1、SSS X X X(CMO) k k k555#sss1986-2005BS X(1986 M)? 2 7v1.Xa1;a2;:;an L , T i d, * x1 +x2 +xn = 1 id L x1;x2;:;xn; Ta1x1 +a2x2 +anxn a1x21 +a2x22 +anx2n . h 5# I 5. : 5 :0 6 xi 6 1;xi x2i 0;xi x2i(i = 1;2;:;n).(1) ai 0(i = 1;2;:;n),5A a1x1 +a2x2 +anxn a1x21 +a2x22 +anx2n;(2)5 iBai 0;ai a1 0(i = 2;3;:;
2、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 +(1x1)x22 x2n)= (a1)(1x1)2 x22 x2n)= (a1)(x2 +xn)2 x22 x2n) 0KB x2;x3;:;xn 0;(x2 +xn)2 = x22 +x2n + P26i x22 +x2n.I 5 : i1 6 i 14(ai +aj).) ai +aj
3、 0, d.8.2. ABC,BCH AD = 12,A sLAE = 13, !BCH LAF = m,m IS = | H,AsY ,?: !O4ABC, !AB AC,B .5OF OAE = DAE.1? sinFAEsinDAE = FEDE ADAF . DE = pAE2 AD2 = 5;FE = FDDE = pAF2 AD2 DE = pm2 122 5 0. ) m 13,OA Npm212255 12m 1.13 2028119 H,A.3. !z1;z2;:;zn , jz1j+jz2j+jznj = 1:p: n ,Ai , l16. : !zk = xk +yki(
4、xk;yk 2R;k = 1;2:;n)| zks FX,Y. jxkjjykj,5|zkb X; jykjjxkj,5|zkb Y. ABF l12. !X.|X s FA,B. xk 0,5|zkb A; xk 6 0,5|zkb B. ABF l14. !A.5Pzk2Ajzkj 14,Pzk2Apx2k +y2k 14.7zk 2 A,x2k y2k,px2k +y2k 6p2xk.) Pzk2Axk 14p2.)j Pzk2Azkj = j Pzk2Axk +i Pzk2Aykj Pzk2Axk 14p2.74p2 16.A l16.8.6: !zk = xk +yki(xk;yk
5、2R;k = 1;2:;n)5jzkj = px2k +y2k jxkj+jykj.)nPk=1jxkj+jykj 1.)j Pxk0xkj+j Pxk0ykj+j Pyk 1.ABl14, !B,5j Pxk0xkj 14.)j Pxk0zkj = j Pxk0xk +i Pxk0ykjj Pxk0xkj 14 16.8.4.X: HP1P2P3P4 ABCH .p: 4P1P2P3;4P1P2P4;4P1P3P4;4P2P3P4, B v4ABC sB. : f :(1) HH ;(2) HH .(1) !P1;P4AB ,P2;P3AC ,P1;P2sYAP4;AP3 .|BMP4,CM
6、P3, ABC2hl,B f(2).(2) !P1AB ,P2AC ,P3;P4BC ,P3P4C .(2.1) P1P2 k BC, !AP1AB = AP2AC = ,P1P2 = BC.P1P2BC (1)h,h ABCBCH.) S4P1P2P3 = (1)S4ABC 6 14S4ABC.(2.2) P1P2 BC, !P1BC vP2BC .VP2T BCLABE,P1P4D.5S4P1P2P3;S4P4P2P3BvS4DP2P3,9vS4EP2P3.(2.1)S4EP2P3 6 14S4ABC.5S4P1P2P3;S4P4P2P3Bv14S4ABC.8.5. ?1,1,2,2,.,
7、1986,1986t B, P 1WC“1 , 2WC“2 ,., 1986WC“1986 . h F .: ?.L ! VS“ ,|X z I|1,2,.,3972.n H, nI| M;n H, nI| .711986W993 , B 2k + 993I| .(k 2 N) 13972W1986 ,k = 496:5. . ?1 p “B.6. iZ T, B . p:BiBH1p3 , . :(1) i 2 A;Bs , !O , !A;O . I nAOBH AOC;AOD, C;DBA;O ,5 5i.5BCDHp3 .(2)5 2 ( , I n i 1, L,2%T%,5 B%
8、. , 1 . iBH1 .8.3=S X(1987 M)v1. !n1 , pZzn+1 zn 1 = 0 1 sA1Hq n+2 V$6“. :6jn+2 H, 7z = ei3 = 12 +p32 i;z6 = 1;jzj = 1.) zn+1 zn 1 = ei3 ei3 1 = (12 p32 i)(12 p32 i)1 = 0.) zn+1 zn 1 = 0 1.zn+1 zn 1 = 0 1ei = cos +icos .5zn+1 zn 1 = (cos(n+1) cosn 1)+i(sin(n+1) sinn ) = 0.) cos(n+1) cosn 1 = (2sin 2
9、n+12 sin 2 +1) = 0.sin(n+1) sinn = 2cos 2n+12 sin 2 = 0.) cos 2n+12 = 0;sin 2n+12 = 1;sin 2 = 12, ! 2 = .(1)sin = 12,sin(2n+1) = 1. = 2k + 62k + 56 ;k 2Z.(2n+1) = (2l + 32)(l 2Z). ) (2n+1)(2k + 16) = 2l + 32; 2n+16 = 2t+ 32;n = 6t+4(t 2Z).(2n+1)(2k + 56) = 2l + 32; 5(2n+1)6 = 2t+ 32;5j4t+3;t 3 (mod
10、5)(t 2Z).!t = 5s+3,5n = 6s+4,96jn+2.(2)sin = 12,sin(2n+1) = 1.A (1). 6jn+2.8.2.H1 ABCHns,Vs HL,| s l ,tl ,i O B b B L .X:(1)A;B;C b sYa;b;c:(2) H Kl F +, FM b M.k p:(1)bKv bKl WK .(2) 9S.:(1) 4 BL M # b WB H, BL . M,5 M.5 KvKl.a;b;cM,A M,K A 0.a;b;c ,Kv Kl ACA;B;C ,K 1.a;b;c M M, !a = b c,Kl c,Kv CL
11、 AB i .n H,CKAB,K p32 .n H,CKAB_ H 12n,K 12q3+ 1n2.(2)| 23; 43, 9 5i(2).| MF 9 (2), (a + b + c.(1)s (a+b+c. 12(n+1)(n+2),) S = 13(12(n+1)(n+2)(a+b+c) = 16(n+1)(n+2)(a+b+c).3. Q81 , mB1 , 1 B % ;,YV1 m,mA$ mHq : mB,A ;B,i mC,C ;B,A ;C.T 4?5 mB , p: mB ; m. :L ! mA, Oi m ;A.!B ;A ;QKB,B m,Ai mC PC ;B,
12、 OimD PB ;D,D ;C.B ;A,C9 ;A, OC ;B ;V .C 1B ;B, OC ;A,B |. A ; .4.B 1 =, ib, k:N =,B VT , HsY H,i O V0.64. : V|0.64100169 +“(“ 0). 1 ABC,AB |A1;B2,AC |A2;C1,BC |B1;C2, PAA1 = AA2 =BB1 = BB2 = CC1 = CC2 = 313AB. A1C2;A2B1;B2C1A0;B0;C0.(1) 4AB2C1;4BC2A1;4CA2B1B c , 6 VsY “2 , (1013)2 +2 “2 = 100169 +“
13、.(2) +AA1A0A2;BB1B0B2;CC1C0C2 , 6BB,5 V H613AB B “ . 2( 613)2 +“ 0)K.5. !A1A2A3A4 B 8, S1;S2;S3;S4sY A1;A2;A3;A4 o o, M M. TiBO, o VTBr oS1;S2;S3;S4M M, VTBR o 8 M M, p: 8 8. : !Siri(i = 1;2;3;4),5AiAj = ri +rj(1 6 i 2i;bj 2j 1.* 1987 = a1 +a2 +am +b1 +b2 +bn.) 1987 2+4+2m+1+3+2n1 = m2 +m+n2.5!s = 3
14、m+4n,m = 13(s4n),13(s4n)(13(s4n)+1)+n2 6 1987.s2 8ns+25n2 +3s12n91987 6 0.s2 +(38n)s+25n2 12n91987 6 0. Y T = (38n)2 4(25n2 12n91987) = 26(198714 n2) 0.s 6 12(8n3+6q198714 n2).!f(n) = 8n+6q198714 n2;f0(n) = 86n(198714 n2)12,n . 4n = 35 H,f(n)Kv280+6q76214.s 6 12(280+6q76214 3),s 2N;s 6 221.s = 221;n
15、 = 35;m = 27. |2;4;:;52;60;1;3;:;69198735 27 , 3m+4nKv221.6 S X(1988 M)Zv1. !a1;a2;:;an , L , r1;r2; ;rn L , T Tr1(x1 a1)+r2(x2 a2)+rn(xn an) 6qx21 +x22 +x2n qa21 +a22 +a2n L x1;x2; ;xn , pr1;r2; ;rn.: 7xi = 0(i = 1;2;:;n),(r1a1 +r2a2 +rnan) 6pa21 +a22 +a2n.) (nPi=1riai)2 nPi=1a2i.7xi = 2ai(i = 1;2;
16、:;n),r1a1 +r2a2 +rnan 6pa21 +a22 +a2n.) (nPi=1riai)2 6nPi=1a2i.) (nPi=1riai)2 =nPi=1a2i.Cauchy T, (nPi=1r2i)(nPi=1a2i) (nPi=1riai)2,nPi=1r2i 1. 7xi = ri(i = 1;2;:;n),nPi=1r2i nPi=1riai 6snPi=1r2i snPi=1a2i.nPi=1riai =snPi=1a2i,nPi=1r2i 6snPi=1r2i,nPi=1r2i 6 1.)nPi=1r2i = 1,Cauchy T |Hqr1a1 = r2a2 =
17、= rnan. 4ri = aipa21 +a22 +a2n(i = 1;2;:;n).2. !C1;C2,C2 C12, HA1A2A3A4 =C1, !A4A1LC2B1, A1A2LC2B2, A2A3LC2B3, A3A4LC2B4.k: HB1B2B3B42( HA1A2A3A4).i | Hq. : !O, OB1;OB4;OA4, !C1R,C22R. HB4A4OB1,Ptolemy ,OA4 B1B4 +OB1 A4B4 OB4 A4B1.RB1B4 +2RA4B4 2RA4B1,B1B4 2A4B1 2A4B4. B1B2 2A1B2 2A1B1,B2B3 2A2B3 2A
18、2B2,B3B4 2A3B4 2A3B3.MFB1B2 +B2B3 +B3B4 +B4B1 2(A1A2 +A2A3 +A3A4 +A4A1). HB1B2B3B42( HA1A2A3A4).| HOAiBiBi+1 ,Ai+1AiO = Bi+1BiO = BiBi+1O = Ai1AiO,) Ai+1Ai = Ai1Ai,(i = 1;2;3;4;A5 = A1;A0 = A4;B5 = B1).) A1A2A3A4 +, = M H, A1A2A3A4Z.3.K L a1;a2; ;an, TBak;ak+1; ;ak+l1 (v1988, * SBHo Cp,iakSH Co Chp(
19、 T Ban 1988, * B9C).L ! iBH C, : 8 VT Ch ( 9Av71988. : : !ak;ak+1;:;ak+m1 ( VT Ch,ak+m ?T Ch,k +m1 = n,5ak;ak+1;:;ak+m1 (v1988. :m B ,E,m = 1 H, !ak ChBH Cak;ak+1;:;ak+l1.l = 1,ak 1988,A .5l 1,ak;ak+1;:;ak+l1 (v1988,ak+1 Ch, ak+1;:;ak+l1 (v1988,ak 1988, .!lm H ( (m 2), !ak ChBH Cak;ak+1;:;ak+l1.1 6
20、l 6 m H,ak;ak+1;:;ak+l1 (v1988,B ,L !ak+l;:;ak+m1 (v1988, .l m H,ak+m Ch,ak+m;ak+m+1;:;ak+l1 (v1988, ak;ak+1;:;ak+l1 (v1988, A 9 .8 , B ,E, .! Chai1;ai1+1;:;ai1+j11;ai2;ai2+1;:;ai2+j21;:;aik;aik+1;:;aik+jk1,j1;j2;:;jk 1 Oim+1 im +jm(m = 1;2;:;k1;k 1). :aim;aim+1;:;aim+jm1 (v1988(m = 1;2;:;k). Ch (9v
21、1988.8.4.(1) ! L a;b;c (a2 +b2 +c2)2 2(a4 +b4 +c4):p:a;b;cB HH.(2) !n L a1;a2; ;an (a21 +a22 +a2n)2 (n1)(a41 +a42 +a4n)n 3. p:t B HH. :(1) , !c a+b,52(a4 +b4 +c4)(a2 +b2 +c2)2= a4 +b4 +c4 2a2b2 2b2c2 2c2a2= (a+b+c)(a+bc)(b+ca)(c+ab) 0.) a;b;c H.(2)n = 3(1) f ,n 3 H, i HH, !a1;a2;a3.5 ( T(n1)(a41 +a4
22、2 +a4n) a4 +b4 +c4,(a2 +b2 +c2)2 2(a4 +b4 +c4).(1),a1;a2;a3 H, . t B HH.5. 8AiBiCiDi(i = 1;2;3),VBi;Ci;DiT fii;fli; i(i = 1;2;3),sY AiBi;AiCi;AiDi 3, VTf(f(n). , T,f(f(n) 3, VTf(f(f(n),. Tf(f(f(n) = 2; kf,kSnop. TlnV Un, k i1 n(n 3), pln.i F .: !n = 2k m(m ).k = 0,n ,f(n) = 2;ln = 1.k 0, I n l2k+1 ,
23、 (ny0,2k+1 - n Ol2k+1 t = 2p q(p 6k;q ),qjn;2pjn;gcd(q;2p) = 1,tjn,) f(n) = 2k+1,f(f(n) = 3;f(f(f(n) = 2;ln = 3.5 |Kltjn,tA ,5tAB y0“n.) f(n) = t;f(f(n) = 2;ln = 2.8 ,ln =8:1; n 2; n = 2k m(m ) l2k+1 “n3; n = 2k m(m ) l2k+1 (“n9 S X(1989 M)gS S/v1.1 , i “A;B, K MF, B m, m 1 .AjV U|“A I HZ_ jm “(j =
24、1;2;:).p:i1 k, PL(Aj B) 12L(A)L(B). L(X)V UF“XM. : “E HZ_ jm “:Ej, L(Aj B) =L(ABj).!b1;b2;:;bnFB ,X O (m,yN2mXj=1L(Aj B) =2mXj=1L(ABj)=2mXj=1L(A(ni=1bji )=2mXj=1nXi=1L(Abji )=nXi=12mXj=1L(Abji )=nXi=1L(A(2mj=1bji )yL(bi) = m, 2mj=1bji z ,V7L(A(2mj=1bji ) = L(A).)P2mj=1 L(Aj B) = nL(A), iBk;1 k 2m, P
25、L(Aj B) n2mL(A) = 12L(A)L(B):2. !x1;x2; ;xn (n 2). Ox1 +x2 +xn = 1. p:nXi=1xip1xi 1pn1nXi=1pxi: : !x1 x2 xn,51p1x1 1p1x2 1p1xnChebyshev TnXi=1xip1xi 1n nXi=1xi! nXi=11p1xi!= 1nnXi=11p1xi10Cauchy TnXi=1p1xi! nXi=11p1xi! n2nXi=1p1xi 6vuutn nXi=1(1xi) =pn(n1)nXi=1xip1xi 1nnXi=11p1xi nnPi=1p1xi npn(n1)
26、=r nn171pn1nXi=1pxi 61pn1vuutn nXi=1xi =r nn1)nXi=1xip1xi 1pn1nXi=1pxi:3. !S ( 1 “),fVSS , iz 2 S,lf(1)(z) = f(z);f(2)(z) = f(f(z); ;f(k)(z) = f(f(k1)(z). Tc 2 S, Pf(1)(c) 6= c;f(2)(c) 6=c; ;f(n1)(c) 6= c;f(n)(c) = c.5cfn . !m v11 , flf(z) = zm,k9 f1989- .:An = fz 2 Sjz fn g,Bn = fz 2 Sjfn(z) = zgfn
27、“,A An Bn,f1(z) = zm,) fn(z) = zmn) fn(z) = z , zmn = z,jzj = 1;) zmn1 = 1;jBnj = mn 1. Bn;An /:(1) kjn,5Bk Bn;Y L , 7n = kq, c 2 Bk;fk(c) = c,5fn(c) = fkq(c) = fk(fk(fk(c)| z q= c.) c 2 Bn;Bk Bn.(2)Bk Bn = Bgcd(k;n);gcd(k;n)knKv .(1),Bgcd(k;n) Bk;Bgcd(k;n) Bn;) Bgcd(k;n) Bk Bn.Q, !c 2 Bk Bn,fk(c) =
28、 c;fn(c) = c, !k 3,5|(x1;x2;:;x30)(x1;:;xi +1;xi+1 1;:;x30).f(x1;:;xi +1;xi+1 1;:;x30)f(x1;x2;:;x30)= (xi +1+xi+1 1)X16j 0f9v, , ii;j 2f1;2;:;29g;i 2;xj+1xj 2,|xixi+1,xj+1xj+1 1,f9v.f |Kv H,x1;x2;:;x30M # KB 2, (1.T (1,1989 = x1 +(x1 +1)+(x1 +29) = 30x1 +435,x1 , .!xt+1 xt = 2;1 6 t 6 29,51989 = x1
29、+x2 +x30 = 30x1 +(1+2+t1)+(t+1+30) = 30x1 +465t.30x1 t = 1524;x1 = 51;t = 6.N HF sY51,52,.,56,58,59,.,81.6. !f : (1;+1) ! (1;+1) /Hq: i L x;y 1;#u;v 0;f(xuyv) 6 f(x) 14uf(y) 14v:k “f f.: 7x = y;u = v = t2(t 0),5f(xt) 6 (f(x)1t .xtx,1tt,5f(x) 6 (f(xt)t.) f(xt) = (f(x)1t .!f(e) = c;c 1,5f(x) = f(e) 1l
30、nx = c 1lnx.6,f(x) = c 1lnx(c 1) H,f(xuyv) = c 1ulnx+vlny;f(x) 14uf(y) 14v = c 14ulnx+ 14vlny.Cauchy T,(ulnx+vlny)( 14ulnx + 14vlny) 1.) 1ulnx+vlny 6 14ulnx + 14vlny.) f(xuyv) 6 f(x) 14uf(y) 14v. pf f(x) = c 1lnx(c 1).13S X(1990 M)u 3 vI 1.j HABCD,ABCD , flO1VA,B OHCDM MP,flO2VC,D OHABM MQ, flO1flO2
31、ME,F.p:EF sL PQ sA1Hq BC k AD. :s s .(1)EF sPQ 1HqPC PD = QAQB.!EFPQK,LPQflO1;flO2sYJ;I.* PC PD = PI PQ;QAQB = PQQJ, KQKI = KE KF = KP KJ.) KQ(KP +IP) = KP (KQ+QJ);KQIP = KP QJ.) KP = KQ , IP = QJ , PC PD = QAQB.(2)BC k AD 1HqPC PD = QAQB.!ABDCS.BC k AD , SDSC = SASB.7SP2 = SASB;SQ2 = SC SD.) PC PD = QAQB , (SC SP)(SP SD) = (SB SQ)(SQSA), (SC +SD)S
