历届中国数学奥林匹克(全国中学生数学冬令营)试题解答

(CMO)1986-2005(1986)1.a1;a2;:::;an,,x1+x2+¢¢¢+xn=1x1;x2;:::;xn;a1x1+a2x2+¢¢¢+anxna1x21+a2x22+¢¢¢+anx2n..::06xi61;xi¡x2i0;xix2i(i=1;2;:::;n).(1)ai0(i=1;2;:::;n),a1x1+a2x2+¢¢¢+anxna1x21+a2x22+¢¢¢+anx2n;(2)ai0,a10.a1;a2;:::;an,ai+a10;ai¡a10(i=2;3;:::;n).)a1x1+a2x2+¢¢¢+anxn¡a1x21¡a2x22¡¢¢¢¡anx2n=a1(x1¡x21)+a2(x2¡x22)+¢¢¢+an(xn¡x2n)a1(x1¡x21)+(¡a1)(x2¡x22)+¢¢¢+(¡a1)(xn¡x2n)=(¡a1)(x21¡x22¡¢¢¢¡x2n¡x1+x2+¢¢¢+xn)=(¡a1)(x21¡x1+(1¡x1)¡x22¡¢¢¢¡x2n)=(¡a1)((1¡x1)2¡x22¡¢¢¢¡x2n)=(¡a1)((x2+¢¢¢+xn)2¡x22¡¢¢¢¡x2n)0x2;x3;:::;xn0;(x2+¢¢¢+xn)2=x22+¢¢¢+x2n+P26ij6nxixjx22+¢¢¢+x2n.:16ij6n,xi=xj=12,xk0.12(ai+aj)14(ai+aj).)ai+aj0,..2.ABC,BCAD=12,\AAE=13,BCAF=m,m,\A,,?:O4ABC,ABAC,\B.OFBC,,\C,\OAB=\OBA=12(180±¡\AOB)=12(180±¡2\C)=90±¡\C.AD?BC,)\CAD=90±¡\C,)\OAB=\DAC.,\C\OAB=\DAC.AE\BAC,\BAE=\CAE.)\OAE=\DAE.(F;DE).\A,O;ABC,\FAE\OAE=\DAE;\A,O;F,\FAE=\OAE=\DAE;\A,O;ABC,\FAE\OAE=\DAE.1sin\FAEsin\DAE=FEDE£ADAF.DE=pAE2¡AD2=5;FE=FD¡DE=pAF2¡AD2¡DE=pm2¡122¡50.)m13,\Apm2¡122¡55£12m1;\Apm2¡122¡55£12m=1;\Apm2¡122¡55£12m1.13m2028119,\A;m=2028119,\A;m2028119,\A.3.z1;z2;:::;zn,jz1j+jz2j+¢¢¢+jznj=1::n,,16.:zk=xk+yki(xk;yk2R;k=1;2:::;n)zkX,Y.jxkjjykj,zkX;jykjjxkj,zkY.12.X.XA,B.xk0,zkA;xk60,zkB.14.A.Pzk2Ajzkj14,Pzk2Apx2k+y2k14.zk2A,x2ky2k,px2k+y2k6p2xk.)Pzk2Axk14p2.)jPzk2Azkj=jPzk2Axk+iPzk2AykjPzk2Axk14p2.4p26,)jPzk2Azkj16.A16..:zk=xk+yki(xk;yk2R;k=1;2:::;n)jzkj=px2k+y2kjxkj+jykj.)nPk=1jxkj+jykj1.)jPxk0xkj+jPxk0xkj+jPyk0ykj+jPyk0ykj1.14,,jPxk0xkj14.)jPxk0zkj=jPxk0xk+iPxk0ykjjPxk0xkj1416..4.:P1P2P3P4ABC.:4P1P2P3;4P1P2P4;4P1P3P4;4P2P3P4,4ABC.::(1);(2).(1)P1;P4AB,P2;P3AC,P1;P2AP4;AP3.BP4,CP3,ABC2,(2).(2)P1AB,P2AC,P3;P4BC,P3P4C.(2.1)P1P2kBC,AP1AB=AP2AC=¸,P1P2=¸BC.P1P2BC(1¡¸)h,hABCBC.)S4P1P2P3=¸(1¡¸)S4ABC614S4ABC.(2.2)P1P2BC,P1BCP2BC.P2BCABE,P1P4D.S4P1P2P3;S4P4P2P3S4DP2P3,S4EP2P3.(2.1)S4EP2P3614S4ABC.S4P1P2P3;S4P4P2P314S4ABC..5.1,1,2,2,...,1986,1986,11,22,...,19861986..:.,1,2,...,3972.n,n;n,n.11986993,2k+993.(k2N¤)139721986,k=496:5...6.,.:1p3,.:(1)2A;B,O,A;O.AOAOC;AOD,C;DA;O,.BCDp3.(2)2,1,,2,.,1.1..3(1987)1.n,zn+1¡zn¡1=01n+26.:6jn+2,z=ei¼3=12+p32i;z6=1;jzj=1.)zn+1¡zn¡1=e¡i¼3¡ei¼3¡1=(12¡p32i)¡(¡12¡p32i)¡1=0.)zn+1¡zn¡1=01.zn+1¡zn¡1=01eiµ=cosµ+icosµ.zn+1¡zn¡1=(cos(n+1)µ¡cosnµ¡1)+i(sin(n+1)µ¡sinnµ)=0.)cos(n+1)µ¡cosnµ¡1=¡(2sin2n+12µsinµ2+1)=0.sin(n+1)µ¡sinnµ=2cos2n+12µsinµ2=0.)cos2n+12µ=0;sin2n+12µ=§1;sinµ2=§12,µ2='.(1)sin'=12,sin(2n+1)'=¡1.'=2k¼+¼62k¼+5¼6;k2Z.(2n+1)'=(2l+32)¼(l2Z).)(2n+1)(2k+16)=2l+32;2n+16=2t+32;n=6t+4(t2Z).(2n+1)(2k+56)=2l+32;5(2n+1)6=2t+32;5j4t+3;t´3(mod5)(t2Z).t=5s+3,n=6s+4,6jn+2.(2)sin'=¡12,sin(2n+1)'=1.¡''(1).6jn+2..2.1ABCn,,,,.:(1)A;B;Ca;b;c:(2),.:(1).(2)S.:(1),.,..a;b;c,,0.a;b;c,A;B;C,1.a;b;c,a=bc,c,AB.n,CAB,p32.n,CAB12n,12q3+1n2.(2)23¼;43¼,(2).(2),a+b+c.(1)a+b+c.12(n+1)(n+2),)S=13(12(n+1)(n+2))(a+b+c)=16(n+1)(n+2)(a+b+c).3.,,,,A:B,AB,C,CB,AC.4,:.:A,A.BA,B,CCB,DBD,DC.BA,CA,CB.CB,CA,B.A.4.1,,:,,,0.64.:0.64100169+(0).1ABC,ABA1;B2,ACA2;C1,BCB1;C2,AA1=AA2=BB1=BB2=CC1=CC2=313AB.A1C2;A2B1;B2C1A0;B0;C0.(1)4AB2C1;4BC2A1;4CA2B1,2,(1013)2+2£2=100169+.(2)AA1A0A2;BB1B0B2;CC1C0C2,,613AB.2(613)2+100169+.(3)AA1A0A2;BB1B0B2;CC1C0C2,,AA1A0A2,B1B0C0C2,B,613ABAA1A0A2,613ABBB1B0B2;B1B0C0C2.,(613)2+(813)2+=100169+..:A;B;C;A0;B0C0100169.100169+(0).5.A1A2A3A4,S1;S2;S3;S4A1;A2;A3;A4,.O,rS1;S2;S3;S4,R,:.:Siri(i=1;2;3;4),AiAj=ri+rj(16ij64).OA2A3A4O1,OA2A3,A3A4,A4A2,O14A2A3A4.O14A2A3A4,OA2A3B,O1A2A3.BA3=12(A2A3+A3A4¡A2A4)=r3,OB=R.r,A3O=r+r3,(r+r3)2=r23+R2;r3=R2¡r22r.r,A3O=r¡r3,(r¡r3)2=r23+R2;r3=r2¡R22r.r1;r2;r4,.6.mn1987,mn,3m+4n?.:ma1a2¢¢¢am,nb1b2¢¢¢bn.)ai2i;bj2j¡1.*1987=a1+a2+¢¢¢+am+b1+b2+¢¢¢+bn.)19872+4+¢¢¢+2m+1+3+¢¢¢+2n¡1=m2+m+n2.5s=3m+4n,m=13(s¡4n),13(s¡4n)(13(s¡4n)+1)+n261987.s2¡8ns+25n2+3s¡12n¡9£198760.s2+(3¡8n)s+25n2¡12n¡9£198760.¢=(3¡8n)2¡4(25n2¡12n¡9£1987)=26(198714¡n2)0.s612(8n¡3+6q198714¡n2).f(n)=8n+6q198714¡n2;f0(n)=8¡6n(198714¡n2)¡12,n.n=35,f(n)280+6q76214.s612(280+6q76214¡3),s2N¤;s6221.s=221;n=35;m=27.2;4;:::;52;60;1;3;:::;6919873527,3m+4n221.6(1988)1.a1;a2;:::;an,r1;r2;¢¢¢;rn,r1(x1¡a1)+r2(x2¡a2)+¢¢¢+rn(xn¡an)6qx21+x22+¢¢¢+x2n¡qa21+a22+¢¢¢+a2nx1;x2;¢¢¢;xn,r1;r2;¢¢¢;rn.:xi=0(i=1;2;:::;n),¡(r1a1+r2a2+¢¢¢+rnan)6¡pa21+a22+¢¢¢+a2n.)(nPi=1riai)2nPi=1a2i.xi=2ai(i=1;2;:::;n),r1a1+r2a2+¢¢¢+rnan6pa21+a22+¢¢¢+a2n.)(nPi=1riai)26nPi=1a2i.)(nPi=1riai)2=nPi=1a2i.Cauchy,(nPi=1r2i)(nPi=1a2i)(nPi=1riai)2,nPi=1r2i1.xi=ri(i=1;2;:::;n),nPi=1r2i¡nPi=1riai6snPi=1r2i¡snPi=1a2i.nPi=1riai=snPi=1a2i,nPi=1r2i6snPi=1r2i,nPi=1r2i61.)nPi=1r2i=1,Cauchyr1a1=r2a2=¢¢¢=rnan.ri=aipa21+a22+¢¢¢+a2n(i=1;2;:::;n).2.C1;C2,C2C12,A1A2A3A4C1,A4A1C2B1,A1A2C2B2,A2A3C2B3,A3A4C2B4.:B1B2B3B42(A1A2A3A4)..:O,OB1;OB4;OA4,C1R,C22R.B4A4OB1,Ptolemy,OA4£B1B4+OB1£A4B4OB4£A4B1.R£B1B4+2R£A4B42R£A4B1,B1B42A4B1¡2A4B4.B1B22A1B2¡2A1B1,B2B32A2B3¡2A2B2,B3B42A3B4¡2A3B3.B1B2+B2B3+B3B4+B4B12(A1A2+A2A3+A3A4+A4A1).B1B2B3B42(A1A2A3A4).OAiBiBi+1,\Ai+1AiO=\Bi+1BiO=\BiBi+1O=\Ai¡1AiO,)Ai+1Ai=Ai¡1Ai,(i=1;2;3;4;A5=A1;A0=A4;B5=B1).)A1A2A3A4,,A1A2A3A4.3.a1;a2;¢¢¢;an,ak;ak+1;¢¢¢;ak+l¡11988,,ak(an1988,).,:71988.::ak;ak+1;:::;ak+m¡1,ak+m,k+m¡1=n,ak;ak