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0 10 20 30 40 50 60 70 80 90 100

1

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10

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............................................................................................................................................................................................................................................................................................................

Marks (out of 100)

Num

ber

ofStu

den

tsMarks on Test 1

U<?| ∗ |ÇDAO8a7Q,2nAO8WJ9A 32 AO.A?3AO8GKD.U,3P8aJQAY.6AO0QPIl9P5AD:4AOP5A

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86Ta0tAG.1.6Ac8CADP<8s<:,O>3AD797QA8?Ó 4j.L,9@254UAO7/20Q7QA7Q,25ASJ9A

100.6,3P8BJQA).1<lQP5,9msE3<O467QA

AOlQP5AG0/3AÅ

im| ∗|u9.j0Q8w25<DP5J«J3<-798).~Ó <D797 A-A/K¥r<-P4UA=8CA=P5AD7QJpmx,3:zl325Ae¡/0Ó AG.1.6A=<e@QAO8*,/417JLÓ 0Q7QAk7Q,2nA'JQA100

.U,3P8JQAS.1<kl9P5,9msE3<D417QAAOlQP5AG0/3A'lt,/0QPB¡30tA'8s<k:=,D>3AO797QA3KJ3<-7982n,/0325AD8.6AD8

ADl9P5AO0/3AO8GK8*,/4o2

A-gG<O.UA <

90T3VWPI8s4QAO.j.UAw,9@254UAO7/2I0Q7QAw7Q,25A)JQA

70J3<-798.6<lQP5,9msE3<O467QA

ADl9P5AO0/3A/Kt8s<':=,D>3AO797QA'8CADP<

ADgG<D.6A<

87T;ÇD,3P8C¡30Ó AO.j.UAS<O0QP<S25ADP5:z417

A.6<lQP5,9msE3<O467QA

AOlQP5AG0/3A3K9mx,3:=@346AD7=JLÓ

ADl9P5AO0/3AO8<O0QP<52xCAG.1.6A

A-mxP4j25AD8Å

Ð~ÐÍ/ÌdÌ?ÍË ÐÌdÌ1 Ò ∗|kÇ/Ó AO798*AG4UgG79<D7/2nAJQAr¥r<DP546Ad41798*mxP4j2).6<7Q,2nABQ79<D.6AJQAO8

32 AO.A?3AO8DT9Ç/Ó AD798CAO46gG79<-7/25AYm5<D.6mn09.UAB.1<:

A-J/46<D7QAJQAB.6<mn.6<D8C8CAAG2;,9@Q254UAO7/2I0Q7QAw7Q,2nA)J9A

80T.1.6A)m5<D.6mn09.UAY<D0Q8s8s43.Ó

AG25AD7QJ/0tAJ9AD8a7Q,25AD8?K8*,/4o2.6<)J34jÔ

AOP5AD7QmnAAD7/2nP5AY.1<)7Q,25AY.1<wl9.j0Q8WE3<D0325A)AG2.1<w7Q,2nA.6<wl9.j0Q8W@3<-8s8*A/K/A?2,9@/2546AD7/2

40T/.j.UAwm5<D.6mn09.UAAD73Q7.6<k:=,D>3AO797QAJ9A'.1<mn.1<-8s8*A'AG2B,9@/2546AD7/2

58T

AD:=,37/2nP5ADPB¡30tAS.Ó AD798CAO46gG79<-7/25Ak<mx,3:z:z418a0Q7QASADP5P5AO0QP©T

TI¦~7Y<BgG<-:=A3K"!<?346ADP<D7QJ$#;,/.6<D7QJ9<25<-f-A250QPn798;m5<D.j.1417Qg,/032QFSE9,/.6Aa730Q:=@QAOP8DT_ E9AatP8n2730Q:=@QAOPmn<O.1.6A-Jw:0Q8x2@QAW<FSE9,/.6AW730Q:@9ADP@9AG2¢FSADAD7w<-7QJY467Qmn.j0tJ3417Qg1<D7QJ

9T/<msE=730Q:@9ADPBmn<O.1.6A-J=<Dh125ADP2xE9AtP8n2W:z0Q8n2@9A'<FSE9,/.6AS730Q:@9ADPFSE/4UmsEz468

125,

10gGP5AO<O25ADPI2xE3<D7k2xE9AlQP5A*/4U,/0Q8W730Q:@9ADPam5<D.j.UADJT

DNU<?|; ∗ | _ E9AtP8n2;2541:A2xE9AwgG<D:A468l9.1<G>3ADJLKD2xE9AYltAOP8*,37FSE9,'mn<O.1.18I2xE9AY730Q:=@QAOP

154682xE9ASF)46797QAOP?T DÆ-l9.1<D417=FSE-> !<G/4UAOPE3<-8w<SF)467973417Qgk8n2nP<O25A-g*>e4jhE9AzgO,9AO8tP8n2<-7QJ=m5<D.j.68

4T

i@-| ∗ | _ E9A=8*ADmx,37QJe2546:=A2xE9AgG<-:=Az468)l9.1<G>3ADJLK2xE9A=lQADP8C,37dFSE9,mn<O.1.18w2xE9A730Q:@9ADP50418I2xE9ABF)46797QAOP?T¦ih!<?346ADPIgO,9AO8tP8n2*K/E9,-FJQ,9AO8aE9A)g?0Q<DP<-7/25A-A2xE3<O2IE9AF)41.j.3F)467QÅ

ÉGÊQËÌ?Í9ÎxÏÐÍ ËÐcÑ2 Ò ∗|¦~7p2xE9AdgG<-:=AdJQAO8*mxP4U@9A-Jp467r@|nKI2xE9A=2<-P5gOA?2730Q:@9ADPF<-8

50T¤3,3P;FSE3<G2aJ/41ÔADP5AO7/2/<D.j0tAD8B,/hQ2xE9Aw2<-P5gOA?2730Q:@9ADP418a4j2g?0Q<DP<-7/25A-ADJ2xE3<G2

#;,/.6<D7QJ9<YF)4j.1.E3<?3A<YF)467973417Qg)8n2nP<O25A-g*>k41h!<?346ADPgO,9AD8aQP8x2xÅT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

!<?346ADPWAG2 #3,3797QASlQ<DP2546mn46lQAD7/2 <0Q7 AG0dJ9<D798B.UAD¡30tAG.msE3<mn0Q7LK32n,/0QP <2n,/0QPCK<-797Q,37QmnA=0Q7«730Q:ADP5,p¡30Ó 41.,/0rAO.j.UA=<emsE9,/468x4TÇDA=lQP5AO:z46ADPw730Q:

ADP5,pJQ,/4o2 A?2nP5A=0Q7AD7/2546ADPWJQA

1<9TRE3<-¡30tA'730Q:

ADP5,=8s0t@38

AD¡30tAO7/2J9,/4j2 AG2nP5A0Q7eAD7/2546ADPW¡3094AD8n2BJ9A

1<10JQA)l9.j0Q8¡/0tA).UA)730Q:

ADP5,lQP

A-mADJQAO7/2?T

U<?|; ∗ |ÇD,3P8aJ9AY.1<wl9P5AD:4AOP5AwlQ<DP2546A3KD.6<wlQADP8C,3797QA)¡/094Q<D797Q,37QmxAOP<w.UAY730Q: ADP5,158CADP<wJ

A-mn.1<-P

A-AYgG<gG79<D7/2nA9TDÆ-l9.j4U¡/0tADP¡30tA$!<?346ADP;<0Q7QAY8x2nP<G2

A-g?46AYgG<gG79<D7/2nAY8?Ó 4j.

,/0tAlQP5AO:z46ADPAD7z<-797Q,37mx<-7/2.6A)730Q:AOP5,

4T

i@-| ∗ |ÇD,3P8J9AB.1<YJ9AO0ÆD4AO:Al9<-P¢254UA/KO.1<lQADP8C,3797QA¡3094/<D797Q,37QmxAOP<.6A730Q: ADP5,508*AOP<J

ADmn.6<DP

ADAgG<gG79<D7/2nA9T+O4!<?346ADP ,/0tABlQP5AO:z46ADPCK-mn,3:=:=AD7/2ltAG032xx4j./8?Ó <D8C8x0QP5ADPJQASgG<-gG7QADP9Å

Ð~ÐÍ/ÌdÌ?ÍË$ ÐÌdÌ2 Ò ∗ | a<D798w.6<zlQ<DP2546A@-|xKL.6Az7Q,3:=@[email protected]

A?25<O4j2

50T a0tAG.1.6AD8=8*,37/2k.UAO8/<O.UAG0QP8dJ/0 7Q,3:@3P5A-smn4U@/.UAr¡/094YlQAO0/3AO7/2=<D8C8x0QP5ADP <

#3,3797QA)0Q7QA8n2nP<O2ADg?4UAgG<-gG79<-7/25AS8x4!<?346ADP ,/0tAl9P5AD:4UAOPQÅ¯

T¦~7S2xE9A)J/46<-gGP<-:dKABCD

468<w8*¡/0Q<-P5Aw<D7QJS2xE9A)mn,9,3P5J34179<O25AD8a,/hA<-7QJ

D<-P5A)<-8W8*E9,-F7TU<?|k ∗ | _ E9ArlQ,/467/2 P

E3<-8emn,9,3P5J34179<O25AD8(10, 0)

T!+-E9,-F 2xE3<G2k2xE9A«<-P5AO<%,/h2nP541<-7Qg?.6APCB

41810T

i@-| ∗ |u3,/417/2 E(a, 0)418p,37 2xE9A

x <©Æ-4188x0tmsE 2xE3<G22nP46<D7Qg?.UA CBE.j4UAO8AD7/2541P5AO. >«,/0328s46JQA=8*¡/0Q<-P5A

ABCDTp¦ihI2xE9A<DP5AD<e,/hI2xE9A2nP541<-7Qg?.6A=418SA-¡/0Q<D.;2n,d2xE9A<-P5AO<',/ht2xE9A)8*¡/0Q<-P5A/KFSE3<G2418a2xE9A/<D.j0tAS,/h

aÅ

im|a ∗|W+-E9,-F 2xE3<G2W2xE9AOP5Ak418Y7Q,lt,/417/2F,37=2xE9A

x <?ÆD468hi,3PFSE/4UmsEz2xE9A<DP5AD<,/ht2nP541<-7Qg?.6AABF

468BAD¡30Q<O.Q25,'2xE9A<DP5AD<S,/hL8C¡30Q<DP5AABCD

TÉGÊQËÌ?Í9ÎxÏÐÍeË~ÐÑ

3 Ò ∗ | G418<)lQ,/467/2,37S2xE9AY.j467QAYlQ<D8C8x467QgB2xE3P5,/0tgOES2xE9AYlt,/417/258

M(0, 8)<D7QJ

N(3, 10)8x0tmsE2xE3<G2 4DCG

.146AD8AD7/2541P5AO. >,/03258x4UJ9A)2xE9A'8C¡30Q<DP5AQT¦ih2xE9A)<DP5AD<,/h 4DCG468aA-¡/0Q<D.32n,2xE9Aw<-P5AO<,/hQ2xE9Aw8*¡/0Q<-P5A/K3JQA?2nAOPn:467QA2xE9A)mx,9,3P5J/4U79<O25AD8B,/h

GT

`

-

6

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ss

s s

s

D (1, 8) C

A (1, 4) B

P (10, 0)

ABCDAO8x2I0Q7m5<-P5P

AAG2.6AD8Bmn,9,3P5JQ,3797

ADAD8BJQA

AAG2aJ9A

D8*,37/2I467QJ/4U¡/0

A-AO8DT

U<?| ∗ |BÇDAlQ,/467/2 P<SlQ,/0QPWmn,9,3P5JQ,3797

ADAD8

(10, 0)T¥%,37/2nP5AOPW¡/0tA.UAw2nP46<D7Qg?.UA

PCB<0Q7QA)<D41P5ASJ9A

10T

i@-|Y ∗ |Y+-,/4o2Y0Q7«lt,/417/2 E(a, 0)K;8x0QPY.Ó <?Æ-AeJ9AD8<@38*mn418C8CAD8?K;J9Ae:z<-734ADP5Ae¡/0tA.UAk2nP46<D7Qg?.UA

CBE8C,/4j28x4j250

Azmx,3:zl9.AG25AD:=AD7/2 <z.~Ó AxÆD2

AOP546AO0QPYJ/0«mn<DPnP

A

ABCDT+O4.Ó <D41P5ASJ/0z2nP46<D7Qg?.UAAD8n2

ADgG<D.6A <).~Ó <O46P5AJ30mn<DPnP

A3K9¡/0tAO.j.UAAD8n2.1<w/<O.UAG0QPaJ9A

aÅ

im| ∗ | AO:,37/2nP5AOPB¡/0Ó 4j.7LÓ AsÆD468n2nAS<D0tmn0Q7lt,/417/2 FK98s0QP.Ó <?Æ-A'J9AD8<@38*mn418C8CAD8?Klt,/0QP.6A-¡/0tAO.Q.Ó <D41P5ASJ/02nP541<-7Qg?.6A

ABFAD8n2

A-gG<O.UA <).Ó <D41P5ASJ/0mn<DPnP

A

ABCDT

Ð~ÐÍ/ÌdÌ?ÍË ÐÌdÌ3 Ò ∗|)+-,/4o2 G

0Q7%lt,/417/2S8x0QP.1<J9P5,/4o2nAc¡/094lQ<D8C8CAklQ<DPW.UAO8Ylt,/417/258M(0, 8)

AG2N(3, 10)

KLJQAk:z<-734ADP5A¡30tA'.UA2nP541<-7Qg?.6ADCG8*,/4o2B8x4j250

Akmn,3:=l3.A?2nAO:AO7/2 <k.~Ó AxÆD2

AOP546AO0QPBJ30cmn<DPnP

AQT

AG25ADP5:z417QADPW.6AD8Ymn,9,3P5JQ,3797

ADAD8JQA

GK38C<-msE3<-7/2a¡/0tA).~Ó <O46P5AJ302nP541<-7Qg?.6ASAO8x2

A-gG<O.UA <).Ó <D41P5ASJ/0em5<-P5P

A9T

TI¤/,3P92xE9Aa8CAG2,/h-730Q:@9ADP8 1K10K100 FSAam5<-7w,9@/2<D417 7

J/468n25467Qm52t730Q:@9ADP8<-8)25,25<O.68S,/hW,37QAe,3Pw:=,3P5AeAO.6AD:=AD7/28',/h2xE9A8*A?2?T _ E9AD8CA=25,25<O.68<-P5A1K10K100

K1 + 10 = 11

K1 + 100 = 101

K10 + 100 = 110

K/<-7QJ1 + 10 + 100 = 111

T _ E9A ÐYÌ Î ,/ht2xE/468W8CAG2I4682xE9A8x0Q: ,/hQ2xE9AD8CAw25,25<O.68?K/417k2xE/468Bm5<-8CA3K

444T

U<?| ∗ |,-F±:z<-7>J3418x25417Qm52730Q:@9ADP8:z<G>@9A=,9@/2<D417QA-Jc<D8)<=8x0Q: ,/h,37QA,3P:=,3P5AzJ34jÔAOP5AD7/2730Q:@9ADP8YhUP5,3: 2xE9A8CAG2 1K10K100

K1000 Å¬R<D.6mn09.6<G2nA'2xE9Alt,-FSAOPxx8s0Q: hi,3PI2xE/418W8*A?2?T

i@-|I ∗| WAG25ADP5:z417QAw2xE9AlQ,-FSADPns8x0Q: ,/hQ2xE9A8CAG21K10K100

K1000

K10 000

K100 000

K1 000 000 T

ÉGÊQËÌ?Í9ÎxÏÐÍcË~Ð'Ñ4 Ò ∗ |IRL,3798s46JQAOPI2xE9A8CAG2 1

K2K3K6K12K24K48K96 Ta),-F:=<D7>zJ/41ÔADP5AO7/2I25,25<O.68<-P5Aw7Q,-F lQ,38C8x4U@/.UA41ht<w25,25<O.9418WJQAGt7QADJ<-82xE9Aw8x0Q:!,/h

1,3P:,3P5AAO.6AD:=AD7/28,/hL<8CAG2xÅ

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

` M

2<-7/2;JQ,3797 AB.Ó AD798CAD:[email protected] 1K10K100 K,37ltAG032,9@2nAO7346P 7 2n,2<D0ÆwJ/468n25467Qm528AD7<-JQJ/4j2546,37979<-7/2a0Q77Q,3:=@9P5A',/0el3.10Q8J9AkmnAG2WAD798CAD:[email protected],2<D0Æk8C,37/2

1K10K

100K1+10 = 11

K1+100 = 101

K10+100 = 110

KA?21+10+100 = 111

TQÇO<ÎGÐ dÌ GÏ6ÎsÎ Í Ì J9AwmxA?2AO798*AO:@/.UAwAO8x2;.1<w8C,3:=:=AwJ9AwmxAO8I2n,2<D0ÆtT/.j.UAYAD8n2 A-gG<O.UA<444

TU<?| ∗ | 2<-7/2JQ,3797 Aa.Ó AD798CAD:[email protected] 1

K10K100

K1000 KDmn,3:@/4UAO7)ltAG032xC,37),9@Q2nAO7346PJ9A2n,2<D0ÆJ3418x25417Qm5258;AD7w<-JQJ/4j2546,37979<-7/20Q7)7Q,3:@3P5AB,/0)l9.j0Q8JQABmnAG2AO798*AO:@/.UAIÅRL<O.Umn09.6ADPI.6<8*,3:z:ADsl309468s8C<D7QmxAJ9ASmnAG2aAO798*AO:@/.UA9T

i@-|I ∗| AG25ADP5:z417QADP.1<8*,3:z:ADsl309468s8C<D7QmxAJQAw.Ó AD798CAD:[email protected]

K1000

K10 000

K100 000

K1 000 000 T

Ð~ÐÍ/ÌdÌ?ÍË ÐÌdÌ4 Ò ∗|a+-,/4j2W.~Ó AO798*AO:@/.UA 1

K2K3K6K12K24K

48K96 TaR,3:=@346AD7%ltAG032xC,37 ,9@2nAO7346P'JQAe25,25<O0Æ«J3418x25417Qm5258AD7¬<-JQJ/4j2546,37979<-7/2S0Q77Q,3:@3P5AS,/0=l9.j0Q8J9ASmnAG2aAO798*AO:@/.UAIÅ

Va0QP'8*ADmx,37QJ%4o2nAO:2xE/418z:,37/2xEr468'2xE9A2002

b%T OT¨Q.10Q7QJ9,37v¥r<G2xE9AD:z<O2546mn8R,37/25AD8n2?T)¥>z2xE3<D7Qf8gO,,/0322n, W,374UJ9A-,/032B,/h¥%AO:,3P46<O.ªB734 3ADP8x4j2~>=hi,3Phi,3PxF<-P5J/467Qg2xE9A:z<O25ADP46<O.Q25,:=AQT

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Ä T ∗ |a¤-4o3A>3AD<DP8B<gO,s<-7QA?2IF<D8,37QA8s4 ÆD2xEz,/hE9AOPa:=,2xE9ADPCÓ 8B<-gOAQTW¦~72xE/41Px2nADAD7'>3AD<DP8B8CE9AYF)4j.1.t@QAE3<D.jhE9ADP:=,2xE9ADPCÓ 8W<gOA9T;bcE3<O2418As<-7QA?2*Ó 8WlQP5AO8*AO7/2a<-gOA-ÅTI ∗|¦ih a + b + c = 0

K3lQP5,O3Aw2xE3<G2a3 + b3 + c3 = 3abc

T¯T ∗| \ mnADP¢25<O467kP5ADm525<D7Qg?.UA)E3<D8<-P5AO< ` <-7QJkJ341<gO,379<O.Q,/hQ.UAO7Qg©2xE 2

√5T;bcE3<G2468a4o258WltAOP541:A?2nAOPxÅ

T¤417QJz<O.1.tlQ,38s4o254o3A)730Q:=@QAOP8x8x0tmsEk2xE3<G2

xx√

x = (x√

x )x TS TT<O2546,379<D.j4j£sAY2xE9AJQAO7Q,3:z4179<O25,3P3U 1

√2 +

√3 +

√6

T

` TWu3,/467/28

A<D7QJ

B<DP5A,37k2xE9A)lQ<DP<@9,/.6<

y = 2x2 + 4x − 2T _ E9AS,3P4Ug?4174682xE9AY:z46JQxlt,/417/2I,/h32xE9A.1417QAY8CA-gG:=AD7/2 ,/4673417Qg

A<-7QJ

BT¤-467QJ2xE9A.6AD7Qg©2xE',/h32xE/468.1417QA8CA-gG:=AD7/2?T

T¦ihlog125 2 = a

<-7QJlog9 25 = b

K/t7QJlog8 9

467k25ADP5:=8B,/ha<-7QJ

bT

TSu3,/467/2P.j4UAO8417=2xE9AStP8n2¡/0Q<J3P<-7/2,37=2xE9AS.j467QA

y = 2xTSu3,/467/2

Q418<lt,/417/2B,372xE9A.j467QA

y = 3x8s0tmsEk2xE3<G2

PQE3<-8W.UAO7Qg©2xE

5<D7QJz468BlQADP5ltAO7QJ346mn09.6<DP25,2xE9A).1417QA

y = 2xT¤417QJk2xE9AlQ,/467/2

PT

T¤/,3PFSE3<O2mn,37QJ34o254U,3798W,37a<-7QJ

b4182xE9AY.1417QA

x + y = a25<D7QgOAD7/22n,S2xE9Amn46P5mn.6A

x2 + y2 = bÅÄ ® T¦~7 4ABCK*FSAaE3<G3A

∠ACB = 120JQADgGP5A-AO8GK

AC = 6<-7QJ

BC = 2T_ E9A467/25ADP579<D.-@/468CA-m525,3PL,/h

∠ACB:=A-A?258L2xE9Aa8x4UJ9A

AB<O2Q2xE9AalQ,/467/2

DT WAG25ADP5:z417QA2xE9A).UAO7Qg©2xEz,/ht2xE9A).j467QA)8*ADgG:AO7/2

CDT

XYAxÆD2*K/FSASlQP5AO8*AO7/22xE9A'8C,/.1032546,3798W25,2xE9A2 x0Q734U,3PY¨t<D.6fD<-7c¥r<O2xE9AO:=<G254Um5<D.Va.o>3:=l346<-J=iM-N-N ?|;2xE3<G2<-l9ltAO<-P5ADJ467k2xE9A WADmxAO:@9ADPa )418C8x0tAS5 <U3-*|sTÄ TI #0tgO,38x.6<?341<G|auQP5,O3Aw2xE3<G2I2xE9A730Q:=@QAOP

11 . . . 111︸︷︷︸

1997

22 . . . 222︸︷︷︸

1998

5

468W<ltAOP5hiADm52a8C¡30Q<DP5AQT c Ï 3Ð 9Ë6Ï~ÐÍ T

11 . . . 111︸︷︷︸

1997

22 . . . 222︸︷︷︸

1998

5 = 11 . . . 111︸︷︷︸

1997

×101999 + 22 . . . 222︸︷︷︸

1998

×10 + 5

=101997 − 1

9· 101999 + 2 · 101998 − 1

9· 10 + 5

=1

9

(103996 − 101999 + 2 · 101999 − 20 + 45

)

=1

9

(103996 + 2 · 5 · 101998 + 25

)

=1

9

(101998 + 5

)2

= (33 . . . 33︸︷︷︸

1997

5)2T

T¬UHIP5ADA-mnA?|ÇDA?2È0Q8 mn,3798s46JQAOP < mn,373AsÆ ltAO7/25<-gO,37

ABCDEK^F)4o2xE

AB = AE = CD = 1K∠ABC = ∠DEA = 90 <-7QJ BC + DE = 1

T¤417QJk2xE9A<DP5AD<S,/ht2xE9AlQAD7/2<gO,37T

`] c Ï 3Ð 9Ë6Ï~ÐÍ TaP<GF 2xE9A^J341<gO,379<O.68

AC<-7QJ

ADT +O467QmnA

∠B = ∠E = 90 <D7QJAB = AE

KFSAmn<D7 mxP5AO<O25A<c2nP541<-7Qg?.6AchUP5,3:2xE9Ad2nP46<D7Qg?.UAO8ABC

<-7QJAEDE3<G/467Qg'<O.j254o250tJQA

AB = AE = 1<D7QJc@3<-8CA

BC + DE = 1T _ E9Ak<-P5AO<z,/h2xE/4687QA?F 2nP541<-7Qg?.6A418 1

2

T _ E/4187QA©F).o>=mxP5AD<G2nADJ2nP541<-7Qg?.6A418W2xE9AO7mx,37QgGP0tAD7/225, 4ACD E9AD7QmnA3K2xE9Aw2n,2<D.t<DP5AD<S,/ht2xE9A)ltAO7/25<-gO,37z4181T

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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....................................................................................

¯T \ .6@9<D7346<?|w¤417QJp<D.j.IlQ<O46P8',/hlQ,38s4o254o3Az417/2nADgOADP8

(x, y)2xE3<G2w8C<G25468xhj>c2xE9Ahi,/.1.6,-F)467QgwA-¡/0Q<O2546,37 U

xy = yx−y T c Ï 3Ð 9Ë6Ï~ÐÍ T¦ih

(x, y)468)<8*,/.j03254U,37p,/h2xE9Ag?4o3AO7«A-¡/0Q<O2546,37p<D7QJ

x 6= 1K2xE9AO7

x > yTV2xE9ADPF)468CA3KQFSAFS,/09.UJdE3<G3A

yx−y ≤ 1KtFSE/4j.UA

xy > 1T'VB@D/4U,/0Q8x.o>K

(1, 1)468Y<8*,/.j03254U,37=,/ht2xE9Ag?4o3AO7AD¡30Q<G254U,37LK3<-7QJk2xE/418W4182xE9AS,373.o>8*,/.j03254U,37SF)4j2xE

y = 1TBÇDA?2

x > y ≥ 2T _ E9AD7'FSA,9@/2<D417k2xE9ASA-¡/0Q<O2546,37

(x

y

)y

= yx−2y T+O467QmnA x

y> 1

KtFSAzgOAG2x − 2y > 0

<D7QJ xy

> 2T \ .68C,K x

y∈

TVa0QPYA-¡/0Q<O2546,37<@9,D3ASm5<-7=@9AYFP4j225AD7z<D8x

y= y

x

y−2 T

+O467QmnAy

x

y−2 ≥ 2

x

y−2 K-FSA,9@/2<D417 x

y≤ 4

T;bcAmx,37Qmn.j0tJQAw2xE3<G22 < x

y≤ 4

T•

¦ih xy

= 3K2xE9AO7

y = 3Kx = 9

T•

¦ih xy

= 4K2xE9AO7

y = 2Kx = 8

T¤4179<D.j.o>K2xE9A)8*A?2a,/hL8C,/.1032546,3798B,/ht2xE9Ag?4o3AO7AD¡30Q<G254U,37418HU

(1, 1)K(8, 2)

K(9, 3) T

`G[ T¨Q09.6gG<-P46<?|;ªB8x467QgY,373. >2xE3P5A-AwJ346g?4j28amn<D7,37QABFP54o2nA

162xE3P5ADA-sJ346g?4j2I730Q:=@QAOP8GK8x0tmsE)2xE3<O27Q,)2¢FS,S,/h32xE9AO:<DP5Awg?4 3417QgW2xE9AY8s<-:=AP5AD:z<D417QJQAOPIJ34 346JQADJk@D>

16Å

c Ï 3Ð 9Ë6Ï~ÐÍ TÇDAG2Y0Q8S8x0QlQlQ,38*Az2xE3<G2FSAmn<D7cFP4j25AeJQ,-F716

8x0tmsEp730Q:=@QAOP8DTr¦U2Y468Smn.UAO<-P2xE3<O28,/ht2xE9AD8CA730Q:@9ADP8:0Q8x2a@QASA*3AO7=<-7QJk2xE9ASP5AO8x2:z0Q8n2W@9AS,9JQJT _ E9ADP5AGhi,3P5A3K2xE9AzJ/4Ug?4o258Ymn<D797Q,2@QAk<O.1.A*3AD7c,3PB<O.1.,9JQJTÇDAG2W0Q8wAsÆD<D:z417QAS2xE9Amn<D8*AFSE9AOP5A'2xE9Ag?4o3AO7«J346g?4j28<DP5A2¢FS,A*3AD7<-7QJp,37QA,9J9JTe _ E9Aem5<-8CAkF)4o2xEd2¢FS,,9J9JpJ346g?4j28<D7QJ,37QAA*3AD7dJ346g?4j2W418Y8x46:41.1<-P©T |cÇDAG2

e1Ke2K<D7QJ

oJQAO7Q,2nAS2xE9A2¢FS,eA*3AD7dJ346g?4j28Y<D7QJ2xE9Ak,9JQJJ346g?4j2*KQP5AD8sltADm5254o3AG.o>TWbcAkm5<-7FP4j25AkAxÆD<m525. >

9,9JQJ2xE3P5A-ADCJ/4Ug?4o2W730Q:@9ADP8F)4j2xE'2xE9Ag?4o3AO7J/4Ug?4o258U

e1e1oKe1e2o

Ke1oo

Ke2e1o

Ke2e2o

Ke2oo

Koe1o

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Kooo

T¦ihFSAeP5A©FP54o2nAz2xE9AD8CAe<-8

a1oK

a2oK

. . .Ka9o

KFSE9ADP5Aa1K

a2K

. . .K

a9<DP5A=2¢FS,9J346g?4j2W730Q:@9ADP8hi,3Pn:=A-Je@D>z2xE9AStP8n2a2¢FS,J/4Ug?4o258Y,/hL2xE9A'730Q:[email protected]*K2xE9AD7

aio − ajo418J34 [email protected]@D>

164jh<-7QJ=,373.o>41h

ai − aj418J34 [email protected]@D>

8T¨Q0322xE9ADP5A<-P5AS,373.o>k2xE3P5ADA)2¢FS,9sJ346g?4j2a,9J9J=730Q:=@QAOP8<-:=,37Qg

a1Ka2K. . .

Ka9KFSE9ADP5AO<-8hi,/0QP<-P5AYP5AD¡30941P5A-J2n,',9@25<O467Shi,/0QPIJ/41ÔADP5AO7/2,9J9J'P5AD:z<D417QJQAOP8IFSE9AO7J/4o/4UJ9A-Jk@D>

8T_ E9AOP5AOhi,3P5A/KFSAmn<D797Q,2FP54o2nAJQ,-F7

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¤4179<D.j.o>K-FSAl9P5AD8CAD7/22xE9A<D798nFSAOP8a25,'2xE9A2002

¨tP4j25418*ER,/.j0Q:@/46<'RL,/.1.6A-gOAO8x0Q734U,3PBw46gOE=+-msE9,9,/.;¥r<O2xE9AO:=<G254Um58RL,37/2nAO8x2*KLuQP5AO.j46:4679<DP~> L,/0Q7QJ2xE3<G2a<-l9ltAO<-P5ADJ467k2xE9A WADmxAO:@9ADP

2002468s8s0tA5 <U3 ] - C|CT TI@ Á TI@ Ã TI@ WTIm T< ¿ TIA TI@TIm

TIA ;Â TIJ TIJ IÁ TIA IÃ T< WTIm

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< 1(M 63%H" 2 %$3R=9 24. $0j R /. . ' . ?>A@ P)B)8 T _ E9AAO.6A-m52nP5,37346ma<[email protected]_ E9A¥r<?>QE9AO: 9J/4j25,3Pd418r+-E3<?F7 HI,9J3417ÈiR<D41P5417QA be41.18*,37 +-ADmx,37QJ3<-P~>È+-msE9,9,/.o|sT_ E9A \ 8s8s418x2<-7/2¥r<G>9E9AD: 3J34o2n,3P418 C,9E37eHP<-7/2¥%m?ÇD,/0tgOE/.1417eiªB734 3ADP8x4j2~>,/hIXYA©F¨tP50Q7985F)4UmsfG|CT _ E9A,2xE9ADPB8x2<DÔ«:=AD:=@QAOP8w<DP5Adu/<O09.V22<GF<?>d a<O.UE9,/0Q8x4UAªB734 3ADPn8s4o2>-|<D7QJcÇO<-P5P~> 4UmnASUªB734o3AOP8s4o2>=,/htbd<O25ADP.U,9,-|sT

C DFEHGJI/K LNMPORQTS(I/KVU

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[email protected]<-8CAd467Qmn.j0tJQAc417¬<D.j.Bmx,3P5P5AD8slt,37QJ9AD7QmnAd>3,/0QPS79<-:=A3KW8CmsE9,9,/.KWgGP<J9A3KBmn4o2>KlQP5,O3417QmxA',3P8x2<O25A3K9<D7QJmx,/0Q7/2nP~>TbcAS<DP5A'AO8ClQA-mn41<D.j.o>.6,9,9f-417Qgwhi,3P8*,/.j03254U,3798WhUP5,3:E/4UgOE%8*msE9,9,/.W8n250tJQAO7/258OT u9.6AD<D8*Ad8*AO7QJ«>3,/0QP'8C,/.1032546,3798'25,«2xE9Al9P5,[email protected]:z8417r2xE/468A-J/4j2546,37)@D>kB¬Ds53©T;+-,/.1032546,3798;P5ADmxAG4o3ADJw<Dh125ADPt2xE/4182541:AIF)41.j.-@QAWmx,3798x4UJ9ADP5ADJ,373.o>k41ht2xE9AOP5A)4182546:=AS@9AOhi,3P5A)l90t@/.146mn<G254U,37=,/ht2xE9A)8*,/.j03254U,3798OT3<-msEcl9P5,[email protected]: 418g?4 3AD7d467 37Qg?.j468CEd<-7QJ«¤/P5AO7QmsEKL2xE9A=,/yemn41<D..6<D7Qg?0Q<gOAO8,/hRL<D79<J3<tT¦~7'468s8s0tAO8WM/K ] K9/K/<D7QJ=/K/7Qg?.1418*E)F)4j.1.9l9P5A-mnA-J9AS¤/P5AO7QmsEK/<-7QJS467'418C8x0tAD8W/K[ K ` K3<D7QJ 3Kt¤/P5AO7QmsE'F)4j.1.tl9P5A-mnA-J9A 37Qg?.j468CET_ E9AA-J/4j25,3Pa2xE3<-7Qf-8 CAD<D7Q?¥r<DP5m _ AOPnP4UAOP<D7QJpY4UJ9AD:4j28s0c+D<ADf-4;,/h2xE9AªB734U3ADP8x4j2~>=,/h¥%,37/2nP5AO<D.Qhi,3PI2nP<D798s.1<O2546,3798B,/ht2xE9A)lQP5,9@/.UAO:=8OT

Ä ® Ä T Ð Ð-ÎGÌ [zË Ì % [ Ì 3Ë )\T¤417QJ=2xE9Ak8C:z<D.j.UAO8x2/<O.10tA,/hk8x0tmsE=2xE3<G2

k!AD7QJ38BF)4j2xE

100£sADP5,38OT ^] ÐQËÌ Ò

k! = k(k − 1)(k − 2) · · · (3)(2)(1) T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T_ P5,/0/3ADP.6<=l9.j0Q8ltA?254j25Ak/<O.UAG0QPwJQAk25AO.j.UA=¡30tA

k!Q73468s8*Az<G3ADm

100£AOP5,38DT

_] ÐQË~Ì Ò k! = k(k − 1)(k − 2) · · · (3)(2)(1) T

`-` Ä ®) T Ð Ð-Î?Ì [ Ì Q [Ì W ÌdÐ Ï Í/ÏDÌ ÎxÏ¢Ë-[eÐ ] Ì

¢ÐDÍ Í 9ËGÐ Í Î ]9T+O0QlQlQ,38*Aa2xE3<G2ABCD

468;<BlQ<DP<D.j.UAG.U,9gGP<D:È<D7QJY2xE3<O2GA

KGB

KGC

KO<-7QJGD<-P5AY2xE9ASmxAO7/2nP5,/4UJ38,/h 4BCD

K 4ACDK 4ABD

K3<D7QJ 4ABCK/P5AD8sltADm5254o3AG.o>T

uQP5,D3AY2xE3<O23UM9T

GAGBGCGD418W<Sl9<-P<O.1.6AO.6,9gGP<-:

9T[GAGBGCGD] =

1

9[ABCD]

KDFSE9ADP5A[ABCD]

4182xE9Aw<-P5AO<S,/hABCD

TT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

a<-7980Q7kl9<-P<O.1.AO.6,9gGP<-:z:A

ABCD,37'8s0Ql9lt,38CAw¡30tA

GAKGB

KGC

A?2GD8*,37/2.6AD8kmxAO7/2nP5AD8zJ9AgGP<?34o2

AcP5AO8ClQA-m5254jhU8JQAO8'2nP46<D7Qg?.UAO8

BCDKACD

KABD

A?2ABC

T¥%,37/2nP5ADPa¡/0tA<UM9TGAGBGCGD

AO8x2I0Q7=l9<-P<O.1.AO.6,9gGP<-:z:A

9T[GAGBGCGD] =

1

9[ABCD]

K9, 0[ABCD]

JAO8s46gG7QA).~Ó <O46P5AJQA

ABCDT

Ä ®)¯ T Ð Ð-ÎGÌ [zË Ì % [ Ì 3Ë )\T+-,/.o3Awhi,3PnU

1001/n × 1002/n × 1003/n × · · · × 1002003/n = 1000T

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T TAO8*,/0tJ3P5AlQ<DPP<DlQlQ,3P2 <

nU

1001/n × 1002/n × 1003/n × · · · × 1002003/n = 1000T

Ä ® T Ð Ð-Î?Ì [ Ì Q [Ì W ÌdÐ Ï Í/ÏDÌ ÎxÏ¢Ë-[eÐ ] Ì ¢ÐDÍ Í 9ËGÐ Í Î ]9T+O0QlQlQ,38*A2xE3<O2

ABCD418k<cl9<-P<O.1.6AO.6,9gGP<-:<D7QJp2xE3<O2

OAK

OBK

OCK<D7QJ

OD<-P5A=2xE9Acmn41P5mn0Q:mnAD7/2nP5AO8,/h 4BCD

K 4ACDK 4ABD

K<D7QJ 4ABCKP5AD8ClQA-m5254 3AO. >T

uQP5,D3AY2xE3<O23UM9T

OAOBOCOD468W<l9<-P<O.1.6AO.6,9gGP<-:

9TYlQ<DP<D.j.UAG.U,9gGP<D:=8ABCD

<D7QJOAOBOCOD

<-P5A)8s41:z4j.6<DP] T

AOBCOD418W<lQ<DP<D.j.UAG.U,9gGP<D:

[ TOABOCD

418W<Sl9<-P<O.1.6AO.6,9gGP<-:9TYlQ<DP<D.j.UAG.U,9gGP<D:=8

AOBCOD<-7QJ

OABOCD<-P5A)8s41:z4j.6<DP?T

` a<-798W0Q7=lQ<DP<D.j.

AG.U,9gGP<D:=:=A

ABCD,37=8x0QlQlQ,38*AP5AO8ClQA-m5254 3AD:=AD7/2W¡/0tA

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KOC

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8C,37/2=.6AD8dmnAD7/2nP5AO8cJQAO8cmxAOP5mn.UAO8cmn41P5mx,3798CmxP54o258dJQAO82nP541<-7Qg?.6AD8BCD

KACD

KABD

<-7QJABC

T¥%,37/2nP5ADPa¡/0tA<UM9TOAOBOCOD

AD8n20Q7zlQ<DP<D.j.AG.U,9gGP<D:=:=A

9T.UAO8WlQ<DP<D.j.AG.U,9gGP<D:=:=AD8

ABCDA?2

OAOBOCOD8*,37/28*AO:@/.6<[email protected]

] TAOBCOD


[ TOABOCD


9T.UAO8WlQ<DP<D.j.AG.U,9gGP<D:=:=AD8

AOBCODAG2

OABOCD8*,37/28*AO:@/.6<[email protected]

Ä ® S T Ð Ð-Î?Ì [aÍ Ì ÏË 5 D Ì?Í ©Ï ÌÏ 5 Ð9Ð D Ì?Í ?Ï ~Ì ]9T+O0QlQlQ,38*Aa2xE3<O22xE9ABP5,9,258,/h

P (x) = x3−2kx2−3x2+hx−4<DP5AJ3418x25417Qm52*K<-7QJk2xE3<G2

P (k) = P (k + 1) = 0T$WA?2nAOPn:467QAY2xE9AY/<O.10tA,/h

hTT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T TVW78x0QlQlQ,38*A)¡/0tAw.6AD8P<mn417QAD8WJ9A

P (x) = x3 − 2kx2 − 3x2 + hx − 48C,37/2J3418x25417Qm52nAO8GK9A?2a¡/0tA

P (k) = P (k + 1) = 0T _ P5,/0/3ADP.1<Y/<O.UAG0QPaJQA

hT

Ä ® T Ð Ð-ÎGÌ [zË Ì % [ Ì 3Ë )\T\4@D>

48*¡/0Q<-P5AzE3<-8w<-7c<DP5AD<=,/h

168C¡30Q<DP5A0Q734o258w<-7QJd<=lQADP46:=AG25ADP,/h

160Q734j28DT _ E3<G2418GKQ2xE9A<-P5AO<z<-7QJdlQADP46:=AG25ADP<DP5A730Q:AOP546mn<O.1. >cA-¡/094o/<O.UAO7/2Y14UgG7Q,3P467Qg0Q734j28,/hL:AO<-8x0QP5AD:=AD7/2¢|sT \ P5Aw2xE9ADP5A<-7>=,2xE9ADPP5ADm525<D7Qg?.UAO8aF)4o2xE467/25A-gGP<O.J/46:=AD7Q8s46,3798;2xE3<O2;8CE3<-P5AW2xE/468Il9P5,3ltAOP2~>QÅ¦ih9lt,[email protected],-F^2xE3<G2L>3,/0kE3<G3Ahi,/0Q7QJS<O.1.8s0tmsEAsÆD<D:=l3.UAO8DT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T TªB7Sm5<-P5PABJQA

4lQ<DP

4<W0Q7QAB<D41P5AJ9A

160Q734o2

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AO8n|sT #<-n2xn41.JLÓ <D032nP5AO8'P5A-m52<-7Qg?.6AD8'JQAeJ/46:=AD798x4U,3798AD7/254ADP5AO8lt,38s8

ADJ9<D7/2wmxA?22nA=l9P5,3lQP4

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>3,/0Q41lk2xE9A7QA?F25,3lm5<-P5JkhU<mnA)0Ql,37k2xE9AY25<[email protected]#;,/0pmn,37/2546730tAk2xE/418wlQP5,9mnAD8s8w0Q7/254j.<O.1.m5<-P5J38)<DP5AhU<-mxAk0Qlp,37e2xE9A'25<[email protected]=¤-467QJe2xE9A,3P5JQAOPw,/h2xE9A=mn<DP5J98w467d2xE9A=JQADmsf4jh¢KFSE9AD7d2xE9A=l9P5,9mxAO8C8w418)ltAOP5hi,3P5:ADJLKL2xE9Am5<-P5J38gOAG2.1<D46J=,/032a,37k2xE9AY25<[email protected])417k2xE9AS,3P5J9ADP

1K2K3K. . .K25T

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+G25ADl ªB7QJ3418x2nP4U@/032nADJ=mn<DP5J98 ¤/,/0Q7QJz8*AD¡30tAO7QmxA

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11K4K16K25K5K12K23K6K17K13K7K20K22K8K14T

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T Ð Ð-Î BÌ - C ¢Ë~Ì [-Í 5 Ë Ì?Í9Î ED TVW7w<WJQAG0ÆBl94A-mnAD8JQAa:=,37979<D46AQTÇ/Ó 0Q7QABAO8x2t0Q7QAal34ADmxAWJÓ 0Q7)J9,/.1.1<-Pt7Q,3Pn:z<D.6AWA?2.Ó <D032nP5AY0Q7QAYhU<D0Q8s8*AYl34ADmxA)JÓ 0Q7zJ9,/.1.1<-PCK<G3ADmJQAG0Æ)hU<mnAD8OT3VW7 AG225Aw<O0=E3<D8C<DP5JmsE3<mn0Q7QAJ9AD8l34ADmxAO8IJ3<-798IJ9AO0ÆB2541P5,/46P8J34jÔ

AOP5AD7/28DTa0tAO.6¡30Ó 0Q7'AO7/2nP5AJ9<D798;.6<msE3<D:@3P5AAG2B,/0/9P5A0Q7dJ9AD8B2541P5,/46P8A?2W<-lQADP ms, Fo2W0Q7QA'l34ADmxA/Km,2

AhU<-mxA9Ta0tAO.j.UA'AD8n2a.6<'l9P5,9@9<[email protected]

A¡30tASmnAG225Al34ADmxA)8*,/4o2amnAO.j.UA <SJQAG0ÆShU<-mxAO8Å3Ð 9Ë6Ï~ÐÍQÌGÐ9Ì Ë QÏ jÏUÍ3Î-Ï 9Ë ÌdÐÍLË T+O4,37P5A-gG<DP5JQAz.6AD8

2l34ADmxAO8SJ9A=:,37979<O4UA/KL41.><=<D0c2n,2<D.

3hU<-mxAO8DT;+O0QPwmnAD82nP5,/468WhU<mnAD8?K94j.Q>AO7<kJQAG0Æ¡3094L<-l9lQ<DP2546AD797QAO7/2<.6<'l34ADmxA)2nP0t¡30

ADAQT \ 41798s4~K9lQ<DP.6<J

AGt734o254U,37=J9Al9P5,9@9<[email protected]

A3K9,37z<<U

P (l34ADmxAY2nP50t¡/0

A-A

) =

H m5<-8ahU<G3,3P<[email protected] 25,25<O. =

2

3

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<=" ) $>*?5A

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¯T Ð Ð-Î?Ì [zË Ì ¬ [ Ì 9Ë (\IT\ mn41P5mn09.6<DPWl9<O2xEz4188s0QP5P5,/0Q7QJQADJd@D>

178x25ADl9l9417QgS8n2n,37QAO8730Q:@9ADP5ADJ

0K1K2K

. . .K16Tc+D<D.j.o>8n25<DP28',37p8n2n,37QA

0<D7QJp:=,D3AO8

18x25ADlc2n,8n2n,37QA

1K2xE9AD7

48n2nAOlQ82n,p8n2n,37QA

5K;2xE9AD7

98x25ADl98)2n,p8n2nAOl

14<D7QJrmn,37/2546730tAO8S4172xE9Ahi,/.j.U,-F)417QgzlQ<G22nAOPn70Q7/2541.<O2W.1<-8n2B8*E9Ak:=,D3AO8

20022 8n2nAOlQ8<-7QJe8x25,3lQ8Y125,eP5AO8x2¢|sTwbcE3<O2B8x25,37QA'468Y+D<D.j.o>8x2<-7QJ/467Qg),37'FSE/41.6A)8*E9A)P5AD8n258CÅ3Ð 9Ë6Ï~ÐÍ [ ÌÊ Í Ì I Ë Í ÎGË QÌ?ÍË BÉ Ð~Ì GÐ-ÎGÌ Í ÐDÏ6Î tÌ n5 Ë ÐÍLË BÌ TÇDAG20Q8atP8n2Q7QJ=E9,-FÈ:z<-7>z8n2nAOlQ8B+D<O.1. >=E3<D8W:=<-JQA#U

12 + 22 + 32 + · · · + 20022 =2002 × 2003 × 4005

6= 2 676 679 005

8x25ADl98DT+O467QmnAWE9ADP9F<O.Ufa418;m>3mn.146mn<O.K*2xE9AaP5AO8s46J30tAW,/hG2xE9Aa730Q:@9ADPL,/h-8x25ADl98;,37wJ/4o/468x4U,37@D>17F)4j.1.tg?4o3Aw0Q8a2xE9A)lt,38x4j2546,37'FSE9AOP5AS+D<O.1. >=AO7QJ98BE9ADPF<O.UftT+O467QmnA

2 676 679 005 = 17 × 157 451 706 + 3KL+-,3ltE/46AAD7QJ9A-J0Qlc,372xE9A2xE/46P5Jz8n2nAOl;T

/&#" : : &: $/!@'" " $>@?5A S T Ð Ð-Î?Ì [ D- [ Ì tÌ 3Ð3Ì [ /ÐË D Ì~Í

DÌ n5/ Ì T\ 7zAG.1.j46l98*ABF)4j2xE':z< ,3PI<©Æ-418AB

<-7QJ'hi,9mn4F<D7QJ

F ′ 41841798*mxP4U@9A-J'467k<mn41P5mn.UAF)4j2xEkJ/46<D:A?2nAOPAB

<-7QJkmxAO7/2nP5ACTP418<wlt,/417/2,37S2xE9AwAG.1.j46l98*AY<D7QJ

D418<wlt,/417/2,37S2xE9Awmn41P5mn.UAY8C,)2xE3<O2P<-J34j0Q8

CD@3418*ADm5258

FPT+-E9,-Fv2xE3<G2.j467QA

DP46825<D7QgOAD7/2;25,2xE9ASAO.j.141lQ8CAQT

3Ð 9Ë6Ï~ÐÍ [ aÍ Ì©ÏiÎ Ï ÎGË QÌ?ÍË 11 D QÌ 3 ÎsÏ Ì Ì Î Í Ï

]Y Ë6Ï~ÐÍ Ð ~Ì/Ì D Ë6Ï ;Ð Í/Ï 9TÇDAG2I2xE9AAO.j.141lQ8CAE3<G3AA-¡/0Q<O2546,37 x2

a2+

y2

b2= 1

T _ E9AO7z4o258ahi,9mn4t<-P5AF (√

a2 − b2, 0)<D7QJ

F ′(−√

a2 − b2, 0)T

¨A-m5<D0Q8CAYlt,/417/2P418.6,9mn<G2nADJk,372xE9AwAG.1.j46l98*A/KD4j28amn,9,3P5J34179<O25AD8mn<D7@9AwAsÆ-lQP5AO8C8CA-J'<D8

P (a cos t, b sin t)T \ .18*,K2xE9Alt,/417/258

A<D7QJ

BE3<G3ASmx,9,3P5J/4679<G2nAO8

A(−a, 0)<-7QJ

B(a, 0)T _ E9Amn41P5mn.UA)F)4j2xEeJ/46<D:A?2nAOP

ABE3<D82xE9AkA-¡/0Q<O2546,37

x2 + y2 = a2T'ÇDA?20Q8mn,3798s46JQAOP

MK2xE9A:4UJ9slQ,/467/2a,/h

FPT¦U2aE3<-8Bmx,9,3P5J/4679<G2nAO8

M

(

a cos t +√

a2 − b2

2,

b sin t

2

) T_ E9ASmn,37QJ34o254U,37k2xE3<O2

CD@/468CA-m528

FP418BA-¡/094o/<O.UAO7/22n,'2xE9A8n25<G2nAO:AO7/22xE3<G2

D418W2xE9A467/25ADP8CA-m52546,37e,/h2xE9AS.j467QA

CMF)4j2xE2xE9A'mn41P5mn.UA

x2 + y2 = a2 TYXY,-FK94o2468BAO<-8>'2n,k8*ADAw2xE3<G2DE3<D8Bmx,9,3P5J/4679<G2nAO8

D

(

a · a cos t+√

a2−b2

√

(a cos t+√

a2−b2)2+(b sin t)2, a · b sin t

√

(a cos t+√

a2−b2)2+(b sin t)2

) T

/ be4j2xE9,/032I.6,38C8a,/hLgOAO7QADP<O.14o2>KDFSAw<-8s8s0Q:=A2xE3<O2

P418467S2xE9AE3<O.1hixl9.1<-7QA

y > 0U<D7QJ2xE9ADP5AGhi,3P5A3K38*,'468

D|CTI+O46:zl9.j41hj>/467Qg2xE9AJQAO7Q,3:z4179<O25,3P8>346AO.6J98

D

(

a · a cos t +√

a2 − b2

A, a · b sin t

A

) KFSE9ADP5A

A = a + cos t√

a2 − b2T

\ .1.tFSAE3<G3AS25,e8CE9,-F7Q,-F4182xE3<G2W2xE9A'.1417QADP

4182<-7QgOAO7/2W2n,z2xE9AAO.j.141lQ8CAF)4j2xEAD¡30Q<G254U,37 x2

a2+

y2

b2= 1

TY¦U2a468aFSAG.1.6Cf-7Q,-F7z2xE3<G22xE9A'8s.6,3ltA',/h2xE9A25<D7QgOAD7/22n,z2xE9AzAG.1.j46l98*A'<O2a2xE9Aklt,/417/2

P (a cos t, b sin t)468 −b cos t

a sin t

T _ E/0Q8GK4o2F)41.j.8x09yemnA2n,8CE9,-F2xE3<G23Ua · b sin t

A− b sin t

a · a cos t +√

a2 − b2

A− a cos t

= −b cos t

a sin t

K

,3P*K9AD¡3094 /<D.6AD7/25. >LKa cos t +

√a2 − b2

A− cos t

a sin t

A− sin t

= − sin t

cos t

T ~M©|

XY,-FK2xE9Aw.UAGh12xCE3<D7QJz8x4UJ9AS,/hL~M©|Imn<D7@9A8x46:zl9.j41tA-J<D8a cos t +

√a2 − b2

A− cos t

a sin t

A− sin t

=a cos t +

√a2 − b2 − A cos t

a sin t − A sin t

=a cos t +

√a2 − b2 − a cos t − cos2 t ·

√a2 − b2

a sin t − a sin t − cos t sin t ·√

a2 − b2

=1 − cos2 t

− cos t sin t

= − sin t

cos t

KFSE/4UmsE=418B2xE9A'P4UgOE2xsE3<-7QJe8x4UJ9Ak,/h;~M©|CT _ E9AOP5AOhi,3P5A/K

DP468417QJQADA-Jz25<D7QgOAD7/2a25,z2xE9AAO.j.141lQ8CAQT

A .A&I $0-"& $/" 2 " 2 3A

/DM SLS T Ð Ð-Î BÌ D BÌOÏ Ì QÌ % [ Ì T _ P5,/0/3AOP.6<w8*,3:z:AwJ9AD8

2002lQP5AO:z46ADP8a25ADP5:AO8J9A).1<8s094o2nA)8s094 /<-7/25A1K2K2K3K3K3K4K4K4K4K5K5K5K5K5K. . .T

3Ð 9Ë6Ï~ÐÍ [;Ð9Ì Ë QÏ jÏUÍ3Î-Ï 3Ë ÌdÐÍË T¦i.DhU<D032AO7)l9P5AD:4UAOPJAG25ADP5:z417QADPL<G3ADm.1<BP

AOlA?254j2546,37J9AB¡30tAG.-7Q,3:@3P5AB8GÓ <-P5P A?2nA.6<'8x094j25A.6,3P8*¡/0Ó AG.1.6A'AD8n2 <S.1<'lQ,38s4o254U,37

2002TVW7=P5AD:z<-P5¡/0tAk¡/0tAkmsE3<-¡30tAS7Q,3:@3P5AAD8n2IP

ADlAG2AY8C<B/<D.6AO0QPIJQAhi,/468OT/VW7msE9AOP5msE9AY.UAYl3.10Q8agGP<D7QJ

n2nAG.Q¡30tA n∑

k=1

k ≤ 2002,/0@346AD7w¡30tA n(n+1)2

≤ 2002¡3094

A-¡/094o/<O032<

n2+n−4004 ≤ 0TGVW7),9@/2546AD7/2

n =

62¡/094J9,3797QA n(n+1)

2= 1953

TRA¡30943AO032J/46P5A¡30tAzJ3<-798.UAO82002

l9P5AD:4UAOP87Q,3:@3P5AD8?K.UAO8)7Q,3:@3P5AD8SJQA1<

62<-l9lQ<DP<D418C8CAD7/2*<O0rmn,3:=l3.UA?2A?2w¡30Ó 41.><O0QP<AD798x094j25A

49hi,/468.6</<D.6AO0QP

63TÇO<)8*,3:z:AwJQAO8

2002lQP5AO:z46ADP8a7Q,3:=@9P5AO8W<)JQ,37QmI.6</<D.6AO0QP

62∑

k=1

k2 + 49 · 63 =62 · 63 · 125

6+ 49 · 63 = 84 462

TI'" : )J' 7:K13L M!/#" $ ) , :-$ )+&, -N.0#132547698, : 047;3(&'0" $

<=" ) $>*?5A S T Ð Ð-Î?Ì [ Ì ] Ë#[ Ð DÌ TuQP5,D3AY2xE9A)46JQAO7/254j2~>

(∑

sin A)2

−(

1 +∑

cos A)2

= 4 cos A cos B cos CK

FSE9ADP5Aw2xE9A)8s0Q:z8B<DP5ASm>3mn.146ma<D7QJA + B + C = π

T3Ð 9Ë6Ï~ÐÍ [ GÐ-Î BÌ OÏ1Î B Dn Ì Ð Í Í GÐ-Î BÌkÉ3Ð © /Ì Í/ÏDÌ ÎxÏ¢Ë Ë

tÐ1ÏË Ì Í/Ï ¢ QÌ Ë -Í [ Ds Ì ~ÐÍ ÏUÍ TbcASmn.1<D41:2xE3<O2*K/4jhA + B + C = π

K2xE9AO7cos2 A = 1 − cos2 B − cos2 C − 2 cos A cos B cos C

T iM?|¦~7zhU<-m52*K38s417QmxA

B + C = π − AK-FSAE3<G3A

cos2 A = cos2(B + C) = (cos B cos C − sin B sin C)2

= cos2 B cos2 C + (1 − cos2 B)(1 − cos2 C)

−2 sin B sin C cos B cos C

= 1 − cos2 B − cos2 C

+2 cos B cos C (cos B cos C − sin B sin C)

= 1 − cos2 B − cos2 C + 2 cos B cos C cos(B + C)

= 1 − cos2 B − cos2 C − 2 cos A cos B cos CK

<-8Bmn.6<O46:=A-JT

/DXY,-FÈFSA'l9P5,D3AS2xE9AS4UJ9AD7/254o2>lQP5,3lQ,38*ADJT _ <-f-417QgS417/2n,=<mnmx,/0Q7/2~M©|<-7QJ=2xE9AhU<m52I2xE3<G2

cos A = sin B sin C − cos B cos Cim>3mn.146m|xK-FSAE3<G3A

(∑

sin A)2

−(

1 +∑

cos A)2

=∑

sin2 A −(

1 +∑

cos2 A)

=∑

(sin2 A − 1) + 2 −∑

cos2 A

= 2(

1 −∑

cos2 A)

= 4 cos A cos B cos CK

FSE9ADP5A<-gG<D417z<D.j.92xE9A<-@Q,O3A8x0Q:=8W<-P5Am¢>3mn.j4UmGT /&#" : : &: $/!@'" " $>@?5A

_ E/418=:=,37/2xEK \ 7QJ9P5AG4w¦~8C:z<D4j.F)46798z<«mx,3l-> ,/hYHIP<Oh 9¡%hUP5,3: u3ADJ9<-gO,9g?0tADP~>+-,/h12¢F<-P5A9TcRL,37QgGP<O2509.1<O2546,3798 \ 7QJ9P5AG4 Z ADADlp8*AO7QJ3417Qg'>3,/0QPwl9P5,[email protected]:z8<-7QJ8*,/.j0t254U,3798OT

LOTS'EJD_U LJD M?D O

3±ÇO<-8n2+-AOl325AD:=@QAOP*K3¦@QADgG<-7)2xE/468mn,/.10Q:z7)F)4o2xE<8*E9,3P¢2J/468Cmn0Q8C8x4U,37<@9,/032E9,-F:=<G2xE9AD:z<O2546mn8Y468)@9AD<O032541h609.;<-7QJcE9,-F2xE/468)@9AD<O032>cmn<D7<DP5418*A'hUP5,3: 2xE9AS/<-P4UA?2>c,/hF<G>38w,/hl9P5,D/467Qg':=<G2xE9AD:z<O2546mn<O.8x2<O25AD:=AD7/28DT'¦;F)4j.1.8x2<-P¢2W2xE/418Y7QA?F8*msE9,9,/.>3AO<-PF)4j2xE=<kmn032nA'l9P5,9,/hL2xE3<O2W7Q,2B,373. ><DP5A'730Q:=@QAOP8Y@QAO<D03254jh609.K@/032a2xE9A*>=<-P5AS<D.18*,467Q2nAOP5AD8n25467Qg

,- %D,$&%5@ T \ .1.tlQ,38s4o254o3Aw417/2nADgOADP8W<-P5Aw467/25ADP5AO8x25417Qg3T#$s3HC T ¤41P8x2*KFSAe8x0QlQlQ,38*Ahi,3P)<cmn,37/2nP<J/4Um52546,37p2xE3<G2w2xE9ADP5AdAxÆ-418x28'<G2).6AD<D8x2,37QA0Q73467/25ADP5AO8x25417Qglt,38x4j254 3A417/2nADgOADP©T _ E9AO7LK<-:=,37Qg<O.1.;8s0tmsEe417/2nADgOADP8?K;,37QA=,/h;2xE9AO::z0Q8n2W@9Aw2xE9A8s:=<O.1.6AD8n2?TWX,-F2xE/468a467/25A-gOAOP4683AOP~>467/25ADP5AO8x25417Qgw417QJQADA-JK98s417QmxAw4j24182xE9A8C:z<D.j.UAO8x2,37QASF)4j2xEe7Q,d,2xE9ADPB417/2nAOP5AD8n25467Qg'lQP5,3lQADP¢254UAO8 _ E/418)mx,37/2nP<-J346m5258),/0QP<-8s8s0Q:zl32546,37T;bcA:0Q8x2mx,37Qmn.j0tJQAw2xE3<G2<D.j.tlt,38x4j254 3Aw417/2nADgOADP8W<-P5Aw467/25ADP5AO8x25417Qg3T

bcAzE3<G3A7Q,2P5AD<O.1. >dlQP5,O3A-Jc<D7>/2xE/417QgzE9AOP5A3K@QADmn<O0Q8*A'2xE9Ak25ADP5: 467/25ADP5AO8x2x467QgÓGE3<D87Q,2@9A-AO7JQAGt7QADJTXYA*3AOP2xE9AG.UAO8C8?KG2xE/418468;<FS,37QJQAOP5h609.AsÆD<D:=l3.UAB,/h3E9,-F^<*lQP5,9,/h@D>=mx,37/2nP<-J346m5254U,37 WFS,3Pnf8OTW¨t<-8x4Um5<D.j.o>K->3,/0=<D8C8x0Q:AY2xE9A7QA-gG<G254U,37=,/hQFSE3<G2>3,/0 <DP5Ae2nP~>/467Qg=2n,«lQP5,O3Ac<-7QJ%8CE9,-F 2xE3<G2)2xE/468k<D8C8x0Q:=l/254U,37r.6AD<-J98S25,r<pmn,37/2nP<J346m5254U,37T _ E/468k8CE9,-F82xE3<O2w2xE9Ad<-8s8s0Q:zl32546,37«468ShU<D.18*A/K;FSE/4UmsE2xE9AD7«46:zl9.j4UAO8S2xE3<G22xE9AS,3P4Ug?4179<D.t8n25<G2nAO:AO7/24682nP0tAQTa¥%,3P5AgOAD7QAOP<D.j.o>K2xE/468a468Wf7Q,-F7z<D8W<-7 C417QJ341P5A-m52lQP5,9,/h T

/ ]_ E9A7QAsÆO2B2xE9A-,3P5AO:#¦FS,/09.6Je.146f-A'2n,cmn,3798s46JQAOPB4673,/. 3AD8w<z730Q:=@QAOPBFSE/46msEp¦t7QJ'¡/094j25A417/2nAOP5AD8n25467Qg3T _ E9AlQP5,9,/h3418,37QAY,/h2xE9A:=,38x2hU<-:=,/0Q8AxÆD<-:zl9.6AD8,/h9l9P5,9,/h@D>=mx,37/2nP<-J346m5254U,37T

,- %D,$&%5 T √2418a46P5P<O2546,379<D.¢T

#$s3HC T \ 8C8x0Q:Akhi,3P<mn,37/2nP<J/4Um52546,37d2xE3<G2 √2418wP<O2546,379<D.¢T _ E9AD7LKFSAzm5<-7FP54o2nA

√2<-8

a/bFSE9AOP5A

aKb<DP5Az467/25A-gOAOP8YF)4j2xEc7Q,mx,3:z:,37pJ34 3418*,3PYgGP5AO<O25ADP2xE3<-7

1<-7QJb 6= 0

j2xE/4684682xE9AzJQAGt734o254U,37c,/h<zP<O2546,379<D.730Q:@9ADP|CT=X,-F FSAJQ,e<.j4j225.6A<D.6gOA-@3P< U√

2 =a

b

2 =a2

b2

2b2 = a2

_ E/4188*E9,-F82xE3<O2a2 468aA*3AO7TaX,-FK8s417QmxAY<D7,9J9Jk730Q:=@QAOPI8*¡/0Q<-P5ADJ'418<-gG<D417,9JQJKFSAk:0Q8x2Bmx,37Qmn.j0tJQAS2xE3<G2

a468YA*3AO7

2xE3<G2W468?K

a = 2nhi,3PB8C,3:Ak,2xE9AOPa467/25A-gOAOP

nT+O0t@98n254j250325417Qgwhi,3P

a<-@Q,O3A3K-FSAgOAG2I2xE9Awhi,/.1.6,-F)467Qg8U

2b2 = (2n)2

2b2 = 4n2

b2 = 2n2

_ E/418:=AD<D798W2xE3<O2b2 418<D.18*,z<D7eA*3AO7730Q:@9ADPFSE/46msEKt@D>z2xE9AS8C<D:AS<DP5g?0Q:AO7/2B<D8<@9,D3A/K/8*E9,-F8I2xE3<G2

b418WA*3AO7T;bcA)E3<?3Aw7Q,-F8*E9,-F7S2xE3<G2

a<-7QJ

b<-P5Aw@Q,2xEkA*3AD7<-7QJzE9AO7QmxA/K9@9,2xEJ/4o/468x4U@/.UA)@D>

2T _ E/468Wmn,37/2nP<J/4Um5282xE9A)<D8C8x0Q:=l/254U,37S2xE3<O2

a<D7QJ

bE3<G3A7Q,)mx,3:z:,37SJ/4o/468C,3PgGP5AD<G2nAOPL2xE3<D71T _ E9ADP5AGhi,3P5A3K?F)4j2xE9,/032<-7>wh60QP2xE9AOPFS,3PxfFSA):z0Q8n2amn,37Qmn.10tJ9AY2xE3<O2 √

2418W7Q,2<P<O2546,379<D.Q730Q:@9ADP©T _ E/468?K9,/hLmx,/0QP8CA3K46:zl9.j4UAO82xE3<O24o2:z0Q8n2a@QAw46P5P<O2546,379<D.~K3<D8WP5A-¡/0946P5ADJT

_ E9AwhU<-m522xE3<G2 √246841PnP<G254U,379<O./F<-8Wf-7Q,-F7'2n,S2xE9A)HIP5ADA-f-8W:z<-7>>3AD<DP8B@9A-hi,3P5Ad:=<G2xE9AD:z<O2546mn46<D798'0Q8*ADJ/<-P46<[email protected],«P5ADl9P5AD8CAD7/20Q7Qf-7Q,-F7¬¡30Q<D7/254j2546AD8OT¬bcA:z46gOE2BFS,37QJ9ADPE9,-F2xE9A*>A*3AOPWFSAO7/2<@9,/032lQP5,O3417Qgk8s0tmsEd<zP5AD8x09.j2WF)4o2xE9,/032YAxÆ/lQP5AO8C8x467Qgw2xE9ASP5AD8x09.j28<O.UgOAD@9P<O4Um5<D.j.o>TY)ADP5A468<D7<D.o2nAOPn79<G2nA'gOA-,3:=AG2nP4UmWlQP5,9,/hL2xE3<G2JQ,9AO8B7Q,27QADA-JS/<-P46<[email protected]¦U2468W<-gG<D417z<Sl9P5,9,/h@D>=mn,37/2nP<J/4Um52546,37T

#$s3HC TBbcA'f7Q,-F@D>z2xE9A=u/>/2xE3<-gO,3P5AD<D7 _ E9A-,3P5AO:!2xE3<O2FSA'mn<D7dmn,3798x2nP0tm52W<kP4UgOE2<-7Qg?.6A-J)2nP541<-7Qg?.6AWF)4j2xE8x4UJ9A.UAO7Qg©2xE381K1<-7QJ √

2T¦ih/FSA<D8C8x0Q:AW2xE3<O2 √

2468IP<O2546,979<D.~K32xE9AO7FSASm5<-7zt7QJ=<S8s41:z4j.6<DPI2nP46<D7Qg?.UAw2xE3<O2WE3<D8<O.1.t417/2nADgOADPa8x4UJ9A.6AD7Qg©2xE38j>3,/0mn<D7:z09.o2546l3.o>c<D.j.2xE9Az8s46JQAO8@D>d2xE9A=J9AD7Q,3:4679<G2n,3P

b41h √

2 = a/b|sTd¦~7pl9<-P¢254Umn0t.6<DP*KDFSA)msE9,9,38CA2xE9A)8s:=<O.1.6AD8n2I8s0tmsES2nP46<D7Qg?.UA

ABCF)4o2xE'417/2nADgOADPI.6AD7Qg©2xE38IFSE/4UmsE'4188s41:z4j.6<DP2n,S2xE9AS,3P546g?4679<O.T

XY,-F FSAS:=<-f-A2xE9AShi,/.1.6,-F)467Qgmx,3798n2nP50tm52546,3798HU$aP<?F<kmn46P5mn.6AkmnAD7/2nP5ADJ<O2AF)4j2xE'P<-J34j0Q8WA-¡/0Q<D.325,k,37QA),/h92xE9AY8*E9,3P¢2nAOPI8s46JQAO8DT _ E/468467/25ADP8CA-m5282xE9A)E->3lt,25AD730Q8CA<O2

DTIRL,3798x2nP0tm52I<Y.1417QAhUP5,3:

DlQADP5ltAO7QJ346mn09.6<DP25,2xE9AwE->9lQ,2nAO730Q8*ABFSE/4UmsE':=A-A?258

BC<G2

ET

/ [

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................

.............................................................................................................................................................................................................................

.

..

.

..

.

.

..

.

..

.

.

.....................................................

....................................................................

A

BC

D

E

bcAS7QADA-J,373. >=:z<f-A<-P5g?0Q:=AD7/28<-@Q,/032IFSE/46msE=8x4UJ9AD8W4172xE9A'J341<gGP<D: :z0Q8n2E3<G3Az417/2nADgOADP.UAO7Qg©2xE38DTc¤-46P8n2*KAD = AC

2xE/0Q8GKAD

418)<-7c467/25A-gOAOP?T _ E9AO7dFSAf7Q,-F 2xE3<O2DB = AB−AD

418<D.18*,Y<-7)417/2nADgOADPCKD8s417QmxAW4o2L4682xE9AJ/41ÔADP5AO7QmxA,/h2¢FS,467/25A-gOAOP8DTY+O417QmxA∠DBE = 45 <D7QJ

∠EDB = 90 K3FSAkf-7Q,-F∠DEB = 45 KFSE/4UmsEc:=AD<D798Y2xE3<O2

DB = DET _ E9ADP5AGhi,3P5A3K

DEE3<-8w417/2nADgGP<D.;.UAO7Qg©2xEc<-8FSAO.j.T\ .68C,K

DE<-7QJ

CE<DP5A25<D7QgOAD7/28)25,d2xE9Amn41P5mn.UA=mxAO7/2nP5A-J<O2

A E9AO7QmxA/K2xE9A*><-P5A2xE9Ad8C<D:A.UAO7Qg©2xET _ E/0Q8GK

CEE3<-8S467/25A-gGP<O..UAO7Qg©2xET ¤-4679<O.1. >LK;FSAdmn<D7%7Q,-F 8CA-A2xE3<O2

EB = CB − CE418B<D7z467/25A-gOAOP*K98x467QmnA)4o258W.6AD7Qg©2xEzmn<D7@QASAxÆl9P5AD8s8*ADJz<-8a2xE9AJ34jÔAOP5AD7QmnA,/h;2¢FS,d417/2nADgOADP8OTd)AD7QmnA3K 4BDE

418)8x46:41.1<-PW2n, 4ACBTbcA=E3<?3A8*E9,-F72xE3<O2 4BDE

E3<-8417/2nADgOADP8x4UJ9A.UAO7Qg©2xE38DT _ E/468;mn,37/2nP<J/4Um528;,/0QP<-8s8s0Q:zl32546,372xE3<O2 4ABC468a2xE9AS8C:z<D.j.UAO8x28s0tmsE2nP46<D7Qg?.UA9TbcAS:z0Q8n22xE9AOP5AOhi,3P5A'mn,37Qmn.10tJ9A2xE3<G2

√2418a46P5P<O2546,379<D.¢TR.6AD<DP5. >LKG2xE/468;4687Q,2¡/094j25A<D8AG.UADgG<-7/2<-8;2xE9ABQP8x2lQP5,9,/h5TXYA*3AOP2xE9AG.UAO8C8?KD4o2418734UmnAY2n,f7Q,-F 2xE3<G28x0tmsE'2xE/467QgG8Bm5<-7z@QA)l9P5,D3AO7'F)4j2xE9,/0322xE9A)<D.6gOA-@3P<D46m25,9,/.682xE3<G2FSASE3<?3ASmn,3:AY2n,zJ9ADlQAD7QJ=,37T

#$s8)7%5@" C¢,$J+/,M 2 2 $O+_ E9AOP5Ac<DP5Ad:=<D7>%,2xE9ADPl9P5,[email protected]:z8S2xE3<O2Smn<D7¬@9Ad8*,/. 3A-Jp0Q8s417Qge<D7r417QJ341P5A-m52lQP5,9,/h5TwAOP5A'<-P5A2xE3P5ADA':,3P5A'l9P5,[email protected]:z8WFSE9,38*ASlQP5,9,/hU8B.UAO7QJz2xE9AD:z8*AG.o3AO8B25,2xE/4682nADmsE3734U¡/0tAQT

M9T'uQP5,D3A2xE3<G241hr418<-7'41PnP<G254U,379<O.9730Q:@9ADPCK/<-7QJ'4jh

a<-7QJ

b<-P5AYP<O2546,379<D.9730Q:=@QAOP8wF)4o2xE

b 6= 0K2xE9AD7c2xE9AAsÆ-lQP5AO8C8x4U,37

a + brP5ADl9P5AD8CAD7/28S<-741PnP<G254U,379<O.730Q:@9ADP©T

9T'uQP5,D3AW2xE3<G22xE9ADP5A<-P5AB4173t734o2nAG.o>:=<D7>Sl9P541:AO8DT _ E/418468I<D7Q,2xE9ADPhU<-:=,/0Q80Q8*A,/h417QJ341P5A-m52lQP5,9,/h5T ÏUÍLË Ò ¨A-g?417=@D>z<-8s8s0Q:467Qg2xE3<O2I2xE9ADP5A)<-P5A,373.o>kt734o2nAG.o>:=<D7>T;bcE3<G2am5<-7'>3,/0=8s<G>k<@9,/032I2xE9A)730Q:@9ADP>3,/0gOA?2FSE9AO7'>3,/0=:z09.o2546l3.o>2xE9AD:±<D.j.92n,9gOA?2xE9ADP<-7QJz<-JQJ1Å

] T'uQP5,D3Az2xE3<G2e418S<-746P5P<O2546,379<D.I730Q:@9ADP©T«XY,25ADUw2xE/468)468S¡3094o2nAeE3<-P5JT#;,/0:=<?>k0Q8*AY2xE9A)hU<-m52I2xE3<O2

e =∞∑

n=0

1

n!

T )

/D ROJK D S RI/M?U )O O LNM,ORO2U D D

S D )O O+I/M?D I/KI 5U

'$

_ E9AOP5A468w<-7,/.6Jcl9P5,[email protected]: 2xE3<O2YgO,9AD8)@D>c:z<-7>c79<D:AO8@/032wgOAO7QADP<O.1. >cP0Q7988*,3:=AG2xE/417QgY.146f-Aw2xE/418HU\ gGP5,/0Ql%,/h

n:=AD7rAD7/25ADP)<dP5AO8x2<D0QP<D7/2<D7QJrmsE9A-msfe2xE9AG46PE3<G258OT _ E9AE3<O2xsmsE9A-msf-AOP;418<@38*AO7/2xs:467QJ9A-JS<-7QJ'J3418x2nP4U@/032nAO8I2xE9AYE3<O28@9<-msfY25,)2xE9A:AO7¬<O2'P<D7QJQ,3: FSE9AD7r2xE9A*>%.6AD<?3AQT bcE3<G2S468'2xE9AclQP5,9@3<@/41.j4j2~>LK

PnK2xE3<O2I7Q,k:z<-7zgOA?258WE/418W,-F7zE3<G2@9<-msfQK/<D7QJzE9,-F JQ,9AO8

Pn@9A-E3<?3A)<-8

n<-l9lQP5,3<-msE9AD8a4673Q734j2~>QÅ_ E9A<D798nFSAOP4182xE3<O2

Pn = 1 − 1

1!+

1

2!− 1

3!+ · · · + (−1)n 1

n!=

n∑

j=0

(−1)j 1

j!

K ~M©|FSE/4UmsE<-l9lQP5,3<-msE9AD8

1/e<D8

n<-l9lQP5,3<-msE9AD8W4173t734o2>T

_ E/418Wl9P5,[email protected]:4680Q8x0Q<D.j.o>S25<-msf-.6A-J'0Q8s417Qg2xE9Aw417Qmn.10Q8x4U,37QsAsÆ-mn.j0Q8s46,37l9P5417Qmn46l3.UA9T¦U2mn<D7S<D.18*,@9A8*,/. 3A-JS@D>SJQA*3AG.U,3l3467QgB<P5ADmn0QP8s4 3AP5AG.6<G254U,3798CE/46lS<D:,37QgPnKPn−1

K<-7QJPn−2

T _ E/468lQ<DltAOPS417/2nP5,9J30tmnAD8z<pJ/41ÔADP5AO7/2'<-l9lQP5,3<-msEKW0Q8x467Qgd<d25A-msE37346¡30tAmn<O.1.6A-J DÏUÍÐ Ï tÏ6Í DÌ ÎxÏÐÍ T¦ih<8*AD¡30tAO7QmxA,/h730Q:=@QAOP8

b0Kb1Kb2KTGTOTn468YJQAGt7QADJe41725ADP5:=8w,/h;<-7Q,2xE9AOP8*AD¡30tAO7QmxAS,/hL730Q:=@QAOP8

a0Ka1Ka2KTGTOTs@D>k2xE9Awhi,3Pn:09.6<

bk =k∑

i=0

(ki

)

aiK ~©|

2xE9AD7e2xE/468YP5AO.1<O2546,3798*E/41lmn<D7@QAk4673AOP25A-Jd<-7QJ2xE9Az730Q:=@QAOP8aiP5AG2nP4UA*3ADJ@D>e2xE9Ahi,/.1.6,-F)467Qghi,3Pn:09.6<3K9mn<O.1.6A-J'2xE9A QÏ6ÍLÐ Ï ~Í DÌ ÎsÏ~ÐÍ Ð DU

ak =k∑

i=0

(−1)k−i

(ki

)

biT ] |

bcAF)4j.1.QP8x2Y0Q8CAz2xE9A=hi,3Pn:09.6<e ] |B25,8*,/. 3A,/0QPwlQP5,9@/.UAO:pT _ E9AO7cFSAkF)41.j.JQAOP54 3A=2xE9A=hi,3Pn:09.6<d@D>c2¢FS,pJ34jÔAOP5AD7/2):=AG2xE9,9J38GK2xE9A=tP8n2w0Q8x467Qgz4173t734o2nA8CADP4UAO8<-7QJk2xE9A)8*ADmx,37QJk0Q8s417Qgw.j467QAO<-P<O.UgOAD@9P<QT "!$#&% ')()*

c© +-, , . 0/-1$/"2 % /-1435/6*6( 7)89/6*:% ;</-=0>-$;:% 7:*&!

/ ` ¾D½ 3½ ½ ½ t¾ ¼/½ _ E9A8CAG2k,/h<O.1.WlQ,38C8x4U@/.UAclQADP5:z032<O2546,3798=,/hW2xE9ApE3<G258=m5<-7 @QAJ34 346JQADJ%467/25,8s0t@38*A?258Y<-8hi,/.1.6,-F8HUa<8s0t@38*A?2

S0mx,3:zlQP468CA-Jd,/h2xE9AlQADP5:z032<O2546,3798FSE9ADP5Ak7Q,37QA,/h92xE9AY:=AD7gOA?258aE/418a,-F7kE3<O2I@9<-msfQK<w8x0t@98CAG2

S1mn,3798s418x25417QgY,/h32xE9AYlQADP5:z032<O2546,3798FSE9ADP5A 0Q8n2),37QA=,/h2xE9A=:AO7pgOAG28SE/468,-F7pE3<G2*K;<-7QJ8C,,37LK0Qlc25,c<8x0t@98CAG2

Snmx,3798x468n25467Qg),/hltAOPn:0325<G254U,3798FSE9ADP5A<O.1.,/hQ2xE9A:AO7=gOAG2I2xE9AG46P,-F7=E3<O28DTR,3798x4UJ9ADPQ2xE9AW8s0t@38*A?2

S2KGhi,3PLAsÆD<D:=l3.UA9TI¦ih-2xE9A2¢FS,Y:=AD7FSE9,wgOAG2t2xE9ABmn,3PnP5ADm52E3<O28B<-P5A:z<-7 H M<D7QJ=:z<-7 H /K/2xE9AO72xE9A730Q:@9ADPa,/hLlt,[email protected]<DPnP<D7QgOAD:=AD7/28B418

Dn−2KO2xE9AY730Q:=@QAOP,/htJ9ADP<D7QgOAD:=AD7/28hi,3P2xE9Aw,2xE9AOP

n−2E3<O28DT+O417QmxA2xE9AOP5AY<-P5A

(n2

) lQ,38C8x4U@/41.j4j2546AD8ahi,3P2xE9ASl9<D41PW,/h:=AD7kFSE9,=gOAG22xE9AG46PW,-F7E3<G258?K32xE9A730Q:@9ADPW,/hltAOPn:0325<G254U,3798W417k2xE9A8CAG2

S2468 |S2| =

(n2

)

Dn−2T

\ lQl3.o>/467QgI2xE9Aa8s<-:=A.U,9g?46mt2n,YAD<-msEY8s0t@38*A?2SiK*FSAW,9@/2<D417 |Si| =

(ni

)

Dn−iT_ E9Aw25,25<O.t730Q:@9ADPa,/hLlQADP5:z032<O2546,3798,/ht2xE9AE3<O28W418

n! = |S0| + |S1| + |S2| + · · · + |Sn|

=

n∑

i=0

(ni

)

Dn−i =

n∑

j=0

( nn − j

)

Dj =

n∑

j=0

(nj

)

DjK

FSE9ADP5AWFSAYE3<G3AYmsE3<D7QgOA-J2xE9A417QJQAxÆ),/hQ8x0Q:=:z<O2546,37ShUP5,3:i25,

j = n− i<-7QJ)2xE9AD7

0Q8*ADJk2xE9A)hU<-m52I2xE3<O2 ( nn − j

)

=n!

j!(n − j)!=(n

j

) TXY,-F FSAw0Q8CAS@/467Q,3:46<O.Q4673AOP8s46,37=jF)4o2xE

an = Dn<D7QJ

bn = n!|U

Dn =n∑

i=0

(−1)n−i(n

i

)

i! =n∑

i=0

(−1)n−i n!

i!(n − i)!i!

= n!

n∑

i=0

(−1)n−i 1

(n − i)!= n!

n∑

j=0

(−1)j 1

j!

T¤4179<D.j.o>K38x467QmnA

Pn = Dn/n!KFSA,9@/2<D417=~M©|CT

½ I ½ ¾ ¼/½ 3½ ½ ¼G½ bcAF)41.j.J9ADP4o3AY2xE9Aw4673AOP8s46,37hi,3P5:z09.1<' ] |hUP5,3: hi,3Pn:09.6<S~©|;0Q8x467QgY4173t734o2nA8*AOP546AD8OT,-FSA*3AOP*K-8s417QmxA2xE9A468s8s0tAY,/htmx,373AOP5gOAD7QmnA),/h32xE9AY8*AOP546AD8;F)4j.1.9@9A4UgG7Q,3P5ADJLK2xE9A<-P5g?0Q:=AD7/2I2xE3<O2FSAF)41.j.g?4 3ASE9AOP5A)468W7Q,2amn,3:=l3.UA?2nAG.o>P4UgO,3P5,/0Q8OTbcA)417/2nP5,9J30tmnASAsÆ-lt,37QAO7/2546<O.gOAD7QAOP<O25417Qgwh60Q7Qm5254U,3798W<-8ahi,/.j.U,-F8U

A(x) =

∞∑

k=0

ak

k!xk K B(x) =

∞∑

k=0

bk

k!xk T

+O0t@98n254j250325417Qgwhi,3PbkhUP5,3: i?|nKFSAgOA?2

B(x) =∞∑

k=0

(k∑

i=0

(ki

)

ai

)

1

k!xk =

∞∑

k=0

k∑

i=0

ai

i!(k − i)!xk T

/-¦~7/2nAOP5msE3<-7Qg?417Qgw2xE9AS,3P5J9ADP,/hL8x0Q:=:z<O2546,37z<-7QJz8x46:zl9.j41hj>/467QgK

B(x) =

∞∑

i=0

∞∑

k=i

ai

i!(k − i)!xk =

∞∑

i=0

∞∑

k=i

(

aixi

i!

)(

xk−i

(k − i)!

)

=∞∑

i=0

aixi

i!

( ∞∑

k=i

xk−i

(k − i)!

)

=

[ ∞∑

i=0

ai

i!xi

]

∞∑

j=0

xj

j!

T

bcAdP5ADmx,9gG734o£sA=2xE9Ad8s0Q:z8S467p2xE9A.6<D8x2AsÆ-lQP5AO8C8x4U,37«<@9,D3Ae<-8A(x)

<D7QJexKP5AD8sltADm5254o3AG.o>T _ E9ADP5AGhi,3P5A3K9FSAkmn<D7e7Q,-FÈFP4j25A

B(x) = A(x)ex KQhUP5,3:±FSE/4UmsEFSAgOAG2A(x) = e−xB(x)

T _ E9AO7A(x) =

∞∑

j=0

(−1)jxj

j!

[ ∞∑

i=0

bi

i!xi

]

=

∞∑

j=0

∞∑

i=0

(−1)jbi

j!i!xi+j T

bcASm5<-7zP5ADx417QJQAxÆk6.6AG225417Qgk = i + j

|2n,zgOA?2A(x) =

∞∑

k=0

k∑

i=0

(−1)k−ibi

(k − i)!i!xk =

∞∑

k=0

(k∑

i=0

(ki

)

(−1)k−ibi

)

1

k!xk T

_ E9Amx,9AGyemn46AD7/2;,/hxk467Y2xE/468;hi,3P5:z09.1<:z0Q8n2;@QAa2xE9AB8C<D:AB<D82xE9Amx,9AGyemn46AD7/2

ak/k!467k2xE9A)41734j2541<D.9hi,3Pn:09.6<SJQAGt73417QgA(x)

T _ E/0Q8?K-FSAS,9@25<O467= ] |sT ½ I ½ ¾ ¼/½ 3½ ½ ¼'º)¾ ¼ bcA7Q,-F <-8s8s0Q:=AB2xE3<O22xE9A8*AD¡30tAO7QmxAO8P5AO.1<O25A-J'@D>2xE9Ahi,3Pn:09.6<w~©|<-P5A Í/Ï¢Ë~Ì8*AD¡30tAO7QmxAO8

a0Ka1KQTOTGT?K

an<-7QJ

b0Kb1KtTOTGT?K

bnT _ E/418m5<-8CA418I<D.j.-2xE3<O2LFSA7QA-ADJQADJhi,3Pa,/0QPlQP5,9@/.UAO: ,/ht2xE9AE3<O28DT | _ E9AD7=~©|Imn<D7@9AP5ADmn<D8x2417/2n,k:=<G2nP54 Æhi,3Pn:±<-8

b0

b1

b2TTTbn

=

1 0 0 . . . 01 1 0 . . . 01 2 1 . . . 0TTT TTT TTT T T T TTT(n0

) (n1

) (n2

)

. . .(nn

)

a0

a1

a2TTTan

T

VW7d2xE9AzP546gOE2xCE3<D7QJ8s46JQAk468w<-7n + 1

@D>n + 1

:z<O2nP4 ÆTFSE9,38*AP5,-F8)<DP5A2xE9AtP8n2

n + 1P5,-F8w,/hWu/<D8*m5<D.~Ó 82nP46<D7Qg?.UA/KQ.j.UADJ467=F)4o2xE=£sADP5,9AO8DT _ E9AAD7/2nP~>e417dP5,-F

i<D7QJemx,/.j0Q:=7

j468

T (i, j) =

(ij

) K3FSE9AOP5Ai<D7QJ

jgO,=hUP5,3:

025,

nUP<G2xE9ADP2xE3<-7

2xE9Amn,373AD7/2546,379<D.730Q:@9ADP467Qg'8n25<DP25417QgS<G21|sT _ E9AS4673AOP8s46,37hi,3P5:z09.1<z ] |8n25<G2nAO8

2xE3<O2t2xE9AW:=<G2nP54 ÆTE3<D8;<D7w4173ADP8CA

XFSE9,38*AWAO7/2nP546AD8<-P5A

X(i, j) =

(ij

)

(−1)i−j T_ E/418W418FSE3<O2FSAF)41.j.tl9P5,D3A9TÇDAG2M = TX

T _ E9AzgO,3<D.4182n,8*E9,-F 2xE3<O2M = I

Kt2xE9A'4UJ9AD7/254o2>e:=<G2nP54 ÆTbcA:0Q8x28*E9,-F 2xE3<O2W2xE9AzJ341<gO,379<O.AG.UAO:AO7/258w,/hM

<DP5A1<D7QJ2xE9Az,/ÔsJ341<gO,379<O.AO.6AD:=AD7/28B<DP5A

0T

/_ E9ASAO7/2nP~>417zP5,-F

a<D7QJ=mn,/.10Q:z7

b,/h

M418

M(a, b) =n∑

i=0

T (a, i)X(i, b) =n∑

i=0

(ai

)(ib

)

(−1)i−b T [ |¦ih

i > aKFSASE3<?3A (a

i

)

= 0KFSE/41.6A)41h

i < bK/2xE9AD7 (i

b

)

= 0T _ E/0Q8GK98C,3:Aw25ADP5:=8

467)2xE9A<-@Q,O3A8s0Q: <-P5AB£sAOP5,KD417kgOAO7QADP<O.T¦~7SlQ<DP2546mn09.6<DP*KGFSA8*ADAB2xE3<G2M(a, b) = 0FSE9AD7

a < bT

_ E9AJ341<gO,379<O.AO7/2nP546AD8S,/hM

,9mnmn0QPFSE9AO7a = b

Tp¦~7c2xE/418'm5<-8CA3K2xE9A,373.o>7Q,37Qn£sAOP5,'2nAOPn: 467k2xE9A)8s0Q:±417= [ |;418Whi,3Pi = a

T;bcASE3<?3AM(a, a) =

(aa

)(aa

)

(−1)a−a = 1T

_ E/0Q8?K/2xE9AJ341<gO,379<O.LAG.UAO:AO7/258B,/hM

<DP5A1K3<D8BJQAO8s41P5A-JT

XY,-F mn,3798s46JQAOPa > b

T _ E9AD7= [ |I@QADmx,3:=AD8M(a, b) =

a∑

i=b

(ai

)(ib

)

(−1)i−b =

a∑

i=b

a!

(a − i)!(i − b)!b!(−1)i−b K

FSE/4UmsEk4682xE9A)8C<D:A)<D8b!

a!M(a, b) =

a∑

i=b

1

(a − i)!(i − b)!(−1)i−b T

XY,-FK-FSAw.6AG2j = i − b

6<-7QJ=E9AO7QmxAi = j + b

|CT _ E9AD7=,/0QPaAD¡30Q<G254U,37418b!

a!M(a, b) =

a−b∑

j=0

1

(a − b − j)!j!(−1)j T

¤4179<D.j.o>K-FSAw.UA?2m = a − b

<-7QJz:09.j2541l9. >=@9,2xE8s46JQAO8B@D>m!

25,zgOA?2m!

b!

a!M(a, b) =

m∑

j=0

m!

(m − j)!j!(−1)j =

m∑

j=0

(mj

)

(−1)j T_ E9AkP4UgOE2xsE3<-7QJe8s46JQAkA-¡/0Q<D.18

0@D>e<SFSAG.1.6Cf-7Q,-F7dmx,3:=@34179<O25,3P546mn84UJ9AD7/254o2>K@QADmn<O0Q8*AW4o2P5AOlQP5AO8*AO7/258

(x+y)m K?FSE9AO7 x = 1<D7QJ

y = −1T _ E9ADP5AGhi,3P5A3K

M(a, b)468£sAOP5,K3<-7QJ'2xE9AlQP5,9,/h418Bmx,3:zl9.6AG25AQTAD@9<k)<G2xE9,/032*K38x250tJ9AD7/2bcAO8x2nP4UJ9gOAS+-msE9,9,/.Qhi,3PH41P5.18u/<D8C<-JQAO79<9K9R \ K9ª+ \ NMM

[email protected]

/ON R ; #

$ $'$S&9$

Ð DÍ3Ï Ë6ÏÐÍ3Î 9Ð3ËLË Ï1Î Ð =ÍÎ Ð 9ÌaÎ?ÌGÍLËLËÐ ÐÌ?ÎsÎGÐ iÉ CkÐ9Ð Ð Ì - Ë dÌ?ÍËÐ ¬ Ë Ì Ë6Ï Î Í 3Ë Ë6Ï6ÎGË6Ï Î Í3Ï DÌ ÎsÏË#[Ð - [ D [ 9Ì Ë Í =] (]

ÇO<-8n2L730Q:=@QAOPQFSAWgG<G3A2nAO7wlQP5,9@/.UAO:=8hUP5,3: 2xE9Aa8*E9,3P¢25.1418x2thi,3PQ2xE9AB ¦~7/25ADPn79<O2546,379<D.;¥r<O2xE9AO:=<G254Um5<D.Va. >9:zl941<JTbcAS8n25<DP22xE/418730Q:=@QAOPF)4o2xEk2xE9ASP5AD:z<D4173467QgMMclQP5,9@/.UAO:=8,/hW2xE9Ad8*E9,3P¢25.1418x2?T ¥>«2xE3<-7Qf-8zgO,«25, \ 7QJ> ÇG4j0KWRL<D79<J/46<D7 _ AD<D:ÇDAD<-JQAOPI2n,S2xE9Ak¦5¥%V 467=+-AD,/09.K/hi,3Pmn,/.1.6A-m525417Qg2xE9AO:pT®Y®Y®

! " #$ "

Oµ!

&% ('!)

+* ¼ -¾U½ -,/. ¼ ¾

ÄYÄ T( 3Ì?ÍË6Ï6Í

)ÇDAG2

ABCD@9A=<mn,373AsÆe¡30Q<-J9P41.1<O25ADP<O.F)4j2xE

AB7Q,2lQ<DP<D.j.UAG.W2n,

CDK<D7QJ¬.UA?2

Y@QAc2xE9Aplt,/417/2z,/h417/2nAOP8*ADm5254U,37 ,/hB2xE9A«lQADP5ltAO7QJ346ms09.6<DP=@/468CA-m525,3P8e,/h

AB<D7QJ

CDT ¦ih

X418<rlt,/417/241798s46JQA

ABCD8s0tmsE¬2xE3<G2

∠ADX = ∠BCX < 90 <D7QJ∠DAX = ∠CBX < 90 K)lQP5,O3Ar2xE3<G2

∠AY B = 2∠ADXTÄ T

(Ì" D DÎ

)¤-467QJe<O.1.l9<D41P8w,/hh60Q7Qm52546,3798

f<D7QJ

ghUP5,3: 2xE9Ak8CAG2,/h;P5AO<D.730Q:@9ADP825,'4o258CAO.jht8s0tmsE2xE3<O2

f(x + g(y)

)= xf(y) − yf(x) + g(x)

hi,3PI<O.1.9P5AD<O.730Q:@9ADP8x<D7QJ

yTÄ ¯ T

(iÍ-Ï

)ÇDAG2

O@9A2xE9A=mn46P5mn0Q:=mxAO7/2nP5A<D7QJ

H2xE9A,3P¢2xE9,9mxAO7/2nP5A,/h<D7<mn0325AB2nP46<D7Qg?.UA

ABCTauQP5,O3AW2xE3<O2L2xE9AOP5AYAxÆ-418x2lt,/417/258

DKEKD<-7QJ

F,378s46JQAO8

BCK

CAK<-7QJ

ABKP5AD8sltADm5254o3AG.o>K8s0tmsE2xE3<G2

OD + DH = OE + EH = OF + FH<-7QJk2xE9Aw.1417QAD8AD

KBE

K/<-7QJCF

<-P5Amx,37Qmn0QP5P5AD7/2?TÄ T( 5 Í

)_ AD7=gG<-7QgG8n2nAOP8B<-P5A8n25<D7QJ3417Qg,37z< Q<O28x0QP5hU<-mxA9T _ E9ASJ3418x2<-7QmnAD8@QA?2¢FSA-AO72xE9AD: <DP5A<D.j.J3418x25417Qm52?T+O46:09.j2<-7QAD,/0Q8s. >«AD<-msE«,/hI2xE9AD: 8CE9,9,258<G2Y2xE9A,37QA'<D:,37Qgw2xE9A',2xE9ADPa73467QAwFSE9,z418W2xE9A'7QAO<-P5AO8x2?T \ 2a.6AD<D8x2WE9,-F :z<-7>gG<-7QgG8n2nAOP8F)41.j.@QA)8*E9,2<O2xÅÄ S T

( Ì Í

)\ 7Q,37QCAO:=l/2>k8*A?2

A,/htP5AD<O.Q730Q:@9ADP8a418Wm5<D.j.UADJk<

B3 8*A?2I4jh2xE9ASmx,37QJ/4j2546,3798

a1Ka2Ka3Ka4Ka5Ka6 ∈ A

<D7QJa1 + a2 + a3 = a4 + a5 + a646:zl9. >c2xE3<G22xE9A=8*AD¡30tAO7QmxAO8

(a1, a2, a3)<-7QJ

(a4, a5, a6)<-P5Az4UJ9AD7/2546mn<O.0Ql25,<ltAOPn:0325<G254U,37T'¤/,3PB<8CAG2

X,/h;P5AD<O.730Q:@9ADP8?K.UA?2

D(X)J9AD7Q,25AS2xE9AJ34jÔAOP5AD7QmnA8*A?2 |x − y| : x

Ky ∈ X TBuQP5,D3AY2xE3<O241h A = 0 = a0 < a1 < a2 < · · · <D7QJ

B = 0 = b0 < b1 < b2 < · · · <-P5A)4173t734o2nA8CA-¡/0tAD7QmnAD8,/hP5AO<D.730Q:=@QAOP8aF)4o2xED(A) = D(B)

K3<-7QJ4jhA468W<

B3 8*A?2*K2xE9AO7

A = BT

Ä T

( Ì ] Ì-Ë Ì ÍÎ

)¦~7)2xE9Al3.6<D7QAWFSA<DP5AYg?4 3AD7)2¢FS,Smn41P5mn.UAO8467/25ADP8CA-m52x467Qg)<O2

X<-7QJ

YTuQP5,O3A)2xE3<G22xE9ADP5ASAsÆD468n2hi,/0QPlt,/417/258B8s0tmsEk2xE3<G2hi,3PWA*3AOP~>mn41P5mn.UA2n,/0tmsE/417Qgk2xE9Ak2¢FS,cg?4o3AO7pmn46P5mn.6AD8w<O2

A<-7QJ

BK<-7QJc:ADAG25417Qg'2xE9Az.j467QA

XY<G2

C<-7QJDKAD<-msE',/h/2xE9A.j467QAO8

ACKAD

KBC

K-<D7QJBD

l9<-8s8*AO82xE3P5,/0tgOE',37QAY,/h/2xE9,38*Ahi,/0QPlt,/417/258OTÄ T( DÎsÎsÏ

)¤3,3PS<clt,/. >97Q,3:46<O.

PF)4j2xE%J3418x25417Qm52SP5AD<O.Bmx,9AGyemn46AD7/28GK.UA?2

M(P )@QA2xE9Aa8CAG2,/h-<D.j.DlQ,/.o>37Q,3:z41<D.182xE3<G2m5<-7w@QAa,9@/2<D417QA-JhUP5,3:

P@D>YlQADP5:z0325417Qg4j28wmn,9AOymn4UAO7/258OT¤417QJe<O.1.L417/2nADgOADP8

nhi,3PFSE/46msEz2xE9ADP5AAxÆ-418x28<klt,/. >97Q,3:46<O.

P,/hJQADgGP5A-A

2000F)4o2xESJ/468n25467Qm52P5AO<D.3mn,9AOymn4UAO7/258I8x0tmsEw2xE3<G2

P (n) = 0<-7QJYFSAm5<-7'gOA?2hUP5,3:!<-7>

Q ∈ M(P )<SlQ,/.o>37Q,3:z41<D.

Q′ 8x0tmsE'2xE3<O2 Q′(n) = 0@D>417/2nAOP5msE3<-7Qg?417Qg<O2:=,38x2,37QAl9<D41Pa,/hLmx,9AGyemn46AD7/28,/h

QTÄ T

( DÎsÎsÏ

)ÇDA?2

A1A2 . . . An@9Ad<cmn,373AsÆclt,/. >3gO,37LK

n ≥ 4T uQP5,D3A2xE3<O2

A1A2 . . . An418zm>3mn.146m'4jh<D7QJ ,373. >r41hAD<-msEp3ADP¢2nAxÆ

Aim5<-7¬@QAc<-8s8s46gG7QA-J%<lQ<O46P

(bi, ci),/hP5AO<D.L730Q:=@QAOP88*,k2xE3<O2

AiAj = bjci − bicjhi,3PW<O.1.

i<D7QJ

jF)4o2xE

1 ≤ i < j ≤ nTÄ T

(Í3ÏËÌÏ6Í QÐ

)ÇDAG2

aKbKL<-7QJ

c@QAlQ,38s4o254o3A'417/2nADgOADP8w8x0tmsE2xE3<G2

c > 2b > 4aTeuQP5,D3A'2xE3<O2B2xE9AOP5A=AxÆ-418x28w<zP5AD<O.730Q:=@QAOP

λ8s0tmsE2xE3<G2B2xE9Ak2xE3P5ADA730Q:@9ADP8

λaKλbK3<-7QJ

λc<O.1.E3<?3Aw2xE9AG46PIhUP<m52546,379<D.tl9<-P¢258a467k2xE9A)417/2nAOP~/<D. (1

3, 2

3

] T®T(Í3ÏËÌÏ6Í QÐ

)\ h60Q7Qm52546,37

F418JQAGt7QADJchUP5,3: 2xE9A=8CAG2w,/h7Q,37Q7QA-gG<G254o3A 467/25A-gOAOP8¬25, 4o258CAO.jh8s0tmsE 2xE3<G2*Kehi,3P%A*3AOP~> 7Q,37Qs7QADgG<O254 3A417/2nADgOADP

nK

F (4n) = F (2n) + F (n)K

F (4n + 2) = F (4n) + 1Kq<D7QJ

F (2n + 1) = F (2n) + 1T uQP5,D3Ac2xE3<O2*KBhi,3PzAO<msE lQ,38s4o254o3A467/25A-gOAOP

mKB2xE9A730Q:@9ADPa,/h467/25A-gOAOP8

nF)4o2xE

0 ≤ n < 2m <D7QJF (4n) = F (3n)

418F (2m+1)

T Ä T(Í3ÏËÌÏUÍ QÐ

)_ E9A25<D7QgOAD7/28a<O2

B<D7QJ

A2n,2xE9A)mn41P5mn0Q:mn41P5mn.UAw,/h<-7k<mn0325A2nP541<-7Qg?.6A

ABC:=A-A?2;2xE9A25<D7QgOAD7/2I<O2

C<G2

T<D7QJ

UKP5AD8sltADm5254o3AG.o>T _ E9A.1417QAD8

AT<D7QJ

BC:=A-A?2<G2

PKL<D7QJ

Q4682xE9Ak:z46JQxlt,/417/2,/h

AP 2xE9A'.1417QAD8

BU<-7QJCA

:ADAG2<G2RK3<D7QJ

S4682xE9A):z46JQxlt,/417/2a,/h

BRT

U<?|WuQP5,D3AY2xE3<O2∠ABQ = ∠BAS

Ti@-| WAG25ADP5:z417QA3K/417k2nAOPn:z8,/hP<O2546,38B,/hL8x4UJ9Aw.UAO7Qg©2xE38GK2xE9AY2nP541<-7Qg?.6AD8ahi,3P;FSE/46msE'2xE/468<-7Qg?.6A)468W<:z<?ÆD46:0Q:pT

XYAxÆD2tFSA250QPn7Y2n,w8C,/.1032546,3798@D>,/0QPLP5AD<-JQAOP82n,wl9P5,[email protected]:z8I,/hD2xE9A§4UA?2n79<-:=AD8CA¥r<O2xE9AO:=<G254Um5<D.R,3:zltA?254j2546,37M-N-N3 MBU3M ` /TÄ Tw¦~7e<'l9.1<-7QA/KQ.UA?22xE9ADP5Ak@9Akg?4o3AO7e<kmn46P5mn.6A)F)4o2xEmxAO7/2nP5AOK/F)4j2xE=P<J/410Q8

R<-7QJS<YlQ,/467/2P41798s46JQAW2xE9AYmn46P5mn.6A3K

OP = d < RT \ :,37QgB<O.1.3mx,373AxÆ)¡30Q<-J9P41.1<O25ADPn<D.18

ABCDK-46798CmxP546@QADJk417'2xE9Amn41P5mn.UAw8x0tmsES2xE3<O2;2xE9AO41PJ/46<-gO,379<D.18

AC<D7QJ

BDmn032AD<-msEp,2xE9AOPw,3P¢2xE9,9gO,379<D.j.o>c<G2

PKJ9AG25ADP5:z417QA2xE9A,37QAO8FSE/4UmsEE3<?3A2xE9AgGP5AO<O25AD8n2ltAOP541:A?2nAOPW<-7QJz2xE9Ak,37QAO8WFSE/46msEeE3<?3A2xE9A'8s:=<O.1.6AD8n2WlQADP46:=AG25ADP©TRL<O.Umn09.1<O25A2xE9AO8*AltAOP541:A?2nAOP8W467k25ADP5:=8B,/h

R<-7QJ

dT

M3Ð DÌ [ Ð 9 dÌ ÎsÎsÏ 3Ë 5 Î9Ð -5 Í Ì Í Q Ï1ÎOË~Ð Ì

~Ì[ jÏ ¢ËÐÍÐ Ì 3Ì Ï6ÎGËÐ CkÌ GÏDÌ ÎsÎsÏ ÎÎ?Ð 3Ë6ÏÐÍ ; Ë~Ì[zË ÌSÌ-Ï¢Ë~Ð Î ÇDAG2

ABCD@QA<)¡/0Q<J3P54j.6<G2nAOP<D.98s<O25418shj>/467Qga2xE9Awg?4 3AD7kmx,37QJ/4j2546,3798GK-<-7QJS.6AG2

pJQAO7Q,2nA)4o258WlQADP46:=AG25ADP©T _ E9AD7p2 = (AB + BC + CD + DA)2

= AB2 + CD2 + BC2 + DA2 + 2(AB · CD + AD · BC)

+ 2(AB · AD + CB · CD) + 2(BA · BC + DA · DC)T

XY,-FAB2 + CD2 = BC2 + DA2 = 4R2 T ~M©|

u32n,/.6AD:'>LÓ 8 _ E9AD,3P5AD: g?4 3AD8a0Q8AB · CD + AD · BC = AC · BD

T\ .68C,K-F)4o2xE8C,3:AFS,3Pnfk,37QA,9@/2<D41798

AC2 + BD2 = 8R2 − 4d2 TAO7QmxA/K2AC · BD = (AC + BD)2 − 8R2 + 4d2 T ~©|

_ E/0Q8?K2(AB · CD + AD · BC) = (AC + BD)2 − 8R2 + 4d2 T ] |

¤0QP¢2xE9ADP5:,3P5A/K2(AB · AD + CB · CD) = 4R · AC

K [ |<-7QJ

2(BA · BC + DA · DC) = 4R · BDT i?|

ªB8s417Qg)~M©|xK9 ] |xK9 [ |xK3<-7QJ=i?|;417=,/0QPaAxÆl9P5AD8s8s46,37hi,3Pp2K-FSAgOAG2

p2 = (AC + BD)2 + 4R(AC + BD) + 4d2 TR,3798CA-¡/0tAD7/25. >LK2xE9Ac:z<?ÆD46:0Q: UP5AO8ClQA-m5254 3AO. >LKa:467341:z0Q:'|,/h

pmn,3PnP5AO8ClQ,37QJ98'25,2xE9Ak:z<?ÆD46:0Q: UP5AO8ClQA-m5254 3AO. >LK:467341:z0Q:'|a,/h

AC + BDK3FSE/4UmsEKQ467z/4UA©F ,/h;~©|xKmx,3P5P5AD8slt,37QJ38B25,2xE9AS:z<?ÆD46:0Q: UP5AO8ClQA-m5254 3AO. >LKQ:467341:z0Q:'|,/h

AC · BDTYXY,25417Qg2xE3<O2

2AC · BD = 8R2 − 4d2 − (AC − BD)2K

FSAmn,37Qmn.10tJ9Ae2xE3<O2)2xE9Ad:=<©Æ-41:z0Q:qUP5AO8ClQA-m5254 3AO. >LK:z417346:0Q:k|),/hpmn,3PnP5AO8ClQ,37QJ982n,z2xE9A:467341:z0Q: UP5AO8ClQA-m5254 3AO. >LK:=<©Æ-41:z0Q:'|W,/h |AC − BD| T'¦U2Bhi,/.1.6,-F8B2xE3<O2 p468S:=<©Æ-41:z4o£sA-JdFSE9AD7

AC = BDKI<-7QJ

p468S:z417346:4j£sADJcFSE9AO7

AC = 2R<D7QJ

BD = 2√

R2 − d212xE9A:z<?ÆD46:0Q: <D7QJS:z417346:0Q: lt,[email protected]©2xE38Ihi,3P;<YmsE9,3P5J2xE3P5,/0tgOE

P|sTAO7QmxA/K

p2max = 16R2 − 4d2 + 8R

√

4R2 − 2d2K

<-7QJp2min = 16R2 + 16R

√

R2 − d2T

É-ÏËÐ ÎÍÐQËÌ Ò bcAm5<-7d2xE9AO7p,9@/2<D417d2xE9Ahi,/.1.6,-F)467QgAsÆ-lQP5AO8C8x4U,3798Yhi,3P pmax<-7QJ

pminU

pmax = 2(√

2R +√

2R2 − d2)

=

(√√2R − d +

√√2R + d

)2 Kpmin = 2

√2R

(√

R + d +√

R − d) T

TeÇDA?2B2xE9AOP5A=@9A=g?4 3AD7c<'FSE9,/.6A730Q:@9ADPn > 1

K7Q,2YJ34 [email protected]@D>1997

TR,3798x4UJ9ADP2¢FS,8CA-¡/0tAD7QmnAD8,/h730Q:@9ADP8 ai<-7QJ bj

JQAGt7QADJ=@D> Uai = i +

ni

1997

i = 1

K2K3K. . .

K1996

|LKbj = j +

1997j

n

j = 1

K2K3K. . .

Kn − 1

|T¨3>'<DPnP<D7Qg?467QgB2xE9A730Q:@9ADP8a,/h32xE9AO8*AB2¢FS,S8CA-¡/0tAD7QmnAD8417S467QmxP5AO<-8x467Qg,3P5J9ADPCKOFSAwgOA?22xE9A8*AD¡30tAO7QmxA

c1 ≤ c2 ≤ · · · ≤ c1995+nT

uQP5,D3AY2xE3<O2ck+1 − ck < 2

hi,3PA*3ADP~>k = 1

K2K. . .

K1994 + n

T3Ð DÌ [ Ð Q dÌ ÎsÎsÏ 3Ë 5 Î9Ð -5 Í Ì Í 9Ï~Ì n Ì

Ð Í9Î ËÌ*ÏUÍ tÐÍLËÐDÏ1ÎGÌ -5 Í Ì CkÌ GÏDÌSË ÌYÎ?Ð 3Ë6ÏÐÍeÐ Ð Í9Î ËÌ*ÏUÍ ¤41P8x27Q,2nA2xE3<O2 ai<-7QJ bj

<-P5A2¢FS,417QmxP5AD<D8s417Qg<-P4j2xE3:=AG2546mn<O.D8*AD¡30tAO7QmxAO8GKF)4j2xEzJ/41ÔADP5AO7QmxAα = 1 +

n

1997

<-7QJβ = 1 +

1997

n

K/P5AD8sltADm5254o3AG.o>TÇDAG2

i ∈ 1K

. . .K1996 <D7QJ

j ∈ 1K

. . .K

n − 1 K<D7QJr8s0Ql9lt,38CA2xE3<O2ai = bj

T _ E9AD7ni = 1997j

T+O417QmxA1997

418)l9P541:Az<D7QJn 6≡ 0 (mod 1997)

KFSAdJ9A-J/0tmxA=2xE3<O2gcd(n, 1997) = 1

T%¤/P5,3:#HI<O0Q8C8?Ó 8 _ E9AD,3P5AD:dK;FSAz2xE9AO7%E3<?3Ai ≡ 0 (mod 1997)

K-FSE/4UmsEk418a46:zlt,[email protected]¦U2hi,/.1.6,-F82xE3<G2ai 6= bj

TN!0 $" 2/. "% T

n < 1997T

bcASAO<-8x41. >8CA-AY2xE3<G2α < 2 < β

K ~M©|<-7QJ

a1 < b1K ~©|

FSE/4UmsEk46:zl9.j4UAO82xE3<G2c1 = a1

TW¥%,3P5AD,D3AOP*K a1996

bn−1

=1996n

1997(n − 1)> 1

T _ E9AO7LKbn−1 < a1996

K ] |FSE/4UmsEk46:zl9.j4UAO82xE3<G2

c1995+n = a1996T

%5*5 . ¤3,3PaA*3ADP~>j ∈ 1

K. . .

Kn − 2 K/2xE9ADP5A'AxÆ-418x28 i ∈ 1

K. . .

K1996 8x0tmsE2xE3<O2

bj < ai < bj+1T

Ð9Ð +O0Ql9lt,38CA3K^hi,3P 2xE9A l30QPnlQ,38*A ,/h mx,37/2nP<-J346m5254U,37LK 2xE3<G2 2xE9ADP5A AxÆ-418x28j ∈ 1

K. . .

Kn − 2 8x0tmsEd2xE3<O22xE9A=417/2nAOP~/<D. [bj , bj+1]

JQ,9AO8S7Q,2)mn,37/25<O467p<D7>,/h2xE9AaiÓ 8OT¬ÇDAG2

p@9Az2xE9AdgGP5AO<O25AD8n2w417QJQAxÆc8x0tmsEc2xE3<G2

ap < bjU8x0tmsEp<

pJ9,9AD8

]AsÆD468n2*Kt8s417QmxA

a1 < b1 ≤ bj|sT _ E9AD7

p < 1996U8x467QmnA

bj < bn−1 < a1996|nK<D7QJ

ap < bj < bj+1 < ap+1TY¦U2ahi,/.1.6,-F8a2xE3<O2

α = ap+1 − ap > bj+1 − bj = βKFSE/4UmsEzmx,37/2nP<-J346m5258B~M©|CT _ E/0Q8?K/2xE9Aw.UAO:=:z<)468Wl9P5,D3ADJT

¦U2Whi,/.j.U,-F8WhUP5,3:!2xE9AS.6AD:z:=<2xE3<O2*KQhi,3PWA*3ADP~>k ∈ 1

K. . .

K1994 + n K3FSA<-P5Aw467=,37QA,/ht2xE9Aw2xE3P5ADA)hi,/.1.6,-F)467Qgwm5<-8CAD8U

6<G|ck = ai

<-7QJck+1 = ai+1

hi,3P8C,3:AiT _ E9AD7

ck+1 − ck = α < 2T

i@-|ck = ai

<-7QJck+1 = bj

hi,3Pt8*,3:=Ai < 1996

1hUP5,3: ] |U|9<-7QJY8C,3:Aj ≤ n−1

T_ E9AO7bj < ai+1

<D7QJck+1 − ck < ai+1 − ai = α < 2

Tim|

ck = bj<-7QJ

ck+1 = aihi,3Pa8C,3:A

i > 11hUP5,3: i?|U|I<-7QJz8*,3:=A

j ≤ n − 1T_ E9AO7

ai−1 < bj<-7QJ

ck+1 − ck < ai − ai−1 = α < 2T

¦~7AO<msEzm5<-8CA3KDFSASE3<?3Ack+1 − ck < 2

K3<D8BJQAO8s41P5A-JT %(6*! D' /. "% T

n > 1997T

_ E/418Bm5<-8CAY468aAD8s8*AO7/2546<O.1. >S2xE9Aw8C<D:Aw<D82xE9AwQP8x2mn<D8*A#UL8s41:=l3.o>'417/2nAOP5msE3<-7QgOAnF)4j2xE

1997<D7QJ

aF)4j2xE

b6<-7QJz<O.68C,

αF)4o2xE

β|sT¯

T),-F±:=<D7>dh60Q7Qm5254U,3798f : ∗ → ∗ <DP5Ak2xE9ADP5Ak2xE3<G2Y8x46:09.j2<-7QAD,/0Q8s. >8C<G25468xhj>'2xE9Aw2¢FS,'hi,/.1.6,-F)467Qgwmx,37QJ/4j2546,3798HU

64 |f(1) = 1

K64j4o|

f(n) · f(n + 2) =(f(n + 1)

)2+ 1997

hi,3P<O.1.n ∈ ∗ Å

∗ J9AD7Q,25AD82xE9A8CAG2,/hL<O.1.tlQ,38s4o254o3Aw417/2nADgOADP8OT |3Ð DÌ [ Ð 9 dÌ ÎsÎsÏ 9Ë Î9Ð -5 Í Ì 9 Ï6ÎGËÐ Ì

5 Ì[ jÏ ¢ËÐÍ Ð ~Ì/Ì Ï1ÎOË~Ð Í Ë Q Í ÎsÏ Î 9Ð-Î Ë ÌGÍ3Î D ÌDÌ Ì CkÌ GÏDÌ'Ë ÌYÎ?Ð 3Ë6ÏÐÍ Í Ð dÌ?ÍË1ÎwÐ 9Ð-Î _ E/418418GKDAD8s8*AO7/2546<O.1. >LKD,37QA,/h2xE9ABlQP5,9@/.UAO:=8I,/h2xE9A

3

¨t<O.UfD<D7¥r<O2xE9AO:=<G254Um5<D.Va.o>3:=l346<-JeiM-N ` |CTw¦U2F<D8lQP5,3lQ,38*ADJe@D>p¨Q09.6gG<-P46<QTYAOP5A)FSA'g?4o3A4j2W<D8<'.6AD:z:=<F)4j2xEe2xE9A=8s<-:=A=l9P5,9,/ha68s.j4UgOE25. >:=,9J34jADJp@D>d2xE9AADJ34o2n,3P85|a2xE3<G2F<D8Sg?4 3AD7<Dh125ADP2xE9ASmx,3:zltA?254j2546,37=@D>'2xE9Aw2nAO<-:z82xE3<O2aE3<-JzlQ<DP2546mn46l9<O25A-JT %5*5 .tT \ 8*AD¡30tAO7QmxA)418BJQAGt7QADJ=@D>

a1 = aKa2 = b

K9<D7QJan+2 =

a2n+1 + c

an

Kn = 1

K2K. . .

KFSE9ADP5A

aKbKc<-P5ASP5AO<D.730Q:@9ADP8<-7QJ

c > 0T _ E9AO7<O.1.

an (n ≥ 1)<-P5A)467/25A-gOAOP8

41hL<D7QJ=,373.o>k4jhaKbK3<D7QJ a2 + b2 + c

ab

<DP5A)417/2nADgOADP8OT Ð9Ð Ò ¦ih a = 0

KL2xE9AO7a3

418w7Q,2YJQAGt7QADJT _ E/0Q8?Ka 6= 0

Tz+O41:z4j.6<DP5. >LK41hb = 0

K2xE9AD7a4468B7Q,2BJQAGt7QADJT _ E/0Q8GK

b 6= 0TY¦U2ahi,/.j.U,-F8W417QJ30tm5254 3AO. >z2xE3<G2

an 6= 0KQhi,3P<D.j.

n ≥ 1TW¥%,3P5A)lQP5ADmn468CAO. >LK9A*3AOP~>k25ADP5: AsÆD468n258a<-7QJ418W7Q,37Qn£sAOP5,T

¨3>k2xE9A)P5A-mn0QP5P5AD7QmnAwFSAwQ7QJLK/hi,3P<O.1.n ≥ 2

Kan+2an − a2

n+1 = c = an+1an−1 − a2n

K

[<-7QJ=E9AO7QmxA/K

an+2 + an

an+1

=an+1 + an−1

an

T_ E9AOP5AOhi,3P5A/K3hi,3P<D.j.

n ≥ 1K-FSASE3<?3A

an+2 + an

an+1

=a3 + a1

a2

=b2+c

a+ a

b=

a2 + b2 + c

ab

K<-7QJ=E9AO7QmxA/K

an+2 =a2 + b2 + c

ab· an+1 − an

T¦ih

aKbK a2 + b2 + c

ab

<DP5AS417/2nADgOADP8?K92xE9AO7e<D7eAD<D8>z467QJ/0tm5254U,378CE9,-F8W2xE3<O2an

468<D7467/25A-gOAOPhi,3P<O.1.n ≥ 1

TR,373AOP8*AG.o>K<-8s8s0Q:=A2xE3<O2

an ∈ Khi,3P<O.1.n ≥ 1

T _ E9AD7a1K

a2 ∈ 46:zl9.j4UAO8S2xE3<O2aK

b ∈ T ¥%,3P5A-,O3ADPCKIFSAcE3<?3Ac ∈ K8x467QmnA

c = aa3 − b2T

_ E9AOP5AOhi,3P5A/K a2 + b2 + c

ab

418'P<O2546,379<D.¢TrbdP4j25A a2 + b2 + c

ab=

p

q

K;FSE9ADP5Ap ∈ K

q ∈ ∗ <-7QJ gcd(p, q) = 1TS¤3,3P

s ∈ ∗ K9FSAF)4j.1.lQP5,O3A'467QJ/0tm5254o3AG.o>d,37 s2xE9Ahi,/.1.6,-F)467QgYlQP5,3lQ,38s4o254U,37

P (s)Uqs | an

K/hi,3P<D.j.n ≥ s + 1

T¤3,3P <D.j.

k ≥ 1Kp2xE9AP5A-mn0QP5P5AD7QmnA

ak+2 =p

q· ak+1 − ak

g?4 3AD8pak+1

q= ak+2 + ak

K<-7QJ%E9AO7QmxA/Kq | (pak+1)

T +O467QmnAgcd(p, q) = 1

K4o2hi,/.1.6,-F82xE3<G2q | ak+1

T _ E/0Q8?Kq | an

hi,3P<O.1.n ≥ 2

K3<D7QJP (1)

4682nP0tAQT+O0QlQlQ,38*Aa2xE3<G2

P (s)E9,/.UJ38hi,3P8C,3:A

s ≥ 1T _ E9AO7

qs | anhi,3PL<D.j.

n ≥ s+1TR,3798x4UJ9ADP<-7>

k ≥ s + 1TI+O467QmnA

ak+2 =p

qak+1 − ak

K-FSAE3<G3Aak+2 + ak

qs=

pak+1

qs+1

T¨3>2xE9A417QJ30tm52546,37E->3lt,2xE9AO8s418GK

qs | ak<D7QJ

qs | ak+2 E9AO7QmxA/K

qs | (ak+2 +ak)T_ E9AOP5AOhi,3P5A/K

qs+1 | (pak+1)T+O467QmnA

gcd(p, qs+1) = 1K3FSA',9@/2<D417

qs+1 | ak+1T_ E/0Q8?K

qs+1 | anK3hi,3P<D.j.

n ≥ s + 2T _ E/418BlQP5,O3AD8

P (s + 1)<-7QJ=mx,3:zl9.6AG25AD8a2xE9A467QJ/0tm5254U,37T

XY,-F .6AG2s ≥ 1

@QAp<-Pn@34o2nP<-P~>T bcA«E3<G3Ac = an+2an − a2

n+1

T±¤3,3Pn = s + 1

K2xE/468>346AO.6J98c = as+3as+1 − a2

s+2

TIªB8x467QgP (s)

Kqs | as+1

Kqs | as+3

Kqs | as+2 ==⇒ q2s | (as+3as+1 − a2

s+2)

==⇒ q2s | cT

_ E9AOP5AOhi,3P5A/Kq2s ≤ c

hi,3P<D.j.s ≥ 1

T ~M©|

¦ih;FSA8s0Ql9lt,38CA

q > 1K2xE9AD7

lims→+∞

q2s = +∞ KtFSE/46msEcmx,37/2nP<-J346m5258)~M©|CT _ E/0Q8GKq = 1

<-7QJ a2 + b2 + c

ab

468W<D7z417/2nADgOADP©T _ E9A).6AD:z:=<)468Wl9P5,D3ADJTbcAw250QP57=7Q,-F 2n,'2xE9Aw46734o2546<O.tlQP5,9@/.UAO:pT+O0QlQlQ,38*A

f : ∗ −→ ∗ 468W<D7>h60Q7Qm52546,37=8x0tmsE'2xE3<O2 f(1) = 1<-7QJ

f(n + 2)f(n) = (f(n + 1))2 + 1997K hi,3P<O.1.

n ∈ ∗ TÇDAG2

b = f(2)T)+O417QmxA

f(n)418<-7=467/25A-gOAOPWhi,3PW<D.j.

nK9FSAkE3<?3A1hUP5,3: 2xE9AS.UAO:=:z<G|

2xE3<O2b ∈ <-7QJ 12 + b2 + 1997

1 · b∈ T _ E/0Q8?K b2 + 1998

b∈ T _ E9AD7 1998

b∈ K<-7QJk2xE9AOP5AOhi,3P5A/K

b | 1998T _ E/0Q8?K

f(2) = b418W<lt,38x4j254 3AJ34 3418*,3P,/h

1998T

R,373AOP8*AG.o>KL.UA?2b@QA<zlt,38x4j254 3AzJ/4o/468C,3P,/h

1998T WAOQ7QA

f : ∗ −→ @D>f(1) = 1

Kf(2) = b

K<-7QJf(n + 2) · f(n) = (f(n + 1))2 + 1997

Tw+O467QmnAf(1) 6= 0

<-7QJf(2) 6= 0

KtAO<msEf(n)

AsÆD468n258B<D7QJz468B7Q,37Qn£sAOP5,=U<D8B4172xE9ASlQP5,9,/h,/h2xE9A).UAO:=:z<G|sTaXY,-Fb ∈ K3<D7QJ

12 + b2 + 1997

1 · b=

b2 + 1998

b= b +

1998

b468I<D7417/2nADgOADPCK-8s417QmxAb | 1998

TW¨3>w2xE9AB.UAO:=:z<9Kf(n)

468I<D7417/2nADgOADPhi,3P<D.j.n ∈ ∗ T_ E/0Q8?K

f : ∗ −→ ∗ T \ 7cAO<-8>=467QJ/0tm5254U,37e8CE9,-F82xE3<G2 f(n) > 0hi,3PBA*3ADP~>

nK8s417QmxA

f(1) > 0<D7QJ

f(2) > 0T _ E/0Q8?K

f : ∗ −→ ∗ K<-7QJcFSAd,9@25<O467r<D7<J3:z418C8x4U@/.UA)8CA-¡/0tAD7QmnAQT_ E9AS<@9,D3A'J/468Cmn0Q8C8x4U,37=P5A*3AO<D.18W2xE3<G22xE9AS730Q:@9ADPW,/hh60Q7Qm52546,3798W2xE3<O2a8C<G25468xhj>@Q,2xE'mx,37QJ/4j2546,3798a14o|<D7QJ1414 |L417S2xE9AYl9P5,[email protected]: 418I2xE9AY8s<-:=AY<D8I2xE9AY730Q:=@QAOP,/hQlQ,38s46254o3AJ34 3418*,3P8I,/h

1998T+O467QmnA

1998 = 2 ·33 ·37 KO2xE9A730Q:=@QAOP,/h9lt,38x4j254 3AJ/4o/468C,3P8,/h1998

418(1 + 1) · (3 + 1) · (1 + 1) = 16

TB~¦U2a418f-7Q,-F7z2xE3<G22xE9AS730Q:=@QAOPB,/hJ34 3418*,3P8B,/hpa1

1 · · · par

r

418(a1 + 1) · · · (ar + 1)

T | T6<G|¤417QJc<O.1.;lQ,/.o>37Q,3:z41<D.18),/h;.UAO<-8n2YJ9A-gGP5ADA3KtF)4j2xEeP<O2546,379<D.mn,9AOymn4UAO7/258?K8s0tmsE'2xE3<O2

f(3√

3 +3√

9) = 3 +3√

3T

i@-| W,9AD82xE9AOP5ASAsÆD468n2<lQ,/.o>37Q,3:z41<D.3F)4o2xEk467/25A-gOAOPamn,9AOymn4UAO7/258W8s0tmsE'2xE3<O2f(

3√

3 +3√

9) = 3 +3√

3Å

3Ð DÌ [ 9ÏÌ n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì U<?|WÇDAG2a = 31/3 T %5*5 . ¦ih

αKβKγ ∈ 8x0tmsE'2xE3<O2

αa2 + βa + γ = 0K2xE9AO7

α = β = γ = 0T

Ð9Ð ÇDA?2αKβKγ ∈ 8x0tmsEk2xE3<G2

αa2 + βa + γ = 0T ~M©|

`+O0QlQlQ,38*A2xE3<G2

α 6= 0T+O417QmxA

a418<YP5,9,2,/h32xE9AY¡/0Q<J3P<O2546m

αx2 +βx+γ = 0KOFSA

:z0Q8n2E3<?3Aa =

−β ±√

∆

2α

K3FSE9ADP5A∆ = β2 − 4αγ ≥ 0

T)XY,25A2xE3<G2 √∆ /∈ U8x467QmnA

a /∈ |sT _ E9AD7−24α3 = β3 + 3β∆ ±

√∆(3β2 + ∆)

T¦U2;hi,/.1.6,-F82xE3<G2

3β2 +∆ = 0KOFSE/46msE.UAO<J382n,

β = ∆ = 0T _ E9AD7)FSAwgOA?2

α = 0K<=mn,37/2nP<J/4Um52546,37T _ E9ADP5AGhi,3P5A3K

α = 0KL<D7QJcA-¡/0Q<O2546,37c~M©|W@9A-mn,3:AO8

βa + γ = 0T+O467QmnA

a /∈ K-FSAJQADJ30tmnAw2xE3<O2 β = γ = 0K3<D7QJk2xE9A).6AD:z:=<)468Wl9P5,D3ADJT

ÇDAG2f ∈ [x]

8s0tmsEe2xE3<O2f(a + a2) = a + 3

Tc¦ihfE3<D8SJQADgGP5A-AeM/K2xE9AD7

f(x) = αx + βhi,3PI8C,3:A

αKβ ∈ T _ E9AD7LK α(a + a2) + β = a + 3

T _ E9AO7k2xE9A.UAO:=:z<B41:=l3.146AD82xE3<O2α = 0

<-7QJα = 1

K©FSE/4UmsEY468mn.6AD<DP5. >w41:=lQ,38C8x4U@/.UA9T _ E9ADP5AGhi,3P5A3Kfmn<D797Q,2E3<?3AJQADgGP5A-AzM9T'¦ih

fE3<D8wJ9A-gGP5ADA=/KQ2xE9AD7

f(x) = αx2 + βx + γhi,3P8*,3:=A

αKβKγ ∈ T _ E9AD7LK3hUP5,3: 2xE9A.6AD:z:=<3K

f(a + a2) = a + 3418AD¡3094 /<D.6AD7/22n,

α + β = 03α + β = 16α + γ = 3

T_ E9A)0Q7346¡30tA8C,/.1032546,37z,/ht2xE/418B8>38x25AD: 418

α = 12

Kβ = −1

2

K3<-7QJγ = 0

T¦U2hi,/.j.U,-F82xE3<O2t2xE9AOP5Aa418<a0Q734U¡/0tAWlt,/. >97Q,3:46<O.f,/hD.UAO<-8n2LJQADgGP5A-ABE3<?3417QgP<O2546,379<D.-mn,9AOymn4UAO7/2588s0tmsE'2xE3<O2

f( 3√

3 + 3√

9) = 3 + 3√

3K379<-:=AO. >

f(x) = 12x2 − 1

2xT

i@-| _ E9A<D798nFSAOPI468W7Q,T+O0QlQlQ,38*A/KLhi,3Pa2xE9Al30QPnlQ,38*Az,/hImn,37/2nP<J/4Um52546,37LKt2xE3<O2W2xE9AOP5AzAsÆD468n258

P ∈ [x]8s0tmsE2xE3<O2

P (a + a2) = a + 3TvÇDA?2

P (x) =n∑

i=0

αixi KFSE9AOP5A αi ∈ hi,3P

AD<-msEiT^bcA:0Q8x2'E3<G3A

n ≥ 3Ka467p346A?F ,/h,/0QPS8*,/.j03254U,37«25,¬6<G|sTÈXY,25Ad2xE3<G2

(a + a2)3 = 12 + 9(a + a2)T _ E9AO7

P (a + a2) =

2∑

i=0

αi(a + a2)i +(12 + 9(a + a2)

)n∑

i=3

αi(a + a2)i−3 T¦U2hi,/.j.U,-F82xE3<G2I2xE9Alt,/. >97Q,3:46<O.

Q(x) =

2∑

i=0

αixi + (12 + 9x)

n∑

i=3

αixi−3

8C<G25468xAO8Q(a + a2) = P (a + a2) = a + 3

K'FSE9ADP5AQ ∈

[x]F)4o2xE

deg Q(x) = deg P (x) − 2TzXY,-F FSAzm5<-7d<-l9l9. >2xE9A8C<D:AP5AO<-8C,373467Qg2n,

Q467l9.1<mnAk,/h

PT \ 7AD<D8>z467QJ/0tm5254U,37=.UAO<J38W2n,z<klQ,/.o>37Q,3:z41<D.

R,/h;J9A-gGP5ADA'<G2W:,38n2

2KF)4j2xEd417/2nADgOADPYmx,9AGyemn46AD7/28GKFSE/4UmsEc8s<O25418stAD8

R(a + a2) = a + 3Td¤/P5,3: U<?|FSA:z0Q8n2aE3<?3A

R(x) = 12x2 − 1

2xKDFSE/4UmsEJQ,9AO8W7Q,2E3<?3Aw467/25A-gOAOPmx,9AGyemn46AD7/28DT _ E/468468W<Smx,37/2nP<-J346m5254U,37T _ E9ASmx,37Qmn.j0Q8s46,37hi,/.j.U,-F8OT

/S T uQP5,D3A2xE3<O2*Khi,3P=A*3AOP~>^lQ,38s4o254o3Ap417/2nADgOADP

nKB2xE9ADP5A%AxÆ-418x28<rlt,38x4j254 3A467/25A-gOAOP

k8x0tmsE'2xE3<O2

19k − 97468BJ/4o/468x4U@/.UA@D>

2nT

3Ð DÌ [ 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï1ÎOË~Ð Í ~Ð-Î %- /Ð -Ï6Î 9Ï Ì Î D Ì-Ì Ì CkÌOÏ DÌ'Ë ÌÎGÐ 9Ë6Ï~ÐÍdÐ ¬D5 3Ð -Ï1Î bcA=F)4j.1.JQAGt7QAd@D>«467QJ/0tm5254U,37r<d8*AD¡30tAO7QmxAc,/halt,38x4j254 3A467/25A-gOAOP8 kn∞

n=18s0tmsE'2xE3<O22n | (19kn − 97)

K/hi,3P<O.1.n ∈ ∗ T+O467QmnA

198 − 97 ≡ 0 (mod 64)K9FSA'.UA?2

k1 = k2 = · · · = k6 = 8T'¤3,3P

n ≥ 6K98x0QlQlQ,38*AY2xE3<G2

kn ∈ ∗ 8s<O25418stAD8 2n | (19kn − 97)T$WAGt7QA

kn+1 =

kn41h

2n+1 | (19kn − 97)K

kn(2n−5 + 1)41h

2n+1 - (19kn − 97)T

bcAaF)41.j.lQP5,O3AB2xE3<G22n+1 | (19kn+1 −97)

T _ E/418468I,9@D346,/0Q84jhkn+1

418J9AOQ7QA-JS@D>2xE9AQP8x2m5<-8CAB467)2xE9ABhi,3P5:z09.1<Y<-@Q,O3AQT¦~7)2xE9A8*ADmx,37QJSmn<D8*A/K-8x467QmnA2n | (19kn −97)<-7QJ

2n+1 - (19kn − 97)KQFSA':z0Q8n2E3<G3A

19kn − 97 = 2n(2m + 1)hi,3PW8*,3:=A

m ∈ T _ E9AO7

19kn+1 − 97 = 19kn+1 − 19kn + 19kn − 97

=(

19kn·2n−5 − 1)

19kn + 2n(2m + 1)T

XY,-FKD467',3P5JQAOP25,SlQP5,O3A2xE3<G22n+1 | (19kn+1 −97)

K-4o2;418AD7Q,/0tgOE)2n,lQP5,O3A2xE3<O219kn·2n−5 − 1 = 2n(2x + 1)

K3hi,3P8C,3:Ax ∈

TbcA)8n25<DP2a@D>khU<m525,3P5417Qg8U19kn·2n−5 − 1 =

(19kn − 1

)·(19kn + 1

)·(192kn + 1

)·

·(

1922kn + 1)

· · ·(

192n−6kn + 1) T

+O467QmnA2n | (19kn − 97)

KIFSE9ADP5An ≥ 6

KFSAcgOAG2)2xE3<G232 | (19kn − 1)

<D7QJ64 - (19kn − 1)

T \ .18*,KQhi,3Pv = 1

K2K. . .

K9FSAE3<G3A19vkn + 1 ≡ 0 (mod 2)<-7QJ=68s417QmxA

kn468BA*3AO7|

19vkn + 1 ≡ 2 (mod 4)T _ E9AOP5AOhi,3P5A/K

19kn·2n−5 − 1 = 25 · 2 · 2 · · · 2︸︷︷︸

n−5 ·(2x + 1) = 2n(2x + 1)

Khi,3P8C,3:A

x ∈ T

bcA)E3<G3AwlQP5,O3A-JS2xE3<O22n+1 | (19kn+1 − 97)

T _ E9AY467QJ/0tm5254U,37k418Wmn,3:=l3.UA?2nA9T

bcAk7QAsÆO2W250QPn72n,8*,/.j03254U,3798Y@D>dP5AO<J9ADP82n,lQP5,9@/.UAO:=8w,/h2xE9A _ 0QPxf-A*> _ AD<D:+-AO.6A-m52546,37 DÆD<D:z4179<O2546,37hi,3P2xE9A38 ¦5¥%V MDNN/ MBU3M ` M ` N /TÄ TS¦~7e<'2nP46<D7Qg?.UA

ABCFSE/4UmsEeE3<D8Y<kP546gOE2<D7Qg?.UA'<O2

AKt.6AG2

HJQAO7Q,2nAS2xE9Ahi,9,2B,/hL2xE9A'<O.j254o250tJQA'@QAG.U,37Qg?417Qg)25,2xE9AkE->3lt,25AD730Q8CAQT+-E9,-FÈ2xE3<O22xE9AS8s0Q: ,/hL2xE9AP<J/414D,/hO2xE9A417Qmn46P5mn.6AD8,/hG2xE9AI2nP541<-7Qg?.6AD8

ABCKABH

KG<D7QJAHC

468AD¡30Q<O.?2n, |AH| T

3Ð DÌ [ GÌ Í QÌ aÍ ÏÌ ©Ê Ì DÍLÌ -5 Í Ì Ð Q dÌ

ÎsÎsÏ 3Ë 5 Î9Ð -5 Í Ì SÐ aÍLÐDÍ Ð QÌ *Ê Í Ì Ï 5 Ì" Ë Ï Ì Ð/ÌGÍ -5 Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ jÏ ¢ËÐÍÐ ~Ì/Ì Ï1ÎOË~Ð D Ì-Ð \L Ì[ Í QÌ?Í ~Ð-Î ¬D5 3Ð -Ï1Î 9Ï Ì Î GD Ì-Ì Ì CkÌOÏ DÌ'Ë ÌÎGÐ 9Ë6Ï~ÐÍdÐ ¬D5 3Ð -Ï1Î _ P541<-7Qg?.6AD8

HBAKHAC

KABC

<-P5A8s41:z4j.6<DP?TÇDAG2r1Kr2Kr@QAw2xE9ASP<-J34j4L,/h2xE9A)467Qmn41P5mn.UAO8B,/ht2xE9AO8*AY2nP541<-7Qg?.6AD8?K3P5AO8ClQA-m5254 3AO. >T _ E9AO7

r1

AB=

r2

AC=

r

BC=

r1 + r2 + r

AB + AC + BC

T_ E/0Q8?Kr1 + r2 + r =

r(AB + AC + BC)

BC=

2[ABC]

BC=

BC · AH

BC= AH

TT _ E9A8CA-¡/0tAD7QmnAD8 an∞

n=1

K bn∞n=1

<DP5ApJ9AOQ7QA-Jr2xE3P5,/0tgOEa1 = α

Kb1 = β

KL<D7QJan+1 = αan − βbn

Kbn+1 = βan + αbn

hi,3PB<D.j.n ≥ 1

T),-F:=<D7>zlQ<O46P8(α, β)

,/hLP5AD<O.730Q:=@QAOP8B<-P5AY2xE9ADP5A8x0tmsE'2xE3<O2a1997 = b1

<D7QJb1997 = a1

Å3Ð DÌ [ Ð Q dÌ ÎCÎxÏ 3Ë 5 Î9Ð Í Ì Ï 5 Ì" Ë Ï Ì

Ð/ÌGÍ -5 Í Ì 9ÏÌ n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì 9 Ï6ÎGËÐ Ì 5 Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï1ÎOË~Ð Í Ë 9 Í ÎxÏ Î 9Ð-Î Ë ÌGÍ3Î D ÌDÌ Ì CkÌOÏ DÌ'Ë ÌÎGÐ 9Ë6Ï~ÐÍ [ Ë Ï Ì _ E9AOP5Ak<-P5A

19998s0tmsElQ<O46P8?K79<D:AG.o>

(0, 0)<-7QJ=2xE9A'lQ<O46P8

(cos θk, sin θk)FSE9ADP5Aθk =

π

3996+

2kπ

1998

k = 0

K1K. . .

K1997

|T_ ,)l9P5,D3Aa2xE/468P5AO8s09.o2*K?FSABP5AD:z<-Pnf2xE3<G2*KOhi,3P<D.j.

n ≥ 1KG2xE9Amn,3:=l3.UAxÆ730Q:@9ADP

an+1 + ibn+1468Bg?4 3AD7=@D>

an+1 + ibn+1 = (α + iβ)(an + ibn)K

8*,d2xE3<O2an + ibn = (α + iβ)n U8x467QmnA

a1 + ib1 = α + iβ)TcbcAF)41.j.E3<?3A

a1997 = b1<D7QJ

b1997 = a141h<D7QJe,373. >z4jh

(α + iβ)1997 = i(α − iβ)TYÇDAG225417Qg

z = α + iβK;FSAcE3<G3A2xE9AdA-¡/0Q<O2546,37

z1997 = izKI25,r@QAd8C,/.o3ADJ«hi,3P

z ∈ T\ 7 ,9@D346,/0Q8z8*,/.j03254U,37%418z = 0

T \ 7> 7Q,37Q5£sADP5,%8*,/.j03254U,37z418z7QA-mnAD8s8C<DP54j.o>¬,/h:,9J/09.10Q8

1KL417=FSE/4UmsEdmn<D8*A

z = 1/z<-7QJ=FSAE3<G3A

z1998 = iTS+O467QmnA'2xE9A8*,/.j0t254U,3798B,/h

z1998 = i<DP5Aw2xE9A

1998mx,3:zl9.6AsÆS730Q:@9ADP8

exp(i(

π3996

+ 2kπ1998

)) F)4o2xEk = 0

K1K. . .

K1997

KFSAE3<?3Aw2xE9A<D797Q,/0Q7QmxADJ=P5AO8s09.o2?T T _ E9ASA-J9gOA

AE,/hL<'mx,373AxÆklQAD7/2<gO,37

ABCDEFSE9,38CAY3AOP2546mxAO8W.146AS,372xE9A0Q734j2mn41P5mn.UAlQ<D8C8CAD82xE3P5,/0tgOE2xE9AWmxAO7/2nP5AW,/hD2xE/418;mn41P5mn.UA9TI¦ih |AB| = a

K |BC| = bK

|CD| = cK |DE| = d

<D7QJab = cd = 1

4

Kmx,3:zl90325A |AC| + |CE| 467z2nAOPn:z8Y,/haKbKcKdT

-N3Ð DÌ [ Ë Q Í ÎsÏ Î 3Ð-Î Ë ÌGÍ3Î D ÌDÌ Ì Í D Ì-Ð \L Ì[ Í

QÌ?Í CkÌ OÏ DÌ Í Î ÏËÌ ÇDAG2 |AC| = xK |CE| = y

K |AD| = pK |BE| = q

T _ E9A=<-7Qg?.6AD8ABE

KACE

KADE

<-P5AdAD<-msE90 K8C,2xE3<G2 a2 + q2 = x2 + y2 = p2 + d2 = 4

T¨3>¬u32n,/.6AD:'>LÓ 8 _ E9AD,3P5AD:dKdx + 2c = py

FSE9AO7QmxA/Kd2x2 + x + 4c2 = p2y2 T_ E9AOP5AOhi,3P5A/K

x = (4 − d2)y2 − 4c2 − d2x2 = 4y2 − 4c2 − d2(x2 + y2)

= 4y2 − 4c2 − 4d2 T\ 79<D.6,9gO,/0Q8s. >LK-2xE9AP5AG.6<G254U,37ay + 2b = qx

.6AD<-J9825,y = 4x2 − 4a2 − 4b2 TR,3798CA-¡/0tAD7/25. >LK

x + y = 16 − 4(a2 + b2 + c2 + d2)T

S TpuQP5,O3A2xE3<G2*Khi,3PYAD<-msEl9P541:Az730Q:@9ADPp ≥ 7

K2xE9AOP5AAsÆD468n258)<=lt,38x4j254 3A467/25A-gOAOPn<D7QJS467/25A-gOAOP8

x1Kx2K. . .

KxnKy1Ky2K. . .

KynFSE/4UmsES<DP5AY7Q,2IJ34 [email protected]@D>

pK38x0tmsEk2xE3<G2

x21 + y2

1 ≡ x22 (mod p)

Kx2

2 + y22 ≡ x2

3 (mod p)K

TTTx2

n−1 + y2n−1 ≡ x2

n (mod p)K

x2n + y2

n ≡ x21 (mod p)

T3Ð DÌ [ Ð Q dÌ ÎCÎxÏ 3Ë 5 Î9Ð Í Ì Ï 5 Ì" Ë Ï Ì

Ð/ÌGÍ -5 Í Ì Í 9 Ï6ÎGËÐ Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï6ÎGËÐ CkÌOÏ DÌ 5 Ì[ ÎBÎGÐ 9Ë6Ï~ÐÍ dÐ-Ï Ì [zË ÌÌ-ÏËÐ Î bcASmn,3798s46JQAOP2¢FS,zmn<D8*AO8HU

p ≡ 1 (mod 4),3P

p ≡ 3 (mod 4)T

U<?|p ≡ 3 (mod 4)

T _ E9AO7p = 4k + 3

hi,3P8*,3:=Aw467/25A-gOAOPkjFSE9AOP5A

k > 08s417QmxAp ≥ 7

|nK9<D7QJ'FSA,9@98CADP~3AY2xE3<O212 + k2 ≡ (k + 2)2 (mod p)

T ~M©|+-AG225417Qg

x1 = 1Ky1 = k

K<D7QJx2 = k + 2

K3FSAkE3<?3Ax2

1 + y21 ≡ x2

2 (mod p)T+O0QlQlQ,38*A7Q,-F 2xE3<O2LFSA<-P5Ag?4 3AD7

x2i +y2

i ≡ x2i+1 (mod p)

hi,3P;8C,3:Ai ≥ 1

TbcAF)41.j.mx,3798n2nP50tm52L417/2nADgOADP8yi+1

<D7QJxi+2

8x0tmsE2xE3<O2x2

i+1+y2i+1 ≡ x2

i+2 (mod p)TbcA)QP8x2:z09.o2546l3.o>z~M©|I@D>

x2i+1

25,S>346AO.6Jx2

i+1 + k2x2i+1 ≡ (k + 2)x2

i+1 (mod p)T

_ E9AO7kFSAmsE9,9,38*Ayi+1 ≡ kxi+1 (mod p)

<D7QJxi+2 ≡ (k + 2)xi+1 (mod p)

T+O467QmnA3K/hi,3PI<-7>l9P541:A

pK-2xE9ADP5A)<DP5A)<)t734o2nAw730Q:@9ADPa,/hL¡/0Q<J3P<O2546maP5AD8x4UJ/0tAD8?KA*3AD7/250Q<O.1. >pFSA=F)4j.1.aE3<G3A

xj ≡ xi (mod p)hi,3P8C,3:A

j > iT bcAdmn<D7«2xE9AD7P5A-n.6<-@QAG.

xi<D8

x1<D7QJ

yi<-8

y1K/<-7QJ=@9A-g?4172xE9A)lQP5,9mnAD8s82xE9ADP5A9T

-N ¤3,3PAsÆD<D:=l3.UA/K/41h

p = 13K2xE9AO7

p = 4k + 1hi,3P

k = 3T;bcA)8x2<-P¢2F)4o2xE

12 + 92 ≡ 112 (mod 13)K

<-7QJzl9P5,9mxADA-Jk25,zgOA?2I2xE9A)hi,/.1.6,-F)467Qgwmn41P5mn094j23U112 + 82 ≡ 42 (mod 13)

K42 + 102 ≡ 52 (mod 13)

K52 + 62 ≡ 32 (mod 13)

K32 + 12 ≡ 72 (mod 13)

K72 + 112 ≡ 122 ≡ 12 (mod 13)

Ti@-|

p ≡ 1 (mod 4)T _ E9AD7

p = 4k + 1hi,3P8C,3:Aw417/2nADgOADP

kjFSE9AOP5A

k > 18s417QmxAp ≥ 7

|nK9<D7QJ'FSA,9@98CADP~3AY2xE3<O212 + (3k)2 ≡ (3k + 2)2 (mod p)

T ~©|Va0QPl9P5,9mxAO8C8a418W8x46:41.1<-P;2n,kl9<-P¢2aU<?|xK9,373. >'2xE/4182546:=AFSA):09.j2541l9. >z~©|I@D>

x2i+1

<D7QJmsE9,9,38*Ayi+1 ≡ 3kxi+1 (mod p)

<D7QJxi+2 ≡ (3k + 2)xi+1 (mod p)

T TIH4 3AD7z<D7467/25A-gOAOP

n ≥ 2K/t7QJk2xE9A):z417346:z<D.3/<O.10tA,/h

x51

x2 + x3 + · · · + xn

+x5

2

x1 + x3 + · · · + xn

+ · · · +x5

n

x1 + x2 + · · · + xn−18s0t@ A-m52a2n,x2

1 + x22 + · · · + x2

n = 1K9FSE9ADP5A

x1Kx2K. . .

Kxn

<DP5AklQ,38s4o254o3A'P5AO<D.730Q:@9ADP8OT3Ð DÌ [ Ð Q dÌ ÎCÎxÏ 3Ë 5 Î9Ð Í Ì Ï 5 Ì" Ë Ï Ì

Ð/ÌGÍ -5 Í Ì 9ÏÌ n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì 9 Ï6ÎGËÐ Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï1ÎOË~Ð n [ -ÏUÍ Í3Ï DÌ ÎsÏË#[ Ð 9Ì Ë É dÐÍLËÐÍ Í Ð-Î %- /Ð -Ï6Î 9Ï Ì Î D ÌDÌ Ì CkÌ$GÏDÌ)Ë ÌBÎGÐ Ë6ÏÐÍeÐ Ð Í3Î Ë~Ì©Ï6Í

¥%,3P5AgOAO7QADP<O.1. >LK?FSAF)4j.1.-Q7QJw2xE9AB:z417346:z<D.O/<O.10tA,/h n∑

i=1

(xp

i

S − xri

) 8s0t@ A-m522n,

xs1 + xs

2 + · · · + xsn = 1

K3FSE9ADP5AS = xr

1 + xr2 + · · · + xr

n

<-7QJpKrKs<-P5Alt,38x4j254 3AYP5AD<O.Q730Q:=@QAOP8W8x0tmsES2xE3<G2

r ≤ s ≤ p2

T _ E9A)g?4o3AO7klQP5,9@/.UAO:468I2xE9Aw8sltADmn46<O.mn<D8*Ap = 5

Kr = 1

Ks = 2

T\ lQl3.o>/467Qg2xE9ASR<D0tmsE->9C+-msE-F<DP£¦~7QA-¡/0Q<D.j4j2~>LK-FSAE3<G3A

(n∑

i=1

(S − xri )

)(n∑

i=1

xpi

S − xri

)

≥(

n∑

i=1

xp/2i

)2 KF)4j2xEA-¡/0Q<D.j4j2~>c41h<D7QJp,373.o>c4jh2xE9ADP5AAxÆ-418x28<=lt,38x4j254 3A=P5AO<D.730Q:@9ADP

λ8x0tmsEd2xE3<G2

S − xri = λx

p/2i

T

-NM¨3>k2xE9Au3,-FSADPn?¥%AO<-7e¦~7QAD¡30Q<O.14o2>=68s417QmxA

p/2 ≥ s|nK

(n∑

i=1

xp/2i

)2

= n2

(n∑

i=1

xp/2i

n

)2

≥ n2

(n∑

i=1

xsi

n

)p/s

=n2

np/s

KF)4j2xEzAD¡30Q<O.14o2>k41hL<D7QJ=,373. >4jh

p = 2s,3P

x1 = x2 = · · · = xn =1

n1/s

T \ .18*,Kn∑

i=1

(S − xri ) = (n − 1)

n∑

i=1

xri = n(n − 1)

n∑

i=1

xri

n

≤ n(n − 1)

(n∑

i=1

xsi

n

)r/s

=n(n − 1)

nr/s

K@D>d2xE9Au3,-FSAOPx©¥%AD<D7r¦~7QA-¡/0Q<D.j4j2~>pU8x467QmnA

s ≥ r|CTd)ADP5A/KAD¡30Q<O.14o2>p,9mnmn0QP8)4jh<D7QJ

,373.o>4jhr = s

,3Px1 = x2 = · · · = xn =

1

n1/s

T+O467QmnA n∑

i=1

(S − xri ) > 0

KQFSAE3<G3A

n∑

i=1

xpi

S − xri

≥

(n∑

i=1

xp/2i

)2

n∑

i=1

(S − xri )

≥

n2

np/s

n(n − 1)

nr/s

=n(r+s−p)/s

n − 1

T

9¡/0Q<D.j4j2~>d,9mxmn0QP8WFSE9AD7x1 = x2 = · · · = xn =

1

n1/s

T _ E9AOP5AOhi,3P5A/Kt2xE9AkAsÆ-lQP5AO8*8s46,37z,37S2xE9A)P4UgOE2I8s46JQAw<-@Q,O3Aw4182xE9Aw:z417346:z<D./<D.j0tA,/h92xE9Aw8s0Q:±,37'2xE9AY.6AOh12I8s46JQA/K8s0t@ A-m52I2n,xs

1 + xs2 + · · · + xs

n = 1T

+-AG225417Qgp = 5

Kr = 1

K3<D7QJs = 2

KFSAwQ7QJk2xE3<O2I2xE9A:467341:=<O.3/<D.j0tA)4172xE9Ag?4o3AO7=l9P5,[email protected]: 468 1

n(n − 1)

T

bcAk7Q,-F 250QP572n,P5AO<J9ADP8?ÓQ8*,/.j03254U,3798Y,/h;l9P5,[email protected]:z8w,/h2xE9ARLE/41.6AD<D7«¥r<G2xE9A-:=<G254Um5<D.Va.o>3:=l346<-J98MDNN [ N/ 5 M<U3M ` N M 3TÄ TH;4o3AO7e2xE3P5A-Az8n2nP<D46gOE2.1417QAD8467c<zl9.1<-7QA/K2xE3<O2Ymn,37Qmn0QP<O2lt,/417/2OKmx,37Q8s46JQAOPL2xE9AW2xE3P5ADAmx,3798CA-mn03254 3A<D7Qg?.UAO8@9AG2¢FSADAD7w2xE9AD: oFSE/4UmsEKD79<O250QP<O.1. >LKD<-JQJ)0Ql)25,

180 |sTaÇDA?2 P@QAW<BlQ,/467/24672xE9ABl3.6<D7QAW7Q,2,37w<-7>),/hD2xE9AD8CAa.1417QAD8?KO<D7QJw.6AG2

AKBKC@QAB2xE9AhiADAG2I,/h/2xE9AYlQADP5ltAO7QJ346mn09.6<DP8aJ9P<?F7'hUP5,3:

P2n,w2xE9A2xE3P5ADA.j467QAO8DT+-E9,-F 2xE3<G22xE9A)467/25ADP579<D.t<-7Qg?.6AD8B,/h 4ABC

<-P5AA-¡/0Q<D.92n,'2xE9,38CA@QA?2¢FSA-AO72xE9Ag?4o3AO7.1417QAD8OT

-N3Ð DÌ [ 9 Ï6ÎGËÐ Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï1ÎOË~Ð Í

D Ì-Ð \L Ì[ Í QÌ?Í CkÌ OÏ DÌ Í Î ÏËÌ

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................B

A

O

P

C

........................................................................

........................................................................

. . . . . . . . . . . . . . . . . .......................................................

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

_ E9AdlQ,/467/28OK

CK

PK

AK

B.146Ac,37p2xE9Acmn46P5mn.6AcE3<G/467Qg

OP<-8kJ341<-:=AG25ADP©T_ E9AOP5AOhi,3P5A/K

∠ABC = ∠AOC<-7QJ

∠ACB = ∠AOBT \ 8<«mx,3798CA-¡/0tAD7QmnA3K

∠BAC468BAD¡30Q<O.Q25,'2xE9Aw2xE/41P5J=mn,3798*ADmn03254o3A)<-7Qg?.6A<G2

OT

T

..............................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................... ................................................................

A B

CD

E

F G

H

.....

.....

.....

.....

........ ..... ..... ..... ..... ..... ..... .....

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ABCDEFGH468k<pmn0t@QA,/hA-J9gOA

2TÈÇDAG2

M@9A2xE9Aw:4UJ9slQ,/467/2I,/hBC

<D7QJN2xE9AY:z46JQxlt,/417/2,/h

EFTIRL,3:=l3032nA2xE9A)<DP5AD<,/h92xE9A)¡30Q<-J9P41.1<O25ADP<O.

AMHNT

3Ð DÌ [ Ð9Ì Ë QÏ jÏUÍ3Î-Ï 3Ë ÌdÐÍË 9ÏÌ n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ Í Ì Q Ï1ÎOË~Ð Ì ~Ì[ jÏ ¢ËÐÍ Ð Ì 3Ì Ï6ÎGËÐ Í D Ì-Ð \L Ì[ Í QÌGÍ CkÌ GÏDÌSË ÌYÎ?Ð 3Ë6ÏÐÍeÐ QÏ jÏUÍ3Î-Ï +O467QmnAemn.UAO<-P.o> −−→

NH =−−→AM

K2xE9A=hi,/0QPwlQ,/467/28AKMKHK<-7QJ

N<-P5Amx,9l9.1<-79<DP?T)+O417QmxA 4ENH

K 4FNAK 4BMA

Kt<-7QJ 4CMH<-P5AkP4UgOE2a2nP46<D7Qg?.UAO8FSE9,38*A8x4UJ9AD8BE3<G3A).UAO7Qg©2xE38

1<D7QJ

2KFSA8CA-Aw2xE3<G22xE9AO8*Aw2nP46<D7Qg?.UAO8B<-P5ASmn,37QgGP50tAO7/2<-7QJ

AM = MH = HN = NAT a0Q<J3P54j.6<G2nAOP<D.

NHMA468I2xE/0Q8a<)PxE9,3:=@30Q8OT

ÇDAG20Q8at7QJ'2xE9A).6AD7Qg©2xEz,/h4j28BJ341<gO,379<O.68AH

<-7QJNM

T¨3>¬<DlQl3.o>/467Qg2xE9A%u/>/2xE3<-gO,3P5AD<D7 _ E9A-,3P5AO:q417¬P546gOE22nP541<-7Qg?.6A 4EHC

KIFSAE3<G3AEC =

√EH2 + HC2 = 2

√2T^XY,2546mn467Qgz2xE3<G2

NM = ECK;FSAcgOA?2

NM = 2√

2TW¨3>S2xE9AY8C<D:AYP5AD<D8*,373417QgKDFSAw8CA-A2xE3<O2

AC = 2√

2T \ l9l9. >3417Qg2xE9Au/>/2xE3<-gO,3P5AD<D7 _ E9A-,3P5AO:±417 4ACH

KDFSASgOA?2AH =

√AC2 + HC2 = 2

√3T

XY,-FK2xE9Ad<DP5AD<c,/hW<cPnE9,3:@/0Q8'468kE3<D.jh2xE9Adl9P5,9J30tm52',/h2xE9A.UAO7Qg©2xE38,/ha4j28J341<gO,379<O.68OTAO7QmxA/KFSA,9@25<O4672√

6<D82xE9A<DP5AD<QT

-N ]¯T^H;4o3AO7%<d2nP<DltA?£s,/4UJ

ABCDKFSE9ADP5A

AB<-7QJ

DC<-P5AdlQ<DP<D.j.UAG.K<D7QJ

AD = DC = AB/2K9JQA?2nAOPn:467QA

∠ACBT

3Ð DÌ [ Ð9Ì Ë QÏ 1Ï6Í9Î%-Ï 3Ë ÌdÐÍË Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï1ÎOË~Ð Ë 9 Í ÎxÏ Î 3Ð-Î Ë ÌGÍ3Î D Ì-Ì Ì D Ì-Ð \L Ì[ Í QÌ?Í aÉ D C Í CwÏ Ï D ÏÌ Í3Ï DÌ ÎsÏË#[ C ËÌ Ð9Ð ,] CkÌ OÏ DÌ'Ë ÌÎGÐ 9Ë6Ï~ÐÍdÐ QÏ 1Ï6Í9Î%-Ï ÇDAG2

M@9A)2xE9A:z46JQxlt,/417/2W,/h

ABK9<-7QJz.6AG2

E@9A)2xE9ASlQ,/467/2W,/h467/25ADP8CA-m52546,37,/h

AD<-7QJ

BCT=+O417QmxA

DC = AB/2<D7QJ

DC‖ABKFSAz8*ADA=@D> _ E3<D.6AD8Y2xE3<G2

D418a2xE9A:z46JQxlt,/417/2W,/h

AE<-7QJ

C418a2xE9A:z46JQxlt,/417/2W,/h

BET _ E/0Q8GK

MKDK9<D7QJ

C<DP5A2xE9Az:4UJ9slQ,/467/28S,/h;2xE9A=8x4UJ9AD8),/h 4ABE

Tc¨3>e2xE9Ac¥«46JQxlt,/417/2 _ E9A-,3P5AO:cKAMCD

418B<l9<-P<O.1.6AO.6,9gGP<-:±<-7QJMC = AD = AB/2

T¦~7 4ABC

KMC

418Y<:ADJ341<-7=FSE/46msE468wE3<O.1h2xE9A'.UAO7Qg©2xEd,/h2xE9A8x4UJ9AAB

TAO7QmxA/K 4ABCE3<-8W<P546gOE2a<D7Qg?.UA)<O2

CT _ E3<G2468?K

∠ACB = 90 T Tz¦~7c<=mn46P5mn.6Az,/hIP<-J34j0Q8

1<DP5A=J3P<GF7c8x4 Æ=A-¡/0Q<D.;<DP5mn8),/hImn46P5mn.6AD8?KP<-J34j0Q8SM/Kmn03225467Qg2xE9AS,3P4Ug?4179<D.mn41P5mn.UA)<-8a417k2xE9A)tg?0QP5AQTR<D.6mn09.6<G2nAY2xE9A)8*E3<-JQADJz<-P5AO<tT

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....................................................................................................................

A

B

O

3Ð DÌ [ Ð9Ì Ë QÏ 1Ï6Í9Î%-Ï 3Ë ÌdÐÍË Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï6ÎGËÐ Í É D C Í CwÏ Ï D ÏÌ Í3Ï DÌ ÎsÏË#[ C Ë~Ì ~Ð9Ð ,] CkÌ OÎGÌQÏ jÏUÍ3Î-Ï ÎWÎGÐ 9Ë6Ï~ÐÍ +O467QmnA)@Q,2xE'<-P5m58al9<-8s8s417QgB2xE3P5,/0tgOE

A<-7QJ

B<-P5Aw,/htP<J/410Q8

1K-2xE9A*>k<DP5Aw8>3::A?2nP546m<@9,/032I2xE9Aw.j467QA

ABT)AD7QmnA3K2xE9Aw<-P5AO<SAD7Qmn.6,38*ADJ@D>'2xE9AD8CAY2¢FS,k<DP5mn8468Wmn032467E3<O.1h@D>k2xE9A.j467QA8CA-gG:=AD7/2

ABTW¦ihL.1417QA8*ADgG:AO7/258B<-P5ASJ9P<?F72xE3P5,/0tgOE=<O.1.93AOPx254UmnAD8Y,/hL2xE9AE9AxÆD<gO,379<O.8n25<DP*K3FSAkgOAG2W<P5ADg?09.6<DPE9AxÆD<gO,3741798*mxP4U@9A-J=467=2xE9Akmn41P5mn.UA9T¤/P5,3: 2xE/468FSAm5<-7SAD<D8s4j.o>mn<O.Umn09.1<O25AB2xE9A<DP5AD<Y,/h/2xE9AW25,25<O.DFSE/4j25A@Q,3P5J9ADPCKDhi,3P4j2F)41.j.@QAw2¢F)46mxAY2xE9A)<-P5AO<'@9AG2¢FSADAD7k2xE9ASmn41P5mn.UA)<D7QJk2xE9ASE9AxÆD<gO,37T

_ E9AcE9AxÆD<gO,37LÓ 8k<-P5AO<d468'8x4 Æe2541:AO8S2xE9Ae<-P5AO<,/hW<-7%AD¡3094j.6<G2nAOP<D.2nP46<D7Qg?.UAd,/h8s46JQA1K79<-:=AO. >

6 ·√

34

= 3√

32

T _ E9Admn46P5mn.6AdE3<-8'<DP5AD<πr2 = π

T AO7QmxA/KI2xE9AE9AsÆD<-gO,379<D.8n25<DPE3<-8W<-P5AO<π − 2

(

π − 3√

32

)

= 3√

3 − πT

_ E/0Q8?K-8x467QmnAW2xE9A8CE3<J9A-J<DP5AD<YE3<D8,373. >5/6

2xE9A<-P5AO<Y,/h2xE9AE9AxÆD<gO,379<O./8x2<-PCK4j28B<DP5AD<)468 56(3

√3 − π)

T

-N [S T ¦~7 P4UgOE22nP46<D7Qg?.UA

ABC2xE9Ar<D.o254j250tJ9A

hc = CD418dJ3P<GF7 25, 2xE9AE->9lQ,2nAO730Q8*A

ABT%ÇDA?2

PKP1KP2

@QA2xE9AP<J/414,/hI2xE9Aemn41P5mn.UAO8)46798CmxP546@QADJ4172xE9A2nP541<-7Qg?.6AD8ABC

KADC

KBCD

K/P5AD8sltADm5254o3AG.o>TI+-E9,-F2xE3<O2P + P1 + P2 = hc

T3Ð 9Ë6Ï~ÐÍ ÍÐ-ÎGÌ Ë6ÏÐÍ Ð GÌ Í QÌ aÍ ÏÌ ©Ê Ì DÍLÌ -5 Í Ì Ï 5 Ì" Ë Ï Ì ;Ð3Ì?Í Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ jÏ ¢ËÐÍ Ð ~Ì/Ì

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62xE3<G2W<DP5A.6AD8s82xE3<-7

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3Ð DÌ [ Ð9Ì Ë QÏ jÏUÍ3Î-Ï 3Ë ÌdÐÍË 9ÏÌ n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï6ÎGËÐ D Ì-Ð \L Ì[ Í QÌ?Í Í É - C Í CwÏ Ï D Ï~Ì Í3Ï DÌ ÎsÏË#[ C Ë~Ì ~Ð9Ð ,] CkÌ GÏDÌ'Ë ÌYÎ?Ð 3Ë6ÏÐÍeÐ Í _ E9A):09.j2541l9.6AD80Q7QJ9ADPmn,3798s46JQAOP<O2546,37<-P5A2xE9A)730Q:=@QAOP8

6n1 ≤ n ≤ 166

|sT_ E9AG46PLlQP5,9J/0tm52L468P = 2166 ·3166 ·(166!) T _ E9AWlQP46:=AWhU<-m52n,3P4j£x<G254U,37),/h P mn,37/25<O46798AsÆD<-m525.o>

40 5Ó 8?K38s417QmxA

∞∑

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5n

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=

⌊166

5

⌋

+

⌊166

25

⌋

+

⌊166

125

⌋

= 33 + 6 + 1 = 40T

_ E9AOP5AOhi,3P5A/KP468)J34 [email protected]@D>

1040K;@30327Q,2w@D>

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PAD7QJ38F)4j2xE

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x = 1 1 1 . . . 1︸︷︷︸

n

T+-E9,-F2xE3<O2*K/4jh

x418W<Sl9P541:A/K2xE9AD7

n468W<lQP46:=AQT

3Ð DÌ [ 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï1ÎOË~Ð Í Ë 9 Í ÎxÏ Î 9Ð-Î Ë ÌGÍ3Î D ÌDÌ Ì CkÌDÎ?Ì ~Ì[ Î ÏËÌ XY,25Aw2xE3<G2x = 1 1 1 · · · 1

︸︷︷︸

n = 1 + 10 + · · · + 10n−1 =

10n − 1

10 − 1=

1

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TXY,-FK/4jh

n468Bmn,3:=lQ,38s4o2nA/K38s<G>

n = n1n2jFSE9AOP5A

n1Kn2 > 1

|nK2xE9AD7x =

10n − 1

9=

(10n1 − 1

9

)

(1 + 10n1 + 102n1 + · · · + 10(n2−1)n1)T

+O467QmnA 10n1 − 1

9

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x418Bmx,3:zlt,38x4j25AQT

AO7QmxA)4jhx418WlQP46:=A3K

n:0Q8x2a@9Al9P541:A9T

-N TWÇDAG2

x@QA)<S730Q:=@QAOP8s0tmsE'2xE3<G2

x +1

x= −1

TR,3:zl90325A

x1994 +−1

x1994

T3Ð DÌ [ Ð9Ì Ë QÏ jÏUÍ3Î-Ï 9Ë ÌdÐÍLË 9ÏÌ n Ì Ð Í9Î ËÌ*ÏUÍ tÐÍ ËÐDÏ1ÎGÌ Í Ì 9 Ï6ÎGËÐ Ì 5 Ì[ jÏ ¢ËÐÍ Ð Ì 3Ì Ï1ÎOË~Ð Ë 9 Í ÎxÏ Î

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TI+O417QmxA1/j = j

<D7QJj2 = j

<D7QJj3 = 1

K-FSAJQADJ30tmnAw2xE3<O2• ¦ih x = j

K2xE9AD7x1994 − 1

x1994= j2 − 1

j2= j − j = −i

√3T

• ¦ih x = jK-2xE9AD7

x1994 − 1

x1994= −i

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√3T

_ E9AO7x1994 − 1

x1994= ±i

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TWÇDAG2ABCD

@QA)<-7m × n

P5ADm525<D7Qg?.UA/K-F)4j2xEmKn ∈

TIR,3798x4UJ9ADP<P<G>,/hL.146gOE22xE3<G2a8x2<-P¢258WhUP5,3:AK9418P5A3ADm52nADJ=<O2a<D7<D7Qg?.UAS,/h

45 ,37=<-7Q,2xE9AOPa8s46JQAS,/h2xE9AP5A-m52<-7Qg?.6A3K3<D7QJ=gO,9AD8B,37zP5A3ADm525467QgY417k2xE/468F<G>TU<?|I+-E9,-F2xE3<O2I2xE9A)P<G>SF)41.j.Qt79<O.1. >zE/4j2<Y3ADP¢2nAxÆTi@-|I+O0QlQlQ,38*A

m<-7QJ

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1T WAG25ADP5:z417QAw2xE9A730Q:@9ADPa,/hLP5AtA-m52546,3798a0Q7QJQAOP5gO,37QA'@D>'2xE9AP<?>=@QAGhi,3P5A)4o2aE/4o258W<w3AOP25AsÆtT

3Ð DÌ [ Ð9Ì Ë QÏ jÏUÍ3Î-Ï 3Ë ÌdÐÍË 9ÏÌ n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ Í Ì Í 9 Ï6ÎGËÐ Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï6ÎGËÐ CkÌ OÎGÌ 5 Ì[ ÎwÌ5Ê Í Ë6ÏÐÍ U<?| OÆD25AD7QJ'2xE9AP5ADm525<D7Qg?.UAw467z<O.1.J/46P5ADm5254U,379825,khi,3P5:±<'R<-P¢2nAO8s41<-7zgGP546JTI _ E9AAsÆD<D:=l3.UAw467k2xE9A)[email protected],-F E3<-8

m = 3<D7QJ

n = 2T |

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................................

................. . . . . . . . . . . . . . . .

0

2

4

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ab

b

c

c

d

d

-N `_ E9Ad<-m5250Q<D.lQ<G2xE«418':z41PnP5,3P5ADJ%@D>r<c8n2nP<D46gOE2).1417QA=F)4o2xErA-¡/0Q<O2546,37

y = xT_ E9AcP<?>¬,/hW.146gOE2wF)41.j.WA*3AD7/250Q<O.1. >¬E/4j2)2xE9A3AOP25AsÆ«mn,3PnP5AO8ClQ,37QJ3417Qg2n,p2xE9A=3ADP¢2nAxÆ

(mn, mn)417k2xE9ASAxÆD25AD798x4U,37T

i@-| _ E9AP<?>zmxP5,38C8CAD8m − 1

.j467QAO8a417=,37QAJ/46P5ADm5254U,37<-7QJn − 1

417k2xE9A,2xE9ADP64172xE9AAsÆO2nAO798s46,37|CT _ E/0Q8GK2xE9A730Q:=@QAOPW,/hP5AtA-m52546,3798W418m + n − 2

TÄ ® TWÇDA?2a@9A<79<O250QP<O.730Q:=@QAOP?T+-E9,-F2xE3<G2I2xE9ASAD¡30Q<G254U,37

x2 − y2 = a3

<D. F<G>38BE3<-8a467/25A-gOAOP8C,/.1032546,3798ahi,3Px<-7QJ

yT

3Ð DÌ [ 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï1ÎOË~Ð D ÌDÐ \ Ì[ Í QÌ?Í Ë Q Í ÎsÏ Î 3Ð-Î Ë ÌGÍ3Î D ÌDÌ Ì Í%É - C Í CwÏ Ï D ÏÌ Í/ÏDÌ ÎxÏ¢Ë-[ C Ë~Ì ~Ð9Ð ,] CkÌ OÎGÌSË ÌÎGÐ 9Ë6Ï~ÐÍdÐ 3Ð-Î ÇDAG2

x =a(a + 1)

2

Ky =

a(1 − a)

2

TIR.6AD<DP5. >LKxKy ∈ Ta¥%,3P5A-,O3ADPCK

x2 − y2 = (x − y)(x + y) =a2 + a − a + a2

2· a2 + a + a − a2

2

= a2 · a = a3 T

XYAxÆD2FSA:,O3A25,S8*,/.j03254U,379825,SlQP5,9@/.UAO:=8,/h32xE9AS¥r<?> Mw730Q:=@QAOP,/h32xE9AÐ ÍLÌ <-7QJS2xE9A) \ 0Q8x2nP46<D7¥r<O2xE9AO:=<G254Um58WVa. >9:zl941<JM-N-N3 M@U- ] M ] /TÄ TWÇDAG2

a@9A<)/Æ-ADJ'FSE9,/.6A)730Q:@9ADP©T

WAG25ADP5:z417QAW<D.j.-8C,/.1032546,3798xKyKz467FSE9,/.UAa730Q:@9ADP82n,2xE9Aa8>98n2nAO:È,/h/AD¡30Q<-254U,3798

5x + (a + 2)y + (a + 2)z = aK

(2a + 4)x + (a2 + 3)y + (2a + 2)z = 3a − 1K

(2a + 4)x + (2a + 2)y + (a2 + 3)z = a + 1T

3Ð DÌ [ 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï6ÎGËÐ GÐ-Î BÌ GÏ6Î B Dn Ì Ð Í/ÏDÌ ÎxÏ¢Ë Ë tÐjÏ¢Ë Ì Í3Ï QÌ Ë -Í [ -x Ì"ÐÍ ÏUÍ ~Ð-Î ¬D5 3Ð -Ï1Î 9Ï Ì Î D ÌDÌ Ì Ì ] Ë-[ Ð DÌ CkÌ GÏDÌ 5 Ì[ Î Ï¢Ë~Ì _ E9A8CA-mn,37QJz<D7QJk2xE/41P5J=A-¡/0Q<O2546,3798g?4 3A

(a − 1)2(y − z) = 2(a − 1)T

bcE9AD7a = 1

KW2xE9A«,3P4Ug?4179<D.B8>98n2nAO: P5A-J/0tmxAO8z25,%2¢FS,%417QJQAOltAO7QJQAO7/2=AD¡30Q<G254U,37985x + 3y + 3z = 1

<D7QJ6x + 4y + 4z = 2

K;hUP5,3: FSE/46msEdFSA,9@/2<D417«<e,37QA-lQ<DP<-:=AG25ADP8CAG2Y,/hFSE9,/.UADs730Q:=@QAOP8*,/.j03254U,3798x = −1

Ky = 1 + t

Kz = 1 − t

KFSE9ADP5At ∈ Z

T

-N/bcE9AD7

a 6= 1K-FSASE3<?3A

y − z =2

a − 1

Ta¦ihyKz<-P5AFSE9,/.UA)730Q:@9ADP8?K/2xE9AD78*,468I2xE9AG46PIJ34jÔAOP5AD7QmnA3KOFSE/4UmsE':=AD<D798I2xE3<O2

a − 1 | 2KP5AD8n2nP546m525467QgB2xE9AYlt,[email protected],

a = −1Ka = 0

Ka = 2

Ka = 3

T/. "% @ T

a = −1T _ E9AA-¡/0Q<O2546,3798@9A-mn,3:A

5x + y + z = −1K

2x + 4y = −4K

2x + 4z = 0K

g?4o/467Qgw<8*,/.j03254U,37x = 0

Ky = −1

Kz = 0

T/. "%P T

a = 0T _ E9ASA-¡/0Q<O2546,3798@9A-mn,3:A

5x + 2y + 2z = 0K

4x + 3y + 2z = −1K

4x + 2y + 3z = 1K

g?4o/467Qgw<8*,/.j03254U,37x = 0

Ky = −1

Kz = 1

T/. "% XQT

a = 2T _ E9ASA-¡/0Q<O2546,3798@9A-mn,3:A

5x + 4y + 4z = 2K

8x + 7y + 6z = 5K

8x + 6y + 7z = 3K

g?4o/467Qgw<8*,/.j03254U,37x = −6

Ky = 5

Kz = 3

T/. "% W T

a = 3T _ E9ASA-¡/0Q<O2546,3798W417Qmn.10tJ9A

5x + 5y + 5z = 3K

FSE/4UmsEzA*346JQAO7/25.o>zE3<-8W7Q,SFSE9,/.6A-x730Q:@9ADP8C,/.1032546,3798GK/8x467QmnA5 - 3

TTÇDA?2

K@9A'<klQ,38s4o254o3AwFSE9,/.6AS730Q:@9ADP©T _ E9AS8*AD¡30tAO7QmxA an : n ≥ 1 418JQAGt7QADJk@D>

a1 = 1<-7QJ

an4182xE9A

n 79<G250QP<D.3730Q:@9ADPIgGP5AD<G2nAOP2xE3<-7 an−1FSE/46msE468Bmn,37QgGP50tAO7/22n,

n:=,9J309.6,

KT

U<?| WAG25ADP5:z417QA<D7AxÆl3.146mn4j2Ihi,3Pn:09.6<)hi,3PanT

i@-|bcE3<G24182xE9AP5AD8x09.j2I41hK = 2

Å3Ð DÌ [ 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[

1Ï Ë~ÐÍ Ð Ì 3Ì Ï6ÎGËÐ Ë 9 Í ÎxÏ Î 3Ð-Î Ë ÌGÍ3Î D Ì-Ì Ì Í n5 [ -Ï6Í Í/ÏDÌ ÎxÏ¢Ë-[dÐ 9Ì Ë É dÐÍLËÐÍ CkÌ GÏDÌË Ì Ï¢Ë~Ì ÐÐ Í9Î ËÌ*ÏUÍ U<?|WÇDAG2

n@9A<lQ,38s4o254o3Aw417/2nADgOADP©T

+O467QmnAan ≡ n mod K

K2xE9A=QP8x2Y467/25A-gOAOPFSE/4UmsEc468SgGP5AO<O25ADP2xE3<D7an

<D7QJmx,37QgGP0tAD7/225,n + 1

:,9J/09.U,K468

an + 1T _ E/0Q8?K-2xE9A

(n + 1)79<O250QP<O.Q730Q:@9ADP

-NgGP5AD<G2nAOP2xE3<D7

anFSE/46msE'418Wmx,37QgGP0tAD7/225,

n:=,9J309.6,

K418

an+1 = an + 1 + nKT+O0Q:=:467QgY2xE9AO8*A)P5AO.1<O2546,3798GKDFSAgOAG2I2xE3<G2*K/hi,3PaA*3AOP~>467/25A-gOAOP

n ≥ 1K

an = a1 + n − 1 + K

n−1∑

i=1

i = n +n(n − 1)

2KT

i@-|¤3,3PK = 2

K-FSA)41:=:=A-J/46<G2nAG.o>zE3<?3Aan = n2 hi,3P n ≥ 1

T TWAG25ADP5:z417QA'<O.1.¡/0Q<J3P50Ql3.UAO8

(a, b, c, d),/hP5AO<D.L730Q:=@QAOP88C<G25468xhj>3417Qgw2xE9AA-¡/0Q<O2546,37

256a3b3c3d3 = (a6 + b2 + c2 + d2)(a2 + b6 + c2 + d2)

×(a2 + b2 + c6 + d2)(a2 + b2 + c2 + d6)T

3Ð DÌ [ 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ -5 Í Ì Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï1ÎOË~Ð Í n5 [ -ÏUÍ Í3Ï DÌ ÎsÏË#[Ð 9Ì Ë É dÐÍLËÐÍ CkÌ GÏDÌÐ Í9Î ËÌ*ÏUÍ Î ÏËÌ ÇDAG2

(a, b, c, d)@9A<'¡30Q<-J9P0Ql9.6Ak,/hLP5AD<O.730Q:@9ADP88s<O25418shj>/467Qg2xE9ASA-¡/0Q<O2546,37T¦ih,37QA,/ht2xE9A730Q:=@QAOP8W468£sAOP5,K2xE9AD7z<O.1.t<-P5A

0T¤/P5,3:±7Q,-F ,37LK-FSA)8s0Ql9lt,38CAw2xE3<G27Q,37QAW,/hO2xE9Ahi,/0QP730Q:=@QAOP8418

0T;+O417QmxA2xE9AaP4UgOE2xsE3<-7QJY8x4UJ9A418lQ,38s4o254o3AI2xE9AOP5Aa:z0Q8n2@QA)<D7=A*3AO7z730Q:=@QAOPa,/h7QADgG<O254 3A)P5AD<O.68a<-:=,37QgG8x2

aKbKcKdT _ E9AD7

(a, b, c, d)418a<8*,/.j03254U,374jh<D7QJd,373.o>4jh

(|a|, |b|, |c|, |d|) 468<8*,/.j03254U,37T _ E/0Q8GK9FSAk:=<?>e8s0Ql9lt,38CA2xE3<O2aKbKcKd<DP5AlQ,38s4o254o3A9T

¤/P5,3:2xE9A \ ¥ HW¥ ¦~7QAD¡30Q<O.14o2>Ka6 + b2 + c2 + d2 ≥ 4(a6b2c2d2)1/4 K

<-7QJ=8x46:41.1<-P.o>khi,3P2xE9AS,2xE9ADPI2xE3P5A-A)hU<m525,3P8,372xE9ASP4UgOE2xsE3<-7QJ=8s46JQAS,/ht2xE9A'AD¡30Q<-254U,37T _ E/0Q8GK(a6 + b2 + c2 + d2)(a2 + b6 + c2 + d2)

×(a2 + b2 + c6 + d2)(a2 + b2 + c2 + d6)

≥ 256(a6b2c2d2)1/4(a2b6c2d2)1/4 × (a2b2c6d2)1/4(a2b2c2d6)1/4

= 256a3b3c3d3 KFSE/4UmsE417QJ346mn<G2nAO82xE3<O2W2xE9Ag?4o3AO7cA-¡/0Q<O2546,37e4182xE9AzAD¡30Q<O.14o2>dm5<-8CA,/h2xE9A \ ¥Ha¥¦~7QA-¡/0Q<D.j4j2~>T _ E9ADP5AGhi,3P5A3K

a6 = b2 = c2 = d2 = a2 = b6 = c6 = d6 2xE3<O2418GKa = b = c = d = 1

T_ E9AO72xE9A8*,/.j03254U,3798<-P5A

(0, 0, 0, 0)<D7QJ

(ε1, ε2, ε3, ε4)FSE9ADP5A

εi = ±1hi,3P

i = 1K2K3K4<D7QJ ∏4

i=1 εi = 1T

-N-NS T;bcA)JQAGt7QA2xE9Awhi,/.1.6,-F)467QgY,3lQADP<G254U,37SFSE/46msEF)41.j.t@QAw<-l9l9.j4UADJ'25,k<P5,-F ,/h@9<DP8@QAG467Qgw8x4j250Q<G2nADJz8x4UJ9A-s@D>Qx8s46JQA,37zlQ,38s4o254U,3798

1K. . .

KNU

3<-msEz@3<-P8s4o250Q<O25A-J<G2<D7=,9J9JQx730Q:@9ADP5ADJ=lQ,38s4o254U,37'468a.6AOh12<D8a468?KDFSE/41.6AAD<-msE@9<DP)<O2w<-7rA*3AO7Qs730Q:=@QAOP5A-JrlQ,38s4o254U,37468SP5AOl9.1<mnA-J«@D>2¢FS,«@3<-P8OT \ h12nAOPY2xE3<G2*KI<D.j.@9<DP8F)4j.1.t@9A)l90328x4UJ9A-s@D>Qx8s46JQAY4678s0tmsEk<YF<G>'2xE3<G2<O.1.t@9<DP8ahi,3Pn:±<7QA©FÈP5,-F <D7QJ<-P5A)8s4o250Q<O25A-JzU8x4UJ9A-s@D>Qx8s46JQA©|I,37zlQ,38s4o254U,37981K. . .

KMT

¤/P5,3: <-746734o2546<O.730Q:@9ADPa0 > 0

,/h@3<-P82xE9AOP5A,3P546g?4679<G2nAO8wU@D>d8x0tmxmnAD8s8s4 3A<-l9l9.j4Um5<O2546,37%,/h2xE9Ae<@9,D3ADCJ9AOQ7QA-J%,3lQADP<G254U,37|<d8CA-¡/0tAD7QmnA3K an : n ≥ 0 ,/h79<O250QP<O.730Q:=@QAOP8GKFSE9AOP5Aan

468Y2xE9A730Q:=@QAOP),/ha@3<-P8<Oh12nAOP)E3<?3417Qgz<-l9l9.j4UADJc2xE9A,3ltAOP<O2546,37n2546:=AD8OT

U<?|WuQP5,D3AY2xE3<O2hi,3P<D.j.n > 0

FSASE3<?3Aan 6= 1997

Ti@-|$WA?2nAOPn:467QAY2xE9A79<G250QP<D.730Q:=@QAOP8a2xE3<O2am5<-7=,373. >=,9mnmn0QP<D8

a0,3P

a1T

3Ð DÌ [ 9ÏÌ n Ì Ð Í9Î ËÌ*ÏUÍ tÐÍË~ÐDÏ6Î?Ì -5 Í Ì Í Q Ï1ÎOË~Ð Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï6ÎGËÐ CkÌ OÏ DÌ Ð Í3Î Ë~Ì©Ï6Í ÎWÎGÐ 9Ë6Ï~ÐÍ U<?|ÇDA?2

n@9A'<k7Q,37Qs7QADgG<O254 3AS467/25A-gOAOP?TBbcE9AD7=2xE9Ak,3lQADP<G254U,37=418<DlQl3.146A-Jz25,<P5,-F ,/h

an@3<-P8?K-2xE9AY25,25<O.Q730Q:@9ADP,/h@9<DP8467QmxP5AO<-8CAD8W@D> an

2

41han

418WA*3AO7LK3<D7QJ@D> an − 1

2

4jhan

418B,9JQJT _ E/0Q8?K3hi,3PA*3ADP~>n ≥ 0

K

an+1 =

3an

2

4jhan

468BA*3AO7LK3an − 1

2

4jhan

468B,9J9JTÇDAG2

p@9AY<w79<O250QP<O.Q730Q:=@QAOPF)4j2xE

p ≡ 2 mod 3TWÇDA?2

n ≥ 0@9AY<-7S467/25A-gOAOP?T+O0QlQlQ,38*A

an+1 = pT'¦ih

an468wA*3AO7LKt2xE9AD7

3an = 2an+1 = 2p ≡ 1 (mod 3)K<mx,37/2nP<-J346m5254U,37LKtFSE/41.6A4jh

an468),9JQJKL2xE9AO7

3an = 2an+1 + 1 = 2p + 1 ≡ 2(mod 3)

K<mx,37/2nP<-J346m5254U,37T _ E/0Q8?Kan+1 6= p

Tw+O467QmnA1997 ≡ 2 (mod 3)

KtlQ<DP2U<?|;468Wl9P5,D3ADJTi@-|bcAkE3<G3Ak8CA-AO72xE3<G2W41h

p ≡ 2 (mod 3)KQ2xE9AD7

pm5<-7d,373.o>e,9mxmn0QPW4172xE9A8*AD¡30tAO7QmxA<D8

a0T

/. "% @ Tp = 9k

KF)4o2xEk ∈ ∗ T¤3,3P

a0 = 4kKIFSAdE3<G3A

a1 = 6k<D7QJ

a2 = 9kT _ E/0Q8?K

pmn<D7¬,9mnmn0QP417«2xE9A8*AD¡30tAO7QmxA<D8

anF)4o2xE

n ≥ 2T

/. "%P Tp = 9k + 1

K-F)4j2xEk ∈

T¤3,3Pa0 = 4k + 1

KFSAE3<G3Aa1 = 6k + 1

<-7QJa2 = 9k + 1

T _ E/0Q8?Kpmn<D7,9mnmn0QP467k2xE9A8CA-¡/0tAD7QmnA<D8

anF)4j2xE

n ≥ 2T

/. "% XQTp = 9k + 3

K-F)4j2xEk ∈

T+O0QlQlQ,38*AW2xE3<O2L2xE9AOP5AAsÆD468n258<-7)417/2nADgOADPn ≥ 1

8s0tmsEY2xE3<O2an+1 = p

TI¦ihan

468IA*3AD7LK2xE9AD73an = 2p = 2(9k + 3)

K<D7QJE9AO7QmxA/Kan = 6k + 2 ≡ 2 (mod 3)

F)4o2xEn > 0

K<wmn,37/2nP<J/4Um52546,37T¦ihan

468,9JQJKD2xE9AO73an = 2p+1 = 2(9k+3)+1 ≡ 1

(mod 3)K<zmx,37/2nP<-J346m5254U,37T _ E9AOP5AOhi,3P5A/K

pm5<-797Q,2,9mxmn0QPB417e2xE9A8CA-¡/0tAD7QmnAz<D8

anF)4j2xEn ≥ 2

T

]/. "% W T

p = 9k + 4K-F)4j2xE

k ∈ T¤3,3P

a0 = 4k + 2KFSAE3<G3A

a1 = 6k + 3<-7QJ

a2 = 9k + 4T _ E/0Q8?K

pmn<D7,9mnmn0QP467k2xE9A8CA-¡/0tAD7QmnA<D8

anF)4j2xE

n ≥ 2T

/. "%S Tp = 9k + 6

K-F)4j2xEk ∈

T¤3,3Pa0 = 4k + 3

KFSAE3<G3Aa1 = 6k + 4

<-7QJa2 = 9k + 6

T _ E/0Q8?Kpmn<D7,9mnmn0QP467k2xE9A8CA-¡/0tAD7QmnA<D8

anF)4j2xE

n ≥ 2T

/. "% ZtTp = 9k + 7

K-F)4j2xEk ∈

T+O0QlQlQ,38*AW2xE3<O2L2xE9AOP5AAsÆD468n258<-7)417/2nADgOADPn ≥ 1

8s0tmsEY2xE3<O2an+1 = p

TI¦ihan

468IA*3AD7LK2xE9AD73an = 2p = 2(9k + 7) ≡ 2 (mod 3)

K;<mn,37/2nP<J/4Um52546,37Td¦ihan

418,9JQJK2xE9AD73an = 2p + 1 = 2(9k + 7) + 1

K3<-7QJzE9AD7QmnA3Kan = 6k + 5 ≡ 2 (mod 3)F)4j2xE

n > 0K<wmx,37/2nP<-J346m5254U,37T _ E9ADP5AGhi,3P5A3K

pm5<-797Q,2I,9mnmn0QP;4172xE9A8*AD¡30tAO7QmxA<-8

anF)4j2xEn ≥ 2

T¦U2hi,/.j.U,-F82xE3<O2I2xE9A)79<O250QP<O.730Q:@9ADP8a2xE3<O2am5<-7=,373.o>z,9mnmn0QP<-8

a0,3P

a1<-P5A2xE9,38*Amx,37QgGP0tAD7/2I2n,

2K3K5K7K9,3P

8 (mod 9)T

T ÇDA?2n@QA^<^/Æ-ADJ 79<O250QP<O.'730Q:@9ADP©T WAG25ADP5:z417QA^<D.j.'lQ,/.o>37Q,3:z41<D.18

x2 + ax + bK-FSE9AOP5A

a2 ≥ 4bK38s0tmsE'2xE3<G2

x2 + ax + bJ/4o/4UJ9AD8

x2n + axn + bT

3Ð DÌ [ Ï 5 Ì Ë Ï ~Ì ;Ð3Ì?Í -5 Í Ì 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ Í Ì 9 Ï6ÎGËÐ Ì ~Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï6ÎGËÐ Ë 9 Í ÎxÏ Î 9Ð-Î Ë Ì?Í9Î D Ì-Ì Ì Í n [ -ÏUÍ Í/ÏDÌ ÎxÏ¢Ë-[^Ð 9Ì Ë É dÐÍLËÐÍ CkÌ GÏDÌ Ë Ï ~Ì Î ÏËÌ ¦ih

n = 1Kt<O.1.Llt,/. >97Q,3:46<O.68

x2 + ax + b<-P5A'8C,/.1032546,3798DTBbcAwF)41.j.8x0QlQlQ,38*A

n > 1hUP5,3: 7Q,-F ,37Te+O467QmnA

a2 ≥ 4bK2xE9AOP5AAsÆD468n2YP5AD<O.I730Q:@9ADP8

x1Kx2

U7Q,27QA-mnAD8s8C<DP54j.o>eJ/468n25467Qm52¢|8s0tmsEz2xE3<G2x2 + ax + b = (x − x1)(x − x2)

T¦U2Whi,/.j.U,-F82xE3<O2x2n + axn + b = (xn − x1)(x

n − x2)T

XY,-FKt41hx2 + ax + b

J34 346JQAO8x2n + axn + b

Kt2xE9AD7x1Kx2

<-P5AkP5,9,258Y,/hx2n + axn + b

K;8C,e2xE3<G2xn

1 = x1,3P

xn1 = x2

K;<D7QJxn

2 = x2,3P

xn2 = x1

T_ E9AOP5AOhi,3P5A/Kx1<-7QJ

x2:[email protected],37QgY25, −1, 0, 1 TXY,-F FSAmsE9ADmsf2xE9AlQ,38C8x4U@/.UA)mn<D8*AO8HU

• ¦ih x1 = x2 = 0K2xE9AD7

a = b = 0<D7QJ

x2 + ax + b = x2 J/4o/4UJ9AD8x2n + axn + b = x2n T

• ¦ih x1 = 0Kx2 = −1

K2xE9AD7a = 1

Kb = 0

<-7QJx2 + ax + b = x(x + 1)J34 346JQAO8

xn(xn + 1),373. >4jh

n418B,9JQJT

• ¦ih x1 = 0Kx2 = 1

K2xE9AO7x2 + ax + b = x(x − 1)

J/4o/4UJ9AD8xn(xn − 1)

T• ¦ih x1 = −1

Kx2 = −1

KD2xE9AO7x2 +ax+ b = (x+1)2

J/4o/4UJ9AD8(xn +1)2,373.o>k41h

n418B,9JQJT

• ¦ih x1 = 1Kx2 = 1

K2xE9AO7x2 + ax + b = (x − 1)2

J/4o/4UJ9AD8(xn − 1)2

T• ¦ih x1 = −1

Kx2 = 1

K2xE9AD7x2 + ax + b = x2 − 1

J/4o/4UJ9AD8x2n − 1

T

] M¦~7)mn,37Qmn.10Q8x4U,37LK©hi,3P

n,9JQJY

n > 1|nK©2xE9Aa8C,/.1032546,3798<DP5A

x2Kx(x+1)

Kx(x−1)

K(x + 1)2

K(x − 1)2

Kx2 − 1

<-7QJ=hi,3PnA*3AD7LK92xE9A'8C,/.1032546,3798<-P5A

x2 K x(x − 1)K

(x − 1)2Kx2 − 1

TÉ-ÏËÐ Î Ð dÌGÍLË Ò Z .6<D:fD467plt,/417/258S,/0322xE3<G2Y41h2xE9Aemx,37QJ/4j2546,37 a2 ≥ 4b468AO.j46:4679<G2nADJLK32xE9A£sAOP5,38

x1Kx2

m5<-7e@QA'mx,3:zl9.6AsÆmn0t@QA'P5,9,28,/h0Q734j2~>LKt<O.1.6,-F)467Qg<-7Q,2xE9AOPlt,[email protected]>Kx2 + x + 1

K3lQP5,O346JQADJn468W7Q,2<:z09.o2546l3.UA,/h

3T

XYAxÆD2pFSA :,O3A 2n,P5AD<-JQAOP8GÓ'8*,/.j03254U,3798¬hi,3P%lQP5,9@/.UAO:=8^,/h=2xE9A ¦~8s.6AD7/£sfD<+G25<-ADP,/hUP,/4Uf-AOlQl9734¤/P<D:E3<O.UJ38C8Cf,/.6<D7QAD:z<'M-N-N/ MDNN ` M<U/ ] ]-] /TÄ T RL<O.Umn09.1<O25Ad2xE9A<DP5AD<p,/hW2xE9AP5ADg?4U,37%417%2xE9Acl9.1<-7QAJQA?2nAOPn:467QADJ @D>r2xE9A467QAD¡30Q<O.14o2>

|x| + |y| + |x + y| ≤ 2T

3Ð DÌ [ Ð9Ì Ë QÏ jÏUÍ3Î-Ï 3Ë ÌdÐÍË 9ÏÌ n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ Í Ì Q Ï1ÎOË~Ð Ì ~Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï1ÎOË~Ð GÐ-Î BÌ OÏ1Î B -n Ì Ð Í3Ï DÌ ÎsÏË Ë tÐjÏ¢Ë Ì Í3Ï QÌ Ë DÍ [ Ds Ì ~ÐÍ ÏUÍ Í n5 [ -Ï6Í Í/ÏDÌ ÎxÏ¢Ë-[Ð 9Ì Ë É dÐÍLËÐÍ CkÌ GÏDÌdË ÌÎ?Ð 3Ë6ÏÐÍ%Ð Ï -n Ì Ð ÇDAG2 A = (x, y) ∈ 2 : |x| + |y| + |x + y| ≤ 2 @QA2xE9Akg?4 3AD7P5A-g?46,37TbcAkmn.6<O46:±2xE3<G2 A E3<D8<DP5AD<

3T _ ,=lQP5,O3AS2xE/418GK/FSA'8n25<DP2B@D>P5A-m5<D.j.1417Qg)2xE3<G2ahi,3PW<D.j.

a ∈ K |a| = | − a| T¦U2Ihi,/.j.U,-F8I2xE3<G2 A 418a8>3:=:=AG2nP4Um;F)4j2xE'P5AO8ClQA-m5225,'P5AtA-m52546,372xE3P5,/0tgOES2xE9A),3P4Ug?417TAO7QmxA/K4o2I8s09ymxAO8I2n,S4173AD8n254UgG<G2nA2xE9Awg?4o3AO7'467QAD¡30Q<O.14o2>k,373.o>FSE9AD7y ≥ 0

T¦~7k2xE9A)QP8x2¡30Q<-J9P<D7/2*KFSE9AOP5A

x ≥ 0<D7QJ

y ≥ 0K-FSA)E3<G3A

x + y ≥ 0K3<D7QJ2xE9A417QA-¡/0Q<D.j4j2~>=@QADmx,3:=AD8

x + y + x + y ≤ 2T _ E9ADP5AGhi,3P5A3K/2xE9Al9<-P¢2W,/h A 467k2xE/468¡30Q<-J9P<D7/24682xE9AY2nP541<-7Qg?.6A

AOB = (x, y) ∈ 2 : x ≥ 0Ky ≥ 0

Kx + y ≤ 1 T

¦~72xE9Ak8*ADmx,37QJd¡/0Q<J3P<-7/2*KQFSE9AOP5Ax ≤ 0

<D7QJy ≥ 0

K9FSAE3<?3A'2¢FS,=lt,[email protected]<G|x + y ≥ 0

,3Pi@-|x + y ≤ 0

T¦~7zmn<D8*A)6<G|nK2xE9AY467QAD¡30Q<O.14o2>@QADmx,3:=AD8−x + y + x + y ≤ 2

K3<-7QJ'FSAE3<?3Aw2xE9Aw2nP46<D7Qg?.UABOC = (x, y) ∈ 2 : x ≤ 0

K0 ≤ y ≤ 1 K

¦~7mn<D8*ASU@|nK32xE9A)417QA-¡/0Q<D.j4j2~>=@QADmx,3:=AD8 −x + y − (x + y) ≤ 2KQJQA?2nAOPn:4673417Qgw2xE9A2nP541<-7Qg?.6A

COD = (x, y) ∈ 2 : −1 ≤ x ≤ 0Ky ≥ 0 T_ E9AO8*AS2nP46<D7Qg?.UAO8wmn<D7c@9Ak8*ADAD7467=2xE9A'g?0QP5A,37=2xE9Ak7QAsÆO2l9<gOA9TBLAtA-m525417Qg2xE9AO:2xE3P5,/0tgOEk2xE9AS,3P4Ug?417LK-FSA)8*ADAw2xE3<G2 A 4682xE9ASE9AxÆD<gO,37

ABCDEFKDF)4j2xE<DP5AD<

3T

]

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A

BC

D

E F

O

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T +O0QlQlQ,38*Ac2xE3<O2

aK

bK<D7QJ

c<-P5Ac2xE9Ac2xE3P5A-ApP5,9,258=,/h2xE9Aplt,/. >97Q,3:46<O.

p(x) = x3 − 19x2 + 26x − 2TIRL<O.Umn09.1<O25A1

a+

1

b+

1

c

T3Ð DÌ [ D ÍÐQË 5 ÎGË QÌ?ÍË GÏ CwÏ6Í9ÎGËÐÍ x5 Ï Ï 5 Ð9Ð Í Ë6Ï Ë6Ï CkÌ?ÎGËÌ Í Í Ï 5 Ð9Ð D [ ¬Ds OÎÉ Í /Ì D ÍLÌ©Î ÎGË QÌ?ÍË Ð Í/ÏDÌ ÎxÏ¢Ë-[ 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ

Í Ì 9 Ï6ÎGËÐ Ì ~Ì[ jÏ ¢ËÐÍ Ð Ì 3Ì Ï6ÎGËÐ GÐ-Î BÌ GÏ6Î B Dn Ì Ð Í/ÏDÌ ÎxÏ¢Ë Ë tÐ1ÏË Ì Í/Ï ¢ QÌ Ë DÍ [ Ds Ì ~ÐÍ ÏUÍ ÉjÏ s 9ÌË D CkÌ©ÎOË~Ì Í Í Ï 5 Ð9Ð - [ CkÌOÏ DÌ Ë ÌÎGÐ 9Ë6Ï~ÐÍdÐ ÍLÐQË S Í Ë6Ï ¦ih

aKbKc<DP5Aw2xE9A)P5,9,258B,/ht2xE9A)lt,/. >97Q,3:46<O.K-2xE9AD7

(x − a)(x − b)(x − c) = x3 − 19x2 + 26x − 2T ~M©|

DÆ-lQ<D7QJ3417Qg>/4UAG.UJ38HU(x − a)(x − b)(x − c)

= (x2 − (a + b)x + ab)(x − c)

= x3 − (a + b)x2 + abx − cx2 + (a + b)cx − abc

= x3 − (a + b + c)x2 + (ab + bc + ac)x − abcT i?|

¤/P5,3:!~M©|<-7QJ=i?|FSAgOA?2a + b + c = 19

Kab + bc + ac = 26

Kabc = 2

T_ E/0Q8?KFSAE3<?3A 1

a+

1

b+

1

c=

bc + ac + ab

abc=

26

2= 13

T

]]¯T \ mn,/.1.6A-m52546,37z,/h

52467/25A-gOAOP8W468Wg?4o3AO7T+-E9,-F2xE3<G2<-:=,37QgG8x22xE9AD8CA730Q:=@QAOP8W4o2418Wlt,[email protected],kQ7QJk2¢FS,k8s0tmsE'2xE3<G2

100J34 346JQAO8BAO4o2xE9ADP2xE9AO41P8x0Q:!,3PI2xE9AG46PJ34jÔAOP5AD7QmnAQT

3Ð DÌ [ Ï 5 Ì Ë Ï ~Ì ;Ð3Ì?Í -5 Í Ì 9Ï~Ì n Ì Ð Í3Î Ë~Ì©Ï6Í tÐÍLËÐDÏ1ÎGÌ Í Ì 9 Ï6ÎGËÐ Ì ~Ì[ 1Ï Ë~ÐÍÐ Ì 3Ì Ï6ÎGËÐ SÐÍ [OÏ Ï ÎGË QÌGÍLË OÏ CwÏUÍ3ÎOË~ÐÍ x5 Ï Ï 5 Ð9Ð D [ n [ -ÏUÍ Í/ÏDÌ ÎxÏ¢Ë-[ Ð 9Ì Ë É dÐÍË~ÐÍ Í Ð-Î %- /Ð -Ï6Î 9Ï Ì Î D ÌDÌ Ì CkÌ GÏDÌ'Ë ÌYÎ?Ð 3Ë6ÏÐÍeÐ -ÏUÍ

_ E/418lQP5,9@/.UAO: <DlQlQAD<DP5A-JY<-8<>L0Q8s8s41<-7wVa.o>3:=l346<-JYlQP5,9@/.UAO: ¡3094o2nAa8*,3:=A2541:A<gO,TIªB73hi,3P¢250Q79<O25AO. >LKt¦J9,7Q,2aE3<?3A<P5AOhiAOP5AD7QmnAQT3<-msE¬,/ha2xE9Ad730Q:@9ADP8zmn<D7 @9AAsÆ-lQP5AO8C8CA-J«467«2xE9Adhi,3P5:

100a + bKIFSE9AOP5A

−49 ≤ b ≤ 50T+O467QmnAB2xE9ADP5A<-P5A,373.o>

51lt,[email protected]/<D.j0tAD8hi,3P |b| K-<O2.6AD<D8x22¢FS,,/h2xE9AD: :0Q8x2@9AS2xE9A'8C<D:A9T¦ih2xE9Akmx,3P5P5AD8slt,37QJ/467QgS730Q:@9ADP8wE3<G3Ak,3lQlQ,38s4o2nA'8x4UgG798<-8s8*,9mn41<O25A-J=F)4j2xE

bKL2xE9AO7d2xE9Az8x0Q: ,/h;2xE9Az730Q:@9ADP8w468)J34 [email protected]=@D>

100 4jh;2xE9A*>E3<G3Aw2xE9A)8C<D:A)8x4UgG798?K2xE9AD7k2xE9ASJ/41ÔADP5AO7QmxAw468BJ/4o/468x4U@/.UA@D>

100T

_ E9ApP5AO8s09.o2k418k2nP50tAp:=,3P5A«gOAD7QAOP<D.j.o>¬4jhWFSApP5AOl9.1<mnA100

@D>n<-7QJ

52@D>

bn/2c + 2T

Tw14o|+-E9,-F 2xE3<O22xE9Ak8x0Q: ,/hL2xE9AJ/4Ug?4o258Y,/hA*3AOP~>417/2nADgOADPW:z09.o2546l3.UAk,/h99KhUP5,3:

1 · 990Ql25,<-7QJ417Qmn.10tJ/467Qg

100 · 99K3418

18T

64j4o|)+-E9,-F 2xE3<G22xE9A8s0Q: ,/hW2xE9ApJ/4Ug?4o258z,/hA*3ADP~>%417/2nADgOADP':09.j2541l9.6Ap,/hW2xE9A730Q:@9ADP10n − 1

K;hUP5,3:1 · (10n − 1)

0Ql25,<-7QJ417Qmn.10tJ/467Qg10n · (10n − 1)

K468n · 9

T3Ð DÌ [ Ï 5 Ì Ë Ï ~Ì ;Ð3Ì?Í -5 Í Ì Q Ï1ÎOË~Ð Ì ~Ì[ jÏ ¢ËÐÍ

Ð ~Ì/Ì Ï6ÎGËÐ n5 [ -Ï6Í Í/ÏDÌ ÎxÏ¢Ë-[Ð 9Ì Ë É dÐÍË~ÐÍ Í Ë6Ï Ë6Ï CkÌ©ÎOË~Ì Í Í Ï 5 Ð9Ð D [ CkÌOÏ DÌ'Ë ÌÎGÐ 9Ë6Ï~ÐÍdÐ -ÏUÍ

64 |99n = 100n − n

T _ E/0Q8?K4jhn468'<d8s417Qg?.UADCJ/4Ug?4o2730Q:@9ADPCKI2xE9AcJ/4Ug?4o258P5AD8x09.j25417QgwhUP5,3: 8x0t@/2nP<-m525467Qg

nhUP5,3:±2xE9A

3 J346g?4j2a730Q:=@QAOP n00<DP5A

n − 1K9K9<D7QJ

10 − nK/hi,3PI<8x0Q:!,/h

18T¦ih

n418a<

2 J/4Ug?4o2730Q:@9ADP abF)4j2xE

b = 0K-2xE9AJ/4Ug?4o258W,/h

100n − n<DP5A

a − 1K9K10 − a

Kt<-7QJ0K3FSE/4UmsE=8x254j.1.L8s0Q:!25,

18 4jh

b 6= 0KQ2xE9AO72xE9ASJ/4Ug?4o258W,/h

ab00 − ab<-P5A

aKb − 1

K10 − a − 1

K3<-7QJ10 − b

K/<gG<O4678s0Q:z:z417Qg2n,18TI~¦~7k<Y8x46:41.1<-PF<?>4j2hi,/.1.6,-F8;2xE3<O22xE9AY8x0Q:,/h32xE9AYJ/4Ug?4o258,/h Ì DÌ [ :z09.o2546l3.UA,/h

99418

18|CT

64j4o|BuQP5,9mxADA-J/467Qg)467<'8s41:z4j.6<DPIF<G>=<D8417e64 |xKQ4o2ahi,/.j.U,-F8a2xE3<O22xE9A'8x0Q: ,/hL2xE9AJ346g?4j28S,/h;2xE9A=730Q:=@QAOP99 . . . 9m

KLFSE9AOP5A2xE9AOP5A=<DP5An 9

Ó 8?K418,9@25<O467QADJp@D>8x0t@Q2nP<m525417QgmhUP5,3:

m00 . . . 0TS¤3,3PBAsÆD<D:=l3.UA/Kt41h

m4182xE9A

2 J346g?4j2B730Q:@9ADP ab7Q,2AD7QJ/467Qgw467

0K/2xE9AD72xE9A8s0tmnmxAO8C8x4o3ASJ/4Ug?4o258B<DP5A

aKb − 1

K9K9K. . .

K9K10 − a − 1

K<-7QJ10 − b

TAO7QmxA/K/2xE9A)8s0Q:!,/ht2xE9ASJ/4Ug?4o258a4189nT

]G[S T _ E9A)8*AD¡30tAO7QmxA an 418BJQAGt7QADJ=@D> a1 = 1

<-7QJK3hi,3Pn ≥ 1

Kan+1 =

an

1 + nan

T¤417QJ

a1996T

3Ð DÌ [ Ï 5 Ì" Ë Ï Ì Ð/ÌGÍ Í Ì Ð9Ì Ë QÏ 1Ï6Í9Î%-Ï 3Ë ÌdÐÍË 9Ï~Ì n Ì Ð Í9Î ËÌ*ÏUÍ tÐÍË~ÐDÏ6Î?Ì -5 Í Ì 9 Ï6ÎGËÐ Ì 5 Ì[ jÏ ¢ËÐÍÐ ~Ì/Ì Ï6ÎGËÐ n5 [ -Ï6Í Í/ÏDÌ ÎxÏ¢Ë-[Ð 9Ì Ë É dÐÍË~ÐÍ Ë6Ï Ë6Ï CkÌ?ÎGËÌ Í Í Ï 5 Ð9Ð - [ SÌ©Ï6Í /ÌGÍ /Ì©Ï \ Ì Ë Ì jÏUÍ D Ì Í [ dÌDÌGÍ O Ë 9Ð dÌ Ë Ì ] Ì-Ë Ì ÍÎ ÍÉ D C Í CwÏ Ï D Ï~Ì Í3Ï DÌ ÎsÏË#[ C ËÌ Ð9Ð ,] CkÌ OÏ DÌË ÌYÎ?Ð 3Ë6ÏÐÍeÐ Ë Ï ~Ì ¨3>z<D746:z:ADJ341<O25A)417QJ30tm52546,37LK

an > 0hi,3P<D.j.

nT

XY,-FK 1

an+1

=1 + nan

an=

1

an+ n

hi,3P<O.1.nT¦U2hi,/.1.6,-F82xE3<G2

1

a1996

− 1

a1

=1995∑

k=1

(1

ak+1

− 1

ak

)

=1995∑

k=1

k =1995 × 1996

2

T+O467QmnA

a1 = 1KFSA¬J9A-J/0tmxAr2xE3<G2 1

a1996

= 1 +1995 × 1996

2= 1991011

T_ E9AOP5AOhi,3P5A/K

a1996 =1

1991011

T T¦~7'<8*¡/0Q<-P5AY@9,9,9f-mn<D8*AW2¢FS,)46JQAO7/254Um5<D.3@Q,9,9f-8<DP5Al3.6<-mxADJS<-8I8*E9,-F74172xE9Ag?0QP5A9T+O0QlQlQ,38*AY2xE9ASE9AO46gOE2a,/ht2xE9AS@9,9,9f-mn<D8*Aw418

1T,-F 2xE/4UmsfS<-P5AY2xE9AS@Q,9,9f-8*Å

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3Ð DÌ [ D ÍLÐQË ÎOË QÌGÍLË GÏ CwÏ6Í9ÎGËÐÍ s5 Ï Ï 5 Ð9Ð D [ K;Ð9Ì Ë QÏ jÏUÍ3Î-Ï 9Ë ÌdÐÍLË 9 Ï6ÎGËÐ Ì 5 Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì Ï1ÎOË~Ð n [ -ÏUÍ Í3Ï DÌ ÎsÏË#[ Ð 9Ì Ë É dÐÍLËÐÍ Í dÌ-Ì?Í@ O Ë 9Ð dÌ Ë Ì ] Ì-Ë Ì ÍÎ LCkÌ OÏ DÌQÏ jÏUÍ3Î-Ï ÎWÎGÐ 9Ë6Ï~ÐÍ _ E9A8C,/.1032546,37k468a4j.1.j0Q8x2nP<G2nADJ=@D><'J/46<-gGP<-:!<G2I2xE9AAD7QJT

0tAz2n,d2xE9A=<@/0Q7QJ9<D7QmxAd,/hP4UgOE2w<D7Qg?.UAO8S<D7QJ«mx,3:zl9.6AD:=AD7/2<-P~>p<-7Qg?.6AD8?KFSAE3<G3A2¢FS,lQ<O46P8,/hmx,37QgGP0tAD7/2t2nP46<D7Qg?.UAO8HU 4ABI ∼= 4FGJK 4CJB ∼= 4HIG

T\ .1.Lhi,/0QP,/h2xE9AO8*AS2nP46<D7Qg?.UAO8<-P5Ak8x46:41.1<-PCK8x467QmnAS2xE9A*>dE3<?3Amn,3PnP5AO8ClQ,37QJ3417Qg'<-7Qg?.6AD8A-¡/0Q<D.¢T

] +O467QmnA

GI = BJ = 112xE9A=@Q,9,9feE9AG4UgOE2¢|nKLFSA=E3<?3A

HI = CJ = cos θ<-7QJHG = BC = sin θ

T+O417QmxAAH = CF = 1

KtFSAz¡30946msf-. >cmx,37Qmn.j0tJQAk2xE3<G2AI = JF = 1 − cos θ

T¦~7 4ABI

KDFSAwE3<G3Asin θ =

1 − cos θ

BI

KOFSE/4UmsEkg?4o3AO8BI =

1 − cos θ

sin θ

K-2xE9AF)4UJ2xEd,/h2xE9Az@9,9,9f=BI = GJ = CD = FE

|sT'¦~7d<8x46:41.1<-PahU<-8CE/4U,37LK9FSA'Q7QJ2xE3<O2AB =

cos θ(1 − cos θ)

sin θ

T+O467QmnA

1 = AD = AB + BC + CD =cos θ(1 − cos θ)

sin θ+ sin θ +

1 − cos θ

sin θ

KFSASgOA?2

cos θ − cos2 θ + sin2 θ + 1 − cos θ

sin θ= 1

K2 sin2 θ

sin θ= 1

Ksin θ =

1

2

TAO7QmxA/K

θ = 30 K3<D7QJk2xE9AS@9,9,9f)F)46J/2xEk418 BI =1 − cos 30

sin 30 = 2 −√

3T

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

AB C

D

EFGH

I

J

1

cos θ

1 − cos θ

sin θ

sin θ

cos θ(1 − cos θ)

sin θ

1 − cos θ

sin θ

θ

θ

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.................

_ E3<G2Lmx,37Qmn.j0tJQAO82xE/468L418C8x0tAa,/hO2xE9A Ð ÍÌ ©Tu9.UAO<-8CAf-A-AOlw8CAD7QJ/467Qg:AaVa. >9:zl941<Jmx,37/25AD8n258W<-7QJS>3,/0QP734UmnA8C,/.1032546,3798DT

]-`

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K<-7QJ<-7=AGÔADm52

yK9mn<D7'FSAwt7QJ'2xE9ASm5<D0Q8CA

xÅkVWPI41h9FSAf7Q,-F2xE9Amn<O0Q8*A

x<D7QJeAOÔA-m52

yKm5<-7FSA'mn,3:A0QlzF)4o2xE2xE9A':=,9JQAG.

KÅ _ E9AD8CA'<DP5A Ï6Í DÌ Î?ÌlQP5,9@/.UAO:=8OTX,28x0QPnl9P5418s417Qg?.o>KO4673AOP8*ABl9P5,[email protected]:z8mn,3798x254o25032nAB<8x4UgG734jm5<-7/2msE3<-l/2nAOP467k2xE9ASE/418x25,3P~>z,/hL8*mn46AD7QmnA3K3AD7Qg?417QA-AOP5417Qg)<-7QJ:=<G2xE9AD:z<O2546mn8OT

\ 8<z3AOP~>p8s41:=l3.UAAxÆD<-:zl9.6A3KImn,3798s46JQAOP2xE9Az.1417QAD<DPY417/2nAOPnlQ,/.6<G254U,37lQP5,9@/.UAO:cK346A?FSADJ@D>2xE9AB<O032xE9,3P<-82xE9AB,/.UJ9AD8n2l9P5,[email protected]:467):z<O2xE9AO:=<G254Um58DT _ E9AJ/46P5ADm52lQP5,[email protected]:!418a2n,zmn<O.Umn09.1<O25A)2xE9AY/<D.j0tAD8,/h<.1417QAD<DPh60Q7Qm5254U,37T _ E9A4173ADP8CASl9P5,[email protected]:±468a25,JQA?2nAOPn:467QA<).1417QAD<DPh60Q7Qm52546,37hUP5,3:!8C,3:AJ3<O2<SlQ,/467/28(xi, yi)

T_ E9A)<O032xE9,3P@3P546A/>zJ/468Cmn0Q8C8CAD8<730Q:=@QAOP,/hLmn.6<D8C8x4UmI4173ADP8CA)l9P5,[email protected]:z8ahUP5,3:E/468n2n,3P~>K417Qmn.10tJ/467Qg \ P5msE/41:ADJQAO8GÓ¨t<O2xE 125A-msE37346mn<O.1. >LK *7Q,37QJQAO8x2nP0tm5254o3AA*/<D.j0Q<254U,37 s|nK32xE9Aw2¢FS,9C@9,9J> Z AOl9.6ADPal9P5,[email protected]:!<-7QJKQ:=,3P5ASP5ADmxAO7/25.o>KQmx,3:zl90325A-J25,3:,9gOP<-lQE->) <J9,37Y2nP<D798shi,3P5:k|sT _ E9AO8*AI3AOP~>YP5AD<-J9<[email protected]/4UgG7QA?22nAO88CE9,/09.UJ41:=l9P5AD8s80Qlt,372xE9AwP5AO<J9ADP2xE9AY41:=lQ,3P2<-7QmnA),/h94673AOP8*AYl9P5,[email protected]:z8417S2xE9AwE/468n2n,3P~>k,/hQ:=<G2xE9AD:z<O2546mn8<-7QJ8*mn46AD7QmnA3KDFSE/46msE8CE9,/09.UJz@9A)P5AD<D8*,37zAD7Q,/0tgOE'25,'467Qmn.j0tJQAY2xE9AO: 467k0Q7QJQAOP5gGP<J/0t<O25AADJ30tm5<O2546,37T),-FSA*3ADPCKG2xE9AOP5AW418<B:=,3P5AW41:=lQ,3P2<-7/2lt,/417/2:=<-JQAB@D>Y2xE9AB<O032xE9,3P4672xE9Az8CA-m52546,37cAD7/254o25.UADJ C [ OÌ /5 ~Í DÌ ÎGÌ Ð~ÌzÎ K79<D:AG.o>K2xE3<G2B25AD<-msE/467Qg4673AOP8*Al9P5,[email protected]:z8B7Q,/0QP5418*E9AO8B<'E3<-@34o2a,/h *4673AOP8*Aw2xE/417Qf-417Qg 467z8n250tJQAO7/258OTÇD,9,9f467Qg,373.o><O22xE9AYJ/46P5ADm52l9P5,[email protected]: 41825,:z418C8I:0tmsE',/h/2xE9AWFSE9,/.6Al34Um5250QP5A9T _ AD<-msE/467Qg,373.o>J341P5A-m52a:A?2xE9,9J98B468?K9417=hU<m52*K9417/2nAG.1.6A-m5250Q<O.1. >z.141:z4o25467Qg3TB _ E/418P5A*/4UA©FSADPa<gGP5ADAD8FSE9,/.UADE9AD<DP25A-J/.o>SF)4j2xE'2xE9A)<D032xE9,3PCÓ 8B8CAD7/2541:AO7/258OT |

] _ E9Ac7QAsÆO2Shi,/0QP'msE3<Dl325ADP8z,/ha2xE9A@Q,9,9fp<DP5AJQA*3,25A-J«25,r4173ADP8CAclQP5,9@/.UAO:=8467^l9P5A-m5<D.6mn09.10Q8?Kwm5<D.6mn09.10Q8?KYJ34jÔAOP5AD7/2541<D.YAD¡30Q<G254U,3798<D7QJ .j467QAO<-P<O.UgOAD@9P<3KYP5AO8ClQA-ms254o3AG.o>T /<msEpmsE3<-l/2nAOPE3<D8S8x4 Æ *:,9J/09.UAO8 S2xE3<G2mn,3798s418x2),/hU64 |<-7467/2nP5,9J/0tm5254U,37hi,3P2xE9A46798n2nP50tm525,3PIUg?4o/467QgB2xE9Awmn,/0QP8*A.UA*3AG.Q,/h32xE9AYlQP5,9@/.UAO:68n|,/htmn,37QmxAOPn7LK/gO,3<D.18GK:=<G2xE9AD:z<O2546mn<O.<-7QJ8Cmn4UAO7/2541tm@9<-msf-gGP5,/0Q7QJeP5A-¡/0946P5ADJd,/hL2xE9A'8x250tJ9AD7/2*Kt<-7QJ7QADmxAO8*8C<DP~>)E3<-P5J-F<-P5A -8C,/h12¢F<-P5A©|xKD1414 | \ m5254 34o254UAO8;iAxÆ-ADP5mn418*AO8;<D7QJmn,3:=l30325<G254U,37985|xKO<D7QJ1414j4o|XY,25AD8w<-7QJ«¤-0QP2xE9AOPLAD<-J3417Qg3T \ 7985FSADP8w<-7QJd<J-346mxAkhi,3P8CAO.6A-m525A-JdlQP5,9@/.UAO:=8w<-P5Ahi,/0Q7QJ)417 \ lQlQAD7QJ/4 Æ \ T \ l9ltAO7QJ34 ÆS¨'mx,37/2<D417988C,/0QP5mxAmn,9JQAO8;hi,3P¥ \_ Ç \ ¨SP5,/0325417QAD82xE3<O2Ymn<D7p@QA0Q8CA-Jdhi,3P8C,3:Az,/h2xE9A=mn,3:=l30325<G254U,3798w417d2xE9A=@9,9,9fTz _ E9AD8CA=8C,/0QP5mxAmx,9J9AD8Bmn<D7z<D.18*,@QAJQ,-F73.6,3<J9A-JkhUP5,3: 2xE9A)<D032xE9,3PCÓ 8FSAD@98x4j25AQT |¤3,3PAxÆD<-:zl9.6A3Ka2xE9Ap:,9J/09.UA«AO7/254j25.6A-J Q Ì 417^RLE3<-l/2nAOP=/K iÍ DÌ Î?Ì

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T<-7QJ

fTUVW7QAF)4j.1.7QA-ADJ _ ,3P5P546mxAG.1.j4Ó 8zÇO<GFKJ/468Cmn0Q8C8CA-Jc467d2xE9AeAO<-P.146ADP:,9J/09.UA Ï¢ËË ~Ì GÏ Ë T | _ E9AkJ/46P5ADm52WlQP5,9@/.UAO: 418W2n,zQ7QJ

Tg?4o3AO7

fU \ 730Q:@9ADP,/h3h60Q7Qm52546,3798

f<-P5A8x0tgOgOAD8n2nADJ)467w2xE9A \ m5254o/4j2546AD8;8*ADm5254U,37T;+G250tJ9AD7/28<DP5A<D.18*,w<-8Cf-A-J2n,<-8CmxAOP2<D417d8C,3:A':=<G2xE9AD:z<O2546mn<O.lQP5,3lQADP¢254UAO8w,/h

TT _ E9A'4173ADP8CAl9P5,[email protected]: 41825,t7QJ

fg?4o3AO7

TT _ E/468)417Qmn.10tJ9AD8w2xE9Amn.1<-8s8s46m' ~Ì Î [ e,3PBF<G2nAOPxsmn.U,9msflQP5,9@/.UAO:6Q7QJ3417Qg

f8*,S2xE3<O2

T (y)468a.j467QAO<-PI467

y|sT

_ E9AOP5Ad<DP5Ae<D.18*,«,3l9lt,3P¢250Q734j2546AD825,rAxÆl3.U,3P5A:,3P5AeP5AO<D.j468n254Um<D8ClQA-m528k,/h2xE/4684673AOP8*AlQP5,9@/.UAO:cKLhi,3PAxÆD<-:zl9.6A3KQFSE9AD7e2xE9AzJ3P<D417Qn2541:Akh60Q7Qm5254U,37T468Y7Q,2Yg?4 3AD7467mn.6,38*ADJShi,3Pn:dK/@3032IP<O2xE9AOP417S2xE9AYhi,3P5: ,/hJ3<O2<)lQ,/467/28W1hi,3PIAsÆD<D:=l3.UA/K/8n2n,3lF<O25msEAsÆ-ltAOP541:AO7/2585|CT _ E9A8x250tJ9AD7/2468W<-8Cf-A-Jk25,zAsÆ-l9.6,3P5A8*,3:=A<DlQl9P5,?Æ-41:=<G254U,37z8CmsE9AD:=AD8U730Q:=ADP4Um5<D./467/25A-gGP<G254U,37S<D7QJ'J34jÔAOP5AD7/2541<O2546,37|0Q8x467Qga2xE9A¥ \_ Ç \ ¨kl9P5,9gGP<-:z8lQP5,9346JQADJT _ E9AB<mnmn0QP<m>S,/h-2xE9AD8CAB8*msE9AO:AO8;4682n,)@9AAxÆD<-:467QADJ)<D.6,37QgF)4o2xEY2xE9AAGÔADm5258,/hL7Q,/468CAw467k2xE9AJ9<G25<QT

_ E9AwmsE3<-l/2nAOP,37S4673AOP8*AYl9P5,[email protected]:z8467S.1417QAD<DP<O.UgOAD@9P<Y467Qmn.j0tJQAO8lQP5,9@/.UAO:=8417AO.6AD:=AD7/2<-P~>=25,3:,9gGP<DltE->KgGP<?341:A?2nP~>e<-7QJ4173ADP8CA/4U@3P<O2546,37T _ E9Ak:=<G2xE9AD:z<O2x4Um5<D.:A?2xE9,9J98'0Q8CA-Jp2xE9AOP5Ae467Qmn.j0tJQAdl9P5, A-m52546,3798,37/2n,rE->3ltAOPnl3.6<D7QAD8?K:=<G2nP54 Æ<D7QJgOAD7QAOP<D.j4j£sADJz4673AOP8*AO8GK9<-7QJAG4UgOAO7/<D.j0tAS<-79<O.o>38s418DTB¤/,3PWAxÆD<-:zl9.6A3K/2xE9AS:=,9J309.6A'AD7Q254j25.6A-J DÇ/Ó \ _ u3,/0QP'Ç/Ó \ P¢2AsÆD<D:z417QAD8w2xE9Ae /Ì Ï w Ì ÐÍ9ÎGË Ë6Ï~ÐÍ ËÌ 5 Í/Ï /Ì \ _ |Y<O.UgO,3P4j2xE3:dK;FSE/4UmsEp0Q8CAD8'8x0tmxmnAD8s8s4 3Ad,3P2xE9,9gO,379<O.lQP5, ADm5254U,37982n,p<-l9lQP5,?ÆD4U:=<G2nA8C,/.1032546,3798S,/h.1417QAD<DP)A-¡/0Q<O2546,3798DT \ 8s41:=l3.UA25,3:,9gGP<DltE->plQP5,9@/.UAO: 468w2xE9AD7mx,3798x4UJ9ADP5ADJT#;,/0S:z46gOE2@9A<@/.UAa2n,wt7QJ8C,3:A,/h2xE9AB4673AOP8*Al9P5,[email protected]:z8;2xE3<O2<DP5AW2nP5AD<G2nADJ4672xE/418@Q,9,9f'8*m5<O225ADP5ADJ2xE3P5,/0tgOE9,/032a8*,3:=A8x2<-7QJ3<-P5JtP8n2xW<-7QJ=8CA-mn,37QJQ>3AD<DPamn<O.Umn09.10Q8<D7QJw.1417QAD<DP<O.UgOAD@9P<W25AsÆO258OTI,-FSA*3AOP*K©>3,/0Sl9P5,9@9<-@3. >FS,37LÓ 2Lt7QJY2xE9AO8*AWlQP5,[email protected]:=8Y2nP5AO<O25A-JF)4o2xEe2xE9A=m5<-P5Az<D7QJlQ,38C8x4U@/.o>2xE9A=JQAOl32xEe2xE3<G22xE9A*>cP5A-mnAO4 3A467e2xE/468@Q,9,9ftT«X,3PBF)41.j.>3,/0pQ7QJp<D7>FSE9AOP5AeAG.68CA=8s0tmsE<emn,3:=l9P5A-E9AO798s4 3Aemn,/.1.6A-m52546,37p,/h

C4673AOP8*A2xE/467QfD467Qg B0Q7QJ9ADP,37QA)mn,D3AOP?T _ E/468W@9,9,9fYFS,/09.UJk8CADP~3Aw<-8</<D.j0Q<@/.UAwP5A-8*,/0QP5mnA)7Q,2,373.o>'467#;AD<DPM)<-7QJ=R<D.6mn09.10Q8a<-7QJdÇG417QAD<DP \ .6gOA-@3P<Smx,/0QP8CAD8W@3032<D.18*,

] 467=mn,/0QP8*AO8,37=J/41ÔADP5AO7/2546<O.A-¡/0Q<O2546,3798B<D7QJdXYA©Fw2n,37341<-7z:=A-msE3<D734Um58DT

¥>',373.o>'mn,3:=l3.6<O467/2;<@9,/0322xE/418@9,9,9fY468mx,37QmnADP57QA-JwF)4j2xEw2xE9A730Q:@9ADP467Qg,/h8*ADm5254U,3798a417 \ lQlQAD7QJ/4 Æ \ K 3Ì"Ì ËÌ aÍ9Î YÌ Î Í ?Ï Ì U;+-ADm5254U,37A.m.n

4172xE9A\ lQlQAD7QJ/4 Æmn,3PnP5AO8ClQ,37QJ982n,:=,9J309.6A(m−1).n

4172xE9Aa:=<O46725AsÆO2?TI¦Q7QJ2xE3<O2¦/:z0Q8n2<D.1:,38n2a<D. F<G>38gO,=@9<-msfS2n,k2xE9A _ <@/.UA',/hRL,37/2nAO7/258W25,zP5AG250QP57=hUP5,3: <'8C,/.1032546,3725,2xE9AB<DlQl9P5,3lQP46<G2nAB:=,9J309.6AW417Y2xE9AW:=<O467Y25AsÆO2?TI¦FS,/09.UJ)E9,3ltA2xE3<O2h603250QP5AA-J/4j2546,3798,/h2xE9AS@9,9,9fwFS,/09.UJ<G2.6AD<D8x2<-JQJ'2xE9A)8x2<-P¢25467Qgwl9<gOA,/hQ2xE9A:=,9J309.6AY2n,S2xE9A8x0t@/254o25.UAO8,/ht2xE9A \ lQlQAD7QJ/4 Æhi,3PAxÆD<-:zl9.6A 1Ï 1Ï -ÏUÍ [ 6l;T9N ` |CT,-FSA*3AOP*K2xE/418a468W@3032<k:467Q,3PW@/.UAO:z418*E467FSE3<G2a418<S0Q734U¡/0tA'<-7QJE/4UgOE/. >P5ADmx,3:z:AO7QJQADJd@9,9,9fQKtmn.6AD<DP5. ><.1<@9,/0QPa,/h.6,D3A@D>z<D7=AsÆ-ltAOP2417k2xE9A)tAO.6JT

Q ~ÌGÍ OÏ6Í 5 ÏUÍLËÌ Î?Ì Î@D>¨ADP579<-P5J9, LA-m5<-:<-7 +D<D7/2n,38?Kl90t@/.1418*E9ADJ^@D>^+G25ADP.1417Qg u90t@/.1418*E/417QgpR,TUKw¦~7QmGTUKM-N-N3¦+9¨X -` N-*3-D 9K98C,/h12nmn,D3AOP*K9N ] l9<gOAO8GK9ª+DNtTjN

Ì ©ÏÌ YÌ [ # Í/ÏDÌ ÎxÏ¢Ë-[Ð C ËÌ Ð9Ð C Ë~Ì ~Ð9Ð ÍLË D ÏÐ T_ E/418;@Q,9,9fW468<mn,3:=l341.1<O2546,37w,/h:z<O2xE9AO:=<G254Um5<D.-gG<D:AO8;<D7QJwl303£5£n.UAO8FSE/4UmsE2xE9A<D032xE9,3PE3<D8wAG4j2xE9AOPmxP5AD<G2nADJd,3Pmn,/.1.6A-m525A-Je,D3AOPa2xE9A>3AO<-P8OTS)AJ34 346JQAO82xE9A'lQP5,[email protected]:=8a467/25,khi,/0QPam5<O25A-gO,3P4UAO8HU \ P54o2xE3:A?254Um5<D.;u903£5£n.UAO8GK3HIA-,3:=AG2nP4Um5<D.;u903£5£n.6AD8?KLÇD,9g?46mn<O.u903£5£n.UAO8GKD<D7QJ \ .6gOA-@3P<D46mWu903£5£n.6AD8OT _ E9Al903£5£n.6AD8P<-7QgOABhUP5,3: 8x46:zl9.6A<O.UgOAD@9P<O4UmFS,3P5JlQP5,9@/.UAO:=82n,8C,3:A)P5AG.6<G254o3AG.o>zJ/41ymn09.j2amx,3:=@34179<O25,3P541<D.tl9P5,[email protected]:z8DT3<-msE8CA-m52546,37zE3<-8M [ ,3PM¡30tAO8x2546,3798DTa¥r<D7>z,/ht2xE9A)l9P5,[email protected]:z8W<-P5A)l9P5,9@9<-@3. >zhU<D:z4j.141<-P2n,P5AD<-JQAOP8,/hL8x0tmsEz<-P¢254Umn.6AD8OTW¦~72xE9A)hi,3P5A©FS,3P5JLK/2xE9A<D032xE9,3PaAxÆl3.6<O467982xE3<O2:,38n2,/hQ2xE9A)l9P5,[email protected]:z8W<DP5A,37QAD82xE3<O2E9AE3<D8Bmn,/.1.6A-m525A-JK/<-7QJz,373. ><)hiA©F <-P5A,3P546g?4679<O.T _ E9A8*,/.j03254U,3798l9P5,D/4UJ9A-J)hi,3PL2xE9Al9P5,[email protected]:z8IP<-7QgOABhUP5,3: 8s41:=l3.UAB<-7985FSADP82n,):,3P5Amn,3:=l3.UA?2nAJ/468Cmn0Q8C8x4U,3798I,/h/2xE9Al9P5,[email protected]:T¦~7S8*,3:=Am5<-8CAD8?KG2xE9A<D032xE9,3P8s0tgOgOAD8n258a/<DP541<O2546,3798,372xE9A'l9P5,[email protected]:dK,3PI2xE9A'l9P5,[email protected]:z8B467/2nP5,9J/0tmxAS0Q730Q8x0Q<D.L:=<G2xE9A-:=<G254Um5<D.t25ADP5:=88x0tmsE=<-8 ©AxÆ-,254UmW730Q:=@QAOP8 T _ E9Ak@9,9,9f'468BlQP5,9@3<@/.o>=8s094o25<[email protected]<-8<P5AD8C,/0QP5mxAhi,3PWE/4UgOE=8CmsE9,9,/.,3Pa8CAD7346,3PWAG.UAO:AO7/25<DP~>8CmsE9,9,/.t25AD<-msE9ADP8OT _ AO<msE9AOP8Ym5<-70Q8*AS2xE9A'lQP5,9@/.UAO:=8417=2xE9AO41Pmn.6<D8C8sP5,9,3: <D8wAO79P546msE3:AO7/2*K,3PW<-8<zm5<O2<D. >98n2a25,8x2<-P¢2J3418*mn0Q8s8s417Qg),2xE9ADP:=<G2xE9AD:z<O2546mn<O.mx,37QmnADl/258OT\ .j2xE9,/0tgOEeADJ30tm5<O25,3P8wmn<D7e<D. F<G>38Q7QJe<kl9.1<mnA,37=2xE9AG46PB@Q,9,9f-8*E9AG.1hhi,3P2xE/4682>3ltA,/hmn,3:=l341.1<O2546,37LK Q ~ÌGÍ OÏ6Í Ï6ÍË~Ì ÎGÌ Î 4687Q,2S<-7 AxÆD2nP<-,3P5J34179<-P~>¬mx,/.6.UADm5254U,37%,/hWlQP5,9@/.UAO:=8k<D7QJr8*,/.j03254U,3798OT _ E9Ac8C,/.1032546,3798'8*ADm5254U,37%mn,/09.UJ%E3<?3Ac@QADAD7AsÆ-lQ<D7QJQADJz8s46gG7341tmn<D7/25.o>T¦JQAO<D.j.o>k4o2FS,/09.6Jk467Qmn.j0tJQA):,3P5A)lQP5,9@/.UAO:x8*,/. 3417Qg25A-msE9734U¡/0tAD8?KG:,3P5A/<DP541<O2546,3798Lhi,3P92xE9AlQP5,9@/.UAO:=8?KG<D7QJY:=,3P5Aamx,3:zl9.6AG25A8*,/.j03254U,3798LP<O2xE9AOP2xE3<-7 0Q8n2W<-7985FSADP8OT \ 8<k8CAG2B,/h;mn.1<-8s8s46mB:z<O2xE=lQP5,9@/.UAO:=8W2xE/468@Q,9,9f'468<J9A-¡/0Q<O25A3K@303241hQ>3,/0=<DP5A).6,9,9f-417QgYhi,3P:,3P5A>3,/0kF)41.j.t7Q,2Q7QJ4j2E9ADP5A9T

] N JS)I/K I 5D/ME LNMPOROO :GJI RI DS PE

variance ≤ (M − x)(x − m) & C

2 $s#'(M 6 2 01, TbcE9AD7QA*3AOP¦/lQP5,O3A-JY<D7417QA-¡/0Q<D.j4j2~>0Q8x467Qgm5<D.6mn09.10Q8:k>B25AD<-msE9ADP<O.oF<?>98L<-8Cf-A-J:A/K ©+-A-Ak4jh>3,/0pmn<D7cl9P5,D3Ak4j2aF)4o2xE9,/0320Q8x467Qgkmn<O.Umn09.j0Q8 T=¨E3<O2541<z<-7QJ a<?3418 M gG<G3AY<w7QA?F @9,/0Q7QJ'hi,3P2xE9AB/<-P46<D7QmxA/K/<D7QJS2xE9AG46Pl9P5,9,/hQ0Q8*ADJkmn<O.Umn09.j0Q8DT _ E9Aw,9@ ADm52,/h;2xE/468w7Q,2nA4182n,dl9P5AD8CAD7/2<-7AG.UAO:AO7/25<DP~>clQP5,9,/h,/h;2xE9AO41PP5AO8s09.o2WF)4j2xE9,/0320Q8x467Qgmn<O.Umn09.j0Q8DT¦~7'2xE/468al9<-lQADPFSAw<-8s8s0Q:=A2xE3<G2

x1K. . .

Kxn

<DP5AwP5AD<O.Q730Q:@9ADP8OT LADmn<O.1.32xE3<G22xE9AY/<-P46<D7QmxA,/ht2xE9AD8CA)730Q:@9ADP8W468BJ9AOQ7QA-Jk25,z@9Aσ2 =

1

n

n∑

j=1

(xj − x)2KFSE9ADP5A

x =1

n

n∑

j=1

xjT

ÇDAG2M = maxx1

Kx2K. . .

Kxn <-7QJ m = minx1

Kx2K. . .Kxn K<D7QJp8*A?2

R = M − mT _ E9Aemn.1<-8s8s46mn<O.@Q,/0Q7QJ«,37c2xE9A/<-P46<D7QmxA/Kf-7Q,-F725,8x250tJ9AD7/28',/h8x2<O25418x2546mn8?K468

σ2 ≤ R2

4

T ~M©|¦~7 M '¨E3<O2541<<-7QJ a<?3418WlQP5,O3A-J

σ2 ≤ (M − x)(x − m)T ~©|

¦U2W418AD<D8>2n,=8CE9,-FÈ2xE3<G2a2xE9ASP546gOE2E3<D7QJ8s46JQA',/h;i?|I418B.UAO8C8W2xE3<D7d,3PWA-¡/0Q<D.t2n,2xE9AP546gOE2E3<D7QJ8s46JQA',/h;~M©|U<D8WFSAwF)41.j.JQ,zhi,/.j.U,-F)417Qgw2xE9ASlQP5,9,/h;@9AO.6,-FY|CT _ E/4188CE9,-F82xE3<O2I2xE9A7QA©F @Q,/0Q7QJ418W8*E3<DPnlQADP2xE3<-7k2xE9ASmn.1<-8s8s46mn<O.@9,/0Q7QJT 7%5B% 24. $O+#$s3HC T

H4 3AD7r2xE9Alt,3l309.6<G254U,37x1K

x2K

. . .K

xnKW7Q,2nAd2xE3<G2'4o258S/<-P46<D7QmxA/KW:AO<-7LK:=<©Æ-41:z0Q:dK/<D7QJk:467341:z0Q:<-P5A467/<DP541<-7/20Q7QJ9ADPI<wlQADP5:z032<O2546,37z,/h32xE9A

xjÓ 8DTbcA/K2xE9ADP5AGhi,3P5A3K9AD8n25<[email protected]?|<D8C8x0Q:z417Qg

x1 ≤ x2 ≤ . . . ≤ xnT ] |

XY,-F ] |;41:=l3.146AD8M = xn

<D7QJm = x1

T _ E/0Q8GK-FSAF)41.j.tlQP5,O3Aw2xE9Aw467QAD¡30Q<O.14o2>σ2 ≤ (xn − x)(x − x1)

T [ |LA-m5<D.j.Q2xE3<G2

nσ2 =

n∑

j=1

x2j − nx2 T¦~7QA-¡/0Q<D.j4j2~>= [ |;468BAO8x2<@/.1418*E9ADJ=@D>8*E9,-F)417Qg

n(xn − x)(x − x1) − nσ2 ≥ 0T ~©|

"!$#&% ')()*c© +-, , . 0/-1$/"2 % /-1435/6*6( 7)89/6*:% ;</-=0>-$;:% 7:*&!

] M _ E9A).6AOh12E3<-7QJz8x4UJ9AS,/hL~©|Imn<D7@9AYFP4j225AD7z<D8

nx(xn + x1) − nx1xn −n∑

j=1

x2j =

n∑

j=1

xj

(xn + x1) −n∑

j=1

x1xn −n∑

j=1

x2j

=

n∑

j=1

(xjxn + xjx1 − x1xn − x2j )

=n∑

j=1

(xn − xj)(xj − x1)T

_ E9A.6<D8x2AsÆ-lQP5AO8C8x4U,37p468'mn.UAO<-P.o>rgGP5AO<O25ADPY2xE3<-7%,3PA-¡/0Q<D.I2n,c£sADP5,KI8s417QmxAzFSAdE3<?3A<-8s8s0Q:=A-J«2xE9AxÓ 8<-P5Ac7Q,37QsJQADmxP5AD<D8s417Qg3T _ E/418zmx,3:zl9.6AG25AD8S2xE9Acl9P5,9,/h,/h [ |w<D7QJE9AD7QmnA),/hti?|<D8FSAG.1.¢TXY,25A2xE3<G2FSAwE3<?3AwA-¡/0Q<D.j4j2~>S4672xE/418417QA-¡/0Q<D.j4j2~>S41hQ<D7QJk,373.o>41h<O.1.8C<D:=l3.UAw/<O.10tAO8<-P5A'A-¡/0Q<D.t2n,=AO4o2xE9ADP

x1,3P

xn 2xE3<O2468?K941h<-7QJ,373.o>z4jhL2xE9AOP5A<-P5A)<O2:=,38x22¢FS,8s<-:zl9.6AY/<O.10tAO8DT

_ ,l9P5,D3Aw2xE3<G2I2xE9AS@9,/0Q7QJ468W8CE3<-P5ltAOPI2xE3<-7R2/4

K-FSA,9@98CADP~3A14(xn − x1)

2− (xn − x)(x − x1)

= x2 − x(x1 + xn) + x1xn + 14(xn − x1)

2

= x2 − x(x1 + xn) + 14(xn + x1)

2

=

(

x − x1 + xn

2

)2

≥ 0K

FSE/4UmsElQP5,O3AD82xE9ASmn.1<D41:pT¤3,3P,2xE9ADPB0Q8*AGh609.IAG.UAO:AO7/25<DP~>clQP5,9,/hU8),/hI8n25<G25468n254Um5<D.467QAD¡30Q<O.14o254UAO8GK8CA-A -KFSE/4UmsEP5A-mnAO4 3A-J'2xE9ASHAD,3P5gOAzu

,/.o>/<)<?F<-P5JT

6, 9(=):%('7(%5<% 2 " Q_ E9AW<O032xE9,3PQF)418*E9AO8L2n,wAxÆl9P5AD8s8E/418gGP<G254j250tJ9A2n,aP?TORt.14jÔ,3P5JzT9bd<-gG7QADPFSE9,z:z<J9AS8CA*3ADP<O.8x2~>3.j468n254Uma8x0tgOgOAD8n254U,3798B<D7QJm5<-P5AGh609.1. >msE9ADmsf-A-J2xE9A<D.6gOA-@3P<9K<D7QJc25,¬uQP5,/hiAO8C8C,3Pbd<DPnP5AO7 u/<-gOAFSE9,c<D.18*,msE9ADmsf-A-Jd2xE9A<O.UgOAD@9P<QT _ E9A<D032xE9,3P<O.68C,S2xE3<-7Qf-8a2xE9AP5AGhiADP5ADA<D7QJk2xE9ASADJ34o2n,3PIhi,3P<D746:zlQP5,O3A-Jzl9P5AD8CAD7/2<O2546,37T %3C %$&% D63%H"M < AD7QJ3P<p¨E3<G2546<<-7QJcRE3<D7QJ3.6ADP a<G/468?K 9ÌËË~Ì 9ÐDÍcÐÍrË Ì - Ï Í Ì K_ E9A \ :AOP546mn<D7d¥r<O2xE9AO:=<G254Um5<D.¥%,37/2xE/.o>K

107~ |nKLXY, [ K3lQlT ] ] ] /3T

'bd<DPnP5AO7¬u/<gOAz<-7QJ§YTUX'TI¥«0QP¢2>K?] Ì D ÍLÌ©ÎCÎ Ì Ë6Ï~ÐÍ9Î dÐÍ dÌ Î Ì©ÎÐ Ì?ÍË ËÌ?ÍQÌ?Í [ Í -Ï1Î Ì ÎxÏÐÍ Ò - Ë w Í - Ë 3K _ E9A _ FS, #;AO<-PR,/.6.UADgOAk¥r<O2xE9AO:=<G254Um58 C,/0QP579<D.13iM-N/©|xKLX,=3K3<D7QJ

14~MDN ] |nKLXY,=M9T§IA-J/09.6<zXSTL¥«0QP2~>

/:AOP54o250Q8)uQP5,/hiAO8C8C,3P,/h¥r<G2xE9AD:z<O2546mn8W<D7QJ=+G25<G25468n254Um58u3AO797Q+G25<G2nA/K)<DPnP468C@30QP5g¥«4UJ9J3.6AG25,-F7LKLu \ K9ª+ \ M /[email protected]

] M-M )K JS)I MPMPI (Q S PE M I/M )O OJM

f(X2)

3s(*$ k" wB(

ÇDAG2k@9Ae<-7>ptAO.6JLKI<-7QJp.6AG2

f(X)@9Ae<-7«<-Pn@34o2nP<-P~>«lQ,/.o>37Q,3:z41<D.,/h

k[X]FSE/4UmsEw468;46P5P5A-J/0tmn4U@/.UAB417k[X]

T \ FSAO.j.Usf7Q,-F7)P5AD8x09.j2;,/h/bd<E/.6AD7QsRL<DltAG.1.j4368*ADA M -Kl;T-M©|AO8x2<@/.1418*E9AO8)7QADmxAO8C8s<-P~>c<D7QJ8x09yemn46AD7/2wmn,37QJ34o254U,3798Yhi,[email protected]>,/hf(g(X)

) 467k[X]

KFSE9AOP5Ag(X)

418<-7>lt,/. >97Q,3:46<O.,/hk[X]

T _ E9Al9P5,9,/ha,/h2xE/468WP5AD8x09.j2418W7Q,2aAG.UAO:AO7/25<DP~>=@QADmn<O0Q8*Aw4j20Q8CAD82xE9AY2xE9A-,3P~>=,/htAG.UJzAsÆO2nAO798s46,3798DT¦~72xE/418B8CE9,3P2<-P¢254Umn.6AYFSAAO8x2<@/.1418*EK-F)4o2xE<Y3ADP~>=AG.UAO:AO7/25<DP~>zlQP5,9,/h¢K37QADmxAO8*8C<DP~>'<-7QJ'8x09yemn46AD7/2Imx,37QJ/4j2546,3798hi,3P2xE9AYP5A-J/0tmn4U@/41.j4j2~>k,/h

f(X2)467

Z[X]KOFSE9AOP5A

ZJQAO7Q,2nAO8w<-7c<DPx@/4j2nP<DP~>d0Q7346¡30tAkhU<m525,3P54o£x<O2546,37cJQ,3:z<D417T \ 8Y<-7d41:=:=A-J/46<G2nAmx,3798CA-¡30tAO7QmxAWFSAY,9@25<O467S<8s41:=l3.UA8s09ymn4UAO7/2Imn,37QJ34o254U,37hi,3PL2xE9A46P5P5A-J/0tmn4U@/41.j4j2~>',/hf(X2)467

Z[X]T

,- %D,$&%5 @Q ÌËf(X)

9Ì Í [=ÍLÐÍ Ì Ð Ð-[-ÍÐ Ï ;Ï6ÍZ[X]

Ì ¢Ð ÐaÏ6ÍÎOË Ë~ÌdÌ?ÍË1Î D ÌSÌOÏ ~ÌGÍLË14o|

f(X2)Ï6Î Ì Ï ÌÏUÍ

Z[X]

64j4o|f(X)

Ï6Î Ì Ï ÌYÏUÍZ[X]

Ð Ë Ì ÌkÌÊ-Ï1ÎOË Ð-[-ÍÐ Ï 6ÎG(X)

H(X)

Ï6ÍZ[X]

Í -Í/Ï¢ËuÐ

Z (Ë Q ËtÏ6Î Í'Ï6Í DÌ Ë6Ï ~ÌÌ"ÌdÌGÍLË;Ð

Z \ 0)Î 5Ë Q Ë

uf(X) = G2(X) − XH2(X)T

?|

#$s3HC T;bcAYQP8x2I8s0Ql9lt,38CAY2xE3<G21414 |4682nP0tAQT¦U2468Wmn.6AD<DP;2xE3<O2f(X2)

[email protected][X]

41hf(X)

468OT _ E9AO78s0Ql9lt,38CA2xE3<G2f(X)

468W46P5P5A-J/0tmn4U@/.UA467Z[X]

T _ E/0Q8?K(?)468B2nP50tA)F)4j2xE

H(X) 6= 0T \ 8<zmn,3798*AD¡30tAO7QmxA/K14o|hi,/.j.U,-F8?K@9A-m5<D0Q8CA

G(X2)<D7QJ

XH(X2)E3<?3ASJQADgGP5A-AO8,/hJ/468n25467Qm52lQ<DP54o2><-7QJ

uf(X2) =(G(X2) − XH(X2)

)(G(X2) + XH(X2)

) TXY,-Fv8x0QlQlQ,38*AW2xE3<O264 |468;2nP0tAQT \ 8C8x0Q:A

f(X)[email protected]

Z[X]i,2xE9ADPF)468CAaFSA<-P5AJ9,37QA?|sT _ E9AD7

f(X2) = g(X)h(X)KOFSE9AOP5A

g(X)Kh(X) ∈ Z[X]<-P5A)7Q,20Q734j28,/h

ZTIRL,/.1.6A-m525417Qg)A*3AO7=lQ,-FSADP8a467

g(X)<-7QJ

h(X)K-FSA,9@/2<D417

g(X) = G(X2) + XL(X2)K

h(X) = H(X2) + XT (X2)K~M©|

hi,3P8C,3:A)lQ,/.o>37Q,3:z41<D.18GKLKHK/<-7QJ

T417

Z[X]TAO7QmxA/K

f(X2) = G(X2)H(X2) + X2L(X2)T (X2)

+ XG(X2)T (X2) + XL(X2)H(X2)T ~©|

"!$#&% ')()*c© +-, , . 0/-1$/"2 % /-1435/6*6( 7)89/6*:% ;</-=0>-$;:% 7:*&!

] M-hi,3P8C,3:A)lQ,/.o>37Q,3:z41<D.18

GKLKHK/<-7QJ

T417

Z[X]T

bcAmn.1<D41:!2xE3<O2L(X)T (X) 6= 0

TYbcA'lQP5,O3AS2xE/468Y@D>dmn,37/2nP<J/4Um52546,37T)+O0QlQlt,38CAkhi,3PAsÆD<D:=l3.UAS2xE3<O2L(X) = 0

j2xE9Azmn<D8*AT (X) = 0

418Y<-79<O.U,9gO,/0Q85|CT _ E/0Q8?KFSASE3<?3Af(X2) − G(X2)H(X2) = XG(X2)T (X2)

T ] |¨,2xEp8x4UJ9AD8S,/h2xE/418'AD¡30Q<O.14o2>p<-P5Az£sAOP5,p@QADmn<O0Q8*A/K,2xE9AOP¢F)418*A/K2xE9A*>rE3<?3AeJQADgGP5A-AO8,/hJ34jÔAOP5AD7/2'lQ<DP54o2>T _ E/0Q8?KT (X) = 0

FSE9AD7QmnA3Kf(X2) = G(X2)H(X2) 2xE3<O2B418GK

f(X) = G(X)H(X)KtFSE/46msEdmx,37/2nP<-J346m52582xE9A<D8C8x0Q:=l/254U,372xE3<G2

f(X)[email protected])467Z[X]

T¦U2mn<D7)@QAa<-8s8s0Q:=A-J2xE3<G2t2xE9ABgGP5AO<O25AD8n2mx,3:z:,37wJ34 3418*,3PL,/h

G(X)<-7QJ

L(X)K8C<?>

D(X)K418AD¡30Q<O.2n,

1K;@9A-m5<D0Q8CA3K;,2xE9AOP¢F)418*A/KFSAzmx,3798x4UJ9ADPB2xE9AhU<-m52n,3P4j£x<G254U,37

f(X2) = g ∗(X)h∗(X)KF)4o2xE

h∗(X) = D(X2)h(X)<D7QJ

g ∗(X) =g(X)

D(X2)=

G(X2)

D(X2)+ X

L(X2)

D(X2)

KFSE9ADP5A=8x0tmsEc<emn,37QJ34o254U,37c418)8C<G25468xADJTdXY,25A2xE3<G2467p,3P5J9ADPB2n,cP5AOl9.1<mnA

g(X)@D>

g∗(X)KFSA7QADA-Jc2n,pf-7Q,-F±2xE3<O2

g∗(X)4687Q,2w<0Q734o2,/h

ZT«¦ih4j2FSAOP5A<=0Q734j2*K2xE9AD7

L(X2) = 0hUP5,3: ~M©|FSE/46msEr41:=l3.146AD8

L(X) = 0TÈ¨Q032)2xE/468k.6AD<-J98S2n,r<mx,37/2nP<-J346m5254U,37LK3<D8F<D8W8*E9,-F7k467k2xE9A)lQP5ADmxADJ3417Qg)lQ<DP<gGP<DltET

XY,-FK/hUP5,3:!~©|xK-341<w2xE9A8s<-:=A<DP5g?0Q:AO7/20Q8CA-Jk467= ] |nKFSAgOAG2G(X)T (X) + L(X)H(X) = 0

K [ |<-7QJ

f(X) = G(X)H(X) + XL(X)T (X)T

\ 8W<'mn,3798*AD¡30tAO7QmxA/KL(X)f(X) = G(X)L(X)H(X) + XL2(X)T (X)

T¨3>0Q8x467Qg) [ |2xE/418@9A-mn,3:AO8

L(X)f(X) = −T (X)(G2(X) − XL2(X)

)FSE9AD7QmnA3K

L(X)468S<cJ/4o/468C,3P),/h

T (X)@QADmn<O0Q8*A

G(X)<-7QJ

L(X)<-P5Admn,3lQP46:=Alt,/. >97Q,3:46<O.68OT _ E/0Q8GK

f(X) = M(X)(G2(X) − XL2(X)

)

hi,3PW8*,3:=AM(X) ∈ Z[X]

T¨Q032FSA'E3<G3A'<D8C8x0Q:ADJz2xE3<G2f(X)

[email protected][X]

T _ E9ADP5AGhi,3P5A3KM(X)

418W<0Q734o2a,/hZK3<D7QJ

(?)hi,/.j.U,-F8OT

2 !$s-o . $O+H@Q Ì-Ëf(X)

9Ì Í [ Ð#[DÍLÐ Ï YÐZ[X]

Ï 5 Ï6Î'Ï n Ì Ï ~Ì=ÏUÍZ[X]

ÎsÎdÌSË 9 Ëf(X)

Q ÎÌ -Ï6Í Ð9Ì c Ï~ÌGÍLËA Í ÐÍ3ÎOË ÍLËIËÌ

C ~Í -ÏË6ÏÐÍeÎ Ð-Î?ÌzË Q Ë

uAÏ6ÎwÍÐQË Î D Ì)Ï6Í

ZÐ Ì /5 DÍ3ÏË

uÐ

ZÐ Ë 9 Ë

ACÏ6ÎBÍÐQË Î - ÌÏ6Í

Z Ì?Í

f(X2)Ï6ÎaÏ n Ì Ï ~ÌÏ6Í

Z[X]

] M ] %5 . $ t ¦ih

f(X)418wJ9AG25A-m525A-Je<-846P5P5A-J/0tmn4U@/.UA/46<S2xE9AFSAO.j.Usf7Q,-F7 468CAD798n2nAG467LÓ 8RLP4j25ADP4U,37 U8CA-A 5/Kl9l;TB ` DC ` C|xKaFSE/46msE <O.68C,«FS,3Pnf8417

Z[X] 9Ë Ë6Ï1Î Ë Í-Ï1Î |nK4o2)hi,/.1.6,-F846:z:ADJ341<O25AO. >p2xE3<G2

f(Xm)[email protected]

Z[X]hi,3P)<D7>lt,38x4j254 3A417/2nADgOADP

mT),-FSA*3ADPCK;,/0QPP5AO8s09.o2WFS,3Pxf-8Y467m5<-8CAD8FSE9ADP5A 468CAD798n2nAG467LÓ 8RLP4j25ADP4U,37468W4179<-l9l9.j4Um5<@/.UA9T \ 8W<-7AxÆD<-:zl9.6AS,/ht2xE/468?KFSASmn,3798s46JQAOPI2xE9Alt,/. >97Q,3:46<O.

f(X) = 3X2 + 2X + 4 ∈ [X]

K-FSE/4UmsEk468BmxAOP2<D4173.o>[email protected])467 [X]

TªB8C467QgSRL,3P5,/.1.1<-P~>M3K/FSAS7Q,25A)2xE3<O2AC = 12

<D7QJ ± 3<-P5AS7Q,2a8*¡/0Q<-P5AO8B467 T¤/P5,3:AO4o2xE9ADP,/h2xE9AO8*A'2¢FS,hU<-m5258BFSAzE3<?3A'2xE3<O2

f(X2m)

= 3X2m

+ 2X2m−1

+ 441846P5P5A-J/0tmn4U@/.UA)417

[X]hi,3P<D7>zlt,38x4j254 3Aw467/25A-gOAOP

mT

%3C %$&% D63%H"5MkuTU¥ T9R,9E37LK /Ì 3K-§I,/.¢Tt/K,C,9E37zbe4j.UA*> +-,3798GKQMDN3-9T5Yb%T Z TLXB4UmsE9,/.18*,37LK iÍË Ð Ë6Ï~ÐÍdËÐ -ÎOË 3 Ë 3Ì 5/K/be41.6A*>LK9M-N-NNQT

X<G25<O.146,TQH;0tADP8CAD7/£/<O4UgªB734o3AOP8s46J9<-J=R \ 3R ¨Q0tAD7Q,38 \ 46P5AO8\ P5gOAD7/254179<[email protected]

] M [

!"$#%&'%()+*, .-/!021), $&34"56&#+ 78:9! 7;0=<>!?;A@CB EDF4"&#'B!/%0(G 08HB!0B!!I0"JKB6L!M)N%OP ,Q$#%&'%7;7; I'@33 I02D&#QRS/EQ6&#T ; D#(RU#I7V#= W!E @%34X#$ C&#YO[ZQ#= ,%X!I+/(RX&#%!2! !0&#S \= 7IU]=7 4$^( FFRU#@(( Y_ W$ @O)`a3/,W

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81

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(e/2R_%&P%8 % 7=4)+ X +!^

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3<-msE)lQP5,9@/.UAO: 468g?4o3AO7)467 37Qg?.j468CE)<D7QJk¤P5AD7QmsEKG2xE9A,/ymn46<O.-.1<-7Qg?0Q<-gOAD8I,/h3RL<D79<J3<tT¦~7'468s8s0tAO8aM3K ] K33K-<-7QJz/K 37Qg?.j468CE)F)4j.1.3l9P5A-mnA-J9AS¤P5AD7QmsEK<-7QJS417S468s8s0tAO8W/K [ K ` K<D7QJ9Kt¤/P5AO7QmsE'F)4j.1.QlQP5ADmxADJQA /7Qg?.1418*ET

¦~7e2xE9Az8C,/.1032546,3798Y8*ADm5254U,37LKt2xE9AlQP5,9@/.UAO: F)4j.1.;@QAzg?4 3AD7d417e2xE9Ak.6<D7Qg?0Q<gOAz,/h2xE9AlQP46:z<-P~>khiAD<G250QP5A-Jz8C,/.1032546,37T_ E9AA-J/4j25,3Pa2xE3<-7Qf-8 CAD<D7Q?¥r<DP5m _ AOPnP4UAOP<D7QJpY4UJ9AD:4j28s0c+D<ADf-4;,/h2xE9AªB734U3ADP8x4j2~>=,/h¥%,37/2nP5AO<D.Qhi,3PI2nP<D798s.1<O2546,3798B,/ht2xE9A)lQP5,9@/.UAO:=8OT T-RL,3PnP5ADm5254U,37T Ð Ð-Î?Ì [ GÐ-Î BÌ GÏ6Î B -n Ì Ð Í Í GÐ-Î BÌÉ /Ð © 3Ì Í3Ï DÌ ÎsÏË Ë tÐjÏ¢Ë Ì Í3Ï QÌ Ë DÍ [ -x Ì"ÐÍ ÏUÍ TÇDAG2

n@9A<7Q,37Qs7QADgG<O254 3A)417/2nADgOADP©T$WA?2nAOPn:467QA

n∑

k=0

tanh(2k)

2 + 2 sinh2 (2k)


+-,/4j2n0Q7=AD7/2546ADP7Q,37z7

A-gG<G2541h5TQR<D.6mn09.UAOPn∑

k=0

tanh(2k)

2 + 2 sinh2 (2k)

T

] M- T9R,3P5P5A-m52546,37T Ð Ð-Î?Ì [ D *ÎsÏ6ÍË1ÎsÏ Î Ì?ÎsÎ ÐÍ/Ï(-Ï D ÌDÌ Ì TH4 3AD7 4ABC

F)4j2xE8x4UJ9AD8aKbKcK3l9P5,D3Aw2xE3<G2

3(a4 + b4 + c4

)

(a2 + b2 + c2)2 +

ab + bc + ca

a2 + b2 + c2≥ 2


¥%,37/2nP5ADPa¡/0tA3K9J9<D798W0Q7k2nP541<-7Qg?.6AABC

JQAm,2AD8

aKbKcK

3(a4 + b4 + c4

)

(a2 + b2 + c2)2+

ab + bc + ca

a2 + b2 + c2≥ 2

T¯ TWR,3P5P5A-m52546,37T Ð Ð-ÎGÌ [ n [ -Ï6Í Í3Ï DÌ ÎsÏË#[ Ð

9Ì Ë É dÐÍË~ÐÍ T+O0QlQlQ,38*Aw2xE3<G2xKyK3<D7QJ

z<DP5AP5AO<D.t730Q:=@QAOP8DTBuQP5,O3Aw2xE3<G2

(x3 + y3 + z3

)2+ 3(xyz)2 ≥ 4

(y3z3 + z3x3 + x3y3

) TWAG25ADP5:z417QAw2xE9ASm5<-8CAD8B,/hLA-¡/0Q<D.j4j2~>TT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+O4xKyAG2

z8*,37/2JQAO8B7Q,3:=@9P5AO8BP

A-AG.68?K3:,37/2nP5AOPa¡/0tA

(x3 + y3 + z3

)2+ 3(xyz)2 ≥ 4

(y3z3 + z3x3 + x3y3

) TAG25ADP5:z417QADP.6AD8Bm5<-8B, 0z4j.3>z<

A-gG<O.14o2

AQT

S Ä ? T Ð Ð-ÎGÌ [ C Ë Ì ÍÐDÎ Î1Ï6ÍLÌ?Í [=Í ÎxÏ ~Í9Í9Î OÎOË Ï QTÇDAG2

mKnK/<D7QJ

N@9Aw7Q,37Qx7QA-gG<G254o3AY417/2nADgOADP8a8x0tmsES2xE3<G2

m + n ≥ 2N + 1TÇDAG2

K = m + n − N − 1TBuQP5,O3Aw2xE3<G2

∞∑

j=0

(−1)j N + 1

N + 1 + j

(N

j

) [(K − j

m

)

+

(K − j

n

)]

=

(m + n

m

)

(2N + 1

N

)T

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T+-,/4j2

mK

nKAG2

NJ9AD8 7Q,3:@3P5AD8 AO7/254UAOP8 7Q,37 7

ADgG<O254jhU8 25AO.18 ¡/0tA

m + n ≥ 2N + 1K9A?28*,/4o2

K = m + n − N − 1T¥%,37/2nP5AOPa¡/0tA

∞∑

j=0

(−1)j N + 1

N + 1 + j

(N

j

) [(K − j

m

)

+

(K − j

n

)]

=

(m + n

m

)

(2N + 1

N

)T

] M ` S

T Ð Ð-ÎGÌ [ OÐ-Î ÏÐ 3Ì*Ï Ï-[ Î -Ï Í T¦~7 4ABC

KGFSAYE3<?3AAB < AC

T _ E9A417/2nAOPn79<O.Q@/468CA-m525,3P,/h∠BAC

:=A-A?258BC

<G2DTaÇDAG2

P@9A)<D7k467/25ADP4U,3PIlt,/417/2,/hQ2xE9AY.1417QA)8CA-gG:=AD7/2

ADK/<-7QJk.6AG2

E<D7QJ

F@9Aw2xE9A)417/2nAOP8*ADm5254U,3798B,/h

BP<D7QJ

CPF)4o2xE

AC<D7QJ

ABK3P5AD8sltADm5254o3AG.o>T

uQP5,D3AY2xE3<O2 P E

P F<

AC

AB


a<-798w0Q7e2nP541<-7Qg?.6AABC

K,37c<AB < AC

TÇO<@3418C8CA-m52nP4UmnAk467/2AOP546AO0QP5A=JQA.Ó <-7Qg?.6A

BACmx,/0QlQA

BCAD7

DT9+-,/4o2

P0Q7zlQ,/467/2I417/2

ADP4UAG0QPJ30z8*ADgG:AO7/2

ADK9A?28*,/4o2

EAG2

F.UAO8a467/25ADP8CA-m52546,3798WP5AD8sltADm5254o3AO8BJQA

BPA?2

CP<?3A-m

ACA?2

ABT

¥%,37/2nP5ADPa¡/0tA P E

P F<

AC

AB

T S

¯T Ð Ð-ÎGÌ [ OÐ-Î ÏÐ 3Ì*Ï Ï-[ Î -Ï Í T¦~7 4ABC

K;FSAcE3<?3AAC = 2AB

T _ E9A25<D7QgOAD7/28k<O2A<-7QJ

C25,p2xE9Amn46P5mn0Q:=mn46P5mn.6AS,/h 4ABC

:ADAG2<G2PT

uQP5,D3AY2xE3<O2I2xE9Aw.1417QABP

@/468CA-m5282xE9A<DP5mBAC

i,/ht2xE9Amn46P5mn0Q:=mn46P5mn.6A?|sTT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T Ta<-798W0Q72nP541<-7Qg?.6A

ABCK9,37=<

AC = 2ABTÇDAO8a2<-7QgOAO7/2nAO8AD7

AAG2aAD7

C<D0mxAOP5mn.UAmn46P5mn,3798*mxP4j2J302nP541<-7Qg?.6AABC

8*Amn,/0QltAO7/2WAO7PT

¥%,37/2nP5ADPY¡/0tA.6<=J3P5,/4j25ABP

J/4o/468CA.~Ó <DP5mBAC

UJ30pmxAOP5mn.UA=mn41P5mx,3798CmxP54o2¢|WAD7JQAG0Æ'lQ<DP2546AD8ADgG<D.6AD8OT

S T Ð Ð-ÎGÌ [ OÐ-Î ÏÐ 3Ì*Ï Ï-[ Î -Ï Í T+O0QlQlQ,38*A2xE3<G2M

<-7QJN<-P5A2xE9Aw:4UJ9slQ,/467/28a,/h92xE9Aw8x4UJ9AD8

AB<D7QJ

CD,/h¡30Q<-J9P41.1<O25ADP<O.

ABCDK/P5AD8sltADm5254o3AG.o>T

uQP5,D3AY2xE3<O2AN2 + DM2 + BC2 = BN2 + CM2 + AD2 TT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+O0QlQlQ,38*,3798B¡30tAM

AG2N8C,37/2.6AD8WlQ,/467/28B:41.j4UAG0=P5AO8ClQA-m5254jhU8J9AD8Bm,2nAO8

ABAG2CD

JÓ 0Q7¡/0Q<J3P54j.6<G2ADP5AABCD

T¥%,37/2nP5ADPa¡/0tA

AN2 + DM2 + BC2 = BN2 + CM2 + AD2 T SLS T Ð Ð-Î?Ì [ aÍË Ì Î Ë Ï Ð -Ï6Î Í Ï ~Ð Ï

Ë ÍË6Ï Í3Ï DÌ ÎsÏË#[ Ð ¢ ËÐÍ TH4 3AD7%2¢FS,%lt,/417/258B<-7QJ

CKat7QJr2xE9A.6,9mn0Q8=,/hB2xE9AplQ,/467/2

A8x0tmsE%2xE3<G22xE9ApmnAD7/2nP5A,/hW2xE9Ac73467QADslQ,/467/2'mn46P5mn.6A,/h 4ABC

.j4UAO8z,37r2xE9Ad417/2nAOP546,3P'@3418*ADm52n,3P,/h∠CAB

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T2<-7/2BJQ,3797

AkJQAG0ÆzlQ,/467/28

BA?2

CK32nP5,/0/3AOPW.6AS.j4UAG0cJ/0dlt,/417/2

A25AO.¡30tAS.UAmxAO7/2nP5AdJ/0%mxAOP5mn.UAeJ9AD8SNxlt,/417/258'J/0p2nP541<-7Qg?.6A

ABC8*,/4o2w8x4j250

A8x0QPY.6<e@3418C8CA-m52nP4UmnA467/2

AOP546AO0QP5ASJ9A).~Ó <D7Qg?.UA

CABT

] M S T Ð Ð-ÎGÌ [ B Î D /Ï n Ï Í/ÏDÌ ÎxÏ QÌ D LÏ~Ð Ð ÐÍLÐ

ÏUÍ TÇDAG2

ak =qk − 1

q − 1

K-FSE9AOP5Aq468W<P5AD<O.730Q:=@QAOP*K

q 6= 1Ta¤/,3P417/2nADgOADP8

n ≥ 0<-7QJk ≥ 1

KaJ9AOQ7QACn,k

<D8Shi,/.1.6,-F8HUCn,1 = 1

KC0,k = 0

hi,3Pk ≥ 2

K<D7QJCn,k =

n−1∑

j=0

ajk−1

aj+1k

Cj,k−1hi,3P

n ≥ 1<-7QJ

k ≥ 2T

+-E9,-F2xE3<O2Cn,k = −(q − 1)k−1

k∑

i=1

(

qi − 1

qk − 1

)nqi

.

..

..

..

..

..

..

..

..

.

...

..

..

..

..

...

..

..

..

..

q−k, q........................................

i

.

..

..

..

..

...

..

..

.

....

..

..

..

..

..

..

.

..

..

..

q, q........................................

i−1........................................

q, q........................................

k

KFSE9ADP5A .

..

.

..

..

...

..

..

..

.

...

..

..

..

..

...

..

..

.

..

.

a, q........................................

0 = 1<D7QJ .

..

.

..

..

...

..

..

..

.

...

..

..

..

..

...

..

..

.

..

.

a, q........................................

i = (1 − a)(1 − aq) · · · (1 − aqi−1)hi,3P

i ≥ 1TT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+-,/4j2q0Q77Q,3:=@9P5ASP

ADAO.J/41Ô

ADP5AO7/2BJQAkM'AG2

ak =qk − 1

q − 1

Tu3,/0QPWJQAO8AD7/2546ADP8n ≥ 0

AG2k ≥ 1

KQ,37JAOQ734j2

Cn,kl9<-P

Cn,1 = 1K9l9<-P

C0,k = 08x4

k ≥ 2K9AG2lQ<DP

Cn,k =n−1∑

j=0

ajk−1

aj+1k

Cj,k−18x4

n ≥ 1A?2

k ≥ 2T

¥%,37/2nP5ADPa¡/0tACn,k = −(q − 1)k−1

k∑

i=1

(

qi − 1

qk − 1

)nqi

.

..

..

..

.

...

..

..

..

.

...

..

..

..

..

...

..

..

..

..

q−k, q........................................

i

.

..

..

..

..

...

..

..

.

....

..

..

..

..

..

.

..

..

..

..

q, q........................................

i−1........................................

q, q........................................

k

K, 0 .

..

.

..

..

...

..

..

..

.

...

..

..

..

..

...

..

.

..

..

.

a, q........................................

0 = 1A?2 .

..

.

..

..

...

..

..

..

.

...

..

..

..

..

...

..

.

..

..

.

a, q........................................

i = (1 − a)(1 − aq) · · · (1 − aqi−1)8x4

i ≥ 1T

S T Ð Ð-ÎGÌ [ OÐ-Î ÏÐ 3Ì*Ï Ï-[ Î -Ï Í TÇDAG2O@9AY<D7S467/25ADP4U,3P;lt,/417/2I,/h 4ABC

K-<-7QJS.6AG2DKEKFK/@9A2xE9A467/25ADP8CA-ms254U,3798B,/h

AOKBO

KCO

F)4o2xEBC

KCA

KAB

K3P5AO8ClQA-m5254 3AO. >T+O0QlQlQ,38*A2xE3<O2

P<-7QJ

Q<DP5A=lQ,/467/28S,37d2xE9Az.1417QA=8*ADgG:AO7/258

BE<-7QJ

CFKP5AD8sltADm5254o3AG.o>K38s0tmsE'2xE3<G2 BP

P E=

CQ

QF=

DO

OA

TuQP5,D3AY2xE3<O2

PF ‖ QETT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+-,/4j2O0Q7^lt,/417/2417/2

ADP4UAG0QPzJ30 2nP46<D7Qg?.UA

ABCKWA?2z8C,/4j2

DK

EAG2

FK.UAO8467/25ADP8CA-m52546,3798J9A

AOKBO

A?2CO

<?3A-mBC

KCA

A?2AB

K3P5AD8sltADm5254o3AO:AO7/2?TVW7c8s0Ql9lt,38CAz¡30tA

PA?2

Q8C,37/2JQAO8wlt,/417/258w8x0QP.6AD8Y8*ADgG:AO7/258

BEAG2

CFKP5AD8sltADm5254o3AO:AO7/2*K/25AO.18B¡30tA BP

P E=

CQ

QF=

DO

OA

T¥%,37/2nP5ADPa¡/0tA

PF ‖ QET

] M S T Ð Ð-ÎGÌ [ OÐ-Î ÏÐ 3Ì*Ï Ï-[ Î -Ï Í T+O0QlQlQ,38*A'2xE3<G2

P468Y<-7e417/2nAOP546,3Plt,/417/2,/h 4ABC

KL<-7QJ2xE3<G2DKEKF<-P5A2xE9A417/2nAOP8*ADm5254U,3798,/h

APKBP

KCP

F)4j2xEBC

KCA

KAB

KDP5AD8sltADm5254o3AG.o>T+O0Ql9lt,38CA2xE3<O2AE + AF

BC=

BF + BD

CA=

CD + CE

AB

TRE3<DP<m525ADP4j£sAY2xE9Alt,/417/2

PTT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

+-,/4j2P0Q7 lQ,/467/2'417/2

ADP4UAG0QPJÓ 0Q7%2nP541<-7Qg?.6A

ABCAG2k8C,/4j2

DK

EA?2

F.UAO8467/25ADP8CA-m52546,3798;P5AD8sltADm5254o3AO8JQA

APKBP

A?2CP

<G3ADmBC

KCA

AG2AB

TD+O0Ql9lt,38C,3798¡30tAAE + AF

BC=

BF + BD

CA=

CD + CE

AB

TRL<DP<m52

AOP5418*AOP.6A)lt,/417/2

PT

S ? T Ð Ð-Î?Ì [ Ð 9 dÌ ÎCÎxÏ 3Ë 5 Î9Ð Í Ì TuQP5,D3A2xE3<O2 ∑

ab

c(c + a)≥

∑

a

c + a

KFSE9ADP5AaKbKcP5ADl9P5AD8CAD7/2Y2xE9A

2xE3P5A-A8x4UJ9AD8B,/hL<w2nP46<D7Qg?.UA9TT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T¥%,37/2nP5ADP¡/0tA ∑

ab

c(c + a)≥

∑

a

c + a

KL, 0aKbKcP5AOlQP

AD8CAD7/25AD7/2B.UAO8

2nP5,/468Bm,2AO8BJLÓ 0Q72nP541<-7Qg?.6AQT

®T Ð Ð-Î?Ì [ Í Ð-Î Ð Ð dÌ Ð B- 3Ì Í/ÏDÌ ÎxÏ QÌ

3 QÐ1Ï 3 QÐjÏ ÏUÍ T¦~7 4ABC<-7QJ 4A′B′C′ K?2xE9Aa.6AD7Qg©2xE38;,/hD2xE9AW8x4UJ9AD88s<O25418shj> a ≥ b ≥ c

<D7QJa′ ≥ b′ ≥ c′ T ÇDAG2 ha

<-7QJha′

J9AD7Q,25Ae2xE9Ae.UAO7Qg©2xE38,/ha2xE9Ad<D.o254j250tJ9AD8S25,p2xE9A,3lQlQ,38s4o2nA)8s46JQAO8ahUP5,3:A<D7QJ

A′ K3P5AO8ClQA-m5254 3AO. >TWuQP5,O3Aw2xE3<G26<G|

bb′ + cc′ ≥ aha′ + a′ha i@-|bc′ + b′c ≥ aha′ + a′ha


a<-798).6AD8Y2nP541<-7Qg?.6AD8ABC

AG2A′B′C′ KL.UAO8).6,37Qg?0tAO0QP8SJQAO8Sm,2 AO88s<O25418shi,37/2

a ≥ b ≥ cAG2

a′ ≥ b′ ≥ c′ T3+-,/4o2 haA?2

ha′

.1<w.6,37Qg?0tAO0QPJ9AD8aE3<D0325AO0QP8418C8x0tAD8aJ9AD88*,3:z:A?258AA?2

A′ TL¥%,37/2nP5ADP¡30tA6<G|bb′ + cc′ ≥ aha′ + a′ha i@-|bc′ + b′c ≥ aha′ + a′ha

T

] MDN Ä T Ð Ð-Î?Ì [ dÌ-Ì?Í@ O Ë 9Ð dÌ Ë Ì ] Ì-Ë Ì ÍÎ T_ E9Amn46P5mn.6A

Γ(P, r)417/2nAOP8*ADm52582xE9A)8s46JQA

AB,/h 4ABC

<G2A3

<-7QJB3K2xE9A8s46JQA

BC<G2

B1<D7QJ

C1K3<-7QJk2xE9A)8s46JQA

CA<G2

C2<-7QJ

A2T

H4 3AD7%2xE3<O2 |A3B3| : |B1C1| : |C2A2| = |AB| : |BC| : |CA| KJQA?2nAOPn:467QAw2xE9Aw.U,9mn0Q8B,/hPTT T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T

ÇDAzmxAOP5mn.UAΓ(P, r)

mx,/0QlQA.6Azm,2A

ABJ30d2nP46<D7Qg?.UA

ABCAO7

A3A?2

B3Kt.UAm,2

A

BCAD7

B1AG2

C1K9AG2I.UAm,2

A

CAAD7

C2AG2

A2T

378x0QlQlQ,38C<D7/2¡/0tA |A3B3| : |B1C1| : |C2A2| = |AB| : |BC| : |CA| KJA?2nAOPn:467QAOP.UAw.146AO0=JQA

PT

T Ð Ð-Î?Ì [ Ï B -[ ÌGÍ Ì 5 ÎGÐ Ð Í3Ï QT

_ E9A8CA-¡/0tAD7QmnA xn 468BJ9AOQ7QA-J=@D> (1 +1

n

)n+xn

= eT

6<G| uQP5,O3AY2xE3<O2 xn 468Bmn,373ADP5gOAO7/2*K9<D7QJ=JQA?2nAOPn:467QAw4j28W.j46:4j2?Ti@-|?WA?2nAOPn:467QAY2xE9A<D8>9:zl325,254UmWAxÆl9<-798x4U,37=,/hQ2xE9A8CA-¡/0tAD7QmnAQT

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T TÇO<S8x094j25A xn AD8n2aJ AGt7346Al9<-P (1 +

1

n

)n+xn

= eT

6<G| ¥%,37/2nP5AOPa¡/0tA xn mx,373AOP5gOASAG2I2nP5,/0/3AOP8C<).j46:4j25AQTi@-|?_ P5,/0/3AOP.UAJ

A?3AG.U,3l9ltAO:AO7/2a<-8>3:=l/2n,2546¡30tAJQAw.6<8x094j25AQT

¯T Ð Ð-Î?Ì [ Ï B -[ ÌGÍ Ì 5 ÎGÐ Ð Í3Ï QT+O0QlQlQ,38*A2xE3<G2

aKbKc<-P5A=mx,3:zl9.6AsÆe730Q:=@QAOP8S8x0tmsEd2xE3<G2 |a| = |b| = |c| TuQP5,D3AY2xE3<O2

∣∣∣∣

ab

a2 − b2

∣∣∣∣+

∣∣∣∣

bc

b2 − c2

∣∣∣∣+

∣∣∣∣

ca

c2 − a2

∣∣∣∣

≥√

3T

T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T TVW7!8x0QlQlQ,38*A ¡/0tA

aK

bAG2

c8*,37/2 JQAO8^7Q,3:@3P5AD8 mn,3:=l3.UAxÆ-AD8 25AO.18 ¡/0tA

|a| = |b| = |c| TL¥%,37/2nP5ADPa¡/0tA∣∣∣∣

ab

a2 − b2

∣∣∣∣+

∣∣∣∣

bc

b2 − c2

∣∣∣∣+

∣∣∣∣

ca

c2 − a2

∣∣∣∣

≥√

3T

] v

] Ð ÐÌ Ï6ÎÌ DÌ Ì ÍLÌ?ÍË -[ Ð-Î?Ì ÌrÌ-Ï¢Ë~Ð Ï1Î [DÎ Ì Î?ÌvË~Ð ÐÍ3ÎsÏ QÌ ¢Ð jÏ ¢ Ë6Ï~ÐÍ=ÍÌ ÎGÐ 9Ë6Ï~ÐÍ9ÎYÐ ÍÌ Ï6Í9ÎxÏ Ë1ÎwÐÍ ÎOË Ð~ÌzÎ bcA=<Dlt,/.6,9g?468CAhi,3Pw,3:z4o225467Qg'2xE9A=79<D:A=,/h \ X L Q¦B+/¦5¥ ¦VYX)K8x250tJ9AD7/2*KR,9,3lQADPªB734U,37'hi,3P \ J/<D7QmxAO:AO7/2,/ht+-mn46AD7QmnAw<D7QJ \ P¢2*K9XYA©F #;,3PnfQK9X#BK/ª+ \ hUP5,3: 2xE9A.1418x2,/ha8*,/. 3ADP8',/hB-Me<D7QJ%-M9K<D7QJ2xE9Ae79<D:Ad,/h¥ ¦R;LÇ«¨ \_\ ¦nÇ3Ç K L,/0tAD7LK¤/P<D7QmxA)hUP5,3:2xE9A).j468n2a,/hL8C,/.o3AOP8B,/h-DNtTY

? T 5MDNN/ U M Ð Ð-Î?Ì [ D Ï 9Ì jÏUÍ É Í3Î ?Ï ~Ì ]

ªB8*AS<D7>8CA-¡/0tAD7QmnA3K ck Kt,/h 0 Ó 8<D7QJ 1Ó 8B25,J9AOQ7QA'<k Ì ÌË6ÏË6ÏÐÍ 5 Ì?ÎxÏ6ÎGË ÍËÎGÌ3Ì?Í Ì

s = sk 467QJ/0tm5254o3AG.o><-8ahi,/.j.U,-F8LUM9Ts1 = c1

Ks2 = 1 − s1 9Thi,3P

n ≥ 2K3.6AG2

L = maxi ≥ 1 : (sm−i+2, . . . , sm, sm+1)

= (sn−i+2, . . . , sn, 0) m < n,

L′ = maxi ≥ 1 : (sm−i+2, . . . , sm, sm+1)

= (sn−i+2, . . . , sn, 1) m < n.68*,W2xE3<O2L4182xE9A:=<©Æ-41:=<O.G.6AD7Qg©2xEY,/hG2xE9AI25<O41.6s8CA-¡/0tAD7QmnAW,/h

(s1, s2, . . . , sn, 0)2xE3<O2<O.6P5AO<J->=,9mnmn0QP8W417(s1, s2, . . . , sn)

K3<D7QJz8x46:41.1<-P.o>khi,3PL′ |xK3<D7QJ

sn+1 =

041h

L < L′,1

41hL > L′,

cn41h

L = L′.( ¤/,3PAsÆD<D:=l3.UA/K/41h

ci = 0hi,3P<O.1.

iK2xE9AD7

s = (0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0,

1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, . . .))

uQP5,D3A,3PaJ/468slQP5,O3Aw2xE3<G2smx,37/2<D41798A*3AOP~>=@34179<-P~>'FS,3P5JT

3Ð 9Ë6Ï~ÐÍ [ ~Ì Í Ð B Í3Ï DÌ ÎsÏ QÌ 3Ì?ÍLÐ-Î Ï Ì©Î /ÌGÍÐ-Î Ï Ì?Î /ÌGÍLË6ÏUÍ dÐ-Ï Ì'Î 1Ï Ë #[ [Ë ÌSÌ-Ï¢Ë~Ð Î TbcAF)4j.1.3lQP5,O3A2xE3<O2A*3ADP~>@/4679<DP~>FS,3P5J,9mnmn0QP84173t734o2nAG.o>':=<D7>S2546:=AD8417

sT¤3,3P<-7>8>9:=@Q,/.

σ ∈ 0, 1 K-FSAFP54o2nA σhi,3P2xE9AS,3l9lt,38x4j25A8>3:@9,/.T

bcAwtP8n28*E9,-F 2xE3<O22xE9AFS,3P5J98B,/ht.6AD7Qg©2xE=M)<DlQlQAD<DP4173t734o2nAG.o>:z<-7>'2546:=AD82xE3<O2468?Kt2xE9Az8>3:@9,/.68

0<D7QJ

1AD<-msEc<DlQlQAD<DP4673Q734j25AO. >,/h125AD7d467

sT 3<-msEd8>9:=@Q,/.<-l9ltAO<-P8<O2;.UAO<-8n2I,37QmnA3K/@9A-m5<D0Q8CA3K@D>J9AOQ734j2546,37LK

s8x2<-P¢258;F)4j2xE

01,3P

10T3+O0Ql9lt,38CA2xE9Ac8>9:=@Q,/.

σ<DlQlQAD<DP8 0Q8n2SQ734j25AO. >%:z<-7>«2541:AO8GK;2xE9Ad.1<-8n2)2546:=A@9AO417Qgd<G2

skT

] -M_ E9AO7

sk+i = σhi,3PW<D.j.

i ≥ 1T¦~7elQ<DP2546mn09.6<DP*K

s2k+2 = σT¨Q032a2xE/468Ymx,37/2nP<-J346m52582xE9AkJ9AOQ734j2546,37,/h

sKQ@QADmn<O0Q8*A)2xE9AS8x2nP467Qg

sk+2 . . . s2k+2 = σk+1 <O.6P5AO<J->,9mxmn0QP8467s1 . . . s2k+1

<-8sk+1 . . . s2k+1

KLFSE9AOP5AD<D82xE9AFS,3P5Jσkσ

JQ,9AO8S7Q,2,9mxmn0QP)<D8<k8s0t@38x2nP467Qg',/hs1 . . . s2k+1

TtbcAkmn,37Qmn.10tJ9AS2xE3<G2AO<msEkFS,3P5Je,/h.6AD7Qg©2xEdM'<DlQlQAD<DP84673Q734j25AO. >=,/h125AD7417sT

XY,-F 8x0QlQlQ,38*A2xE9AOP5Ap468=8C,3:AdFS,3P5J^,/h.UAO7Qg©2xE^gGP5AO<O25ADP'2xE3<D7vM2xE3<G2z<DltltAO<-P8=,373.o>%Q734j25AO. > ,/h12nAO7T _ E9AO7%2xE9ADP5Ac468z<p8CE9,3P25AD8n2S.UAO7Qg©2xEn ≥ 2

hi,3P'8s0tmsEFS,3P5J98OT;RLE9,9,38*A8*,3:=A'FS,3P5J,/h.UAO7Qg©2xEn2xE3<G2Y<DlQlQAD<DP8S,373. >dQ734j25AO. >,/h12nAO7LK8s<G>

t = d1 . . . dnK;<D7QJp.UA?2

k ≥ 0@QAz2xE9Ae730Q:@9ADP,/h2546:=AD8)2xE3<O2

t<DlQlQAD<DP8DT _ E9AFS,3P5J

d1 . . . dn−1<-l9ltAO<-P8Y4673Q734j25AO. >,/h125AD7LK@9A-m5<D0Q8CA4o258Y.6AD7Qg©2xEe418Y.UAO8C82xE3<-7

nTÇDAG225417Qg

t∗ = d1 . . . dn−1dnK©FSA8CA-AW2xE3<G2

t∗ :0Q8x2;<DlQlQAD<DP4173t734o2nAG.o>S,/h12nAO7LKD8s417QmxAt<DlQlQAD<DP8,373.o>kQ734j25AO. >=,/h125AD7T

+O0QlQlQ,38*Ak = 0

T _ E9AD7 2xE9AFS,3P5JtJQ,9AO8e7Q,2=<-l9ltAO<-Pz<O2z<D.j.417

sTÇDA?22xE9A'tP8n2a2¢FS,<DlQlQAD<DP<-7QmnAD8w,/h

t∗ AO7QJe<O2BlQ,38s4o254U,3798 i<-7QJ

jKtP5AO8ClQA-m5254 3AO. >T _ E/0Q8GK

si−n+1 . . . si = t∗ j2xE9ASQP8x2a<DlQlQAD<DP<-7QmnA?|<D7QJ sj−n+1 . . . sj = t∗ 12xE9AS8*ADmx,37QJ<-l9ltAO<-P<D7QmxA©|CT _ E9AhU<-m52B2xE3<G2sj = dn

mn,37/2nP<J/4Um528Y2xE9AzJQAGt734o254U,37c,/hsK@9A-m5<D0Q8CA2xE9A)FS,3P5J

t∗ <D.1P5AD<-J>,9mxmn0QP8B417 s1 . . . sj−1<-8

si−n+1 . . . siK/FSE9AOP5AD<D8W2xE9A)FS,3P5J

tJ9,9AD8W7Q,2a,9mnmn0QP467

s1 . . . sj−1T

+O0QlQlQ,38*Ak ≥ 1

T _ E9AO7p2xE9AFS,3P5Jt<DlQlQAD<DP8'<G2w.UAO<-8n2),37QmxA=467

sTaÇDAG2Y2xE9A.6<D8x2<DlQlQAD<DP<-7QmnA=,/h

tAO7QJd<O2lQ,38s4o254U,37

pT _ E/0Q8GK

sp−n+1 . . . sp = tK<D7QJe2xE/418Y4182xE9A.6<D8x2;<-l9ltAO<-P<D7QmxAw,/h

t417

sT/RL,3798*AD¡30tAO7/25.o>KD2xE9AOP5AY<-P5AY7Q,wFS,3P5J38417

s,/h9.6AD7Qg©2xE

p+12xE3<G2AO7QJwF)4o2xEY2xE9A8n2nP5417Qg

tTVW7w2xE9A,2xE9ADP;E3<D7QJLKG2xE9AOP5A:0Q8x2;@QA8*,3:=AaFS,3P5J,/h.UAO7Qg©2xE

p + 1AO7QJ3417QgS417

t∗ 2xE3<O2,9mnmn0QP8Y:=,3P5AS2xE3<D7d,37QmxAS467 sK@9A-m5<D0Q8CAS2xE9AOP5A<-P5A=4173t734o2nAG.o>:z<-7>p<DlQlQAD<DP<-7QmnAD8',/h

t∗ 467 s<D7QJ«,373.o>cQ734j25AO. >p:z<-7>dFS,3P5J38S,/h.UAO7Qg©2xE

p + 1T _ E/0Q8?KQ2xE9ADP5A:0Q8x2AxÆ-418x2a2¢FS,e8x2nP467QgG8

si−p . . . si<D7QJ

sj−p . . . sjKF)4j2xE

i < jK/2xE3<G2aP5AOlQP5AO8*AO7/22xE9A8C<D:AYFS,3P5J

wAD7QJ/467Qgw417

t∗ T9bcAE3<G3A sj = dnT¨Q0322xE9A=J9AOQ734j2546,37p,/h

sP5A-¡/0946P5AO8Y2xE3<O2

sj = dnK@9A-m5<D0Q8CAk2xE9Ak25<O41.6s8CA-¡/0tAD7QmnA

wAD7QJ/467Qg467t∗ <D.1P5AD<-J>,9mxmn0QP8417 s1 . . . sj−1

<D8si−p . . . si

KOFSE9ADP5AO<-82xE9ADP5Aw<-P5Aw7Q,FS,3P5J98a467s,/ht.UAO7Qg©2xE

p + 1AD7QJ/467QgY467

tT3bcAE3<G3A)<'mn,37/2nP<J/4Um52546,37T

\ mn,37/2nP<J/4Um52546,37cmx,3:=AD8hUP5,3: <-8s8s0Q:467Qg2xE3<G2<SFS,3P5Jd<DlQlQAD<DP8Y417s,373.o>e<t734o2nA730Q:=@QAOPI,/h/2546:=AD8OT _ E9ADP5AGhi,3P5A3KA*3AOP~>wFS,3P5JS<-l9ltAO<-P8I4173t734o2nAG.o>S:z<-7>)2541:AO8DT

!#"%$&('*)+-,,/.10324-+(.'. 56/'7/!08+89:;4-+<.'. 54=)+-,> !:.@?10(/BADC0<4=)!FE '7.G21, > !H.;I11J%+-'KL1B1 M4=)N53.1080<.OD+-'6!:P,12/034Q$R1S/T

a U b U c V SW@X;W@YHW@S Z/[/TL\^];S_LS[1`ba[ W@c`-S/T r ∈ dfe \8THg r ≥ 2

hgSWar(b + c) + br(c + a) + cr(a + b) ≥ 2

3r−12

(ab + bc + ca)r+12

i(Pj

k 4Q4=)*'J . 5)+-,,/.10324-+(.'B)l J@%4=)7l!: !#" $m `-S1[1_L` n*S op@[1`8\8T(n%gX`-ca\bW%Xp@_qTHgS1X;_LS1rs\bt

r = 1 h7g\-a`-S1[ cauTLXlTHgSW[TLp@_[`o;pGSaHTL\-X;W v g;[Twg;[ xxGSWat=X;_

r ∈ (1, 2) y

] - Ì ¢Ð ÐaÏ6ÍwÏ1Î ÎGÐ 9Ë6Ï~ÐÍcË~Ð=Ë Ï1Î 3Ì©ÎOË6Ï~ÐÍ 3Ð 9Ë6Ï~ÐÍ [ 3 ÎsÏ Ì /Ï ËÐ sÌ Í/ÏDÌ ÎxÏ¢Ë-[Ð ÐDÏ~Ì?ÎGË6Ï Ð Í3Ï QTbcAzF)4j.1.lQP5,O3A0Q8s417Qgz2¢FS,«J/41ÔADP5AO7/2:=AG2xE9,9J382xE3<O2~M©|Y<O.68C,«E9,/.6J98hi,3P)<D.j.

r ∈ (1, 2)T

% 29- 8' nT3bcAYF)4j.1.QlQP5,O3AS~M©|;hi,3Pr ≥ 1

@D>z8CE9,-F)467Qg2xE3<G2I2xE9A)h60Q7Qm52546,37f(x) =

1

x + 1ln

[ax(b + c) + bx(c + a) + cx(a + b)

6

]

468Y467QmxP5AO<-8x467Qg,37[1, ∞)

2xE3<O2468?K@D>c8CE9,-F)467QgS2xE3<G2f ′(x) ≥ 0

,37[1, ∞)

T _ E9AP5A-¡/0946P5ADJz417QA-¡/0Q<D.j4j2~>k2xE9AO7zhi,/.j.U,-F8hUP5,3:f(r) ≥ f(1)

T_ E9A)417QA-¡/0Q<D.j4j2~>

f ′(x) ≥ 0418BA-¡/094o/<O.UAO7/2I2n,

A ≥ CK-FSE9AOP5A

A = a(b+c)ax(x+1)b(c+a)bx(x+1)c(a+b)cx(x+1) KC =

[ax(b + c) + bx(c + a) + cx(a + b)

6

]ax(b+c)+bx(c+a)+cx(a+b) TbcAYF)4j.1.Q8*E9,-F2xE/418B@D>zlQP5,O3417Qg2xE3<O2

A ≥ B ≥ CKDFSE9ADP5A

B = a(b+c)axx+a(bx+cx)b(c+a)bxx+b(cx+ax)c(a+b)cxx+c(ax+bx) T_ E9A417QA-¡/0Q<D.j4j2~>

A ≥ B468AD¡3094 /<D.6AD7/2;25,

ln A ≥ ln BT _ ,'l9P5,D3A2xE9A.1<O225ADP2nP50tAYFSA):0Q8x28CE9,-F2xE3<G2

g(x) ≥ 0hi,3P<D.j.

x ≥ 1K-FSE9AOP5A

g(x) = [(b + c)ax − a(bx + cx)] ln a + [(c + a)bx − b(cx + ax)] ln b

+ [(a + b)cx − c(ax + bx)] ln cT

bcASE3<?3Ag′(x) = (b + c)ax ln2 a + (c + a)bx ln2 b + (a + b)cx ln2 c

− (axb + abx) ln a ln b − (bxc + bcx) ln b ln c

− (cxa + cax) ln c ln a

= ab(ln a − ln b)(ax−1 ln a − bx−1 ln b)

+ bc(ln b − ln c)(bx−1 ln b − cx−1 ln c)

+ ca(ln c − ln a)(cx−1 ln c − ax−1 ln a)T

¤3,3Px ≥ 1

K/4j2AD<D8s4j.o>khi,/.j.U,-F82xE3<O2g′(x) ≥ 0

TQRL,3798*AD¡30tAO7/25.o>Kg(x)

418W417QmxP5AD<D8s417Qg,37[1, ∞)

2xE3<O2418GKg(x) ≥ g(1) = 0

T_ ,l9P5,D3Aw2xE9Aw467QAD¡30Q<O.14o2>

B ≥ CKDFSA)P5A?FP4j25Aw2xE/418W417QA-¡/0Q<D.j4j2~>z<D8ahi,/.1.6,-F8>U

xx1

1 xx2

2 xx3

3 xx4

4 xx5

5 xx6

6

≥(

x1 + x2 + x3 + x4 + x5 + x6

6

)x1+x2+x3+x4+x5+x6 KFSE9ADP5A

x1 = axbKx2 = abx

Kx3 = bxc

Kx4 = bcx

Kx5 = cxa

Kx6 = cax

T_ E/418f7Q,-F7)467QAD¡30Q<O.14o2>whi,/.1.6,-F8;hUP5,3: CAO798*AO7LÓ 8a¦~7QAD¡30Q<O.14o2><-l9l9.j4UADJw2n,Y2xE9Amx,373AxÆh60Q7Qm5254U,37h(x) = x ln x

T

] ] % 29- 8' ¢nTQ+O417QmxASiM?|;418BE9,3:,9gOAO7QA-,/0Q8?K-FSASm5<-7zP5AOhi,3P5:z09.1<O25Aw2xE9A)lQP5,9@/.UAO:jF)4o2xE9,/032.U,38s8B,/hgOAO7QADP<O.14o2>-|<-8ahi,/.j.U,-F8LU¦ih

aKbKc<-P5AlQ,38s4o254o3Az467/25A-gOAOP8'8x0tmsEc2xE3<O2

ab + bc + ca = 3<D7QJ

r ∈ (1, 2)K2xE9AD7

ar(b + c) + br(c + a) + cr(a + b) ≥ 6T i?|

¤/P5,3:ab + bc + ca = 3

KDFSAY:z<G>FP4j25Aa(b + c) = 3 − bc

Kb(c + a) = 3 − ca

Kc(a + b) = 3 − ab

K3<D7QJ=~©|I@QADmx,3:=AD8ar−1(3 − bc) + br−1(3 − ca) + cr−1(3 − ab) ≥ 6

K,3Par−1 + br−1 + cr−1 ≥ ar−1br−1cr−1 · (ab)2−r + (bc)2−r + (ca)2−r

3+ 2

T+O467QmnA

0 < 2 − r < 1KD2xE9Ah60Q7Qm5254U,37

f(x) = x2−r468mx,37Qm5<G3A9T _ E/0Q8?K/@D> CAD798CAD7LÓ 8¦~7QA-¡/0Q<D.j4j2~>LK-FSASE3<?3A

(ab)2−r + (bc)2−r + (ca)2−r

3≤(

ab + bc + ca

3

)2−r

= 1K

<-7QJ4o28x09yemnAD82n,l9P5,D3Aw2xE3<G2ar−1 + br−1 + cr−1 ≥ ar−1br−1cr−1 + 2

T ] |¨A-m5<D0Q8CA2xE9Aa417QA-¡/0Q<D.j4j2~>4688>9:z:A?254UmnA3KsFSAa:=<?>Y<-8s8s0Q:=A2xE3<O2

a ≥ b ≥ cKCF)4o2xE9,/032<-7>.6,38C8W,/hgOAO7QADP<O.14o2>TLXY,-F FSAFP54o2nA ] |<D8

ar−1 + br−1 − 2 ≥ (ar−1br−1 − 1)

(3 − ab

a + b

)r−1 T+-AG225417Qg

x =√

abKt4j2hi,/.j.U,-F8hUP5,3: 2xE9A \ ¥%CHa¥ ¦~7QA-¡/0Q<D.j4j2~>e2xE3<O2

a + b ≥ 2x<-7QJar−1 + br−1 ≥ 2xr−1 T+O417QmxA a ≥ b ≥ c

<D7QJab + bc + ca = 3

46:zl9. >1 ≤ x <

√3K/4o28x09yemnAD82n,l9P5,D3Aw2xE3<G2

2(xr−1 − 1) ≥ (x2r−2 − 1)

(3 − x2

2x

)r−1 T4o/467QJ/467Qg)@D>k2xE9A)7Q,37Qs7QADgG<O254 3A)hU<m525,3P

xr−1 − 1K2xE/418a.6<D8x2I467QAD¡30Q<O.14o2>z@QADmx,3:=AD8

2 ≥ (xr−1 + 1)

(3 − x2

2x

)r−1 K,3P

2 ≥(

3 − x2

2

)r−1

+

(3 − x2

2x

)r−1 T_ E/418W417QA-¡/0Q<D.j4j2~>4182nP50tAS@9A-m5<D0Q8CA

1 ≥ (3 − x2)/2 ≥ (3 − x2)/(2x)hi,3P

x ≥ 1T

] [

T 5 MU [9` M z<D7QJ U [ Ð Ð-Î?Ì [ C Ë Ì ÍLÐOÎ

ÎjÏUÍÌGÍ [=Í ÎsÏ iÍQÍ3Î OÎOË Ï QT_ E9A8*AD¡30tAO7QmxAY,/h/h60Q7Qm5254U,3798?K J(n) = J(n, w) K n = 0K1K. . .

KO468J9AOQ7QA-J<-8ahi,/.1.6,-F8LUJ(0) = a

KJ(1) = w + b

KJ(n + 1) =

J(n) (J(n) (w J(n) − 1) − J(n − 1))

J(n − 1) (w J(n) + 1) + J(n)

hi,3Pn > 0

T6<G|z+-E9,-F2xE3<O2*K/4jh

a = 0K2xE9AD7k2xE9A8CA-¡/0tAD7QmnASmx,3798x468n258W,/hLlt,/. >97Q,3:46<O.68OTi@-|=+-E9,-F 2xE3<O2L2xE9ADP5AAxÆ-418x28<lQ<O46P

(a, b),/h37Q,37Qn£sAOP5,w467/25A-gOAOP8I8s0tmsEY2xE3<O2<D.j.-2xE9A

J(n)<-P5A)lt,/. >97Q,3:46<O.68IF)4j2xEk417/2nADgOADPmx,9AGyemn46AD7/28DT

3Ð 9Ë6Ï~ÐÍ [ C Í YÌ*Ï ÎOË QÌGÍLË t Í -Í 5 Í Ë Ì Ï ~Ì 5 Ð9Ð ËË 35 ÌeË~Ð -Í Í ] Ð Í/ÏDÌ ÎxÏ¢Ë-[ Q ÍOÎ 9 ÏUÍ QT_ E9ASg?4 3AD7zP5ADmn0QPnP5AO7QmxA)468BAD¡3094 /<D.6AD7/2I2n,J(n + 1)J(n − 1)wJ(n) + J(n + 1)J(n − 1) + J(n + 1)J(n)

= w(J(n)

)3 −(J(n)

)2 − J(n)J(n − 1)K

,3PwJ(n)

((J(n)

)2 − J(n + 1)J(n − 1))

=(J(n + 1) + J(n)

)(J(n) + J(n − 1)

) T+-AG225417Qg

an = wJ(n)K-FSA,9@/2<D417

an(a2n − an+1an−1) = (an+1 + an)(an + an−1)

T ~M©|U<?| \ 8C8x0Q:AY2xE3<G2

a = 0TÇDA?2

x = wJ(1) = w(w + b)T %5*5 . ÇDA?2 bn @QAk2xE9Az8*AD¡30tAO7QmxA=JQAGt7QADJp@D> b0 = 0

Kb1 = x

K<D7QJchi,3P<D.j.n ≥ 1

Kbn+1 = (x − 2)bn − bn−1 + x

T ~©|_ E9AO7LK3hi,3P<D.j.

n ≥ 1K

b2n − bn+1bn−1 = xbn

T ] | Ð9Ð Ò _ E9A=lQP5,9,/h418S@D>c467QJ/0tm5254U,37«,37 n

TbcE9AD7n = 1

K;A-¡/0Q<O2546,37« ] |B418w2nP0tA3K8s417QmxAb0 = 0

<-7QJb1 = x

TX,-FK98s0Ql9lt,38CA'A-¡/0Q<O2546,37 ] |I418a2nP50tAhi,3Pn = k ≥ 1

T_ E9AO7b2

k+1 − bk+2bk =((x − 2)bk + x − bk−1

)bk+1

−((x − 2)bk+1 + x − bk

)bk

= (x − 2)bkbk+1 + xbk+1 − bk−1bk+1

− (x − 2)bkbk+1 − xbk + b2k

= xbk+1 + b2k − bk−1bk+1 − xbk = xbk+1

T_ E9AOP5AOhi,3P5A/K/2xE9A).6AD:z:=<SE9,/.6J98B@D>417QJ30tm52546,37T

] -¤/P5,3:!~©|<-7QJ= ] |nKQ,37QAm5<-7'3ADP41hj>'2xE3<O2

bn(b2n − bn+1bn−1) = (bn+1 + bn)(bn + bn−1)

TR,3:zlQ<DP5417QgY2xE/418a25,zAD¡30Q<G254U,37=iM?|<D7QJz7Q,25467Qg2xE3<G2

a0 = wJ(0) = wa = 0<D7QJ

a1 = wJ(1) = w(w + b) = xK

FSASmn,37Qmn.10tJ9Aw2xE3<O2an = bn

hi,3P<O.1.n ≥ 0

T)AD7QmnA3Kan+1 =

(w(w + b) − 2

)an − an−1 + w(w + b)

T¦U2418AD<D8>25,S8*E9,-F @D>S467QJ/0tm5254U,372xE3<O2

an468<Ylt,/. >97Q,3:46<O./467

wT _ E9AOP5AOhi,3P5A/K

J(n)468W<lt,/. >97Q,3:46<O.Q467wK38x467QmnA

w468W<)hU<-m52n,3P,/h

anhi,3P<O.1.

nT

i@-| _ <-f-417Qg(a, b) = (1, −1)

KGFSAYE3<G3AJ(0) = 1

<-7QJJ(1) = w −1

T _ E9AO7a0 = w

<D7QJa1 = w(w − 1)

TÇDA?2 cn @QA2xE9AS8*AD¡30tAO7QmxAkJQAGt7QADJe@D> c0 = wK

c1 = w(w − 1)K3<-7QJhi,3P<D.j.

n ≥ 1K

cn+1 = (w − 2)cn − cn−1 + wT

ªB8s417Qg2xE9A8C<D:Aw417QJ30tm5254 3AlQP5,9,/hL<D8a467k2xE9Aw.UAO:=:z<S<-@Q,O3AYFSAm5<-7=AD8n25<[email protected]

n − cn+1cn−1 = wcnK

hi,3PL<O.1.n ≥ 1

12xE9AW,373.o>wJ/41ÔADP5AO7QmxAa4172xE9AalQP5,9,/hU84184672xE9Aa41734j2541<D.O8x25ADl|CT _ E9ADP5AGhi,3P5A3Kan = cn

hi,3P<D.j.n ≥ 0

K-FSE/4UmsE:=AD<D798a2xE3<G2*K/hi,3P<O.1.n ≥ 1

K-FSAE3<G3Aan+1 = (w − 2)an + w − an−1

T¦U2468kAD<D8>p2n,«8*E9,-F 2xE3<G2

an468'<clt,/. >97Q,3:46<O.IF)4o2xE«417/2nADgOADPSmx,9AGyemn46AD7/28'hi,3P<D.j.

n ≥ 0T9+O467QmnA

w468<whU<-m52n,3P,/h

anhi,3PI<D.j.

n ≥ 0KOFSAw8CA-A2xE3<G2

J(n) = J(n, w)468<SlQ,/.o>37Q,3:z41<D.3F)4o2xEk467/25A-gOAOPamn,9AOymn4UAO7/258ahi,3P<O.1.

n ≥ 1T

w%L1 /G!H. > . ,/1J ?+ 4<.! ';Aq';+(I1/!3,:+<4 . 5l+b5=Al+b5=A&=, !LP0

(W THgSMx`b[ W@SM[ _LSZ\^];SW [ W S`8`b\bx@a S e \^THg \8TaT e X t=X#\ M [1W@cN U [ W@c T e XxGX\bWTLa U A [ W@c B U X;W\8T U a X%THg;[/T AB ‖ MN

v \8THg X;W;`^n [ W p@WrB[ _ S ca#T#_[1\-ZgTHY S1c@ZS U HX;Wa#T#_LpLTF[Nc\-[1r ST#S_uXt THgS#\b_#` S

ABNM

!.102;4-+<.' ?"$# + L)0&%7;4 /+b080<PAQ.G2/';A(' !L1'#v S a g;[1`8`)#X;WaHT#_pLTTHgS c;\b[ r S/TLS1_ [` X;W@ZTHgS r\-W@X;_l[+*1\-alXt THgS S`8`b\bx@a Si e g\,:gM\-a THgS xGS_#x@S1W@c\-#p`b[ _ V \-a S"LTLX;_wXt V XTHg MN

[ W@cABj =t

AM[ W@c

BNr S1S/T THgS%S`8`b\bx@a S*[ Z/[1\bW [/TC[ W@c

D U _LS1a:xGS.LTL\^];S/`^n U l x@[1a a S1aDTHg;_LXpGZgBTHgS\bWT#S_HYaS.LTL\ X;WKx@X\-WTa%XtAN e \^THg BM U Xt DM e \8THg CN U [1W@c Xt AM e \8THg BNi [/TDr X;a#TDX;W@S*Xt e g\,:g/HXp`-ct [1\8`@T#XBS0* \baHTDcpGS TLXBx[ _[`b`-S`cS21@W;\-W@Zl`8\-W@Sa j:43 XT#STHg;[T V n an;r r S/T#_(n U l x@[1a a S1a THg;_LXpGZg THgS5#S1WT#_LSXtqTHgS$#\-_#`-S ABNM e gX;a Sc;\b[ r S/TLS1_ e S [1_LSlTLX5HX;Wa#T#_LpLT

] `

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xT /Ð 3Ë6ÏÐÍ [ DÐ-Î Ï~Ð 3Ì*Ï Ï-[ Î -Ï Í TbcA<-8s8s0Q:=Az2xE3<O2DKEKFKGKHK<-7QJ

I<DP5Admn,37Qm¢>3mn.j4Um)<-7QJp.1<@9AO.j.UADJp8*,2xE3<O2;2xE9AY8C¡30Q<DP5AD8<-P5A

ABHIKBCDE

K-<-7QJCAFG

T3+O417QmxAFKGKHK<D7QJ

I<DP5Amx,37Qm>3mn.146m©KFSAE3<G3A

∠HIG = ∠HFGTQ+O467QmnA

∠HIA = ∠AFG = 90 K-FSASgOA?2∠AIG = ∠AFH

T¦ih2xE9AO8*Ak<D7Qg?.UAO8Y<DP5Ak7Q,2a£sADP5,K3FSAkE3<G3A 4AIG ∼ 4AFHi@9A-m5<D0Q8CA∠IAG = ∠FAH

|CT _ E/418)41:=l3.146AD8AI : AG = AF : AH

2xE3<O2468?KAB :

√2AC = AC :

√2AB

K;FSE/46msE«2xE9AO7%41:=l3.146AD8S2xE3<O2AB = AC

TVW72xE9Az,2xE9ADPE3<D7QJLKt41h∠AIG = ∠AFH = 0

KQ2xE9AO7I, A, G

<-P5Amx,/.j.1417QAD<DP*K<D8w<DP5AF, A, H

T¦~7k2xE/418m5<-8CA3K∠BAC = 45 T+O46:41.1<-P.o>K;8s417QmxADKEKFKGmx,37Qm>3mn.146m©KLFSAdE3<G3AeAO4o2xE9ADP

AC = BC,3P

∠ACB = 45 TQRL,3:@/4673417QgY2xE9AO8*Amx,37QJ/4j2546,3798FSASgOA?2I2xE9A)hi,/.j.U,-F)417QgYhi,/0QPmn<D8*AO8DT Î?Ì

AB = AC<D7QJ

AC = BCT _ E/0Q8GK 4ABC

468BAD¡3094j.6<G2nAOP<D.¢T Î?Ì

AB = AC<-7QJ

∠ACB = 45 T _ E9AO7 4ABC468<D7468C,38*mnAO.6AD8)P4UgOE22nP541<-7Qg?.6AYF)4o2xEP546gOE2<-7Qg?.6A<G2

AT

Î?Ì ∠BAC = 45 <D7QJ AC = BC

T _ E9AO7LK<D8467rR<-8CAd3K 4ABC468S<D7468C,38*mnAO.6AD8aP546gOE2I2nP541<-7Qg?.6AYF)4o2xEP546gOE2<-7Qg?.6A<G2

CT

Î?Ì ∠BAC = 45 <-7QJ

∠ACB = 45 T _ E9AO7 4ABC418W<-7468C,38*mnAO.6AD8aP4UgOE22nP541<-7Qg?.6AYF)4o2xEP546gOE2<-7Qg?.6A<G2

BT

bcA)mn,37Qmn.10tJ9AY2xE3<G2 4ABC468aAO4o2xE9ADPAD¡3094j.6<G2nAOP<D.t,3P;4j2418a<D7k418*,38CmxAG.UAO8P4UgOE22nP541<-7Qg?.6AQT _ E9Amx,373AOP8*A/K41h 4ABC418WAO4o2xE9ADPI<-7zAD¡3094j.6<G2nAOP<D.92nP46<D7Qg?.UA),3PI<-7k468C,38*mxAG.UAO8YP4UgOE2W2nP46<D7Qg?.UA2xE9AD7

DKEKFKGKHKt<-7QJ

I<-P5Amn,37Qm¢>3mn.j4Um©KQ418Y8n2nP<D46gOE25hi,3PxF<-P5JT

É-ÏËÐ Î Ð dÌ?ÍË Ò +-AG46:4o>/<cl9P5,D/4UJ9A-J«2xE9AcJQA?25<O41.18'hi,3P)2xE9Acmx,373AOP8*A/[email protected]<G3A2xE9AD:hi,3P;2xE9A)P5AO<J9ADP467z,3P5J9ADP;2n,':=<-f-A)P5,9,3:hi,3PI<-7<O.j25ADP579<O254 3Aw8*,/.j03254U,37T¢nT Ð DÏ6Í Ë6ÏÐÍÐ3ÎGÐ 9Ë6Ï~ÐÍ9Î$ [ 9 Ï6ÎGËÐ Ì ~Ì[ 1Ï Ë~ÐÍ Ð ~Ì/Ì

Ï6ÎGËÐ Í dÌDÌGÍ D ¢Ë 9Ð dÌ" Ë Ì ] ÌË Ì ÍÎ T¦ihDKEKFKGKHKI<-P5A'mn,37Qm¢>3mn.j4Um©K/2xE9AO7z2xE9A'mnAD7/2nP5A',/hmn46P5mn.6A

DEFGHI:z0Q8n2.146A,37)2xE9AltAOPnlQAD7QJ/4Umn09.1<-P@/468CA-m525,3P;,/hDE

KO<-7QJ'E9AO7QmxA/K-,37)2xE9AltAOPnlQAD7QJ/4Umn0t.6<DP;@/468CA-m525,3P,/hBC

TDªB8s417Qg2xE9A8s<-:=A<DP5g?0Q:AO7/2hi,3PFG

jF)4o2xEAC

|xK©FSAmn,37Qmn.10tJ9A2xE3<O22xE9AmnAD7/2nP5A':0Q8x2B@QA2xE9Akmn46P5mn0Q:=mxAO7/2nP5AO,/h 4ABC

T _ E/0Q8GK32xE9Akl9P5,3lt,38CA-JlQP5,9@/.UAO:±468BAD¡3094 /<D.6AD7/2I25,zmsE3<-P<-m52nAOP54o£n467Qg2xE9,38CAw2nP46<D7Qg?.UAO8W8s0tmsE'2xE3<G2OD = OF = OH

T ~M©|ÇDAG2

M@QAB2xE9A:z46JQxlt,/417/2I,/h

BC<-7QJ

M ′ 2xE9A:4UJ9slQ,/467/2I,/h DETtÇDA?2

R@QA

]] M2xE9AP<J/410Q8B,/ht2xE9ASmn41P5mn0Q:=8CmxP546@QADJ=mn41P5mn.UA,/h 4ABC

T/bcASE3<?3AOD2 = OM ′2 + M ′D2 = (OM + MM ′)2 + M ′D2

= (R cos A + 2R sin A)2 + (R sin A)2

= R2(3 + 2 sin 2A − 2 cos 2A)K

be4j2xE8x46:41.1<-Pmn<O.Umn09.1<O2546,3798ahi,3POF

<-7QJOH

K-FSAmn<D7zP5A?FP4j25ASiM?|<D8sin 2A − cos 2A = sin 2B − cos 2B = sin 2C − cos 2C

T ~©|VW7QAlQ,38C8x4U@/41.j4j2~>khi,3PA-¡/0Q<D.j4j2~>418

∠A = ∠B = ∠CT

+O0QlQlQ,38*A2xE3<G2∠A 6= ∠B

T3+O417QmxAsin 2A − cos 2A = sin 2B − cos 2B

mn<D7@QA FP54o22nAO7 <D8sin(2A − 45) = sin(2B − 45)

KdFSA mn,37Qmn.10tJ9A!2xE3<G2(2A − 45) + (2B − 45) = 180 Ta¦~7«2xE/418kmn<D8*A C = 45 TWX,-F mx,3798x4UJ9ADP<-7Qg?.6AD8

A<-7QJ

CT 4j2xE9AOP

A = C (= 45),3P

A 6= CKQ467zFSE/4UmsEemn<D8*A2xE9Ak8C<D:A<-P5g?0Q:=AD7/2W<D8<-@Q,O3A0Q8s417Qg~©|g?4 3AD8W0Q8

B = 45 TL¦~7eAG4j2xE9AOPWmn<D8*A/K 4ABC418<D7468C,38*mnAO.6AD8aP546gOE2I2nP541<-7Qg?.6AQT

bcAYmx,37Qmn.j0tJQAB2xE3<O22xE9AP5AO:,25AB3AOP2546mxAO8,/h32xE9A8*¡/0Q<-P5AO8<-P5AYmx,37Qm>3mn.146m4jhQ<D7QJ,373.o>k4jh 4ABC418BAG4j2xE9AOP<-7zA-¡/0941.1<O25ADP<O.92nP541<-7Qg?.6A,3P<D7418*,38CmxAG.UAO8aP546gOE2I2nP46<D7Qg?.UA9T

<? + + $ /&'%$6 , ! + I $5!*'%"- 3" $>@? . + A& ? + I 3 6 + ?5A + I$ < %: $ 23 ; 2%: : 1&1 : + .7 ! 5I$ 5(0'0 : &7: ' E$ -& ' , $ '" : A" ;5; .! I&! $ 9#" %$ ! $ 5I 9" &! + 66 + $3"('%&" $ : :-"-?3 - $/? : % $ ? + I* 5+ $9<= : " :-7:-"- &'3 $ I"(A.02& 7$ ; $6:-**" 3" : , 5%:7:" $ 9 , "$56$ 9I ! 3$/;3 ?3 ':-"-E?3 - $ : " $/6$ 5I " 2*13 019A

$5'7: ' 3"3: I'" : $51 *2" &'- 2&1&19: " 2* ' , &" ' , " : * A7: &' " 2&%""-23 &'- & 47"-:-: '%" :-&A :-"-2 : * 3 " E&'"* ' " 2 :&' &3" "-: -7$/ '%"50 E"-23 : 04, - : 23"@" : 0 @:-"-2 : * ' E " 2 2&019"5'

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lL1*1 G!:. > . , JM?"%7!P2 H !1)@O /!LA#/ .!+-0(F'+<I1!,H+34-. 51O53.G21'J 0- '7JA4!4 E/.)' ,:A, ZS1W@S_[1`8\H[/TL\ X;WBXt%p;T#W[ r U opGS1a#TL\ X;W

p@x@x@X;aSTHg;[T AF

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DC= g

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EA= h

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[ a[x@_LX;x@X;_TL\-X;W XtGTHgS[1_LS1[*Xt 4ABC

]]

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A B

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QR

3Ð 9Ë6Ï~ÐÍ [ C Ë Ì ÍÐDÎ Î1Ï6ÍLÌ?Í [=Í ÎxÏ iÍQÍ3Î DÎGË Ï 9T_ E9Awmx,37/25AD7/2,/h32xE/468l9P5,[email protected]: 418<3ADP~>ShU<-:=,/0Q8a,37QA417QJQADA-JK-4j28<-7985FSADP;418g?4o3AO7@D>'2xE9A)hi,/.1.6,-F)467Qg*U

,M 24- " (- %D!$&%5 T CwÏË Ë ÌwÍÐQË Ë6ÏÐÍdÐË Ì Ð~Ì YÌ 9 DÌ Ò[P QR]

[ABC]=

(ghi − 1)2

(gh + g + 1)(hi + h + 1)(ig + i + 1) Í[DEF ]

[ABC]=

ghi + 1

(g + 1)(h + 1)(i + 1)

T Ì Ì

[XY Z] Ì Ì©ÎGÌ?ÍË1Î)Ë Ì - Ì ÐË Ì Ì

XY Z

%5 . $ T \ 82¢FS,zmn,3798*AD¡30tAO7QmxAO8,/ht2xE/4182xE9A-,3P5AO:cK-FSAE3<G3A<U %KG . " ,- %D,$&%5 T _ E9A'2nP<D7983ADP8s<D.18 AD

KBE

Kt<-7QJCF

<-P5Amn,37Qmn0QPnP5AO7/241hL<D7QJ=,373. >4jhghi = 1

T % D%G . MD" O(- %D!$&%5 T _ E9A2xE3P5ADAYlQ,/467/28

DKEKD<-7QJ

F,372xE9AwUAsÆO2nAO7QJQADJ|8s46JQAO8B,/h 4ABC

<DP5ASmn,/.1.j467QAO<-PI4jhL<-7QJz,373.o>k41hghi = −1

T %3C %$&% D63% T5M3T \ T9¥%AO.o£x<f9K iÍË Ð Ë6Ï~ÐÍË~Ð D Ì-Ð dÌ-Ë [ KC,9E37Sbe41.6A*>S<-7QJk+-,3798?K9XA?F #;,3PnfQKM-N ] T

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t2K2xE3<O2kJQ,%7Q,2'417/2nAOP8*ADm52S2xE9Ac.j467QA8*ADgG:AO7/2

O1O2:ADAG2I<G2

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tc2xE3<G2JQ,9AO8417/2nAOP8*ADm52;2xE9AY.j467QA8*ADgG:AO7/2

O1O2:=A-A?2582xE9Aw2<-7QgOAO7/258

t1<D7QJ

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∠A4682xE9Aw.6<DPxgOAD8n2<-7Qg?.6AQTtÇDAG2

P@QA<-7>S417/2nAOP546,3PlQ,/467/2I1hi,3PIAsÆD<D:=l3.UA/KO2xE9A467QmnAD7/2nP5Aw,/h 4ABC

|8s0tmsEz2xE3<G2a2xE9A'ltAOPnlQAD7QJ/4Umn09.1<-P8wJ3P5,3lQlQA-JhUP5,3:P25,z2xE9A'8s46JQAO8Y,/h 4ABC

417/2nAOPx8*ADm52I2xE9A8s46JQAO8B<G2<-7467/25ADP4U,3PlQ,/467/2a,/hAD<-msEz8x4UJ9AQT _ E9Aw2xE3P5A-AS¡/0Q<J3P54j.6<G2nAOP<D.18W467/25,FSE/4UmsE'2xE9AD8CA)ltAOPnlQAD7QJ/4Umn09.1<-P8B8s0t@9J34 346JQA 4ABC<DP5A<O.1.mn.6AD<DP5. >zmx,37Qm>3mn.146mGT

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..

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..

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......

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..

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.....

A

B C

B′

C′

B′′

C′′

XY,-F mx,3798n2nP50tm52a<-7>z.1417QAB′C′ FSE9ADP5A B′ .146AD8,37 AB

<-7QJC′ .j4UAO8,37 AC8s0tmsE2xE3<G2

∠AB′C′ = ∠ACB<-7QJ

∠AC′B′ = ∠ABCT \ .18*,pmx,3798n2nP50tm52w2xE9A.1417QA

B′′C′′ lQ<DP<D.j.UAG.25, BCFSE9ADP5A

B′′ .146AD8,37 AB′ <-7QJ C′′ .j4UAO8,37 AC′ Ti+-ADA2xE9AYJ/46<-gGP<-: <-@Q,O3AQT |LR.6AD<DP5. >LK?2xE9AY¡/0Q<J3P54j.6<G2nAOP<D.18B′C′CB

<-7QJB′′C′′C′B′ <-P5A@Q,2xEpmx,37Qm>3mn.146mGT¤-0QP2xE9AOPn:=,3P5A3K0Q8x467Qgk2xE9A<-P5g?0Q:=AD7/2w4172xE9A=QP8x2wl9<-P<-gGP<-lQEcFSAmn<D7c8x0t@QJ/4o/4UJ9A 4AB′′C′′ 417/2n,z2xE3P5A-Amx,37Qm>3mn.146mY¡30Q<-J9P41.1<O25ADP<O.68OT _ E/418wg?4o3AO80Q8w<mn.6<D8C8B,/h8*,/.j03254U,379825,'2xE9A)lQP5,9@/.UAO:pT

XY,25AB2xE3<O22xE9A730Q:=@QAOP;33AB467)2xE9AlQP5,9@/.UAO: 8x2<O25AD:=AD7/2mn<[email protected]<-mxADJ'@D><-7>417/2nADgOADPgGP5AD<G2nAOPI2xE3<-7=,3PA-¡/0Q<D.92n,3T

? I !@; + .A + $ ?% : " ?3 - $ 3: " $ !I(!I + < $ 3 * : $%. 13 " 2*13 019A

E % "# % % ! & ')( ! % ' ''6' ! % % ! % ' ' % %% % ' & % 3 & % % " & ! % ! & &

ABC3 ! " % ! ' ! ''6' !

;3(2&13@ %& , 23 1 " & @"-2%: '%" :-9A

luL1B 1 G!:. > . , J ?"E/. , *,2/+-, * 9 %71!#!:!:. AF'+<I1!,H+34-4G.10b+<4"L'+ 3BJ@ ' 4-032 ' 1A/!#!1,:,A ! > /+ 'q ][`bp@[/T#S

∞∑

n=1

tan−1

(f2

n+1

1 + fnf2n+1fn+2

)

e gS1_LS fn\-aqTHgS

nth ! \ V X;W["0#\W;p@r V S_ i THg;[/Tq\-a U f0 = 0 U f1 = 1[1W@c U t=X;_ n ≥ 2 U

fn = fn−1 + fn−2

j:

]] 3Ð 9Ë6Ï~ÐÍ [ Ï GÐ Í3ÎGÐÍ Ì DÌ ËÐÍ TbcASE3<?3A

f2n+1 = (fn+2 − fn)fn+1 = fn+2fn+1 − fn+1fn

TªB8s417Qg2xE9Aw2nP546gO,37Q,3:A?2nP546m46JQAO7/254j2~>

tan−1

(x − y

1 + xy

)

= tan−1 x − tan−1 yK

FSA:z<G>SFP54o2nAtan−1

(

f2n+1

1 + fnf2n+1fn+2

)

= tan−1

(fn+2fn+1 − fn+1fn

1 + fn+2fn+1 · fn+1fn

)

= tan−1(fn+2fn+1) − tan−1(fn+1fn)T

ÇDAG2S@QAw2xE9A)8s0Q: FSA)8*ADA-ftT _ E9AO7

S = limk→∞

k∑

n=1

(tan−1(fn+2fn+1) − tan−1(fn+1fn)

)

= limk→∞

(tan−1(fk+2fk+1) − tan−1(f2f1)

) K8s417QmxAI2xE9A8s0Q: 25AO.6AD8Cmx,3lQAD8OT-X,-Fr2xE9Aal9P5,9J30tm52

fk+2fk+1417QmxP5AD<D8*AO8tF)4j2xE9,/032@Q,/0Q7QJ<-7QJ

f2f1 = 1TAO7QmxA/K

S =π

2− π

4=

π

4

T <? + + $ /&'%$6 , ? I !@; + .A + $?% : "

?3 - $ 3: " $ ! =6 $ 3 $*< $ 9I @ !$EA + 9I + 3$ 5%:7:% ? + ? + $ /'% : 7$/ 3" : & ?* 7A" + II$ , 2 ;0 3 &7$ ?$ 9I .! + ! $;3#" 07$ < $/ 9I + .( ! 9I0$/ (0' : 3 7:' E$ &04 ' , $ '" : ! $5;3 ?3 '%:" E?3 % $ : 3" $96$" 9I 3 " 213 019A

* 19: 3" &'" "-2" "-2313 - , &E- 01&1 : , " : "-23E 0'0" : A 3 0$ " % &)& % ' $?3 % < %"-25A.A . + (4) (2001)

$ 1&1AA

l G!H. > . ,/1J ?" E.! 1/'"A.0<4?. P0bA 4=)74=)7/!0- '7J,1

m \b_#` SaΓ1(O, R)

[1W@cΓ2(I, r)

T#Xp:g `b\bW@St[/T

D U@e gS1_LS R > r[1W@c

O[ W@cI`b\-SlX;W*THgSa [1r Sa:\-c@SlXt

t hgSlx@X\-WT

A\-aw[1WnMx@X\-WTQX;W

Γ1

hgSTL[1W@ZS1WTaT#XΓ2THg;_LXpGZg

A\bWT#S_aS.LT

t[T

B[1W@c

C U _LS1a:xGS.LTL\^];S/`^n S1W@XTLS*THgS%\bW_[c\b\FXt4ABD

[1W@c 4ACD V n r1[1W@c

r2 U _LS1a:xGS.LTL\^];S/`^n gX e THg;[T r1 + r2

\ba HX;Wa#TL[1WTw[ aA][ _\ Sa X;W

Γ1

' . /'4L S];S1_[` a X`^];S_aqW@XTL\-#S cTHg;[/TGTHg\baFx_LX V `-S1r [1W@c\^TLaqaX`8p;TL\ X;Wg;[/];SD[ xxGS[ _LS1cV S/t=X;_LS \-W m _Lp**[ a %@_LX V ` Sr 2320

B; ; ! ;

]]D`I- <(? + + $ /&'%$36 , 6 ?I?! + &! !I !/$7A A + E: 4

:- 6&(: $&0 & :$@I13: + .( ! 9I0$ 5(0'0 : &7: ' E$ & ' , $/ '" : !I ! I + < $ 3 * : $. 13 ; + %+ AE!*!9$ :- 5%:7:" $ < : %$5?$ 5I ! $;3 ?3 '%:" ?3 - $" : " $/6 0$ 5I 3 "-23 13 019A. &'*2&*""-2""-23 , : : "- '% : " 2 19: "

A % 0 "-23 , " 2 , : , -

Γ2

0: E%&%"-23" %""E"-23 , : ,

Γ1

13 5"E"-23 , " 3"5%"D

A

? G!:. > . , J ? C/034=)7/!wE '7.G21,:AF!L,208+ '7/';61 '@,:+32 BA&='@';,?!P2 L"A k 2,4=!+-@R1S/T

A U B U [1W@c C V STHgS[ W@Z`-S1a Xt[lT#_L\b[ W@Z`-S gX e THg;[T∑

1

tan(

A2

)+ 8 tan

(π−A

4

)3 ≤ 9√

3

11

!;.10324-+(.' ?" # 1'F2P0 %F';+<4<. A, 3 ! ' + 2 !#!+ 1'J+80b+(. '!#' *1'J@ 9HA,. 6 !H. ('7. A4! > +-'( .J +q1J ?"B4=)7* J +<4<.!

)

S21@W@SlTHgSlt-p@W LTL\ X;WfX;W

[0, π] V n f(π) = 0 U [ W@ct=X;_ 0 ≤ x < π Uf(x) =

1

tan(

x

2

)

+ 8 tan3(

π − x

4

)

hgSW #` S[ _`^nf(x) > 0

t=X;_ [1`8`x ∈ [0, π]

7v S%ag;[`b`qx@_LX];S*THg;[TDTHgSMr [ * \brBp@rXtGTHgS t-p@W LTL\-X;WJ(x, y, z) = f(x) + f(y) + f(z)

X1];S_QTHgSHX;rBx@["LTwaSTT = (x, y, z) ∈ d 3 : 0 ≤ x U y U z ≤ π x + y + z = π

\-a S1o;p@[`@T#X 9√

3

11≈ 1.41713

p@_wx@_LXXt4#X;Wa:\baHTa Xt 1];Sx[ _3TLa [1_L\b[TL\-X;W Xt

f(x)X;W

[0, π]

;n c\-_LS.LT)HX;rBxp;T[TL\-X;W U e S1@W@cMTHg;[/T

f ′(x) = −(f(x)

)2[1

2sec2

(x

2

)

+ 24 tan2(

π − x

4

)

sec2(

π − x

4

) (

−1

4

)]

=1

2

(f(x)

)2[

12 tan2(

π − x

4

) (

1 + tan2(

π − x

4

))

− 1 − tan2(

x

2

)] i(Pj

S/Tr = tan

(x

2

) U s = tan(

π − x

4

) [ W@ct = tan

(x

4

) h7gS1W t=X;_x 6= π e S

g;[/];Sr =

2t

1 − t2U [ W@c s =

1 − t

1 + t

`8\-r\-W[/TL\-W@Zt Ue S ZS/T r =

1 − s2

2s

\-W #Sf(x) > 0

t=X;_x ∈ [0, π) U a S/TTL\bW@Z f ′(x) = 0

n;\-S`-ca

12s2(1 + s2) = 1 +

(1 − s2

2s

)2

=(1 + s2)2

4s2X;_(1 + s2)(48s4 − s2 − 1) = 0

]]

+-,/.o/467Qg48s4 − s2 − 1 = 0

K3FSAkgOA?2s2 =

1 +√

193

96

T _ E/0Q8GK92xE9Ak,373. >lt,38x4j254 3AP5,9,2468

s0 =

√

1 +√

193

96

TI)AD7QmnA3Kξ0 = π − 4 tan−1(s0) ≈ 1.64076

468Y2xE9A,373.o>rmxP54o254Um5<D./<O.10tAd,/hf(x)

,37(0, π)

T+O417QmxAf(0) = 0.125

Kf(π) = 0

K<-7QJf(ξ0) ≈ 0.640475

K/FSAmx,37Qmn.j0tJQAY2xE3<O2maxf(x) : 0 ≤ x ≤ π = f(ξ0) = M

T¢nT§<D.j0tAD8B,/h

J(x, y, z),37

∂TK-2xE9AS@9,/0Q7QJ9<DP~>,/h

TT\ 2I2xE9AY3AOP2546mxAO8B,/h

∂TK-FSAE3<?3A

J(π, 0, 0) = J(0, π, 0) = J(0, 0, π) = 0.25T

¦~7k2xE9Aw417/2nAOP546,3P,/h∂T

K/,37QA,/hQ2xE9A2xE3P5A-Amn,9,3P5J34179<O25AD8468£sADP5,K<-7QJ'2xE9AOP5AOhi,3P5A/K/hi,3P<D.j.ξ ∈ (0, π)

K-FSAE3<G3AJ(x, y, z) = f(0) + f(ξ) + f(π − ξ) ≤ 0.125 + 2M < 1.406 <

9√

3

11

T¢xT§<DP541<O2546,37=,/h

f ′(x),37

[0, π]T

41ÔADP5AO7/2546<G25467Qg)iM?|FSA,9@/2<D417LK3<Dh125ADP8*,3:=A)8s41:=l3.14jm5<O2546,3798GKf ′′(x) = f(x)

[f ′(x)A(x) + 1

2f(x)B(x)

] KFSE9ADP5A

A(x) = 12 tan4(

π − x

4

)

+ 12 tan2(

π − x

4

)

− tan2(

x

2

)

− 1K

<-7QJB(x) = A′(x) = −12 tan5

(π − x

4

)

− 18 tan3(

π − x

4

)

− 6 tan(

π − x

4

)

− tan3(

x

2

)

− tan(

x

2

) TbcAz8*A?2

f ′′(x) = 02n,d8*AO<-P5msEehi,3PY8C,/.1032546,3798Y467

(0, π)T;ªB8s417Qg'2xE9A=JQAGt734o254U,37,/h

f(x)K/2xE9AhU<-m522xE3<G2

f(x) > 0K3<-7QJk2xE9ASP4UgOE2WE3<D7QJ=8x4UJ9AS,/hiM?|2n,8s0t@38x254o25032nA)hi,3P

f ′(x)K-FSA<DP5A).6A-J'2n,S2xE9ASAD¡30Q<G254U,37 U

[

12 tan4(

π − x

4

)

+ 12 tan2(

π − x

4

)

− tan2(

x

2

)

− 1]2

=[

8 tan3(

π − x

4

)

+ tan(

x

2

)] [

12 tan5(

π − x

4

)

+ 18 tan3(

π − x

4

)

+6 tan(

π − x

4

)

+ tan3(

x

2

)

+ tan(

x

2

)] T ~©|¦~7=2nAOPn:z8Y,/h2xE9A)/<-P46<[email protected]

rKs ∈ (0, 1)

417/2nP5,9J30tmnA-JeAD<DP5.j4UAOP*KtA-¡/0Q<O2546,37d~©|@QADmx,3:=AD8(12s4 + 12s2 − r2 − 1)2 = (8s3 + r)(12s5 + 18s3 + 6s + r3 + r)

T ] |

]] +O0t@98n254j250325417Qg

r =1 − s2

2s

<D7QJ1 + r2 =

(1 + s2)2

4s2

417/2n,z ] |nKFSAY2xE9AD7=E3<?3A(

12s4 + 12s2 −(1 + s2

)2

4s2

)2

=

(

8s3 +1 − s2

2s

)(

12s5 + 18s3 + 6s

+

(1 − s2

) (1 + s2

)2

8s3

)

,3P (1 + s2

)2 (48s4 − s2 − 1

)2=(16s4 − s2 + 1

) [

8s3(12s5 + 18s3 + 6s

)

+(1 − s2

) (1 + s2

)2] T

+O46:zl9.j41hj>/467QgK-FSA,9@/2<D4174s2(s2 + 1)(192s8 + 388s6 − 60s4 − 39s2 + 1) = 0

T [ | É Ò _ E/418kmn<D7%@9AdmsE9A-msf-ADJr<-7QJc3AOP54jADJ%AO<-8x41. >0Q8s417Qg¥ \ uÇ «,3P)8C,3:Ae,2xE9ADPmx,3:zl90325ADP<O.UgOAD@9P<S8>38x25AD:T

9¡/0Q<O2546,37 [ |E3<D8IAxÆD<m525. >2¢FS,w8C,/.1032546,3798467(0, 1)

KG79<D:AG.o>@Us1 ≈ 0.595805<-7QJ

s2 ≈ 0.157625Ka>/4UAG.UJ/467Qgc2xE9Aphi,/.1.6,-F)467Qgd2¢FS,^mxP4j2546mn<O.W/<D.j0tAD8e,/h

f ′(x)U

ξ1 = π − 4 tan−1(s1) ≈ 0.992276<D7QJ

ξ2 = π − 4 tan−1(s2) ≈ 2.51623KP5AD8sltADm5254o3AG.o>T \ .18*,K-hi,3P;2xE9Aw<-P5g?0Q:=AD7/22n,k@QAY0Q8*ADJk417du/<DP2W¦6§%@QAG.U,-FKOFSAw7QA-ADJ'25,t7QJd2xE9Ak/<O.10tA

ξ3417

(0, π)8x0tmsEe2xE3<O2

f ′(ξ3) = f ′(0) = 23128

T _ E/468mx,37QJ/4j2546,37.UAO<J38a25,'2xE9Awhi,/.1.6,-F)467QgwA-¡/0Q<O2546,37467sU

(s2 + 1)(48s4 − s2 − 1) =23

64(16s4 − s2 + 1)2,3P

(1 − s2)(5888s6 + 2080s4 − 169s2 − 87) = 0T

_ E9A 8CA-mn,37QJ hU<m525,3P 417 2xE9A A-¡/0Q<O2546,37 <-@Q,O3A E3<D8 < DÍ3Ï 3Ì lt,38x4j254 3A!P5,9,2s3 ≈ 0.439150

FSE/4UmsEd>346AO.6J98'<e0Q7346¡30tAξ3 = π − 4 tan−1(s3) ≈ 1.48641

T¦~7Qmx,3P5lt,3P<G25467QgY<O.1./2xE9AY4173hi,3Pn:z<O2546,37,9@/2<D417QA-Jk<-@Q,O3A3KOFSA)m5<-7k8s0Q:z:=<DP54o£sA2xE9A/<P541<O2546,37=,/hf ′(x)

,37[0, π]

417k2xE9ASmsE3<[email protected],-F U

x 0 · · · ξ1

≈ 0.99· · · ξ3

≈ 1.49· · · ξ0

≈ 1.64· · · ξ2

≈ 2.51· · · π

f ′(x) 23128

≈ 0.449 23128

0 ≈ −0.526 −0.5

T \ 7QA-mnAD8s8C<DP~>kmx,37QJ/4j2546,37Shi,3P2xE9AYP5AG.6<G254o3AYAsÆO2nP5AD:z<),/hJ(x, y, z)

2n,',9mnmn0QP467k2xE9A)417/2nAOP546,3P,/h _ T+O0QlQlQ,38*AJ(x, y, z)

<G225<O46798z<«P5AO.1<O254 3A«AxÆD2nP5AO:z0Q: <O2S2xE9A417/2nAOP546,3Pklt,/417/2(x0, y0, z0)

,/hTT _ E9AO72xE9A:=AG2xE9,9JS,/hÇO<gGP<D7QgOA:z09.o2546l3.146ADP8I<-8s8s0QP5AO8;2xE3<O22xE9AOP5AAsÆD468n258W<

λ0 ∈ 8x0tmsEk2xE3<G2Lx(x0, y0, z0, λ0) = Ly(x0, y0, z0, λ0) = Lz(x0, y0, z0, λ0) = 0

KFSE9ADP5A

L(x, y, z, λ) = f(x) + f(y) + f(z) − λ(x + y + z − π)T _ E/0Q8?KtFSAE3<G3A

f ′(x0) = f ′(y0) = f ′(z0) = λ0F)4j2xE

0 ≤ x0Ky0Kz0 ≤ π

8x0tmsEd2xE3<G2x0 + y0 + z0 = π

T

]] NbcAAsÆ-l9.6,3P5AW2xE9Alt,[email protected];hi,3PL2xE9AO8*Amn,37QJ34o254U,3798;25,@QA8C<G25468xADJS@D>)0Q8x467Qg2xE9AS4173hi,3Pn:z<O2546,37e,372xE9A)/<DP541<O2546,37,/h

f ′(x)25,.U,9,9fk<G22xE9ASlQ,38C8x4U@/.UA).U,9m5<O2546,37,/h2xE9AY/<D.j0tA

λ0417k2xE9AP<-7QgOA,/h

f ′(x)T

¦ihf ′(ξ2) ≤ λ0 < 23

128= f ′(ξ3)

K32xE9AD7f ′(x0) = f ′(y0) = f ′(z0) = λ0FS,/09.UJd41:=l3.o>2xE3<O2

x0 + y0 + z0 > 3ξ3 > πKQFSE/4UmsEe468w<mn,37/2nP<J/4Um52546,37TI¤-0QPx2xE9ADP5:,3P5A/KL4jh

λ0 = f ′(ξ1)Kt2xE9AO7

x0 = y0 = z0 = ξ146:zl9.j4UAO82xE3<O2

3ξ1 = πK<-7Q,2xE9AOPwmn,37/2nP<J/4Um52546,37TIAO7QmxA/KtFSAz:z0Q8n2wE3<?3A 23

128≤ λ0 < f ′(ξ1)

K467eFSE/46msEmn<D8*Af ′(x) = λ0

E3<D8wAxÆD<m525. >=2¢FS,8C,/.1032546,3798467[0, ξ3]

T _ E9AOP5AOhi,3P5A/Kt417c,3P5J9ADPWhi,3Pf ′(x0) = f ′(y0) = f ′(z0) = λ0

2n,zE9,/.UJK9<O2.UAO<-8n22¢FS,z,/hx0Ky0K3<-7QJ

z0:z0Q8n2@QA2xE9AY8s<-:=A3KD<-7QJ2xE9AYl9P5,[email protected]:7Q,-F P5ADJ30tmnAD8I25,t7QJ/467QgB2xE9A:=<©Æ-41:z0Q: /<D.j0tAw,37

[0, ξ3],/hQ2xE9A8x467Qg?.6AY/<DP541<@/.UAwh60Q7Qm5254U,37

J(x) = 2f(x) + f(π − 2x)T

TL¥r<©Æ-41:z0Q: /<D.j0tAS,/hJ(x) = 2f(x) + f(π − 2x)

,37[0, ξ3]

T+O467QmnAJ ′(x) = 2f ′(x) − 2f ′(π − 2x)

KOFSAY8*ADA2xE3<G2J ′(x) = 0

4jht<D7QJk,373.o>41hf ′(x) = f ′(π − 2x)

TQªB8x467Qg)iM?|nK/2xE/418W418BA-¡/094o/<O.UAO7/2I2n, U(f(x)

)2[

12 tan4(

π − x

4

)

+ 12 tan2(

π − x

4

)

− tan2(

x

2

)

− 1]

=[f(π − 2x)

]2[

12 tan4(

x

2

)

+ 12 tan2(

x

2

)

− cot2 x − 1]

,3P[

cot x + 8 tan3(

x

2

)]2[

12 tan4(

π − x

4

)

+ 12 tan2(

π − x

4

)

− tan2(

x

2

)

− 1]

=[

tan(

x

2

)

+ 8 tan3(

π − x

4

)]2

·[

12 tan4(

x

2

)

+ 12 tan2(

x

2

)

− cot2 x − 1] T i?|

\ gG<D4170Q8s417Qgr = tan

(x

2

) <D7QJs = tan

(π − x

4

) K-FSAE3<G3Acot x =

1

tan 2(

x2

) =1 − tan2

(x2

)

2 tan(

x2

) =1 − r2

2r=

−s4 + 6s2 − 1

4s(1 − s2)

T+O0t@98n254j250325417Qg)467/25,i?|I<D7QJ8x46:zl9.j41hj>/467QgKFSAS,9@/2<D417z2xE9Ahi,/.j.U,-F)417QgAD¡30Q<G254U,37z467

sKFSE9ADP5A

s ∈ (0, 1)K

(4 − 17s2 + 30s4 − 17s6 + 4s8)2(48s6 + 47s4 − 2s2 − 1)

= s2(1 − s2 + 16s4)2 ·[12(1 − s2)6 + 48s2(1 − s2)4 − s2(−s4 + 6s2 − 1)2 − 16s4(1 − s2)2

] K,3P*K9,37h60QP¢2xE9ADP8x46:zl9.j41hj>/467Qgw<D7QJhU<-m52n,3P467QgK

4(s2 + 1)(1 − 3s2)(192s18 − 212s16 − 1617s14 + 4406s12

− 5404s10 + 3258s8 − 1136s6 + 154s4 + 15s2 − 4) = 0T

VW7%2xE9A417/2nAOP~/<D.(0, 1)

KI2xE9Ac.6<D8x2kAD¡30Q<G254U,37 E3<D8=AsÆD<-m525.o>r2xE3P5ADAp8*,/.j03254U,3798?K79<-:=AO. >LKσ0 = 1√

3

Kσ1 ≈ 0.521949

<-7QJσ2 ≈ 0.477039

>346AO.6J3417Qgk2xE9Az2xE3P5ADA

]G[ mxP54o254Um5<D.3/<O.10tAO8,/h

J(x)467

(0, ξ3)U

x0 = π − 4 tan−1(σ0) =π

3

Kx1 = π − 4 tan−1(σ1) ≈ 1.21738

K<-7QJx2 = π − 4 tan−1(σ2) ≈ 1.36115

T¨3>=J/46P5ADm52amx,3:zl9032<O2546,3798GK-FSAwt7QJ'2xE3<O2

J(x0) =9√

3

11

KJ(x1) ≈ 1.41514

K <-7QJJ(x2) ≈ 1.41615

T+O467QmnA

J(0) = 2f(0)+f(π) = 0.25<-7QJ

J(ξ3) = 2f(ξ3)+f(π−ξ3) ≈ 1.4116KFSAzQ79<D.j.o>mx,37Qmn.j0tJQAk2xE3<O22xE9A=:z<?ÆD46:0Q: /<O.10tA=,/h

J(x),37

[0, ξ3]K<-7QJmx,3798CA-

¡30tAO7/25.o>«,/hI2xE9A=h60Q7Qm52546,37J(x, y, z)

,O3ADPTKL468 9

√3

11

KFSE/46msE418<O22<D417QA-Jc41h<D7QJ,373.o>k41hx = y = z =

π

3

T 23 * : , ( , " '%" :-9A

Ku E/. , * ,2/+-, * /9 %7 !#!H/!H. '7J E121 ' E/. , * 6;.9 2AF'+<I1!,H+34-4(G.10b+<4"L'+ 3BJ@ ' 4-032 ' 1A(%71! #P0(.'@A ! > /+ 'q

=t 1

1 + a+

1

1 + b+

1

1 + c6= 2 U cS/TLS1_LrB\bW@S[`b`q_LS1[`FW;p@r V S_a x U y U z a:p:gTHg;[T

0 = (1 + 2a2)x2 + (1 + 2b2)y2 + (1 + 2c2)z2

+ 2xy(ab − a − b) + 2yz(bc − b − c) + 2zx(ca − c − a)

!;.10324-+(.' ?"$#K+ L)7P0&%7;4 /+b080<PAQu.G2;'A&'! ' HhgS*Z\ ];S1W S1o;p@[/TL\ X;W/L[ W V S e _L\^TT#SWB[ a

(ax + by − z)2 + (ax − y + cz)2 + (−x + by + cz)2 = 0

Twt=X`8` X e a THg;[/T \8TaaX`8p;TL\ X;Wa[1_LSlTHgX;a SXtGTHgS*gX;r XZS1W@S1Xp@a`8\-W@S[ _ ana#T#Srax + by − z = 0 Uax − y + cz = 0 U

−x + by + cz = 0

hgS c@ST#S_#r\-W[1WTXtTHg\-a ana#T#Sr \-a*S1o;p@[`uT#X1 − ab − bc − ca − 2abc

hgSHX;W@c\8TL\-X;W

1

1 + a+

1

1 + b+

1

1 + c6= 2

\-alS1o;p\ ][1`-S1WTTLX1 − ab − bc − ca − 2abc 6= 0

hgS_LSt=X;_LS U x = y = z = 0\-a THgS X;W;`^nMa X`bp;TL\-X;WBXtTHgSgX;r XZS1W@S1Xp@aan;aHTLS1r [1W@c%THgS X;W;`^nMaX`8p;TL\ X;WXt@THgSZ\^];SW S1o;p@[/TL\ X;W

]G[ M G ? -I !; + =.A + $ ?% : " ?3 % $ 3: " $ . +

A & ? + I$&*" 3E& < %: 5%:7:" $ 2 ; 2: : 1&1 : 3@ (! /< # + &!9$6& 4 %:- & - I%" %" $36 , $ " .! + ! $;3#" -7$ < $ 9I ?@ + I A < + $ - I%" %" 5%:7:" $%I % $ $ 5I + .( ! 9I0$ (0' : 3 7:' E$ &04 ' , $ '" : A ;5; .! I&! $ 9%#"$! $ 9I +&+ 4 " 3$" /90$@?2: & / &! + 66 + $"('%&" $ : :-"-?3 - $ ? : % $ ; !I + A I ! 9II&! &! $ " 27$ , 3 " 2@13 019A

L1B; / G!H. > . ,/1J ?$#K+ L)7P0 %74-+80b0(AQu.G2;'A&' !L1'# W THgS `8\-W@S a S Z/r S1WT

AB U `-S/T C U D V S aHp:g THg;[/T AC

CB=

BD

DA=

1

3

\-a#TL\-W LTQx@X\-WTa

M1 U M2 U M3`8\ SlX;W[#\b_#` Sx@[1a aH\-W@Z THg;_LXpGZg

B[1W@c

C[ W@cM[ _LSa:p:g THg;[T

∠M1BC = 2∠M1CB U ∠M2BC = 2∠M2CB U [ W@c ∠M3AD =2∠M3DA

gX e THg;[T 4M1M2M3\-a S1o;p\8`-[/T#S_[1`

!;.10324-+(.' ?" . ,)+<. !;+ +-1A #O 1, " +bAE > 'q R1ST 4ABC V Sl[NT#_\-[1W@Z` S e \^THg ∠B = 2∠C ` ST D V STHgSt=XXTXtwTHgS x@S1_LxGSW@c;\,#p`-[1_ t _LX;r

ATLX

BC [1W@cK`-S/T M V S THgS rB\-c@YHxGX\bWT*Xt BC

hgSWAB = 2DM

G!:.. 5 Fv S a S/T

α = ∠C hgSW

∠B = 2α R1S/T

N V STHgS r\ cY:x@X\-WTlXtAC

h7gS1WNM

\bax@[1_[1`8` S/`T#XAB U NM = 1

2AB U [ W@c ND = NA = NC

hgp@a U ∠NDC = ∠NCD = α

\-W #SNM‖AB U ∠NMC = ∠ABC = 2α

hgS_LSt=X;_LS U ∠MND = ∠NMC − ∠NDC = 2α − α = α = ∠MDN U aXTHg;[T

DM = NM = 12AB THg;[T \ba U AB = 2DM

R1S/T 4ABC V S [T#_L\b[ W@Z`-S e \8THg AB < AC ` ST D V STHgSlt=XXTXtTHgS%xGS_#x@S1W@c\-#p`b[ _Dt _LX;r

AT#X

BC [ W@c `-S/T M V S THgS%rB\-c@YHxGX\bWT Xt BC =t

AC = 2DM U THgS1W ∠ABC = 90 + 12∠ACB

G!:.. 5 R1S/T

N V SBTHgS rB\-c@YHxGX\bWT Xt AC hgSW

NM\bax@[1_[1`8` S/`TLX

AB U[ W@cND = NA = NC

\bW HSDM = 1

2AC = DN UQe SKc@S1c;p#S THg;[/T

∠NMC = 90 + 12∠NDM

\bW HSND = NC U e S X V T[1\bW

∠NDM = ∠NCD = ∠ACB U aX THg;[T ∠NMC = 90 + 12∠ACB

\-W #SNM‖AB U e Sg;[/];S ∠NMC = ∠ABC

hgp@a U ∠ABC = 90 + 12∠ACB

3 X e e S TLp@_LW T#X THgS X;_\ Z\bW[1` x@_LX V ` Sr sv S [ a:a:p@r S THg;[/T

M1[ W@cM2

`8\ S X;W THgS r [(X;_*[ W@c r\-W@X;_*[1_#aBC U _LS1a:xGS.LTL\^];S/`^n hgSW M3

`8\ SaX;W THgS rB[(X;_ [1_BC

v S aSTα = ∠M1CB

[1W@cβ = ∠M2CB

h7gS1W∠M1BC = 2α

[ W@c∠M2BC = 2β

\-W #S∠M1CM2 + ∠M1BM2 = 180 Ue Slg;[/];S (α + β)+ (2α +2β) = 180

hgp@aα + β = 60

v Sa S/TAB = 4a THgS1W

AC = a U CD = 2a U [ W@c DB = a R1ST

T V STHgSNt=XXTXt;THgSNxGS_#x@S1W@c\-#pGY`-[1_Dt _LX;rM3

T#XAC

@v S*a S/Tx = TC

R1STM V STHgS*rB\-c@YHxGX\bWT Xt AD U THgS1W

CM = 12(CD − AC) = 1

2(2a − a) = a

2

\-W #S∠M3AD = 2∠M3DA U;e Sg;[/];S i V n ];\b_TLpGSBXtu`-S1rBr [ Pj M3A = 2TM = 2(x + 1

2a)

= 2x + a \-W #S

M3T ⊥ AB U e S*ZST M3B2 − M3A2 = TB2 − TA2 hgp@a Ue Sg;[];S

M3B2 = M3A2 + TB2 − TA2

= (2x + a)2 + (x + 3a)2 − (a − x)2

= 4x2 + 12ax + 9a2 = (2x + 3a)2

]G[ AO7QmxA/K

M3B = 2x + 3aT

ÇDAG2N@9Az2xE9A=:4UJ9slQ,/467/2w,/h

CB 2xE9AD7

TN = TC + CN = x +3a

2

K,3P2TN = 2x + 3a = M3B

TB+-A?2γ = ∠M3BC

Tw¨3> .UAO:=:z<¬/KFSArgOA?2∠M3CB = 90 + 1

2γTQ+O417QmxA

∠M3CM2 + ∠M3BM2 = 180 K-FSAE3<G3A(90 + 1

2γ + β

)+ (γ + 2β) = 180

2xE3<O2W468?K180 + 3(γ + 2β) = 360 KQhUP5,3:!FSE/46msEzFSAkgOAG2

γ + 2β = 60 T _ E9A-P5AOhi,3P5A/K∠M3M1M2 = ∠M3BM2 = γ + 2β = 60 KFSE/46msE¬41:=l3.146AD8k2xE3<G2

∠M1M3M2 = ∠M1CM2 = α + β = 60 TAO7QmxA/K 4M1M2M3468BA-¡/0941.1<O25A-P<D.¢T

- ; ++ A"E!*!9$ :- 5%:7:" $" < : %$*?$ 9I ! 3$@;3 ?3 '%:" ?3 - $ : " $/6$ 9I 3 " 2@13 019A

/Bi HX;_L_LS"LTLS c \bW q !j G!:. > . , J ? # + L)0%74-+80b0(AQu.G2;'A&'! ' H p@x@x@X;aSlTHg;[/T

A U B U C [1_LSlTHgS [ W@Z`-S1a Xt[lT#_L\b[ W@Z`-S %@_LX];STHg;[T

8(cos A + cos B + cos C) ≤ 9 + cos(A − B) + cos(B − C) + cos(C − A)

≤ csc2(A/2) + csc2(B/2) + csc2(C/2)

!;.10324-+(.' ?",/+ .).G2A4G.10 " ' . 21';+<4 ' .10b0(6AQCl+-'q4(/!)l;I1'A4',A ! k \-W #S

f(x) = csc2 x\ba HX;W];S0* X;W

(0, π) @JM$ \b_LS"LT&#X;r x;p;TL[/TL\ X;WaQagX eTHg;[T

f ′′(x) = 2 csc4 x + 4 csc2 x cot2 x > 0 U1e S*g;[];S

csc2

(A

2

)

+ csc2

(B

2

)

+ csc2

(C

2

)

≥ 3 csc2

(A + B + C

6

)

= 12

≥ 9 + cos(A − B) + cos(B − C) + cos(C − A)

! X;_THgS `-StbT \bW@S op@[1`8\8T(n U W@XTLS1@_aHTTHg;[T

cos A + cos B + cos C

= 2 cos

(A + B

2

)

cos

(A − B

2

)

+ 1 − 2 sin2

(C

2

)

= 1 + 2

[

cos

(A − B

2

)

− cos

(A + B

2

)]

cos

(A + B

2

)

= 1 + 2

[

2 sin

(A

2

)

sin

(B

2

)]

sin

(C

2

)

= 1 + 4 sin

(A

2

)

sin

(B

2

)

sin

(C

2

) i(Pj

]G[9]\ .68C,K

cos(A − B) + cos(B − C) + cos(C − A)

= 2 cos

(A − C

2

)

cos

(A + C − 2B

2

)

+ 2 cos2(

C − A

2

)

− 1

= 2 cos

(C − A

2

) [

cos

(A + C − 2B

2

)

+ cos

(C − A

2

)]

− 1

= 4 cos

(C − A

2

)

cos

(C − B

2

)

cos

(A − B

2

)

− 1~©|

¤/P5,3:!~M©|<-7QJ=i?|nKFSA)8CA-AY2xE3<O2I2xE9Aw.UAGh12417QA-¡/0Q<D.j4j2~>468BAD¡3094 /<D.6AD7/2I25,8 sin

(A

2

)

sin

(B

2

)

sin

(C

2

)

≤ cos

(A − B

2

)

cos

(B − C

2

)

cos

(C − A

2

) KFSE/4UmsEYE3<D8@QADAD7Y8CE9,-F7Y<G2t.6AD<D8x292xE3P5ADA2541:AO8lQP5A*/4U,/0Q8x.o>417wRLP0Æ>U©/ 5MDN MLU ]] -K [ - 5 U [-[ [-[ MK3<-7QJ-M 5 ] U3M-M-N M- 3T

= -I + 6 + 9I @ %)?$ 5%:7:" I7$EI7$ 9 : = %%: & < G) + ? + $ 3% $ / %: ; + + /! I + $ ;33" : $*6 , I&?! A ! $ ' '( 5%:7:" $5<=" (0$ $ 9I .! + ! $;3#" -7$ < $ 9I '" : + G.( ! 9I0$ 5(0'0 : &7: ' E$ & ' , $ '" : < I A @< $" :-7:-"- -&#" 0$ + 3" %$% +&+ 4 3$ 90$9?2%: 9" &! + 66 + $ "('%&" $ : %:" ?3 % $ ? : % $ ; !I + A I ! 5II! ! 3$ *"-23&7$ , "-23*13 019A 2 * 3 : , ( , " '%" :-9A

<="3 ':&' %3 * 0'-" " 2 , 7: , & / "-: , -3 '0 :-" :- E! A 9" " "3 A3! , &' $ :-&" :"

( ) : " 2 '" : %"(' %&% , E E& &' : "-2%: &

(§ A $ 1A

)A-13#" : , '0 $ 3 : $'$ % , :" /( " '&4

:-"-0$s2 ≥ 16rR − 5r2

$5 " 2 &' + '- -3 '0 :-"-0$2r ≤ R

$*2 r

3R

&" "-23: :'@ , : , ' %: ' "-: -

ABC$3 (19 , " : 0$3

s&":-" E: 19: " A

9 , :&' 3 " *2: , 2 0'0" "-2"*2 &" 0: 3 : 1 - $23 23 "-2"/"-23 " : 3 '0 :-"- , & &E"

8F (a, b, c) ·∑

cos A ≤ 1 + 8F (a, b, c) +

∑

cos(A − B)

( ∗

*2

F (a, b, c) = F =

(

1 +(√

a −√

b)2

8√

ab

)(

1 +(√

b − √c)2

8√

bc

)(

1 +(√

c − √a)2

8√

ca

)

?% 0$F (a, b, c) ≥ 1

A I0: , :-" : * "-2" ∑

cos A > 1$ , , '&G" 2&%"

8(F − 1) ·∑

cos A ≥ 8(F − 1)

*2: , 2: 1 : (∗)

"-2"

8∑

cos A ≤ 8 − 8F + 8F ·∑

cos A ≤ 9 +∑

cos(A − B)

]G[-[ Ä ? T 5 >U ]-] Ð Ð-Î?Ì [YÉ -D D Ð9Ð Ï Ì 9 ÌdÐ Ï Í3Ï

DÌ ÎsÏË#[ 9ËGÐ Í Î ] tTH4 3A'<'lQP5,9,/h;@D>3A-m525,3P8B2xE3<G22xE9A':ADJ341<-798,/h<S2nP541<-7Qg?.6AkE3<?3A'<mn,3:=:=,37lt,/417/2,/hQ417/2nAOP8*ADm5254U,37 UO<)l9P5,9,/h¢K3E9,-FSA*3AOP*K* "#$0=&= &7 ' &( 5 *:OT

_ E9A=3ADm52n,3P)lQP5,9,/hU8k,/h2xE/418kP5AO8s09.o2YF)4j2xEcFSE/46msE¬¦<D:#hU<D:z4j.141<-P)<D798nFSAOPw2xE9A¡30tAO8x2546,37zlt,38CA-J'2xE/468F<G> UuQP5,D3AI2xE3<G2Q2xE9A:ADJ341<-798,/h 4ABC

417/2nAOP8*ADm52<G2 1

3

(−→OA +

−→OB +

−→OC

) KFSE9ADP5A

O4682xE9A,3P546g?467T_ E9AlQP5,9,/h¢KL,/hmn,/0QP8*A/Kt2xE9AO7c<-:=,/0Q7/258w8x46:zl9. >25,d8CE9,-F)467Qg2xE3<O2W2xE/468wlQ,/467/2B468w,37AD<-msEz:=A-J/46<D7T

3Ð 9Ë6Ï~ÐÍ [ 9 Ï6ÎGËÐ Ì ~Ì[ 1Ï Ë~ÐÍ Ð Ì 3Ì Ï6ÎGËÐ TÇDAG2A@9A2xE9Aw,3P4Ug?417LK<D7QJ'.6AG2 −→

b<D7QJ −→c @9AYP5ADl9P5AD8CAD7/2<O254 3AD8a,/h92xE9AB3ADm52n,3P8

AB<D7QJ

ACT _ E9A':4UJ3lt,/417/258

LKMKQ<D7QJ

N,/h8s46JQAO8

BCKCA

KQ<D7QJAB

E3<?3Alt,38x4j2546,37Y3ADm52n,3P8 12(−→b +−→c )

K 12−→c <-7QJ 1

2

−→bKOP5AO8ClQA-m5254 3AO. >T _ E9Aa3A-m525,3P;AD¡30Q<G254U,3798,/h

ALKBM

K/<-7QJCN

<-P5A)<-8ahi,/.1.6,-F8LUAL : −→r =

1

2s(−→

b + −→c)

BM : −→r =−→b + t

(1

2

−→c − −→b)

CN : −→r = −→c + u(

1

2

−→b − −→c

) T_ E9AO8*A2xE3P5ADA.1417QAD8l9<-8s8L2xE3P5,/0tgOEY<Wmx,3:z:,37YlQ,/467/2t4jh-<-7QJw,373. >4jhO2xE9ADP5AWAxÆ-418x23/<D.j0tAD8,/h

sKtK3<D7QJ

u8s0tmsE'2xE3<O2

1

2s(

−→b + −→c ) = (1 − t)

−→b +

1

2t−→c = (1 − u)−→c +

1

2u

−→bT

+O467QmnAz2xE9A3ADm52n,3P8 −→b<-7QJ −→c <-P5Az.1417QAD<DP5. >c467QJ9ADlQAD7QJ9AD7/2*K2xE9A=<@9,D3A=8>98n2nAO: ,/h3A-m525,3PaAD¡30Q<G254U,3798a468BAD¡3094 /<D.6AD7/2I2n,

1

2s = (1 − t) =

1

2uK

1

2s =

1

2t = (1 − u)

T_ E9AS,373. >z8C,/.1032546,37k468

s = u = t = 23

T =- <? + + $ /&'%$ 6 , 6 ?-I&?! + &! !I !9$

7A A + E: :6(: $ %& :-$I13: < A 9:-" $9!*A3?%: &'(:&3 + A 6( )& ($ $3I0130: + I II + I @< ! + $E?. ?"-: , ' E$" 2 " 2 3 ?* + I A < + $ - I%" %" / 5%:7:" $9IE% $ $ 5I =. + &A& ? + I$3*" 3 &< %: :47:-"-0$ 2 ; 2: : 1&1 : 3 3 (! < # + !/$ 6( %:- & - I%" %" $ 6 , $ " 0 ! II &A $ :-7:-"- ?3 % ?019 3 "$I0%3$ I + .7 ! 5I$ 5 40' : 3 7:' E$ -& ' , $3 '" : A ;5;E.! I&! $9#" %$! $ 5I + !6&6 + A" 9$ &$ ? $ 9I < I A *< $ :-7:-"- &#" $ + 3" %$

]G[ 9" + A/< + I$I %" !3" $ !/3" 0$ $ 9I + ? I%; + ? $ !""&474': , 5%:7:" $ < %& ' 0$/ ! $ ;3 G?3 ':-"- ?3 % $ : 3" %%$/6 0$ 9I A&<=" " 2 0'%E:" " '" :* 7: E: @" E" 2 %&%%A

9 -"2&9 , " " '%" :-&51/'% : 2 :- + A " 2$ ! ! )& & % & )! & ! &)' )&)& ! " ' $ + $ 9 : %$ : / $ + A < A;" " %$ 3 , " 0$!/ : 9$ %&%$ A

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O U M U B U F [ _LSX;WB[#\-_#`-S Ω e \^THg OF[ a[*c;\b[ r S/TLS1_ S1W #S U

∠OFB = ∠EMB WTHgSlXTHgS_Qg;[1W@c U EB ·ED = EC2 = EO ·EM THgp@a U

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ΓT#XBg;[];S*S1o;p@[/TL\ X;W

x2 + y2 = 1 U [1W@c` ST

T =

(1 − t2

1 + t2,

2t

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V S[x@X\-WTDX;WMTHgS#\-_#p@rt=S1_LSW HS h7gS*S op@[TL\-X;WaXt@THgSlT[ W@ZSWTD[/T At = a

UTHgS T[ W@ZSWT[/TCt = c

U [1W@cTHgS `b\bW@S BD t _LX;r

t = bTLX

t = d U _LS1a:xGS.LTL\^];S/`^n U

]G[3`<-P5A

tA : (1 − a2)x + 2ay = 1 + a2 tC : (1 − c2)x + 2cy = 1 + c2

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¦ih2xE9AO8*A)<-P5Amx,37Qmn0QP5P5AD7/2*K2xE9AD7det

1 − a2 2a −(1 + a2)1 − c2 2c −(1 + c2)1 − bd b + d −(1 + bd)

= 0T

_ E3<G2418GK−2(a − c)(ab + ad + cb + cd − 2ac − 2bd) = 0

T+O467QmnA

a 6= cKDFSA)JQADJ30tmnAY2xE3<O2

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b<-7QJ

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τ√

3 ;WBp;TL\b`8\-a:[ WTw`b[%an;rST#_L\-S%c@Sax nY_[ r\ cS1a U X;WxGS/p;T)L[1`,#p` S_` S/p@_wg;[p;T#S/p@_DSWMt=X;_HYr [1WTp@W T#_\-[1W@Z` S _LS.LTL[1W@Z` S HX;rBxGX;aS c p@W@Sg;[1p;TLSp@_*c@S T#_\-[1W@Z` SS op\b`b[TS_[1`DS/T p@W@S cS1r\ Y`-[1_LZSp@_Q[`b`b[ WT cS`b[ V [1aSlc@SN`-[ g;[1p;TLSp@_ cp/XTS[1p HSWT#_LS*cp _LS.LTL[1W@Z` S

R[ g;[p;T#S/p@_ c@S `b[ x n;_[ r\ cS S1a#T c@X;W √

3τ2 − 1x@[1_wx nTHg;[ZX;_LS

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VW7z<'J9,37Qm

§I,/.10Q:=Au/>9P<D:z46JQA=

¨t<D8*A · E3<O032nAG0QP3

=4τ

√3τ2 − 1

3AG2;§I,/.10Q:=ASVBm52<ADJ9P5A=

8τ√

3τ2 − 1

3≈ 11,2962

0Q734o2A 2 T

ÇO<JQAG0Æ-4AD:=Amn,373g?0QP<G254U,37dAO8x2ahi,3Pn:A-AkJQA2nP5,/468B2

A?2nP<ADJ9P5AO8DT¦i.>eAD7e<'0Q7¡3094aAD8n2P

A-g?09.j4UAOPSJÓ <DP A?2nA

2τ¡/0tAd7Q,/0Q8'<-l9l AG.UAOP5,3798§I,/.j0Q:A \ AG2

22A?2nP<ADJ9P5AO8hi,3Pn:

AD8kmsE3<-mn0Q7%JQA

22nP46<D7Qg?.UAO8

A-¡/0941.1<O2

AOP<D0ÆcAG2

22nP541<-7Qg?.6AD8S418*,9mAG.UAO8'¡/0tAd7Q,/0Q8<-l9l AG.UAOP5,37983,/.10Q:=A

BT

37 03254j.1418C<D7/2e.1< 8>9:AG2nP4UA J/02

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A<G3ADm0Q7QAeE3<D0325AO0QPJQAz2nP546<-7Qg?.6A

A-¡/0941.1<O2

AOP<D.AG2Y0Q7p2546ADP8'JQAeE3<D0325AO0QPJQAz2nP546<-7Qg?.6A

A-¡/0941.1<O2

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A)<D0mxAO7/2nP5ASJ9ASgGP<G/4j2

AJQAw.6<S@3<-8CAQT

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6

3

TWÇO<c@9<D8*AAG2<-7/2w0Q7p2nP46<D7Qg?.UA

AD¡3094j.6<G2

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3T \ 46798x4K.UA3,/.j0Q:A

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√2

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BKt,37=03254U.1418*AOP<¬.6A J

AG25ADP5:z4179<-7/2dJ9A RL<?>3.6A*>Q©¥%AD7QgOAOPdo3,/46P 5M*|sTX0Q:

ADP5,25,3798r.6AD8«8*,3:z:A?258rJ30È2

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dijJ9<D798v.UA J

AG25ADP5:z4179<-7/2P5ADl9P

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AD8BJ/0z2

A?2nP<ADJ9P5A9T

288V 2 =

∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 1 1 1

1 0 d212 d2

13 d214

1 d221 0 d2

23 d224

1 d231 d2

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1 d241 d2

42 d243 0

∣∣∣∣∣∣∣∣∣∣∣∣∣

T

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RAG.6<SJQA*/4UAO7/2288V 2 = 384τ4 − 128τ2 ¡/0948CA8x46:zl9.j41tAAO7

§I,/.j0Q:AB =

2

3τ√

3τ2 − 1T

\ 46798x4Q.6<Smn,373g?0QP<G254U,37<)0Q7k3,/.j0Q:AY2n,2<D.J9A2τ3

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A 2 T

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a = p(q2 + r2) U b = q(p2 + r2) U c = (p + q)(r2 − pq)

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wa =

√

bc

(

1 − a2

(b + c)2

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U wb =

√

ac

(

1 − b2

(a + c)2

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U

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wc =

√

ab

(

1 − c2

(a + b)2

) T+O0t@98n254j250325417QgS2xE9Ak<@9,D3A/<O.10tAO8Yhi,3Pa2xE9Ak8s46JQAO8w,/h;<AOP5,37346<D72nP541<-7Qg?.6A<D7QJd8x46:=l9.j41hj>/467Qg)g?4o3AO8

wa =2qr(r2 − pq)(p + q)

pr2 + 2qr2 − pq2

√

p2 + r2K

wb =2pr(r2 − pq)(p + q)

2pr2 − p2q + qr2

√

q2 + r2K

wc =2pqr

r2 + pq

√

(p2 + r2)(q2 + r2)T

_ E9AJ9AD7Q,3:4679<G2n,3P8)<DP5Azlt,38x4j254 3A4jhr2 > pq

T | _ E9A2xE3P5ADAz<-7Qg?.6A=@3418*ADm52n,3P8F)41.j.E3<G3ABP<O2546,379<D.-.6AD7Qg©2xE38;41hp2+r2 <-7QJ q2+r2 <DP5ABltAOP5hiADm52;8C¡30Q<DP5AD8OT¦ih2xE9AB8s46JQAO8I,3P<-7Qg?.6A'@/468CA-m525,3P8B<DP5Aw2xE9AD7=P<G254U,379<O.L@/032a7Q,2417/2nADgGP<D.~KFSASm5<-7z8*m5<D.6A)0Ql2xE9A'AO7/2546P5A2nP541<-7Qg?.6A@D><-7<-l9lQP5,3l9P541<O25AzhU<m525,3P25,c:=<-f-A2xE9AD: <O.1.;417/2nADgOADP8OTc¦U2Y418)7Q,2)E3<DP5J2n,ct7QJp8*,3:=A<mnmxAOl32<@/.UA/<D.j0tAD8)hi,3P

pKqKrU

p = 7Kq = 32

Kr = 24

jF)4o2xEr2 − pq = 352 > 0

<-8WP5A-¡/0946P5ADJ|nKQ:z<fD467Qgp2 + r2 = 252

<D7QJq2 + r2 = 402

T_ E9AO8*A/<D.j0tAD8>346AO.6Ja = 11200

Kb = 20000

Kc = 13728

TbcA=8s0t@38x254o25032nA2xE9AO8*Ak/<O.10tAO8S,/h

aKbKc467/25,d2xE9Azhi,3P5:z09.1<-8<-@Q,O3Azhi,3P2xE9A<-7Qg?.6A«@/468CA-m525,3P8z<-7QJ gOAG2'P<O2546,379<D.B730Q:=@QAOP8kF)4j2xE¬JQAO7Q,3:z4179<O25,3P8,/h

527K779

K<-7QJ1T+-mn<O.1417Qg@D>¬<hU<-m52n,3Pk,/h

529 · 779>/4UAG.UJ38z<p8C,/.1032546,37r25,r2xE9Al9P5,[email protected]: U

a = 4 597 969 600Kb = 8 210 660 000

Kc = 4 635 797 024

K/FSE/4UmsE'>346AO.6J=<-7Qg?.6A@3418*ADm52n,3P8B,/hwa = 256 658 688

Kwb = 75 963 888

Kwc = 5 517 563 520

T K " 2 13 019$ *2 1 2&% 7:-& -%"-2

315 409 500$

388 584 504$

426 433 644A 6& " 2: "-: - "-23 : 3" ( : , "

312 405 600$

278 555 200$

375 350 976AGE &'7$" 2 : &%: ' "-23 " : 0 : E : "

r = 104 085 135A

lNL1B; G!:. > . , J ?"$# = !! ,4=!1A&%7 ';6 /0<.!HA&('7J + S1_\^];SN[a S/TXta:\-c@S`-S1W@ZPTHg*S * x@_LSa aH\ X;Wat=X;_qTHgSNt [1rB\8`^n*Xt S1_LX;W T#_L\b[ W@Z`-S1a

ABC\-W e g\-:gBTHgS%W;\-W@S1Y:x@X\-WT #S1WT#_LS V

`b\-S1aNX;W a:\-c@SBC

i S1_LX;W T#_\-[1W@Z` Sg;[ a \-WTLS ZS_laH\ cS1a [ W@c \-WTLS ZS_l[1_LS1[ js S1S x_LX V `-S1r "" i x@_\b` jL1 Kq B; 1; !;.10324-+(.' ?"$#K+ L)7P0&%7;4 /+b080<PAQu.G2;'A&'! ' H! _LX;r x@_LX V ` Sr e S/W@X e THg;[/T V

\-aX;W THgSB`8\-W@SBC

\8t [ W@c X;W;` n \8tcos(B − C) = 0 THg;[T \ba U B = C +

π

2

X;_C = B +

π

2

v S e \8`b`ucS/TLS1_LrB\bW@STHgS aH\ cS1aBXt THgS S1_LX;W T#_\-[1W@Z` SaBa [/TL\-aHt8n;\bW@ZB = C +

π

2 THgX;aS a:[TL\ba:t8n\-W@Z

C = B +π

2

[ _LSX V T[1\bW@S c V n\-WTLS1_:g;[1W@Z\-W@Z b[1W@c

c

R1S/TK V S THgS [1_LS1[ XtDa X;r S a:p\^TL[ V `-S 4ABC

hgSW THgSKR[ e Xt \bW@S1an;\-S`-ca b

cos C=

c

sin C=

abc

2KU a XMTHg;[/T cos C =

2K

ac

[ W@csin C =

2K

ab

S1W #S U

]

4K2(b2 + c2) = a2b2c2K<D7QJdFSA8*ADAz2xE3<G2

b2 + c2468<elQADPhiA-m52)8C¡30Q<DP5A3K8s<G>

b2 + c2 = λ2 T¤0QP¢2xE9ADP5:,3P5A/Kb2 + c2 − a2

2bc= cos A = cos

(π

2− 2C

)

= sin 2C = 2 sin C cos C =8K2

a2bc

E9AD7QmnA3Ka2(b2 + c2 − a2) = 16K2

T _ E/0Q8GKb2 + c2 − a2 = µ2

hi,3P8C,3:Alt,38x4j254 3A467/25A-gOAOPµT'XY,25AS2xE3<O2

2λK = abc<D7QJ

4K = µaKL8C,z2xE3<O2

λµ = 2bcT _ E9AD8CAP5AD8x09.j28AO<-8x41. >k.UAO<J'25,

a2 = λ2 − µ2 K (b − c)2 = λ(λ − µ)K

(b + c)2 = λ(λ + µ)T

XY,-FK/hUP5,3:λ2 = a2 + µ2 KDFSASE3<?3ASAG4j2xE9AOP

1|

a = 2dmnKµ = d(m2 − n2)

Kλ = d(m2 + n2)

K9,3P2|

a = d(m2 − n2)Kµ = 2dmn

Kλ = d(m2 + n2)

Khi,3PB8C,3:A'lQ,38s4o254o3AS417/2nADgOADP8

dKmKn8x0tmsEz2xE3<O2

mKn<-P5Akmx,3l9P541:A/KL,/h,3l9lt,38x4j25AlQ<DP54o2>K9<D7QJ

m > nT

¦~7mn<D8*ABiM?|nK(b−c)2 = 2d2n2(m2+n2)

K©FSE/46msE)m5<D.j.68hi,3P2(m2+n2) = k2hi,3PL8C,3:AalQ,38s4o254o3A467/25A-gOAOP

kT¨Q032t2xE/46841846:zlt,[email protected]

m2+n2 418;,9J9JT _ E/0Q8GKFSAY:0Q8x2I@9A417kmn<D8*AYi?|nKOFSE/4UmsE.UAO<J382n,(b− c)2 = d2(m−n)2(m2 +n2)

K<D7QJ(b + c)2 = d2(m + n)2(m2 + n2)

TAO7QmxA/K

m2 + n2 = k2 hi,3P8*,3:=AYlt,38x4j254 3A467/25A-gOAOP kT _ E/468ag?4o3AO8

b = dkm<-7QJc = dkn

<-7QJSFSA:z<G>zmx,37Qmn.j0tJQAY2xE3<O2I2xE9A8x4UJ9AD8aKbKc<DP5ASg?4 3AD7=@D>

a = d(m2 − n2)K

b = dkmK

c = dknFSE9ADP5A(m, n, k)

418z<pl9P541:z4o254o3A¬u/>2xE3<gO,3P5AO<-7%2nP46l3.UApoF)4j2xEm > n

|<D7QJd<lt,38x4j254 3Aw467/25A-gOAOP?T

R,373AOP8*AG.o>KY8x0QlQlQ,38*A2xE3<O2aK

bKY<D7QJ

c8s<O25418shj>%2xE9AD8CArP5AG.6<G254U,3798OT ¤/P5,3:

a

b − c=

m + n

k

K<-7QJ b + c

a=

k

m − n

KI<D7QJ(m − n)2 < k2 < (m + n)2

KFSAcAO<-8x41. >rJ9A-J/0tmxA2xE3<O2b − c < a < b + c

K<D7QJABC

418'<m5250Q<O.1. >«<e2nP46<D7Qg?.UA9Tr)ADP5,37LÓ 8Shi,3P5:z09.1<dg?4o3AO84K2 = d4m2n2(m2 − n2)2

FSE9AO7QmxA/KK418S<D7467/25A-gOAOPIi@9A-m5<D0Q8CA

m,3P

n468A*3AD7|sTAO7QmxA/K 4ABC

468I<S)ADP5,372nP46<D7Qg?.UA9T¥%,3P5A-,D3AOP*KI2xE9AdP5AO.1<O2546,37 a2 + c2 − b2

2ac= −2K

ab

418AO<-8x41. >%msE9ADmsf-A-Jr<-7QJr:AO<-798S2xE3<G2cos B = − sin C

2xE3<O2418GKB = C +

π

2

<D7QJV468B,37

BCT

] ÐQËÌ TV

418 ,37 2xE9A .1417QA Î?ÌdÌGÍLËBC

4jh=2xE9A<-JQJ/4j2546,379<D.=mx,37QJ/4j2546,37cos(A−B)·cos(C−A) > 0

E9,/.6J98i@9A-m5<D0Q8CAVKO8s0Ql9lt,38CA-J),37w2xE9Aa.1417QA

BCKDE3<D8

(0, b cos(C − A), a cos(A − B))hi,3PD Ì mn,9,3P5J34179<O25AD8P5AO.1<O254 3A25,

(A, B, C)|sT_ E9Aemn,37QJ34o254U,37p:z<G><D.18*,@9AFP4j225AD7p<D8

cos(C − B) + cos(3A − π) > 0,3P

] -M

cos(3A) < 0T _ E/418I>/4UAG.UJ38

C <π

6

,3Pcos C >

√

3/2T¤P5,3: 2xE9AwP5AO8s09.o258<@9,D3A/K4j2418BAD<D8>'2n,8CA-AY2xE3<G2I2xE/468F)41.j.@9Aw2xE9Amn<D8*Aw41h

m2 > 3n2 T ? -I !; + .A + $/?% : " ?3 % $ 3: " $" ! $5;3

?3 '%:" ?3 - $ : " %%$ 6 0$ 9I ! @ 3$ /' , 2 " $ / : "-2313 019A

6& 1 : , :"/ 1 - , "m

'9" " 2 * u2 − v2

2uv (

*23 u$v @ %" :- *13: 197:" :@: "7$

v < u < (√

2+1)v)$

n '0"" 2@ A :

d: #:" ( 197:-" :-*: 3" %A 23

a = d|u4 − 6u2v2 + v4| ( b = d(u4 − v4)(

c = 2duv(u2 + v2)

23 : uv(u2 − v2)|u4 − 6u2v2 + v4| A

I"-(0$9: &:-" :- " 13 %:-: '%" :- " 2: @13 -" (" :"- %" 3" 13 $ 2 " 2&%"" 2 - @: 13 019#" : ':-03"

1 & ! & $ ! & % " & % ! ',&

BC

2 |B − C| =

π

2

3 tan B tan A = −1

4 OA‖BC

5 AH

! ' % &)& % % % "#! #" "#! #" 3 4ABC

6 AH

! ' % &)& % % % & ! & $ ! & % " ! "# 3 4ABC

7 BC

! ' " % 'AH

8 AN = 1

2OH

9 AC

(BC

% ! ' " %∠OCH

lNL1Bu1 G!:. > . , J ? # = ! !11,/4=!A(%71'610(.!:PAQ&='J +-@hgSx@X\-WTaO(0, 0) U A(1, 0) U B(0, 1)

[1_LS Z\ ];S1W R1S/TD(a, 0) U E(1−a, a) U

F (0, 1 − a) V S][ _\-[ V `-S x@X\-WTaMX;W THgS a:\-c@Sa%Xt 4OABi0 < a < 1

j R1STPc@SW@XT#S THgS xGX\bWT*Xt#X;W #p@_#_LSW HS Xt THgS #\-_#`-S1a

ODF U DEA U [ W@c BFE

S/TLS1_LrB\bW@SlTHgS `-X#p@a XtP

' . ?1+ '4-+(.'B. 5;,/.10324-+(.', ?.@?!4 %@+80b+-', " +-A4 24=!H/ .'4 A ' 1'J G 4<! QCM.. A(%@+(.10 F'+<I1!,H+34-1A ,1$#K+ !;JA ' k A ! k

7S"L[1p@a SP\-aDX;x@x@X;a:\^T#SNTHgSl_\ ZgT [1W@Z` Sl[T

OX;W%THgS#\b_#` S

ODPFi e g\-:g\-rBx`8\ SaFTHg;[T

FD\-aFTHgSc;\b[ r S/TLS1_ j Ue S a S SDTHg;[/T (a

2,1 − a

2

) rBp@a#T V SDTHgS HSWT#_LS gS1W #S U \8Ta S op@[TL\-X;WB\bax2 − ax + y2 − (1 − a)y = 0

i(Pj! X;_ #\-_#`-S AED U \^TLa #S1WT#_LS K

\-a7THgS#X;_#W@S_Xt [1WN\-a X;a #S`-S1a_L\-ZgTGT#_\-[1W@Z` SKNM

e \8THg%_\ ZgTQ[1W@Z` S[TFTHgSlr\ cY:x@X\-WT M

(1 + a

2, 0

) XtDA Ue g\b`-S N

i e gS1_LSNTHgSxGS_#x@S1W@c\-#p`b[ _ V \-a S"LTLX;_ KNXt

AEr S STLa

OAj \-a

(1 − a, 0) S1W #S U K

\ba(

1 + a

2,3a − 1

2

) U [ W@c%THgS#\-_#`-Sl\-a

x2 − (1 + a)x + y2 − (3a − 1)y = −a i(Pj

] -¤3,3Pw2xE9Ae.6,9mn0Q8,/h

PFSAcAO.j46:4679<G2nA

a@D>r8s0t@2nP<m525417Qgd~©|YhUP5,3:q~M©|25,%gOA?2

a =x − 2y

1 − 4y

K2xE9AD7z8s0t@38x254o250325467Qg)@3<msf467/25,z~M©|CTTL,/0325467QA<O.UgOAD@9P<).UAO<J38a25,(

x − 3

8

)2

+

(

y − 3

8

)2

=1

32

] |E9AD7QmnA3K

P.j4UAO8B,37k2xE9Amn46P5mn.6AYF)4o2xEzmxAO7/2nP5A (3

8, 3

8

) <-7QJzP<-J34j0Q81/(4

√2)T

bcAp7QAsÆO2k4173AD8n254UgG<G2nAeFSE3<O2klt,3P¢254U,37 ,/hB2xE9Armn41P5mn.UAp ] |4182nP<-mxADJ^@D>P<D8

aP0Q798'hUP5,3:

025,

1TvXY,25A2xE3<O2)2xE9AdlQ,/467/2

U(25, 1

5

) ,37«2xE9Ad:ADJ341<-7p2xE3P5,/0tgOEB8C<G25468xAO8SAD¡30Q<G254U,37«~©|Bhi,3Pw<O.1.

aK8*,e2xE3<O2

U418S,37p<D.j.2xE9Amn46P5mn.6AD8

AEDT%¨3>8>9:z:A?2nP~>LK3,37QA)m5<-7k<-P5g?0tA2xE3<O2;2xE9Awlt,/417/2 (1

5, 2

5

) ,37S2xE9Aw:=A-J/46<D7'2xE3P5,/0tgOEA.j4UAO8,37

BFEhi,3P<O.1.//<O.10tAO8B,/h

aT¦U2hi,/.j.U,-F8I2xE3<O22xE9Aw.6,9mn0Q8Wmn<D7=@9AJQAO8*mxP4U@9A-J<D82xE9A8*A?2B,/hlQ,/467/28U,2xE9ADP2xE3<-7

E|FSE9ADP5AS<kmn41P5mn.UA)4672xE9ASltAO7Qmn41.,/hmn41P5mn.UAO8W2xE3P5,/0tgOE

A<-7QJU467/25ADP8CA-m528W2xE9A'mn,3PnP5AO8ClQ,37QJ3417QgSmn46P5mn.6A',/h2xE9ASlQAD7Qmn4j.t2xE3P5,/0tgOE

B<D7QJ

VT¦U2468SAD<D8s46ADPCKIE9,-FSA*3AOP*K25,p,9@/2<D4172xE9Amx,9,3P5J/4679<G2nAO8',/h

P<-8w2xE9AlQ,/467/2),2xE9ADP2xE3<-7

D(a, 0)8s<O25418shj>/467Qgw~M©|<-7QJ=i?|8x46:09.j2<-7QAD,/0Q8s. > U

P =

(4a2 − 5a + 2

16a2 − 16a + 5,

4a2 − 3a + 1

16a2 − 16a + 5

) TVW7QAw8s41:=l3.o>kmsE9ADmsf8I2xE3<O2

P = UFSE9AD7

a = 0KP468I2xE9AY:z46JQxlt,/417/2,/h

ABFSE9AD7

a = 12

K3<D7QJP = V

FSE9AD7a = 1

T _ E/0Q8GKP85FSA-AOlQ8B,/0322xE9A<DP5mW,/ht2xE9Amn46P5mn.6A ] |<@9,D3Aw2xE9Aw.1417QA

UVhUP5,3:

U2n,

V<D8

agO,9AO8ahUP5,3:

025,

1T

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[email protected]. · t t!

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