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R

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P O1

OP

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sin α =

1 2 y 2

sin β

¢

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m
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v∈V

d(v) = (n − m)m + m(n − m) = 2m(n − m)

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2|E| =

P

f

d(v)

ß@‚ G347ID4

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|E|

m=

2

n`

n

|E| =

‰µ

m=

 2  n

CPk

4

CPk

2  n − 1

4

n n

2

(n − 1)

CP8$4-‡B4M:y¢ CP8$=3;<; f

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ä

ab + bc + ca =

ß56\à a b

+

3

b c¢ + c a

ß5Jà

4

a b c f + + b+1 c+1 a+1

c² "¯  ± «  ±

s f
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a=b=c=

1 3

CKLNH32 TNA+GB6A

3

(a − b)2 + (b − c)2 + (c − a)2

= 2(a2 + b2 + c2 − ab − bc − ca) = 2[(a + b + c)2 − 3(ab + bc + ca)]
f = 2 1 − 3 · 31 = 0

˜™š Œ G347ID47ku=BID4>¢ a b

sæs f
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+

b c + =3 c a





ä >ãèyêKç5è

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7é7ê

i¯ 

6M:<;

-±

b c 3f a + + = b+1 c+1 a+1 4

® (¯

—ä‰ç Bãä 

a=1−b−c

—èBç Mãäæé*ç 

CP:>AE=



ab + bc + ca =

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b =

cf

1 3

1 3

− b(1 − b)




c

= −3 b2 + 32 b −

ç5ä+ã\é

>ãè  é



4 ¥74\A

(1 − b − c)b + bc + c(1 − b − c) = 31
f c2 + (b − 1)c + 31 − b(1 − b) = 0

=BI

 CK: i 4

1 ¢‚ 3

¬



   

¢

ì

1 9




= −3 b −

f


1 2 3

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a b c + + =3 b c a

6:<;

c =

a =

3

a b c 3f + + = b+1 c+1 a+1 4

3



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AC


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E

BD

3π f <α< 7 7



7é Mè

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ã\ä

γ

δ

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α

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α

4ECD

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AC

6:<;

4π 7

β

B

ABCD

2(α + β) + 3(γ + δ) = 2π

α+β =

β

A

ä B

ã f

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2γ

2δ

CP8

f

TBCK472K;38

α + β = 2(δ + γ)

f

BD

ß à CK8

π − (α + β) =

3π f 7

f

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α + 2β + γ = π

=BI

πf  γ =α− 7

CP: i 4

β+γ =

γ>0

¢‚ 4„8944OA+GB6A

`

7

πf α> 7

− α +γ =

7

7


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x + cx + d = 0



b

d

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a+c ac b+d bd

k

d 6= 0

c + d = −1

k

¢A+G34M:

b = 1

¢‚ > G C i G TBCK472K;38 ¢A+G34M:

d=0

−a ¢ −b ¢ −c f d

= = = =

¢

¢6:<; i =B:38-4M’B?r47:>AD2ˆT ¢

√ √ 2¢ a=± c=∓ 2 2

2a ac b

¢

2a = −c ac = −1 √ f 6M:<; d=± 2−1

¢6:<;

¢

= −c ¢ = −b ¢ = −c

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(a, b, c, d) ∈



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(0, 0, 0, 0) (1, 2, −2, 0)

¢



±

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µ

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éã  O_rã  7å

ä

¢

¢

¢

x1 x2 . . . x n

xn+1 = x1

:<4 i 4M8*896MIDC@2ˆT"C@k

f  G3=‚

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¯  ª •®
oµ

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n J<4$:<=B:<†*:<4M¥67ADC ‡B4ID4M672>:B?
f

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i=1

xi xi+1 ≤ n

Œ² "¯  ± «     ± n≥5


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C@k

¢

=1

2



2 X

i=1

¢

i=1

n ∈ {1 2 3 4}

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n=2 x 1 + x2 = 2 f 47: i 4>¢ x1 + x 2

x1 x2 ≤

¢

4

xi xi+1 = x1 + x2 ≤ 2

f

xi = n

¢zJB?BAa:<=A

 :<4M’B?<672PCˆAŠT ¢3‚ 4=3J>A‰6MCP:

f

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i=1

3(x1 x2 + x2 x3 + x3 x1 ) ≤ (x1 + x2 + x3 )2

xi xi+1 = x1 x2 + x2 x3 + x3 x1 ≤

¢

(x1 + x2 + x3 )2 f = 3 3

4

f  ’>?<6M2@C@A€TFG3=>2K;38CPky6M:<;F=B:B2 TC@k ‘ 4+òMA i =B:38+C5;34MI fOx Œ 1G34M= f ŸBTNA+G34 g › › : x2 = x3 = 1

x 1 + x2 + x3 + x4 = 4  :<4’>?<6M2@C@A€Ty¢‚ 4 ;34;>? in4R= A+GB64A √ x 1 x2 + x 2 x3 + x 3 x4 + x 4 x1

= ≤

47: i 4>¢ 4 X

i=1

p

(x1 + x3 )(x2 + x4 ) (x1 + x3 ) + (x2 + x4 ) 2

xi xi+1 = x1 x2 + x2 x3 + x3 x4 + x4 x1 ≤ 22 = 4

= 2

f

f

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i=1

xi xi+1 =

n>4

n n n 4 f · = ·n > ·n=n 2 2 4 4


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c + d = 13

¢

f =BI c − d = −4 c−d = 7 ‘ = ‚¢8*CP: i 4 6M:<; GB6\‡B4NA+G34F896ML“4NH<6MIDCˆAŠTV6:<; ¢ ¢ ¢ ¢ c ∈ {0 1 . . . 9} f ‚ 4 4768+CP2 TN;<4Mc;B? −i 4RdA+GB67A c + d6:<; c=0 d f= 4 Œ G347: (a, b, c, d) = (2, 0, 0, 4)

˜

 ï ô


fŒ G34aH3ID=3JB2K4ML°8+A‰67AD4MLF4M:>ACP8$4MIDID=B:<4=>?<8 f ›w6MH32K4 ¥\C ‡B4M8 34! = 295 232 799 cd9 604 140 847 618 609 643 5ab 000 000

f

Œ 3 G 4QID4M6M8-=B:BCP:<¥“6J3=M‡B4 i 6M:…J<4Q6;36MHBAD4;V‚ =BID;ƒku=BIX‚ =BID;VAE=ƒH¢ f (a, b, c, d) = (2, 0, 0, 3)

¯ f0Œ G34$AEIDCP6:<¥\2K4 ¢‚ G347ID4 ¢GB6M8 i CKI i ?CP:>A ?32K6MI0k5ID=BL AE= ABC L“4M4A‰8 6¥AB 6MCP: 67A A BC S P X f ¢B6:<; L“4M4A‰8 6¥6MCP:N67A AC  G3=‚ BX S C@ky6Q :<;N=B:B2ˆT"CPk CP8‹6;>CK6ML“4\AE47I=>k f A+GB67A

(¯ ±±    5 ´   oµ  Jµ     (¯ J±   1w²·  ®
   1 µ ‚µ ¶Jµ
± BX = CX


 å Mã 

 åæê 

PQ

S

ã ä+ã Mè  åPå  ãè <  è ã-å ê yã aãê  ãä‰å èé "ã 7ç Mã Mè  åPå é$é å BêKç è f0Œ 3  ?
∠XCB

‚ G34M: i 4B¢

A

Q

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

= ∠XBC = ∠QBC = ∠QAC `

f



ãã\è



X

 Aku=>2P2K=‚a8A+GB67A

AQ k¢B‚ BC G>C i G““ L 476:38$A+GB6A ◦ ∠P AQ = PQ CK8‹6 ;BCP6LF4AD90 4MIz=>k f S Œ G34a6MID¥\?AzCP8‹ID4-‡B4MI‰8+C5J>254 f

D

B

C

P

á f ¡M4\A ¢ ¢ J34H<=B8*CˆADCˆ‡B4aID4M672y:B?
° ° ¸¬ (¯     
 3 ¶  Ž›•«¬ ¸   t«o µ ³/¬     ‡ ® J± $¯ ª  1² ªE ® ¬
 X_  8  ´ µ °     3 ³   ¯    ª    ´ ® µ
   ( ¯ ¨­   ²Jµ 8µ   ¬    "µ ¸  ® $¯  x yz + xy z + xyz


 å Mã

ê‰ê€ã ä+â+ã-å Mãä 3ê€ãä‰å 

≤

3

€ç â yã-å 3ê 7ç@åPå€ã >ãè ä MèyâEã çŠã\äEä*ã äEèBé \ê€ã_çKè 7çPé èBé ä MèyâEã  é ã 7Pç é   äEä+ãä 0è>ç Mã\ä‰é+çæê 3ê åPçŠêã âDè>çŠâ <ã Bê å Mè è 7çKè  ã *ä ã Mè  åPå <ãè  rã  7å ä-ê  Mè   Mä  è RçPå ä‰ç  Mä‰çŠã\ä 0è>ç Mã\ä‰é+çæê "ã <ä‰éê 7ç Mãê yãXé å 3êKç è
Bê çPå@åŠã ID=BL A+G34 • 6M? i GT i G‚a6MI‰Þ  :<4M’B?<672PCˆAŠT6:<;"A+G34aGT3H<=A+G34M8+CK8\¢‚ 4aB G 6‡B4



x2 yz + xy 2 z + xyz 2 = xyz(x + y + z)

√ f 1 1 ≤ xyz(12 + 12 + 12 ) 2 (x2 + y 2 + z 2 ) 2 = 3 xyz p £ :QA+G34F=A+G347IXGB6:<; ¢ 1 ¢y?<8+CK:<¥ A+G34 g › ›

= 13 (x2 + y 2 + z 2 ) ≥ 3 x2 y 2 z 2 3  :<4’>?<6M2@C@A€T G347: i 4>¢ X f Œ 3 G  4 < ; 7 4 * 8 P C D I  4 F ; K C < : M 4 B ’ < ? 7 6 P 2 ˆ C Š A F T u k =>2P2K=‚a8CPLFLF4;>CK6AE42ˆT 1 √ ` xyz ≤ 3 3

4

JMT i =BLFJBCP:BCK:<¥XA+G34RAæ‚ ="ID4M8+?32@A‰8 f

´ ­  $¯

o« ³R¬

µ

aã <ê O㠝ç Mã ê yãOé  å BêKç  è 
  MäEä*ã\ä   ID=BL A+G34„CK;<47:>ADC@A€T 6:<; 2 = a2 + b2 + c2 + 2(ab +2@25bc + ca) A+G34O‚ 472@25† ¤ :<=‚a:"CK:<4M’B?<672PCˆ(a AŠT +2b + c) > ¢ ˆ C  A u k > =  = a ‚  8 + A B G 7 6 A 2 2 a + b + c ≥ ab + bc + ca

3(ab + bc + ca) ≤ (a + b + c)2

f

ß à

˜ M
 4A‰ADCP:<¥

¢

a = xy b = yz

¢B6:<;

c = zx

CP:Fß à—TBCK472K;38

3(x2 yz + xy 2 z + xyz 2 ) ≤ (xy + yz + zx)2 ≤ (x2 + y 2 + z 2 )2 = 1

k5ID=BL ‚ > G C i G ‚ 4a¥74\A 2 x yz + xy 2 z + xyz 2 ≤ ‘ =ADC i 4O+A GB67A4M’B?<672PCˆAŠTNG3=>25;B8‚ G34M:

1f 3

x=y=z=

1 √ 3

¢

f

ë f ¡M4A 6M:<; J34 CP:>AE4M¥74MI‰8O¥ID4767AD4MIA+GB6: f •—=B:38+C5;34MI6: ID4 i AD6M:<¥\?32K6MI m=>CK:>A‰8CPn ¥IDCK;Q=>k0H< :NA+G34HB2K6M:<4 f  =BLF4 =>kyA+G34718-4 Hr=>CP:>AD86MID4 i =>m 25=>?<×ID4Mn;“ID4M;FCK:F8*? i G 6X‚a6TaA+GB67AŽ:<= A+GBID4M4„ID4M;"Hr=>CP:>AD86MID4XkA+G34‡B4MIæADC i 4M8=>k 6RIDCK¥7GA+†*6M:<¥\254M;AEI‰CK6M:<¥\254XAæ‚ = =>k—‚ G3=B894"8+C5;34M8X6ID4"H36I‰672P2K472 AE=NA+G34"8+C5;34M8O=>k—A+G34¥IDCK; fÙ 4\AE47IELCK:<4 A+G34¥ID4767AD4M8EA Hr=B8*8*CKJB2K4X‡>6M2@?r4 =>k f

ª ¬ k

°

³R¸    "µ


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ê‰êŠã

ä è â+ã ¢3A+GB6AA+G34¥ID4M6AE478+A‹Hr=B8*8*CKJB2K4

p = m+n f ‡>6M2@?r4 > = k CK8 k m+n−2 f trôMöï  f Œ G3p4= ¥IDCK4;ƒCK8 ¢Ž6:<;VA+G34QLF6_òCPLN?kID4;”Hr=>CP:>AD8 CP8 i 2K4M6MID2 T fŒ G>2?<× f 8\¢A+2G34aID4M8+?32@AzG3=>25;B8‹ku=BI 2=2+2−2 p=2 f trôMöï f g 898+p?<≥ f L“4 4 A+GB67A‹A+G34ID478*?32ˆAXG3=>25;B8Oku=BI$6M2@2 ¢ 8+? i G“A+GB6A m p = m+n f n>1 ¡M4A?<8 i =B:38+C5;34MI6 ¥IDCK;‚„C@A+G QC@A+GQ:<=“2K=B898R=>kŽ¥747:<4MI‰67p2PCˆ+ AŠT 1¢<‚ =4m LN6+Tn6M898+?2P?2K=>?kB‚ G3=B8-4O m8+C5≥ ‚„ ¥\254≥ Aæ‚ = ;34M836ID4OH<6MI‰6M2@2542>AE=aA+G34O8*CK;<478 =>k0A+G34N¥I‰C5;ß56J<JBID4-‡BCP67AD4;:<=QID4M; AD67CK:38O6A m ≥ n 254768EA—Aæ‚ =aID4M; Hr=>CP:>AD8\¢7A+G347:6A0254768EA—Aæ‚ = =>k>A+G347L' ‚ =>?32K;GB6\‡B4X6OID4M; Hr=>CP:>A=B:aA+G34 896ML“4 i =>2@?C i G ‚ =>?325;N¥\Cˆ‡B4„6 ID4M; AEI‰6;BC i ADC5=B: f Œ G>?<8\¢MA+G347ID4O4*òMCK8EAD8Ž62PCP:<4‹‚„CˆA+Ga:<=aL“=BID4‹A+GB6:=B:<4XID4M; Hr=>CP:>A fÙ 472K4ADCP:<¥A+G>CK8 2PCP:<4B¢y‚ 4“=3JAD67CK:V6 i =>25=>?? × =B:NGTBHr=A+G3478*CP8$47:38*?CP:>AD8„CP:SA+G>CP8 ¥IDCK; f“Œ G>?<8¢—A+G34MID4F6ID4F6ARLF=B8+A (m − 1) + n − 2 ID4;NH<=>CK:>A‰8CK:"A+G34„CP:BC@ADCP6M2r¥IDCK; f (m −•—1) =B:+ ‡B47nI‰8-− 42ˆT 2¢3+ JMT 1i =>=25=>m ?CP:>AD8zCK:A+G34RrI‰8EAID=‚'6M:<;"A+G34OrI‰8EA i =>2@?CP:>Az‚ G>C i GFCP8CP:J<=A+GNA+G34 rI‰8EA$ID=‚ 6:<;NA+G342@?25=>?CK:>A‰8$6M:<;N:<=ID4M; rg0Œyf Œ G347ID47ku=BID4$A+G34OLN6\òMCKL?−kCK:>A‰8ŽCK8 ¢‚ G>C i G 4M:<;B8 m+n−2 A+G34„CK:<;>? i ADC5=B:N6M:<;"A+G34aH3ID=3=>k f



7

7

7

7

í f CP:<;N672P2r89=>2P?BADCK=B:38CK:NH<=B8*CˆADCˆ‡B4OCK:>AD4¥747I‰8

¢

¢

a b c

AE= A+G34 4’>?<67ADCK=B:

(¯ °   ª X±± ¬   
  °    3 ³/ ¸        Oµ    ‰¬  (¯         ¨µ


 å Mã 

a!b! = a! + b! + c!

ã  Mé*é*ç@å êuä M
é  ä ä è â+ã  è ç€ãäEä+ã äEèBé \ê€ã_çKè 7çPé èBé ê‰ê€ã ä é å 3êKç è
7é9é+çPå k ¢A+G347:V‚„CˆA+G3=>?BA„2K=B898=>k‹¥747:<4MI‰672PCˆAŠT ¢ ;BC ‡BCK;<4 af 6=4M: bi 4B¢ CK8‹:<=Az;BC ‡BCP8*CKJBa2K4a>JMT

ä è â+ã



b!

c! = a!b! − a! − b!

 è MèyâEã

f”Œ G34M:

è "ã

ä <ã 7ç Mã ê  ã

;3=34M8 :<=A

b fŒ G34M: a! ¢>6:<; a! a>c

˜˜M™ ¢ ‚ G>C i GCPLFHB2PCK4M8XA+GB6A

a!b! a!AE=+i b!  AzCP8$4M=6M8‰T *i ¤ c! G34 + A+GB6 < AŽA+2a! G>CK8$¥\+C ‡Bb! 4M8‹:<=8-=>2@?BADC5=B:38 f

(a! − 1)(b! − 2) < 2

f

Œ G347ID47ku=BID4>¢M‚ 4XLN?<8EAGB6\‡B4 f£ ??<67ADCK=B:J34 i =BL“478 a!CK28X= f a= G34M: i 4 6M:<;“A+G>?<8 IDCˆAEb4 ‚ G347ID4 6:Q2a! CP:>AE4M+¥74Mc!I f ` a! | c! a < c c = a + k k > 0 Ù Cˆ‡>CK8+C5=B:JMT fŒ G347: :<=‚hTBCK472K;38 ¢ a! > 2 f + i 4>¢‚ + ‚ G>C i G"CKLNH32@C547a! 8zA+GB6A 6Ma! :<; = 2 + (a 47: 1)(a 4„2) L?<·8+·A· GB(a 6\‡B+ 4 k) 3 | a!

a>2



(a + 1)(a + 2) · · · (a + k) ≡ 1 (mod 3)

f

Œ G347: ßu=A+G347I悄CP8-4 CP8‹;>Cˆ‡>CK8+C5J>254RJMT à f k ¢ 3 f k k = 1 ¢ i G+ ¢3‚ G>C(a TBCK1)(a 472K;38 + 2) · · ·A+(a G347:+ k) ‚ " 4 ¥74kA < 3 ` A+GB67A (a, b, c) k =¢r=B 2I ‚ 4¥74A a! = a2+ 3 ¢>‚ G>C i GNCPaLF= HB2PCK34M8 G347: =i 4>(3, ¢ 3, 4) ¢ ` a! = a + 3a + 4 a | 4 a = 1 a = 2 f ACK8$4768‰TAD= i G34 i*¤ A+GB67AŽA+G3478-4 i 6M8-478k56MC@2 f a = 4Œ G>?<8\¢>A+G34a=B:B2ˆT89=>2P?BADCK=B:CP8 f (a, b, c) = (3, 3, 4)

7

‘ =‚ ‚  4 2K=3= ¤ 6AO89=>2P?BADCK=B:38Ok5ID=BL¾=>??<:<;”˜F=>kA+G34SŸrI‰C@ADCP8-G ›w67A+G347LF6ADC i 6M2 £ 2ˆTBLFHBCK6;“˜M™™˜ ˜™™  ¥\Cˆ‡B47:F6AD˜M™™šFìB˜
˜ š  f



Ø f  =BIO4M6 i GSCP:>AE4M¥74MI ¢0254\A ;34M:<=AD4A+G34N2K6MID¥74M8EAOHk f n Ù 4AD4MIDLNCP:<4a6M2@23AEIDCPH32K4M8 n ¢ > ¢ 1 =>k—;>CKp(n) 8EADCK: i AHr=B8+C@ADC ‡B4RCK:>AD4¥747I‰8$8*67ADCP8*k@T>CK:<¥ x y z

ßPCˆà

¢

¢

x y z

ßKC@Cˆà

6ID4RCK:N6MIDCˆA+GBL“4\ADC i H
(¯

f

 °

ª ¬ 

p(xyz) ≤ 3

3³/¸   Oµ

çŠã\äEä*ã äEèBé \ê€ã_çKè 7çPé  èBé ê‰ê€ã  ä MèyâEã ¡M4A ¢ ¢y6:<; J<4N;BCP8+ADCP: i AXH<=B8*CˆADCˆ‡B4CK:>AD4¥747I‰8R896ADCK8+k@TBCP:<¥ßPCˆà6M:<;VßPCPC à f S4 ;<4M;B? i 4Rk5ID=BxL°yßKC@CˆàyA+GB6A z ¢>6M:<; ‚ G347ID4 ¢ ¢ a b¢ c e × 3d × 3f ¢ ¢ ¢B6:<; 6ID4„:<=B:<x†*:<= 4M¥267ADC × ‡B4„3CP:>AE4My¥7= 4MI‰8 2f Q CˆA+G3=>?BA2K=B89z8‹= =>k—2¥747:< 4MI‰672PCˆAŠT ¢‚ 4„aLN6bT c68*8*d?6M2K4M:>AŽAE=


 å Mã  

x
f

x + z = 2y f‘ =AD4RA+GB6A  x y z  8*67ADCP8*r4M8‹ß à6M:<; ¡M4A δ = gcd(x, y, z) , ,

ß à

δ δ δ

p



x y z × × δ δ δ



≤ p(xyz) ≤ 3

f

Œ > G ?<8\¢  x y z  CP8"6S8-=>2@?BADC5=B: f ›…=BID4=7‡B4MI9¢ CP8"6S8-=>2@?BADC5=B:”ku=BI , , (mx, my, mz) δ δ δ ¢‚ G34MID4 6M:<; 6MID4 :<=B:<†+:<4¥6ADCˆ‡B4aCP:>AE4M¥74MI‰8 fŒ G347ID47ku=BID4>¢>‚ 4 4M6 i G = 2 p × 3q q fŒ G3p47: f LF6\TN6Mm 898+?< L“4OA+GB67A δ=1i  ID=BL¿ß à+¢‚ 4a ;<4M;B? 4„A+GB6A ace6M:<=; 0 = GB6\bdf ‡B4RA+G34„896ML“4„H<6MIDCˆAŠT f trôMöï



x

z

M6 :<; 6MID4 =3;3; f Œ G3x 47:Qß àzID4;>? i 478‹AD= b ß58+CK: i 4 à f k f c+1 d ¢‚„CˆA+G f ¢7A+G347:„A+G34$254kPA08*CK;<4=>k3A+G>+ CK8Ž34’>?<= 67AD2 CK=B:aCP8× ;>Cˆ3 ‡>CK8+C5J>254JMT f > ¢\‚ b> G C i GRku=BI ix4M8 < z

b>0

f

3

d>0

˜˜ Œ 3 fƒŒ G>?<8 G 47: ¢0‚ G>C i G i =B:>AEI‰6;BC i AD8"=>?ADC5=B:VA+GB6A ¢ δ LF=6\TV 1J<4NID4_‚aIDCˆbA‰AE47=:V6M08 f AXku=>2@25=‚a8A+GB67AOß à‹ k5ID=BL ‚ δ > G≥ C i GF3‚ 4F;34;>? i 4"A+GB67A f ŸAE4M¥74MI8-=>2@?BADC5=B: c+1 f 2 − 3 = 1 fŒ G>CK82K4M6;38zAE= f =>k m CP8 n



2

−3 =1

f

(m, n) = (2, 1)

(x, y, z) = (1, 2, 3)

M6 ID4 4-‡B47: f Œ G3x 47: Lz?<8+AJ34=3;3;Q6M:<; ßu=A+G347I悄CP8-4 à f  ID=BL¶ß à+¢3‚ 4 fOŒ G>?<8\¢ydß ≥ àzID1 4;>? i 478AE= a 1 =e y > x d f ‘ =AD4 A+GB6A ;<4M;B? i 4A+GB6A y f  CK: i 4RA+G34aI‰C5¥7GAz8+C5;34 2=>kyß+à2CK8$;>=Cˆ‡>2CK8+C5× ¢389b== A+GB6fA = 0  J>253 4aJMT JB?BAz:<=A z > y ≥ 3 e ≥ 2 2=‚a:Fß58944 fŒ G347:“ß àCP8 d f ¤ JMT ¢A+G>CP8ku=BI i 478 < Ÿ B ? Ž A @ C Ž A K C  8 ‚  4 P 2 2 < : 3=>k − 2e−1 = 1   4àŽA+GB67AA+G34=B:B2ˆaT“= CP:>AE14M¥74MI$89=>2P?BADCK=B:38R 6MID4 =BI 3m − 2n ==B1I (m, n) = (1, 1) f fŒ G>CK82K4M6;38zAE= (m, n) = (2, 3, 4) 16) gi+= i =B(2, ¤ ¢z) ID;BCP:<3) ¥zAE=XA+G34‹CK:BCˆADCK672ID47(x, LF6MI y, A+G34 ;<478*CPID4;OAEI‰CKHB2547(x, 86My, ID4‹z) A+G3=B= 894z(2, ‚ G>9, C i Ga GB6\‡B4 =B:<4 =>krA+G34Rku=>2P2K=‚„CK:<¥Xku=BIELN8$ßu=BIA+G347CPIzH<4MIDLN?BA‰67ADCK=B:38Eà+ì trôMöï

6M:<;

(2p × 3q , 2p+1 × 3q , 2p × 3q+1 )

¢

(2p+1 × 3q , 2p × 3q+2 , 2p+2 × 3q )

‚ G34MID4

 ï_óæï *ïî

p

6:<;

_ï7ö

(2p+1 × 3q , 2p × 3q+2 , 2p+4 × 31 )

q

¢ ¢

6MID4a:<=B:<†+:<4¥6ADCˆ‡B4„CP:>AE4M¥74MI‰8 f











   f  CK4MIDH3CP:38 ¤ CŠ¢ \˜ Q ™ H3ID=3JB2D4MÝ LF4M8 ;<4NA+G04Mž =BIDCK4w4ž 2D47ž L“47:>AD67CKID4“;<478 :<=BLFJ3ID478 M¢
4\AzHrJ f
f H<J f



¯ f ¡M4\A f J<4z6zAEIDCP6:<¥\2K4B¢\6:<;X2K4A J<46Hr=>CP:>Ay=B: 8+? i GA+GB6A AB 4AD Œ 3 G 4"GB6M2@ABC ku†+2@CK:<4 CP8O;BI‰6‚a:=B:NA+G34"8*6D LF48+C5;34"=>k 68 ¢<8+A‰6IæADCK:<¥a k5ID=BL = AB M6 :<; AB C f kŽA+G34 i CKI i ?k ‚„CˆA+G ¢—‚ G34MID4 I i 2K4Q> = k θ DA θ = ∠ACB f L“4M4A‰8zA+G34 GB6M2@ku†+2@CK:<4 6A ¢>8-G3=‚A+GB6A

° 1² (•¯ ® oµ  $¯ ‰`¬ ¬  ·        1 µ

ABC

`

P

oµ   X± 

P B = 2P D

Bê çPå@åŠã Bã\è ä MèyâEã è ã  ä+ã 
  Mè  åPå  <ã\è 3ê 7ç@åPå€ã é$é å BêKç è ¡M4A J34ƒA+G34w8-4 i =B:<;nH<=>CK:>AF=>kOCK:>AD4MI‰894 i ADCK=Bh : =>kXA+G34”2@CK:<4 ‚ CˆA+GœA+G34 „ DP i CKI i < ? L i KC I i Q 2K4 ß58944Rr¥\?

 å Mã   ç€â yã-å
 "ãç Mã 

6:<;

∠ADQ = ∠BDP = π − ∠ADP = π − ∠ACB = ∠AQB

f¿Œ G34MID4ku=BID4B¢

∠QAD 6M:<; = ∠BAQ 6ID4 +8 CKLCP2P6I f 4ADQ 4AQB Œ G347: AD ¢ AQ f  CP: i 4 = AB = 4AD AQ

AB

f Ÿ2@25=‚a8SA+GB67A AB 6M= 6:<; ID4O6M2AQ 2P8-=a8*CPLNC@2K6MI-¢ 4P 4AQB 8*CP: i 4 DB ßPk5ID=BL 6J3=M‡B4_à ∠BDP = ∠AQB f 6:<; ∠DP B8*CPLN =C@2K6M∠QP  ID=BL A+G>CP8 IDCˆAŠT B 6M:<; = A+G3∠QAB 4°ID42K6† ADC5=B: ¢‚ 4 i =B: i 2@?r;<4QA+GB6A AB = f 2AQ P B = 2P D

A

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

Q

θ

D

P

θ

B

C

`

˜˜˜ á f ¡M4A

f :

ßK6à  G3=‚

 

d>0

J<4„6 H<4MIDLN?BA‰67ADCK=B:“=>k
→



=>k 6M2@2rHr=B8+C@ADC ‡B4RCK:>AD4¥747I‰8 f

A+GB67A‹A+G347ID4"CP8O6:6I‰C@A+GBLF4ADC i H3ID=3¥ID4M8*8*CK=B: ¢ ¢ ¢<‚ G347ID4 a f a + d a + 2d ¢38+? i GA+GB67A f (a) < f (a + d) < f (a + 2d)

ßuJà›”?<8+A A+G347ID4XJ34$6:„6MIDCˆA+GBL“4\ADC i H0 f (a) < f (a + d) < · · · < f (a + 2003d)   g Hr47IEL?BAD6ADC5=B:S=>k CK8R6N=B:<4†DAE=3†*=B:<4"kK?<: i ADC5=B:F‚ G3=B8-4"CPLF6¥74"CP8XA+G34 ‚ G3=>254=>k A+GB67AzCK8\¢<6akK?<: i ADC5=B:Fk5ID=BL AE= 8*? i G"A+GB67Azku=BI6M2@2 A+G347ID4aCK8$6 ?<:BCK’B?r4 a` m∈ f 8*? i GA+GB67A



  (¯  ±±

   ª =¬  ° ³R¸     ¸   ª     O¯  ¨¸ •® oµ   (¯  ‰· o­ X³ µ

n∈

f (n) = m

 å

ä è â+ã  è ç     "ã 7ç Mã   é $ä‰çŠêŠã  — f Œ f ŸBTRA+G34$H3CK¥74=B:2K4 ß56\à¡M4A M 6 < : ; 3 G 47: a = f −1 (1) m = f (a + 1) m≥2 H?r4M: i 4 m−1 i 6M:3:<=AJ<4LF=B:<=AE=B:BC i 672P2 Ta;34 i ID4768+CK:<¥ f i f 74 : i 4>¢A+G34MID4$4*òMCK8EAD8 ‚„Cˆ{f A+G (a + 2 )}i=0 8*? i GXA+GB67A k k+1 k 0 ≤ k ≤ m−2 ) f f (a+2 ) < f (a+2 M¡ 4A k fŒ G347:
 å Mã   €ç ãäEä+ã  Eä èBé \ê€ã_çKè   è>çæê   @å åŠã Bã RçKè—ê€ãä

d=2

çKé  è3é Mãè




ê‰êŠã

f (a) < f (a + d) < f (a + 2d)

uß Jà ‘ = f  :œk56 i A-¢A+G34MID4”;3=34M8N:<=A:<4 i 47898*6I‰CP2 Tœ4*òMCK8EA"8*? i GÉ6:œ6I‰C@A+GBLF4ADC i f HkŽ2547:<¥_A+G f”Ù 4r:<4 C@k i i+1 4 A+G34CP:>AE47I€‡>f6M(n) 4(3iku)=B− nT„− Œ G347: CP8;<4 i ID4M6M8*CP:<¥X=B:„ 2 i = i+1 I06: CK:>1 AD4¥7473I ≤ nf ¢ j ¢ ¢ Ž ¢ M 6 < : ; 3 ≤ a < 3j+1 3k ≤ a + d < 3k+1 f3Œ l G3≤ < 3l+1 4MID4aku=B+ ID4B2d ¢ m m+1 ¢‚„C@A+G



≤ a + 3d < 3

3

0≤j
2d = (a + 3d) − (a + d) > 3m − 3k+1 ≥ 3k+2 − 3k+1 = 2(3k+1 )

Œ G>?<8\¢

d > 3k+1

¢‚ G>C i G i =B:>AEI‰6;BC i AD8zA+G34„k56 i AŽA+GB67A

d < a + d < 3k+1

f

f

ë f ¡M4\A J34 6 kK?<: i ADCK=B:Fk5ID=BL°A+G34a8-4\A‹=>k—:<=B:<†*:<4M¥67ADC ‡B4aCK:>AD4¥747I‰8$CK:>AD=C@A‰8-42Pky8*? i G A+GB67A-¢>ku=BfI6M2@2 ¢ ßPCˆà ßKC@Cˆà =‚

(±¯ ±

n≥0
2
2 ¢B6M:<; f (2n + 1) − f (2n) = 6f (n) + 1

f (2n) ≥ f (n)

f

LN6:T:B?
6MID4RA+G347ID4„CK:"A+G34RCKLN6¥74a=>k

°    J ±±  

     ¸      O ¯  ¨¸ •® oµ   (¯   2003

f



ª µ

 å

ã 7é9é+çPå êuä 7é  Mä ä è â+ã  è ç     
  å@åŠã Bã RçKè—ê€ãä Mãè "ã 7ç Mãê yãXé å 3êKç è    KC : i 4 CP8z=3;3;y¢J<=A+G 6M:<; f (2n + 1) − f (2n) f (2n + 1) + f (2n) LN?<8EAJ<4 =3;36f ;y¢3(n) JMTNßK+ C à f1 k ¢>A+G347:
 å Mã     è>çæê 

f (2n + 1) − f (2n) ≥ 3

6:<; 2f (2n) =

f (2n + 1) + f (2n) ≤

1 3

6f (n) + 1




= 2f (n) +

1 3



¢ f (2n + 1) + f (2n) − f (2n + 1) − f (2n) ≤ 2f (2n) + 31 − 3

˜˜  ‚ G>C i GOCP86 i =B:>AEI‰6;>C i ADCK=B: f—Œ G347ID47ku=BID4>¢ ¢76:<;„ßKC à? i 4780AD= f =>f2ˆ‡>(2n+1)−f CK:<¥‹A+G34X8‰T38EAE(2n) 47LS¢‚ =4R1¥74\A f f (2n) = 3f (n) f 6:<(2n ; + 1) + f (2n) = 6f (n) + ku=B1I6M2@2 f ‘(2n + 1) = 3f (n) + 1 n ≥ 0 =‚‚ 489G3=‚¢rJMTFCP:<;B? i ADCK=B:Q=B: ¢BA+GB67AC@kyA+G34J36M8-4 ID4MH3ID4M894M:>A‰67ADCK=B:Q=>k

2 n f  CK: i 4 CP8 ¢A+G34M: k k n ak 2 ¢<+ 2 AE+ f (n) ak4a3A+GB6 +A · · · + a61:<3;N+A+G3a4 0i 2K67CKL CP8 f‹g 89= A+G34· ·i ·2P6M+CPL a1CK8‹ ID?r4a0ku=BI 8+?
n = a k 2 + · · · + a 1 2 + a0

= f (2(ak 2k−1 + · · · + a1 ) + a0 ) ¢ = 3f (ak 2k−1 + · · · + a1 ) + a0 = ak 3k + · · · + a1 3 + a0 ‚ G34MID4XA+G34O2K6M8+AŽ8EAE47H"CK8JMTA+G34OCP:<;B? i ADCK=B:NGT3H<=A+G34M8+CK8 f CK:3672P2 Ty¢8*CP: i 4 7 ¢ C@A—ku=>2P2K=‚a80A+GB6A 6 CP8A+G34$2P6ID¥7478+A0:B? 2003CP: 1093 :B?k 3f + CP:A+G34„ CKLN6¥74 =·A+·G3·4M+ ID4a36+ ID4 1 = 7 f 2 = 128 2003 =>k f f f (n)



‘ 4+òMAŽ‚ 4„AD?2@?BADC5=B:38‹k5ID=BL =>?2547LF8X=>k A+G34 y67Þ+6 ¤ G ‘ 6ADC5=B:3672›w6A+G34MLN67ADC i 672 £ 2ˆTBLFHBCK6;“˜M™™˜†9˜™™  ¥\Cˆ‡B47:NCP: D˜M™™šFìB˜ š ˜   f

¯ f ß
  è 3ãê 7å@çŠã à g :<¥\25478 B 6M:<; C =>k0AEI‰CK6M:<¥\254 ABC 6ID4"6 i ?BAE4 f  C5;34 =>k—ID4 i A‰6:<¥\2K4 J<425=B:<¥8AE=N894¥LF4M:>A ¢3H<=>CK:>A‰8 6M:<; J3472K=B:<¥ KN KLM N AE=S894¥LF4M:>A‰8 6M:<; ¢0ID4M8*Hr4 i ADCˆ‡B42ˆT f ¡M4A BC J34A+G34NCP:>AE47L I‰8-4 i ADC5=BM :SHr=>CP:>AR=>k ACf A+G34;>CK6¥7=B:36M2P8AB =>k ¡M4\A J34„A+G34„CP:>AE47O I‰8-4 i ADC5=B:NHr=>CP:>A‹=>k 2PCP:<4M8 6M:<; KLM N C f 1AD4MI‰894 i AD1CK=B:“H<=>CK:>A‹=>ky2@CK:<478 6M:<; ‡B4„A+GB6A MN B1 CO LK 2PCP:<4M8 ¢ ¢B6M:<; 6ID4 i =B: i ?A f

µ °

3¯



oµ   X± 1² •®
oµ

AO BB1

CC1


 å3êKç  è    ã  ä+ã 

è 7å@å

<ã\è

A ... ......... ... .. ... .... .... .... . . . . . .... .... .... .... .. .. .. .... .. .. .. .... . . .. . . . . . .. ..... . . . . . . 2 ..... .. .. ... . . . . .. . . .. ... ... .. . . . . . .. .. ... .... ... .... .. .... . . .. . . . . . .. .. .. . . . . . . . . .. .. .. .. . . .. . . . . . . .. . .. .. . . . . . .. . . . .. . .. .. . . . . . . . . .. .. ..... . . . . .. . . . .. ..... .. . . . . . .. . ... ..... . .. . . . .. .. ..... . . . . . . .................................................................................................................................................................... . . . ..... .. ................... . . . . . .. .. . . .. ............. ........... . . . . . 1..... ..... ... ... ... ................... .............. .. .......... ........ .. .... ....... .... ................. .... . . . . . . . . . . . . . . . . . . . . . ... .. . .................. . .. .. 1 ............. .................. ... ... .. ..... .... .... ................ .. ......................... .. ... .... .... .... .................... .. ... .. .... .... .. ..... .... ......... ............. .............................. . .. . . . . ... . . . . . . . . . . . . .. ................... ....... ........ . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ........ ........... ... .. ....... ....................... . . . . . . . . . . . . .. . ........ .......... .. ...... ......... . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . ........ ........... . . ........ . . . . ........ . . . . . . . . . . . . . . . . . . . . . . ........ .. .......... .... .. .... ...................... . . . . . . . . . . . . . . . . . . .. .. ..................................................................................................................................................................................................................................................................................

B

p

u q

L *  P   B

M

C

v

O

u+v

B

r

K

s

N t

C

M¡ 4A2@CK:<4 LF44\A 6:<; 67A 6M:<; ¢ID4M8*Hr4 i ADCˆ‡B42ˆT f ¡M4\A ß5:<=A BB1 LN 8-G3=‚a:CP:"A+G34a;B CP6¥I‰6ML"àJ<4O A+G34„Hr=>CP:>OA A=>k CP:>P AE47I‰8-4 i ADB C5=B2:N=>k 2@CK:<4 6M< : ; C2 f S4 OA ?<8-4aA+G34ku=>2@25=‚„CP:<¥a:<=AD6ADC5=B:—ì ¢ ¢ CC1 ¢ ` AL = p LB = q BK = r KN = LM =s f ¢

N C = t ` LP = u P O = v

˜˜  CK: i 4 p s f = q r‘ + =‚ t

4ALM ∼ 4ABC

¢0‚ 4QGB6‡B4 p

=

s

p+q q ` = r+s+t r+t

A+GB67ARCP8¢

‚ 4w6H3H32 T'›…47:<472P6M?<8\§ Œ 3 G 4=BID47L AD=œA+G34ƒAEIDCP6:<¥\2K4†DAEI‰6:38‰‡B47I‰89672OH<67CKI‰8 6M:<; DA =N¥74A

(4LKN ; B1 C)

(4LKN ; P B)

LO t LB1 u r+s f · = = · ON s + t B1 K u + 2v r

•—=B:3894’>?r4M:>AD2 Ty¢ u + 2v u

v 1 = u 2

= 

r+s s+t ` · r t

(r + s)(s + t) −1 rt

g HCK:<¥›…4M:<42K67?<8§ Œ G34=BID47L OB2 B2 A

=

v u

·

A+GB6ACK8\¢

q p+q

=

AD=A+G34aH36MCPI



=

s(r + s + t) f 2rt

(4OAL; B2 B)

s(r + s + t) 2rt

·

r+t r+s+t

¢‚ 4a=3J>A‰6MCP:

=

s(r + t) f 2rt

ŸBTF8‰T3LNL“4\AEI€Ty¢ ;BC ‡BCK;<478 CP:NA+G34 98 6ML“4aI‰67ADCK= A+GB67AzCK8\¢BA+G34 Hr=>CP:>AD8 C82¢ OA B2 i =>CP: i C5;34 fŒ G>?< i =B: i ?
6:<;

®6µ °   3¯
 (¯  °  ƒ¬       ª ¬  ¸   Oµ   $¯ `¬ ·    1µ

á f ß CPk 2

C2

AO BB1

CC1

B2 (= C2 )

7é  ã à  CK:<;„A+G34XLN6\òMCKLN6M2>6M:<;LCK:BCPLF672‡>672P?r478z=>k>A+G34X8+?
a + b2 ≤ c ≤ 1

a+b+c

°

3³

å Mã  ç€â yã-å Bê çPå@åŠã Bã\è ä è â+ã  è çŠã\äEä*ã äEè3é\êŠã-ç5è 7çPé  èBé ê‰ê€ã ä MèyâEã "㠝ç Mã 3ê 7ç@åPå€ã é$é å BêKç è √ 6:<;A+G34aLCK:BCPLF6723‡>672P?r4RCK8 Œ G34aN f L 6\òMCKLN6M2>‡>6M2@?r4„CP8 1+ 2 − 21 √  ?2@25=‚a80A+GB≤ 6:<; ¢ a +b ≤¢>cku=B≤ I8-1=BLF4„ID4M672 r:B?<= LFJ<47aI + 67A r ≤ c a = r cos θ b = r sin θ

θ   √ π f a + b = r(cos θ + sin θ) = r 2 sin +θ 4 √ √ f
 CK: i 4 6M:<; ¢r‚ 4NGB6‡B4 π −1 ≤ sin + θ ≤ 1 r ≥ 0 −r 2 ≤ a + b ≤ r 2 4 Œ G347: √ √ ¢ a+b+c ≤ c+r 2 ≤ 1+ 2 √ f—g 2K89= ¢ 8*CP: i 4 6M:<; c≤1 r≤ c≤1 √ √ √ a+b+c ≥ c−r 2 ≥ c− 2 c =



√ 1 c− √ 2

2

−

1

≥ −

1 f 2

2 √ Œ G>?<8\¢ 1 f − ≤a+b+c ≤1+ 2 ›…=B2ID4=7‡B4MI9¢ CPk—‚ 4 ‰A 6 ¤ 4 6:<; ¢rA+G347: 2 a = b = √12 c = 1 a + b2 ≤ c ≤ 1 √ ¢ ‚ G>C@254 ‚„C@A+G 6:<; M 6 < : ; ¢r‚ 4F3 = J>A‰6MCP: a+b+c = 1+ 2 a = b = − 21 c = 21  f Œ i f  6 < : ; 3 G a 4 3 H D I 3 = > =   k K C 8 B = F L B H 5 2 \ 4 E A 4 a2 + b 2 ≤ c ≤ 1 a + b + c = − 21

˜˜ ë f ß A+GB67A

®ƒµ °   3¯

7é  ã à¡M4AyAæ‚ =„894’>?r4M: i 4M8=>kBID4M672>:B?AD4¥747I ¢

a0 = b 0 = 0

an = a2n−1 + 3

•—=BLNH<6MID4RA+G34a:B?
ª ¬ 

6M:<;

°

a2003

n

6M:<; b2003

bn = b2n−1 + 2n

f

f

³R¸    "µ

ç€ãäEä+ã äEè3é\êŠã-ç5è çKé  è3é ê‰êŠã  ä è â+ã  =BI ¢r‚ 4FGB6\‡B4 2 ¢—8*CP: i 4 n fŒ G>CP8Oku=>2P2K=‚a8„JMTQCK:<;>? i ADC5=B:V=B: an > 22 fÉg n i M6 n:<; ≥ 1 u k B = I < :  = + A 3 G M 4  I 7 4  6 ‰ 8 w T P C < : B ; ? A D K C B = É : 9 8 3 G  = a ‚ 8A+GB6A 2 = 3 n−1 Œ G>?<8¢>ku=BnI ≥ 1  ku=BaI n > af ¢ ‚ 4 GB6\‡B4


 å3êKç  è 

a1 2n > n + 3

n≥3

a2n

n≥3

> 22

n

> 2n+3

f

ß à

fŒ G34 i 2P6MCPLÂCP8OAEID?r4ku=BI S4 i 2K67CKL A+GB67A ku=BIX672P2 ¢ ≥JMTF 3 CK:<;>? i AD5C =B:y¢B‚ 468*8*?n CK:<¥ GB67= AA+G334 b3¥\C = i 2K67CKLÂaG33=>2K= faÚ 8+CK:<¥„A+G34"CP:<;B? i DA CK=B:SGTBHr=A+G3478*CP8O6:<;ß à+¢ ;38147 ku=BI$8-=BLF4 ‡B4M: 72 n ≥ 3 ‚ 4 =3JAD67CK:  
= 2 b2n + 2n+1 = 21 (2bn )2 + 2n+3
¢ < 21 a2n + a2n < a2n + 3 = an+1

2bn+1

‚ G>C i GN4M:<;B8A+G34RCK:<;>? i ADC5=B: f ŸBT…?<8+CK:<¥“A+G34 i 2P6MCPL 6M:<; i G34 i*¤ CK:<¥“A+G34SrI‰8EA ku4\‚ i 6894M8\¢‚ 4V;<4M;B? i 4SA+GB6A f :”H36IæADC i ?32P6I9¢ ku=BIR672P2 ¢—‚„CˆA+Gƒ4’>?<6M2@C@A€TVC@kz6:<;”=B:B2 TVC@k b n ≤ an n = 0 f n ≥ 0 a2003 > b2003

Œ = i =BLNH32K4AD4OA+G>CK8‹:B?2@?BADC5=B:38AE="H3ID=3JB2K4MLN8$=>k A+G34 Ú¤ I‰76 CK:BCP6:›w6A+G34MLN67ADC i 672 £ 2 T3LNH3CP6;   =BIDL¿¥\Cˆ‡B47:NCP: D˜M™™šFìB˜  ˜
 f

Ø f  CK:<;“6M2@2—ID476M2 8*? i GNA+GB6AA+G34ku=>2@25=‚„CP:<¥a8‰T38EAE47L¶=>k4M’B?<6ADC5=B:38OGB68X6?<:BCK’B?r4 k 8-=>2@?BADC5=B:—ì



x2 + y 2 kx − y

= 2k2 f = 2k

¢

°  ª  ¸ (¯  O µ ƒ ¬   $¯ `¬     · ~ ­  X³   µ ¬


 å Mã  

°

3³

ç€â yã-å Bê çPå@åŠã Bã\è ä è â+ã  è çŠã\äEä*ã äEè3é\êŠã-ç5è 7çPé  èBé ê‰ê€ã ä MèyâEã "㠝ç Mã 3ê 7ç@åPå€ã é $ä‰çŠêŠã Œ G34O8‰T38EAE47L GB68z6O?<:BC5’>?r4R8-=>2@?BADC5=B: CPk<6:<;"=B:B2ˆT CPkBA+G34O2@CK:<4 ‚„CˆA+G"4’>?<67ADCK=B: ` fRŒ G>CK8 CK8A‰6:<¥747:>A‹AE=NA+G34 i CKI i 2K4 ‚„C@A+GQ4M’B?<6ADC5=B: 2 2 kx − y = 2k γ x + y = 2k2 i =B:<;>C@ADCK=B: i 6M:aJ<4‹8+A‰67AD4;„4’>?3Cˆ‡>672547:>AD2ˆTR6M80ku=>2P2K=‚a87ìBA+G34‹I‰6;>CP?<8 √ =>k CP84’>?<6M2 2|k| γ AE=OA+G34=BIDCK¥\CK:y¢\‚ G>C i GRCP8 |k · 0 − 0 − 2k| f—Œ G>?<8\¢‚ 4=3J>A‰6MCP: AE=OA+G34X;>CK8EAD6M: i 4$k5ID=BL `

A+G34„ku=>2P2K=‚„CK:<¥ i =B:<;BCˆADC5=B:F=B:

k

√

ì

2k2 =

 =>2ˆ‡>CK:<¥Oku=BI

k

¢‚ 4Rr:<;A+GB67A

¢

¢

4k2 1+

k ∈ {0 1 −1}

f

k2

f

1 + k2

˜˜Mš ¯ f 1CP?<8 S r =>k C@A‰8CK:38 i IDCKJ<4M; i CPI i 254>¢A+G34M: S

°

√ f ≥ 3 3

°  ª  ¸ (¯ ê‰ê€ã    ä è â+ç€ãâ y ã- MåŽè -¬ 3_rê 7ãç@ åP倝ã å  G´µ °  ä-ê ä  MèyâE ã¯M ã\ä  çŠã\ª äEä*ã ® ¬
oäEèBµ é "\ê€ã ã_çKè ç$ ¯Mã ° 7çPé ä-ê èB‰é3· é³ é  å3êKç  è ›  ±  ç —ã  Nê  ã ã çæê  ä)µ
 å Mã

r2

 >ãè

4

Ù 4M:<=AD4FA+G34Q8+C5;34F2547:<¥_A+GB8"=>kA+G34FAEI‰CK6M:<¥\254QJMT ¢ ¢Ž6:<; 6:<;SA+G34Q8-47LNCK† a AE= b A+G34“Hrc=B8+C@ADC ‡B4F:B?
s−a s−b

Ú 8*CP:<¥ =3J>A‰6MCP:

s−c

(s − a)(s − b)(s − c) ≤

(s − a)(s − b)(s − c) ≤

Œ G347:y¢ S 2 r4

≥ 27

¢B6:<;FG347: i 4>¢ S r2

s`

s3 f 27

ß à

4MID=B:y§ 8  =BIEL?32K6N6M2K=B:<¥ ‚„CˆA+GVß à‹6:<;QA+G34 ¤ :<=‚a:ku=BIEL?32K6 S 2 = s(s − a)(s − b)(s − c) ≤

3

s4 27

≤

S = rs

¢ ‚ 4

S4 f 27r 4

√ f ≥3 3

ë f ¡M4\A J<4F6“ID4M672Ž:B?‡B4 i B = :3894 i ?BADC ‡B4AE47IELN8 > = kA+G34FCP:Br:BCˆAE4 8-4M’B?r47: i α 4 ¢ ¢ ¢ ¢ ¢ 6DI 4"I‰6ADC5=B:3672 f < 1 ID=M‡B4 A+GB67A 7å@å A+G34RAE47IELN8$sin =>krA+αG34asin 894’>2α ?r4M: i sin 4a63α ID4„I‰67.AD.CK=B. :36Msin 2 f nα . . .


 å3êKç  è 

 

ô f ¡M4A

vï

ßPCˆà

³R¸    "µ

ç€ãäEä+ã äEè3é\êŠã-ç5è çKé  è3é ê‰êŠã  ä è â+ã J34a6M:CK:>AD4¥747Iz8*67ADCP8*k@T>CK:<¥R6M:TF=>k2P2K=‚„CK:<¥Xku=>?C@ADCK=B:387ì

p


sin(pα) = sin (p + 1)α = 0 `
ßKC@Cˆà sin(pα) = sin (p + 2)α = 0 `
Kß C@CPC à cos(pα) = cos (p + 1)α = 0 `
f ßKC ‡à cos(pα) = cos (p + 2)α = 0

ΠG347:

ª

°

ª ¬ 



¢

f

¢

sin α ∈ {−1 0 1} ä  
‰f ßPCˆà  ?
6:<;

0 = sin (p + 1)α

Œ G>?<8\¢ 6:<;

f

sin α = 0 ßKC@Cˆà  ?
0 = sin (p + 2)α







= sin(pα) cos α + cos(pα) sin α = ± sin α

f


fRŒ G34M: sin(pα) = sin (p + 2)α = 0 cos(pα) = ±1

= sin(pα) cos(2α) + cos(pα) sin(2α) = ± sin(2α)

f

˜˜  Œ G>?<8\¢

sin(2α) = 0

¢‚ G>C i G"CPLFHB2PCK4M8zA+GB6A

¢

¢

sin α ∈ {−1 0 1}

f

ßKC@CPC à6M:<;FßPCˆ‡à6ID4 i =B:38+C5;34MID4M;CP:N6 8+CKLCP2P6I‚a6\T f

 ?AE4M¥74Msin(pα) I n>0



fRŒ G34MID44*òMCK8EAD8 8+? i GFA+GB6A 6:<; sin(pα) 6= 0
f p ∈ {n + 1, n + 2, n + 3} cos (pŒ G3 −47: 1)α 6= 0 trôMöï

cos α =



sin (p − 1)α + sin (p + 1)α 2 sin(pα)


i 6:VJ34F4+òH3ID4M8*8-4M;68R6NHr=>2 T3:<=BLCK6720CP: KC 8RI‰6ADC5=B:3672 f  CK: i 4 cos (p − 1)α α
CP8"I‰67ADCK=B:36M2‹ßK6cos ‚„C@A+GwI‰6ADC5=B:3672 i =34
i CK4M:>A‰8¢CˆAaku=>2@25=‚a8aA+GB67A :<;wCP8 cos (p − 1)α :<=B:<†EÞ*47ID= ¢3JMTFGT3H<=A+G34M8+CK8Dà fŒ G34MID4ku=BID4B¢
sin(pα) − sin (p − 1)α cos α
sin α = cos (p − 1)α

CK8‹I‰6ADC5=B:3672 f i 6:J34"4*òH2 T<† ¡M4A J<46Hr=B8+C@ADC ‡B4 CP:>AE4M¥74MI f  CK: i 4 :<=BLNCP6M2AD8\sin(kα) ¢>C@AŽku=>2P2K=‚a8zA+GB6A CK8‹I‰6ADC5=B:3672 f sin α

sin(kα)


f  =BI 476 i G f ¢ ¢ ¢9‚ 4GB6‡B4 p ∈ {n+1 n+2 n+3} sin(pα) cos (p−1)α Œ G347:y¢0JMTQA+G34HBC5¥74M=B:254H3IDCP: i CKHB254>¢04C@A+G347I ku=BI‹Aæ‚ =N‡>6M= 2@?r4M08
sin(pα) =>k ¢ ¢ =BI ku=BIAæ‚ =="8*0? i Ga‡>672P?r478$=>k f p ∈ {n + 1 n + 2 n + 3} cos (p − 1)α = 0  :F47CˆA+G34MI i 6894B¢M‚ 4RLF6\T"?<8-4OA+G34O2547LFLN6RAE=;34;>? i 4OA+GB67A f p : ¢ ¢ sin α ∈ {−1 0 1} H<6MI‰ADC i ?32K6MI-¢ CK8‹I‰6ADC5=B:3672Š¢>6:<;"A+G34 i =B: i 2@?<8*CK=B:ku=>2@25=‚a868CP:F•y6M8-4 f

trôMöï

sin α

í f‹Ù =34M8zA+G347ID4 4*òMCK8EA6 :B?
q∈



6:<;N6aH
3p + 7 p = 2 · 5 q 

°  ¸ (¯ ê‰ê€ã  ä Mèyç€âEâ ãOyµ ã-åƒ"¬ ã Bê ç$ ¯MçPå@ã`åŠã ¬  3 ê 7ç@BåPå€ã\ãè · é~ ¯Mã\ä ä‰é+è ç â+ ã è1 µ è


 å Mã  

ª ¬  çŠã\äEä*ã

p∈



äEè3é\êŠã-ç5è

8*? i GA+GB6A

 ° 7 çPé  èBé3³

Œ G347ID4„CK8‹:<=8+? i GH<67CKI f g 898+?krA+G34Rku=BIEL G 7 4 : > G < ? q 2·5

‚ G34MID4

p

2 · 5q = 3p + 7p = (3 + 7) · A = 2 · 5 · A

¢

f

A = 3p−1 − 3p−2 · 7 + 3p−3 · 72 − · · · − 3 · 7p−2 + 7p−1  A„ku=>2@25=‚a8RA+GB6A ßK8*CP: i 4 à f  ? ¢1‚ 4aGB6‡B4 A > 1 :<; 5 3 ≡ −2 (mod 5) 7 ≡ 2 (mod 5) A ≡ 2p−1 + 2p−1 + 2p−1 + · · · + 2p−1 ≡ p · 2p−1 (mod 5)

f

˜˜ g 8X6ID478*?32ˆA-¢

f  CK: i 4 ¢ 6M:<;FA+G34=B:B2 T“Hr=B8*8*CKJBC@2PCˆAŠTNCK8 p=5 q ¢A+G34„Hk CP8 i =BLNH32K4AD4 f

p · 2p−1 ≡CP8‹0:<= (mod A=>krA+G35) 4„ku=BIDL 3 + 7 = 2 · 52 · 341 5

5

² f CP:<;N672P2
8+? i GA+GB67A

→

f :

f xf (x) + f (y)

ku=BIz672P2

6M:<;

2·5

f




= x2 + y

°  ¸®
(o¯ µ     $ ¯    Ž„·G¬ ¸         ’  µ ª     ± ±


 å Mã

x∈

y∈

ª

¬    O ° ¯  ¨3¸ ³

 äEèBé \ê€ã_çKè  ç€â yã-å 3ê 7ç@åPå€ã >ãè ä MèyâEã çŠã\äEä*ã 7çPé  èBé ê‰ê€ã ä è â+ã  è ç å è>çæê    åPå€ã >ã Rç5èyêŠã\ä  Mãè  "㠝ç Mã é‹é å 3êKç è  AaCK84768‰TƒAD=V‡B47IDC@k@T”A+GB6AJ<=A+G 6:<; 8*67ADCP8*k@TƒA+G34 kK?<: i ADC5=B:3672y4’>?<67ADCK=B: f S4O‚„C@2P22@−x ?BADC5=B:38 f J3468-=>2@?BADC5=B: f  4\A‰ADCK:<¥ CP:ƒA+G34QkK?<: i ADC5=B:3672‹4M’B?<6ADC5=B:…¥\C ‡B4M8 ¡M4A
f ku=BI6M2@2 f 4M: i 4B¢>kux=BI= 6M2@2 0 ¢ ¢

f f (y) = y

y∈

x y∈

x2 + y = f xf (x) + f (y)

Œ G>?<8\¢

f (x)2 = x2

ku=BIz672P2

x∈





f = f f (x)f f (x) + f (y) = f (x)2 + y

f k



f (1) = 1

¢>A+G347:


2 = f (1 + x)2 = f 1f (1) + f f (x)
2 ¢ = 12 + f (x) = 1 + 2f (x) + x2 f CKLCP2P6I‰2ˆT ¢>C@k ku=BI6M2@2 ¢A+G34M: 6:<;"A+G>?<8 f (x) = x x∈ f (1) = −1 1 + 2x + x2

1 − 2x + x2

6:<;"A+G>?<8

f (x) = −x


2 = f (−1 + x)2 = f 1f (1) + f f (x)
2 ¢ = 12 + f (x) = 1 + 2f (x) + x2

ku=BI6M2@2

x∈

f

Œ GB6A i =BLNH32K4AD4M8RA+G>CK8 :B?kA+G34   äEè ãä f  ‚„C@2P2J<4‚ =BI ¤ CP:<¥“=B:VA+G34   äEè ãä J<4M¥\CK:3:BCP:<¥CP:–›w6T ¢‹89=w8-47:<;ÉLF4“TB=>?2P?BADCK=B:386M:<; ¥74M:<47I‰6M2@C@Þ+6ADC5=B:38‹89=3=B:   4MH>AE47L“J34MI

˜˜








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Œ G34  :>AD4MID:367ADCK=B:36M2 Œ =>?A"=>k Œ =‚a:38CK8"6VLN67A+G347LF6ADC i 6M2H2547L“† 8-=>2 ‡BCP:<¥ i =BLFH<4ADCˆADC5=B:–ku=BIQG>C5¥7Gh8 i G3=3=>2„8EAD?r;<47:>AD8k5ID=BL AD=‚a:38QA+GBID=>?r¥7G3=>?BA“A+G34 ‚ =BID2K; fƒŒ G34“rI‰8EA Œ =>?A =>k Œ =‚a:38aAD=3= ¤ H32P6 i 4“CK: ?<8*8*CP6QCP: 
M
M
™3¢ 6:<;…C@A"GB6M8F¥ID=‚a:…AD=wA+G34VH<=>CK:>A„‚ G34MID4AD=‚a:38k5ID=BL 6M2@2$ID4M¥\C5=B:38F=>k‹A+G34Q ‚ =BI‰25; H<6MI‰ADC i CKH367AD4 f—Œ G34 Œ =>?ArAD6 ¤ 4M80HB2K6 i 4‹476 i G‹TB4M6MI 6:<; i =B:38+CK8EAD80=>k7Aæ‚ =X8+A‰6¥74787ì g ?BAD?C i ?32@A JB?BA2547898$Hr=>CP:>AD8 6M:<;“6M: g O2K4-‡B472€¢rLF=BID4;BC i ?32ˆA$6M:<;“LF=BID4H<=>CK:>A‰8 fŒ G347ID46ID4 Aæ‚ =a‡B4MI‰8+C5=B:38$=>ky` 476 i G"H<6MHr47I-¢ +?<:BCK=BI6:<;  47:BC5=BI fŒ G34  4M:BCK=BIH<6MHr47IŽCK8zCK:>AD4M:<;34; ku=BI 8+AD?r;34M:>A‰8"k5ID=BL A+G342P68EA„Aæ‚ =VTB476I‰8N=>kG>CK¥7G…8 i G3=3=>2‹ßKCP:œ• 6:36;36B¢‹¥I‰6;34M8  6:<; ˜\àE¢Ž6:<;SA+G34 +?<:BCK=BI„H36H<4MIOCK8aCP:>AE47:<;<4M;ƒku=BIR8+AD?r;34M:>A‰8 k5ID=BL¾25=‚ 47I„¥I‰6;34M8 f  :<;BC ‡BCK;B?<672Ž8 i =BID478R6ID4FJB6894;V=B:QA+G34FJ34M8EAO=>kA+G34ku=>?CP:>AD8k5ID=BL A+G34OAE=BHA+GBID4M4aH3ID=3JB2K4MLN8=B:F4M6 i GH<6MHr47I-¢B6:<;N6MID4a8 i 67254M;FJB6894; =B:A+G34a8+AD?r;34M:>A-§ 8X¥I‰6;<4„254-‡B42 fg AE=‚a:y§ 8$8 i =BID4„CP8J36M8-4M;F=B:A+G34a8 i =BID4M8k5ID=BL°A+G34 J<478+A$H36H<4MI‰8Xk5ID=BL¿A+G34 AE=‚a:y¢B‚ G34MID4aA+G34":B?kH36H<4MI‰8X?<894;NAE=Q;34AD4MIDLNCP:<4 A+G348 i =BID4aCP8XJ36M8-4M;Q=B:A+G34aAD=‚a:y§ 8Hr=BHB?32K6ADC5=B: f  AD?r;<47:>AD8‹‚ G3=F4*ò i 44M;“6 i 4MIæAD67CK: LNCP:BCKL?k i CK4M: i 4M8 f Œ G>CP8J3=3= ¤"i =B:38*CP8+A‰8$=>k 894 i ADCK=B:38¢B4M6 i G i =B:>AD67CK:BCP:<¥OA+G34a47CK¥7GAH36H<4MI‰8‹k5ID=BL =B:<4 Œ =>?A-¢7ku=BIrA+G34 Œ =>?A‰8 
A+GBID=>?r¥7Ga˜  ¢ i =M‡B47IDCP:<¥A+G34TB4M6MI‰8 M

 AE=…˜™™>˜ f  6 i Gw8-4 i ADC5=B: i =B:>AD67CK:38"8-=>2@?BADC5=B:38aAD=w672P2‹=>kzA+G34SH3ID=3JB2K4MLN8k5ID=BL A+GB6A Œ =>?A fŒ G34MID4$CK8Ž6X8*LF672P2>6ML“=>?<:>A=>k3;B?kBH2547LF8\¢8+CK: i 4X6ku4_‚ H2547LF86MHk6‹H<6MHr47I f—Œ G347ID4‹6ID4 ˜ ˜R;BC 047ID4M:>AH2547LF86M2ˆAE=3¥74\A+G34MI0CK:aA+G34RJ3=3= ¤ fyg 2PCP8+A=>krš>˜O¥74M:<47I‰6M23ID4ku4MID47: i 478 CK8¥\Cˆ‡B47:a67A A+G344M:<;a=>kA+G34J<=3= ¤ G3=‚ 4-‡B4MI9¢M:<=‚ G34MID4‹CP:RA+G34J3=3= ¤ CK8—A+G>CK8LF6AE47IDCP6M2 ` i C@AD4; f Œ G34$H2547LF8CP:RA+G>CP8ŽJ<=3= ¤ 6MID4$:<=A—ku=BI A+G34$LF6A+G34MLN67ADC i 672P2 TRk56MCP:>A06AG3476IæA f Œ G3=>?r¥7G472K4MLF4M:>A‰6I€TßPCK:„A+GB67A—A+G34-T;<=a:<=AID4’>?3CKID4 ¤ :<=‚„2K4;3¥74O=>k i 672 i ?32@?<8¢M2@CK:<476I 6M2K¥74JBI‰63¢ =BI‹=A+G34MI‹6MID4M6M8X=>k06;‡>6: i 4;“LF6A+G34MLN67ADC i 8Dà+¢3A+G34-TF6M2@2—ID4M’B?3CPID46 i 4MIæAD67CK: ;<4M¥ID44=>k‹LN67A+G347LF6ADC i 6M2z89=BHrG>CP8+ADC i 6ADC5=B: f  ‡B47:ƒA+G344M6M8*CK4M8EA„H2547LF8ID4M’B?3CPID4 8-=BLF4"CP:<¥74M:B?3CˆAŠT f Œ G34L“=BID4N;>C i ?32@AH2547LF8RID4M’B?3CPID4N6NG>C5¥7GQ2K4-‡B472=>kCP:38*CK¥7GA 6:<;„CKLN6¥\CP:367ADCK=B:RAE=„89=>2ˆ‡B4 f g 2KLF=B8+A04-‡B4MI€TaH36H<4MI i =B:>A‰6MCP:38Ž6A02K4M6M8+A0=B:<4X¥74=BLF4AEI€T H2547LS¢M:<=B:<4=>k‚ G>C i GRCK: i 2@?r;<4786X;BCP6¥I‰6ML f£ A+G347I—H2547LF8 i =7‡B4MI—6MID4M6M88*? i G 68R:B??<:>ADCP:<¥Nß5JB?BAX:<=AX8EAD6M:<;36MID; i =BLFJBCP:367AD=BIDC i 8Dà+¢—6M:<; 25=3¥\C if  =BLF4"H2547LF8O6MID4;BC i ?32ˆA‹AD= i 2K6M898+CPk@T ¢ 68$A+G34-T i =BLFJBCP:<4=B:<4=BI$L“=BID4 8+A‰6:<;B6ID;ƒH2547L A€T3H<4M8 f ›w6M:TƒID4’>?3CKID4NA+G34“H3ID=3=>k=>kz6“ID4M8+?32@ARI‰6A+G34MIXA+GB6M:”6 8*CPLFHB254 i 672 i ?32P67ADCK=B: f  4-‡B4MI‰6720H2547LF8O6M898+?k8+A‰6:<;B6ID;¥6LF4M8 8*? i G68 i G347898\¢ i G34 i*¤ 4MI‰8\¢36M:<; i 6MID;38 f

7

 

 

7

˜  ™ Œ G34Vku=>2@25=‚„CP:<¥“A+GBID44ƒH3ID=3JB2K4MLN8F89G3=>?325;œ¥\Cˆ‡B4V8-=BLF4Sku442‹ku=BI A+G34V8+A€TB2K4ƒ=>k H2547LF8$4M: i =>?<:>AE47ID4;CK:"A+G>CP8J3=3= ¤ ì

3

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oM}Mð +îBô NïîMõ  æðBî @}
 mðõ+ð "î   

jvï >ï~   g IDCK¥7GA+†*6M:<¥\254M;AEI‰CK6M:<¥\254aLN6;34 =>k—H36H<4MICK8ku=>25;34;N6725=B:<¥R6 8+AEI‰67C5¥7GA2PCP:<4a8-= A+GB67ArA+G34z‡B47I‰AD4*òX6A A+G34‹IDCK¥7GA—6:<¥\2K4 i =>CK: i CK;<478—‚„CˆA+G„=B:<4$=>kA+G34$=A+G34MI<‡B4MIæADC i 4M8 =>krA+G34OAEIDCP6:<¥\2K4a6M:<;N6 ’B?<6;3I‰CP2P67AD4MI‰672rCK8$=3JAD67CK:<4M; f ß56\à ßuJà

3

SGB6ACP8NA+G34”I‰67ADCK=œCP:>AE=w‚ G>C i GÉA+G34w;BCP6¥7=B:3672K8“=>kXA+G>CP8Q’B?<6;3I‰CP2P67AD4MI‰672 ;>Cˆ‡>C5;34 476 i GN=A+G347I ߀˜aH<=>CK:>A‰8Eà  Œ G>CK8R’>?<6;BIDC@2K6AE47I‰6M2—CK8 i ?BA6725=B:<¥aC@A‰8X25=B:<¥7478+A;>CK6¥7=B:36M2 f  CK:<;FA+G346MID4M6 =>kŽA+G34Q8*LF672P2K4M8EARH3CK4 i 4=>kzH<6MHr47IOA+GB67AOCK8=3JAD67CK:<4M;ƒC@kA+G34Q6MID4M6=>kA+G34 ߀˜aH<=>CK:>A‰8Eà =BI‰C5¥\CP:36M23AEI‰CK6M:<¥\254RCK8 f

:8 8 i 74:4 4  1

oM}Mð +îBô NïîMõ 7ïî P}  mðõ*ð î
nvï >ï~ > g L“=B:<¥a6894A$=>k i =>CP:38¢r6M2@2yC5;34M:>ADC i 672yCK:Q6MHCK:38 6:<; 6MID4‹k56 ¤ 4 fyg 2N :TOAæ‚ =RID476M2 i =>CK:38GB6‡B4A+G34$896ML“4‚ 47CK¥7GA 2N GB6‡B42A+G34R8*6LF4‚ 4C5¥7GA-¢7‚ G>C i G CK8;>C—4MID47:>AŽk5ID=BL A+G34‚ 4C5¥7GA=>kr6RID4M672 i =>CP: f =‚ i 6M:“=B:<4 ;BC ‡BCK;<4RA+G34 i =>CK:38‹CP:>AE="Aæ‚ =N¥ID=>?k04M’B?<672rAE=A‰6M23‚ 4C5¥7GAJMT ?<8*CP:<¥R6JB6M2P6: i 4a6AzLF=B8+A ADCKLF4M8\¢>C@k 4

ß56\à ßuJà

N = 16 `

3 7458

N = 11 

ß  H<=>CK:>A‰8Eà

8 3 i



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4AD??k A+G344*ò7AE47:<;<4M;wH3ID=3JB2K4ML 6ML“=>?<:>A‰8aAE=ƒ89=>2ˆ‡>CK:<¥F6M: :<=B:<†*8+CK:<¥\?32P6I„8‰TB8+AD4ML =>ky2@CK:<476I4M’B?<6ADC5=B:38$LF=3;B?32K= ¢‚ G347ID4 CP8A+G34RADm =AD67× 2 :Bm ?ky2K= i*¤ 4MI‰8CK:A+G34 i =BIDIDCK;<=BI f SG>C@254zA+G34$8‰TB8+AD4ML GB26M86‹?<:BC5’>?rm489=>2P?BADCK=B:Rku=BIy6:TR8*?rJB8-4\A0=>kA+G34‹2K= i*¤ 4MI‰8 ßu=BI06:Ta8+?rJ3894A=>kA+G348+AD?r;34M:>A‰8EàE¢Mr:<;>CK:<¥‹CˆA—A+G>CP80‚a6T„CP8AD4;>C5=>?<8\¢7AD=a8*6TRA+G34$2K4M6M8+A f 47: i 4ku=BI‰A+G ¢y‚ 4"‚„CP2@268*8*?kŽ25= i*¤ 47I‰8a6M:<;SA+G34 :B?kz8+AD?r;34M:>A‰8 6ID4FCP:Br:BCˆAE4 fSŒ G34“ID4M8+?32@A‰8aCP: Œ G34M=BID4MLN8 ¢˜B¢6:<;  G3=>25; ku=BIA+G34 i 6894F=>k6M:TS<:BC@AD4N:B?kŽ2K= i*¤ 4MI‰8„6M:<;V8EAD?r;<47:>AD8 8+CKLNH32 TC5¥:<=BID4 ` 6;3G34MID4"AD=QA+G34 6:TS:B?CK:<¥ CP:S6:TVm =>kA+G34N894A‰8X‚ 4N;BCP8 i ?<8*8 f S4 m i =B:‡B47:>ADC5=B:FA+GB67AA+G348-4\A =>k:36AD?Az6:<;2K= i*¤ 4MI8-4\AD8 f 0  =BI i 4MIæAD67CK:"89H<4 i CP6M238*?rJB8-4\AD8=>kA ‚a6T"AD=N6MH2 ‡BCP:<¥R6a2K6MID¥74„2PCP:<4M6MIz8‰T38EAE47L fŒ =‚a6MID; A+G>CK84M:<; ¢>;347<:<4$A+G34 é+ç Mè Bê ä+ã =>k36X:367AD?k36M2@2>Hr=B8+C@ADC ‡B4 4*òHr=B:<47:>AD86MH?A$AE=N‚aIDCˆAE4L“47L“J34MI‰8„=>kA+G348*CK¥:367AD?kŽ6N:B?k CK8zA+G34 {1 47LFH>2} AŠT` 8-15 4\A-¢ = f 3 5 {1} 1 ∅  A CK8N4-‡>C5;34M:>AaA+GB6AaA+G34S8-’>?<6ID4k5ID44V:B?A‰6MCP:<4;"CK:"A+G34„8-4\A ‚ G3=B8-4‹8*CK¥:367AD?A‰6MCP:<4;RCP:RA+G34$894A {1} ¢ ¢ ¢ ¢76:<;OA+G34 i ?rJ34M86ID4zA+G3=B8-4 68-4\. A . . } ¢ ¢ ¢ f  :S¥747:<4MI‰672Š¢ :B?A‰6MCP:<4;“{2 CP:FA+4G34" . .:B.?<}L“J34MI‰8 ‚ G3=B894 ¥\Cˆ‡B47:É6S8+?rJ3894A =>k ¢‚ 4Q254\A ;34M:<=AD4QA+G34{3 8-4\A6=>k‹96M2@2
¢7A+G348-4\A f0Œ G34Xσ(A) 8*CK¥:367AD?A‰A 6MCP:<4;aCP: 894A=>k389’B?<6MID47k5ID4M4X:B?k89’B?<6MID4M8CP8 ¢ ¢ ¢ ¢3A+G348-4\A$=>k i ?rJ34M8CP8 ¢ ¢ ¢ ¢r6:<;“8-= {26:4 } =>k =B: f"£ k i =>?σ¢—LN T8+6?rJ3.89.4.A‰8„ 6ID4:<=A =>k6M:T>σA+G>{3 CP:<¥ f 6  9=BIX.4+.òM.6}LNH32K4B¢ A+G34 f Hr=‚ 47I‰8=>k ßu=BIz=>ky6M:TN89H<4 i C@ i HkrA+G34„ku=BIDL ku=BIz6M:T 2 σ(A) A  ?k>8+AD?r;34M:>A‰8CK8Ž=>kA+G34‹ku=BIEL ku=BI—89=BL“4$8+?rJ3894A =>k f  8A+G347ID46: 4254M¥6:>A ‚a6\TOAE= i GB6MI‰6 i AD4MI‰C@Þ*4A+G34894A0=>σ(A) k>2K= i*¤ 4MI‰80A+GB6A ‚„CP2@2>J<4‹254A kPA=BH<4M: JMTQA+G34M894N8EAD?r;<47:>AD8  :<;344M;A+G347ID4CP8  S4:<44M;V=B:<4L“=BID4N;<4r:BCˆADC5=B:CP:V=BID;<47I AE="8EAD6AE4XA+G34„ID478*?32ˆA f Cˆ‡B47:N6„894A =>k :36AD?254\A J<4R6M2@2<:367AD??<6M20AD=6: ã Mãè :B?CK:<¥ à$=>k 0 f  =BIz4*òM6MLFHB254>¢>CPk A+G34 472K4MLF4M:>A‰8$=>k  ¢ ¢ ¢ ¢>A+G347:











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A

e(A)



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σ e(A)



f

¢

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¢

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A = {3 6 9 . . . }

¢

¢

¢

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= {1 2 6 7 8 12 13 14 18 19 20 . . . } f ¢ ¢ = {n ∈ | n ≡ 0 1 2 (mod 6)}

¡M4A


ID47LF67CK:F=BA Hr47: ⊆f



f

 kO8+AD?r;34M:>A‰8

σ(A)

6ID4”;BCP89H367A i G34; ¢$A+G34M:œ2K= i*¤ 4MI‰8

‚„CP2@2—ID47LF67CK:Q=BHr47:“C@k6M:<;=B:B2 TFCPk06M:=3;<;“:B?k08+AD?r;34M:>A‰8 ä  
¡M= i*¤ 47I AE=>? i G”CˆA A+GB6Aam CP8¢ŽC@k GB6M8"6:…=3;<;w:B?k$;BC ‡BCP8-=BI‰86ML“=B:<¥NA+G34:B?A$LF6MI i G>CK:<¥¢ 6M:<; CK:FA+G348+AD` ?r;34M:>A$8-4\A f  m ?A‰8 8*?k αmi CK: 6MID4X6M2@2B:B?m k>A+G3=4Xkup=B1IDL p2 γ ·¢·‚ · G3p47kID4 pi

σ(A)

pi

γ∈A

˜  i 6:<; ¢C@k—4M6 i G e(A) ` i 89∈ f krA+G>CP8CP8zA+α CK8‹¥ID4MA6AE47IŽA+GB6M:Fm=BI∈4’>σ?<6Me(A) 23AD= LF4MLFJ<47I‰8$=>k ¢>kuα =BI =BL“ 4 G34 i 2a A a ≥ 0 i i i 6M8-4>¢3A+G34M:FA+G34:B?k;BC ‡BCP8-=BI‰8X=>k 6LF=B:<¥„A+G348+AD?r;34M:>A‰8‹‚ G3=‚„C@2P2yLN6I i G m=3;<; :B?kæ¢>8+??<6M2 AD= L“47L“J34MI‰8O=>k fŒ G347:FA+α G34" :B?<e(A) LFJ<47I=>k;>α Cˆ‡>iCK89=BI‰8O=>k 1LN6I i G i =B:>A‰6MCP:3A m¥ 6LF=B:<¥A+G34$8+AD?r;34M:>2a A‰80i‚ −G3=„ 8A+G34$k56 i AE=BI ¢MLN6 ¤ CK:< 2a − 1 + 1 = 2a i i C@A4-‡B47: fŒ G>?<8¢>2K= i*¤ 4MI ‚„C@2P2 J34 i 2K=B8-4M; f m

 4O:<=AE4A+GB67AA+G34OHk =>kBA+G>CK8zID478*?32ˆACK8z6R8EAEI‰6MCK¥7GADku=BIæ‚a6MID;¥74M:<47I‰6M2@C@Þ+6ADC5=B: S =>kA+G34N8EAD6M:<;36MID;Sku=BIDLF672H3ID=3=>kz=>kA+G34N8-=>2@?BADC5=B:“AE=“A+G34F=BIDCK¥\CK:36722K= i*¤ 4MIH2547L ß58944>¢0ku=BIR4+òM6LNH32K4B¢   à fFŒ G>CK8„ID4M8+?32@AOLF6 ¤ 4M8a4M6M8‰TQ‚ =BI ¤ =>k894-‡B4MI‰672CP:>AE47ID4M8EADCK:<¥ H2547LF8$ß56ML“=B:<¥XA+G34ML A+G34 =BIDCK¥\CK:3672ˆà f



O O O O

   

f k6M2@2 A+G348+AD?r;34M:>A‰8O6MID4"894M:>A$;<=‚a:FA+G34"GB672P2€¢3‚ G>C i GF25= i*¤ 47I‰8XID47LF67CK:

*} ~Pï =BHr47:  *} ~Pï





f  kA+G348-’>?<6ID478Ž6ID4$8-47:>A;<=‚a:RA+G34XGB672P2€¢\‚ G>C i GR25= i*¤ 47I‰8ŽID47LF67CK: =BH<4M:

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*} ~Pï

A ∪ (A + 1)

f  G3=‚¿A+GB6A

6ID4FCP:‡B4MI‰894Q=BHr47I‰67ADCK=B:38 fŒ GB67AOCK8\¢8-G3=‚°A+GB6A

6M:<;

e ku=BI6f:T8-4\A =>ky:36AD?



1à+ì



o BïM} *ï

σ f (A)

f

¡M4\A



f

 kO2K= i*¤ 4MI‰8


LN?<8EAJ<4aA ;BCP89⊆ H367A i G34; f

4M8EAD6JB2@CK89G$A+G34ku=>2@25=‚„CP:<¥ß@‚ G>C i GXLF6 ¤ 4M8

σ(A)

6MID4VAE=œID4MLN6MCP:–=BHr47:y¢X8+AD?r;34M:>A‰8

t }
*}~ˆ~ˆô  f0Œ G34O8-4\A=>kCK:<¥X8EAD?r;<47:>AD8CK8ŽCK:aA+G34OCPLF6¥74R=>k CPk<6:<;"=B:B2ˆT C@k3A+G34 σ 8-4\A=>k 2K= i*¤ 4MI‰8254kPA=BH<4M:CP8CK:"A+G34„CPLF6¥74a=>k f



ΠG34M=BID4ML

O 
O  

σ

˜„LF6 ¤ 4M82PCK¥7GA‚ =BI ¤ =>k
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˜ 

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m



mS = {ms | s ∈ S}

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ª

n ∈ lS



mS





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f„‘ =AE4aA+GB6A ¢ ¢ ‚„C@A+G LF6\TQJ34 r:BCˆAE4 ä  
¡M4\A {l6\1T"lAD2= ¤ . 4M. 4M. H“ } 4+òM6 i AD2 lT" 1 A+< i*¤ 47·I‰·8‹· CP: =BHr47:CK8‹L 6M8‹ku=>2@25=‚a8ì rI‰8EA = ICP:Br:BCˆAE4 fL£ :<=4O‚a B G34Rl225= < L25= i*¤ 47I fg kPAD4MIA+G34-TƒGB6\‡B4FLN6I i G34M;y¢=B:B2ˆT 8-47:<;V8+AD?r;34M:>A‰8 CP8 =BH<4M: fFŒ G34M: l S l‚„ 1 CP2@2yAD=>? i GQ6:T=>k 8-47:<;8EAD?r;<47:>AD8 1 f‘ =AE4 A+GB6A:<=B:<4N=>k—A+G348EAD?r;<47:>AD8XCK: l2K2= Si*¤ 4MI‰8  f  CK: i 4 A+G34„CP8‹8-4 i =B:<; li 26S;3ID4a=>ky8EAD?r;<47:>AD8$GB68 l − 1 l < l LF6MI i G34; ¢ 2=B:B2 TN2K= i*¤ 4MI‰8 6M:<; 1 ‚„C@2P2y2J34=BHr47: f  k=B:<4O‚ 4MID4RAE= i =B:>ADCK:B?r4aCP:NA+G34 l 1 fQŒ G34M: k5689G>C5=B:y¢H?32K;”J34Q=BH<4M: f”‘ =‚¿2K4A L f  ?nAO∈ =>kH3ID4 i CP8-42ˆT n ∈ lS f k l ∈ =>k—A+G348-4\AD8 ¢rku=BI CP8O4-L‡B4M:y¢A k ‚„C@2P2GB6\‡B4LFlS l ∈ L k 6MI i G34;6:S4-‡B4M::B?kADCPL“478 f„Œ G>CP8„GB6M8XA+G348*6LF4N4 04 i AX6M8 n 8+AD?r;34M:>A :<=ALN6I i G>CP:<¥R67A672P2 f k CP8‹=3;<; ¢A+G347:N8EAD?r;<47:>A ‚„CP2@2rGB6‡B4„LN6I i G34M; 6:N=3;<;:Bn?k6:<;A+G>CK8‹kGB68zA+G34„8*6LF4a4 04 i A68n8+AD?r;34M:>A LF6MI i G>CK:<¥ n ÷5?<8+A=B: i 4 f



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i M6 :aku=BIDLN?32P67AD4 Œ G34=BID47L  ’>?3C@AD4 4 GB6\‡B4 i G3=B894M:NG34MID4RAD=H3ID4M894M:>A  CP:FGB6:<; ¢B8-4-‡B47I‰6M2 =A+G34MI i 6M8-478 i G f

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*} ~Pï f 25= *i ¤ 47I‰8=BH<4M:

1=p



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˜  š g 8‹6„<:36M2 =3JB8-47I€‡>67ADCK=B:y¢M‚ 4„:<=AE4OA+GB6AŽA+G347ID4 i 6:NJ<4„:<=":<=B:<†947LFH>AŠT8*?rJB8-4\A = krA+G34„8+AD?r;34M:>A‰8A+GB6A ¤ 4M4MH38$H3ID4 i CP8-42ˆTA+G34„2K= i*¤ 4M‰I 8 > =BHr47:‚ G34M:"A+G34 GB6M2@2ˆ‚a6\T A A CK8CP:Br:BCˆAE4 f





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ª

A

f •y=B:38*CK;<47I$25= i*¤ 47I f ¢ ¢ ‚„CˆA+G ä  
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O 



*} ~Pï  fŒ G34„6J3=M‡B4„H3ID=3=>krk56MC@2K8zCP:6 i =BIEI‰C5;3=BI‚„C@A+G"6Rr:BCˆAE4R:B?kr25= i*¤ † M4 I‰8 f  G3=‚ A+GB67AA+G347ID4CP8X6894A CK:Q6M:T“<:BC@AD4 i =BIEI‰C5;3=BIz‚ G347ID4"8EAD?r;<47:>AD8 ‚„CP2@2 f  kA+G347A ID4S6MID4L“=BID4FA+GB6:”Aæ‚ =ƒ2K= i*¤ 4MI‰8 CK:ƒA+G34SGB6MA 2@2ˆ‚a6\Ty¢ 25476‡B4S=BH<4M:”25= i*¤ 47I‰8 A+G34MID4O‚„C@2P2 J34aLN6:T8*A ? i G894A‰8 f S4"2K4M6\‡B4A+G34ID476;34MI‚„C@A+G=B:<4"<:36M20H2547L f  =BI$6:T8+?rJ3894A =>kA+G34 A f :367AD?
O 





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k=0

n−1 X n−1 X k=0 j=1

= n det A +

n−1 X

αj

j=1

‘ =‚¢—8+CK: i 4 ‚ 4 GB6\‡B4

αj ω kj +

ωn = 1

6M:<;

n−1 P

ω kj =

k=0

n−1 X

n−1 X

ω kn det B

k=0

n−1 X

ω

kj

+

k=0

1 − ω nj = 0 1 − ωj

n−1 X

ω kn det B

f

k=0

ku=BI

¢

¢

¢

j = 1 2 ... n − 1

det(A + ω k B) = n(det A + det B)

¢

f

k=0

ΠG347:

n−1 P

det(B + ω k A) = n(det B + det A)

¢B6M:<;"A+G34aID478*?32ˆAku=>2P2K=‚a8 f


@
U H !,77I[T?9?J<E57"983& (XPU* ]12.]K  1 k=0



 ŽO p?  #"%$&'$()* +- #Gk/0UHzŽ…4<)?HC,u0A7Un>"0B* „;G"0B)" ‚€HZ0Bv)"? (J0ƒ=>-Ch ;=3)?"73$$C‹ XY PL]5L S@V
\^y\] SAM`]\]3‰ya [ ^V`M5\^ZM 1’ M KL?M 2 x c > [ ^/1MJPL S@^ZM`L]3eZN \R Y PL]5L SUVQ^ [ MP\]5fc/(x) M [ V8P =[Y x MJ−P\cM ftanh @ S

V  c  L J a 5 ]  L  \ % V @ S  ^ _ [−x0 , x0 ]   [ ] [  [ [ Z [ g [
S@VQMJ(x) √  P ‹ L T J V A S 5 M S  e ‹ L ] M J M  P / L L  f ? \ 5 M S ^ ˆ x0 = ln c + c2 − 1 x
S@= L\a%P x ∈ (−x , x ) MJPL
P [ ]5S  [ ^ZM5\R#R@S@^L T\ViVJSU^_MJP] [ f#_P x,cosh ^ZM`Lc]V8La5M5V f (x) MJPL
_?]\T#P [Zg f0\M:M0Y
[/[ MJPL]oT [ SU^ZMV Y SAMJP!\ N Via`SUV%Vi\V x1 (x) \^c x2 (x)  L #^Lb\ g f^a5M5S [ ^ \V g|[ RAR [Y V  g : (−x0 , x0 ) → 

 g(x) = x + c2 tanh x1 (x) + c2 tanh x2 (x)  ] [ eL [ ]hcZSUV%T] [ eLuMJP\?M g|[ ]o\RAR  g(x) > 0 x ∈ (0, x0 )   *0B=B$ "  (oH6$=3) [ V [ RAfM5S [ ^V Y L]5Lp]5LaJL?SŠeLc g|[ ]6MJPZSUVFT] [ RUL PL^aJL  SAM']5L‡\SU^V N [ T#L^ 'W PLuT] [ T [ ViL] LRAS>L8eLVoMJP\M'MJPLba [ ^ 3La5M5f]5LuS@VoM`]f#L  VJSU^a`LpMJPL]5LuS@Vh\‡TR>L LTS@]5SUa`\R6L8eZS>cL^a`L  N

 !"#%$'&)(+*-, *.%/0123!4-.65879/;:)=< (?>@*4BA+/CEDGFH59I/(J.+5K>L2M*N#&O79*5@/.%>P5PC Q *RS4F!5P4# TU%V p > 1 !WX#Y[Z U%V a b c d ∈ 1/√p, √p  \#]^-_ UMV+` W V a Wb p + 2√ ≤ a + b + c ≤ 1 + 2√√p 1+p 1+ p a+b b+c c+a 1+p 1+ pc √ b aLd 3 p  p 3 b c d 1 a + + + + ≤ + √ ≤ √ 3

1+p

1+

3

p

a+b

b+c

c+d

d+a

1+p

1+

3

p

˜ B



 å3êKç  è ä



åŠê 
Mè`¶  é\ã •®
›±

ß56\à Œ G34OAEI‰6:38*Hr=B8+C@ADCK=B:

 ç —ã   ê yã ã çæê  ä f

(a, b, c) 7→ (b, a, c)

CP:"A+G34„CP:<4’>?<6M2@C@A€T

√ 2 p a b c 1 + + ≤ + √ a+b b+c c+a 1+p 1+ p

ß à

¥\Cˆ‡B478A+G34a4’>?3Cˆ‡>672547:>AzCP:<4’>?<6M2@C@A€T

√ 2 p b a c 1 + + ≤ + √ b+a a+c c+b 1+p 1+ p

 CK: i 4

a b c + + = 3− a+b b+c c+a

‚ 4a8944OA+GB6Aß àCK8‹8*67ADCP8*r4;"C@ky6:<;N=B:B2ˆT"CPk a b c + + ≥ 3− a+b b+c c+a

Œ G>?<8\¢>AD=H3ID=M‡B4 ßK6àE¢BCˆACP8‹8*? ¢ ¡M4A

x = b/a y = c/b





f

b a c + + b+a a+c c+b

 ¢

√  √ 2 p 2 p 1 p + + = √ √ 1+p 1+ p 1+p 1+ p

f

i C547:>AŽAE=H3ID=M‡B4 ß à f fŒ G34M:Fß àŽJ<4 i =BLF4M8 ¢36M:<; z = a/c

√ 2 p 1 1 1 1 + + ≤ + √ 1+x 1+y 1+z 1+p 1+ p

f

߀˜_à

‘ =AD4A+GB6A fFŒ =SHAAD= 6M:<; ¢ ¢ H
= ≤

47: i 4>¢

2+x+y 1 − t2 = 1+ 1 + x + y + xy 1 + x + y + t2 1 − t2 1−t 2 1+ = 1+ = 1 + 2t + t2 1+t 1+t

1 1 1 2 1 2 t2 + + ≤ + = + 1+x 1+y 1+z 1+t 1+z 1+t 1 + t2

¡M4A A+GB67A

h

h(t) =

2 t2 + 1+t 1 + t2

CK8$;34 i ID4768+CK:<¥„=B:

(0, 1]

f  CK: i 4

h0 (t) =

f •y=B:38-4M’B?r47:>AD2ˆT ¢>ku=BI

√ h(t) ≤ h(1/ p) =

ß à f

f

−2(1 − t)(1 − t3 ) ¢2@25=‚a8 (1 + t)2 (1 + t2 )2

¢ √ 1/ p ≤ t ≤ 1

√ 2 p 1 + √ 1+p 1+ p

Œ G>CP8$H3ID=M‡B478‹CP:<4’>?<6M2@C@A€TF߀˜_à6:<; i =BLNH32K4AD4M8zA+G34aH3ID=3=>k—=>k—ßK6à f

f

˜ ™ ßuJà Œ G>CP8CK8AEID4M6AE4M;"8+CKLCP2P6I‰2ˆT f0Œ G34AEI‰6:38*Hr=B8+C@ADCK=B: (a, b, c, d) 7→ (b, a, d, c) CK:"A+G34„CP:<4’>?<6M2@C@A€T √ 33p a b c d 1 + + + ≤ + √ a+b b+c c+d d+a 1+p 1+ 3p

ß à

TBCK472K;38zA+G34 4M’B?3C ‡>6M2K4M:>ACP:<4’>?<6M2@C@A€T

 CK: i 4

√ 33p a d c 1 b + + + ≤ + √ b+a a+d d+c c+b 1+p 1+ 3p

a b c d + + + = 4− a+b b+c c+d d+a

‚ 4a8944OA+GB6Aß àCK8‹8*67ADCP8*r4;"C@ky6:<;N=B:B2ˆT"CPk b c d a + + + a+b b+c c+d d+a



b a d c + + + b+a a+d d+c c+b

¢

√  33p p 4− + √ p+1 1+ 3p f p 3 + √ 3 p+1 1+ p

≥

i CK4M:>AŽAD=H
x = b/a y = c/b u = c/d

 ¢



=

Œ G>?<8\¢>AD=H3ID=M‡B4 ß5JàE¢BCˆACK8‹8+? ¡M4A ¢

f

v = d/a

fŒ G34M:Fß àŽJ<4 i =BLF4M8

√ 33p 1 1 1 1 1 + + + ≤ + √ 1+x 1+y 1+u 1+v 1+p 1+ 3p

f

ß à

‘ =AD4XA+GB67A fŒ =HAAD= 6:<; ¢ ¢ ¢ f [1/p, p] :<; v ∈ H¢ 8+CK: i 4 A+GB67A ¢3‚ 4 8-4M4aA+GB6A t≤s ts = 1 ¢B6M:<;"A+G>?<8¢ t ≤ 1 ≤ s f  ¢ ‚ a 4 B G  6 B ‡ 4 2 s2 /u = s2 /p ≤ u ≤ p ‘ =v‚≤ ¢>ku=BpIŽò4; ¢ s /p ≤ u s   ¢ s2 max u + v uv = s2 ≤u≤p p   s2 s2 s2 f = max u + ≤u≤p = p+ u

p

p

Œ G>?<8\¢

1 1 + 1+u 1+v

=

≤ =

2+u+v s2 − 1 = 1− 1 + u + v + uv 1 + u + v + s2 s2 2+p+ 2 s −1 p 1− = 2 s2 s 1+p+ + s2 1+p+ + s2 p p p 1 f + s2 + p 1+p

ßuš\à

˜  Ú 8*CP:<¥OCK:<4M’B?<672PCˆADC5478$ßuš\àŽ6M:<;Fß  à+¢‚ 4a¥74A 1 1 1 2s p 1 1 + + + ≤ + 2 + 1+x 1+y 1+u 1+v 1+s s +p 1+p

¡M4A

g(s) =

kK?<: i ADC5=B:ÉGB6M8"6Q25= i  4 B S G 6\‡B4

 CP: i 4

2(s − p)(s3 − p) ¢A+G>CK8 2 2 2 (s f CK:ƒ+A+G31) 4QCK(s :>AD4M+ I€‡>67p)2

g 0 (s) =

¢‚ G>C i G”CP8 √ 3 p

s =

(1, p)

‚ G34M: i 4B¢

6:<; p 2 1 g(1) = −1 + g(p) = − <0 + < 0` 1;" +A+G3 p47ID47ku=BID4>¢ p+1 p+1
¢B6M:<

max g(s) = g

s∈[1,p]

2s p f + 2 1 + s s 6M2zLF6_òCPLN?
f

√ 3 p

1 1 1 1 + + + 1+x 1+y 1+u 1+v

√ 23p 1 p + √ + p 3 1+ 3p 1+p p2 + p √ f 33p 1 + √ 3 1+ p 1+p

≤ =

Œ G>CP8$H3ID=M‡B478ß à6M:<; i =BLNH32K4AD4M8zA+G34aH3ID=3=>k—=>k—ß5Jà f


H
U Z DBIB![OCH

uQ 5  ?m#"J$&'$(?)*
+-
t;(%0A7B)p… "8=3$; J)C:‚€HZ0Bv)"(J0ƒ=>-u$n7B$03)(?=U0UC E:$k‡;H0U;~6;H<* }X;+"03)D7'sp$(J�UH6)D(?28GC:;"0@(iC:I";H<2J) KL?M n ≥ 3  \^c,RUL?M x  x  . . .  x N L‹T [ VJSAM5S eL‹]5L\RX^f N L]VV%f#a%PMJP\?M 1  [] \^2c n = 4  nT] [ eLuMJP\?M x1 x2 · · · x n = 1 n=3 1 1 1 n  x21 + x1 x2

+

x22 + x2 x3

{ $7ƒG=U03$Hl+-=|4<) &6"%$&'$()?"D &QV%S@^_ MJPL/V%f N VJM5SŠM5fM5S [ ^V _SŠeL^S@^L ˆ f\RASAM3‰ N La [ ,LV

x1 =

+ ··· +

qa

2

a1



x2n + xn x1

qa

x2 =

3

a2



≥

2



. . . xn =

a1 a2 an n + + ··· + ≥ √ √ √ a2 + a 1 a3 a3 + a2 a4 a2 + a n a2 2

d SU^a`L





√ √ a + a3 a + a2 a 1 a3 ≤ 1 ... a n a2 ≤ n 2

a1

+

+ ··· +

‰!MJPL ” \f#a%P‰  d a%P Y \]  ’ ^L ˆ f\R@SŠMB‰  Y L P\eL

a1 + 2a2 + a3



2

a2

(a1 + · · · + an )2

a2 + 2a3 + a4

qa

1

 MJPL

an



 SAMOVJf a`LVOM [ V8P [Y MJP\M an n 

an + 2a1 + a2

≥

4

≤ [a1 (a1 + 2a2 + a3 ) + · · · + an (an + 2a1 + a2 )]   a1 an · + ··· + a1 + 2a2 + a3 an + 2a1 + a2



˜

˜

Œ G>?<8\¢BCˆAz8+?

i 4M8zAE=89G3=‚A+GB6A

4(a1 + · · · + an )2 ≥ n[a1 (a1 + 2a2 + a3 ) + · · · + an (an + 2a1 + a2 )] Œ G>CP8‹CP:<4’>?<6M2@C@A€TCP8‹6:C5;34M:>ADCˆAŠT"ku=BI f  =BI ¢>CˆACP8 n=4 n=3 a21 + a22 + a23 ≥ a1 a2 + a2 a3 + a3 a1

f

¢

‚ G>C i G"CK8zAEI‰?r4B¢3J34 i 6M?<894

2(a21 + a22 + a23 − a1 a2 − a2 a3 − a3 a1 ) f = (a1 − a2 )2 + (a2 − a3 )2 + (a3 − a1 )2 ≥ 0

Œ G>CP8 i =BLFHB254\AE478A+G34„Hk f ’B?<672PCˆAŠTNG3=>25;B8z‚ G347:

xi = 1

ku=BIz672P2 f i

I * X) & 

^E& E&
X
 \ JN%<ENlNd3K" "#+D$9"BPU1R& 33Q 1+p.$= 1R& () B!I.*' 12

^E& M ( MS (') 112 )2nP'.=QB 3FJS 2SU (K



Q 5 ?m#"J$&'$(?)*
+-
t;(%0A7B)p… "8=3$; J)C:‚€HZ0Bv)"(J0ƒ=>-u$n7B$03)(?=U0UC E:$k‡;H0U; KL?M 0 < m < 1  \^c/RUL?M a  b  c ∈ √m, 1/√m   ] [ eLuMJP\?M m  a3 + b3 + c3 + 3(1 + m)abc ab(a + b) + bc(b + c) + ca(c + a)

≥ 1+

2

{ $7ƒG=U03$Hl+- ?$)87 { 254Z7B$(?+)?"ƒzC:9<;-(J0B*)DCE!CQ‚ { j  KL?M f (x) = −2x2 + m2 + 3m
x + m3 − 3m2 − 2m + 2
  [ ]

x≥m

f 0 (x) = −4x + m2 + 3m ≤ −4m + m2 + 3m = m(m − 1) < 0

Y ZP S>a%P!SUTRAS>LVOMJP\?M d fTT [ V8LMJP\M \^c

S@VQcLaJ]5L\V%S@^_    \^c

f (x) a = m b ∈ [m, 1]

c=1

'W PL^

m=a≤b≤c=1

2[a3 + b3 + c3 + 3(1 + m)abc] = = = =

Y PZS>a%P
‰SULRUcV





− (m + 2)[a2 b + ab2 + b2 c + bc2 + c2 a + ca2 ] 2[m3 + b3 + 1 + 3(1 + m)mb] − (m + 2)[m2 b + mb2 + b2 + b + m + m2 ] 2b3 − (m2 + 3m + 2)b2 + (−m3 + 4m2 + 5m − 2)b + (m3 − 3m2 − 2m + 2) (1 − b)[−2b2 + (m2 + 3m)b + (m3 − 3m2 − 2m + 2)] (1 − b)f (b) ≥ (1 − b)f (1) = (1 − b)m(m − 1)2 ≥ 0 a3 + b3 + c3 + 3(1 + m)abc m ≥ 1+ 2 2 2 2 2 2 a b + ab + b c + bc + c a + ca 2





˜   ??BA25=B8*8a=>kz¥747:<4MI‰672PCˆAŠT ¢y‚ 4 a b c  i 6M:F6M898+?
a b f ≤ ≤ 1 c c   ¢‚ 4 GB\6 ‡B4 74 : i 4>¢Bku=BI a a b (m0 , a0 , b0 , c0 ) = , , ,1 m ≤

c c c

m ≤ m 0 = a0 ≤ b 0 ≤ c 0 = 1

6:<;"A+G347ID47ku=BID4>¢

a03 + b03 + c03 + 3(1 + m0 )a0 b0 c0 a02 b0 + a0 b02 + b02 c0 + b0 c02 + c02 a0 + c0 a02

‚ G>C i G i 6M:FJ<4X‚aIDCˆA‰AE47:N68

¢

≥ 1+

m0 ¢ 2

a03 + b03 + c03 + 3(1 + m0 )a0 b0 c0 −1 a02 b0 + a0 b02 + b02 c0 + b0 c02 + c02 a0 + c0 a02   1 3a0 b0 c0 f 0 ≥ m − 02 0 0 02 02 0 0 02 02 0 0 02 2 a b +ab +b c +bc +c a +ca

ŸBT"A+G34 g ›



4

 :<4’>?<6M2@C@A€Ty¢

›

a2 b + ab2 + b2 c + bc2 + c2 a + ca2 6

47: i 4>¢ 1

 CK: i 4

−

≥

√ f 6 a6 b6 c6 = abc

3abc a2 b

ab2

b2 c

2 + + + ¢‚ 4aGB\6 ‡B4 a : b : c = a 0 : b0 : c0

bc2

+

c2 a

+

ca2

≥ 0

f

a3 + b3 + c3 + 3abc

−1 a2 b + ab2 + b2 c + bc2 + c2 a + ca2 a03 + b03 + c03 + 3a0 b0 c0 = −1 a02 b0 + a0 b02 + b02 c0 + b0 c02 + c02 a0 + c0 a02   1 3a0 b0 c0 0 − 02 0 ≥ m 2 a b + a0 b02 + b02 c0 + b0 c02 + c02 a0 + c0 a02   1 3abc 0 = m − 2 2 a b + ab2 + b2 c + bc2 + c2 a + ca2   1 3abc ¢ ≥ m − 2 2 2 2 2 2 2 a b + ab + b c + bc + c a + ca

6:<;"A+G347ID47ku=BID4>¢

a3 + b3 + c3 + 3(1 + m)abc m f ≥ 1+ 2 2 2 2 2 2 a b + ab + b c + bc + c a + ca 2


 
 f !R,a59IBU"$MNeS 23Q1RSE"#3QBPU* 1(l"BP &BP+ 5'(#SU 1RSUPU. $ ..][
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Z ( S () 112 )P'
+QBP SU (Z.Q# FJSUPRP 

˜ á Ø ë

© 7

ª  ° i ¯ ° ° ±   

J±ç5ä+ã   é7ê  <ãèyê •®0è>ç$¯Mã\ä‰é+ç c´ >âæç  è 7å 

D˜M™™šÜì  ™  ¢  ™š  ä   é ã    _ç Mä-êKç5è ã  3ê  è  <ã ã ç€â  ã ç€â rf J<46M:T:<=B:<† i =B:38+A‰6:>A i =BLNH32K4*òNL“=B:BC i Hr> = 2 T3:<=BLCK672 f  G3 = ‚ A+GB6A ¡M4A f A+G34MID4„CP8‹6 Pi =B(z) LFHB254+ò :B?


f

4MHB2K6 i 47L“47:>A f

°

‡¬ 

|w| ≤ 1

w

|P (w)| ≥ 1



s f
 å BêKç  è   ç€â yã-å 3ê 7ç@åPå€ã   >ãè  ä MèyâEã f f  ?AEI‰6MI€Ty¢ ¡M4A n n−1 P (z) = z + a z + · · · + a z + a n−1 1 0 f •—B i i A+GB67A = :38+C5;34MI ku=BIa6M2@2 =BLNH32K4*òƒ:B?
Q(z) = a0 z + a1 z z 6= 0

fŒ G>?<8\¢>CPk

¢A+G34M:

R  Aku=>2P2K=‚a8zA+GB6A 2π 0

Z

2π

Z

it

Q(e ) dt = 0

8*CP: i 4

2π R 0

eikt dt =

0

z

z

=‚ 4-‡B47I-¢


¢ a0 eint + a1 ei(n−1)t + · · · + an−1 eit + 1 dt = 2π


1 e2kπi − 1 = 0 ik

ku=BI672P2

¢

¢

¢

k = 1 2 ... n

f

2π 2π R R ¢36 i =B:>AEI‰6;BC i ADC5=B: f 2π = Q(eit ) dt ≤ |Q(eit )| dt < 2π 0 0 sæs f
 å BêKç  è   <çKè    é7ê  <ãèyê è  Mè      è  —èBç Mãäæé*çŠê   
 ç
yã   MãMâ <è å  Mè  âæçŠã\èyâEã   Mè   3ã_ç  BçKè  ê yã ã çæê  ä f

²

ζ

47: i 4>¢

2π

|Q(eit

 )| dt < 2π f

Q(z) = z P

¡M4A

n

¬       



›±

n = zB z n−1 ¢ P (z) ¢ 6M:<+ ; an−1 ku=BI+ · · · +¢ a1¢ z ¢ + a0¢ 2n k

= −1 ζ 2n−1 X k=0

=1

P (ζ k )

® (¯



f ¡M4A

iπ/n ζ f =‘ e = ‚ ¢ k = 0 1 2 . . . 2n − 1

|ζ | = 1




f„Œ G34M:


≥ P (1) + P (ζ 2 ) + · · · + P (ζ 2n−2 )


− P (ζ) + P (ζ 3 ) + · · · + P (ζ 2n−1 ) n−1 X
2k 2k+1 = P (ζ ) − P (ζ )

ß à

k=0

6:<; n−1 X k=0

P (ζ 2k ) − P (ζ 2k+1 )




=

n−1 X k=0

+

(ζ 2k )n − (ζ 2k+1 )n

n−1 X j=1

aj

n−1 X k=0




(ζ 2k )j − (ζ 2k+1 )j


f

߀˜_à

‘ =AD4 A+GB67A ¢ (ζ 2k )n − (ζ 2k+1 )n = (ζ n )2k − (ζ n )2k+1 = 1 − (−1) = 2 6:<;FG347: i 4>¢ n−1 X k=0

(ζ 2k )n − (ζ 2k+1 )n




= 2n

f

ß à

˜ £ :A+G34a=A+G34MIzGB6:<; ¢Bku=BIz4M6 i G n−1 X k=0

¢

¢

¢

j = 1 2 ... n − 1

(ζ 2k )j − (ζ 2k+1 )j

= (1 − ζ j )




= (1 − ζ j )

8*CP: i 4

¢‚ 4aGB6\‡B4 n−1 X

(ζ 2j )k

k=0

1 − (ζ 2j )n

f

1−

ζ 2j

= 0

¢

ß à

2j n 2n j (ζ =ADCP(ζ  ?rJ38E)ADC@AD?B :<¥aß  )à6M= :<;F1ß àCK:>AD=N߀˜_à6:<;"?<8*CP:<¥aß à+¢‚ 4OA+G347:“GB6\‡B4 2n−1 X k=0

P (ζ k ) ≥ 2n ¢

k5ID=BL ‚ G>C i Ga‚ 4a;34;>? i 4OA+GB67AA+G347ID4„LN?<8EAJ34„8-=BLF4 ¢M‚„C@A+G ¢>ku=BI k 0 ≤ k ≤ 2n − 1 i f ‚ G>C i G ¢ = B N L 3 H K 2  4 D A P C < : O ¥ + A 3 G „ 4 < H D I 3 = > = k k |P (ζ )| ≥ 1


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