2009年北京市中学生数学竞赛初赛-高一-(1)

2009年北京市中学生数学竞赛初赛(高一):G424.79:A:1005-6416(2010)04-0027-03(6,36)1.1[-4,6]y=f(x)y=g(x).f(x)g(x)().(A)[-1,3](B)[-4,3][5,6](C)[-4,2](D)[-4,-1][3,6]12.x=sin12y=cos13z=cos14.xyz().(A)xyz(B)xzy(C)yxz(D)zyx3.!ABC,AB=6,AC=8,BAC=90∀,ADBEBCAC.ADBE().(A)1365(B)32(C)-1365(D)04.log2sin12+log2sin6+log2sin512=().(A)-3(B)-1(C)1(D)35.a,f(x)=(x+a)3x-2+a2-(x-a)38-x-3a.a().(A){0,5}(B){-2,5}(C){-5,2}(D){1,2009}6.f(x)=x2+bx+cx#0;-2,x0,f(4)=f(0),f(1)=-1.f(x)=x().(A)0(B)1(C)2(D)3(8,64)1.b1,b2,∃,b60b2b1=b3b2=∃=b60b59.logb11b50(b1b2∃b60).2.sin(+)=45,cos(-)=310.(sin-cos)(sin-cos).3.ABCDA(1,2)B(2,5)C(7,3)D(5,1),ABCD.4.[x]x.[log36]+[log37]+∃+[log3n]=2009,n.5.!ABC,AB=BCAC,AHAMBC,S!AMHS!ABC=38.cosBAC.6.f(x)=2x2-2x-1g(x)=-5x2+2x+3,y=ax+b.a-b.7.6027abcabc∃abc2009abc91.abc.28.2,ABCD,ABC=120∀,BC=63,PBCC,APCDE,BEDPQ.P,QBP=15∀;,QBP=35∀.P,Q.2720104参考答案1.D.2.B.3124,,x0.4131445,yz0x.3.C.33,A.A(0,0)B(6,0)C(0,8)D(3,4)E(0,4).,AD=(3,4),BE=(-6,4).cosAD,BE=3%(-6)+4%432+42%(-6)2+42=-1365.4.A.log2sin12+log2sin6+log2sin512=log2sin12&12&sin512=log212sin12&cos12=log214sin6=log218=-3.5.C.f(x),f(-a)=f(a),2a&38+a-3a=2a&3a-2+a2.a=08+a-3a=a-2+a2.a=0,f(x)=x(3x-2-38-x).a∋0.8+a-3a=a-2+a2,a=-52,,f(x)=-(5-x)323+x-(5+x)323-xf(x)=(2+x)32+x+(2-x)32-x.6.D.f(4)=f(0),f(1)=-116+4b+c=c,1+b+c=-1b=-4,c=2.f(x)=x2-4x+2,x#0;-2,x0.4,3.41.,{bk},q.bk=b1qk-1.,bkb61-k=b1b60.logb11b50(b1b2∃b60)=logb1b60(b1b60)30=30.2.(sin-cos)(sin-cos)=(cos&cos+sin&sin)-(sin&cos+cos&sin)=cos(-)-sin(+)=-12.3.5,ABCD5,M(1,1)N(1,5)P(7,5)T(7,1),ABCDMNPT!AMD!ANB!BPC!CTD.SMNPT=24,S!AMD=2,S!AND=32,S!BPC=5,S!CTD=2.SABCD=24-2+32+5+2=272.4.3ki3k+1-1,[log3i]=k.3k+1-1i=3k[log3i]=2%3kk.3+4k=22%3kk20093+5k=22%3kk,2835n36-1.3+4k=22%3kk+5(n-35+1)=2009,n=474.5.!ABC,BDACD.AB=AC=a.S!AMHS!ABC=MHBC=38,,MH=38a,HC=MC-MH=12a-38a=18a.BAC=.BCA=.Rt!BCD,DC=acos.!ABC,AC=2DC=2acos.Rt!ACH,cos=HCAC=18a2acoscos2=116.,,cos=14.6.f(x)g(x)y=2x2-2x-1,y=-5x2+2x+3.,7y=5(2x2-2x-1)+2(-5x2+2x+3)=-6x+1.y=-67x+17,a=-67,b=17.,a-b=-1.7.91=7%13,1001=7%11%13,91|1001.abcabc=1001%abc,,91|abcabc.abcabc∃abc2009abc91,abcabc∃abc2009abc=abcabc∃abc2%1004abc%1000+abc,abcabc∃abc2%1004abc91,abc91.91182,910,,abc1092.68.6,BD,!ABDAPF,BFDFFCCQ.DFB=DFP=BFP=120∀,BFE=ECP=120∀.,BFEC.EBC=PFC.DFP=DCP=120∀DFCPPFC=CDPQBP=PFCBCQDQ!BCD.,PBCC,Q!BCDCDC.!BCD63,63%32%23=6,,2%6=12.Q!BCD35∀-15∀=20∀,,Q!BCD40∀,19.,P,QQ1Q2129=43(7).7(李延林)2920104