Solutii Etapa Locala Suceava
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&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD,,,D 1XPHUHOHVXQW D ± D ± D D D îD î îD îD î îD D D D E D î D E îVDX± E
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD,9D ± ^î>î D± @` ± ^î> D± @` ^î> D± @` ± î> D± @ D± D± D± D D ± D ⋅ E F DEF DDD D D E F HOHYLúLE QFL
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD9D
$YHP Q = ⋅ = ⋅ + GHFLFkWXOHVWHLDUUHVWXOHVWH
$YHP = ⋅ = ( + )⋅ = ( ⋅ ) + ( ⋅ )
6HQRWHD] 1 = DDD D
1 = FLIUH
FLIUH
+ =
&XPD≠úLD ⇒D
$WXQFL
+ + + + + , + = + + + + , +=
= + + + + = = + = = ⋅+ = ⋅
( )
&kWHFLIUHDUHQXP UXO 1 = FLIUH
FLIUH
= ⋅ =
FLIUH
1XP U P]HURXULOHDG XJDWHGXS QXPHUHOHQDWXUDOH « ⋅ ⋅ 1XP U PFLIUHOHRE LQXWHGLQQXPHUHOHQDWXUDOHGHRFLIU « 1XP U PFLIUHOHRE LQXWHGLQQXPHUHOHQDWXUDOHGHGRX FLIUH « ⋅ 1XP U PFLIUHOHRE LQXWHGLQQXPHUHOHQDWXUDOHGHWUHLFLIUH « ⋅ 'HFL1DUH⋅ ⋅ ⋅FLIUH
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD9,D Q )RORVLQG IDSWXO F GHVFRPSXQHP ILHFDUH WHUPHQ DO VXPHL GDWH úL YRP = − N (N + Q ) N N + Q
− + − + − + − + − 5HGXFkQG WHUPHQLL DVHPHQHD Q= − ⇒Q= YDORDUHFHYHULILF LQHJDOLWDWHD < < ⇔ Q ∈ 'HRDUHFH [ ≥ ∀[ ∈ 1 ⇒ ∀[ ∈ 1 'DU /
RE LQH
Q=
'LQUHOD LLOH úL ⇒ \ / ⇔ / ⇔ \ / 'DU \ ∀ \ ≠ 'LQUHOD LLOH úL ⇒ \ = ⇔ \
J VLP
( )
\
3HQWUX\ HFXD LDGHYLQH + = ⇔ + = ⇔ = ⇔ = ⇔ [ = ⇔ ⇔[ ' & $ 2 % [ \ ] $ 2E LQHP = = = = $ P $2& °GHFLP VXUDXQJKLXOXLIRUPDW GHELVHFWRDUHOH XQJKLXULORU$2&úL'2%HVWH $ 'LQ $ $ = $ $ ⇒ $ PLMORFSHQWUX [ $ $ ] [
[
$ $ = $ $ = ( + + + ) = ⋅
[
[
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD9,,D [ [ ( + + + ) = ( + + ) + [ [ ⋅ [ ⋅ = ⋅ + ⋅ = [ = 3( 3& ÌQ ∆3)% FRQIRUPWHRUHPHLOXL7KDOHV ⇒ = ∆'2( LVRVFHO ⇒ ∠'2( ≡ ∠'(2 () %& GDU ∠'(2 ≡ ∠3(& RSXVHODYkUI ∠'2( ≡ ∠3(& $& ∠(2& ≡ ∠(3& DXFRPSOHPHQWHFRQJUXHQWH± ⇒ ∆&32 LVRVFHO ⇒ 3& = &2 = ∆'2( ≡ ∆%2) ⇒ () = ⋅ 2( 'LQ úL ⇒ 3( ⋅ %& = 2( ⋅ $& S T U 3XWHPSUHVXSXQHF S ≤ T ≤ U DYHPVXFFHVLY + + = Q ∈ ⇔ T U S S U + T S + U T = QSTU ⇔ T TS + U = SU (QT − S )GHFL T SU (QT − S ) &XP T HVWHSULPúLFXP
(
)
S U VXQWSULPHUH]XOW F DYHPGHDQDOL]DWXUP WRDUHOHFD]XUL S U D T S FXP S T VXQWSULPHGHGXFHP S = T GHXQGH + = Q ∈ VDX S + U = PUS P ∈ U S 'HFL U = PUS − S = S(PU − S )DGLF VDX U S VDX U PU − S FHHDFHFRQGXFHOD U S DGLF
S = T = U E T U VHGHGXFHSULQWUXQUD LRQDPHQWDQDORJF S = T = U
F T QT − S VHRE LQHLPHGLDW T S FDUHVHUHGXFHWRWODSXQFWXO D
K K 0( 0* = = ⇒ 0( = $QDORJ 1) = 01 0' * ( E D * ( *( *( E = ⇒ * ( = * ) = = ⇒ = '1 * ) *) () D + E (D + E )K K KE 0( = *( = ⇒ 0( = (D + E ) (D + E ) * ( ⊥ 01 * ( ⊥ 01
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD
D + E = N ∈ DWXQFL D + E = N D + E + DE DGLF D+E D + E = N DE GH XQGH − N > DGLF VLQJXUD SRVLELOLWDWH HVWH N = FHHD FH
'DF
( − N )(
&ODVDD9,,,D
(
)
(
)
(
)
)
D + E D + E FRQGXFHOD = DGLF D = E ∈ IUDF LDILLQGDúDGDUXQLWDU vQFD]XOGHWHUPLQDW D 6HFRQVWDW XúRUF V N ∈ {}GHFL V ≥ GHRDUHFHSHQWUX D = D = = D = RE LQHP V = GHGXFHPV PLQLPXOYDORULORUOXL V HVWH E 9RPQRWDFXDQXP UXOGHQXPHUH V N FDUHVXQWHJDOHFXúLFXEQXP UXOGHQXPHUH V N FDUH VXQW HJDOH FX GHGXFHP F V = D + E GDU D + E = GH XQGH RE LQHP V = D − − D = D + QXP USDU F 9RP GHPRQVWUD vQ SUHDODELO DILUPD LD GDF H[LVW XQ N ∈ {} SHQWUX FDUH V N = DWXQFL V N + = XQ QXP U V N FDUH HVWH HJDO FX HVWH XUPDW GH XQXO HJDO FX ÌQ DGHY U GDF V N = UH]XOW F (D N D N + D N + )∈ {(−) (− −)} GH XQGH GHGXFHP F D N + + D N + = GHFL V N + = D N + + D N + − D N + = D N + = ÌQ FRQVHFLQ QXP UXO PD[LP GH QXPHUH GH WLS V N FDUH LDX
YDORDUHD HVWH FHOHODOWH QXPHUH OXkQG YDORDUHD GHFL V ≤ × + = 3HQWUX D = D = D = D = D = D = = D P + = D P + = = D = D = D = D = D = D = D = D = = D P + = D P + = = D = D = − RE LQHP V = GHFLPD[LPXOYDORULORUVXPHLHVWH &RQIRUP WHRUHPHL FHORU WUHL SHUSHQGLFXODUH G (0 &' ) = 0' = ⇒ $' =
G (0 %& ) = 0% = ⇒ $% = $4 ⊥ %' $4 = G ($ (0%' )) = $2 XQGH $2 ⊥ 04
04 = = 6HGHPRQVWUHD] F PXFKLLOHRSXVHGLQWHWUDHGUXVXQWSHUSHQGLFXODUH9RPDU WDGHH[HPSOXF %& ⊥ $' )LH + RUWRFHQWUXOWULXQJKLXOXL$%& {+ }= $( ∩ &) XQGH ( ∈ %& úL ) ∈ $% &XP '+ ⊥ ($%& ) úL %& ⊂ ($%& ) RE LQHP '+ ⊥ %& 'LQ $( ⊥ %& úL '+ ⊥ %& DYHP %& ⊥ ($( '+ )DGLF %& ⊥ ($'( )&XP $' ⊂ ($'( ) UH]XOW F %& ⊥ $' $QDORJVHDUDW F $& ⊥ %' úL $% ⊥ '& 6H GHPRQVWUHD] F XQJKLXULOH SODQH FX YkUIXO vQ ' VXQW GUHSWH 9RP DU WD GH H[HPSOX F '& ⊥ $' 'LQLSRWH] DYHP %' ⊥ '& LDUPDLVXVDPDU WDWF $% ⊥ '& 'HDLFLRE LQHPF '& ⊥ (%' $% ) DGLF '& ⊥ ($%' ) FHHD FH LPSOLF '& ⊥ $' ÌQ PRG DQDORJ VH DUDW F $' ⊥ '% 'HFL WULXQJKLXULOH $'% $'& úL %'& VXQW GUHSWH vQ ' úL DSOLFkQG WHRUHPD OXL 3LWDJRUD RE LQHP UHOD LLOH $% = '$ + '% $& = '$ + '& &% = '& + '% SH FDUH GDF OH DGXQ P úLFXP ∆0$4 HVWHGUHSWXQJKLFLVRVFHODYHP $2 =
PHPEUX FX PHPEUX RE LQHP $% + $& + &% = '$ + '% + '& FDUH HVWH HFKLYDOHQW FX $% + $& + &% = '$ + '% + '&
(
) (
(
)
)
$WXQFLLQHJDOLWDWHDGLQHQXQ FDS W IRUPD ($% + $& + &% ) ≤ $% + $& + &% ⇔ $% + $& + &% − $% ⋅ $& − $% ⋅ %& − $& ⋅ %& ≥ ⇔ ($% − $& ) + ($% − %& ) + ($& − %& ) ≥ FDUH HVWH DGHY UDW GHRDUHFH ([ − \ ) ≥ (∀)[ \ (JDOLWDWHDDUHORFGDF $% = $& = %& DGLF WULXQJKLXO$%&HVWHHFKLODWHUDO
(
)
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD,;D &XP [ = [[ ]+ {[}LQHFXD LDGHYLQH [[ ] − ([[]+ {[}− )[[ ] ≤ − {[}VDX
[[] − [[] − {[}⋅ [[] + [[] ≤ − {[} VDX vQF [[] ( − {[}) ≤ − {[} GH XQGH ( − {[})([[] − )≤ &XP − {[}> LQHFXD LDHVWHHFKLYDOHQW FX [[ ] − ≤ GHXQGH [[ ]∈ [− ]∩ = = {− } L 'DF [[ ] = − DWXQFL [ ∈ [− ) LL 'DF [[ ] = DWXQFL [ ∈ [) LLL 'DF [[ ] = DWXQFL [ ∈ [ ) ÌQFRQVHFLQ [ ∈ [− )∪ [)∪ [ ) = [− ) 'LQ [ + \ + ] = [ \ [ > GHGXFHP [ \ ] ∈ () $YHPVXFFHVLY
+ ≥ = + ≥ = [ + \ \ + ] [ + \ + ] + \ \ + ] ] + [ [ + \ + ] + ] + ≥ = GHXQGHGXS vQVXPDUHRE LQHP ] + [ [ + \ [ + \ + ] + [ + + + + + + ≥ + + ≥ VDX − [ − \ − ] [+ \ \+] ]+ [ + [ + \ + ] + [ + \ + ] 2EVHUYD LH'DF [ + \ + ] < LQHJDOLWDWHDQXHVWHvQGHSOLQLW GXS FXPVHSRDWHYHGHDGDF OX P GHH[HPSOX [ = \ = ] = DWXQFL + + = + + = ≠ + + + − − − (∀)D Q Q ≥ WUHEXLHFD D Q V ILHvQSURJUHVLH ⇔ U D Q (D Q − ) ⇔ U D (D − )'HFLYRUILDWkWHD
SURJUHVLLFk LGLYL]RULQDWXUDOLDUH ⋅ = ⋅ ⋅ DGLF VXQWGHSURJUHVLL
$U W PF SHQWUXRULFH U ∈ GLYL]RUDOQXP UXOXL ⋅ H[LVW RSURJUHVLHDULWPHWLF FHVDWLVIDFH FRQGL LDGLQHQXQ )LH D GLQSURJUHVLH ⇒ (∃)N ∈ DVWIHOvQFkW D = + NU
D = ( + NU )
S=
DSDU LQ
SURJUHVLHL
⇔ (∃)S ∈
( + NU ) − = ⋅ + N + N λ ∈ U
U
DVWIHO
vQFkW
( + NU ) = + Sλ
⇔
,$ & $ % $ &$ %$ E + F = + = + = XQGH ,$ & % % & &% %& D FXDEFDPQRWDWOXQJLPLOHODWXULORUWULXQJKLXULORU$QDORJPDLVFULHPvQF GRX UHOD LLGHDFHVWWLS ,$ ,% ,&
D E F úL RE LQHP + + = + + ≥ FX HJDOLWDWH SHQWUX FD]XO XQXL WULXQJKL ,$ ,% ,& E + F D + F D + E HFKLODWHUDO 'LQWHRUHPD9DQ$XEHOúLWHRUHPDELVHFWRDUHLDYHP
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD;D
DE
EF
DE EF + D
$SOLFkQGLQHJDOLWDWHDPHGLLORURE LQHP F + D ≥ F EF DE EF FD DEF E D F D E + ≥ ⋅ ⇒ + + ≥ D + E + F ∑ ∑
≥ E = ⋅ E ⇒
Q + + + = Q + Q + Q + [
(FXD LDVHVFULH I ([ ) =
[
[
Q + Q + + + + = J ([ ) XQGHIúLJGHILQHVFIXQF LLVWULFWPRQRWRQHFXFDUDFWHUGH Q + Q + PRQRWRQLHGLIHULW&XP I () = J ()úLHFXD LDDUHFHOPXOWRVROX LH[ UHSUH]LQW VLQJXUDU G FLQ 6HDSOLF LQHJDOLWDWHDPHGLLORU &D]XO, ] ≥ [
[
$SOLFkQGLQHJDOLWDWHD D + E ≤ D + E (∀)D E ∈ & RE LQHP + ] + + ] + + ] = = + ] + − − ] + + ] ≥ ( + ] ) + − − ] + + ] = ] − ] + + ] =
(
)
= ] ⋅ − ] + + ] ≥ − ] + + ] ≥ − ] + + ] =
&D]XO,, ] ≤
(
) (
)
$SOLFkQGDFHHDúLLQHJDOLWDWHSXWHPVFULH + ] + + ] + + ] ≥ + ] + ] ⋅ + ] + + ] = + ] + ] + ] + − ] − ≥
≥ + ] + ] + ] + − ] − = + ] + − ] ≥ + ] + − ] =
(
) (
)
$VHP Q WRUVHDUDW FDPDLVXVF GDF Q P ∈ DWXQFL + ] Q + + ] P + + ] Q + P ≥ (∀)] ∈ &
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD;,D
'LQ $ = , Q ⇒ $ − , Q = Q ⇒ ($ − , Q ) $ + $ + , Q = Q úLFXPH[LVW −
($ − , Q )
(
⇒ $ + $ + , Q = Q
)
'LQ $ = , Q ⇒ $ + , Q = , Q ⇒ ($ + , Q ) $ − $ + , Q = , Q ⇒ GHW ($ + , Q )⋅ GHW $ − $ + , Q = Q
(
'LQ $ = , Q ⇒ GHW $ =
)
(
)
'LQ ⇒ $ ( $ + , Q ) = − , Q ⇒ GHW $ ⋅ GHW ($ + , Q ) = (− ) úLIRORVLQG ⇒ GHW ($ + , Q ) = (− ) Q
Q
'LQ úL ⇒ GHW ($ − $ + , Q ) = (− ) ⋅ Q D + EF E(D + G ) D E VXPD HOHPHQWHORU PDWULFHL $ HVWH DWXQFL $ = 'DF $ = F G F(D + G ) EF + G Q
6 = D + EF + (D + G )(E + F ) + EF + G = (D + G )(E + F ) + D + DG + G + EF − DG =
= (D + G )(E + F ) + (D + G ) − GHW ($) = (D + G )(D + E + F + G ) − GHW ($) &XP D + E + F + G 6 GHGXFHPF D + E + F + G GHW ($) DGLF D + E + F + G 'HRDUHFH D E F G
VXQWQXPHUHQDWXUDOHVHGHGXFHF D + E + F + G ∈ {}HWF 3UHVXSXQHP F Q Q ⋅ [ Q Q≥ QX DUH OLPLWD (∃)D D > DVWIHO FD SHQWUX R LQILQLWDWH GH Q
(
Q Q
Q
⋅ [ Q > D VDX [ Q >
[Q >
D
Q
)
Q
VDX
Q
[Q >
Q
Q
D
Q
&XP OLP Q →∞
FDUHFRQWUD]LFHIDSWXOF (\ Q )HVWHFRQYHUJHQW
Q
D
Q
= ⇒ 3HQWUX R LQILQLWDWH GH Q
[ = ⇒ ([ Q )QXHPRQRWRQ N − Q (− ) úL OLP [ = E 6HDUDW SULQLQGXF LHF [ Q = ∑ Q Q →∞ N H N =
D 6HFDOFXOHD] [ = [ = [ =
Q
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD;,,D
'HPRQVWU P F RSHUD LD HVWH ELQH GHILQLW SXWHP VFULH $ ∗ % = ($ − , )(% − , ) + , GH XQGH GHGXFHP F GHW ( $ ∗ % − , ) = GHW (( $ − , )( % − , )) = GHW ( $ − , )⋅ GHW ( % − , ) ≠ GHFL RSHUD LD HVWHELQHGHILQLW 6HGHPRQVWUHD] XúRUF RSHUD LDHVWHDVRFLDWLY úLF HOHPHQWXOQHXWUXHVWH ,
3HQWUXVLPHWULFXOHOHPHQWXOXL $ ∈ (0 ∗)SURFHG PDVWIHO SXQHP $ = , + 8 FX 8 ∈ */ GH
XQGHSULQFDOFXOGHGXFHPF $ = 8 − (, + 8 ) = , + 8 − ∈ 0 Q 6HDUDW XúRUF − [ ∈ 0 (∀)[ ∈ 0 úLSULQLQGXF LHPDWHPDWLF F ( − [ ) = − [ (∀)Q ∈ GH XQGHGHGXFHPF [ + ( − [ ) = ∈ 0 Q
D
E
(YLGHQW IXQF LLOH J E úL I D VXQW SULPLWLYDOH úL GLQ UHOD LLOH GLQ HQXQ UH]XOW F I úL J VXQW D
E
GHULYDELOH úL I = DJ E J = EI D 'LQ UHOD LLOH UH]XOW F EI ⋅ I
E D
D
= D ⋅ J ⋅ J E ⇒
D +E E + D + D +E EI D DJ E D I DE = ⇒ (∃)N ∈ DVWIHOvQFkW ⋅ DE = J E + N úLFXP D+E D+E D E + + D E
I ( ) = J ( ) = ⇒ N = ⇒ I
D
E
= J
'LQ úL UH]XOW F I = DI ⇒ I ([ )⋅ H − D[ = (∀)[ ∈ ⇒ (∃)U ∈ DVWIHO vQFkW I ([ ) = UH D[ úL FXP I ( ) = ⇒ U = ⇒ I ([ ) = H D[ úLGLQ ⇒ J ([ ) = H E[ 6HREVHUY F IXQF LLOHJ VLWHVDWLVIDF UHOD LLOHGLQHQXQ FRV [ ≠ úL J ([ ) = [ + QHQXO SH &RQVLGHU P IXQF LLOH I J → I ([ ) = [ [ =
(
)
I DGPLWHSULPLWLYHúLILH ) → RSULPLWLY DVD J GHULYDELO úLFXGHULYDWDFRQWLQX SH vQ QRW SOXV = K([ ) = DGPLWH SULPLWLYH úL ILH + ([ ) = DUFWJ[ + & R SULPLWLY &RQVLGHU P J ([ ) [ + * → GDW GH * ([ ) = J ([ )⋅ ) (+ ([ ))3XWHPVFULH * ([ ) = J ([ )⋅ ) (+ ([ )) + J ([ )⋅ ) (+ ([ ))⋅ + ([ ) = J ([ )⋅ ) (+ ([ )) + J ([ )⋅ I (+ ([ ))⋅ (∀)[ ∈ J ([ ) DGLF * ([ ) = [ ⋅ ) (+ ([ )) + I (+ ([ )) GH XQGH I (+ ([ )) = * ([ ) − [ ⋅ ) (+ ([ )) (∀)[ ∈ &XP * FkWúL [ ⋅ ) (+ ([ ))DGPLWSULPLWLYHUH]XOW F IXQF LD I $ + DUHSULPLWLYH
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD,9D ± ^î>î D± @` ± ^î> D± @` ^î> D± @` ± î> D± @ D± D± D± D D ± D ⋅ E F DEF DDD D D E F HOHYLúLE QFL
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD9D
$YHP Q = ⋅ = ⋅ + GHFLFkWXOHVWHLDUUHVWXOHVWH
$YHP = ⋅ = ( + )⋅ = ( ⋅ ) + ( ⋅ )
6HQRWHD] 1 = DDD D
1 = FLIUH
FLIUH
+ =
&XPD≠úLD ⇒D
$WXQFL
+ + + + + , + = + + + + , +=
= + + + + = = + = = ⋅+ = ⋅
( )
&kWHFLIUHDUHQXP UXO 1 = FLIUH
FLIUH
= ⋅ =
FLIUH
1XP U P]HURXULOHDG XJDWHGXS QXPHUHOHQDWXUDOH « ⋅ ⋅ 1XP U PFLIUHOHRE LQXWHGLQQXPHUHOHQDWXUDOHGHRFLIU « 1XP U PFLIUHOHRE LQXWHGLQQXPHUHOHQDWXUDOHGHGRX FLIUH « ⋅ 1XP U PFLIUHOHRE LQXWHGLQQXPHUHOHQDWXUDOHGHWUHLFLIUH « ⋅ 'HFL1DUH⋅ ⋅ ⋅FLIUH
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD9,D Q )RORVLQG IDSWXO F GHVFRPSXQHP ILHFDUH WHUPHQ DO VXPHL GDWH úL YRP = − N (N + Q ) N N + Q
− + − + − + − + − 5HGXFkQG WHUPHQLL DVHPHQHD Q= − ⇒Q= YDORDUHFHYHULILF LQHJDOLWDWHD < < ⇔ Q ∈ 'HRDUHFH [ ≥ ∀[ ∈ 1 ⇒ ∀[ ∈ 1 'DU /
RE LQH
Q=
'LQUHOD LLOH úL ⇒ \ / ⇔ / ⇔ \ / 'DU \ ∀ \ ≠ 'LQUHOD LLOH úL ⇒ \ = ⇔ \
J VLP
( )
\
3HQWUX\ HFXD LDGHYLQH + = ⇔ + = ⇔ = ⇔ = ⇔ [ = ⇔ ⇔[ ' & $ 2 % [ \ ] $ 2E LQHP = = = = $ P $2& °GHFLP VXUDXQJKLXOXLIRUPDW GHELVHFWRDUHOH XQJKLXULORU$2&úL'2%HVWH $ 'LQ $ $ = $ $ ⇒ $ PLMORFSHQWUX [ $ $ ] [
[
$ $ = $ $ = ( + + + ) = ⋅
[
[
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD9,,D [ [ ( + + + ) = ( + + ) + [ [ ⋅ [ ⋅ = ⋅ + ⋅ = [ = 3( 3& ÌQ ∆3)% FRQIRUPWHRUHPHLOXL7KDOHV ⇒ = ∆'2( LVRVFHO ⇒ ∠'2( ≡ ∠'(2 () %& GDU ∠'(2 ≡ ∠3(& RSXVHODYkUI ∠'2( ≡ ∠3(& $& ∠(2& ≡ ∠(3& DXFRPSOHPHQWHFRQJUXHQWH± ⇒ ∆&32 LVRVFHO ⇒ 3& = &2 = ∆'2( ≡ ∆%2) ⇒ () = ⋅ 2( 'LQ úL ⇒ 3( ⋅ %& = 2( ⋅ $& S T U 3XWHPSUHVXSXQHF S ≤ T ≤ U DYHPVXFFHVLY + + = Q ∈ ⇔ T U S S U + T S + U T = QSTU ⇔ T TS + U = SU (QT − S )GHFL T SU (QT − S ) &XP T HVWHSULPúLFXP
(
)
S U VXQWSULPHUH]XOW F DYHPGHDQDOL]DWXUP WRDUHOHFD]XUL S U D T S FXP S T VXQWSULPHGHGXFHP S = T GHXQGH + = Q ∈ VDX S + U = PUS P ∈ U S 'HFL U = PUS − S = S(PU − S )DGLF VDX U S VDX U PU − S FHHDFHFRQGXFHOD U S DGLF
S = T = U E T U VHGHGXFHSULQWUXQUD LRQDPHQWDQDORJF S = T = U
F T QT − S VHRE LQHLPHGLDW T S FDUHVHUHGXFHWRWODSXQFWXO D
K K 0( 0* = = ⇒ 0( = $QDORJ 1) = 01 0' * ( E D * ( *( *( E = ⇒ * ( = * ) = = ⇒ = '1 * ) *) () D + E (D + E )K K KE 0( = *( = ⇒ 0( = (D + E ) (D + E ) * ( ⊥ 01 * ( ⊥ 01
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD
D + E = N ∈ DWXQFL D + E = N D + E + DE DGLF D+E D + E = N DE GH XQGH − N > DGLF VLQJXUD SRVLELOLWDWH HVWH N = FHHD FH
'DF
( − N )(
&ODVDD9,,,D
(
)
(
)
(
)
)
D + E D + E FRQGXFHOD = DGLF D = E ∈ IUDF LDILLQGDúDGDUXQLWDU vQFD]XOGHWHUPLQDW D 6HFRQVWDW XúRUF V N ∈ {}GHFL V ≥ GHRDUHFHSHQWUX D = D = = D = RE LQHP V = GHGXFHPV PLQLPXOYDORULORUOXL V HVWH E 9RPQRWDFXDQXP UXOGHQXPHUH V N FDUHVXQWHJDOHFXúLFXEQXP UXOGHQXPHUH V N FDUH VXQW HJDOH FX GHGXFHP F V = D + E GDU D + E = GH XQGH RE LQHP V = D − − D = D + QXP USDU F 9RP GHPRQVWUD vQ SUHDODELO DILUPD LD GDF H[LVW XQ N ∈ {} SHQWUX FDUH V N = DWXQFL V N + = XQ QXP U V N FDUH HVWH HJDO FX HVWH XUPDW GH XQXO HJDO FX ÌQ DGHY U GDF V N = UH]XOW F (D N D N + D N + )∈ {(−) (− −)} GH XQGH GHGXFHP F D N + + D N + = GHFL V N + = D N + + D N + − D N + = D N + = ÌQ FRQVHFLQ QXP UXO PD[LP GH QXPHUH GH WLS V N FDUH LDX
YDORDUHD HVWH FHOHODOWH QXPHUH OXkQG YDORDUHD GHFL V ≤ × + = 3HQWUX D = D = D = D = D = D = = D P + = D P + = = D = D = D = D = D = D = D = D = = D P + = D P + = = D = D = − RE LQHP V = GHFLPD[LPXOYDORULORUVXPHLHVWH &RQIRUP WHRUHPHL FHORU WUHL SHUSHQGLFXODUH G (0 &' ) = 0' = ⇒ $' =
G (0 %& ) = 0% = ⇒ $% = $4 ⊥ %' $4 = G ($ (0%' )) = $2 XQGH $2 ⊥ 04
04 = = 6HGHPRQVWUHD] F PXFKLLOHRSXVHGLQWHWUDHGUXVXQWSHUSHQGLFXODUH9RPDU WDGHH[HPSOXF %& ⊥ $' )LH + RUWRFHQWUXOWULXQJKLXOXL$%& {+ }= $( ∩ &) XQGH ( ∈ %& úL ) ∈ $% &XP '+ ⊥ ($%& ) úL %& ⊂ ($%& ) RE LQHP '+ ⊥ %& 'LQ $( ⊥ %& úL '+ ⊥ %& DYHP %& ⊥ ($( '+ )DGLF %& ⊥ ($'( )&XP $' ⊂ ($'( ) UH]XOW F %& ⊥ $' $QDORJVHDUDW F $& ⊥ %' úL $% ⊥ '& 6H GHPRQVWUHD] F XQJKLXULOH SODQH FX YkUIXO vQ ' VXQW GUHSWH 9RP DU WD GH H[HPSOX F '& ⊥ $' 'LQLSRWH] DYHP %' ⊥ '& LDUPDLVXVDPDU WDWF $% ⊥ '& 'HDLFLRE LQHPF '& ⊥ (%' $% ) DGLF '& ⊥ ($%' ) FHHD FH LPSOLF '& ⊥ $' ÌQ PRG DQDORJ VH DUDW F $' ⊥ '% 'HFL WULXQJKLXULOH $'% $'& úL %'& VXQW GUHSWH vQ ' úL DSOLFkQG WHRUHPD OXL 3LWDJRUD RE LQHP UHOD LLOH $% = '$ + '% $& = '$ + '& &% = '& + '% SH FDUH GDF OH DGXQ P úLFXP ∆0$4 HVWHGUHSWXQJKLFLVRVFHODYHP $2 =
PHPEUX FX PHPEUX RE LQHP $% + $& + &% = '$ + '% + '& FDUH HVWH HFKLYDOHQW FX $% + $& + &% = '$ + '% + '&
(
) (
(
)
)
$WXQFLLQHJDOLWDWHDGLQHQXQ FDS W IRUPD ($% + $& + &% ) ≤ $% + $& + &% ⇔ $% + $& + &% − $% ⋅ $& − $% ⋅ %& − $& ⋅ %& ≥ ⇔ ($% − $& ) + ($% − %& ) + ($& − %& ) ≥ FDUH HVWH DGHY UDW GHRDUHFH ([ − \ ) ≥ (∀)[ \ (JDOLWDWHDDUHORFGDF $% = $& = %& DGLF WULXQJKLXO$%&HVWHHFKLODWHUDO
(
)
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD,;D &XP [ = [[ ]+ {[}LQHFXD LDGHYLQH [[ ] − ([[]+ {[}− )[[ ] ≤ − {[}VDX
[[] − [[] − {[}⋅ [[] + [[] ≤ − {[} VDX vQF [[] ( − {[}) ≤ − {[} GH XQGH ( − {[})([[] − )≤ &XP − {[}> LQHFXD LDHVWHHFKLYDOHQW FX [[ ] − ≤ GHXQGH [[ ]∈ [− ]∩ = = {− } L 'DF [[ ] = − DWXQFL [ ∈ [− ) LL 'DF [[ ] = DWXQFL [ ∈ [) LLL 'DF [[ ] = DWXQFL [ ∈ [ ) ÌQFRQVHFLQ [ ∈ [− )∪ [)∪ [ ) = [− ) 'LQ [ + \ + ] = [ \ [ > GHGXFHP [ \ ] ∈ () $YHPVXFFHVLY
+ ≥ = + ≥ = [ + \ \ + ] [ + \ + ] + \ \ + ] ] + [ [ + \ + ] + ] + ≥ = GHXQGHGXS vQVXPDUHRE LQHP ] + [ [ + \ [ + \ + ] + [ + + + + + + ≥ + + ≥ VDX − [ − \ − ] [+ \ \+] ]+ [ + [ + \ + ] + [ + \ + ] 2EVHUYD LH'DF [ + \ + ] < LQHJDOLWDWHDQXHVWHvQGHSOLQLW GXS FXPVHSRDWHYHGHDGDF OX P GHH[HPSOX [ = \ = ] = DWXQFL + + = + + = ≠ + + + − − − (∀)D Q Q ≥ WUHEXLHFD D Q V ILHvQSURJUHVLH ⇔ U D Q (D Q − ) ⇔ U D (D − )'HFLYRUILDWkWHD
SURJUHVLLFk LGLYL]RULQDWXUDOLDUH ⋅ = ⋅ ⋅ DGLF VXQWGHSURJUHVLL
$U W PF SHQWUXRULFH U ∈ GLYL]RUDOQXP UXOXL ⋅ H[LVW RSURJUHVLHDULWPHWLF FHVDWLVIDFH FRQGL LDGLQHQXQ )LH D GLQSURJUHVLH ⇒ (∃)N ∈ DVWIHOvQFkW D = + NU
D = ( + NU )
S=
DSDU LQ
SURJUHVLHL
⇔ (∃)S ∈
( + NU ) − = ⋅ + N + N λ ∈ U
U
DVWIHO
vQFkW
( + NU ) = + Sλ
⇔
,$ & $ % $ &$ %$ E + F = + = + = XQGH ,$ & % % & &% %& D FXDEFDPQRWDWOXQJLPLOHODWXULORUWULXQJKLXULORU$QDORJPDLVFULHPvQF GRX UHOD LLGHDFHVWWLS ,$ ,% ,&
D E F úL RE LQHP + + = + + ≥ FX HJDOLWDWH SHQWUX FD]XO XQXL WULXQJKL ,$ ,% ,& E + F D + F D + E HFKLODWHUDO 'LQWHRUHPD9DQ$XEHOúLWHRUHPDELVHFWRDUHLDYHP
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD;D
DE
EF
DE EF + D
$SOLFkQGLQHJDOLWDWHDPHGLLORURE LQHP F + D ≥ F EF DE EF FD DEF E D F D E + ≥ ⋅ ⇒ + + ≥ D + E + F ∑ ∑
≥ E = ⋅ E ⇒
Q + + + = Q + Q + Q + [
(FXD LDVHVFULH I ([ ) =
[
[
Q + Q + + + + = J ([ ) XQGHIúLJGHILQHVFIXQF LLVWULFWPRQRWRQHFXFDUDFWHUGH Q + Q + PRQRWRQLHGLIHULW&XP I () = J ()úLHFXD LDDUHFHOPXOWRVROX LH[ UHSUH]LQW VLQJXUDU G FLQ 6HDSOLF LQHJDOLWDWHDPHGLLORU &D]XO, ] ≥ [
[
$SOLFkQGLQHJDOLWDWHD D + E ≤ D + E (∀)D E ∈ & RE LQHP + ] + + ] + + ] = = + ] + − − ] + + ] ≥ ( + ] ) + − − ] + + ] = ] − ] + + ] =
(
)
= ] ⋅ − ] + + ] ≥ − ] + + ] ≥ − ] + + ] =
&D]XO,, ] ≤
(
) (
)
$SOLFkQGDFHHDúLLQHJDOLWDWHSXWHPVFULH + ] + + ] + + ] ≥ + ] + ] ⋅ + ] + + ] = + ] + ] + ] + − ] − ≥
≥ + ] + ] + ] + − ] − = + ] + − ] ≥ + ] + − ] =
(
) (
)
$VHP Q WRUVHDUDW FDPDLVXVF GDF Q P ∈ DWXQFL + ] Q + + ] P + + ] Q + P ≥ (∀)] ∈ &
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD;,D
'LQ $ = , Q ⇒ $ − , Q = Q ⇒ ($ − , Q ) $ + $ + , Q = Q úLFXPH[LVW −
($ − , Q )
(
⇒ $ + $ + , Q = Q
)
'LQ $ = , Q ⇒ $ + , Q = , Q ⇒ ($ + , Q ) $ − $ + , Q = , Q ⇒ GHW ($ + , Q )⋅ GHW $ − $ + , Q = Q
(
'LQ $ = , Q ⇒ GHW $ =
)
(
)
'LQ ⇒ $ ( $ + , Q ) = − , Q ⇒ GHW $ ⋅ GHW ($ + , Q ) = (− ) úLIRORVLQG ⇒ GHW ($ + , Q ) = (− ) Q
Q
'LQ úL ⇒ GHW ($ − $ + , Q ) = (− ) ⋅ Q D + EF E(D + G ) D E VXPD HOHPHQWHORU PDWULFHL $ HVWH DWXQFL $ = 'DF $ = F G F(D + G ) EF + G Q
6 = D + EF + (D + G )(E + F ) + EF + G = (D + G )(E + F ) + D + DG + G + EF − DG =
= (D + G )(E + F ) + (D + G ) − GHW ($) = (D + G )(D + E + F + G ) − GHW ($) &XP D + E + F + G 6 GHGXFHPF D + E + F + G GHW ($) DGLF D + E + F + G 'HRDUHFH D E F G
VXQWQXPHUHQDWXUDOHVHGHGXFHF D + E + F + G ∈ {}HWF 3UHVXSXQHP F Q Q ⋅ [ Q Q≥ QX DUH OLPLWD (∃)D D > DVWIHO FD SHQWUX R LQILQLWDWH GH Q
(
Q Q
Q
⋅ [ Q > D VDX [ Q >
[Q >
D
Q
)
Q
VDX
Q
[Q >
Q
Q
D
Q
&XP OLP Q →∞
FDUHFRQWUD]LFHIDSWXOF (\ Q )HVWHFRQYHUJHQW
Q
D
Q
= ⇒ 3HQWUX R LQILQLWDWH GH Q
[ = ⇒ ([ Q )QXHPRQRWRQ N − Q (− ) úL OLP [ = E 6HDUDW SULQLQGXF LHF [ Q = ∑ Q Q →∞ N H N =
D 6HFDOFXOHD] [ = [ = [ =
Q
&21&8568/1$ ,21$/$/5(9,67(,$5+,0('( (WDSDORFDO 6XFHDYD &ODVDD;,,D
'HPRQVWU P F RSHUD LD HVWH ELQH GHILQLW SXWHP VFULH $ ∗ % = ($ − , )(% − , ) + , GH XQGH GHGXFHP F GHW ( $ ∗ % − , ) = GHW (( $ − , )( % − , )) = GHW ( $ − , )⋅ GHW ( % − , ) ≠ GHFL RSHUD LD HVWHELQHGHILQLW 6HGHPRQVWUHD] XúRUF RSHUD LDHVWHDVRFLDWLY úLF HOHPHQWXOQHXWUXHVWH ,
3HQWUXVLPHWULFXOHOHPHQWXOXL $ ∈ (0 ∗)SURFHG PDVWIHO SXQHP $ = , + 8 FX 8 ∈ */ GH
XQGHSULQFDOFXOGHGXFHPF $ = 8 − (, + 8 ) = , + 8 − ∈ 0 Q 6HDUDW XúRUF − [ ∈ 0 (∀)[ ∈ 0 úLSULQLQGXF LHPDWHPDWLF F ( − [ ) = − [ (∀)Q ∈ GH XQGHGHGXFHPF [ + ( − [ ) = ∈ 0 Q
D
E
(YLGHQW IXQF LLOH J E úL I D VXQW SULPLWLYDOH úL GLQ UHOD LLOH GLQ HQXQ UH]XOW F I úL J VXQW D
E
GHULYDELOH úL I = DJ E J = EI D 'LQ UHOD LLOH UH]XOW F EI ⋅ I
E D
D
= D ⋅ J ⋅ J E ⇒
D +E E + D + D +E EI D DJ E D I DE = ⇒ (∃)N ∈ DVWIHOvQFkW ⋅ DE = J E + N úLFXP D+E D+E D E + + D E
I ( ) = J ( ) = ⇒ N = ⇒ I
D
E
= J
'LQ úL UH]XOW F I = DI ⇒ I ([ )⋅ H − D[ = (∀)[ ∈ ⇒ (∃)U ∈ DVWIHO vQFkW I ([ ) = UH D[ úL FXP I ( ) = ⇒ U = ⇒ I ([ ) = H D[ úLGLQ ⇒ J ([ ) = H E[ 6HREVHUY F IXQF LLOHJ VLWHVDWLVIDF UHOD LLOHGLQHQXQ FRV [ ≠ úL J ([ ) = [ + QHQXO SH &RQVLGHU P IXQF LLOH I J → I ([ ) = [ [ =
(
)
I DGPLWHSULPLWLYHúLILH ) → RSULPLWLY DVD J GHULYDELO úLFXGHULYDWDFRQWLQX SH vQ QRW SOXV = K([ ) = DGPLWH SULPLWLYH úL ILH + ([ ) = DUFWJ[ + & R SULPLWLY &RQVLGHU P J ([ ) [ + * → GDW GH * ([ ) = J ([ )⋅ ) (+ ([ ))3XWHPVFULH * ([ ) = J ([ )⋅ ) (+ ([ )) + J ([ )⋅ ) (+ ([ ))⋅ + ([ ) = J ([ )⋅ ) (+ ([ )) + J ([ )⋅ I (+ ([ ))⋅ (∀)[ ∈ J ([ ) DGLF * ([ ) = [ ⋅ ) (+ ([ )) + I (+ ([ )) GH XQGH I (+ ([ )) = * ([ ) − [ ⋅ ) (+ ([ )) (∀)[ ∈ &XP * FkWúL [ ⋅ ) (+ ([ ))DGPLWSULPLWLYHUH]XOW F IXQF LD I $ + DUHSULPLWLYH
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