Revista Olimpiada 2012

43
5/14/2018 RevistaOlimpiada2012-slidepdf.com http://slidepdf.com/reader/full/revista-olimpiada-2012 1/43

description

Revista Olimpiada Matemática Extremadura 2012

Transcript of Revista Olimpiada 2012

Page 1: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 1/43

Page 2: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 2/43

WWA Ciao{afef Ofuboæua`f

=

Bwu}bofer}f =;6=

LFPB @COF]@FI=6 eb Fm}ai eb =;6=) 6;>0; gc}fp

F@BR@GFI)F^RFJF) MF]@F]]CUF) @FPF] EB @Æ@B]BP) ECH MBHAUC) JÁ\C]F) OÁ]AEF) OC]FIBNF) [IFPBH@AF) PFH \A@BHUB EB

FI@ÆHUF]F) PBJR]F EB IBÐH q PA]RBIF( 

[}cmibof)Lb`gf

Vrapvrb>

Lbrjafu>

=

=

Ofr}apao{b}eabu(Eraphb`{r}rphch

Ru}aprp{r}rp)`chjrb|bi)ofuuapae)

 WWA CIAO[AFEF OFUBOÆUA@F

Ïhea`b

0/5

2/62

=5/0<

F}uï`ricebif@chpbnb}febBer`f`aðh

]bif`aðheb@bhu}cp{f}ua`a{fhubpbhifWWCiao{afefOfuboæua`f

[}cmibofpebifpeapuahufplfpbp

FpïlrbifWWCiao{afef

6</=0

IfWWACiao{afef

0:/5=

LFPB FRUCHÐOA@F=2) =< q =: eb ofqc eb =;6=

\fi|b}eb ebi L}bphc 

[c}ufef> Of}ïf @föfef Oc}bhc(

 ABP Echcpc @c}uáp eb Ech Mbhauc(

Lcucj}flïfp> [be}c @c}`gc Pæh`gb~)

 Fhuchac Ocifhc ]cob}c q Ncpá [be}c

 Of}uïh Ic}bh~c(  

Ofvrbuf`aðh> Pb}jac Pfhucp ]cpbii 

Ea}b``aðh> Ncpá [be}c Of}uïh Ic}bh~c 

Eb{( Ibjfi> MF/12/=;;<

APPH> 677</6==1

Page 3: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 3/43

0

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

[}bpbhuf`aðh eb if Bw`of( @chpbnb}f eb Ber`f`aðh q @riur}f

’Bi ibhjrfnb eb if @abh`af‟

Jbhb}f} ahub}áp {c} bi f{}bhea~fnb eb ifp

Ofuboæua`fp bh hrbpu}c firohfec bp rhf {}bc`r/

{f`aðh {b}ofhbhub eb icp ec`bhubp) lrhefobhufi/

obhub bh`foahfef f icj}f} vrb ifp `co{bubh`afp

mæpa`fp ebpf}}ciifefp bh bi frif {rbefh u}fhplb}a}/

pb q jbhb}fia~f}pb f `rfivrab} æomauc eb if |aef

`cuaeafhf eb lc}of cgb}bhub q f}oðha`f(

Bi pfmb} ofuboæua`c bpuæ bh bi c}ajbh eb if

`a|aia~f`aðh( Eb gb`gc) icp ubwucp ofuboæua`cp

oæp fhuajrcp hcp }bochufh f Obpc{cufoaf( Pr/

ob}acp q mfmaichacp ebpf}}ciif}ch `co{ibncp pap/

ubofp eb hrob}f`aðh q lbha`acp q bja{`acp `chua/

hrf}ch f|fh~fhec bh bpuf `abh`af( Ebp{ráp lrb}ch

icp j}abjcp) `coc bh ufhucp cu}cp fp{b`ucp) icp vrb

{rpab}ch ifp mfpbp q bwubheab}ch bpub pfmb} {c} bi

Obeaub}}æhbc) `ch ecp lajr}fp `if|b> Ufibp eb Oa/

ibuc q [auæjc}fp eb Pfocp( Bi {ceb} eb fmpu}f``aðh

lrb rhf eb prp {}ah`a{fibp f{c}uf`achbp( Rh {ceb}

eb fmpu}f``aðh vrb fþh gcq bp |æiaec {f}f bpu}r`/

ur}f} hrbpu}cp {bhpfoabhucp(

Pah eref) ifp Ofuboæua`fp bpuæh bh if mfpb

eb icp f|fh`bp `abhuïla`cp) ub`hciðja`cp q uá`ha/

`cp( Bpuæh {}bpbhubp bh hrbpu}c eïf f eïf(

Ifp Ofuboæua`fp pch hb`bpf}afp {f}f `co/

{}bheb} q fhfia~f} if fmrhefhub q |f}afef ahlc}/

of`aðh vrb hcp iibjf) lf|c}b`b q jbhb}f if `f{f/

`aefe eb {bhpfoabhuc fmpu}f`uc) {b}oaub bh`chu}f}

fhficjïfp q bpufmib`b} `}aub}acp ealb}bh`afibp bhu}b

ea|b}pcp lbhðobhcp q `}bf bi gæmauc eb bhl}bhuf}

{}cmibofp) ucof} aha`afua|fp `cgb}bhubp q }bpci/

|b}icp bla`f~obhub(

@coc |fic} `riur}fi) ifp Ofuboæua`fp fo/

{iïfh if bplb}f `riur}fi ebi ahea|aerc qf vrb ebpf/

}}ciif gæmaucp eb ib`ur}f) obnc}f ifp gfmaiaefebp eb

ah|bpuajf`aðh) {b}oaub }b`fmf} efucp q lf|c}b`b if

ahub}{}buf`aðh eb paurf`achbp ufhuc f ha|bi {b}pc/

hfi `coc `cib`ua|c( If ofqc}ïf eb ifp {}clbpachbp

Page 4: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 4/43

WWA Ciao{afef Ofuboæua`f

5

Bwu}bofer}f =;6=

q u}fmfncp uá`ha`cp f`urfibp }bvrab}bh `chc`a/

oabhucp ofuboæua`cp( Ifp Ofuboæua`fp pch) {c}

bheb) `chpaeb}fefp rhf eap`a{iahf q gb}}foabhuf

rha|b}pfi) bp bi ibhjrfnb eb if `abh`af q eb if uá`/

ha`f q pb gf`b {}b`apc pr ecoahac(

If ah`irpaðh eb ifp `co{bubh`afp mæpa`fp

bh bi `r}}ï`ric) bhu}b biifp if }blb}aef f if ofuboæ/

ua`f) {b}pajrb ahubj}f} icp eapuahucp f{}bhea~fnbp)

ufhuc icp lc}ofibp) ah`c}{c}fecp f ifp ealb}bhubp

eap`a{iahfp) coc icp ahlc}ofibp q hc lc}ofibp) fpï 

`coc lf|c}b`b} if ahubj}f`aðh eb prp f{}bhea~fnbp

f{ia`æhecicp eb ofhb}f blb`ua|f `rfhec }bpriubh

hb`bpf}ac bh ifp ea|b}paefe eb paurf`achbp) `a}/

`rhpufh`afp q `chubwucp( Ufomaáh pch rhf ao{c}/

ufhub gb}}foabhuf f lah eb c}abhuf} icp {}c`bpcp eb

bhpböfh~f#f{}bhea~fnb) fi {b}oaua} eap`}aoahf} b

aebhuala`f} icp `chubhaecp q icp `}aub}acp eb b|firf/`aðh vrb uabhbh `f}æ`ub} ao{}bp`aheamib b ahp{a}f}

ifp eb`apachbp eb obnc}f q c{uaoa~f`aðh }bifua|fp f

ea`gc {}c`bpc(

@coc hc {rbeb pb} eb cu}c ocec) if @ch/

pbnb}ïf eb Ber`f`aðh q @riur}f ebi Jcmab}hc eb

Bwu}bofer}f f{cqf b ah`bhua|f bpub ua{c eb f`ua|a/

efebp) vrb |fh oæp fiiæ ebi u}fmfnc ebpbo{böfec

bh ifp frifp( Pch f`ua|aefebp lc}ofua|fp eb {}aob}

c}ebh vrb }co{bh bi u}aæhjric {}clbpc}/firohc/

frif `chpajrabhec rhf ocua|f`aðh bwu}f {f}f bi

firohfec( [c} ucec biic) vrab}c bw{}bpf} oa fj}f/

eb`aoabhuc f if Pc`abefe Bwu}boböf eb Ber`f`aðh

Ofuboæua`f ’\bhur}f ]bqbp [}cp{b}‟ {c} bi u}fmf/

  nc }bfia~fec {f}f vrb bpuf WWA Ciao{afef Ofub/

oæua`f bh Bwu}bofer}f pbf rhf }bfiaefe(

[f}f `ch`ira}) vrab}c fj}feb`b} q lbia`auf}

eb lc}of orq bp{b`afi f icp ec`bhubp) {c} pr bhur/

pafpoc) bplrb}~c q {c} gfmb} lcobhufec q fhaof/

ec if {f}ua`a{f`aðh bh bpuf Ciao{afef bhu}b prp

  nð|bhbp firohcp( Ufomaáh f{}c|b`gc if c`fpaðh

{f}f ah|auf}ibp q fhaof}ibp f pr lc}of`aðh q f`urf/

ia~f`aðh bh bpuf eap`a{iahf bh f}fp eb fef{uf}pb fifp hb`bpaefebp b ahub}bpbp eb pr firohfec( Bp oa

ebpbc vrb bpuf Ciao{afef pbf rh æomauc aeðhbc eb

`ch|a|bh`af) mrbhfp {}æ`ua`fp) ahub}`fomac eb

bw{b}abh`afp ber`fua|fp q eb f{}bhea~fnb(

U}ahaefe Hcjfibp Mfpf}}fub

Page 5: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 5/43

2

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

Fpï lrb if WW Ciao{afefIf Nrhuf Ea}b`ua|f eb if P(B(B(O( bpureað ifp {bua`achbp eb ic`fiaefebp {f}f pb} pbebp eb ifp ealb}bh/

ubp lfpbp) pajrabhec bi `}aub}ac eb ef} if {cpamaiaefe eb {f}ua`a{f`aðh oæwaof eb bp`cif}bp q {}c`r}fhec `r/

m}a} ucefp ifp ~chfp eb hrbpu}f `corhaefe fruðhcof( Pb f`c}eð lanf} {f}f if lfpb `cof}`fi ifp pajrabhubp

pbebp>

^CHF  [CMIF@AÐH  @BHU]C 

FIMR]VRB]VRB  FIMR]VRB]VRB A(B(P( @FPUAIIC EB IRHF

FIOBHE]FIBNC   \AIIFL]FH@F EB ICP MF]]CP  @( PFH NCPÁ 

F^RFJF#IIB]BHF  IIB]BHF A(B(P( LB]HFHEC ]CMAHF

MFEFNC^  MFEFNC^  A(B(P( PFH NCPÁ 

@Æ@B]BP  @FPF] EB @Æ@B]BP A(B(P(C( \ÏF EB IF [IFUF

@C]AF  \FI\B]EB EBI L]BPHC  A(B(P(C( \FI EB WÆIAOF 

ECH MBHAUC  ECH MBHAUC  A(B(P( NCPÁ OFH^FHC 

OÁ]AEF  OÁ]AEF  A(B(P( PFHUF BRIFIAF 

[IFPBH@AF  [IFPBH@AF  @BHU]C RHA\B]PAUF]AC 

PA]RBIF  PA]RBIF  A(B(P(C( \A]JBH EB FIUFJ]F@AF 

^FL]F  ^FL]F  A(B(P( PRÆ]B^ EB LAJRB]CF 

@coc pbeb eb if lfpb fruchðoa`f pb f`b{uð if {}c{rbpuf {}bpbhufef {c} bi Bw`bibhuïpaoc Fqrhuf/

oabhuc eb If [f}}f 'Mfefnc~!(

Bh Of}~c pb `cijð bh ahub}hbu if }b|apuf eb if WW Ciao{afef Ofuboæua`f {f}f firohcp eb =¾ eb

B(P(C( bh if ea}b``aðh guu{>##zzz(ber`f}bw(bp#zbm#jrbpu#ciao{afefp/ofubofua`fp/bwu}bofer}f q bh if eb if

{æjahf eb if P(B(B(O( ’\bhur}f ]bqbp [}ðp{b}‟(

LFPB @COF]@FI

If lfpb `cof}`fi pb `bibm}ð bi eïf : eb Ofqc f ifp 6;>0; `chlc}ob {}b|bïf if `ch|c`fuc}af(

Icp {fvrbubp vrb `chubhïfh ifp {}rbmfp ao{}bpfp bh rhfp `f}{buaiifp) fpï `coc icp mciïj}flcp) ifp gc/

 nfp eb efucp {b}pchfibp eb icp {f}ua`a{fhubp) ea{icofp eb icp firohcp) {}clbpc}bp) fpï `coc icp `}aub}acp eb

b|firf`aðh lrb}ch bhu}bjfecp f icp `cc}eahfec}bp eb ~chf bh if }brhaðh `bibm}fef bh bi @([(] eb Oá}aef bi5 eb Ofqc f ifp < eb if uf}eb( Ea`gc {fvrbub hc lrb fmab}uc gfpuf bi oapoc ocobhuc bh vrb pb bhu}bjf}ch

ifp {}rbmfp f icp firohcp(

Page 6: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 6/43

WWA Ciao{afef Ofuboæua`f

<

Bwu}bofer}f =;6=

[f}f bi ebpf}}ciic eb ifp {}rbmfp pb lanf}ch ifp pajrabhubp hc}ofp) vrb pb eab}ch f `chc`b} f ucecp icp

{f}ua`a{fhubp fhubp ebi aha`ac eb ifp oapofp>

v ]biibhf} `ch ibu}f `if}f q ibjamib ucecp icpefucp eb if `if|b eb aebhuala`f`aðh(

v [chb} bh bi æhjric pr{b}ac} eb}b`gc eb `fefgcnf ruaia~fef bi hþob}c vrb f{f}b`b bh if `if|beb aebhuala`f`aðh(

v Ruaia~f} rhc c oæp lciacp {c} `fef {}cmibof(

v Ahea`f} bh bi æhjric pr{b}ac} a~vrab}ec ebhu}ceb rh `ï}`ric bi hþob}c eb `fef {}cmibof(

v Pb{f}f} `fef `rbpuaðh ebi {}cmibof `ch rhfiïhbf ea|apc}af(

v Ebufiif} fi oæwaoc ucecp icp {fpcp efecp {f}f}bpci|b} fef bnb}`a`ac(

v Pb {rhurf}æ if {}bpbhuf`aðh q icp}f~chfoabhucp bw{rbpucp bh if }bpcir`aðh eb ifpealb}bhubp rbpuachbp {ifhubfefp(

v Bhu}bjf} ifp gcnfp `ch ifp }bp{rbpufp c}ebhfefp`chlc}ob fi hþob}c ebi {}cmibof(

v [rbebh ruaia~f} `fi`rifec}f(

v Er}f`aðh eb if {}rbmf> ecp gc}fp `cocoæwaoc(

Bi irhbp eïf 1 eb Ofqc `fef `cc}eahfec}) bh|að {c} u}fhp{c}ub r}jbhub c bi obeac vrb bpuaoð oæp

pbjr}c q }æ{aec) ifp {}rbmfp eb pr ~chf {f}f pb} `c}}bjaefp bh if ~chf fpajhfef( Ifp `if|bp aebhuala`fua|fp

vrb {}bpb}|fmfh if aebhuaefe eb icp {f}ua`a{fhubp) vrbef}ch {c} bi ocobhuc bh {ceb} eb icp `cc}eahfec}bp}bp{b`ua|cp(

Rhf |b~ }bfia~fec bi ahub}`fomac) if `c}}b``aðh q if mf}bof`aðh eb ucefp ifp {}rbmfp) lrb}ch bh|af/

efp fi `cc}eahfec} }bjachfi {f}f {}c`beb} f if pbib``aðh eb icp {f}ua`a{fhubp bh if lfpb fruchðoa`f( Ifp

{}rbmfp amfh c}ebhfefp pbjþh if {rhurf`aðh(

Bi eïf =; eb Ofqc pb }brhað if `coapaðh eb b|firf`aðh vrb hcoahð fi {}aob} `ifpala`fec eb `fef ~c/

hf pbjþh }b~f bh if `ch|c`fuc}af f{f}b`aef bh bi E(C(B( Fpaoapoc) pb pbib``achð fi }bpuc eb {f}ua`a{fhubppbjþh `}aub}acp eb {rhurf`aðh q {f}ua`a{f`aðh {c} pbeb) gfpuf `co{ibuf} icp u}bahuf vrb fpapuab}ch f if lfpb

fruchðoa`f vrb pb `bibm}f}ïf bh If [f}}f 'Mfefnc~!(

Page 7: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 7/43

:

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

Icp `ifpala`fecp {f}f if Lfpb Fruchðoa`f lrb}ch icp pajrabhubp>

F[BIIAECP  HCOM]B  @BHU]C 'IC@FIAEFE! 

FOFEC] [C]]FP  NBPÞP  ABP [Á]B^ @COBHEFEC] '[IFPBH@AF! Æ]AFP FMBIAFHF  U]AFHF  ABP PFHUF BRIFIAF 'OÁ]AEF! 

F]]CQC FPBPAC  NRFH  ABP ]CE]ÏJRB^ OCÖAHC 'MFEFNC^! Æ\AIF OF]@BIC  @F]OBH  ABP L]FH@AP@C EB C]BIIFHF 'U]RNAIIC! 

MCIIFP MB@B]]F  FHUCHAC NCPÁ  ABP @FO[CP EB PFH ]CVRB 'MFEFNC^! 

@FI^FP ER]ÆH  ÆHJBI NBPÞP  ABP IRAP @GFOA^C 'ECH MBHAUC! 

@F]]B]C JCH^ÆIB^  L]FH@AP@C NF\AB]  ABP FIFJÐH '@C]AF! 

@C]@GFEC ]CI  MBFU]A^  PFJ]FEC @C]F^ÐH EB NBPÞP '@Æ@B]BP! 

EÏF^ JF]@ÏF  [BE]C NCPÁ  ABPC OF]AFHC MF]MF@AE 'PCIFHF EB ICP MF/

]]CP! 

EÏF^ OF]]ÐH  FIBNFHE]C  ABP AIEBLCHPC PB]]FHC 'PBJR]F EB IBÐH! 

LB]HÆHEB^ LRBHUBP  NCPÁ OFHRBI  HU]F( P]F EB IF J]FHFEF ‐ PUC FHJBI 'IIB/

]BHF!

L]FH@C @FPUFÖC  EABJC  @CIBJAC PFH NCPÁ '\AIIFL]FH@F EB ICP MF]]CP! 

L]FH@C JCH^ÆIB^  FIBNFHE]C  ABP MBHF^FA]B 'GB]]B]F EBI ERVRB! 

JFIIF]EC MC]]BJC  @F]ICP @CIBJAC PFH NCPÁ '\AIIFL]FH@F EB ICP MF]]CP!

JF]@ÏF/@FPAIIFP MF]]CPC  IR@ÏF  PFJ]FEC @C]F^ÐH EB NBPÞP '@Æ@B]BP! 

JAI OF]UÏH  PFORBI  ABPC \FI EB WÆIAOF '\FI\B]EB EBI L]BPHC! 

J]FJB]F OÆP  NCPÁ IRAP  HU]F( P]F( EBI @F]OBH 'MFEFNC^! 

LB]HÆHEB^ MBHÏUB^  OF]ÏF  PFIBPAFHCP O® FRWAIAFEC]F 'OÁ]AEF! 

GB]HÆHEB^ M]ACPC  PAI\B]AC  ABP AIEBLCHPC PB]]FHC 'PBJR]F EB IBÐH! 

GB]HÆHEB^ @B]B^C  FIBNFHE]C PFH @FIAWUC '[IFPBH@AF! 

GCQFP PÆH@GB^  @IF]F ABP []CLBPC] GEB^( [F@GB@C '@Æ@B]BP! 

OF]UÏH UCOÁ  FHE]BF  ABP [Á]B^ @COBHEFEC] '[IFPBH@AF! OCIB]C UBHF  MBIÁH  ABP OAJRBI ER]ÆH 'F^RFJF! 

[F]BNC ]FOCP  BPU]BIIFABP [BE]C EB \FIEA\AF '\AIIFHRB\F EB IF PB/

]BHF! 

]CE]ÏJRB^ OF]@CP  @F]OBIF ABP [Á]B^ @COBHEFEC] '[IFPBH@AF! ]C[B]C J]FJB]F  EFHABI  @CIBJAC PFH NCPÁ '\AIIFL]FH@F EB ICP MF]]CP!

PÆH@GB^ MCIÏ\F]  [BE]C  ABP BWU]BOFER]F 'OÁ]AEF! 

PÆH@GB^ OFR]B]F  OF]@C FHUCHAC  ABP ^R]MF]ÆH 'MFEFNC^!

PÆH@GB^ ]A\B]C  FHF  ABP []CLBPC] GB]HÆHEB^ [F@GB@C '@Æ@B]BP! UÁIIB^ ECOÏHJRB^  ]CMB]UC  ABP []CLBPC] GEB^( [F@GB@C '@Æ@B]BP!

]BPB]\FP 

6¾ JÐOB^ UR]ÁJFHC  OF]UF  PFHUF @B@AIAF '@Æ@B]BP!

=¾ ]CE]ÏJRB^ PFH [BE]C  FI\F]C  ABP []CLBPC] GEB^ [F@GB@C '@Æ@B]BP! 

0¾ EBIJFEC IÐ[B^  NCPÁ ÆHJBI  HU]F( P]F( EBI @F]OBH '\AIIFL]FH@F EB ICP

MF]]CP! 

5¾ JAI MB]HÆIEB^  MBFU]A^  ABP [B]B^ @COBHEFEC] '[IFPBH@AF! 2¾ IF ]BF [FNF]BP  NC]JB  FPRH@AÐH NCPBLAHFP '@Æ@B]BP!

Page 8: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 8/43

WWA Ciao{afef Ofuboæua`f

7

Bwu}bofer}f =;6=

Bpub eïf ufomaáh pb biajað fi `f}ubi vrb {}bpbhuf}æ f if Ciao{afef Ofuboæua`f eb =;6=) fpï `coc f icp

ecp f``ápau( [f}f biic `chufocp `ch if `cifmc}f`aðh ebi {}clbpc}fec ebi A(B(P( Pfh Ncpá eb Mfefnc~( Bi nr}f/

ec bpur|c `co{rbpuc {c} E( Ncpá Ncfvrïh @fpaiifp F}afp) E® Of}ïf [f~ [á}b~ ]ce}ïjrb~ q E( Oajrbi Æhjbi

Oc}bhc ]bechec eb if bp{b`afiaefe eb Ofuboæua`fp) E® @f}obh Of}uïh Pfhfm}af eb @abh`afp Hfur}fibp q E®Ob}`bebp [á}b~ Gr}ufec ebi eb{f}ufobhuc eb Lïpa`f q Vrïoa`f(

Icp f}ubibp pbib``achfecp {b}ubhb`ab}ch f>

JFHFEC] OF]ÏF @FÖFEF OC]BHC ABP ECHCPC @C]UÁP( ECH MBHAUC

F@@BPAU 6¾ NF\AB] OF]UÏH UC]]BP ABP ECHCPC @C]UBP( ECH MBHAUC

F@@BPAU =¾ FHF AHÁP @CPF FJRA]]B ABP NCPÁ OFH^FHC( ECH MBHAUC

LFPB FRUCHÐOA@F

Er}fhub icp eïfp 0) 5 q 2 eb Nrhac) pb `bibm}ð if Lfpb Fruchðoa`f eb if WW Ciao{afef Ofuboæua`f

eb Bwu}bofer}f bh if ic`fiaefe eb If [f}}f 'Mfefnc~!(

If Lfpb Fruchðoa`f pb aha`að bi eïf 0 eb Nrhac( [f}f bi u}fpifec f If [f}}f pb {rpc f eap{cpa`aðh eb

icp {f}ua`a{fhubp ea|b}pcp obeacp eb u}fhp{c}ub( Bi irjf} bibjaec `coc }bpaebh`af er}fhub bi lah eb pbofhf

lrb bi @ch|bhuc Gcp{beb}ïf If [f}}f( @rfhec ucecp icp {f}ua`a{fhubp iibjf}ch) pb ibp f`coceð bh prp }bp/

{b`ua|fp gfmauf`achbp f if |b~ vrb pb ga~c bhu}bjf eb if }bif`aðh eb icp j}r{cp lc}ofecp {f}f if {}rbmf ebi

`a}`rauc ofuboæua`c(

If {}rbmf ebi `a}`rauc ofuboæ/

ua`c eb bpub föc pb {}b{f}ð bh bi bh/

uc}hc ochrobhufi eb If [f}}f) bh prp

{if~fp) lrbhubp) fpï `coc bh if Ajibpaf

eb Hu}f( P}f( eb if Fprh`aðh( @fmb

eb`a} vrb bpuf {}rbmf pajrb pabhec rhf

eb ifp f`ua|aefebp oæp fu}f`ua|fp eb if

Ciao{afef qf vrb pa}|b `coc {rhuc eb

{f}uaef {f}f rhf obnc} }bif`aðh bhu}b

ucecp icp {f}ua`a{fhubp( Feboæp) ufo/

maáh {b}oaub `chc`b} b ahub}`fomaf}

ao{}bpachbp pcm}b bpu}fubjafp vrb }b/

Page 9: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 9/43

1

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

prbi|bh ifp `rbpuachbp vrb pb {ifhubbh bh if {}rbmf ebi `a}`rauc ofuboæua`c(

Bi `}aub}ac pbjraec {f}f lc}of} icp j}r{cp lrb a} }bif`achfhec `ga`c `ch `ga`f eb ic`fiaefebp ealb}bh/

ubp ic oæp eapufhub {cpamibp(

Icp j}r{cp }bpriufhubp lrb}ch>

J]R[C 6

FOFEC] [C]]FP  NBPÞP  [IFPBH@AF F]AFP FMBIAFHF  U]AFHF  OB]AEF 

]C[B]C J]FJB]F  EFHABI  \AIIFL]FH@F EB ICP MF]]CP 'MF! 

J]R[C = 

PFH@GB^ MCIÏ\F] [BE]C OCHUANC 'MF!GCQFP PÆH@GB^ @IF]F @Æ@B]BP

MCIIFP MB@B]]F FHUCHAC NCPÁ \FI\B]EB EB IBJFHÁP 'MF!

J]R[C 0

OF]UÏH UCOÁ FHE]BF OFI[F]UAEF EB [IFPBH@AF '@@!

OCIB]C UBHF MBIÁH F^RFJF 'MF!J]FJB]F OÆP NCPÁ IRAP MFEFNC^

J]R[C 5

UÁIIB^ ECOÏHJRB^ ]CMB]UC @Æ@B]BPPFH@GB^ OFR]B]F OF]@C FHUCHAC MFEFNC^

]CE]AJRB^ OF]@CP @F]OBIF [IFPBH@AF '@@!

J]R[C 2

JAI OF]UÏH PFORBI \FI\B]EB EBI L]BPHC '@@!

[F]BNC ]FOCP BPU]BIIF \AIIFHRB\F EB IF PB]BHF 'MF!F]]CQC FPBPAC NRFH MFEFNC^

J]R[C < 

PFH@GB^ ]A\B]C FHF @Æ@B]BPEÏF^ JF]@ÏF [BE]C NCPÁ PCIFHF EB ICP MF]]CP 'MF!

LB]HÆHEB^ LRBHUBP NCPB OFHRBI IIB]BHF'MF!

J]R[C : 

F\AIF OF]@BIC @F]OBH IF @ROM]B '@@!@FI^FP ER]ÆH ÆHJBI NBPÞP \A\F]BP 'MF!

JFIIF]EC MC]]BJC @F]ICP \AIIFL]FH@F EB ICP MF]]CP 'MF!

Page 10: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 10/43

WWA Ciao{afef Ofuboæua`f

6;

Bwu}bofer}f =;6=

J]R[C 7

LB]HÆHEB^ MBHÏUB^ OF]ÏF OÁ]AEF 'MF!@F]]B]C JCH^FIB^ L]FH@AP@C NF\AB] @C]AF '@@!L]FH@C @FPUFÖC EABJC \AIIFL]FH@F EB ICP MF]]CP 'MF!

J]R[C 1

@C]@GFEC ]CI MBFU]A^ @Æ@B]BPEÏF^ OF]]ÐH FIBNFHE]C LRBHUBP EB IBÐH 'MF!GB]HÆHEB^ @B]B^C FIBNFHE]C [IFPBH@AF '@@!

J]R[C 6;

L]FH@C JCH^ÆIB^ FIBNFHE]C GBIB@GCPF EB ICP OCHUBP 'MF!JF]@ÏF/@FPAIIFP MF]]CPC IR@ÏF @Æ@B]BP

GB]HÆHEB^ M]ACPC PAI\B]AC F]]CQCOCIAHCP EB IBÐH 'MF!

Ebp{ráp eb if `coaef q u}fp rh m}b|b ebp`fhpc) ucef if bw{bea`aðh ciïo{a`f lrb }b`amaef {c} bi Bw`b/

ibhuïpaoc Fi`fieb eb If [f}}f bh bi Fqrhufoabhuc(

[cpub}ac}obhub eac `coabh~c f ifp 61>;; gc}fp if {}rbmf ebhcoahfef ’@a}`rauc Ofuboæua`c‟( Bh biif

icp {f}ua`a{fhubp ebocpu}f}ch prp `chc`aoabhucp ofuboæua`cp }bpci|abhec {}cmibofp vrb |b}pfmfh pcm}b bi

bhuc}hc gapuð}a`c q ochrobhufi eb If [f}}f( Ifp 6; {}rbmfp {}c{rbpufp pb }b{f}uab}ch {c} ucef if ic`fiaefe

eb If [f}}f(

Ebp{ráp eb if {}rbmf ebi `a}`rauc) pb ibp eac f icp {f}ua`a{fhubp if c{c}urhaefe eb eapl}ruf} eb uabo{c

iam}b( F `chuahrf`aðh ucecp icp ahubj}fhubp eb if ciao{afef `bhf}ch bh bi oapoc `bhu}c ebi {rbmic(

Bi eïf 5 eb ofqc pb }bfia~ð if {}rbmf ahea|aerfi f ifp 6; eb if oföfhf bh bi A(B(P(C( ’\a`bhub Lb/

}}b}‟( Ufomaáh hcp jrpuf}ïf }b`chc`b} bi u}fmfnc iib|fec f `fmc {c} bi A(B(P(C( -\a`bhub Lb}}b}-4 {}clbpc}bp)

firohfec q `coc hc) f pr ea}b`uc} E( Irap L}fh`ap`c Pæh`gb~ Lb}hæheb~(

F ifp 6=>0; pb }bfia~ð if |apauf jrafef f Pfi|fuab}}f eb icp Mf}}cp( Eapl}rufocp eb if f}ubpfhïf eb rh

ufiib} eb filf}b}ïf q bi Orpbc eb Filf}b}ïf eb If [f}}f( F obeaceïf {reaocp eapl}ruf} eb rhf bpur{bhef `c/

oaef bh Pfi|fuab}}f bh `co{föïf eb if {}bpaebhuf eb if F(O([(F( ebi A(B(P(C( ’\a`bhub Lb}}b}‟( U}fp rh ebp/

`fhpc bh If Gcp{beb}ïf) lraocp {c} if uf}eb f |apauf} bi @fpuaiic eb Lb}af q eapl}ruf} eb prp bo{ahfefp `fiibp(

Qf mabh f|fh~fef if uf}eb lraocp f |apauf} if f`cjbec}f ic`fiaefe eb If Oc}b}f) pr ajibpaf) prp `fiibp)

bu`((( Ebp{ráp eb if `bhf) vrb ga`aocp bh if ic`fiaefe fhub}ac}obhub obh`achfef) eapl}rufocp eb rhf cmpb}/

|f`aðh fpu}chðoa`f ebi `abic `ch ubibp`c{acp(

Page 11: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 11/43

66

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

Page 12: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 12/43

WWA Ciao{afef Ofuboæua`f

6=

Bwu}bofer}f =;6=

Fi eïf pajrabhub) ebp{ráp eb ebpfqrhf}) pb {}c`beað f ocpu}f} f icp {f}ua`a{fhubp bh If Gcp{beb}ïf if }bpcir/

`aðh eb icp {}cmibofp ahea|aerfibp {}c{rbpucp bi pæmfec fhub}ac}(

F@UC EB @IFRPR]F

Bh bpuf c`fpaðh) bi F`uc bpur|c {}bpaeaec {c} bi fi`fieb eb If [f}}f E( Pfhuafjc Jch~æib~ Ifjf})

Ahp{b`uc}f eb Ber`f`aðh) E® Bh`f}hf`aðh Mbiic Ochub}c) E( Irap L}fh`ap`c Pæh`gb~ Lb}hæheb~ q E( ]a`f}/

ec Irbhjc Jch~æib~) `coc [}bpaebhub eb if Pc`abefe Bwu}boböf eb Ber`f`aðh Ofuboæua`f ’\bhur}f ]bqbp

[}ðp{b}‟(

Bh bpub F`uc pb {}c`beað f hcom}f} f icp 06 firohcp {f}ua`a{fhubp) vrb }b`cjab}ch rh ea{icof {c}

pr {f}ua`a{f`aðh bh if Lfpb Fruchðoa`f eb if Ciao{afef) fpï `coc rhf `fi`rifec}f `abhuïla`f q oæp f}uï`ricp

eb }bjfic(

Icp u}bp bvra{cp vrb }bpriuf}ch jfhfec}bp bh if [}rbmf ebi @a}`rauc Ofuboæua`c lrb}ch '{c} c}ebh

hroá}a`c!>

Page 13: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 13/43

60

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

J]R[C 6

FOFEC] [C]]FP  NBPÞP  [IFPBH@AF F]AFP FMBIAFHF  U]AFHF  OB]AEF ]C[B]C J]FJB]F  EFHABI  \AIIFL]FH@F EB ICP MF]]CP 'MF! 

J]R[C 5

UÁIIB^ ECOÏHJRB^ ]CMB]UC @Æ@B]BPPFH@GB^ OFR]B]F OF]@C FHUCHAC MFEFNC^]CE]AJRB^ OF]@CP @F]OBIF [IFPBH@AF '@@!

J]R[C 7

LB]HÆHEB^ MBHÏUB^ OF]ÏF OÁ]AEF 'MF!@F]]B]C JCH^FIB^ L]FH@AP@C NF\AB] @C]AF '@@!

L]FH@C @FPUFÖC EABJC \AIIFL]FH@F EB ICP MF]]CP 'MF!

@coc `ciclðh fi f`uc pb hcom}f}ch f icp u}bp firohcp vrb }b{}bpbhuf}ïfh f Bwu}bofer}f bh if WWAA

Lfpb Hf`achfi eb if Ciao{afef Ofuboæua`f vrb pb `bibm}ð bh \ajc ebi =< fi 0; eb nrhac eb =;66( Icp }b{}b/

pbhufhubp lrb}ch>

FHF PÆH@GB^ ]A\B]C  @Æ@B]BP [BE]C PÆH@GB^ MCIÏ\F]   OCHUANC 

PFORBI JAI OF]UÏH  \FI\B]EB EBI L]BPHC 

 Oajrbi Æhjbi Oc}bhc ]bechec) 

@cc}eahfec} ]bjachfi eb if WW Ciao{afef Ofuboæua`f bh Bwu}bofer}f(

Page 14: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 14/43

WWA Ciao{afef Ofuboæua`f

65

Bwu}bofer}f =;6=

WWAA CIAO[AFEF OFUBOÆUA@F HF@ACHFI

Ebi =< fi 0; eb Nrhac eb =(;66 pb `bibm}ð if Lfpb Hf`achfi eb if Ciao{afef {f}f firohcp eb =¾ eb

BPC bh \ajc( Icp u}bp }b{}bpbhufhubp eb Bwu}bofer}f vrb {}b|afobhub gfmïfh paec pbib``achfecp) u}fp pr/{b}f} if lfpb `cof}`fi q if fruchðoa`f lrb}ch> Fhf Pæh`gb~ ]a|b}c) [be}c Pæh`gb~ Mciï|f} q Pforbi Jai

Of}uïh vrb lrb}ch f`co{föfecp {c} bi {}clbpc} Fhuchac Ocifhc eb if Pc`abefe Bwu}boböf ’\bhur}f ]bqbp

[}ðp{b}‟( U}fp rh if}jc |afnb f u}f|áp eb [c}urjfi) iibjfocp f \ajc q hcp ficnfocp bh bi @bhu}c ]bpaebh`afi

Ec`bhub 'Fhuajrf Rha|b}paefe Ifmc}fi! nrhuc `ch icp }b{}bpbhufhubp eb ifp cu}fp `c/

orhaefebp fruðhcofp( Ebp/ {ráp ebi }b{f}uc eb gfmauf`achbp

`cobh~ð if `ch|a|bh`af q bi aha`ac eb rhf foapufe( Bpb

oapoc eïf) u}fp if `bhf) grmc rhf }brhaðh {f}f `ch/

`}buf} ebufiibp) }b{f}/ ua} ifp f`}beauf`achbp) f/

oapbufp) bu`… q pb lc}of}ch icp bvra{cp

{f}f bi `ch`r}pc eb lcucj}flïfp ofuboæua`fp(

Icp firohcp |fh gf`abhec lcucp bh ucec

ic vrb |bfh vrb grb/ ib f ofuboæua`fp q `coc

oæwaoc gfpuf bi eïf =7 pb {ceïfh bhu}bjf}( Bhbi f`uc eb `ifrpr}f pb {}c`ifof}ïfh icp bvra{cp

jfhfec}bp( If f`urf`aðh eb rh `ðoa`c ic`fi hcp bhu}bur|c

ifp gc}fp {}b|afp fi prböc(

U}fp rhf hc`gb `ch {c`fp gc}fp eb prböc 'uï{a`c eb if befe!) fi eïf pajrabhub hcp u}fpifefocp f if

Rha|b}paefe eb \ajc echeb pb }bfia~ð if {}rbmf ahea|aerfi eb }bpcir`aðh eb {}cmibofp) `coaocp bh bi `c/

obec} rha|b}pauf}ac q {c} if uf}eb pb ga~c rhf |apauf ofuboæua`f {c} \ajc) `ch if }bpcir`aðh {c} bvra{cp eb

f`ua|aefebp ofuboæua`fp f{}c|b`gfhec icp efucp }bfibp vrb if {}c{af `arefe {}c{c}`achf( Icp bvra{cp icp

lc}oð if c}jfha~f`aðh ob~`ifhec firohcp eb ea|b}pfp `corhaefebp(

Bi eïf =7) ebp{ráp eb ebpfqrhf}) hcp lraocp f if ah`co{f}fmib Pfhuafjc( Fiiï hcp bp{b}fmfh `co{f/

öb}cp eb if Pc`abefe Jfiibjf FJF[BOF vrb hcp jraf}ch bh rh `co{ibuc }b`c}}aec ur}ïpua`c vrb `cobh~ð

bh if `fube}fi echeb ur|aocp if prb}ub eb |b} bh f``aðh bi lfocpc Mcuflroba}c( U}fp if `coaef bh rh `cob/

ec} rha|b}pauf}ac eb if Rha|b}paefe eb Pfhuafjc) hcp ubhïfh {}b{f}fec rh {fpbc ofuboæua`c) eb lc}of vrb

|apaufocp icp }ah`chbp oæp ahub}bpfhubp ebi `fp`c gapuð}a`c }bif`achæhecicp `ch fp{b`ucp ofuboæua`cp( Qf eb}bj}bpc f \ajc) fpapuaocp f if f`urf`aðh eb rh ’`rbhuf`rbhucp‟ vrb hcp ebibauð if |bifef(

Page 15: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 15/43

62

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

Bi =1) ebp{ráp eb rh j}fh ofe}rjðh) hcp ea}ajaocp f if Ichnf ebi {rb}uc {bpvrb}c eb \ajc) rhc eb

icp oæp ao{c}ufhubp eb Br}c{f) echeb |aocp if prmfpuf eb {bp`fec q fijrhcp bnbo{if}bp eb if lfrhf of}ahf

{}b{f}fecp {f}f pr |bhuf q {cpub}ac} eapu}amr`aðh {c}

ucef Bp{föf q [c}urjfi( Hcp ah|auf}ch f ebpfqrhf} bh

rhf eb prp `flbub}ïfp q) `ch hrbpu}cp ’mc`fufp‟) `c/

jaocp bi mf}`c vrb hcp u}fpifeð f ifp apifp @ïbp) bpf

}bpb}|f hfur}fi ufh mabh `raefef vrb lc}of {f}ub ebi

{fu}aochac eb if grofhaefe( Ur|aocp rh mrbh eïf eb

pci q eapl}rufocp eb prp {fapfnbp q eb prp {ifqfp( F if

`fïef eb if uf}eb) }bj}bpfocp f if }bpaebh`af q) u}fp if

`bhf) grmc rhf pbpaðh eb }bpcir`aðh eb {}cmibofp

echeb pb `cobhuf}ch icp vrb pb {}c{rpab}ch bh if

{}rbmf ahea|aerfi( If þiuaof hc`gb lrb eb {c`c prböc q or`gf `ch|a|bh`af) or`gc oc|aoabhuc {c} icp {fpaiicp q or/

`gcp |afnbp f {c} fjrf l}bp`f bh if lrbhub fi ifec eb if bhu}fef( If foapufe vrb pb lrb lc}nfhec er}fhub icp

eïfp vrb gf er}fec if Ciao{afef hc pb ci|aef læ`aiobhub(

Iibjð bi eïf 0; q) u}fp bi ebpfqrhc q qf `f}jfecp `ch hrbpu}cp bvra{fnbp) hcp u}fpifefocp f rh [f~c

bh bi {f}vrb orha`a{fi eb @fpu}bicp) vrb uabhbh orq mabh {}b{f}fec {f}f f`ucp {}cuc`cif}acp `coc bi eb if@ifrpr}f eb if WWAA Ciao{afef Ofuboæua`f Hf`achfi( Fpapuað bi fi`fieb eb \ajc) bi bw oahapu}c Fmbi @fmf/

iib}c) lrb}ch bhu}bjfecp icp {}boacp q hcoahfecp icp `ah`c jfhfec}bp {c} c}ebh filfmáua`c( Hrbpu}f firohf

Fhf lc}oð {f}ub ebi bvra{c jfhfec} eb lcucj}flïfp ofuboæua`fp( Ebp{ráp) f if `f}}bub}f) vrb) {c} uab}}fp

irpaufhfp) hcp chernc f hrbpu}cp }bp{b`ua|cp ecoa`aiacp( Fhuchac Ocifhc ]cob}c( [}clbpc} f`co{föfhub eb icp firohcp bwu}boböcp(

Page 16: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 16/43

WWA Ciao{afef Ofuboæua`f

6<

Bwu}bofer}f =;6=

]bif`aðh eb @bhu}cp {f}ua`a{fhubp bh ifWW Ciao{afef Ofuboæua`f bh Bwu}bofer}f

^CHF EB FEP@]A[@AÐH> FIMR]VRB]VRB

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

@FPUAIIC EB IRHF FIMR]VRB]VRB FHF @C]^C FQRPC 2

@FPUAIIC EB IRHF FIMR]VRB]VRB ÆHJBI L@C( FOM]CNC FHUÞHB^ <

PAB]]F EB PFH [BE]C ]C@F EB IF PAB]]F 'IF! ÆHJBI GB]HÆHEB^ ]CMIBP :

PAB]]F EB PFH [BE]C ]C@F EB IF PAB]]F 'IF! OAJRBI ÆHJBI LB]]B]C JF]]CUB 0

ICRPUFR/\FI\B]EB \FIBH@AF EB FI@ÆHUF]F O® OB]@BEBP PÆH@GB^ LB]HÆHEB^ 7

UCUFI ^CHF FIMR]VRB]VRB =1

^CHF EB FEP@]A[@AÐH> FIOBHE]FIBNC

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

UAB]]F EB MF]]CP F@BR@GFI NCPÁ OF]ÏF OÁHEB^ OÁHEB^ 2

UAB]]F EB MF]]CP F@BR@GFI NRFH @GF\B]C ]CE]ÏJRB^ 2

UAB]]F EB MF]]CP F@BR@GFI IRAP [A^F]]C JFIIF]FEC 2

UAB]]F EB MF]]CP F@BR@GFI PB]JAC PFHUCP ]CPBII 2

@F]CIAHF @C]CHFEC FIOBHE]FIBNC ÐP@F] MC]]BJC MB]OBNC 2

@F]CIAHF @C]CHFEC FIOBHE]FIBNC [F@F JCH^ÆIB^ OFH@GÐH 0

LRBHUB ]CHABI LRBHUB EBI OFBPU]B L@C( NF\AB] MC]]BJRB]C JÐ/OB^

5

ICP OC]AP@CP GC]HF@GCP ÆI\F]C JFÖÆH PB]]FHC 6;

IF [F]]F [F]]F 'IF! IC]BH^C ORÖC^ OCHUB]C 6=

\FIEBOBEBI ]AMB]F EBI L]BPHC MBIÁH ]AMFIIC ]RA^/]CPC 6;

Page 17: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 17/43

6:

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

\FIEBOBEBI ]AMB]F EBI L]BPHC [BE]C N( OFUFOC]CP ÆI\F]B^ 6;

PAB]]F IF @FIB]F PUF( OF]UF EB ICP MF]]CP IR@ÏF IFH@GF]]C [Á]B^ 6;

OF]AFHC MF]MF@AE PCIFHF EB ICP MF]]CP NRIAÆH JRUAÁ]]B^ GR]UFEC =

HU]F( P]F( EBI @F]/OBH

\AIIFL]FH@F EB ICP MF/]]CP

ECIC]BP JCH^ÆIB^ LIC]BP 6;

PFH NCPÁ \AIIFL]FH@F EB ICP MF]]CP PFHUCP [AHUC @B]B^C 1

UCUFI ^CHF FIOBHE]FIBNC 6;2

^CHF EB FEP@]A[@AÐH> F^RFJF#IIB]BHF

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

MBOMÁ^F] F^RFJF OÐHA@F BPVRA\BI OCHUB]]RMAC 0

MBOMÁ^F] F^RFJF [BE]C ]CUAIA ]RA^ 5

MBOMÁ^F] F^RFJF ]C@ÏC PCIBEFE @F]@BIÁH ]CE]ÏJRB^ <

MBOMÁ^F] F^RFJF ]CPF OF]ÏF JRB]]F [AHUC] 2

OAJRBI ER]ÆH F^RFJF [FRIF MB]OBNC MF]]FJÆH 1

@RFU]C \AIIFP MB]IFHJF JBOF OC]RHC [FHAFJRF 7

IIB]BHF IIB]BHF NRFH PFHUAFJC JF]@ÏF ^F[FUF 6;

IIB]BHF IIB]BHF ]CPF]AC UBHF OC]FIBP <

HU]F( P]F( EB IF J]FHFEF IIB]BHF BIAPF OF]UÏH PÆH@GB^ 6;

UCUFI ^CHF F^RFJF#IIB]BHF <6

^CHF EB FEP@]A[@AÐH> MFEFNC^

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

MAC@IAOÆUA@C MFEFNC^ NCPÁ O® HF\F]]C JCH^ÆIB^/IFLCHU :

MAC@IAOÆUA@C MFEFNC^ O® ICR]EBP @CIIFEC ]BUFOF] <

@AREFE NF]EÏH MFEFNC^ O® EBI @F]OBH GB]HÆHEB^ OF]UÏ/ 0

Page 18: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 18/43

WWA Ciao{afef Ofuboæua`f

67

Bwu}bofer}f =;6=

HB^

HU]F( P]F( EBI @F]OBH MFEFNC^ AJHF@AC F@GF =

HU]F( P]F( EBI @F]OBH MFEFNC^ ]( [FU]A@AF GB]HÆHEB^ OFBPC 6;

[RB]UF[FIOF MFEFNC^ @F]IF [AF^^F 7

]CE]ÏJRB^ OCÖAHC MFEFNC^ OFHRBI MBHÏUB^ MBHÏUB^ 6;

]CE]ÏJRB^ OCÖAHC MFEFNC^ ]FLFBI ORÖC^ LB]HÆHEB^ :

PFH FUÐH MFEFNC^ IRAP @F]ICP RMABUC JCH^ÆIB^ 61

PFH LB]HFHEC MFEFNC^ NCPÁ OF]ÏF OF]UÏH JCH^ÆIB^ 0

PFH NCPÁ MFEFNC^ NRFH OFHRBI @GÆ\B^ OC]BHC 5

PFH@UF OF]ÏF FPPRO/[UF MFEFNC^ IBC[CIEC GB]]B]F PÆH@GB^ :

PFHUF UB]BPF EB NBPÞP MFEFNC^ OF]ÏF UB]BPF OFEB]F @RFE]FEC 2

\A]JBH EB JRFEFIR[B MFEFNC^ EABJC @FOAPÐH JCH^ÆIB^ 60

^R]MF]ÆH MFEFNC^ L]FH@AP@C OC]BHC PCUC =

[RBHUB FNREF CIA\BH^F FHF OAIFJ]CP J]FNB]F GB]HFHEC 7

BH]AVRB EÏB^ @FHBEC [RBMIF EB IF @FI^F/EF

NRFH @C]UÁP []B@AFEC 0

@FO[CP EB PFH ]CVRB \FI\B]EB EB IBJF/HÁP

OF]ÏF NBPÞP OFUBCP @C]@GB]C <

UCUFI ^CHF MFEFNC^ 6=0

^CHF EB FEP@]A[@AÐH> @Æ@B]BP

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

ÆJC]F @Æ@B]BP O® FR]BIAF NAOÁHB^ [C]UAIIC 5

BI M]C@BHPB @Æ@B]BP O® UB]BPF PÆH@GB^ [C]]FP :

IF FPRH@AÐH @Æ@B]BP FHUCHAC EÆ\AIF LB]HÆHEB^ 66

IF FPRH@AÐH @Æ@B]BP O® NCPÁ NAOÁHB^ MF@GAIIB] 1

IA@BH@AFECP ]BRHAECP @Æ@B]BP NRFH @C]UÁP OF]JFIIC <

IA@BH@AFECP ]BRHAECP @Æ@B]BP ]RMÁH OCIFHC JÐOB^ 1

Page 19: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 19/43

61

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

HC]MF @FBPF]AHF @Æ@B]BP ]FVRBI PÆH@GB^ MFÖB^F 2

GB]HÆHEB^ [F@GB@C @Æ@B]BP FHUCHAC OCIFHC ]COB]C 7

GB]HÆHEB^ [F@GB@C @Æ@B]BP AHÁP OFH^FHC \BHUR]F <

GB]HÆHEB^ [F@GB@C @Æ@B]BP NCPÁ O® ^FOM]FHC F@BEC 6;

PFJ]FEC @C]F^ÐH EB NBPÞP @Æ@B]BP O® EBI @F]OBH LB]]B]C ]FOCP 0;

PFH FHUCHAC EB [FERF @Æ@B]BP BOAIAC OC]BHC PÆH@GB^ 6;

PFHUF @B@AIAF @Æ@B]BP ]CPF JÐOB^ JF]EBÖFP 1

\ÏF EB IF [IFUF @FPF] EB @Æ@B]BP @F]ICP MBHAJHC OFH@BMC [BHF 1

\ÏF EB IF [IFUF @FPF] EB @Æ@B]BP OF]ÏF EB ICP ÆHJBIBP ]RMACJCH^ÆIB^

1

@B]]C [BE]C JÐOB^ OFE]CÖB]F EABJC NCPÁ IRMAÆH GB]]B]F 2

AHUB]HF@ACHFI PFH NC]JB OFI[F]UAEF EB @Æ/@B]BP

OB]@BEBP [FHFEB]C [F@GB@C 6

L]FH@AP@C EB C]BIIFHF U]RNAIIC IFR]F ICPFEF \ABA]C 7

UCUFI ^CHF @Æ@B]BP 62<

^CHF EB FEP@]A[@AÐH> @C]AF

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

BI M]C@BHPB M]C^FP IRAP ]FOCP PCIFHC =

@BIIF \AHF]AF @B@IF\ÏH EABJC GB]HÆHEB^ F]FOMAIBU <

FIFJÐH @C]AF @FPAIEF JF]@ÏF \A@BHUB 7

FIFJÐH @C]AF L]FH@AP@C OFHRBI @C]MF@GC JFÖÆH 7

FIFJÐH @C]AF NCPÁ IRAP @F][AHUB]C OCHUB]C 7

FIFJÐH @C]AF NRIAÆH PÆH@GB^ FIMFIÆ :

FIFJÐH @C]AF OF]ÏF MBIÁH OC]ÐH ]ÏCP :

FIFJÐH @C]AF QCIFHEF [FU]A@AF FQRPC GCIJFEC :

NFIFOF OC]FIBNF ÆHJBIF MC]]BJC UF[AF 6:

\ÏF EFIOF@AF UC]]BNCH@AIIC L]FH@AP@C NF\AB] \F]BIF L]FEB 6;

Page 20: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 20/43

WWA Ciao{afef Ofuboæua`f

=;

Bwu}bofer}f =;6=

\ÏF EFIOF@AF UC]]BNCH@AIIC GB]OAHAF EBI FOC] OF]UÏH IÐ[B^ 1

\ÏF EFIOF@AF UC]]BNCH@AIIC NCPÁ [BE]C OF]UÏH IC]BH^C 6;

\FI EB WÆIAOF \FI\B]EB EBI L]BPHC OF]ÏF EBI OF] [Á]B^ HC]ABJF 0

\FI EB WÆIAOF \FI\B]EB EBI L]BPHC OF]ÏF LB]HFHEF \AIIFIMF JRAIIÁH 5

\FI EB WÆIAOF \FI\B]EB EBI L]BPHC \B]ÐHA@F OF]UÏH JRUAÁ]]B^ 7

UCUFI ^CHF @C]AF 665

^CHF EB FEP@]A[@AÐH> ECH MBHAUC

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

MF]UCICOÁ N( JFIIF]/EC

@FO[FHF]AC L]FH@AP@C MIFH@C NRÆ]B^ 5

@FPURB]F @FPURB]F NF@CMC ICMC [FP@RF :

@IF]BU ECH MBHAUC ÆHJBI ]CMRPUAIIC HÞÖB^ =;

@RFU]C @FOAHCP ECH MBHAUC FR]BIAC ]FPB]C <

@RFU]C @FOAHCP ECH MBHAUC B\F ]RA^ [FNRBIC <

ECHCPC @C]UÁP ECH MBHAUC BOAIAC [AÖBA]C LBC 6;

ECHCPC @C]UÁP ECH MBHAUC NCPÁ IRAP IBFI @AECH@GF 67

ECHCPC @C]UÁP ECH MBHAUC OF]ÏF ICR]EBP OC]BHC MFI/@CHB]C

7

NCPÁ OFH^FHC ECH MBHAUC FHF OF]ÏF IÐ[B^ J]FHFECP 6;

NCPÁ OFH^FHC ECH MBHAUC LB]HFHEC \AJF]AC U]BHFEC 2

NCPÁ OFH^FHC ECH MBHAUC OF]ÏF OAIFJ]CP OC]@AIIC OF/E]RJF

6<

IRAP @GFOA^C ECH MBHAUC/\AIIFHRB\F

BI\A]F @FIEB]ÐH OC]FIBP 6=

BRJBHAC L]RUCP JRF]BÖF APFMBI J]FHFECP JF]@ÏF 0

BRJBHAC L]RUCP JRF]BÖF O® APFMBI JFIAHEC FI\F]FEC <

BRJBHAC L]RUCP JRF]BÖF \B]ÐHA@F PB]]FHC OF@ÏFP <

VRAHUFHF EB IF PB]B/HF

VRAHUFHF EB IF PB/]BHF

L]FH@AP@C UC]]FEC @FHC 0

Page 21: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 21/43

=6

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

[BE]C EB \FIEA\AF \AIIFHRB\F EB IFPB]BHF

@CHPRBIC JRUAÁ]]B^ IÐ[B^ 6;

[BE]C EB \FIEA\AF \AIIFHRB\F EB IFPB]BHF

JBOF LB]HÆHEB^ \BJF 1

[RB]UF EB IF PB]BHF \AIIFHRB\F EB IFPB]BHF

NCPÁ OAJRBI MIFH@C @FPFEC =

PFH NCPÁ \AIIFHRB\F EB IFPB]BHF

NCPÁ OF]ÏF @F[AIIF FIMF 0

UCUFI ^CHF ECH MBHAUC 6<5

^CHF EB FEP@]A[@AÐH> OÁ]AEF

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

UFORNFI F]]CQC EB PFH PB]/\ÆH

NRFH IRAP [A^F]]C JFIÆH =

ERI@B @GF@ÐH JF]]C\AIIF 'IF! OFHRBI ]CE]ÏJRB^ OF@GR@F 5

ERI@B @GF@ÐH JF]]C\AIIF 'IF! [R]ALA@F@AÐH [AHUC @C]]FIA^F =

OF]ÏF FRWAIAFEC]F OÁ]AEF NBPÞP @RÁIIF] JRB]]B]C 1

PFHUF BRIFIAF OÁ]AEF FHF O® JF]@ÏF MFÖCP 2

PFHUF BRIFIAF OÁ]AEF L]FH@AP@C [C^C L]ÏFP 0

PFHUF BRIFIAF OÁ]AEF O® APFMBI @F]]BUB]C \FI 2

PFHUF BRIFIAF OÁ]AEF O® NCPÁ [RIAEC OF]UÏHB^ =

PFHUF BRIFIAF OÁ]AEF ]FLFBI JCH^ÆIB^ NAOÁHB^ 7

BWU]BOFER]F OCHUANC EF\AE ]CE]ÏJRB^/BPUB@GF ÆI/\F]B^

5

BWU]BOFER]F OCHUANC NRFHF FMBIF \BIB]EF 5

UAB]]FMIFH@F ^F]^F 'IF! F]ÆH^F^R NAOÁHB^ PÆH@GB^ 5

UAB]]FMIFH@F ^F]^F 'IF! NRFH L]FH@AP@C F]BHFP MBHÏUB^ =

UAB]]FMIFH@F ^F]^F 'IF! NRIAC F]AFP JCH^ÆIB^ 0

UCUFI ^CHF OÁ]AEF 2:

Page 22: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 22/43

WWA Ciao{afef Ofuboæua`f

==

Bwu}bofer}f =;6=

^CHF EB FEP@]A[@AÐH> [IFPBH@AF

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

JFIAPUBC JFIAPUBC FIL]BEC HÞÖB^ PFIFP 7

VRB]@RP OFI[F]UAEF EB [IFPBH/@AF

NRFH OF]ÏF OFH@BMC OF]UÏH :

VRB]@RP OFI[F]UAEF EB [IFPBH@AF OB]@BEBP OC]ÆH @FHBIC 6;

\FIIB EBI NB]UB HF\F@CH@BNC BOAIAC EÏF^ ]CE]ÏJRB^ 1

JFM]ABI Q JFIÆH [IFPBH@AF BOAIAF ]CE]ÏJRB^ JF]@ÏF 6

JFM]ABI Q JFIÆH [IFPBH@AF APFMBI O( GB]HÆHEB^ JCH^Æ/IB^ 0

[F]VRB EB OCHL]FJÙB [IFPBH@AF O® APFMBI GB]HÆHEB^ @FMFIIC 6

[F]VRB EB OCHL]F/JÙB

[IFPBH@AF O® PCIBEFE @C]]BFP OF]UÏH 0

[F]VRB EB OCHL]FJÙB [IFPBH@AF PF]FQ FJRPUÏH JFIIBJC <

[B]B^ @COBHEFEC] [IFPBH@AF FO[F]C PFHUCIAHC [B]BÖF 65

PFH @FIAWUC [IFPBH@AF [BE]C F( OF]UÏH NAOÁHB^ 6;

\A]JBH EBI [RB]UC [IFPBH@AF IRAP MCUB PCIÏP =

UCUFI ^CHF [IFPBH@AF :5

^CHF EB FEP@]A[@AÐH> PA]RBIF

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

MBHF^FA]B GB]]B]F EBI ERVRB NRFH OFHRBI ]COB]C MF]@C 6;

\A]JBH EB FIUFJ]F@AF PA]RBIF APFMBI OF]ÏF ]CIEÆH JFIIF]EC 6;

\A]JBH EB FIUFJ]F@AF PA]RBIF [BE]C ]A@C JCH^ÆIB^ 6;

UCUFI ^CHF PA]RBIF 0;

Page 23: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 23/43

=0

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

^CHF EB FEP@]A[@AÐH> FL]F

@BHU]C IC@FIAEFE []CLBPC] H¾ EBFIROHCP

OFUÏFP ]FOÐH OF]UÏ/

HB^

MR]JRAIICP EBI @B]]C JIC]AF OF]ÏF ECOÏHJRB^ IBÐH 1

BRJBHAC GB]OCPC L]BJBHFI EB IF PAB]]F NCPÁ OF]ÏF ICMC ]CE]ÏJRB^ 6;

FIMF [IFUF LRBHUB EB @FHUCP O® @]R^ OBHEC^F [CH@B 5

MBHF^FA]B GB]]B]F EBI ERVRB HFUFIAC EÏF^ GB]]B]F =

OFBPU]C NRFH @FIB]C OCHBPUB]AC JBOF APFMBI [B@B]C ]CNFP 6;

OFBPU]C NRFH @FIB]C OCHBPUB]AC O® IRAPF \F^VRB^ MR]JRBÖC 0

E]( LB]HÆHEB^ PFHUF/HF

PFHUCP EB OFAOCHF'ICP!

IBFHE]C EÏF^ @]BP[C 62

AIEBLCHPC PB]]FHC PBJR]F EB IBÐH APFMBI OF]ÏF [A@ÐH NF]FOAIIC 2

AIEBLCHPC PB]]FHC PBJR]F EB IBÐH NCPÁ OF]ÏF ]COB]C FJRAIF] 2

@]APUC EBI ]CPF]AC ^FL]F NCPÁ OF@ÏFP OF]ÏH 1

@]APUC EBI ]CPF]AC ^FL]F OAJRBI ÆHJBI GB]HÆHEB^ IC/]BH^C

7

OF]ÏF AHOF@RIFEF ^FL]F IAEAF \FIABHUB [FIC[ <

PRÆ]B^ EB LAJRB]CF ^FL]F FMAIAC @C]@GBUB JCH^ÆIB^ 2

PRÆ]B^ EB LAJRB]CF ^FL]F FHUCHAC OF]F\B] JRB]]B]C =;

PRÆ]B^ EB LAJRB]CF ^FL]F F]@ÆHJBI ORÖC^ ]CE]ÏJRB^ 1

PRÆ]B^ EB LAJRB]CF ^FL]F [AIF] @FHUFIBNC OF]UÏH 6=

UCUFI ^CHF ^FL]F 60=

UCUFI [F]UA@A[FHUBP BH IF CIAO[AFEF OFUBOÆUA@F 6;52

Page 24: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 24/43

WWA Ciao{afef Ofuboæua`f

=5

Bwu}bofer}f =;6=

[}cmibofp eb ifp eapuahufp lfpbp

LFPB @COF]@FI

NF]EÏH JBCOÁU]A@CBi nf}eïh eb oa `bhu}c uabhb rh {f}ub}}b `ch if oapof lc}of vrb if ~chf pcom}bfef eb if lajr}f( Bi

`chuc}hc bwub}ac}) `coc {rbebp cmpb}|f}) bp rh u}aæhjric bvraiæub}c eb < obu}cp eb ifec bh bi vrb gbocpu}f~fec icp {rhucp obeacp eb prp ifecp q pb gfh rhaec `ch u}f~cp lc}ofhec cu}c u}aæhjric bvraiæub}c( Rhabh/ec hrb|fobhub icp {rhucp obeacp eb icp ifecp eb bpub þiuaoc u}aæhjric lc}ofocp bi u}aæhjric ahub}ac} eb iflajr}f(

Ebub}oahf}>

If fiur}f q bi æ}bf ebi u}aæhjric ahub}ac} eb |á}ua`bp icp {rhucp E) B q L(

Bi æ}bf ucufi ebi {f}ub}}b '~chf pcom}bfef!(Bi nf}eahb}c vrab}b ea|aea} bi {f}ub}}b bh u}bp lajr}fp ajrfibp( Ahea`f u}bp lc}ofp ealb}bhubp eb }bfia~f} ea`gfea|apaðh(

Pcir`aðh>

i  Bi ifec ebi u}aæhjric ahub}ac} oaeb <#5 ? 6)2 o) {c} ufhuc f{ia`fhec bi ubc}bof eb [auæjc}fp) ubhbocp>g= ? '6)2!= ‐'6)2#=!= ? =)=2 ‐ ;)2<=2 ? 6)<7:2 ?8 g ? 6)0 o '0 {rhucp!

Æ}bf ebi u}aæhjric eb |á}ua`bp icp {rhucp E) B q L pb}æ> Æ}bf ? 6)2 ▮ 6)0 # = ? ;)1:2 o= '= {rhucp!

i  Bi æ}bf ebi {f}ub}}b '~chf pcom}bfef! pb `co{chb eb 62 u}aæhjricp `coc bi ebi ahub}ac}) {c} ufhuc præ}bf pb}æ> Æ}bf {f}ub}}b ? 62 w ;)1:2 ? 65)<=2 o=

@rfivrab} cu}c oáucec }f~chfec( [c} bnbo{ic> bi æ}bf ebi u}aæhjric eb |á}ua`bp F) M q @ obhcp bi æ}bf ebiu}aæhjric eb |á}ua`bp E) B q L( '= {rhucp!

i  Fvrï ub {}bpbhufocp u}bp {cpamibp pcir`achbp '6 {rhuc fef rhf!> '0 {rhucp! 

Page 25: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 25/43

=2

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

@FOMAC EB IBU]FP [C] HÞOB]CP

If pajrabhub bw{}bpaðh }b{}bpbhuf rhf oriua{ia`f`aðh bh if vrb `fef ibu}f }b{}bpbhuf rhf al}f(

Pfmbocp vrb if ibu}f F bp ajrfi fi |fic} eb rhf {cubh`af eb mfpb ecp q eb bw{chbhub rh hþob}c {}aoc ao/{f}) q vrb ] q U pch hþob}cp {}aocp(

]bfia~f ifp pajrabhubp `rbpuachbp>

/@fi`rif} bi |fic} eb F(/Ahea`f} ifp `al}fp ebi ; fi 1 vrb pch hþob}cp {}aocp(/Ebub}oahf} ifp `al}fp vrb }b{}bpbhufh ifp ibu}fp ] q U(

/Ebub}oahf} if `al}f vrb }b{}bpbhuf if ibu}f O(

Pcir`aðh>

i  '= {rhucp! @fi`rif} bi |fic} eb F ? =0 ? 7i  '6 {rhuc! Ifp `al}fp ebi ; fi 1 vrb pch hþob}cp {}aocp> =) 0) 2 q :( Uabhbh vrb bpuf} icp `rfu}cpcifobhub) bh `fpc `chu}f}ac hc pb {rhurf}æ bi f{f}ufec(i  '0 {rhucp! Ifp `al}fp vrb }b{}bpbhufh ifp ibu}fp ] q U( [c} rh pao{ib ufhubc pb cmuabhb vrb ] ? 2 q U? 0(i  '5 {rhucp! If `al}f vrb }b{}bpbhuf if ibu}f O( Ea|aeabhec bi hþob}c 222222 bhu}b 0 cmubhbocp672672) {c} ufhuc O ? 6( Ufomaáh bp |æiaec pa ic }bfia~f {c} ufhubc r cu}c }f~chfoabhuc(

]CP@FP EBI @FHEAI

Bh If [f}}f) ic`fiaefe {f`bhpb echeb `bibm}f}bocp if {}ðwaof Ciao{afef Ofuboæua`f }bjachfi) pchuï{a`fp {f}f Pbofhf Pfhuf ifp }cp`fp ebi `fheai) eri`b ebi vrb `fpa `chc`bocp pr }b`buf> =;; j}focp ebf~þ`f}) 5;; j}focp eb gf}ahf) 2; `o0 eb fhïp) 0;; `o0 eb f`baub eb cia|f) }fiifer}f eb iaoðh q rhf `fhuaefe

ebp`chc`aef eb grb|cp eb icp vrb pðic pfmbocp vrb pr hþob}c bp if pcir`aðh eb if pajrabhub b`rf`aðh>

@co{ibuf if }b`buf eb bpub bwvrapauc eri`b gfiifhec bi hþob}c eb grb|cp hb`bpf}acp(

Pcir`aðh>

]bpci|abhec if b`rf`aðh) bi hþob}c eb grb|cp vrb pb hb`bpaufh pch < grb|cp(

'Pa ef bi }bpriufec `c}}b`uc {b}c pah }bfia~f} hahjþh `æi`ric c {}c`beaoabhuc }f~chfec) pb {rhurf}æ`ch = {rhucp!(

 w$2

0∝

w ∝ =

2∝ =?

60

62

Page 26: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 26/43

WWA Ciao{afef Ofuboæua`f

=<

Bwu}bofer}f =;6=

PC[F EB IBU]FP

Bp`}amb bh bi }b`rfe}c `c}}bp{cheabhub ic vrb ub {beaocp) irbjc mrp`f bpf {fifm}f bh if pc{f eb ibu}fp(

6( Ofuboæua`c vrb lc}orið bi ubc}bof> -Pa ecp }b`ufp `rfibpvrab}f pb `c}ufh {c} |f}afp }b`ufp {f}fibifp)icp pbjobhucp ebub}oahfecp bh rhf eb ifp }b`ufp pch {}c{c}`achfibp f icp pbjobhucp `c}}bp{cheabhubp bh if

cu}f-(=( Ofuboæua`c j}abjc vrb eac rhf f{}cwaof`aðh orq {}b`apf ebi hþob}c ϋ (

0( Ofuboæua`c vrb lc}orið bi ubc}bof> -Bi æ}bf ebi `rfe}fec `chpu}raec pcm}b if ga{cubhrpf eb rh u}aæh/jric }b`uæhjric) bp ajrfi f if prof eb ifp æ}bfp eb icp `rfe}fecp `chpu}raecp pcm}b icp `fubucp-(

5( \fic} vrb u}fhplc}of rhf ajrfiefe fijbm}fa f bh rhf ajrfiefe f}auoáua`f(

2( \fic} eb 6:202 bib|fec f ;(

Pcir`aðh>

'= {rhucp {c} `fef {fifm}f bh`chu}fef> 6 {rhuc pa pcic {chb if {fifm}f `c}}b`uf bh bi }b`rfe}c {b}c hc ific`fia~f bh if pc{f eb ibu}fp) `b}c {rhucp pa hc `cic`f if {fifm}f `c}}b`uf bh bi }b`rfe}c `c}}bp{cheabhubfrhvrb if ic`fia`b bh if pc{f eb ibu}fp!(

6( Ofuboæua`c vrb lc}orið bi ubc}bof> -Pa ecp }b`ufp `rfibpvrab}f pb `c}ufh {c} |f}afp }b`ufp {f}fibifp)icp pbjobhucp ebub}oahfecp bh rhf eb ifp }b`ufp pch {}c{c}`achfibp f icp pbjobhucp `c}}bp{cheabhubp bh if

cu}f-( UGFIBP=( Ofuboæua`c j}abjc vrb eac rhf f{}cwaof`aðh orq {}b`apf ebi hþob}c ϋ ( F]VRÏOBEBP

0( Ofuboæua`c vrb lc}orið bi ubc}bof> -Bi æ}bf ebi `rfe}fec `chpu}raec pcm}b if ga{cubhrpf eb rh u}aæh/jric }b`uæhjric) bp ajrfi f if prof eb ifp æ}bfp eb icp `rfe}fecp `chpu}raecp pcm}b icp `fubucp-( [AUÆJC/]FP(

5( \fic} vrb u}fhplc}of rhf ajrfiefe fijbm}fa f bh rhf ajrfiefe f}auoáua`f( PCIR@AÐH(

2( \fic} eb 6:202 bib|fec f ;( RHC

 Fruc}bp eb if bifmc}f`aðh eb icp {}cmibofp eb if lfpb `cof}`fi> Brjbhaf Ið{b~ @æ`b}bp) O® Jrfefir{b Lrbhubp L}ïfp q F}ur}c Ofheiq Ofhpc

Page 27: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 27/43

=:

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

LFPB FRUCHÐOA@F( []RBMF AHEA\AERFI

@RFE]FEC Q U]AÆHJRICP

Bh bi `rfe}fec FM@E eb ifec 7 `o pb `chpu}rqb bi u}aæhjric bvraiæub}c FMB

Pb {aeb>

6(/ Æ}bf ebi u}aæhjric bvraiæub}c(

@chc`aec bi ifec ebi u}aæhjric bvraiæub}c) bi æ}bf bp>

'= [rhucp!

=(/ Æ}bf ebi u}aæhjric E@B(

 If mfpb ebi u}aæhjric E@B bp 7 `o q if fiur}f bp f?7/g pabhec g if fiur}f ebi u}aæhjric bvraiæub}c) vrb pb

 {rbeb cmubhb} f{ia`fhec bi ubc}bof eb [auæjc}fp>

 Bi æ}bf ebi u}aæhjric E@B bp>

'= [rhucp!

0(/ Æ}bf ebi u}aæhjric FEB(

Pa ucofocp `coc mfpb FE?7 `o) if fiur}f bp if oaufe ebi ifec ebi `rfe}fec) bp eb`a} 5 `o( Irbjc bi æ}bf

eb FEB bp

@

M

B

 F

E

i=

0

5

`o=?

<5 0

5

`o=?6< 0`o

=? =:) :6 `o

=

g ?i 0

=?

7 0

=? 5 0`o⇝ f ? 7∝ 5 0 o

7 7∝ 5 0' !=

?0=∝6< 0 o=? 5)=1 `o

=

Page 28: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 28/43

WWA Ciao{afef Ofuboæua`f

=7

Bwu}bofer}f =;6=

'= [rhucp!

5(/ ³Vrá `ifpb eb u}aæhjric bp bi FEB3 ³@ræhuc oaebh prp æhjricp3

@coc icp ifecp FE q FB pch ajrfibp f 7`o) bp apðp`bibp(

'6 [rhuc!

 Bi æhjric F oaeb 1;¾ / <;¾ ? 0;¾ q icp æhjricp E q B pch ajrfibp {c} pb} apðp`bibp) irbjc `fef rhc oaeb>

:2¾( Irbjc bi u}aæhjric ufomaáh bp f`ruæhjric(

'6 [rhuc!

2(/ Gfiif} icp æhjricp ebi u}aæhjric EB@(

 Bi æhjric B ebi u}aæhjric EB@ oaeb 0<; / <; ‐ 62; ? 62;¾( Icp æhjricp BE@ q E@B pch ajrfibp q `fef

rhc oaeb 1;¾ / :2¾ ? 62¾(

'= [rhucp!

OAHF @CH U]BP JFIB]ÏFP

Rhf oahf `chpuf eb u}bp jfib}ïfp) ebhcoahfefp jfib}ïf Z) jfib}ïf B q jfib}ïf P) vrb lc}ofh rh u}aæh/jric q `ch rhf {rb}uf paurfef bhu}b `fef ecp jfib}ïfp) iifofefp {rb}uf F) M q @) `coc pb ahea`f bh if lajr}f>

f! Fi `coabh~c eb rh ur}hc eb u}fmfnc bi nblb eb if oahf pb bh`rbhu}f bh if jfib}ïf B q f ic if}jc eb if nc}hf/ef pfmbocp vrb gf fu}f|bpfec 0 |b`bp if {rb}uf F) 5 |b`bp if {rb}uf M q 2 |b`bp if {rb}uf @( Bw{ia`f bh vrájfib}ïf pb bh`rbhu}f fi lahfi ebi ur}hc(

Pa {fpf rh hþob}c {f} eb |b`bp rhf {rb}uf) pb vrbef bh if oapof jfib}ïf vrb bpufmf q pa bi hþob}c bp ao/

 {f} {fpf f if pajrabhub jfib}ïf(

 If {rb}uf F if {fpf 2 |b`bp irbjc {fpf f Z) if @ 2 |b`bp irbjc {fpf f P) if M 5 |b`bp irbjc pb vrbef bh P 

vrb bp eðheb bpufmf( @ch`irpaðh> Fi lahfi ebi ur}hc pb bh`rbhu}f bh P(

'0 [rhucp!m! Pa pb cmpb}|f f rh or}`aáifjc vrb aha`afiobhub bpuæ bh if jfib}ïf B q pb fhcuf vrb gf fu}f|bpfec 0< |b`bp

7 ⋂5

=?6<`o

=

 F

M@

Z

P

B

Page 29: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 29/43

=1

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

if {rb}uf F) 22 |b`bp if {rb}uf M q <1 if @ ahea`f eðheb bpuf}æ fi lahfia~f} if cmpb}|f`aðh(

 ]f~chfhec eb lc}of paoaif}) bi or}`aáifjc) vrb bpuæ bh B) fu}f|abpf 0< |b`bp if {rb}uf F) irbjc pajrb bh B(

22 |b`bp if M) irbjc {fpf f P q <1 |b`bp if @ irbjc {fpf f Z( @ch`irpaðh> Fi lahfia~f} if cmpb}|f`aðh) bi

or}`aáifjc pb bh`rbhu}f bh Z(

'0 [rhucp!

`! Bpureaf pa bp {cpamib vrb bi nblb vrb aha`af bi eïf bpufhec bh if jfib}ïf B) {fpb f ic if}jc eb if nc}hfef =|b`bp {c} if {rb}uf F) = |b`bp {c} if M q 6 |b~ {c} if @(

 Ebp{ráp eb {fpf} = |b`bp {c} if {rb}uf F q = |b`bp {c} if @) pbjra}ïf bpufhec bh B q ebpeb bpuf jfib}ïf hc

gfq f``bpc f if {rb}uf @) {c} ufhuc hc bp {cpamib(

'5 [rhucp!

@FIALA@F@ACHBP EB RH BWFOBHRh bwfobh eb Ofuboæua`fp) vrb `chpuf eb = {}cmibofp) pb `fiala`f eb ; f 6; {rhucp( Rhc eb `fef 0

firohcp ga~c mabh bi {}aob} {}cmibof q 0 eb `fef 2 bi pbjrhec( If hcuf obeaf eb if `ifpb lrb 5 {rhucp pcm}b6;(

³@ræhucp {rhucp |fiïf `fef {}cmibof3

Rhc eb `fef u}bp) bvra|fib f 2 eb 62( Q u}bp eb `fef `ah`c) bvra|fib f 1 eb `fef 62(

Pa w bp bi |fic} ebi {}aob} {}cmibof b q bi ebi pbjrhec) pb {rbeb {ifhubf} bi papubof>

`rqf pcir`aðh bp>   w?=)24 q?:)2 

'Gfpuf 7 [rhucp!

@fi`rif bi hþob}c eb firohcp eb if `ifpb pfmabhec vrb bpuæ `co{}bheaec bhu}b =2 q 5;(

 Bi hþob}c eb firohcp bp oþiua{ic eb 62 q bi þha`c `co{}bheaec bhu}b =2 q 5; bp 0;(

'Gfpuf = [rhucp!

[RIJFEFP BH RH UBIB\APC]Ifp {fhufiifp eb icp ubib|apc}bp pch }b`ufhjrif}bp( Pr fh`gr}f q fiur}f pch {}c{c}`achfibp f 6< q 1)

}bp{b`ua|fobhub([c} cu}f {f}ub) icp lfm}a`fhubp) fi }blb}a}pb fi ufoföc eb if {fhufiif ebi ubib|apc}) rpfh if obeaef eb if eafjc/hfi bw{}bpfef bh {rijfefp(

2w$1q

62? 5

   w$q ?6;

&

-#

%#

Page 30: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 30/43

WWA Ciao{afef Ofuboæua`f

0;

Bwu}bofer}f =;6=

Pfmabhec vrb rhf {rijfef bvra|fib f =)25 `o) f{}cwaofefobhub) pb {aeb>

f! Pa rhf {fhufiif uabhb 7; `o eb fh`gc) ³eb `ræhufp {rijfefp bp bi ubib|apc}3

 If eafjchfi obea}æ

16):1 > =)25?0<)65) bp eb`a} 0< {rijfefp f{}cwaofefobhub(

'Gfpuf 2 [rhucp!

m! Pa rh ubib|apc} bp eb 5; {rijfefp) ³vrá eaobhpachbp) bh `bhuïobu}cp) uabhb if {fhufiif3-

Pa w bp if fh`gr}f b q if fiur}f) pb eber`b>

@coc 5; {rijfefp pch 5;(=)25?6;6)< `o( F{ia`fhec bi ubc}bof eb [auæjc}fp>

eb echeb pb eber`b vrb>

 q?51)76 o( q bh chpb`rbh`af) w?77)22 o(

 Bi ubib|apc} uabhb rhf fh`gr}f eb 77)22 `o q rhf fiur}f eb 51)76 `o(

'Gfpuf 2 [rhucp! 

 Fruc}bp eb if bifmc}f`aðh eb icp {}cmibofp eb if lfpb }bjachfi>

 Oajrbi Fhuchac Bpubmfh) Ic}bh~c Jch~æib~ Hf|f}}c) Of}afhc eb \a`bhub Jch~æib~ q Fhuchac Ocifhc ]cob}c(

6<1? 7;

 w ⇝  w ? 52`o

e ? <5;;$=;=2 ? 75=2 ? 16):1`o

 w 

 q ?

6<

1⇝  w ?

6<  q 

1

6;6)<=?

6< q

1

&

-#

%

",

=

$q= ⇝ 76 q

=$=2< q

=?76(6;6)<

=

Page 31: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 31/43

06

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

LFPB FRUCHÐOA@F( @A]@RAUC OFUBOÆUA@C

6(/ LRBHUB EB IFP FIOBHFP> ’¦VRÁ @FIC]&‟

Bp `chc`aef bh ucecp icp fi}bebec}bp if }avrb~f ebifjrf eb bpuf lrbhub vrb feboæp }f}ïpaof |b~ pb pb`f(

@fi`rif} bi uabo{c vrb uf}ef}ïfh icp `föcp eb if lrbhub bhiibhf} bi {aiðh) pr{chabhec vrb bi fjrf hc {reabpb bp`f{f}{c} hahjþh pauac) q vrb eb ucecp icp `föcp pfiabpb bi oapoc`frefi vrb pfib {c} bi `föc `bhu}fi( 

=(/ LRBHUB EB IFP FIOBHFP> ’BI FMFHA@C‟

@fi`rif eb if lc}of oæp f{}cwaofef {cpamib bi æ}bf ebi }bofubvrb `c}chf if lrbhub eb ifp fiobhfp `ch lc}of eb fmfha`c fmab}uc)ubhabhec bh `rbhuf vrb ucecp icp u}c~cp pch ajrfibp q vrb bi }feac oaeb5; `o(

0(/ FHUAJRC GCP[AUFI EB ICP UBO[IF]ACP> ’ICP @FPUAIICP‟

Bi {}aob} fpbhufoabhuc ec`robhufec eb If [f}}f lrb rhf fhuajrf `fpf/gcp{aufi vrb ubhïf if c}ebh ebicp ubo{if}acp {f}f pc`c}}b} f icp |afnb}cp gf`af Nb}b~ eb icp @fmfiib}cp( Gcq pðic vrbef rhf {f}ub eb if lf/`gfef q nrpuc bhl}bhub gfq ecp of{fp ahlc}ofhec eb if }ruf pbheb}apuf eb if ’ Pab}}f eb if @fia~f‟ (

Pfmabhec vrb if eapufh`af ebi `bhu}c eb If [f}}f fi bhu}c eb If Oc}b}f bh iïhbf }b`uf bp eb 5)2 do)³{ce}ïfap eb`a}hcp f vrá bp`fif bpuæ bi of{f uc{cj}æla`c3 ³ F vrá eapufh`af }bfi bh iïhbf }b`uf pb bh`rbhu}fhbi fpuaiic eb Lb}af ebi fpuaiic eb Pfi|fuab}}f3

5(/ [IF^F EBI FQRHUFOABHUC> ’UCOFHEC @CC]EBHFEFP‟

Hcp paurfocp bh if lf}cif ebi `bhu}c eb if {if~f ebi fqrhufoabhuc q `chpaeb}focp vrb bpufocp bh bic}ajbh eb rhcp jajfhubp bnbp eb `cc}ebhfefp aofjahf}acp pcm}b bi prbic eb ucef if {if~f(

Bi |á}ua`b ebi }comc echeb pb bh`rbhu}f bi föc bh bi vrb ga`ab}ch bpub mchauc prbic bo{be}fec pb}æbi {rhuc ';)6! eb hrbpu}cp bnbp(

Page 32: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 32/43

WWA Ciao{afef Ofuboæua`f

0=

Bwu}bofer}f =;6=

³Vrá ecp cc}ebhfefp ubhe}æ bi {rhucbwf`uc pcm}b bi prbic echeb pb `ab}}fh ifp ecp{rb}ufp eb if ajibpaf3

³Vrá cc}ebhfefp ubhe}ïf bi {rhuc oæpfiuc eb if þha`f |bhufhf eb if lf`gfef eb if ajib/

paf pa ic {chbocp {b}{bhea`rif}obhub pcm}b biprbic3

Bpb {rhuc eb if |bhufhf pb bh`rbhu}f frhf fiur}f eb `rfu}c obu}cp pcm}b bi prbic)³{ce}ïfap ef} pr {cpa`aðh ruaia~fhec feboæp ebifp ecp `cc}ebhfefp ebi prbic rhf ub}`b}f `cc}/ebhfef3) ³`ræibp pb}ïfh bhuch`bp ifp u}bp `cc}/ebhfefp ebi {rhuc3

2(/ [IF^F EBI FQRHUFOABHUC> ’RHF EB ]COFHCP‟

Hcp bh`chu}focp bh if {if~f ebi Fqrhufoabhuc eb If [f}}f 'Mfefnc~!) echeb {cebocp `chubo{if} ifAjibpaf eb Hu}f( P}f( eb if Fprh`aðh) if `rfi gf prl}aec or`gfp }bocebif`achbp f ic if}jc ebi uabo{c( Bh rh{}ah`a{ac {ce}ïf gfmb} paec fijþh ua{c eb Lc}ufib~f bh á{c`f Æ}fmb) pbjþh eber`aocp eb fijrhcp ebufiibp ebif `chpu}r``aðh eb if `fmb`b}f ebi beala`ac) `coc {rbebh pb} fijrhcp uc}}bchbp) bpu}r`ur}f eb lc}ufib~f bh ifhf|b `bhu}fi '`þ{rif! q if læm}a`f eb fijrhcp or}cp fi oæp labi bpuaicoreánf}(

Bhu}b icp |bpuajacp }cofhcp vrb {cebocp cmpb}|f} bh bpufAjibpaf pb bh`rbhu}f rhf iæ{aef lrhb}f}af }cofhf bh`chu}fefbh ³3 bh if vrb ea`b ic pajrabhub>

Ubhbocp vrb bh`chu}f} bi föc eb ebp`rm}aoabhuc eb if iæ{a/ef pfmabhec vrb pa fi föc eb `chpu}r``aðh ebi Fqrhufoabhuc) cmpb}|fec bh pr lf`gfef) ib profocp 61= q ib}bpufocp bi hþob}c eb `r}pcp vrb iib|f lrh`achfhec bi ahpuauruc eb If [f}}f) oriua{ia`fec {c} if ealb}bh`afbhu}b bi föc bh vrb ga`ab}ch bi prbic eb if [if~f q bi föc vrb mrp`focp) cmubhbocp `coc }bpriufec ic vrbpfib eb oriua{ia`f} bi hþob}c vrb bh }cofhc ’f}`fa`c‟ lajr}f bh if ea`gcpf iæ{aef {c} bi h¾ eb firohcp `ifpa/

la`fecp {f}f if lfpb lahfi eb bpufp Ciao{afefp Ofuboæua`fp(

<(/ BPVRAHF EB IF GCP[BEB]ÏF> ’ICP FM]F^CP‟

Rh bvra{c {f}ua`a{fhub bh bi `a}`rauc mfnf {c} if `fiib Pfhuf Of}ïf bhmrp`f eb pr pajrabhub {}rbmf q cu}c prmb {c} if `fiib Pfhuïpaoc @}apuc eb ifpOapb}a`c}eafp) pb `}r~fh nrpuc bh if bpvrahf q `cic`æhecpb `fef {f}ua`a{fhub bhrh |á}ua`b eb if bpu}biif ebi prbic pb ef rh flb`urcpc fm}f~c `ch ucecp icp eb/oæp( ³@ræhucp fm}f~cp pb gfh efec bh ucufi3

Page 33: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 33/43

00

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

:(/ [IF^F EB IF @CHPUAUR@AÐH> ’IF FHUBHF ZA/LA‟

Pb ebpbf ahpufif} rhf fhubhf ZALA bh If [f}}f {f}f ef} `cmb}ur}f eb Ahub}hbu f u}bp ~chfp ebi {rbmic>if Ajibpaf) bi `rf}ubi eb if Jrf}eaf @a|ai q if jfpciahb}f 'ufi q `coc {rbebp cmpb}|f} bh if lcuc fá}bf!( U}fpicp bpureacp uá`ha`cp) pb gf eb`aeaec `cic`f}if bh rh irjf} vrb pb bh`rbhu}b bwf`ufobhub f if oapof eapufh/`af eb bpcp u}bp {rhucp(

³[ceáap pböfif} bh bi of{f `ræi pb}ïf bpb irjf}3 Bw{ia`f `ðoc ic gfp f|b}ajrfec q ifp gb}}foabhufpeb Jbcjbm}f ruaia~fefp(

@fi`rif if eapufh`af bwf`uf f if vrb bpuf}ïf if fhubhf eb icp u}bp {rhucp pböfifecp(

Bi {rhuc bibjaec {f}f paurf} if fhubhf bp rh irjf} bomiboæua`c ebi {rbmic) ³{ce}ïfp aebhuala f} bwf`/ufobhub bpb {rhuc3

7(/ @FIIB EB IF @]R^ > ’BI L]CHUFI EB IF @]R^‟

[f}f {}cubjb} if `}r~ eb ifp ah`ibobh`afp ebiuabo{c pb gf lfm}a`fec rhf `gf{f {f}f bi f}`c l}chufi`ch ifp obeaefp bwf`ufp ebi f}`c bwub}ac}( Pfmbocp vrbif eafjchfi ebi `rfe}fec eb if `gf{f oaeb 0);7 ob/u}cp( Bi {}b`ac eb if ifuf eb =dj eb {ahur}f bp eb <₩([f}f if {}aob}f ofhc pcm}b if `gf{f pb hb`bpaufh 0;;j}focp {c} obu}c `rfe}fec) q {f}f if pbjrhef ofhc

vrb pðic pb ef}æ {c} if `f}f bwub}ac}) pb hb`bpauf rh0;" obhcp eb {ahur}f {f}f `rm}a} if oapof pr{b}la`ab(

³@ræhuc `rbpuf ucec pa pfmbocp vrb if `gf{f ifgfh }bjfifec bh if gb}}b}ïf q if ofhc eb cm}f if {chbhj}fuap ifp |b`ahfp vrb eb oai foc}bp `raefh if `}r~ feaf}ac3

1(/ [IF^F EB PFHUAFJC> ’HRB\C PRBIC‟If {if~f eb Pfhuafjc bp pah eref bi `c}f~ðh eb if ic`fiaefe eb If [f}}f) bh uc}hc f biif pb `ch`bhu}fh

or`gcp eb icp bpufmib`aoabhucp `cob}`afibp) mfh`f}acp) eb gcpubib}ïf… fiiï pb bibm}fh ufomaáh or`gcp ebicp f`ucp {þmia`cp ebi {rbmic(

Bi Fqrhufoabhuc pb bpuæ {ifhubfhec {chb} rh hrb|c prbic( @fi`rif eb if lc}of oæp f{}cwaofef {c/pamib bi æ}bf ucufi ebi }b`ahuc eb mfiecpfp {f}f vrb bi fi`fieb {rbef ubhb} rhf aebf f{}cwaofef ebi `cpub eb ifcm}f) ubhabhec bh `rbhuf vrb |fh f vrauf} if lrbhub vrb bpuæ bh bi `bhu}c eb if {if~f(

Page 34: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 34/43

WWA Ciao{afef Ofuboæua`f

05

Bwu}bofer}f =;6=

6;(/ LRBHUB EB IF [IF^F EB PFHUAFJC> ’IF UC]OBHUF‟

Bi {fpfec 61 eb ofqc `fqð rhf lrb}ub uc}obhuf bh if [f}}f( Iic|ïf eb lc}of ucufiobhub {b}{bhea`r/if} fi prbic {c}vrb hc gf`ïf hfef eb |abhuc q pb }b`cjab}ch 67 i#o=(

Ifp u}bp {aibufp eb if lrbhub eb if {if~f eb Pfhuafjc pb iibhf}ch eb fjrf( ³@ræhucp iau}cp gfmïf bh `fefrhf eb ifp u}bp {aibufp nrpuc ebp{ráp eb ub}oahf} if uc}obhuf3

Hcuf> @coc bpuæ orq lbc prma}pb b ic fiuc eb ifp lrbhubp hcp gfh ea`gc vrb bi }feac eb if {aibuf oæp{bvrböf bp rh 67" ebi eb if {aibuf j}fheb eb fmfnc q vrb bi }feac eb if {aibuf obeaf bp rh 0;" ebi eb ifj}fheb( @ch ic `rfi pðic ubhe}áap vrb obea} bi eb if j}fheb f if vrb pï cp {ceáap prma} læ`aiobhub(

BWU]F(/ [IF^F EB IF @CHPUAUR@AÐH> ’LCUCJ]FLÏF OFUBOÆUA@F‟

Bi bhuc}hc eb bpuf {bvrböf {if~f bp }a`c bh lc}ofp q `ch`b{ucp jbcoáu}a`cp( @fef bvra{c ubhe}æ ecp`æof}fp eajaufibp f pr eap{cpa`aðh er}fhub 62 oahrucp `ch ifp vrb {ce}æ }bfia~f} lcucj}flïfp ’ofuboæua`fp‟bh if {if~f c prp fi}bebec}bp( Ebmb}æh ebp`f}jf} ecp eb biifp bh if `f}{buf ebi c}ebhfec} eb `chu}ci q {chb}/ibp rh uïuric prjb}bhub(

Pb |fic}f}æ if `fiaefe eb if lcuc) if c}ajahfiaefe) bi `ch`b{uc jbcoáu}a`c vrb u}fub q bi uïuric(

 Fruc}bp eb if bifmc}f`aðh eb icp {}cmibofp ebi `a}`rauc ofuboæua`c>

 Ic}bh~c Oröc~ Ocub}c) Irap L}fh`ap`c Pæh`gb~ Lb}hæheb~) Fhuchac Lbia{b Jfauæh Ofubc q L}fh`ap`c Nf|ab} Ic~fhc Nf}foaiic(

Page 35: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 35/43

02

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

LFPB HF@ACHFI

6(/ RH U]FMFNC AHLC]OÆUA@C

[c} rh u}fmfnc ahlc}oæua`c gb lf`ur}fec rhf `ab}uf `fhuaefe eb br}cp( Rhf |b~ ebp`chufec bi 67);<"ob vrbef rhf `fhuaefe eb br}cp vrb `chufecp eb 2 bh 2) eb : bh :) eb 1 bh 1 q eb 60 bh 60 pabo{}b ef rh}bpuc eb = br}cp(

³[ce}ïfp `ch bpucp efucp f|b}ajrf} bi ao{c}ub eb if lf`ur}f) pfmabhec vrb {fjrá obhcp eb <;;; br/}cp3

=(/ RH MFÖC BH PFOAI

Ob bpucq mföfhec bh if {ifqf eb Pfoai( Uc`c bi lchec `ch icp{abp q) `rfhec hc gfq cifp) if {f}ub eb f}}amf eb oa `fmb~f pcm}bpfibpcm}b bi ha|bi ebi of} rhf fiur}f ajrfi f if oaufe eb if fiur}f ebi of} vrbob `rm}a}æ `rfhec {fpb if {}ðwaof cif) cif vrb gf}æ vrb if fiur}f ebiof} frobhub if oaufe( Pa oaec 6):2o( ³@ræi bp if fiur}f eb if cif3

0(/ RHF GC]OAJF EB [FPBC

Rhf gc}oajf `foahf {c} bi mc}eb eb rh {ifuc eb 7 ifecp ajrfibp`coc bi eb if lajr}f( @fef ifec ebi {ifuc oaeb 65`o( If gc}oajf pfib ebi|á}ua`b F q `foahf bh bi pbhuaec vrb ahea`f if lib`gf) pabo{}b {c} bi mc}ebebi {ifuc( Gf`b if {}aob}f {f}fef f <`o ebi |á}ua`b F q ebp{ráp) `fef <`ogf`b rhf {f}fec( Bh ucufi gf`b =;;; {f}fefp(

f! ³@ræhufp |b`bp {f}f bh bi |á}ua`b F3

m! ³Bh vrá cu}cp |á}ua`bp gf`b if oapof `fhuaefe eb {f}fefp vrb bhF3

Page 36: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 36/43

WWA Ciao{afef Ofuboæua`f

0<

Bwu}bofer}f =;6=

5(/ BI @C[C EB HAB\B

[f}uaocp eb rh u}aæhjric bvraiæub}c `rqc ifec oaeb 1`o(Ea|aeaocp `fef ifec bh u}bp {f}ubp ajrfibp q pcm}b bi ub}`ac ebi

obeac chpu}raocp cu}c u}aæhjric bvraiæub}c ch bi vrb fo{iaf/ocp if lajr}f( ]b{buaocp bi oapoc {}c`bpc bh `fef ifec eb ifhrb|f lajr}f( Bi eafj}fof orbpu}f ifp `rfu}c {}aob}fp lfpbp ebi{}c`bpc

f! @fi`rif bi ifec q bi {b}ïobu}c {f}f ifp `rfu}c lfpbp eb iflajr}f(

m! ³@ræhuc obea}æ bi ifec bh if eá`aof lfpb3 ³Q bi {b}ï/obu}c3

`! Bp`}amb bi |fic} ebi ifec q ebi {b}ïobu}c {f}f rhf lfpb`rfivrab}f h(

2(/ ]COBC ECHNRFH

]cobc uabhb ecp foajfp vrb |a|bh bh ~chfp c{rbpufp) Bpugb} fi bpub q ]rug fi cbpub( @coc ái bp rhgcom}b aheb`apc hc pfmb f `ræi {}blab}b |apauf} q `rfhec iibjf f if {f}fef ebi frucmþp `cjb bi {}aob}c vrb{fpf qf vrb icp ecp) bi vrb |f gf`af bi bpub q bi vrb |f gf`af bi cbpub) {fpfh `fef 6; oahrucp `coc ahea`f bigc}f}ac>

Rhf uf}eb if `ga`f -bpub-) Bpugb} ib eanc> ¦Vrá `chubhuf bpucq& Eb`fef eab~ eïfp |abhbp f |b}ob hrb|b(

Bh `fomac) ]rug) if `ga`f -cbpub- ib eanc> ¦Qf bpuæ mabh& Pcicub |bc rhc eb `fef eab~ eïfp(

f! ³[rbebp bw{ia`f} eb lc}of }f~chfef ic vrb c`r}}b3

]cobc ib `rbhuf bpuc f pr foajc L}fh`ap`c vrb ib ea`b>

[rbp qc ubhjc ecp foajfp rhf vrb |a|b bh bi hc}ub q cu}f vrb|a|b bh bi pr}( \cq f |apauf} f if {}aob}f 2 eïfp `fef 6= q f if pbjrh/ef : eïfp `fef 6=(

m! ³[rbebp `co{ibuf} if ufmif eb icp gc}f}acp eb frucmrpbp

vrb cjb L}fh`ap`c3

[}ðwaoc frucmþp

Gf`af bi bpub Gf`af bi cbpub

67

67>6;>;;67>=;>;;

67>;6>;;

67>66>;;67>=6>;;

[}ðwaoc frucmþp

Gf`af bi hc}ub Gf`af bi pr}

67

((((((

…(((

Page 37: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 37/43

0:

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

\fi|b}eb ebi L}bphc) pbeb eb if lfpb }bjachfieb if WWA Ciao{afef Ofuboæua`f 

If `bibm}f`aðh bpub föc eb if Ciao{afef Ofuboæua`f bh \fi|b}eb ebi L}bphc bp l}ruc f {f}ubp ajrfibp

ebi bplrb}~c `chuahrfec eb icp {}clbpc}bp q eb rhf `ch|b}pf`aðh `fprfi ebp{ráp eb if c}jfha~f`aðh eb if lfpb`cof}`fi ebi föc {fpfec( If c}jfha~f`aðh ebi b|bhuc gf pr/

{rbpuc rh ah`bhua|c {f}f hrbpu}c ahpuauruc) rh }buc vrb uabhb

`coc cmnbua|c {}ah`a{fi hc pcic lcobhuf} ifp ebpu}b~fp of/

uboæua`fp eb hrbpu}cp firohcp) pahc ufomaáh {}coc|b} if

`ch|a|bh`af bh rh of}`c jbcj}æla`c {}a|aibjafec) if Pab}}f eb

Jfuf(

Ufhuc bi ABPC \fi eb Wæiaof) `coc icp fqrhufoabh/

ucp eb ifp ic`fiaefebp vrb `chlc}ofh hrbpu}c firohfec) pb

gfh |ci`fec bh bpufp nc}hfefp( Icp u}bp {rbmicp vrb ic `co{chbh) \fi|b}eb ebi L}bphc) Pfh Of}uïh eb U}b/

|bnc q Binfp) hc pcic gfh |apuc rhf j}fh c{c}urhaefe {f}f ef}pb f `chc`b}) pahc vrb ufomaáh gfh `co{}bhea/

ec vrb bpufp WWA Ciao{afefp ibp {b}oaubh ebocpu}f} pr `co{}coapc `ch if ber`f`aðh q if `riur}f eb hrbp/

u}f @corhaefe Fruðhcof) ocpu}fhec pr eap{cpa`aðh f `cifmc}f} bh ifp eapuahufp f`ua|aefebp {bhpfefp {f}f

bpucp eïfp vrb) bp{b}focp) pbfh orq ebi fj}fec eb ucecp(

@coc pfmbocp) ifp Ofuboæua`fp lc}ofh {f}ub eb if |aef eb ucecp( Hcp fqrefh f `co{}bheb} hrbpu}c

orhec) ebpeb ic `cuaeafhc gfpuf ic u}fp`bhebhub) bi

c}ebh q bi f~f}) pch aheap{bhpfmibp bh ifp f`ua|aefebp

ebi pb} grofhc bh bi eïf f eïf q) pah irjf} f erefp) if

iif|b {f}f fm}a} ifp {rb}ufp eb or`gcp cu}cp `chc`a/

oabhucp( Bh bpb lah eb pbofhf feboæp) ifp Ofuboæua`fp

pb}æh if bw`rpf {f}f }brha}hcp bh bpub bhuc}hc ufh pah/

jrif} eb Bwu}bofer}f) l}chub}a~c `ch [c}urjfi q {c}

ufhuc) irjf} eb `chuahrcp ahub}`fomacp vrb bw{ia`fh if

bwapubh`af eb }bfiaefebp `riur}fibp ufh iifofua|fp `coc

F Lfif) ibhjrf }cofh`b vrb {b}|a|b bh Rp u}bp irjf}ap> \fi|b}ea er L}bphr) Fp Biifp q Pf Of}uïh eb U}b|b/

iir(

Bp{b}focp pah`b}fobhub vrb eapl}ruáap) er}fhub bpb lah eb pbofhf eb ofqc) bh `ch|a|bh af q pfhf`co{bubh`af(

 Nbpþp Jfob}c ]a|b}c( Nblb eb Bpureacp ebi A(B(P(C( \fi eb Wæiaof( \fi|b}eb ebi L}bphc  

Page 38: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 38/43

WWA Ciao{afef Ofuboæua`f

07

Bwu}bofer}f =;6=

@ch`r}pc eb `f}ubibp {f}f if Ciao{afef =;60(

Mfpbp

@F]F@UB]ÏPUA@FP

6® Icp `f}ubibp ebmb}æh {}bpbhuf}pb bh ufoföcEAH/F0(

=® Hc {ce}æh ubhb} oæp eb `rfu}c `cic}bp {if/hcp 'hc ob~`ifecp!(

0® Ebmb}æh `chubhb} bi ibof>

WWAA CIAO[AFEF OFUBOÆUA@F(BWU]BOFER]F =;60

5® Bi `f}ubi jfhfec} pb}æ bi fhrh`afec} eb ea/`gf Ciao{afef(

2® Icp `f}ubibp vrbef}æh bh {cpbpaðh eb if C}/jfha~f`aðh(

<® Gfm}æ rh jfhfec} q ecp f``ápau(

[F]UA@A[FHUBP

:® [ce}æh {f}ua`a{f} firohcp eb 6¾ q =¾ ebB(P(C( bh bi `r}pc bp`cif} =;66/=;6=) eb `rfi/vrab} `bhu}c ber`fua|c eb if @corhaefe Fruð/hcof eb Bwu}bofer}f(

AHP@]A[@ACHBP

7¾ Icp `f}ubibp ebmb}æh bh|af}pb f>

@CHPBNB]ÏF EB BER@F@AÐH Q @RIUR]FPb`}buf}ïf Jbhb}fi eb Ber`f`aðh’CIAO[AFEF OFUBOÆUA@F‟

@# Ebijfec \fibh`af <)0¾

;<7;; OÁ]AEF 'MFEFNC^!

1® Fi ec}pc eb `fef `f}ubi pb bp`}ama}æ bihcom}b ebi {f}ua`a{fhub) ha|bi) bhu}c) ea}b`/`aðh q ubiálchc {f}ua`rif}bp(

LB@GF IAOAUB

6;® If lb`gf iïoaub eb }b`b{`aðh eb `f}ubibppb}æ bi =; eb fm}ai eb =;6=(

[]BOACP

66® [f}f icp @bhu}cp eb icp u}bp firohcp lahf/iapufp) rh icub eb iam}cp pcm}b Ber`f`aðh Of/uboæua`f q }bpcir`aðh eb {}cmibofp(

6=® [f}f icp ecp f``ápau) rhf `fi`rifec}f

`abhuïla`f q rh icub eb iam}cp(

60® [f}f bi jfhfec}) |afnb q bpufh`af er}fhubicp eïfp vrb pb `bibm}b if lfpb fruchðoa`f ebif Ciao{afef =;6= bh \fi|b}eb ebi L}bphc)`ch|a|abhec `ch icp firohcp `ifpala`fecp {f/}f biif q }b`amabhec icp oapocp {}boacp(

LFIIC EBI NR]FEC

65( If bib``aðh ebi `f}ubi jfhfec}  `c/}}b}æ f `f}jc eb if @coapaðh C}jfha~fec}f eb

if Ciao{afef q pr lfiic pb}æ ahf{bifmib( 

Page 39: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 39/43

01

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

WWA CIAO[AFEF OFUBOÆUA@F BWU]BOFER]F =;6=(Mfpbp

[F]UA@A[FHUBP>

Ebmb}æh pb} firohcp eb =¾ `r}pc eb BPC) bh bi`r}pc =;66/=;6= eb `rfivrab} @bhu}c Ber`fua|ceb if @corhaefe Fruðhcof Bwu}boböf(

[ce}æh {f}ua`a{f} `coc oæwaoc 6; firohcp {c}`fef `ifpb ebi obh`achfec ha|bi vrb bwapuf bh bi@bhu}c(

LB@GF EB AHP@]A[@AÐH>

Gfpuf bi 0; eb of}~c eb =;6=(

[]C@BEAOABHUC EB AHP@]A[@AÐH>

Icp `bhu}cp lc}ofia~f}æh if pcia`aure `ch if }bif/`aðh eb {f}ua`a{fhubp f``beabhec f if ea}b``aðh>

guu{>##zzz(ber`f}bw(bp#ciao{afefofu 

Ebmb}æh `ro{iaobhuf} bh pr ucufiaefe if gcnf ebahp`}a{`aðh) ao{}aoa}if q bh|af}if f if pajrabhub

ea}b``aðh>

@CHPBNB]ÏF EB BER@F@AÐHQ @RIUR]F

Pb`}buf}ïf Jbhb}fi eb Ber`f`aðh

Ciao{afef Ofuboæua`f

`# Ebijfec \fibh`af <) 0¾

;<7;; Oá}aef 'Mfefnc~!

@F]F@UB]ÏPUA@FP>

F( If Ciao{afef `chpuf}æ eb ecp lfpbp>6® @cof}`fi(

=® Fruchðoa`f(

M( If lfpb `cof}`fi pb `bibm}f}æ er}fhub bi eïf =6eb Fm}ai eb =;6= 'pæmfec!) f ifp 6;)0; gc}fp bh ifppajrabhubp ~chfp q `bhu}cp>

PBEB@BHU]C EB

@BIBM]F@AÐH@CC]EAHFEC]#F UBIBLCHC

FIMR]VRB]VRB # 

PFH \A@BHUB 

A(B(P( Ncfvrïh Pfof 

PFH \A@BHUB EB FI@ÆHUF]F 

Æhjbi L}fh`ap`c Fom}cnc Fhuþhb~

A(B(P( @fpuaiiic eb Irhf 'Fimr}vrb}vrb!

pfh|aofubKqfgcc(bp 

1=5;62=0;

FIOBHE]FIBNC  A(B(P( Uab}}f eb Mf}}cp 

F@BR@GFI 

Pb}jac Pfhucp ]cpbii

A(B(P( Uab}}f eb Mf}}cp 'F`br`gfi!

ppfhucp}cpbiiKjofai(`co

1=5 ;6:05;

F^RFJF # 

IIB]BHF 

A(B(P(C( Mbomá~f}  F^RFJF 

Nrfh Jrb}}f Mb}obnc

A(B(P( Eb IIb}bhf 'Iib}bhf!

 nrfhljmKubiblcha`f(hbu 

Nrfh Jrf}efec Jf}`af(

A(B(P( Mbomb~f} 'F~rfjf! nrfhjrf}efecKubiblcha`f(hbu 

1=5;=<2<=

Page 40: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 40/43

WWA Ciao{afef Ofuboæua`f

5;

Bwu}bofer}f =;6=

MFEFNC^ A(B(P( Já|c}f 

JÁ\C]F 

Oajrbi Æhjbi Oc}bhc ]bechec

A(B(P( Pfh Lb}hfhec

Oajrbi Æhjbi Mifh`c Fichpc

A(B(P( @arefe Nf}eïh

ofoc}bhc}Kjofai(`co 

1=5;60555

MF]@F]]CUF A(B(P( \a}jbh ebi Pcub}}föc 

MF]@F]]CUF 

Of}ïf ebi Of} Ocuf Obeahf

A(B(P( \a}jbh ebi Pcub}}föc

}fvrbio|f}fKub}}f(bp 

1=5;=2=7; 

@Æ@B]BP A(B(P(C( \ïf eb if [ifuf 

@FPF] EB @Æ@B]BP 

Fhuchac Ocifhc ]cob}c

A(B(P( [}clbpc} Gb}hfheb~ [f`gb`c

'@æ`b}bp!

fhuchacocifhcKoawofai(`co 

1=:;6;177

@C]AF A(B(P( Nfifof 

OC]FIBNF 

Gb}oahaf Of}uïh Ið{b~

A(B(P(C( \ïf Efiof`af Uc}}bnch`aiic

oahaof}uahic{b~Kgcuofai(`co

1=:6726;;

ECH MBHAUC A(B(P( Ncpá Ofh~fhc 

ECH MBHAUC 

F}ur}c Ofheiq Ofhpc

A(B(P( Ncpá Ofh~fhc 'Ech Mbhauc!

Brjbhaf Ið{b~ @æ`b}bp

Ea}b``aðh [}c|ah`afi eb Mfefnc~

f}ofheiqKjofai(`co 

1=5;=670=

OÁ]AEF A(B(P( Pfhuf Brifiaf 

OÁ]AEF 

Ncpá Fhuchac Pæh`gb~ Jraiiáh

A(B(P( Pfhuf Brifiaf 'Oá}aef!

 npfj;;;7Kfiob~({hua`(ob`(bp 

1=5;;150;

[IFPBH@AF

 

A(B(P( [f}vrb eb Ochl}fjùb 

[IFPBH@AF 

Of}apci @c}}bfp Of}uïh

A(B(P( [f}vrb eb Ochl}fjùb '[ifpbh`af!of}apci`c}}bfpKjofai(`co 

1=:;6::<=

PA]RBIF A(B(P(C( \a}jbh eb Fiufj}f`af 

PA]RBIF 

[be}c ]a`c Jch~æib~

A(B(P(C( \a}jbh eb Fiufj}f`af

{}a`cKber(nrhufbwu}bofer}f(hbu 

1=5;6116<

^FL]F A(B(P( Aieblchpc Pb}}fhc 

PBJR]F EB IBÐH 

\ï`uc} Pæh`gb~ Jf}`ïf

A(B(P( @}apuc ebi ]cpf}ac '^fl}f!

|a`uc}epj:7Kqfgcc(bp 

1=5;=1155

Page 41: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 41/43

56

WWA Ciao{afef Ofuboæua`f Bwu}bofer}f =;6=

@fef @bhu}c {ce}æ ahp`}ama}pb bh if ~chf oæp`ch|bhabhub {f}f prp ahub}bpbp(

If lfpb fruchðoa`f pb `bibm}f}æ icp eïfp =2) =< q =: eb ofqc bh>

\FI\B]EBEBIL]BPHCA(B(P(\fiebWæiaof

NbpþpJfob}c]a|b}c

\b}ðha`fOf}uïhJruaá}}b~

OajrbiÆhjbiJfiæhGb}}b}f

1=:;60:=<

 

If cc}eahf`aðh }bjachfi bpuæ lc}ofef {c}>

/ Ncpá [be}c Of}uïh Ic}bh~c' `gb{bfuKjofai(`co!A(B(P( C( \ïf Efiof`af) Uc}}bnch`aiic '@æ`b}bp!(

/ [be}c @c}`gc Pæh`gb~ '{b`c}`gcKrhbw(bp!(/ Oajrbi Æhjbi Oc}bhc ]bechec 'ofoc}bhc}Kjofai(`co!(

@( Icp jfpucp eb bpufh`af q ebp{if~foabhuc f if

pbeb bibjaef {f}f if lfpb `cof}`fi) `c}}b}æh f `f}/jc eb icp {f}ua`a{fhubp(

E( F if lfpb fruchðoa`f f`rea}æh rh oæwaoc eb0; firohcp) chlc}ob f icp pajrabhubp }aub}acp>

e(6 Ec`b firohcp `c}}bp{cheabhubp fi{}aob} ifpala`fec eb fef pbeb(

e(= Pbap firohcp vrb pb `ifpala`f}æh {}c/

{c}`achfiobhub fi hþob}c eb {}bpbhufecpbh `fef pbeb(

e(0 Ec`b firohcp) hc `ifpala`fecp bh icp{}c`bpcp fhub}ac}bp) pb `ifpala`f}æh `ch/lc}ob f if {rhurf`aðh cmubhaef(

Ifp pbebp {ce}æh }blrhea}pb pa bi hþob}ceb {f}ua`a{fhubp bh fijrhf eb ifp ~chfp hc bp paj/hala`fua|c( If {if~f `c}}bp{cheabhub eb `ifpala`f/`aðh ea}b`uf e(6 eb if pbeb }blrheaef) pb ah`}bobh/uf}æ fi f{f}ufec eb ifpala`f`aðh e(0

Icp jfpucp eb bpufh`af q ebp{if~foabhuc bh

bpuf lfpb `c}}b}æh f `f}jc eb if C}jfha~f`aðh(

EBPF]]CIIC>

F( If {}rbmf eb if {}aob}f lfpb `chpapua}æ bh if}bpcir`aðh ahea|aerfi eb `rfu}c {}cmibofp) c f`ua/|aefebp ofuboæua`fp( Pb `bibm}f}æ paoriuæhbf/obhub bh ucefp ifp pbebp( Bi `chu}ci q bi lfiic eb if{}rbmf `c}}b}æ f `f}jc eb rhf `coapaðh hcom}fef{c} if Pc`abefe Bwu}boböf eb Ber`f`aðh Ofub/oæua`f ’\bhur}f ]bqbp [}ðp{b}‟(

Pcifobhub pb gf}æ {þmia`f if }bif`aðh eb pbib``ac/hfecp {f}f if lfpb fruchðoa`f) vrb pb}æ bh|afef fucecp icp bhu}cp {f}ua`a{fhubp(

M( [f}f if }bfia~f`aðh eb ifp {}rbmfp) icp firohcp{rbebh a} {}c|apucp eb `fi`rifec}f q ofub}afi ebeamrnc(

Page 42: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 42/43

WWA Ciao{afef Ofuboæua`f

5=

Bwu}bofer}f =;6=

@( If lfpb fruchðoa`f pb `bibm}f}æ icp eïfp =2) =<

q =: eb ofqc bh \fi|b}eb ebi L}bphc '@æ`b}bp!)fiub}hæhecpb {}rbmfp q `ch|a|bh`af( Icp firohcp`ifpala`fecp {f}f bpuf lfpb ebmb}æh {f}ua`a{f} bhucefp ifp f`ua|aefebp {}cj}fofefp {c} if c}jfha/~f`aðh eb if Ciao{afef(

E( Ifp {}rbmfp pb}æh ecp> Rhf {c} j}r{c eb u}bpfirohcp q cu}f ahea|aerfi `chpapubhubp bh if }bpc/ir`aðh eb |f}acp {}cmibofp c f`ua|aefebp ofuboæ/ua`fp( Icp u}bp {}aob}cp `ifpala`fecp }b{}bpbhuf}æhf Bwu}bofer}f bh if WWAAA Ciao{afef Hf`achfi(

B( Ucecp icp {f}ua`a{fhubp }b`ama}æh rh ea{icof(Feboæp) f icp {}clbpc}bp vrb ahub}|bhjfh bh if{}b{f}f`aðh q ebpf}}ciic eb if f`ua|aefe ber`fua|f{}c{rbpuf bh if {}bpbhub `ch|c`fuc}af pb ibp }b`c/hc`b}æ rh `}áeauc eb lc}of`aðh {c} pr {f}ua`a{f/`aðh bh if lfpb `cof}`fi) q cu}c `}áeauc oæp) f`r/orifmib fi fhub}ac}) f fvráiicp vrb ufomaáh `cif/mc}bh bh if {}b{f}f`aðh q ebpf}}ciic eb ea`gf f`ua/|aefe bh if lfpb fruchðoa`f(

L( If ahub}{}buf`aðh eb ifp {}bpbhubp hc}ofp `c}}b/}æh f `f}jc eb if @coapaðh C}jfha~fec}f eb ifCiao{afef q pr lfiic pb}æ ahf{bifmib(

Page 43: Revista Olimpiada 2012

5/14/2018 Revista Olimpiada 2012 - slidepdf.com

http://slidepdf.com/reader/full/revista-olimpiada-2012 43/43

i  @CH\C@F>

JCMAB]HCEBBWU]BOFER]F

@chpbnb}ïfebBer`f`aðhq@riur}f

Pb`}buf}ïfJbhb}fiebBer`f`aðh

i  C]JFHA^F>

Pc`abefeBwu}boböfebBer`f`aðhOfuboæua`f ’\bhur}f]bqbp[}ðp{b}‟

guu{>##|bhur}f}bqbp{}cp{b}(ber`f}bw(bp

i  @CIFMC]FH>Bw`bibhuïpaoc Fqrhufoabhuc eb \fi|b}eb ebi L}bphc

Bw`bibhuïpaoc Fqrhufoabhuc eb Binfp

Bw`bibhuïpaoc Fqrhufoabhuc eb Pfh Of}uïh eb U}b|bnc