Monografia de Matematica

 

WHE\GQVEMCM ]GFHKIKJEFC MG IKV CHMGV LCFWI]CM MG EHJGHEGQEC

GVFWGIC CFCMGBEFK ^QKLGVEKHCI MG EHJGHEGQEC MG VEV]GBCV G EHLKQBC]EFC

’EHGFWCFEKHGV‒ Ehgfucfekhgs-Bctgbãfc Aãsefc

FWQVK0 BC]GBC]EFC ACVEFC

MKFGH]G0 LQCHDIEH UCHSWE MECP

CW]KQGV0 QKAGQ] QCBEQGP VCIFGMK. @OCHV @OKHC]OCH FKIICMK VCICV QKHCIM VCIEHCV VKQEC JWEVG^G OCQQU SWEHK IKAC]KH

  Vgtegbarg mgi 1>1>

Cachfcy-^grð

 

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EHMEFG EH]QKMWFFEKH.............. ................. ................ ................. ................ ................. ................ ............ 6 3. EHGFWCFEKHGV................. ................. ................ ................. ................ ................. ................ ..... = 1. CH]GFGMGH]GV................ ................. ................ ................. ................ ................. ................ ... : 1.3 OEV]KQEC.............. ................. ................ ................. ................ ................. ................ .......... ... : 6. ICV EHGFWCFEKHGV K MGVEJWCIMCMGV.......... ................... .................. ................... ................... ................... .......................... ................ : 6.3 Qgjic mg ic subc0......... .................. .................. ................... ................... ................... ................... ................... ................... .................................... ........................... ? 6.1Qgjic mgi prkmuftk0......... ................... ................... ................... ................... ................... ................... .................. ................... ................... ........................ ............... ? 6.= G@GB^IK MG C^IEFCFEKH......... .................. ................... ................... .................. ................... ................... ................... ............................. ................... 5 =. EHGFWCFEKH MG ^QEBGQ JQCMK.............. ................. ................ ................. ................ ...... 3> 7. EHGFWCFEKHGV MG VGJWHMK JQCMK......... .................. ................... ................... .................. ................... ..............................3> ....................3> ................... ................... ................... ................... .................. ................... .................................. ........................ 31 >cF. ‖ 3:>>mF.) mgaemk ci surjebeghtk mg uh prkaigbc gh gi fuci ic rgspugstc pkmâc sgr bãs mg uhc caskiutc, sehk qug pkmâc fkhtghgr uh jrupk mg hðbgrks Ocfg cprkxebcmcbghtg 6:>> cóks, gh Bgskpktcbec y Acaeikhec yc sg scaâch rgskivgr gfucfekhgs mg prebgr y sgjuhmk jrcmk. ^kfk mgspuås tcbaeåh grch uteiezcmcs gh Gjeptk. Gi bktevk grc rgskivgr prkaigbcs rgicfekhcmks fkh ic rgpcrtefeöh mg vâvgrgs, fksgfocs y bctgrecigs. Gi pcpgi cðh hk gxestâc y iks acaeikheks gsfreaâch skarg tcaieiics mg acrrk oðbgmk pcrc qug fuchmk sg sgfcrc qugmcrc rgjestrcmk. Whc ehgfucfeöh gs uhc gxprgseöh bctgbãtefc ic fuci sg fcrcftgrezc pkr tghgr  iks sejhks mg mgsejucimcm4 Veghmk uhc gxprgseöhEhgfucfekhgs-Bctgbãfc cijgarcefc hks mc Aãsefc fkbk rgsuitcmk uh fkh`uhtk gh gi fuci ic vcrecaig ehmgpghmeghtg pugmg tkbcr gi vcikr  fucigsquegrc mg gsg fkh`uhtk fubpieghmk gstc mgsejucimcm. Ic hktcfeöh c 9 a sejhelefc qug c gs bghkr qug a y ic hktcfeöh c 8 a quegrg mgfer qug c gs bcykr qug a. Gstcs rgicfekhgs skh fkhkfemcs fkh gi hkbarg mg ehgfucfekhgs gstreftcs, fkhtrcstchmk fkh c ≭ a (c gs bghkr k ejuci c a y c ≧ a (c gs bcykr k ejuci a), Ic hktcfeöh c88a quegrg mgfer qug c gs bufok bcykr qug a. gi sejhelefcmk mg pugmg vcrecr, rglereåhmksg c uhc melgrghfec ghtrg cbaks ehmglehemc. Vg usc gh gfucfekhgs gh gi uh vcikr bufok bcykr fcuscrc qug ic rgskiufeöh mg ic gfucfeöh crrk`g c ic iuz uh fegrtk rgsuitcmk.

6. ICV EHGFWCFEKHGV K MGVEJWCIMCMGV Ics mgsejucimcmgs skh gxprgsekhgs mg ic lkrbc C9A, mkhmg C y A pugmgh sgr  gxprgsekhgs hubårefcs k cijgarcefcs y gi sâbakik 9 (Bghkr qug) 8 (Bcykr qug)

 

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9 (Bghkr k ejuci qug) 8 (Bcykr k ejuci qug)

 C ic gxprgseöh qug ocy c ic ezquegrmc ig iicbcrgbks prebgr begbark y ic qug ocy c ic mgrgfoc, sgjuhmk begbark.

G`gbpik0 1 ∕ = ‖ 3 93> Gs uhc mgsejucimcm ’fegrtc‒ (gquevcightg c : 93>). Gh gstc ðitebc, gi prebgr begbark gs : y gi sgjuhmk 3>.

6.3 Qgjic mg ic subc0 Ve subcbks (k rgstcbks) uhc besbc fchtemcm c iks mks begbarks mg uhc mgsejucimcm, Katghgbks ktrc mgsejucimcm mgsejucimcm gq gquevcightg uevcightg mgi besbk besbk sghtemk, K ssgc, gc, C9A Ô  C + F 9 A + F pcrc fuciquegr fchtemcm F. Ehgfucfekhgs-Bctgbãfc Aãsefc

Ve subcbks 1 uhemcmgs c iks mks begbarks mg ic mgsejucimcm chtgrekr, tghmrgbks qug : + 1 9 3> + 1, ösgc, 5 9 31 (fegrtk). Qgstchmk 6 uhemcmgs (pkr g`gbpik) c iks mks begbarks, tghmrgbks : ‖ 6 9 3> ‖ 6, k  Vgc, = 9 : (fegrtk).

6.1Qgjic mgi prkmuftk0 Ve buitepiefcbks (k mevemebks) iks mks begbarks mg uhc mgsejucimcm pkr  uhc bes besbc bc fch fchtem temcm cm (meste (mestehtc htc mg fgrk), fgrk), kat katghm ghmrgb rgbks ks ktr ktrc c mgs mgseju ejucim cimcm cm gquevcightg, mgi besbk sghtemk Ve F 8 > y mg sghtemk fkhtrcrek se F 9 >. K sgc, Ô (C∕F9A∕F se F gs uhc fchtemcm pksetevc k C∕F8A∕F se F gs uhc fchtemcm hgjctevc). Ve gh ic mg mgse seju juci cimc mcm m : 9 3> buit buitep epieiefc fcbk bkss ik ikss mk mkss be begb gbar arks ks pk pkrr 1, tghmrgbks qug :∕193>.1, K sgc, 3= 9 1>.

 

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Ve ik ocfgbks pkr (- 1), tghmrgbks : ∕ (-1) 8 3> ∕ (-1), k sgc, - 3= 8 -1> (Kasgrvc qug ik qug chtgs grc ’9‒ sg oc fkhvgrtemk cokrc gh ’8‒4 ic mgsejucimcm oc fcbaecmk mg sghtemk). Whc ehgfucfeöh gs uhc mgsejucimcm ghtrg gxprgsekhgs cijgarcefcs, C ics igtrcs igt rcs mg mef mefocs ocs gxprg gxprgsek sekhgs hgs cijgar cijgarcef cefcs cs igs iic iicbcr bcrgbk gbkss ehf ehföj öjhet hetcs, cs, y mgpghmeghmk mg fuchtcs ocyc, mergbks qug sg trctc mg uhc ehgfucfeöh fkh uhc, mks, trgs, ehföjhetcs.

6.= G@GB^IK MG C^IEFCFEKH 6x ‖ = ≭ 1 Gs uhc ehgfucfeöh fkh uh uhc c ehföjhetc (x). Iks vcikrgs mg ic ehföjhetc(cs) qug ocfgh fegrtc ic mgsejucimcm sg iicbch skiufekhgs mg ehgfucfeöh. ^crc ic ehgfucfeöh chtgrekr, x 2 > gs skiufeöh (pkrqug gs fegrtk qug 6∕> ‖ = ≭ 1), pgrk x 2 6 hk gs skiufeöh (pkr qug hk gs fegrtk qug 6 ∕ 6 ‖ = ≭ 1). Qgskivgr uhc ehgfucfeöh fkhsestg gh ghfkhtrcr tkmcs sus skiufekhgs. Mergbk Mer gbkss qug mk mkss ehg ehgfuc fucfek fekhg hgss skh gquevc gquevcigh ightgs tgs se teg teghgh hgh ics besbcs besbcs skiufekhgs. Ehgfucfekhgs-Bctgbãfc Aãsefc

^kr g`gbpik0  6 ∕ x ‖ = ≭ 1 Gs gquevcightg c0 6∕x≭< K  C x ≭ 1. ^kmgbk ^kmg bkss ut uteieiez ezcr cr ic Qg Qgji jic c mg ic su subc bc y ic Qgji Qgjic c mgi mgi prkm prkmuf uftk tk pc pcrc rc trchslkrbcr uhc ehgfucfeöh gh ktrc gquevcightg bãs, Vghfeiic mg rgskivgr. ^kr g`gbpik0 =x ‖ 6 ≭ 7 + 1x (subchmk 6 c iks mks begbarks) begbarks) =x ≭ ? + 1x (Qgstchmk 1x gh iks mks begbarks) =x ‖ 1 ≭ ? (mevemeghmk pkr 1 gh iks mks begbarks) x ≭ =4 iugjk ics skiufekhgs mg gstc ehgfucfeöh skh tkmks iks hðbgrks mgi ehtgrvcik (-∘,=R. Ve lugsg =x ‖ 6 9 7 + 1x, ic i c skiufeöh sgrâc gi ehtgrvcik (- ∘, =).

 

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Gh ics pã pãjeh jehcs cs sej sejueg ueghtg htgss cpr cprghm ghmgrg grgbks bks c rgs rgskiv kivgr gr mesteh mestehtks tks tep tepks ks mg ehgfucfekhgs.

=. EHGFWCFEKH MG ^QEBGQ JQCMK Whc ehgfucfeöh mg prebgr jrcmk gs uhc mgsejucimcm gh ic qug ic pktghfec mg vcrecaig gs uhk. G@GB^IK0 -

 gs uhc ehgfucfeöh mg prebgr jrcmk. gs uhc gfucfeöh mg

-

prebgr jrcmk.

 , hk gs uhc gfucfeöh mg prebgr jrcmk pkrqug ic vcrecaig sg ghfughtrc mg mghkbehcmkr. -

G@GQFEFEK QGVWGI]K0 Ehgfucfekhgs-Bctgbãfc Aãsefc

G_^IEFCFEKH AQG\G Gi bghks x mgi icmk mgrgfok pcsc c rgtcr ghtkhfgs sejhks ejucigs sg subch, pgrk iigvch gi sejhk mgi bcykr. Gi bktevk mg fcbaek mg sejhk gs pkrqug ic vcrecaig teghg qug sgr pksetevc ghtkhfgs ci bkbghtk mg fcbaecr, fcbaech tkmks mg sejhk.

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7. EHGFWCFEKHGV MG VGJWHMK JQCMK Fuchmk ics gxprgsekhgs mg cbaks icmks skh pkiehkbeks mg jrcmk bghkr k ejuci c 1, uteiezchmk ic sejueghtg lkrbuic x 2

∑a ³

∐ a ∑ = cf . 1

1c

 

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G@GB^IK0

^rebgrk fcifuicbks iks vcikrgs pcrc iks qug sg fubpig ic ejucimcm. ^crc giik, fcbaecbks ic mgsejucimcm pkr uhc ejucimcm. Mg gstg bkmk tghmrgbks uhc gfucfeöh mg sgjuhmk jrcmk gh gstc lkrbc c x + ax + f fuycs rcâfgs mgtgrbehch iks gxtrgbks mg iks ehtgrvciks mg ics skiufekhgs mg ic ehgfucfeöh0 1

Vetucbks ics rcâfgs gh ic rgftc rgci r gci y katghgbks 6 ehtgrvciks0

Gsfkjgbks uh hubgrk ci czcr mg fcmc ehtgrvcik (pkrEhgfucfekhgs-Bctgbãfc g`gbpik, x2-1, x2> yAãsefc x2=) y fkbprkacbks se pcrc cijuhk mg gstks vcikrgs sg fubpig ic ehgfucfeöh. Hk ebpkrtc fuci gsfkjgbks pugstk qug gi sejhk mg ic ehgfucfeöh mg bchteghg fkhstchtg gh fcmc ehtgrvcik.

FKB^QKACBKV0

^kr ik tchtk, ic ehgfucfeöh g vgrelefc gh mks mg iks ehtgrvciks0 Mkhmg iks fkrfogtgs ehmefch qug iks gxtrgbks mg iks ehtgrvciks gstãh ehfiuemks (gs gh giiks mkhmg sg mc ic ejucimcm mg ic ehgfucfeöh).

 x6+x1+6x+

Hktc0 Wh båtkmk glefcz mg pkmgr rgskivgr uhc gfucfeöh mg fuciquegr jrcmk, gs cpiefchmk gi båtkmk Qullehe, gi fuci hks pgrbetg mevemer y lkfciezcr ics rcâfgs mg uh pkiehkbek. Ehgfucfekhgs-Bctgbãfc Aãsefc   Veghmk0 ^(x)2 6x6+36x1-36x+1 \(x)2 x-3 Qgciezcr ic sejueghtg kpgrcfeöh0 (6x6+36x1-36x+1)0 (x-3) 2

 Csâ0 F(x)26x1+3 Mkhmg ^(x) y S(x) skh pkiehkbeks. Gs mgfer, skh cqugiics gfucfekhgs gh ics qug hks cpcrgfg uhc ’x‒ gh gi Ehgfucfekhgs-Bctgbãfc Aãsefc mghkbehcmkr. ^crc rgskivgr gstg tepk mg gfucfekhgs tghgbks qug tghgr gh fughtc qug iks hubgr ubgrcm cmkr krg gs cij cijga garc rcef efks ks,, ci ej eju uci qug ik ikss hub ubå åre reffks, ks, sg sup upre rebg bgh h buitepiefchmk pkr su bâhebk fkbðh bðitepik (k gi prkmuftk mg tkmks giiks).

G`gbpik 3 gfucfeöh rcfekhci0

 _.(_+3) +1_.(_-3) +1_.(_-3) 26(_1-3)_1+_+1_1-1_26 26(_1-3)_1+_+1_1-1_26_1-66_1-6_1 _1-66_1-6_1-_+62>_26 -_+62>_26

:.1 Qgskiufeöh mg ehgfucfekhgs rcfekhcigs. Gh uhc luhfeöh rcfekhci tghgbks ehtgrvciks mkhmg gi vcikr mg ic luhfeöh gs pksetevk y ktrks ehtgrvciks mkhmg gi vcikr mg ic luhfeöh gs hgjctevk. C prekre, gs fkbpiefcmk scagr quå trcbks sgrãh pksetevks y fuãigs sgrãh hgjctevks. Qgskivgr uhc ehgfucfeöh rcfekhci fkhsestg gh katghgr gi rchjk mg vcikrgs mg x fubpich ic mgsejucimcm, gs mgfer, katghgr iks trcbks pcrc iks qug ic luhfeöh sgc pksetevc k hgjctevc, sgjðh sgc ic mgsejucimcm mg ic ehgfucfeöh.

 

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^crc giik, mgagbks rgciezcr iks sejueghtgs pcsks0 3.

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Katghgr ch chtgs iik ks vvc cikrgs m mg g x mk mkhmg iic c llu uhfeöh ffc cbaec m mg g sseejhk, mg mg pksetevc c hgjctevc k vefgvgrsc Qgprgsghtcr ik iks puhtks gh gh iic c rrg gftc rg rgci, tg tgheghmk gh ffu ughtc ssee ssg g fk fkjg k hk pcrc gi rgsuitcmk. Fcifuicr gi sejhk mg fcmc ehtgrvcik.

=.

Gi mg rc rchicjkehgfucfeöh k iks rrc chjks mg mg vvc cikrgs q qu ug fu fubpich iic c m mg gsejucimcm, sg sgrã iic c skiufeöh e hgfucfeöh

1.

G`gbpik rgsugitk mg ehgfucfeöh rcfekhci 3

Ehgfucfekhgs-Bctgbãfc Aãsefc

 

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:.6 Ehgfucfekhgs Errcfekhcigs. Vkh ehgfucfekhgs rcfekhcigs, cqugiics gh ics qug tchtk gi hubgrcmkr fkbk gi mghkbehcmkr skh ehgfucfekhgs pkiehöbefcs fucmrãtefcs k pkiehöbefcs mg jrcmk bcykr c 1Gs uhk mg iks qug trcg bãs fkbpiefcfekhgs, pkrqug uhc ehgfucfeöh rcfe rcfekh khci ci gs uh uhc c gx gxpr prgs gseö eöh h mg tepk tepk lr lrcf cffe feöh öh,, mk mkhm hmg g ic vcre vcreca caig ig gstã gstã gh gi hubgrcmkr y gi mghkbehcmkr. G@GB^IK _∑12>

x2 1

_∑=2>

x2 =

EBCJGH CM@WH]CMC0 ebcjgh.>3

Ehgfucfekhgs-Bctgbãfc Aãsefc

 

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?. EHGFWCFEKHGV FKH \CIKQ CAVKIW]K Whc ehgfucfeöh mg vcikr caskiutk gs uhc fkbaehcfeöh mg mks fkhfg fk hfgptk ptks0 s0 vci vcikrg krgss cas caskiu kiutks tks g ehg ehgfu fucfe cfekh khgs gs ieh iehgc gcigs igs.. ^kr ik tchtk, tchtk, pcr pcrc c rgsk rgskiv ivgr gr uh uhc c eh ehgf gfuc ucfe feöh öh mg vcik vcikrr cask caskiu iutk tk mg mgag agss us uscr cr ik ikss bå båtk tkmk mkss mg rgskiufeöh mg prkaigbcs mg cbacs bctgrecs. Gi vcikr caskiutk mg uh hðbgrk c, rgprgsghtcmk fkbk |c|, gs su vcikr hubårefk (fkh sejhk pksetevk). ^kr g`gbpik0

Ehgfucfekhgs-Bctgbãfc Aãsefc

Hktgbks qug0 

se gi hðbgrk gs pksetevk, su vcikr caskiutk gs gi prkpek hðbgrk4

se gi hðbgrk gs hgjctevk, su vcikr caskiutk gs su kpugstk (hðbgrk fkh sejhk kpugstk, gs mgfer, fkh sejhk pksetevk)4

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se gi hðbgrk gs >, su vcikr caskiutk gs >, cuhqug > hk gs he pksetevk he hgjctevk.

?.3 ^rkpegmcmgs0 Gi vcikr caskiutk segbprg gs bcykr k ejuci qug >, seghmk > söik fuchmk su crjubghtk gs >0

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Gi vcikr caskiutk mg uh prkmuftk gs gi prkmuftk mg iks vcikrgs caskiutks mg iks lcftkrgs0

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3<

 

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\cikr caskiutk mg ic subc0

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^rkpegmcm ebpkrtchtg0 se tghgbks ic mgsejucimcm (bghkr k ejuci)

^kmgbks gsfreaer 

Sug gs ik besbk qug mgfer 

(]eghgh qug fubpiersg cbacs rgicfekhgs). Mefok gh lkrbc mg ehtgrvciks0 Ehgfucfekhgs-Bctgbãfc Aãsefc

Ve ic mgsejucimcm gs (bcykr k ejuci)

^kmgbks gsfreaer 

(Gs uhc uheöh0 teghg qug fubpiersg uhc mg ics mks). Mefok gh lkrbc mg ehtgrvciks0

G`gbpik0

^kr prkpegmcm (7) lcftkrezchmk y sebpielefchmk. -|1x + =| ≧ < Vkiufeöh

 

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|1x + =| ≧ < ⇑ 1x + = ≭∑< ∯ 1x + = ≧ < ⇑

 _ + 1 ≭∑6 ∯ x + 1 ≧ 6

⇑

 _ ≭∑7 ∯ x ≧ 3

Fkbk x gs bghkr k ejuci qug ∑7 k x gs bcykr k ejuci qug 3, gi fkh`uhtk skiufeöh gstcrã mcmk pkr ic uheöh mg gstks ehtgrvciks Iugjk gi fkh`uhtk skiufeöh sgrã0R ∑ ∘, ∑7R ∠ T3, ∘ T

Ehgfucfekhgs-Bctgbãfc Aãsefc

 

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5. FKHFIWVEKHGV0 Whc gfucfeöh gfucfeöh gs uhc prkpugstc prkpugstc mgsejucimcm mgsejucimcm gh ic qug ehtgrveghg ehtgrveghg cijuhc igtrc iicbcmc ehföjhetc. Ic skiufeöh mg ic gfucfeöh gs gi vcikr k vcikrgs mg ics ehföjhetcs qug ocfgh qug ic ejucimcm fegrtc. Qgskivgr gfucfeöh gs ociicr su skiufeöh, k skiufekhgs, k iigjcr csgc ic fkhfiuseöh mg quguhc hk gxestg. ^umebk ^ume bkss vg vgrr gh gi tr trca cac` c`k k ic icss melg melgrg rght htgs gs mg mgse seju juci cimc mcmg mgss qu qug g sg pu pugm gmgh gh prgs prgsgh ghtc tcrr gh bufo bufoks ks pr prka kaig igbc bcss bc bctg tgbã bãtetefk fks, s, pc pcrc rc gs gstc tc hk juec juecbk bkss mg cxekbcs y lörbuics qug sg cpiefch kaiejctkrecbghtg gh ic rgskiufeöh mg gstks prkaigbcs.

3>. Aeaiekjrclâc y Qglgrghfecs0 

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\rchfd gh, Veiv Veivec, ec, Mch Mchegi egic c Bùi Bùiigr igr,, ch chm m Cmr Cmrech echc c Ghj Ghjigr. igr. "Eh "Ehgfu gfucfe cfekhg khgss \rchfdgh, cijgarcefcs. WhcUupchc. gxpgreghfec memãftefc crtefuichmk mevgrsks sestgbcs rgprgsghtcfeöh." Qgvestc mg Gmufcfeöh Bctgbãtefc mg ic WHImg  7 (1>3>)0 77-31) 31).. Qgs Qgskiu kiufeö feöh h mg prk prkaig aigbcs bcs fkh ehg ehgfuc fucfek fekhgs hgs Vã Ehgfucfekhgs-Bctgbãfc Aãsefc fucmrãtefcs. Whc prkpugstc gh gi bcrfk mg ic tgkrâc mg setucfekhgs memãftefcs. \E Fkikquek Ehtgrhcfekhci Ghsgóchzc mg ics Bctgbãtefcs , . Hc Hcpk pkieietc tchk hk,, B. B.,, & Bc Bcrt rtâh âhgz gz,, B. (1 (1>3 >35) 5).. 36 36>= >=-3 -35 5 BC BC]G ]GBÃ BÃ]E ]EFC FC Ehgfucfekhgs.

 

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