=90+2,BMO=OCD.,OBMODC,OBC=ODC.B,O,C,D,OCBD.(2)(1)BO=OD,KO=OC,B,K,O,CD,K,CBD,BK=CD=AB,KBD=CDB=ABD,KABD,AKBD,KC//BD,AKKC,AKC=90.(15),7,x,y,0x-y1+xy33.7a1,a2,a3,a4,a5,a6,a7,-2,271,2,3,4,5,6,7,tani=ai(i=1,2,3,4,5,6,7).-2,266:-2,-3,-3,-6,-6,0,0,6,6,3,3,2.,i,j6,ij,0i-j6,0tan(i-j)tan6,0tani-tanj1+tanitanj33.x=ai,y=aj,,7,x,y,0x-y1+xy33.()23