Solutii Etapa Zonala Ed3 25feb2006



⇒  ⋅ (D + E + F ) =   ⇒ D + E + F =  



&ODVDD,9D , D E [  ,, D ««   E  ,,, LQHPFRQWF  ÌQFRQWLQXDUH  ⋅$OGRLOHD MXF WRU FkúWLJ  vQWRWGHDXQD GDF  UHVSHFW  XUP WRDUHD VWUDWHJLH GDF  SULPXO LD D SLHWULFHOH DWXQFL DO GRLOHD LD E SLHWULFHOH DVWIHO vQFkW D  E   ÌQ DFHDVW  FRQGL LH GXS   DVWIHO GH DSDUL LL YLQH UkQGXO SULPXOXL FDUH DUH GRDUSLHWULFHOHvQJU P MRDU úLQXDUHYRLHV OHLD  &ODVD9D , D &DOFXO PSXWHULDOHOXLúLPDLPLFLGHFkW  =     =    =     =    =    =     =    =    =    =    =  'HDLFLUH]XOW [≤úL\≤ ÌQWUXFkW VLQJXUDVROX LHHVWH[ úL\   E  5HVWXOvPS U LULLOXLDODSDWUXHVWHDFHODúLFXUHVWXOvPS U LULLOXL ODFDUHHVWH XD  úLXD  1XP UXO1QXSRDWHILS WUDW SHUIHFWGHRDUHFHDUHX1   ,,D  $ = ⋅ (D + E + F )  úL % = ⋅ (D + E + F ) ⇒ $ + % =  ⋅ (D + E + F )  úL $ − % =  ⋅ (D + E + F )  ⇒  ⋅ (D + E + F ) =  ⋅ (D + E + F )+  ⇒

, D [ \ ]   E $QXODUHOXQLúL LQHPFRQWF  úLUHVWGHFLH[LVW ROXQ  DDQXOXLvQFDUHFHOSX LQWUHLFRSLLDX]LXDGHQDúWHUHVLWXDW vQDFHDOXQ  ,, D [ E DE  DE  úL 'HFLV   ,,,6HFRQVLGHUDF SHILHFDUHFRORDQ VXPDQXPHUHORUHVWHDFHHDúL    «   $WXQFL FDUHDSDU LQHFHOXLGHDOGRLOHDUkQGSHQWUXF   ⋅FRUHVSXQGH QXP UXOXL ± ,9  PLQXWH

&ODVDD,,,D

62/8 ,,

(GL LDD,,,DHWDSDD,,D]RQDO IHEUXDULH



(



(

)

3HQWUXPLPSDU ⇒  + P HVWHSDU ⇒ 1 =  ⋅ 1 LDUSHQWUXPSDU ⇒ 

E  1 = DD D D D D D  1 =  + 1 =  ⋅  +  P =   + P 

)

 Q =   'DF  Q < N ⇒  Q  +   N −Q =   ⋅  ⇒  ⇒  +   N −Q = 

'DF  Q = N ⇒  ⋅  Q =  ⋅  IDOV

SDU

)

 

,9D  $ = % ∪ & XQGH % = {}úL & = {}VDX % = {} úL & = {}  E $YHP {}FX +  =  +   {}FX +  =  +   «««««««««« {}FX +  =  +   ÌQJHQHUDO N −  + N = N −  + N −  N≥  ⇒ % = {}úL & = {}

(

 1 =  ⋅   ⋅  +   ⋅ P =    ⋅  + P =  ⋅ W   





)

N =  N =  ⇒   ⇒   Q −  ⇒N úLQ  Q −  ⋅   =  = +  = 

(

Q =  ⇒  N − QXDUHVROX LH  = 





=



 ⋅  Q −  N

⇒ 

N

,,,D 'DF  Q > N ⇒   N  ⋅  Q −  N +  =  ⋅  

5 ⇒ $ = ⋅  =  VDX $ = ⋅  =  

&XP 5 < ⋅ (D + E + F )⇒ (D + E + F )⋅ U < ⋅ (D + E + F ) ⇒ U <   GDU U  HVWH UHVWXOvPS U LULLOXLOD[ ⇒ U =  ⇒ 5 = D + E + F   F  D E F FLIUHQHQXOH ⇒ D + E + F ∈ {}5FXESHUIHFW⇒5 VDX

⇒5GLYL]LELOODDEF  ⇒ 5 = (D + E + F )⋅ U ⇒  = [ + U 

 &XP D E F ≠   ⇒GRX GLQQXPHUHOHDEFVXQWHJDOHFXúLXQXOFX ⇒ D ⋅ E ⋅ F =    E 'LQLGHQWLWDWHDvPS U LULLFXUHVW ' = , ⋅ & + 5   ≤ 5 < , ⇒ ⇒ ⋅ (D + E + F ) = ⋅ (D + E + F )⋅ [ + 5 ⇒ 5 = (D + E + F )⋅ ( − ⋅ [ ) ⇒ 





  Q +  ⋅ 

∈1⇔      Q +  ⇒Q +  ≥ ⇒Q ≥  ⇒

) )

  D ∈ 1 ⇒ Q =  N −   N ≤   ⇒ D =

N

=  N

−−N

 'DU D S WUDW





 3HQWUX N =  S +  ⇒  S+ −  −  S −  =  ⋅  S −  S −  = (PXOW +  − S − )  ⇒ S = T ⇒ N = T +  ⇒ N ∈ {} GHFL$FRQ LQHFXEXULSHUIHFWH  ⋅  ,, D  &DOFXO P  +  +  +  +  = =   &XP  ±     vQVHDPQ F H[LVW QXPHUHDF URUGLIHUHQ HVWH]HUR  E &XPDEFVXQWQXPHUHSULPHPDLPDULGHFkW⇒DEFLPSDUH⇒ ⇒   E − D úL   F − E 5HOD LD E = D + F ⇔ E − D = F − E = K  'DF  K = PXOW H[HUFL LXOHVWHUH]ROYDW 'DF  K = PXOW ±   GHRDUHFH E = PXOW ±  ⇒ D VDX F HVWH PXOW GHFL QX HVWHSULPGHFLFD]XODOGRLOHDQXHVWHSRVLELO([HPSOXD E F 

S = PXOW ⇒ N ∈ {}  (N ≤ ) 

SHUIHFW ⇒  N − − N  QXP U SDU ⇒ N LPSDU ⇒ N ∈ {}∪ {}  $úDGDU H[LVW S WUDWHSHUIHFWH  D FXESHUIHFW⇒  N − − N PXOWLSOXGH  3HQWUX N =  S  ⇒   S −  −  S =  S −  −  S = PXOW +  −  −  S ⇒





N −

⇒  N ≤  'HRDUHFH  =  úL  = ⇒  N ∈ {} $∩4∩1 DUH  −  =  HOHPHQWH

  ⇒Q ≥ ⇒ Q ≥  +  &HO PDL PLF QXP U QDWXUDO FX DFHDVW    SURSULHWDWHHVWHQ  Q  E   D = ∈1⇔ Q +  =  N  N ≤ Q  ⇒ Q =  N −  ≤ ⇒  Q +



( (

6H YHULILF  LPHGLDW F  $ = % ∪ &  úL GLIHUHQ D VXPHORU FHORU GRX  VXEPXO LPLHVWH  &ODVD9,D Q+ Q +  Q + Q +  − −  −  Q +  − Q +  Q +   −   −   , D   =  =    =    − −  −   −     −  −  

)





( )

 =     = 

(

)

$

%

$

=  ⇒ &$ < &' < &% 

2

$

)

$



'

)

&D]XOE   & $ ' %      %' = %& − &' =  −  =  −  =  ( − )FDUH QX H GLYL]LELO SULQ

(

&D]XOD   % & $ '      %' =  +  =  ⋅  +  =  ⋅  = PXOW 'HFLDFHVWFD]DUHVROX LH

X  −   =  ⇒ $' GLYL]LELOSULQ⇒VROX LHEXQ 

(

   &  $' = &' − &$ =  −   

 

'HFLDFHVWFD]QXDUHVROX LH  &RQFOX]LHDYHPVLWXD LLOH % − & − $ − ' VDX ' − $ − & − %  

 

 



  

X   +  =  +  =  ⇒ $' QXHVWHGLYL]LELOOD⇒VRO QXFRQYLQH

$

 

( )

=     = 

       ' & ⇒ $' = $& + &' =   +  

,9

 

⇒   −  =  <  ⇒ % ∈ ,QW(< $2$ ) 

=   ⇒  Q + −  <  ⇒Q ⇒

⇒ P(< $2% ) =   −    =  −   = 



'HRDUHFH  =    =   =  ⇒ ⇒ Q +  =  ⇒ Q =    E  P(< $2$ ) =   −  =  

(

  Q + −  < D 



( ) =

,,,D  P(< $2$ )+ P(< $ 2$ )+  + P(< $Q 2$Q + ) =  +   +  +  Q

&ODVD9,,D

Q(Q + ) , D   +  +  +  + (Q − ) = Q  úL   +  +  +  + Q =  $WXQFL             D= ⋅ ⋅  ⇒ D =   ⋅ ⇒D = =  ⇒   ⋅   ⋅   ⋅  ⋅        



(

(

)

)

'HFL  ⋅ DD D Q = 0  + DD D Q − −  ⋅ DQ &XPúLVXQWSULPH

)





QGDFDP < Q 6L + − 6L = P + Q + P − Q ⇒ 6 L + − 6L =   PGDFDP ≥ Q 'HFL 6L + − 6L  HVWH WRWGHDXQD XQ QXP U SDU &XP VXPD LQL LDO  D WXWXURU QXPHUHORUGHSHWDEO HVWH  +  +  +  +  =  ⋅ DGLF QXP USDU

vQWUHHOHUH]XOW    DD DQ ⇔   DD DQ − −  ⋅ DQ   E $SOLFkQGFULWHULXOSHQWUXRE LQHP± ⋅ 'DU  ⋅⇒HVWHGLYL]RUDOOXL ,,,D )LH(PLMORFXOOXL>%&@úL)PLMORFXOOXL>$&@1RW P {7} $( ∩') )0 )*  ⇒$(⊥')2'⊥$%2'PHGLDWRDUH  = = ⇒ 0* $%  )' )%  2' ⊥ $% ⇒ 2' ⊥ 0*   E 2RUWRFHQWUXO ∆'*) ⇒ 20 ⊥ '&  ,9/DILHFDUHSDVLvQDLQWHGHúWHUJHUHDDGRX QXPHUHPúLQILH 6 L + VXPD QXPHUHORU GH SH WDEO  úL 6L VXPD QXPHUHORU GH SH WDEO  GXS  úWHUJHUHD QXPHUHORUPúLQ$WXQFL 6L + − (P + Q ) = 6L + (P − Q )DGLF 



= ( ⋅  + )DD DQ − + ( − )DQ

(

 ⋅ DD D Q =  ⋅ DD DQ −  + DQ =  ⋅ DD DQ − +  ⋅ DQ =

 F   =  =  ⇒SULPHOHGRX FLIUH ,,D  2EVHUY PF   ⋅ +  =  UH]XOW   ⋅  N =  ⋅  N − = ( + )⋅  N − SHQWUXRULFHN≥

GLYL]LELOOD⇒ E ∈5?4

FLIUH

⇒  D =   ∈1  E  E =  − ⇒ E =      E GLYL]LELO OD  GDU E QX H



)

(

) ( )

 

'



&

([ + \ )([ + ] ) ≤ ∑ [ + \ + [ + ] = 

([ + \ + ] )  =    6 ,,,&XWHRUHPDFHORUWUHL 4 SHUSHQGLFXODUHRE LQHP 6$ ⊥  $%&  1  ⇒ 6' ⊥ %&  $' ⊥ %&  3 $SOLF PWHRUHPDFDWHWHLSHQWUX FDWHWD6$vQWULXQJKLXULOH6$% $ úL6$&GUHSWXQJKLFHvQ$úLRE LQHP 6$ = 63 ⋅ 6%  0  ⇒ 63 ⋅ 6% = 64 ⋅ 6&    6$ = 64 ⋅ 6&  % $SOLF PWHRUHPDFDWHWHLSHQWUXFDWHWD6' vQWULXQJKLXULOH6'%úL6'&GUHSWXQJKLFHvQ'úLRE LQHP 6'  = 60 ⋅ 6%   ⇒ 60 ⋅ 6% = 61 ⋅ 6&   6'  = 61 ⋅ 6& 

$WXQFL [ + \] + \ + [] + ] + [\ = ∑

=

    +    −    −  +   −    ⋅ −  = ⋅ = =− = − ∈4                     E  [Q] = −  = −   = −  {Q}= −   = − − − =           ,, [ + \ + ] =  ⇒ [ + \] = [([ + \ + ] )+ \] = [ + [\ + [] + \] = ([ + \ )([ + ] )

(

DWXQFLVXPDQXPHUHORUGHSHWDEO GXS SULPDRSHUD LHGHúWHUJHUHYDILWRW XQ QXP U SDU 5HSHWkQG SURFHGHXO WRDWH VXPHOH 6 L +  úL 6L  YRU IL QXPHUH SDUH úL FXP XOWLPD VXP  QX HVWH DOWD GHFkW XOWLPXO QXP U U PDV SH WDEO  vQVHDPQ F DFHVWDHVWHSDU  &ODVD9,,,D      +     −     +   −   , D  Q = ⋅ − = ⋅ −  =         −     +    

63 ⋅ 6% 64 ⋅ 6& 63 64 = ⇒ = ⇒ 01 34  60 ⋅ 6% 61 ⋅ 6& 60 61

F 2EVHUY PF  0* =

0$ + 0% + 0&  

&′

$

$′

%′

 $ RE LQHP

&





XUPDUH X  + Y = D  úL X + Y  = E  'DF XY∈1 UH]XOW  X  + Y ≥ (X + ) úL

[ = E  − \  5H]XOW  F  H[LVW  X Y ∈ 1 FX [ = X   úL \ = Y   $YHP SULQ

,, 6  QRW P D = [ + \ ∈ 1 úL E = \ + [ ∈ 1 'HFL

\ = D  − [  úL

E  − DF F  − DE $% + $& = 2  GH XQGH (E + D )(E + F ) (F + D )(F + E )

UH]XOW  E  − DF = F  − DE =  ÌQFRQFOX]LHDYHPD E F

'DF  0

∑

D  − EF 0$ = 2  (D + E)(D + F )

  D D  ∑ + 0$  *′ = * ⇒ 0* = 0*′ ⇒   D +F D +E

D  D  ⇒ ∑  + −  0$ = 2  + + D F D E  

0*′ =

0$′ + 0%′ + 0& ′ 0*′ = &RQIRUPSXQFWXOXLDQWHULRUUH]XOW F  



%

&ODVD,;D

 , D 'LQWHRUHPDELVHFWRDUHLUH]XOW F  DF DE %$′ = úL &$′ =  E+F E+F E F 0% + 0&   E  0$′ = E+F E+F





& ′3 ⊥ %′'′ & ′3 ⊥ %'′ E  ⇒ & ′3 ⊥ (%%′'′)⇒ ⇒ (& ′34 ) ⊥ (%%′' )  ′ ′ & 3 ⊥ %% & ′4 ⊥ %'′

'1 ⊥ $′& ⇒ $′& ⊥ ('01 ) ⇒ ('01 ) ⊥ ($$′& ) '0 ⊥ $′&

'0 ⊥ $& ,9D  ⇒ '0 ⊥ ($$′& )  ⇒ '0 ⊥ $′&  '0 ⊥ $$′

'LQ úL RE LQHP

} {( )

S

}

(

S



)



≤  − [  S  

(α





)

 α  + β  +℘

α  + β  +℘

 'HRDUHFH  1XP UXODHVWHLUD LRQDO 

D= + β +℘ 5H]XOW F  D =

)

 

LQkQGFRQWGHLL UH]XOW  I () =    7RW GLQ L  SHQWUX \ = − [  UH]XOW  F  I  = I  [ ⋅ I − [  ∀  [ ∈ 5 úL FRQIRUP DYHP I  [ ⋅ I − [ =  ∀ [ ∈5úLGHDLFLUH]XOW F  I ([ ) ≠   ∀ [∈5  ÌQL OXkQG\ [RE LQHPF  I ( [ ) = I  ([ )∀ [∈5

&ODVD;D  , ÌQL SHQWUX\ UH]XOW F  I  [ = I  [ ⋅ I  ∀ [∈5 



α + β +℘

(

α  + β  + ℘  =  [  + \  + ]  − [\ − \] − ][ 

,91RW P α= [− \  β = \ − ] ℘ = [ − [ 2EVHUY PF  α + β +℘ =  

( − [ ) S ( + [ ) ≤ ( − [ S ) ( + [ S )

S

)(

S

)

           −   +  ≤  − S   + S  VXQWHFKLYDOHQWH [ [   [   [  

(

≤  − [ S  − [  S     'DF  [ ∈ ( ∞ )DWXQFL ∈ () úLRE LQHPGLQ  [ S

( − [ )S ( − [  )

 'DF  [ ∈ [] DYHP ( − [ )S ≤  − [ S     − [  'LQvQPXO LUHDLQHJDOLW LL FX RE LQHP

S

,,,D $YHP ( − [ ) + [ ≤ ( − [ )+ [ =   (∀)[ ∈ []

{( )

X + Y ≥ (X + Y + ) VDX X + Y +  ≤  FRQWUDGLF LH'HFLX VDXY   'DF X DWXQFL[ úL\ QQ∈1  'DF Y DWXQFL\ úL[ PQ∈1 5H]XOW  6 =  Q   Q ∈ 1 ∪ Q    Q ∈ 1 

X + Y  ≥ (Y + )  'HFL Y ≥ X +   úL X ≥ Y +   3ULQ DGXQDUH RE LQHP

[

[

 +  

[



\ 

[

 +  ±   −  =  

(

)

 'LQ \ +

(

)

  −  RE LQHP  \  −  −  \ +  =  FX = \ 



vPS U LUHDFX ] − ]  ] Q − + ] Q −  ] +  + ]

( )

Q−

]+]

Q −

( )

+ Q ] + ] + Q =  ⇔



Q  ⇒ ] Q − ] + Q ]  − ]  + Q ] − ] =   ÌQV  ] ∈ & ? 5 ⇒ ] ≠ ]  ⇒ GXS   

E 'HRDUHFH FRHILFLHQ LL HFXD LHL VXQW UHDOL UH]XOW  F  ] VROX LH D HFXD LHL  ] Q + Q]  + Q] + D =  ⇒ ]  VROX LH D HFXD LHL ⇒  Q  3ULQ VF GHUH   ] + Q ] + Q ] + D = 



∆ =  −   +  −  = −  −  <  FHHDFHDUDW F  \  ∉5



[

 +  −   −   +  −   −   'LQ   = RE LQHP [ = ORJ        

 +  +   −   +  +   −   'LQ   = RE LQHP [ = ORJ        



 ±   

  +    'LQ \ + =  RE LQHP  \  −  +  \ +  =   FX U G FLQLOH \ 

HFXD LDGHYLQH ] +  = ]  −  VDX ]  − ] −  =  FX ]  =

[

    1RW P   = \  úL DYHP \ +  + = \  +   6XEVWLWXLQG \ + = ]  \ \ \ 

[

   ,,D ÌPS U LQGFX[RE LQHP   +  +   =     

GHFL UH]XOW  F  H[LVW  7 =  [ ≠   DVWIHO vQFkW I ([ + 7 ) = I ([ )  ∀  [ ∈ 5 GHFLIHVWHSHULRGLF 

DFHDVWDFRQIRUPL UH]XOW F  I ([ +  [ ) = I ([ ) I ( [ ) = I ([ ) ∀ [ ∈5úL

[ ≠   /XkQG vQ   [ = [   UH]XOW  F  I  [ = I   [ = −  =   úL FX

'LQLL UH]XOW F H[LVW [ ∈5DVWIHOvQFkW I  [ = − &RQIRUP UH]XOW  = ]

Q−N

]+]

+ ]

Q−

Q −



=Q]

Q −

Q −

 Q −

⇒

$QDORJ $ % = $ % ⇔ α  = α  

 



⇔ ( − α )(D − D  ) = ( − α  )(D − D ) ⇔  − α =  − α  ⇔ α = α  

$ % = $ % ⇔ D − αD − ( − α )D = D − α  D  − ( − α  )D ⇔ 

⇔ (α  − α )(D − D ) = (α  − α  )(D − D ) 

+ D ( − α  + α  − )+ D ( −  + α − α  ) =  ⇔

 D + α  D + ( − α  )D ⇔ D ( − α −  + α  )+ D (α  +  − α  − ) 

⇔ D + α  D  + ( − α  )D + D  + α  D + ( − α  )D  = D + αD + ( − α )D  + 

 D + α D + ( − α  )D   D + α  D + ( − α  )D  0     0            0 0  0  0  SDUDOHORJUDP ⇔ P + P = P + P XQGH 0 L (PL ) ⇔

 D + α  D + ( − α  )D   D + αD + ( − α )D  0 L PLMORDFH ⇒ 0      0           

 $QDORJ 0  ≠ 0  ⇒ 0 0  0  0  SDWUXODWHUQHGHJHQHUDW  E  1RW P $L (DL )  DL ∈ & %  % ∈ ($ $ )⇒ % (αD + ( − α )D ) úL % (α  D + ( − α  )D ) %  % ∈ ($ $ )⇒ % (α  D + ( − α  )D ) úL % (α  D + ( − α  )D ) α L ∈ ()

DIO vQVHPLSODQHRSXVHID GH $ $ 

⇒ 0  ≠ 0  $QDORJ 0  ≠ 0   0  ≠ 0   0  ≠ 0   0  ≠ 0  GHRDUHFHVH

,,, D  'DF  0  = 0  ⇒ $ $ % %  SDUDOHORJUDP $ $ $ $  IDOV

⇒  5H ] +  ≤ ]

⋅] + ]

Q −

⋅]+ ]

ÌQV  ] + ] =  5H ] ⇒ Q  5H ] +  ≤ Q ⋅ ]

Q−

Q −

] +  + ]

'HFL Q ] + ] +  ≤ Q ]

≤ ]

Q −

( )

Q−

úLDWXQFLDSOLFkQGPRGXOXOvQ

( )

= − Q ] + ] − Q   

Q −

Q −

⋅ ] N − = ]

Q−

Q ] + ] +  = − Q ] + ] − Q = ] Q − + ] Q −  ⋅ ] +  + ]

ÌQV  ] Q − N ⋅ ]

N −

⇔ ] Q − + ] Q −  ] +  + ]



D − D α  − α  = ∈ 5 ∗ ⇒ $ $ $ $ ⇒IDOVGLQLSRWH]  D  − D  α  − α

(∑ [ − OJ ) ∑ [  −  ≥  ∑ [ −  ∑ [  ∑ [  −  =  (∑ [ ) ∑ [  −  ≥        



=

N =

OQ Q

≤

(

Q

)



Q Q − ÌQV  OQ Q OQ Q 67 &(6$52 OQ (Q + ) OQ(Q + ) OLP OLP = OLP = Q Q→∞ Q OQ Q Q→∞ (Q + )OQ (Q + )− Q OQ Q Q→∞   OQ +  + OQ (Q + )  Q

XQGHUH]XOW F 

OQ Q ≤ Q OQ Q

∑Q N − Q

Q

&ODVD;,D , D 5H]XOW GLQLQHJDOLWDWHDOXL%HUQRXOOLE 5H]XOW WUHFkQGODOLPLW FkQG Q → ∞LQHJDOLWDWHDGHODSXQFWXOSUHFHGHQWF 6HúWLHF  H [ ≥ [ +  [ ∈5  OQ N OQ N Q 3HQWUX [ = OQ N  N ∈ Q VHRE LQH Q N ≥ + 'HFL  + ≤ N ≤ Q Q  Q Q Q OQ Q Q Q N ∈ Q   ÌQVXP P GXS  N ∈ Q  úL VH RE LQH Q + ≤ ∑ N ≤ Q Q Q  GH Q N =

 ≥ −  =     

=

∑ OJ D ≤ − OJ  6 QRW P [ = − OJ D  \ = − OJ E  ] = − OJ F  'HFL ∑ [ ≥  OJ  FHHDFHLPSOLF  OJ  + OJ E + OJ F \ + ] − OJ  [ + \ + ] − OJ  =∑ =∑ −=  ∑ ORJ D EF = ∑ OJ D [ [

,9 'LQ LQHJDOLWDWHD PHGLLORU RE LQHP  ≥  DEF  VDX HFKLYDOHQW

'HFL α  = α ⇒ α  = α  THG

⇒

 D ⇒ E  $YHP α = α   úL α  = α   ⇒   $GHY UDW ⇒ 0 0  0  0   SDUDOHORJUDP  E ⇒ D 'HFL $GHY UDW'DF  α  ≠ α ⇒ α  ≠ α  ⇒ 

)



OQ Q

( ) = 2 3HQWUXD

F 

6H



±DYHP

 Q −

( =

Q −

LQGXF LH

 

Q

( )

ELQRPXOOXL1HZWRQDYHP $

( Q = Q − ( 

úL

DWXQFL

= (D − )Q ((D + ),  − ( )Q = 

= (  úL ( Q = Q − (  IRORVLQG

−   D +  −  =  −  D + 

−

$ 3HQWUX D ∉ {− }DYHP

SULQ

 D − − D − D   D +     $ =   − D D −   − D  = (D − ) −     −  − D  − D D  −     ( D + ),  − ( )  &XP ,  ⋅ ( = ( ⋅ ,  = (D − )( 

($ ) = 

Q

GHPRQVWUHD] 

         −        

$ =   −    GHXQGH $ =      = ( XQGH ( =               −       



3HQWUXD RE LQHP $ = 2 úLLPHGLDW $

Q

D     F $YHP $ =   D   FX GHW ($) = D  − D +  = (D − ) (D +  )   D  

( )

'DF  GHW $ ≠  DWXQFL GHW $ = (GHW $) ≥  úL GHW $ = (GHW $) ≥  







 E DEF  ⇒ GHW $ =  − D − E − F =  DEF − D − E − F ≤  GLQLQHJDOLWDWHD PHGLLORU  'DF  GHW $ =  DWXQFL $ = 2 úL GHW $ =  

(

Q Q Q − H Q − H [Q −  LDU OLP = OLP = OLP =  GHRDUHFH OQ Q Q →∞ Q →∞ Q →∞ OQ Q [Q Q  OQ Q   Q Q OLP [Q = OLP =  ÌQFRQFOX]LH OLP ∑ N − Q  =   Q →∞ Q →∞ Q Q→∞ OQ Q   N =   EF −   − F  − E    ,,D $YHP GHW $ = DEF +  − D − E − F  $ =   − F DF −   − D     − E  − D DE −   

+

RE LQHP $

(

)

Q

Q



(

)

1RW P [Q + − [Q = X Q $WXQFL



(Q + )X Q + + X Q + + X Q = ⇔ (Q + )X Q+ + X Q = ⇒X Q + = −



 X Q   Q+     $FHDVW UHOD LHVHGHVI úRDU  X = − X  X = − X = X    ⋅     X = − X  = − X « X Q − = − X Q −  = − Q − X    ⋅ ⋅  Q Q     XQ = − X Q − = − Q X  X Q + = − X Q = − Q + X  (Q + ) (Q + )  Q + Q+



⇔ (Q + )([Q +  − [Q + )+ ([ Q +  − [Q + )+ ([Q + − [Q ) =  

FRQWUDGLF LHÌQFRQFOX]LH $ = , Q  ,9D 5HOD LDGHUHFXUHQ VHPDLSRDWHVFULH (Q + )([Q +  − [Q+ )+ ([Q +  − [Q = ) ⇔

3ULQvQPXO LUHDUHOD LHL FX % N −  UH]XOW  & S % N − = 2Q FHHDFHFRQVWLWXLHR

& S % + & S %  +  + & NS − % N − = 2Q   

, Q = $ S = (, Q + % )S = , Q + & S % + & S %  +  + & NS − % N − VDX

SURSULHWDWHD % N ≠ 2Q úL % N − ≠ 2Q 'LQFRQGL LD UH]XOW 

WRDWHYDORULOHSURSULLDOHPDWULFHL % = $ − , 1 VXQWQXOHUH]XOW  % Q = 2Q  'DF  % = 2Q UH]XOW  $ = , Q 'DF  % ≠ 2Q V FRQVLGHU P N ∈ { Q}FX

WU $ = λ + λ  +  + λ Q = Q  FHHD FH LPSOLF  λ = λ  =  = λ Q =   'HRDUHFH

  GHGXFHP λSM =   (∀) M ∈ { Q} 'HFL λ M =   (∀) M ∈ { Q} úL

( ) = (D − )  (D + ) ,

Q

 (D − )Q −  (     ,,,6 QRW PFX λ  λ   λ Q ∈&YDORULOHSURSULLDOHPDWULFHL$'LQFRQGL LL

N =

∑

 Q N  &Q (D + )Q−N (− )N N = (D +  − )Q −    N = 

&XP ∑ &QN (D + )Q−N (− )N N − =

Q

Q    (D − )Q  (D + )Q ,  + ∑ & QN (D + )Q − N (− )N N − (     N =  



  Q     Q +    [ + =  $Q XQGH   Q +    [Q +     

Q    Q   Q [Q + ⋅  =  ⋅ [Q + + + ≥     [Q +  [Q +    [Q +



Q

 

∫

∫

∫

( )

,, D  3HQWUX D =    DYHP ,   = ∫



 



G[

∫

=

(

(

)

)

  'DF  D ∈     [    D  DUFWJ D − DUFWJ   'DF  D ∈    ∞  DYHP , (D ) =      −D  −D −D   D − D  −   D + D  −          OQ  DYHP , (D ) =       D −   D + D −   D − D −      D DUFWJ [ − DUFWJD D G[ G[ + (DUFWJD )   $YHP ,  (D ) =      [ + D[ +  [ + D[ +  [−D DUFWJ D DUFWJ [ − DUFWJD D  + D[ G[  &X VXEVWLWX LD 'HFL ,  (D ) = G[ =     [ + D[ +  [ + D[ + 

[  = H(∀) [ ∈ * 

D  H DEVXUG 5H]XOW  D  H úL SULQ XUPDUH [  = [  (∀) [ ∈ * ? {H} 'HFL

&ODVD;,,D , 'DF D ≠HDWXQFLUH]XOW F  D  ∈ * ? {D}'HFL D = D  VDXHFKLYDOHQW

Q→∞

'HRDUHFH OLP $Q = +∞ UH]XOW  OLP [Q = +∞ 

$Q =

 Q + >    

 (− )Q     'LQ [Q + − [Q = X Q GHGXFHP X =  úL [Q + = X  − + −  +    (Q + )    'HDLFL OLP [Q =  −  Q →∞ H       Q +    Q       Q +      [Q + +  + > =  E  [Q + =     [Q +  +    [Q +        [Q  [Q +       



DUFWJ \

D

DUFWJ [

∫−D \  − D\ + G\ = −∫ [  + D[ + G[ 'HFL



G[ = − ∫

D



D  = D ⇒D HIDOVGDF  D  = D − 



[\ ∈ + D  GHFL D  ∈ + D  DGLF  D  ∈ +  VDX D  = D  VDX D  = D −  'DF 

 1RW P XN ([ ) = 3( I ([ + ND ))  [ ∈ 5 'HRDUHFH X N ([ )  DGPLWH SULPLWLYH UH]XOW F IDGPLWHSULPLWLYH'DUIHVWHDGLWLY úLFRQIRUPFXSFWD UH]XOW I FRQWLQX  ,9&XP + D  HVWH VXEJUXS DO OXL * UH]XOW  F  ∀  [ \ ∈ +D RE LQHP

+ (− )Q − & QQ−− 3( I ([ )) = QD I Q − (D )⋅ I ([ )+ β ∀ [∈5

FRQWLQX  'DF  I HVWH QHQXO  DWXQFL H[LVW  D ∈ 5 DVWIHO vQFkW I (D ) ≠   ,GHQWLWDWHD GH PDL VXV GHYLQH   & Q − 3( I ([ + (Q − )D ))− & Q − 3( I ([ + (Q −  )D ))+ 

+ (− )Q − & QQ−− 3([ ) = QD E Q − [ + β ∀ [ ∈5'DF IHVWHQXO DWXQFLIHVWH

∈5 ∆3HVWHXQSROLQRPGHJUDGFHOPXOWQ±$QDORJ ∆ 3 = ∆(∆3 ) YDIL XQSROLQRPGHJUDGFHOPXOWQ±úDPG$SOLFkQGRSHUD LD ∆GHQ±RUL SHQWUXE∈5DYHP & Q− 3([ + (Q − )E )− & Q − 3([ + (Q − )E )+ «

úL D ≠  &RQVWDW PF GDF GHILQLPRSHUD LD ∆3 = 3([ + D )− 3 ([ ) ∀ [∈

) ([ + \ )− ) ([ )− [I (\ ) = ) (\ )− ) ( ) ∀ [\∈5 ) ([ + \ )− ) ([ )− ) (\ )+ ) ( ) I (\ ) =  ∀  [ ∈ 5  ∀  \ ∈ 5 FDUH HVWH [ FRQWLQX vQ\  E &RQVLGHU PSROLQRPXO 3([ ) = D [ Q + D [ Q − +  + DQ ∀ [∈5Q≥

1RW P X′\ ([ ) = ϕ(\ )∀ [∈5$YHP X \ ([ ) = X \ () 'HFL

[ ∈ 5 X′\ ([ ) = I ([ + \ )− I ([ )− I (\ ) =  ∀  [ ∈ 5 ⇒ X \ ([ ) = FRQVWDQW 

∫

D

DUFWJ [ G[ úLSULQXUPDUH  [  + D[ +  [ + D[ +  D D DUFWJ [ G[  ∫ [  + D[ + G[ =  (DUFWJD )∫ [  + D[ +   ,,,D )LH)RSULPLWLY DOXLI'HILQLP X \ ([ ) = ) ([ + \ )− ) ([ )− [I (\ )∀ 

[−D RE LQHP ,  (D ) =  + D[

DUFWJ [ − DUFWJD

\=

 

}

úL FRQVLGHU XQ [ ≠ H  [ ∈ +  [ ⋅ D ∉ + ⇒  [ ⋅ D = D  VDX [ ⋅ D = D −   [ = H ⇒ IDOV5 PkQHF  D  = D − ⇒ D  = H   [ = D −  = H

{

 &RQVLGHU P[∈+úLRE LQHP [ ⋅ D ∉ + DGLF  [ ⋅ D = D VDX [ ⋅ D = D − ÌQ SULPXOFD]RE LQHP[ HvQDOGRLOHD [ = D − = D IDOV  5 PkQH D  ∈ +  D  = D  ⋅ D ∉ + ⇔ D  ∈ D D −  'DF  D  = D ⇒ D  = H