Subiecte Etapa Zonala Ed2

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,  &ODVDD,9D

 



                   





D (IHFWXD LFDOFXOXO >±î î±î @±  E 5HFRQVWLWXL LDGXQDUHD $%&%&& $

S  S 

3HQWUXDFkúWLJDXQFRQFXUVSDUWLFLSDQ LLWUHEXLHV DWLQJ ILHFDUHMDORQQXPHURWDW GHODOD$U WD LSHGHVHQXODO WXUDWFHPXW ULWUHEXLHI FXWHSHQWUXFDWUDVHXO V ILHFkWPDLVFXUW S  































6XPDDWUHLQXPHUHHVWH'DF DG XJ PODILHFDUHDFHODúLQXP URE LQHP úL &DUHVXQWQXPHUHOH" S  3HWUX HVWH PDL PDUH GHFkW 3DXO FDUH HVWH PDL PDUH GHFkW $QD %LDQFD HVWH PDL PDUHGHFkW3DXO6DELQDHVWHPDLPLF GHFkW3HWUXúLPDLPDUHGHFkW%LDQFD &DUHHVWHFHOGHDOWUHLOHDvQRUGLQHDYkUVWHL" S 

1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXGHPLQXWH

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,  &ODVDD9D

         

 3HGRX UDIWXULVXQWvQWRWDOGHF U L'DF GHSHSULPXOUDIWVHPXW F U LSH DO GRLOHD UDIW DWXQFL SH SULPXO UDIW U PkQ GH GRX  RUL PDL PXOWH F U L GHFkW SH SULPXOUDIW&kWHF U LHUDXSHILHFDUHUDIW" S  S  D &DOFXOD L±± E 6XPDWXWXURUQXPHUHORUQDWXUDOHFDUHODvPS U LUHDFXGDXFkWXOHJDOFX S  6 VHDIOHFHOPDLPLFQXP UQDWXUDOGHIRUPD  D D  D N FDUHHVWHGHRULPDL PDUHGHFkWQXP UXO D D  D N  XQGH N ≥  

)LH[\]QXPHUHQDWXUDOHQHQXOHFXSURSULHWDWHDF  Â[Â\Â]  $IOD LFHDPDLPDUHYDORDUHSRVLELO DVXPHL[\]  &DUHVXQWQXPHUHOH[\úL]vQDFHVWFD]"             

  1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

S  S 

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,  &ODVDD9,D

 

D &DOFXOD L  





 



 





E  $U WD L F  QXP UXO Q    FRQVHFXWLYH

 









− 

S 









 HVWH SURGXVXO D GRX  QXPHUH QDWXUDOH S 

1XPHUHOH QDWXUDOH [ \ ] DX SURSULHWDWHD F  [ ± \ ± ]   'HPRQVWUD L F  QXP UXO[Â]\ VHGLYLGHFX S 

 

3HVHJPHQWXO>$%@VHFRQVLGHU SXQFWHOH0úL1FXSURSULHW LOH

$0  $1  = úL =  $%  $%  )LH 2 úL 3 PLMORDFHOH VHJPHQWHORU >01@ úL UHVSHFWLY >$%@ 'DF  23   FP GHWHUPLQD LOXQJLPHDVHJPHQWXOXL>$%@ S 

  

'HWHUPLQD L WRDWH SHUHFKLOH GH QXPHUH QDWXUDOH P Q  úWLLQG F  P  Q   úL S  FkWXOvPS U LULLFXUHVW DOOXLPODQHVWH

        



1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,  



  

       

&ODVDD9,,D  $IOD LWRDWHIUDF LLOHPDLPDULGHFkW FXSURSULHWDWHDF P ULQGQXP U WRUXOFX  XQ QXP U QDWXUDO úL vQPXO LQG QXPLWRUXO FX DFHODúL QXP U IUDF LD QXúL VFKLPE  YDORDUHD S     +  +  +       D  LQkQGFRQWHYHQWXDOGHIDSWXOF QQ± QDU WD LF  )LHVXPD6 

6

E ^6`

  

 XQGH^[`UHSUH]LQW SDUWHD]HFLPDO DOXL[ 

S  S  5HYLVWD$UKLPHGH 

6H FRQVLGHU  WULXQJKLXO $%& vQ FDUH 0 1 3 4 VXQW PLMORDFHOH VHJPHQWHORU >%&@>$0@>%1@UHVSHFWLY>$3@   D  $IOD LUDSRUWXODULLORUSDWUXODWHUXOXL0143úLWULXQJKLXOXL$%& S  E  'DF  vQ SOXV WULXQJKLXO $%& HVWH LVRVFHO GH ED]  %& DU WD L F  WULXQJKLXO 013HVWHLVRVFHO S  D  'HPRQVWUD LF XQSDWUXODWHUFRQYH[úLRUWRGLDJRQDOFXGLDJRQDOHOHSHUSHQGL FXODUH HVWHURPEGDF úLQXPDLGDF SXQFWXOGHLQWHUVHF LHDOGLDJRQDOHORUHVWH HJDOGHS UWDWGHPLMORDFHOHODWXULORUSDWUXODWHUXOXL S  E )LH0XQSXQFWvQLQWHULRUXOXQXLS WUDW$%&'$WXQFLVHJPHQWHOH>0$@>0%@ >0&@úL>0'@VXQWODWXULOHXQXLSDWUXODWHUFRQYH[ S        

1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,  

&ODVDD9,,,D [+ \ ≥ [\ &kQGDUHORFHJDOLWDWHD" 



D  'DF [\∈∞ GHPRQVWUD LF 



E 'DF [\]∈∞ GHPRQVWUD LF [\ \] ][ ![\]

 

 [S S +   S +  −  S −  S S +  ]RULFDUHDUILS∈5 

D  S S +  =



E  6 ∈5?4XQGH6   ⋅  +  ⋅  +  ⋅  +  +  ⋅  





S 

'HPRQVWUD LF 



 

S 

S  S 

ÌQWUXQFHUFHVWHvQVFULVXQSDWUXODWHU$%&'ùWLLQGF $% %& FPGLDJRQD OHOH $& úL %' VXQW SHUSHQGLFXODUH úL $& HVWH QXP U vQWUHJ VH FHUH SHULPHWUXO SDWUXODWHUXOXL$%&' S  5HYLVWD$UKLPHGH  )LH9$%&SXQFWHQHFRSODQDUHúL,,,FHQWUHOHFHUFXULORUvQVFULVHvQ¨9$% ¨9%&UHVSHFWLY¨9$&6 VHGHPRQVWUH]HF ,,, ||$%& GDF úLQXPDLGDF 

9$ + 9% 9% + 9& 9$ + 9& = =  $% %& $&             

1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

S 

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,          

&ODVDD,;D )LHHFXD LDvQQHFXQRVFXWD[ [  + P[ + P  − P −  =  XQGHPHVWHXQSDUDPHWUX

UHDO D  3HQWUXFHYDORULDOHSDUDPHWUXOXLPHFXD LDDGPLWHU G FLQLUHDOH"

   E 6 VHDUDWHF GDF [∈5HVWHRVROX LHDHFXD LHLDWXQFL[∈ −     

)LHQXPHUHOHUHDOH[\] D  6 VHDUDWHF GDF |[||\|≤DWXQFL|[\|≤úL|[±\|≤(VWHDGHY UDWúL

UHFLSURF" S      E 'DF [\]VXQWVWULFWSR]LWLYHúLDXSURSULHWDWHDF [ \ ]   DU WD LF  H[LVW UHOD LLOH     ≥  \ + ] úL ([ + \ + ] ) ≤ + +  [ \ ] [



)LHIXQF LDI>∞ →5I[    +  + [ 



E 6 VHDUDWHF IXQF LDIHVWHVWULFWFUHVF WRDUH



S 

F  3HQWUX FH YDORUL DOH SDUDPHWUXOXL P HFXD LD DGPLWH DPEHOH U G FLQL vQWUHJL GLVWLQFWH" S 

[+ 

S 

D  6 VHDUDWHF H[LVW YDORUL[∈5SHQWUXFDUHI[ ∈4

S 

S 

S 

F  $U WD LF DQXQGHDQ   +  +  +  +  +  úLQ∈1   Q UDGLFDOL

S 



G &DOFXOD LSDUWHDvQWUHDJ DQXP UXOXLDQQ∈1 



)LH D E F QXPHUH UHDOH FX SURSULHW LOH D ≥ E ≥   úL D ≥ F ≥   $WXQFL GHPRQ



D  D  = E  + F  





  

S 

VWUD LF XUP WRDUHOHDILUPD LLVXQWHFKLYDOHQWH

E   D + E + D − E  D + F + D − F =  D + E + F 

1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

S 

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,   

&ODVDD;D D  $U WD LF  OJ  ⋅ OJ  = OJ  ⋅ OJ  úL OJ  ⋅ OJ  ⋅ OJ  > OJ  ⋅ OJ  ⋅ OJ  



E 'DF DEF∈∞ VXQWvQSURJUHVLHJHRPHWULF DWXQFL



F  'DF DEF∈∞ VXQWvQSURJUHVLHDULWPHWLF DWXQFL

  

OJ  E ≥ OJ D ⋅ OJ F 

S 

OJ  E ≥ OJ D ⋅ OJ F 

S 

D  6  VH DUDWHF  H[LVW  R SURJUHVLHJHRPHWULF LQILQLW QHFRQVWDQW GLQ FXEXUL GHQXPHUHQDWXUDOH S  E 6 VHGHPRQVWUH]HF QXH[LVW RSURJUHVLHDULWPHWLF LQILQLW QHFRQVWDQW FX WRDWHHOHPHQWHOHFXEXULGHQXPHUHQDWXUDOH S 



F  $U WD LF [  [ +  [ +  [ ∀ [∈5



)LHI5→5I[  

   

   

S 





S 



[ +  [ + [ + D  6 VHFDOFXOH]HI5  E 6 VHDUDWHF IQXHVWHLQMHFWLY  F  )LHJ>∞ →5J[  I[ SHQWUX[≥$WXQFLJHVWHLQMHFWLY  

6 VHGHPRQVWUH]HF GDF [\]!úL[\\]][ DWXQFL D 

[\]

 + [   + \   + ] 

≤

  

E &kQGVHUHDOL]HD] PD[LPXO"

S  S 

5HYLVWD$UKLPHGH 

     

S  S  S 

1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,   

&ODVDD;,D D  )LH$%∈0& 6 VHFDOFXOH]H7U$%±%$ 

S 

7U$%$ úL7U$%$ 

S 

7U$ úL7U$ 

S 



E 'DF $%∈0& FXSURSULHWDWHDF $  2V VHFDOFXOH]H



F  )LH$∈05 DVWIHOvQFkW$ $, 26 VHFDOFXOH]H



'DF $∈0Q& DWXQFL7U$ VXPDHOHPHQWHORUGHSHGLDJRQDODSULQFLSDO 



D  )LHúLUXOGHQXPHUHUHDOH[Q Q≥FXSURSULHWDWHD[Q   [ Q −  ∀ Q∈1



  







6 VHGHPRQVWUH]HF [∈>∞ 

E 3HQWUXD!VHFRQVLGHU úLUXULOHGHQXPHUHUHDOH

[ Q = Q D

 Q+

 Q

−  Q≥ \ Q = Q D − D 

&DOFXOD L OLP [Q úL OLP \Q  Q →∞

 Q +

S 

Q≥

Q →∞

S 

)LHDEF∈1 6 VHGHPRQVWUH]HF GHWHUPLQDQWXO

     D E +  F +        HVWHQHQXO E +  E D +       F +  D +  F





   + +  + Q≥ ⋅   ⋅  QQ + 



D  6HFRQVLGHU úLUXO[Q Q≥[Q 



 6 VHGHPRQVWUH]HF úLUXO[Q Q≥HVWHFUHVF WRUúLV VHFDOFXOH]HOLPLWDVD



E )LHS≥XQQXP UQDWXUDOúLDQ Q≥XQúLUGHQXPHUHUHDOHFXSURSULHWDWHDF  DN Q≥ N = N

S 

S 

DQ + S = DQ SHQWUXQ≥6HFRQVLGHU úLUXO VQ = ∑ Q

  



 6 VHDUDWHF XUP WRDUHOHDILUPD LLVXQWHFKLYDOHQWH  L  úLUXOVQ Q≥HVWHFRQYHUJHQW  LL DDDS 

1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

S 

&RQFXUVXOGHPDWHPDWLFáDOUHYLVWHL$UKLPHGH (GLêLDD,,D (WDSD]RQDOá;,    

&ODVDD;,,D 6 VHFDOFXOH]H D 

∫



E  ∫



F 



G  ∫

  





      





 +  [ − [  G[ [∈ 

OQ [ + OQ [ OQ [









S 

G[ [∈∞ 

 + VLQ [ G[

S 

∫ VLQ [ + FRV [ +   [ ∈      π

S 

 + FRV [ G[  π  [ ∈     VLQ [ + FRV [ +   

S 



)LHQ≥XQQXP UQDWXUDO

8Q {]∈&]Q }úL+Q {]∈&]Q ]N≠∀ N∈{«Q±}} D  6 VHGHPRQVWUH]HF 8Q⋅ HVWHJUXSDEHOLDQ

E 6 VHGHWHUPLQH8úL+∪{`

S 

S 

F  6 VHGHPRQVWUH]HF 8Q +Q∪{}GDF úLQXPDLGDF QHVWHQXP USULP )LHIJK5→5WUHLIXQF LL6 VHGHPRQVWUH]HF 



D  'DF IDGPLWHSULPLWLYHDWXQFLIXQF LLOH J[  I[ ÂFRV[∀ [∈5úLK[  I[ ÂVLQ[∀ [∈5  DGPLWSULPLWLYH

E 'DF  IXQF LLOH X[   I [  ⋅ FRV [ [ ∈ 5 úL Y[   I [  ⋅ VLQ [ [ ∈ 5 DGPLW 

SULPLWLYHDWXQFLIXQF LDIDGPLWHSULPLWLYH



)LH* ^DDEE`DE∈4DE≠}6 VHGHPRQVWUH]HF 

D  *⋅ HVWHJUXSDEHOLDQ

E ∉*

      1RW  7RDWHVXELHFWHOHVXQWREOLJDWRULL)LHFDUHVXELHFWVHQRWHD] GHODODS  7LPSGHOXFUXRUH

S 

S 

S 

S 

S