1、#(ggg-EEE?2020 ccc 7 ?8. AK.1?. |K.?12nnn. |K.Y?u22ooo. |K.MLu34. “K.ekt48NNN 2019 cGEK) .54?5/#(?0AK1.3 4ABC , O ?%, H R%, D BC ?:, BE,CF p.BO AC = M, B,O,M,P N:?, aq N,Q.K OH,BK AC = U, CK AB = V , PU QV = L. 4BCL 4EFY _q. y: AK ? DY .2. ABCD ?S?o/, ? AC,BD ?u P, DA BC = Q. ePQ AC, E AB :, y: PE C
2、Q.13. 4ABC , O,H O?%, R%, P A u OH ?:, P,A 3BC . E,F O3 AB,AC , BE = PC, CF = PB, K = AP OH.y: EKF = 90.4. : T u 4ABC S, A1,B1,C1 T u BC,CA,AB :, ?(A1B1C1). A2,B2,C2A1T,B1T,C1T ?,?:. y: AA2,BB2,CC2?:u .25. F/ ABCD , AB ? CD, ? AC,BD ?u P, 1L B ACu A ?, 2L C BD u D ?, 3 4BPC ?. y:1,2,3?%3 AD .6. 3 4A
3、BC , BAC = 90, AD BC ?p, P AD :, C ?%, CA ? BP ?u E,F :, B ?%, BA? CP ?u X,Y :, E 3 PB , Y 3 PC .y: 4DEX v 4DY F.38. ? O ?u AF PE, EF BC = S. SD PE D. y:4APD,4BPD,4CPD ?R%?.47. K,L,M,N 3 4ABC ?, KL?AC, MN ?AB, BLAC =E, CN AB = F, MK BC = X. y: AX ? (AEF) .9.P o/ ABCD S:, BPC = 2BAC, PCA =PAD, PDA =
4、PAC, y: PBD = |BCA PCA|.10. ? I S?o/ ABCD Sk: P, 4PAB, 4PBC,4PCD, 4PDA ?. P ?%? , ? PA,PB,PC,PD O?u A1,B1,C1,D1. y: PI,A1C1,B1D1?:.511. 4ABC 9? , : E,F O B,C ?:, I S%, K AI EF ?:, ? T l BAC ?:. ? A ? (AEF) ?,?:O X,S, S0 S u AI ?:. J ? (AS0K) AX ?,?:. y: T,J,I,K o:?.12. ? 4ABC (AB = AC) S% I. ? L C A
5、I u I, AC, ? (ABC)?,?:O Q,D, M,N O AB,CQ ?:. y:AD,BC,MN n?:.613.o/ ADJE ?u B!C, F BC :, G!H 3AB!AC , FG ? CD, FH ? BE, G ?% GB ? 4AGH?u l. y: l L:.14. : P,Q u1o/ ABCD ?, ? 4ABP BCQ ?n?/. PL P Ru PD ? l1, L Q Ru DQ ? l2.y: l1 l2?: u 4ABC L B :?p.715. ?pR?o/ ABCD S?u? ? (O). L: A C? ? (O) ?, BD n?/ .
6、y: 4BOD ? ?.16. ? ABCD ?S?o/. L: A!B ?, CD u: E. L: C!D ?, AB u: F. AE DF ?u: G,BE CF ?u: H. y: 4AGF!BHF!CHE!DGE ?S%3?.817. 4ABC ?S? BC!AC Ou: D!E. ? P S?lDE :, v APE = DPB. AP!BP O? DE u:K!L. y: 2KL = DE.18. ? n?/ ABC ?. ? AB , ? u:, T: C u AB ?. BCA ? ?u?: P!Q. y: ABP = QBC.919. ? 4ABC ?b?n?/. A ?
7、S? 4ABC ?2g?u: D. ? O 4ABC ?%. AOB,AOC ?O AD ? ?u: P,Q. PQ AD ?R?u: R.y: AR ? BC.20. ? 4ABC ?, ? ? KL L AB ?: M (K,C u AB ?). ?L: L M, CK ?u: P Q (Q 3 KP). LQ 4KMQ ?2g?u: R. y: A,P,B,R o:?.1021. ? ABCD ?S?o/, 8/, ?pR?u: H. - M,N O BC,CD ?:. ? MH,NH O AD,AB ?u S,T. y: 3 uo/ ABCD ?: E, v:1) ? EH BES 9
8、 TED;2) BEN = MED.11|K?1. “|1. ? n, XJ3 k p?8 A = a1,a2, ,ak,? 1,2, ,n A eZ?L, k ?. (?MK)2. K? p!q!m!n, m + n + 1 ? km, km+1, k1, k0, k1, , knv:(1) k1 0 k1(2) p km km+1 kn q.?.123. 1,2, ,107/x 2 , #N?1Xe: J a, a 9 a p?kUC?. ek, : ULkg, kCx?2. |A4. XJm?:/ n (z?/:), m?, n1 ?:, n ?kU?. (?MK)135. ? n ?,
9、 T kv x + y /? k / (Tk k ), ?:(1) ? a ?: A!B, o AB a , o3: C, AC!BC a (2) ?n?/, ?k.y: k 2.143. :K (S)7. ?:/!7?, y: 3 n / (?, #Nkn:?): A1A2An, :-% G .8. ? n!k ?, T = (x,y,z) | x,y,z N,1 x,y,z n m?IX n3?:?8. T (3n2 3n + 1) + k :?/, vXe: XJ T : P!Q :, PQ 1uI, KPQ ?k?:. y: 3? k p?N, c1, z:.159. u k m nk
10、:? n n n :? = (x1,x2, ,xk) | xi N,1 xi n,1 i k, ?:Um/, =: ?: P(x1,x2, ,xk)!Q(y1,y2, ,yk), ”?31 j (1 j k) ?, xi= yi(1 i k,i 6= j), xj yj, KT0u P!Q m?:M(z1,z2, ,zk):, zi= xi= yi(1 i k,i 6= j),xj zj yj.?/k p :, P:? k ? fk(p), fk(p) ?.10. ? X = (x1,x2, ,xn) | x1,x2, ,xn Z n mk:?8, a = (x1,x2, ,xn)!b = (
11、y1,y2, ,yn) X, ?l:|a b| = |x1 y1| + |x2 y2| + + |xn yn|.? r ?, a = (0,0, ,0), b = (2,0, ,0), P r(a,b) =u X| |u a| r,|u b| r, y: |r(a,b)| =nPj=0Cjr1Cr1n+rj1. (?MK)164. |O (=)11. 3eZf? 2 , f?/. /?.1815. k n fA n r?, m r?O3 m fp, 3n m r3?. : U?mf?V?5. |4 (?O)16. ? n (n 2), |X| ?, ? X ? n ?f8 B1,B2, ,Bn
12、(7p), 3 X ? n ?f8 Y , v: i = 1,2, ,n, k |Y Bi| 1, p |A| Lk8 A ?.1917. ? X1,X2, ,X1008 S ?p?f8, v: XiXi+1= , Xi Xi+16= S (1 i 99), |S| ?.18. ? X 100 ?8, ke?5?g, n: u?dX ?f8? n ?S? A1,A2, ,An, 3 1 i j 1.2. ? N?N?8. y34O?f : NN, ve?:(1) f(1) = 2;(2) ? n, k n + f(n) = f(f(n).223. e? n ke5, K/?0: ? n ?k?
13、d1,d2, ,dk, 3 1,2, ,k 1,1, vn = 1d1+ 2d2+ + kdk.y: 3?, T 2020 ?.4. 3:8 S = (x,y) |x,y = 1,2, ,101 ?f8 A, ? A?o:. x y 1?F/ ()/). A?U.235. ?8 T = 1,2, ,n. u8 S = s1,s2, ,sk T, s1 s2 sk, Pf(S) = s1 s2+ s3 s4+ =kXi=1(1)i+1si.OPSTf(S).6. |?k 65 If, 3!8 65 If-# 65|, z|? (If). (Juy?|, e(Tk:(1) |?Ip;(2) |?I
14、p.y: 7k|/v?o:?. Lzn:?.e/k3?:, Kd?/?0, K/?0. ?=?259. ? 2 = p1 p2 p3/?:o/, zT/ 108 :. y: 3z?:n:?n?/, ?on?/?.2713. k? (a,b), va | b2,b | a2,(a + 1) | (b2+ 1).14. k? n 1, ?3 1,2, ,n ? p1,p2, ,pn, v pi+ i, i = 1,2, ,n, 9 pi i, i = 1,2, ,n, ? n ?X.2815. ? M ?Ik?:?8, N?N?8. 3A f : N M, v?np? a,b,c, e(a,b,c
15、) 1, K f(a), f(b), f(c) ?16. ? ai,bi,ci, i = 1,2, ,N, 3N ?. ? 1 i N,ai,bi,ci?k. y: 3? x,y,z, ? xai+ ybi+ zci(1 i N) ?k47N .2917. ? 2012 :/ n , ?z?:p. u n :?8, e:?p, K/80. ?8? n ?.18. ?8 M = z C|z| = 1,Re(z) Q. y: E3?n?/, n:3 M .3019. ? m n 2. ve? f : 1,2, ,n 1,2, ,m ?: 3? i 1,2, ,n 1, v f(i) f(i + 1
16、).20. 8?f8. y: u?(m,n), 3f8?n x y ?, G B2. G ?U.24. n n ?kUISz?m/x. zg? 2 2 ?/, ?x/, /, /x. k? n, ?L, ?zxC, ?zCx.33|KMLu1. ? n ?. ?Xe: a1= n, z? k 1, akv0 ak k 1 ? k ? a1+ a2+ + ak?. y: ? n,c? a1,a2, l,.2. ? p p + 2 , p 3. ? an a1= 2, ? n 2kan= an1+pan1n,p dxe L?u x ?.y: n = 3,4,.,p 1, k n | pan1+ 1
17、 .343. ? m ? |m| 2. ? a1,a2, v a1,a2?uz? n k a1,a2,. y: XJ? r s 2 ? ar= as= a1, Kr s |m|.4. (1) ? n,k ?. ? x,y vnk+1 x n.(2) ? n ?, A 1,2,3, ,n, A ?L n. | (1) ?(, |A| ?.?O.355. k? k, ?z? n,(kn)!n!U? 2(k1)n+1?.6. ? q ?kn. k, u: X. 31 n (n = 1,2,.), zJH?o?, X?c? qnl. L?eZ?, q u: ( X),?. q kU?.367. k?
18、a,n, ? n ? (a + 1)n an.8. k p,q ? 3pq1+ 1 ? 11p+ 17p.379. ? n u 1 ?, a1,.,amkL n n p?, m = (n) . y: n ?f p ?K? k, kvp(n) vp mXi=1aki!,p vp(x) L x p f?.10. ? S = A1,A2,.,An, A1,A2,.,An n p?k8 (n 2), v? Ai,Aj S, k Ai Aj S. e k = min1in|Ai| 2.y: 3 x Sni=1Ai, ? x u A1,A2,.,An?nk8.3811. ? X1,X2,.,X1008 S
19、 ?f8. z i 1,2,.,99, k Xi Xi+1= , Xi Xi+16= S. |S| ?.12. ? S n2+ n 1 ?8. S ?k n ?f8a. y: 7ka3 n p? n ?f8.3913. ? A k8, A1,A2, ,An A ?f8, ve?:(1) A k? 0;(2) ? xi Ai(i = 1,2, ,n), k x1+ x2+ + xn 0.y: 3 1 k n 1 i1 i2 ik n, ?Ai1Ai2Aik/ C4. ?.16. ? G 2019 ?, E 8. vXe? n ?: E ?u n, K3 E ? 908 ?f8, v: (1) ?
20、f8?Tk?:; (2) ?f8?.4117. / R ? 2016 ?/, z?/?1u R ?, ?/?:(:. 3?/?, e:(:, S(:, K?. k, ?.18. ? (a1,b1),(a2,b2),.,(a100,b100) ?kSK?. - N Lv 1 i j 100 |aibj ajbi| = 1 ?| (i,j) ?8. N U?.4219. (1) ? Pi(xi,x3i) (1 i 3) - y = x3?n:. (?7.(2) y: 3: ,P2,P1,P0,P1,P2,vXe5: ?n?a,b,c, :Pa,Pb,Pc?=?a+b+c =2020.20.? n
21、K?.?k? (n + 1)28 Si,j1,2,.,2n ( 0 i,j n ?), vXe:(1) ? 0 i,j n, k |Si,j| = i + j.(2) XJ 0 i k n 0 j l n, K Si,j Sk,l.4321. n ?pp?)?3?, ?ph1 h2 8, |V | = n. N?f : V Z /?0, XJ f v:(1)PvVf(v) = |E|;(2) ?eZ:/, Ko3: v, ? f(v) L v ?/?:.? m(G) k/?0N? f ?. y: e V z:?,?:k, K n m(G) n!.24. ? m,n u 1 ?, m n. a1
22、,a2,.,an n L m ?p?, a1,a2,.,anp. y: ? x, 3i (1 i n), ? |aix| 2m(m+1)|x|, p |y| L y ?C?l.4525. ?k 68 ?, z?7. 3? k ? (k,k) (k,k) y3. )l? 136 ?eZ, ?vkp. e,?, KT)?. ? k, ? 68 N?, T)oU?y?o?u k.26. ? S dmk:?8, v5: S ?:m?lp. b? S ?:?I (x,y,z) ?, k 1 x,y,z n.y:|S| /?z:7o/, z?: 108 . y: 3z?:n, ?on?/?.47“Kekt
23、1. ? n 2, a1,a2, ,an 1 v a1a2an= 2n. y:a1+ + an2a1 2an n.2. ? n 2, e a1,a2, ,an? a1a2an= 1. y:a1+ + an n mina1,a2, ,an 1a1+ +1an n.483. m3?8 M: ? M ?8?k:8?4. xn+a1xn1+an(an6= 0) k n , n 3, an2= 0,an1an n + 1, y: ?kum (12,0).495. ? f(x) = (x x1)(x xn), 1 xi 1, i = 1,2, ,n, y:3 a,b v(1) |f(a)| 1;(2) |
24、f(b)| 1;(3) 1 a 0 b 0? n k12ann 2.8. ? c, ?v x1+ x2+ + xn xn+1?S? xnn=1kx1+x2+ +xn cx1+ + xn.519. ? P(z) = az3+ bz2+ cz + d, a,b,c,d C v |a| = |b| = |c| =|d| = 1. y: 3? 1 ?E z ? |P(z)| 6.10. z? d, ?mm I R ? a0, a1, ,a2d1 I, P(x) = x2d+ a2d1x2d1+ + a1x + a0vk.5211. f : Z Z k51) e m,n Z, m 6= n, Kf(m)
25、f(n)mn Z;2) 3? A B 9? N, ?|f(n)| A|n|N, |n| B, n Z.y: f(n) = P(n), n Z, P(x) knX.12. n N, 1,2, ,2n y n ?f8 A,B. e3cn, ?Noy, 3 A ? a1, ,an B ? b1, ,bn?nXi=1(ai bi)2 cn. cn?.532019 cGEK)2019 c 11 ?, #(gGE1?g (?). zgo?K, m? 50 . 0?K9A?). K 1.x L1 1 g?1 x K, K 2.y ?aq.I. KKKKKK 1.1 ?n 2,?,?a1,a2, ,ank n
26、Xi=1aiai1+ ai+1! nXi=1aiai+1! nXi=1a2i, a0= an,an+1= a1.( ekt K)KKK 1.2 n? k0,k1,k2, k0 O ?%, A1A2. P B,C1,C2O? k1 k2, k1 k0, k2 k0?:, A1C2 A2C1?u k0S: D. ? t1,t2O A1,A2?u k0?. y: e t1 k1, t2 k2O, K O,D,B ?.(E?N # K)KKK 1.3 k?X f(x), ? m,n, f(x) m(mod n)k).(u“ ?u K)KKK 1.4 ? n 4. e? m vu?k n : m ?,
27、3?, Tk?:?. m ?.(x1 K)KKK 2.1 : D 3 4ABC ? BC AD ? BAC. 4ABD,4ADC?S%O I1,I2, I1I2O? AB,AC u M,N. y: AD,BN,CM n?:.(E?N # K)54KKK 2.2 kUL?g? a1 a2 n.(u“ K)KKK 2.3 kve? f : Z Z :(1) f(2019) = 2018;(2) f(n f(n) = 0, n Z;(3) 3? t ? n Z, f(x) = n Tk t ).(u“ K)KKK 2.4 ? G r?k, |G| = n, S(G) ?H G ?z:?k4?, ma
28、x|G|=nS(G).( ekt K)II. )KKK 1.1 ?n 2,?,?a1,a2, ,ank nXi=1aiai1+ ai+1! nXi=1aiai+1! nXi=1a2i, a0= an,an+1= a1.yyy. max= 1., 1.? n = 2 , a12a2+a22a1 2a1a2a21+ a22= 1.? n 3 , ? a1= M,ai= 1,i = 2, ,n, M 0, Pni=1aiai1+ai+1(Pni=1aiai+1)Pni=1ai2=M2+2M+1+n32(2M + n 2)n 1 + M2,- M ? 1.55, = 1 ?, du n 2,ai 0,
29、 K nXi=1aiai1+ ai+1! nXi=1aiai+1!nXi=1aiai1+ ai+1(ai1ai+ aiai+1) =nXi=1a2i.n, max= 1.?5?J?“K, ? 45% ?)dK. ?Kk n ? ?Z?, JY 1, ?3 a1,a2, ,an:?4?. y?, I?rnPi=1aiai+1 ai1ai+ aiai+1?.KKK 1.2 n? k0,k1,k2, k0 O ?%, A1A2. P B,C1,C2O? k1 k2, k1 k0, k2 k0?:, A1C2 A2C1?u k0S: D. ? t1,t2O A1,A2?u k0?. y: e t1 k
30、1, t2 k2O, K O,D,B ?.yyy: ? ki?% Oi, ri, tiu Ei(i = 1,2), k0?r. t1k t2, k OA2k O1E1. ( t1,t2R) O,C1,O1?, O1C1E1= O1E1C1= 9012E1O1C1= 9012C1OA2= OC1A2. A2,C1,E1?. n, A1,C2,E2?, E1,B,E2?.3F/ A1E1E2A2, O A1A2:, D ?:.E1BE2B=O1BO2B=r1r2qA1E1=(r + r1)2 (r r1)2= 2rr1, A2E2= 2rr2,56?A1DDE2=E1DDA2=A1E1A2E2=r
31、1r2.5?OA1OA2=DA1 sinODA1DA2 sinODA2,BE2BE1=DE2 sinBDE2DE1 sinBDE1? OA1= OA2, KsinODA1sinODA2=DA2DA1,sinBDE2sinBDE1=BE2 DE1BE1 DE2?sinODA1sinODA2sinBDE1sinBDE2=DA2DE1DE2DA1BE1BE2=r2r1r2r1r1r2= 1=sinODA1sinODA2=sinBDE2sinBDE1? ODA1+ ODA2= A1DA2= E1DE2= BDE1+ BDE2, ODA1= BDE2,ODA2= BDE1.? O,D,B ?. K.?y
32、yy? (eeeQQQLLL)?nnn): ? k1?% O1, k2?% O2. ? A2C1? k1u E1, A1C2? k2u E2. ey: A2OC1B ?A1OC2B ?.5? C1 ?O ?O1? q%. ? O1E1k OA2. d O1E1k t1. q t1 ?O1. ? t1 ?O u E0, K O1E0 t1. d E0 E1-. nk E2t2:. A2A1, ? t2k t1. ?E2BC2= A2E2C2= E2A1E1,57? C2BE1A1?. n A2C1BE2?.du A1A2, ? A2C1A1= 90. kA1E21= E1C1 E1A2= E1B
33、 E2E1MC1 MA2= E1B E2E14A1E1B 4E2E1A1A1C2E1= A1BE1= E2A1E1.?A1O = OC24OE1C2= 4OE1A1OC2E1= 90,? OC1E1A1?, ?k A1C2BE1?. k A1OC2BE1:?. nkA2OC1BE2:?. ?A1OC2BE1 ?A2OC1BE2 ?O, d%nk A2C1, A1C2, BO?:. ? B,O,D ?.?5?J?AK, ? 50% ?)dK. ?K?J:3u? k1 t1?: E1, k3 t2?: E2, uy A1,D,E2 A2,D,E1n:?. |n?ysinODA1sinODA2=si
34、nBDE2sinBDE1l?(; ? ?A1OC2BE1, ?A2OC1BE2, ?O n?%n, ?A2C1, A1C2, BO ?:l?(.KKK 1.3 k?X f(x), ? m,n, f(x) m(mod n)k).): K8?du? x ?H? n ?X, f(x) ?H? n ?X.(n N)ky x Z, |f(x + 1) f(x)| = 1.e3 x0 Z ? |f(x0+ 1) f(x0)| 1. P |f(x0+ 1) f(x0)| = k, Kf(x0),f(x0+ 1), ,f(x0+ k 1),d f(x0) f(x0+ 1) (mod k) , ? k ?X,
35、g! x Z, f(x+1)f(x) = 1. K3 x Z f(x+1)58f(x) = 1 f(x + 1) f(x) = 1.d?n, f(x + 1) f(x) 1 f(x + 1) f(x) 1. f(x) = x + a f(x) = x + a, a Z.?5? J?K, ? 25% dK. K?3uuy f 3? 1, l? f X 1 ?g.KKK 1.4 ? n 4. e? m vu?k n : m ?, 3?, Tk?:?. m ?.): m ? 2n 3.: d 2 : n 2 :|?, v, dm = 2n 4.,: ? m 2n 3 , n 8B5y(.n = 4
36、, m 5, (w,.b?k n :?(, ?k n + 1 :,(1) XJz:? 3.?:|? A1A2Ak, u? A1A2Ak?: A1, d A1?u?u 3, ? A1?: A2,Ai,Aj, , 2 i j BC AD ? BAC. 4ABD,4ADC?S%O I1,I2, I1I2O? AB,AC u M,N. y: AD,BN,CM n?:.59yyy: BI1,CI2?u AD : I, K I S%.y?:(1)? I1I2k BC,d1?59?nII1I1B=II2I2CIDBD=IDCD BD = CD.d, 4ABC ?n?/, AD. d I1I2k BC M,N
37、 u AD . AD,BN,CM n?:.(2)? I1I26k BC, ? I1I2? BC u Q. 4IBC 9? I1I2Q, d Menelaus n,1 =II1I1BBQQCCI2I2I=IDBDBQQCCDDI=BQQCCDBD,BQCQ=BDCD.q 4ABC 9? MNQ, d Menelaus n,AMMBBQQCCNNA= 1.dAMMBBDDCCNNA= 1.d Ceva n?_n BN,CM,AD ?:. ?y!?555. ?AK, ? 78% ?)dK. dKIU I1I2 BC 1?1?, 1?, 1?I I1I2 BC ?u Q, 2( Menelaus n
38、Ceva n?_ny?(.KKK 2.2 kUL?g? a1 a2 n.60yyy: kynnnne 4k + 3 . p ? x2+ y2, x,y Z, K p | x p | y.n?y: e p - x. K3 x ? p ?_ x1 Z, xx1 1 (mod p), l?p | (x1y)2+ 1.P m = x1y, Km2 1(mod p).?d?n9p12, ?1 mp1=m2p12 (1)p12= 1(mod p)g. ? p | x. l? p | y?K.Iy: u? n, 3? k, kn + i, i = 1, ,nUL?. 4k + 3 .k, ? n u n p
39、?p1,p2, ,pn. 5? n,p1, ,pnp, u k |8:kn+1 p1(mod p21).kn+r pi(mod p2i).kn+n pn(mod p2n)dIn|k), ?) k. d, pi|kn + i, ? p2i- kn + i, i = 1, ,n. e kn + i UL?,dn, ? pi?g, kn + i pi?gT 1 g.?555. ?K? J?K, ? 25% ?dK. )|? 4 3 ?g(In3?Y?, .IN?P“, 2015 cK TST ?ke?K:61P b1 b2 dkU?K?|?S?. y? m, 3? n, v bn+1 bn=m.dK
40、?mT, k?EL?, 2L? 4 3 ?fm?L.?, u“1N?y?mT?. k,?e.KKK 2.3 kve? f : Z Z :(1) f(2019) = 2018;(2) f(n f(n) = 0, n Z;(3) 3? t ? n Z, f(x) = n Tk t ).): v? f =k f(n) = n 1, n Z.d ( 3 ) , ? f(x) = 0 ?k) x1 x2 xt. ey t = 1.y. e t 2, f(x) = x2 x1,?k) y1 y2 yt. d ( 2 ) ,fyi f(yi)= 0, i = 1, 2, , t.l? yi (x2 x1)
41、f(x) = 0 ?). =yi x2+ x1 x1, x2, , xt.5?y1 x2+ x1 y2 x2+ x1 0 g. ? t = 1.d ( 1 ) ( 2 ) ,0 = f2019 f(2019)= f(1).? 1 f(x) = 0 ?). ?d ( 2 ) 1 = n f(n). = f(n) = n 1. u?, f(n) = n 1 v.?)?: u?n,d(3),?f(x) = n?)x1,x2, ,xt.d (2),0 = f(xi f(xi) = f(xi n),62 xi n,i = 1,2, ,t f(x) = 0 ? t ). qd ( 1 ) ( 2 ) ,0
42、 = f2019 f(2019)= f(1).= 1 f(x) = 0. ?3 k(1 k t), ? xk n = 1, l?n = f(xk) = f(n + 1).d n ?5, f(n) = n 1,n Z. u?, Tv.?555. dK?N?“K, ? 60% ?)dK. (3)?y? f 7LV?. )B|?5?(; )?|?5f(x) = n ?.KKK 2.4 ? G r?k, |G| = n, S(G) ?H G ?z:?k4?, max|G|=nS(G).555k G r?, e?: A,B, l A ? Bl B ? A 3k.):max|G|=nS(G) =8:3,n
43、 = 3(n + 1)24, n 4P G ?k: A1, A2, , An. A B Ll A ? B ?k.? n = 3 , r G k/, = A1 A2 A3 A1. d,S(G) = 3. e n 4 ?/.k?E.? k =n + 12. Xe? G :A1 A2 A3 Ak1 Ai A1, i ?H k, k + 1, , n. K G r.?v?4, l Ai? Aj(k i, j n, i 6= j) ?L A1, A2, , Ak1g, =d?u k, u4? k(n + 1 k) =(n + 1)24. ? max|G|=nS(G) (n + 1)24.ey: ?: n 4 r G, 7?L(n + 1)24?4?Hk:.? d(i, j) l Ai? Aj?, d?5vk?:.63? k = maxi6=jd(i, j). ”? A1 A2 Ak+1 k ?.qdr5, l Aj? Aj+1(k + 1 j n, An+1= A1), 3Lk ?. , 4L k +k(nk) = k(n+1k) (n + 1)24. ?555. dK?J?|K, ? 2% ?)dK. ), ?:lL?, ? S(G) ?., 2?.7U1, ekt1, 2, ?u2(1. 2004442. u“, 200241)64
