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F�>9bP-�2fAN730P8+ArS \ F/0P8?L

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z417�2fANOf:t8�-/j

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w = 8Lx = 11

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S

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AD : DC = AE : PC = 1 : 2S

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(a, m, n)8c0decF�2rF3<H2

am + 1K/4xz/4TK9AD8

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è S(E�ì��&�Dï��{ð6ï

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é S(HE��|��`�DðTï

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at@PAB2rF9A�730P:q@9ADO�-/jPef46O5ef.6AD8

PiPjPk2rF3<N2�er-37/2{<D417

Pt417q2rF9AH46OV467/25ADO{4T-3OCL�<D7PK�.6AH2

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nLE8r0decFt2rF3<N2 2rF9A�bP-/467/2{8X-/j

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m(S) = f(n)S

ë S(H5Aw��)*' D;��rJì3��1ð )

)±DAH2

n<�7PK

k@9A bP-38c4x254xz3Aæ417/2fADiNADO{8�8c0decFó2rF3<H2

n

2< k ≤ 2n

3

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2f-gb3.6<�erA�AD<�ecFm-/jm

bP<?G�798�-37�<�8+£/0P<�O5Aa-/j^<D7n × n

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� S(��5ïDí )f�

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p<D7PK

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S-/j {0

L1L2L. . . } 8c0decF�2rF3<N2 0 ∈ S

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L�2rF9A~467/25A�iNANO{8n + p

<D7PKn + q

@9AN.6-37Pi�2f-SS

� S(��{ï�í )r�

)ñ/-3O�<�bP-38c4x254xz3A^467/25A�iNANO

nLN.6AH2

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nS ñ�467PKt<N.1.Pbd-38r4p254 z3A~467/25A�iNANO{8B8c0decF(2rF3<H2

d(n)3 = 4nS

� S(� îTï��

)\ F9A~K341<�iHO{<D: 8CF9-�G�8 <X8r2{<D41O5ef<D8+AD�C@3O546ech�G*4x2rF

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njk-3O;GRF/4TecF(4x2J468^bd-38C�[email protected]@/0941.6Kt<D7

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a<�7PKbO5AH.6<H254xz3AH.x>*b9O541:qAV25-

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S� S

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(an) (n = 1L2L3L. . . )

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an > 1997n jk-3O A+z3ADO|>n ∈ ∗ Lu64p4xv

aαn ≤ Un

jk-3O�A+z3ADO|>n ≥ 2

LPGRF9ADO5AUn

468�2rF9A�iHO5AD<H2fAN8r2�ef-3:=:=-37`K34 z3418+-3OB-/j2rF9A�8+A?2^-/jE730P:q@9ADO{8 {ai + aj | i + j = n} S�3��¤ì9î6ð|��í�� 'PïD�=�a�#� 4^ï�� cð �Tï �%�9îk�{ï��H�9�dì�� {ï�í )r��ï��/ï(r�îy��®��î 'E��#��ðyîy�� \ F9ARiHO5AN<N25AD8f2 730P:q@9ADO

α418 1

2

S±DAH2

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n�� ñ�46@P-379<�eref49730P:=@PANO?SV±DAH2

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F2mn > 1997n jk-3OJ<D.p. n ∈ ∗ L/<D7PK�.6AH2 an = 3F2mnS \ F9A*8CA�£/0dAD7PefA

(an)∞n=1

8c<N25418c¡dAD8*er-37PK/4p2546-37gu14xvcSt_À(G*41.p.�b9O5-Dz3A�2rF3<H2�4p2�8C<H25468r¡EAN8*er-37PK/4p2546-37gu64p4xvG*4p2rFα = 1

2

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F−n = (−1)n+1FnS [ .p25ADO579<N254 z3AN. >�LDGRA*ef<D7tKPAH¡d7PA

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i<�7PK

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(

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F2mn = Fmn (Fmn+1 + Fmn−1) = Fmn (2Fmn + Fmn−1) ≤ 3F 2mn

S\ F/0P8?L

an = 3F2mn ≤ 9F 2mn ≤ U2

n

LP<D7PK=¡P79<D.p.x>ELa1/2

n ≤ UnLP<�8�O5AD£30941O5A�K=417er-37PK/4p2546-37=u64p4xvcS

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n

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L GRAgF3<?z3Alog a2n

2n<(

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log an

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2kn<(

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n

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k→∞log a2kn

2kn= 0

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Emily

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Isaac

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Olivia

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Kris

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Gwenyth

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Mike

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Dave

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Bob

Tom

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Alan

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Larry

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Ulm

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Rivera

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Petra

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�! �� n %�'�),),'�* '��7- .�0R1�341�5�6�8 9,;��TF�H��6?E��0(J E�E1 ��5RJ UQE J M�� H�5�B�JSB�� B1��6 L H�1�E�K1�I7J L 1 8�6��1�5�B�F��J L F M � F�0�BGF�I�1 M ��1�5�B�F��J L FGnLsEF5GYiUKJGPS:0Qi>GP2fVc�5�<=Q=UVEF5f : � → � J�PRk�5�PSJ e m

f(x) =

{

5xUgf

x ≤ 12

`5 − 5x

Ugfx > 1

2

n�SPRQ

f1(x) = f(x)9S5�J

fn(x) = f(

fn−1(x)) f�EF:

n ≥ 2n&LM9IHK<�cGHd9IQ=P2Qi>GP�P?NS9;<=Q

�X9SHgc�PTEXff1998

(

4

516 − 1+

4

5125 − 1

) n=�1�E�� B�J1,HO9�;�� J L�� 6?E V F�BGF7J�E ES6 M �41��6�H M�� 0(F�H L 6_n FEF:�PS9;<?>D5�EF5�r?5�PS\R9IQ=U �FP[Ud5XQ�PS\IPS:

k`$pTP�p2UgHdHbHVPRQ

ak = 5k · 4

516 − 1

9S5�J

bk = 5k · 4

5125 − 1

n(¡[P"p2UgHdH�aG:=ES�FPlQi>F9IQ�Ugf1 ≤ k ≤ 1999

9S5�JkUdYb5�E_Q�JFU �FUdY?U e HVP

e m165�EF: e m

125`XQi>GPS5

fk(a0 + b0) = ar + bs` �

Rk �

M Z �

GRF9ADO5Ak ≡ r

u6:q-9K16vfL

0 < r < 16L�<�7PK

k ≡ su6:q-9K

125vrL

0 < s < 125S\ F/418BO5AN8c09.x2î?4xz3AN8^2rF9A*<�7985GRADO �

f1998

(

4

516 − 1+

4

5125 − 1

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= 4

(

514

516 − 1+

5123

5125 − 1

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u1v�êkj

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0 ≤ ` ≤ 123L/2rF9AN7

ak =5k

1 + 5 + · · · + 515<

5k

515≤ 514

515=

1

5

L<�7PK�.146h�A?G*418+A/L

b` < 15

S \ F9AN7ak + b` < 1

2

Su2v

a15 > 12

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515

LEGRF/4TecFa418ef.TAN<�O{.x>(2fO{0dAPS ñ�0PO¤2rF9ADO5:q-3O5A/L

5(1 − a15) = 5 · 515 − 1

516 − 1= 1 − 4

516 − 1= 1 − a0

S,N46:�41.1<�O{.x>EL

b124 > 12

<D7PK5(1 − b124) = 1 − b0

S_À�G*4p.1.J7P-�G§8CF9-�G 2rF3<N2X4pjû

RkvB418~2fO50dA=jk-3OX8+-3:=AqbP-38c4x254xz3At417/2fADiNADO

k ≤ 1998GRF/4TecF�418~7PAN4x2rF9ADO�<=:�09.p2541b9.6At-/j16

7P-3O�<t:�09.p2541b9.6A=-/j125

LE2rF9AD7(Rk′)

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r = 15<�7PK

s = 1248r46:�09.p2{<�7PAD-/0P8c. >�LD8c417PerA

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16<�7PK

125L9<D7PK

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u6<Hvr ≤ 14

<D7PKs ≤ 123

S \ F9AD7k′ = k + 1

L9<D7PKar + bs < 1

2

LPuTO5AN:=<DOrh�u � vkvcS}�AN7PerA/L

fk′(a0 + b0) = f(ar + bs) = 5(ar + bs) = ar+1 + bs+1

S \ F/0P8HLuRk′

v;418 2fO50dA9Suk@�v

r = 15<�7PK

s ≤ 122S \ F9AN7

k′ = k + 2L3<D7PK

a15 + bs > 12

L3u6O5AD:t<�Ofh(u|MwvkvcS}�AN7PerA/Lfk+1(a0 + b0) = 5(1 − a15 − bs) = 5(1 − a15) − bs+1

= 1 − (a0 + bs+1)S

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2

uTO5AN:=<DOrh�u � vTvrL�GRA�F3<?z3Afk′

(a0 + b0) = 5[1 − (1 − (a0 + bs+1))] = a1 + bs+2S

\ F/418^468BuRk′

vcSuke{v

r ≤ 13<�7PK

s = 124S \ F/418�e5<�8CA~468V8r46:�41.1<�O�25-tuT@�vcS

M Z�

ukK�vr = 15

<D7PKs = 123

S \ F/468�e5<�8CA=-9efef0PO{8�-373.x>ajk-3Ok = 623

L;<D8~418�AN<�8r41. >ecF9A�ech�ADK�S \ F9AN7k′ = 626

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2

Lf625(a0 + b0) = 5(1 − b124 − a0) = 1 − (b0 + a1) > 1

2

Lf626(a0 + b0) = 5[1 − (1 − (b0 + a1))] = a2 + b1

S\ F/0P8?L9u

Rk′v;418 2fO50dA9S

uTA?vr = 14

<�7PKs = 124

S \ F/468�e5<�8CAq-9efef0PO{8�-373.x>ajk-3Ok = 1374

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x

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sin y

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cos(

x

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y

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4 sin t − sin 2t − 2t > 0n

LsEF5GYiUKJGPS:jQi>GP�fVc�5�<=Q=UKEF5f(u) = 4 sin u sin v − u sin 2v − v sin 2u

`lpT>GPS:=Pu ∈ [0, π/4)

9;5�JvUdYT9S5W9S: e UgQ�:q9S:�m�k_N;P;J�5Fc�Z e PS:�Ud5

(0, π/4)n�C�U��hPI:=PS5XQ=Ud9IQir

UV5�\lQ}p2UK<�PF`SpTP.\IPRQ

f ′(u) = 4 cos u sin v − sin 2v − 2v cos 2u`

f ′′(u) = 4 sin u(2v cos u − sin v)n

M Z �

© 8c417Pi=ukM?vfL�GRAt8+ADA�2rF3<H2f ′′(u) > 0 �

GRF9AN7PerA/L�GRA=F3<Hz3Af ′ 417PerO5AD<D8c417Pi3S(�3>¬u � vrL

f ′(0) = 4 sin v − sin 2v − 2v > 0 �F9AN7PerA/L

f ′(0) > 0jk-3O

u ∈ [0, π/4)S \ F/0P8HL

f418^467PerO5AN<�8r467Pi~<D8 GRAN.p.d<�7PK=u � v;jk-/.p.T-�G�8?L/8c417PerA

f(0) = 0S

êT2 O5AD:t<D41798^25-�bPO5-Nz3ARukM?v�<D7PK=u � v �u|Mwv=Q�-3798r4TK9ADO

g(t) = 2t cos b − sin tjk-3O

t ∈ [0, π/4)S \ F9AD7

g′(t) = 2 cos b − cos t >√

2 − 1 > 0L

8+-R2rF3<H2g418^467PerO5AN<�8r467Pi3S \ F9AN7qukM?v;jk-/.p.T-�G�8^8r467PefA

g(0) = 0S

u � v \ F9AB:=AH2rF9-9K~468;8c41:t4p.6<DOdG*4x2rFh(t) = 4 sin t−sin 2t−2t

L?GRF9-38CA�KPANO54 z/<N254 z3Ah′(t) = 4 cos t(1 − cos t)

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Kn

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= BKKE

= λ 6= ±1n

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M � ¨

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AD2 + BE2 =2c2

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n

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N9;5�J

BC9IQ

L′ n] :=ES�FP,Qi>F9IQ

�V9 �(

BL

LC

) (

CM

MA

) (

AN

NB

)

(

BL′

L′C

)(

CM ′

M ′A

)(

AN ′

N ′B

)

= 1 �

� e �(

BL

LC

) (

CM

MA

) (

AN

NB

)

≤ 1

8�

��< � [LMN ] = [L′M ′N ′] �% �,E_Q=P *

[XY Z]J�PI5�E_Q�PIY(Qi>GP&9;:=PI9,EXf 4XY Z

n -

��J � [LMN ] ≤ [ABC]

3

n�SEG<�9RQ�P,Qi>GP.a�EXUV5XQ

PpT>GPI5�P��Fc�9IHdU�Q mj>GEXHKJFYbUV5jaG9;:}Q=Y � e � 9;5�J �KJ � n

=�1�E�� B�J1,HO9�;��F�P?JS8 ��1 6?>G>�ES6�0 M 5�B��8�67H�B M �70(J�H�JSBA;��1�E�E6�I�6 M � F�O9,0(J8,I�6 M N �$n�Ic�a�a�EFY�P�Qi>F9RQ

P>F9;Y69;:=PI9SH�<iEGEF:=JXUV5G9RQ�PIY

λ : µ : ν`;pT>GPS:=P

λ + µ + ν = 1n

�SPRQDe P.Qi>GPTf�EGE_Q&EXfsQi>GP�a�PI:�a�PS5�JXUK<�cGHd9;:6fK:=EFZ

PQ=E

BCn"oM>GPI5M` e m�J�PRk�r

5FUgQ=UVEF5M`;pTPT>F9O�FP12aPD = [BCP ] = λ[ABC] = 1

2aha

`Y�E�Qi>F9RQ

PD = λhan�o�>Xc�YR` 4PLL′ ��pT>XUK<?>�UdY.<�HVPS9S:=H m[YiUVZ�UdHd9;:6Q�E ABC

Y?Ud5�<iPQi>GP�<�EF:�:=PIY�a�EF5�JFUd5�\2Y?UVJ�PIY69;:=P.9IHdH�a�9S:q9SHgHKPRH � >F9SY6U�Q=Y&9IHgQ=U�Q=c�J�P29.fK9_<=Q=EF:bEXf λ

Y�Zj9SHgHKPI:Qi>F9;5�Qi>F9IQbEXf

ABC`X9;5�Jj>GPS5�<�PF`XUgQqY&YiUKJGPSY69S:=P2Ud5�Qi>GP.Y?9;ZP2a�:=EFa�EF:qQ=UVEF5hn

¡[P�Qi>Xc�Y&>F9R�FP

BN ′ = PL = λc`

CM = PL′ = λb`

CL′ = PM = µa`

AN = PM ′ = µc`

AM ′ = PN = νb`

BL = PN ′ = νan

M � Z

(a)

(

BL

LC

) (

CM

MA

) (

AN

NB

)

(

BL′

L′C

)(

CM ′

M ′A

)(

AN ′

N ′B

)

=νa

(1 − ν)a

λb

(1 − λ)b

µc

(1 − µ)c

(1 − µ)a

µa

(1 − ν)b

νb

(1 − λ)c

λc= 1

S

(b)_À�F3<Hz3A

(

BL

LC

) (

CM

MA

) (

AN

NB

)

=νa

(1 − ν)a

λb

(1 − λ)b

µc

(1 − µ)c

=ν

(1 − ν)

λ

(1 − λ)

µ

(1 − µ)

SQ�-3798r4TK9ADO�2rF9ARArò�b9O5AD8c8c46-37ν

(1 − ν)+

λ

(1 − λ)+

µ

(1 − µ)

=1

(1 − ν)+

1

(1 − λ)+

1

(1 − µ)− 3 ≤ −3 +

9

3 − λ − µ − ν=

3

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L�2rF9AN41O bPO5-9K/0de52 :�0P8r2 er-3798CA�£/0dAD7/25. >t@PA~<N2 :q-38f2 18

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2rF3<N2V418HLP4pj;<D7PKa-373.x>=4pjP418 2rF9ARefAD7/2fO5-/46K�S

(c)_À*8+ADA~2rF3<H2

[PMN ] =1

2PM · PN · sin ∠MPN

=1

2µa · νb · sin C

u68c417PerA∠MPN = π − ∠PMA = π − C)

=abc

4Rµν

L[PM ′N ′] =

1

2PM ′ · PN ′ · sin ∠M ′PN ′

=1

2µc · νa · sin B =

abc

4Rµν

S\ F/0P8?L

[PMN ] = [PM ′N ′]L�<�7PK8c41:t4p.6<DO5. >

[PNL] = [PN ′L′]<D7PK

[PLM ] = [PL′M ′]L�8+-�2rF3<H2

[LMN ] = [L′M ′N ′]u60P8r467Pi�2rF9A�-9@Dz/4T-/0P8*jT<�e522rF3<N2 bP-/467/2

P468^41798c46KPA�@9-�2rF�2fO{46<D7Pi?.TAN8fvcS

(d)© 8r467Pi�2rF9A~jk-3Of:�09.6<�A�<�@P-Nz3A3L�GRA*8+ADAX2rF3<N2

[LMN ] =abc

4R(λµ + µν + νλ)

S

M �#�

_À�7P-�25A~2rF3<H2 abc

4R

468V<�8r2{<�7PK3<�O5K�jk-3O5:t09.1<�jk-3O[ABC]

L3<D7PK�2rF3<H2

(λµ + µν + νλ) =1

2(λ + µ + ν)2 − 1

2(λ2 + µ2 + ν2)

S_À�F3<Hz3A

(λ2 + µ2 + ν2)(12 + 12 + 12) ≥ (λ + µ + ν)2 = 1@D>(2rF9ARQE<D0decF�>9�,�ecF�G�<�O¤2{¥�ê|7PA�£/0P<D.p4p2|>�L38+-R2rF3<N2J2rF/418V418V<N2 :q-38f2

1

2− 1

2

(

1

3

)

=1

3

S\ F/0P8?L

[LMN ] ≤ 1

3[ABC]

L<�8�ef.6<N46:=A�K�S [ iH<D417qA�£/0P<D.p4p2|>q-9eref0PO{8B4pj�<�7PKq-373. >t41j

λ = µ = νLP8+-(2rF3<N2

P418V2rF9AerAN7/2fO5-/4TK�S

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EXf�Qi>GP.YiUKJGPSYAB

9S5�JBC

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2`FY�>GE;p Qi>F9RQ�Qi>GP �Xc�9;5XQ=U�Q m

AC + λBC>F9SY&9.ZjUd5FUVZ�c�Z pT>GPS5

AC =

(

λ+√

λ2−8

2

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BC`X9;5�Jj9.Z9�N;UdZjc�Z pT>GPI5

AC =

(

λ−√

λ2−8

2

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BCn

=�1�E�� B�J1,HO9�; �71 6 Y[1� F�0R8 M .�1,0�B�F�E6�5 M � � M N =�C,n�SPRQa`

b`

ce PQi>GP�HKPI5�\�Qi>FY�EXf�Qi>GP�YiUKJGPSY�EFaGa�EFYiUgQ=PQi>GP�9S5�\OHKPIY

A`

B`

C`

:=PSY?a�PS<=Q=U��FPRH�msn6�SPRQac = k

n��IUV5�<�Pa2 + b2 = c2 =

k2

a2

`;pTP.\IPRQ

b =

√k2 − a4

a

n

��ff(a) =

√k2 − a4

a+ λa

`0Qi>GPS5f ′(a) =

−k2 − a4

a2√

k2 − a4+ λ

n �lE;p�`$Y�PRQqQ=Ud5�\f ′(a) = 0

`;pTP.>F9R�FPb2 − λab + 2a2 = 0

`_pT>XUV<?>j>F9;Y6:=EGE_QqY

b = a

(

λ ±√

λ2 − 8

2

) n

M �D]

Q�-3:tb90325417PiX2rF9A*8+ADer-37PKtKPANO54 z/<N254 z3A3L�GRA~¡d7PK

f ′′(a) = 2(b2 − 2a2)c2

a2b3

S[ 2

b = a

(

λ +√

λ2 − 8

2

) GRA�F3<Hz3A

b2 − 2a2 = a2(

12λ2 + 1

2λ

√λ2 − 8 − 4

)

> 0S

}�AN7PerA/L�GRA�F3<Hz3A*<�.6-9ef<N.d:�467341:t0P:¬S [ 2b = a

(

λ −√

λ2 − 8

2

) GRARF3<?z3A

b2 − 2a2 = a2(

12λ2 − 1

2λ

√λ2 − 8 − 4

)

< 0S

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λ2 − 8 ⇐=⇒(

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