2009年上海市TI杯高二年级数学竞赛

2009年上海市TI杯高二年级数学竞赛:G424.79:A:1005-6416(2010)04-0030-02(1~46,5~89,60)1.:21431268571410121416911131518=(10-5).2.rR,mM.R=109r,M=330000m.(0001).3.EABCDBC,S!ABES!AECS!ACD.EAB=(1∀).4.Py=2x,O.PO(0001).5.x+3x2-1(0001).6.EFGHABCDABBCCDDA,EGFH=3SABCD.EGFH(1∀).7.n1,x=1+12+#+1n,y=lg2+lg3+#+lgn.[x]=[y]n([a]a).8.30,30.:.∃说明%.(20)1,xOy,A03km/sx.1AN,ON=10km,xON=105&,,O07km/sB.BA,xOB(0001&).(20)(1):(4sin2x-3)(4cos2x-3)=4sin22x-3(xR);(2):28t=04sin2t29-3.(20).,.∃说明%,.(20)(1)1.1x0x1x2x3x4y0y1y2y3y4y=ax2,a,Pi(xi,yi)(i=1,2,3,4)Ai(xi,ax2i)4i=1PiA2i.(2)(1),2,y=ax2(a001).2x01234y0-260-990-2260-398030(20)n,tn=n(n+1)2,t1=1,t2=3,t3=6,t4=10,t5=15##:013605#..(20)n,f(n)3n2+n+1(f(3)332+3+1=31,f(3)=4).(1):n,f(n)∋1,f(n)∋2;(2)n,f(n)=3.1.2604732.02553.20&54(19∀4.08575.(-1492,1785)6.19&28(16∀7.{5,6}8.448548949ts,BC2(2),xOB=.NC=03t,OC=07t,NOC=105&-,ONC=75&.!ONCNCsinNOC=OCsinONC03tsin(105&-)=07tsin75&sin(105&-)=37sin75&105&-=arcsin37sin75&105&-=180&-arcsin37sin75&=80546&=-50546&().,xOB=80546&.(1)(4sin2x-3)(4cos2x-3)=16sin2xcos2x-12(sin2x+cos2x)+9=4sin22x-3.(2)Ak=2k-1t=04sin2t2k-3.k)2,(1)Ak=2k-2-1t=04sin2t2k-34sin2(2k-1-t)2k-34sin22k-12k-3=2k-2-1t=04sin2t2k-1-3-4sin22k-22k-1-3=-Ak-1.A9=A1=(4sin20-3)4sin22-3=-3.abcdef.abcdef=kabcdef,abc1000+def=kabcdef.,abc|def.def=labc(l{1,2,#,9}).,klabc2=(1000+l)abc,klabc=1000+l.,l|1000.l=125.l=1,kabc=1001=71113,k=7,abc=143,,abcdef=143143;l=2,abc500,2kabc=1002,kabc=501=3167,k=3,abc=167,,abcdef=167334;l=5,abc200,5kabc=1005,kabc=201=367,.,143143167334.(1)4i=1PiA2i=4i=1(ax2i-yi)2=a24i=1x4i-2a4i=1x2iyi+4i=1y2i=4i=1x4ia2-24i=1x2iyi4i=1x4ia+4i=1x2iyi4i=1x4i2+4i=1y2i-4i=1x2iyi4i=1x4i23120104=4i=1x4ia-4i=1x2iyi4i=1x4i2+4i=1y2i-4i=1x2iyi4i=1x4i2.a=4i=1x2iyi4i=1x4i,4i=1PiA2i4i=1y2i-4i=1x2iyi4i=1x4i2.(2)24i=1x4i=14+24+34+44=354,4i=1x2iyi=12(-26)+22(-99)+32(-226)+42(-398)=-8824.,(1)a=-8824354∗-249.,y=-249x2.tn+20-tn=(n+20)(n+20+1)2-n(n+1)2=20n+210=10(2n+21),tn+20tn.,20.013605186556815063100.(1)3n2+n+13,f(n)∋1.f(n)=2,3n2+n+110,3n2+n+1=10k+1(k1).,n(3n+1)=2k5k.(n,3n+1)=1,,n=2k,3n+1=5k.,3n+1+4n=42k5k..f(n)∋2.(2)n=8,3n2+n+1=201,,f(8)=3.()2009年全国高中数学联赛甘肃省预赛:G424.79:A:1005-6416(2010)04-0032-04(7,56)1.A={x|x=2n,nN},B={x|x=3n,nN},C={x|x=n2,nN}.A,B,C=.2.(1,2)x2+y2=1.xy.3.PABC,APB=BPC=CPA=90&,S..4.y=f(x)xR,y=f(x))0,f(x+1)=9-f2(x).x[0,1),f(x)=2x,0+x+12;lg(x+31),12+x1.f(1000)=.5.mn.(0,0)(1,-1)(mn)32