2010年北京市中学生数学竞赛高一年级初赛试题及参考解答

2010(36)1.y=2sin(3x-2)(0),,().(A)3(B)4(C)6(D)812.P(a,b)ABCD(),ABCD1.ba().(A)qm,pn(B)pm,qn(C)qm,qn(D)pm,pn3.PABC,PA+PB+PC=2AB.SABC=6,PAB().(A)4(B)3(C)2(D)14.f(x)=ax2+bx+cx=1(2,0),f(-1)f(1)().(A)-3(B)-2(C)2(D)325.ABC,ADBEG,sinC().(A)25(B)35(C)23(D)456.f(x)(-!,0)∀(0,+!),f(1x)+1xf(-x)=4x,f(2)#f(-12)().(A)31.5(B)30.5(C)-30.5(D)-31.5(64,8)1.f(x)=x2-1D,{-1,0,1,3},D.2.log223+log234+log245+log256+log267+log278log32log43log54log65log76log87.33.ABCD,EAB,AB=14,CE=13,DE=15.CF∃DEF,AF,BF.ABF.4.,y=kxy=2x+4-22x-8(x-3)(-3%x%3)(x3)3.k.5.PABC,2PA&&∋+3PB&&∋+6PC&&∋=0∋,PAB,PBC,PCA.56.5,ABCD,(BAD+(ADC=240),EFADBC,EF=7,ABCDA1A2,ABCDA3.A1A2A3.67.sin48+cos438+sin458+cos478.8.ABC,(ABC=100),(ACB=65),ABM,(MCB=55),ACN,(NBC=80).(NMC..1.:(C).y=2sin(3x-2)(0),y=2sin[3(x+)-2]=y=2sin[3x+(3-2)],,00,3-2=k,6.2.:(B).baOP,OC,OA,bapm,qn.3.:(C).ABCO,∗PA&&∋+PB&&∋+PC&&∋=2AB&&∋,+OA&&∋+OB&&∋+OC&&∋-3OP&&∋=2OB&&∋-2OA&&∋,+3(OA&&∋-OP&&∋)=OB&&∋-OC&&∋,3PA&&∋=CB&&∋.+SPAB=13SABC=2.4.:(A).f(x)=ax2+bx+cx=1(2,0),(0,0),c=f(0)=0,f(x)=ax2+bx,f(1)=a+b,f(-1)=a-b.x=1,-b2a=1,ba=-2.f(-1)f(1)=a-ba+b=-3.5.:(B).ABC,BC=a,CA=b,AB=c,GD=x,GE=y,GA=2x,GB=2y.AGBAGEBGD,4x2+4y2=c2,4x2+y2=b24,x2+4y2=a24.c2=a2+b25.,cosC=a2+b2-c22ab=5c2-c22ab=4c22ab,4c2a2+b2=4c25c2=45.(C,sinC=1-cos2C%1-(45)2=35.6.:(D).x=-2,x=12f(1x)+1xf(-x)=4x,f(-12)-12f(2)=-8,f(2)+2f(-12)=2.f(2)=9,f(-12)=-72,f(2)#f(-12)=9#(-72)=-31.5..1.:27.∗f(0)=-1,f(1)=0,f(-1)=0,f(2)=1,f(-2)=1,f(2)=3,f(-2)=3,+0−D;-1,1D;-2,2D;-2,2D.,D3#3#3=27.2.:-6.log223+log234+log245+log256+log267+log278log32log43log54log65log76log87=log2233445566778lg2lg3lg3lg4lg4lg5lg5lg6lg6lg7lg7lg8=log228lg2lg8=log22-2lg23lg2=-213=-6.3.:36.96.BE.BE=x,AE=14-x,ADEBCE,DE2-AE2=AD2=BC2=CE2-BE2,152-(14-x)2=132-x2,x=5.,AD=BC=132-x2=12.SABCD=12#14=168,SCDE=84.DF=y,EF=15-y,CDFCEF,CD2-DF2=CF2=CE2-EF2,142-y2=132-(15-y)2,y=8.4.,SCDF=84#8.415=288.45=47.04.SCDF+SABF=12SABCD=84,SABF=84-SCDF=84-47.04=36.96.4.:(23,2).y=2x+4-22x-8(x-3)(-3%x%3)(x3),.4l1:y=2x1,l2:y=23x2,,l2l1,k23k2,l3:y=kx3.5.:6.2.3.7,PAPBPCA1,B1,C1,PA1&&∋=2PA&&∋,PB1&&∋=3PB&&∋,PC1&&∋=6PC&&∋,PA1&&∋+PB1&&∋+PC1&&∋=0∋.+PA1B1C1,+SPA1B1=SPB1C1=SPC1A1.∗SPAB=SPA1B12#3,SPBC=SPB1C13#6,SPCA=SPC1A16#2,+SPAB.SPBC.SPCA=6.2.3.6.:28.BACDP,(BAD+(ADC=240),(BPC=60).BD,BDG,EG,FG,,GE/BP,GF/PC,(EGF=120),EG=12AB,FG=12CD.EGF,EF2=EG2+FG2-2EGFGcos120)=AB22+CD22+AB2CD2,AB22+CD22+AB2CD2=EF2=72=7.AB2+CD2+AB#CD=4#7=28.A1A2A328.7.:32.sin48+cos438+sin458+cos478=sin48+cos4-58+sin458+cos4-8=sin48+cos458+sin458+cos48=sin28+cos282-2sin28cos28+sin258+cos2582-2sin258cos258=2-12sin282-12sin1082=2-12sin42-12sin542=2-12222-12-222=2-1212-1212=32.8.:25).(BAC=15),MCB,BNM1,CM1CM,CM1=CM.(M1CM=(M1BM=100)-80)=20),(ACM=65)-55)=10),(M1CN=10);CN=CN,M1CN0MCN.,(NMC=(NM1C=(CMB=(BAC+(ACM=15)+10)=25).(北京数学会普及委员会提供)