Centrale Informatique MP-PC-PSI 2017

Thème de l'épreuve Mars Exploration Rovers: mission d'exploration martienne
Principaux outils utilisés algorithmique, recherche de minimum, listes, SQL
Mots clefs numpy, calcul de distances, minimisation, algorithme génétique, algorithme du plus proche voisin, jointure, robot, Mars

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Énoncé complet

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Rapport du jury

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Énoncé obtenu par reconnaissance optique des caractères


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 H i#H2 ALah_ /2b BMbi`mK2Mib 2K#`[mûb- p2+ H2b +QHQMM2b
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r ALnLPJ MQK /2 HBMbi`mK2Mi
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r hun.1a H2 MQK /m ivT2 /MHvb2
 H i#H2 ALhuS /2b BMbi`mK2Mib miBHBbûb TQm` mM ivT2 /MHvb2- /2 +H27 T`BKB`2 
UhunLlJ- ALnLlJV- p2+ H2b
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r ALnLlJ H2 MmKû`Q U2MiB2`V /2 HBMbi`mK2Mi
r Ahn.l_ H /m`û2 biM/`/ /miBHBbiBQM /2 HBMbi`mK2Mi /Mb +2 ivT2 /MHvb2 UMQK#`2 
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/2 DQm` K`iB2MV
 H i#H2 LGu BM/B[mMi TQm` +?[m2 TQBMi /BMiû`i H2b ivT2b /MHvb2b ¨ 2z2+im2` Qm 
2z2+imû2b- /2 +H27
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r 1snLlJ MmKû`Q /2 H2tTHQ`iBQM ¨ H[m2HH2 TT`iB2Mi H2 TQBMi /BMiû`i
r SAnLlJ MmKû`Q /m TQBMi /BMiû`i /Mb H2tTHQ`iBQM
r hunLlJ ivT2 /2 HMHvb2
kyRd@yj@kd ky,Rj,98

S;2 9f3

r Ln.h /i2 /2 HMHvb2 ULlGG bB HMHvb2 M Tb ûiû 2z2+imû2V
r Ln.h_ /i2 /2 i`MbKBbbBQM bm` h2``2 /2b `ûbmHiib /2 HMHvb2 ULlGG bB HMHvb2 M 
Tb ûiû i`MbKBb2V
r Ln_1a `ûbmHii /2 HMHvb2 U/QMMû2 QT[m2 /QMi H bB;MB}+iBQM /ûT2M/ /m ivT2 
/MHvb2V
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MQK#`2 /2 DQm`b K`iB2Mb
/2TmBb H``Bpû2 /m `Q#Qi bm` H THMi2X
AX*XRV ú+`B`2 mM2 `2[mi2 aZG [mB /QMM2 H2 MmKû`Q /2 H2tTHQ`iBQM 2M +Qm`b- bBH v 
2M  mM2X
AX*XkV ú+`B`2 mM2 `2[mi2 aZG [mB /QMM2- TQm` mM2 2tTHQ`iBQM /QMi QM +QMMBi H2 
MmKû`Q- H HBbi2 /2b TQBMib
/BMiû`ib /2 +2ii2 2tTHQ`iBQM p2+ H2m`b +QQ`/QMMû2bX
AX*XjV ú+`B`2 mM2 `2[mi2 aZG [mB /QMM2 H bm`7+2- 2M Ki`2b +``ûb- /2 +?[m2 xQM2 
/ûD¨ 2tTHQ`û2 T` H2
`Q#QiX G xQM2 /2tTHQ`iBQM 2bi /û}MB2 +QKK2 H2 THmb T2iBi `2+iM;H2 [mB 2M;HQ#2 
H2Mb2K#H2 /2b TQBMib /BMiû`ib
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#b+Bbb2 2i /2b Q`/QMMû2bVX
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#b2 /2 /QMMû2b \
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7QBb Q +?[m2 BMbi`mK2Mi
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iQiHBiû /2 H2tTHQ`iBQMX

AA SHMB}+iBQM /mM2 2tTHQ`iBQM , T`2KB`2 TT`Q+?2
pMi /2 /ûK``2` mM2 MQmp2HH2 2tTHQ`iBQM- H2 `Q#Qi /QBi /ûi2`KBM2` mM +?2KBM [mB 
HmB T2`K2i /2 Tbb2` T`
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*?[m2 TQBMi /BMiû`i b2` `2Tû`û T` mM 2MiB2` TQbBiB7 Qm MmH +Q``2bTQM/Mi ¨ bQM 
BM/B+2 /Mb H2 i#H2m /2b
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/QMMMi H2b BM/B+2b /2b
TQBMib /BMiû`i /Mb HQ`/`2 /2 H2m` T`+Qm`bX
AAX 

Zm2H[m2b 7QM+iBQMb miBHBiB`2b

AAXXRV GQM;m2m` /mM +?2KBM
ú+`B`2 mM2 7QM+iBQM /2Mii2
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[mB T`2M/ 2M T`Ki`2 mM +?2KBM ¨ T`+Qm`B` 2i H Ki`B+2 /2b /BbiM+2b 2Mi`2 TQBMib 
/BMiû`i Ui2HH2 [m2
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H2 `Q#Qi TQm` bmBp`2
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HB;M2f+QHQMM2 /m i#H2m /V 2i 2M
pBbBiMi iQmb H2b TQBMib /BMiû`i /Mb HQ`/`2 BM/B[mûX
AAXXkV LQ`KHBbiBQM /mM +?2KBM
ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 MQ`KHBb2`n+?2KBMU+?2KBM,HBbi- M,BMiV @= HBbi,
[mB T`2M/ 2M T`Ki`2 mM2 HBbi2 /2MiB2`b 2i `2MpQB2 mM2 HBbi2 +Q``2bTQM/Mi ¨ mM 
+?2KBM pHB/2- +2bi@¨@/B`2
+QMi2MMi mM2 b2mH2 7QBb iQmb H2b 2MiB2`b 2Mi`2 y 2i M U2t+HmVX SQm` +2H +2ii2 
7QM+iBQM +QKK2M+2 T` bmTT`BK2`
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DQmi2 ¨ H }M H2b ûp2Mim2Hb ûHûK2Mib KM[mMib 2M
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AAX"XRV .ûi2`KBM2` 2M 7QM+iBQM /2 - MQK#`2 /2 TQBMib ¨ pBbBi2`- H2 MQK#`2 /2 
+?2KBMb TQbbB#H2b TbbMi
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AAX"XkV *2i H;Q`Bi?K2 2bi@BH miBHBb#H2 TQm` mM2 xQM2 /2tTHQ`iBQM +QMi2MMi ky 
TQBMib /BMiû`ib \ CmbiB}2`X
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2M +?QBbBbbMi bvbiûKiB[m2@
K2Mi H2 TQBMi- MQM 2M+Q`2 pBbBiû- H2 THmb T`Q+?2 /2 H TQbBiBQM +Qm`Mi2X
AAX*XRV ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 THmbnT`Q+?2npQBbBMU/,MTXM/``vV @= HBbi,
[mB T`2M/ 2M T`Ki`2 H2 i#H2m /2b /BbiM+2b `ûbmHii /2 H 7QM+iBQM 
+H+mH2`n/BbiM+2b U[m2biBQM AXXkV
2i 7Qm`MBi mM +?2KBM /2tTHQ`iBQM 2M TTHB[mMi HH;Q`Bi?K2 /m THmb T`Q+?2 pQBbBMX
kyRd@yj@kd ky,Rj,98

S;2 8f3

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2M +QMbB/û`Mi [m2 +2i
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THmbnT`Q+?2npQBbBM \
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+?QBbBbbMi mM TQBMi /2
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7Qm`MBi Tb Mû+2bbB`2K2Mi H2
THmb +Qm`i +?2KBMX
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BKTQ`iMi- HH;Q`Bi?K2 /m
THmb T`Q+?2 pQBbBM 7Qm`MBi mM +?2KBM [mB T2mi i`2 8yW THmb HQM; [m2 H2 THmb 
+Qm`i +?2KBMX

AAA .2mtBK2 TT`Q+?2 , H;Q`Bi?K2 ;ûMûiB[m2
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7QMi BMi2`p2MB` +BM[ i`Bi2K2MibX
RX AMBiBHBbiBQM
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TQm` H2tTHQ`iBQM ¨
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kX úpHmiBQM
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+Q``2bTQM/Mi ¨ b
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+?2KBMX
jX aûH2+iBQM
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BM/BpB/mbX SHmbB2m`b Kû@
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K2BHH2m`2 MQi2- ûHBKBMiBQM T`
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9X *`QBb2K2Mi
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BM/BpB/mbX
8X JmiiBQM
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KmiiBQM- +2bi@¨@/B`2
mM2 i`Mb7Q`KiBQM HûiQB`2X *2ii2 ûiT2 T2`K2i /ûpBi2` ¨ HH;Q`Bi?K2 /2 `2bi2` 
#HQ[mû bm` mM QTiBKmK
HQ+HX
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ûpQHm2` H TQTmHiBQM- Dmb@
[m¨ i`Qmp2` mM BM/BpB/m [mB `ûTQM/2 m T`Q#HK2 BMBiBHX *2T2M/Mi /Mb H2b +b 
T`iB[m2b /miBHBbiBQM /2b
H;Q`Bi?K2b ;ûMûiB[m2b- BH M2bi Tb TQbbB#H2 /2 bpQB` bBKTH2K2Mi bB H2 T`Q#HK2 
2bi `ûbQHm UH2 THmb +Qm`i +?2@
KBM };m`2@i@BH /Mb K TQTmHiBQM \VX PM miBHBb2 /QM+ /2b +QM/BiBQMb /``i 
?2m`BbiB[m2b #bû2b bm` mM +`Bi`2
`#Bi`B`2X
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mM K2BHH2m` +?2KBM /2tTHQ`iBQM
[m2 +2HmB Q#i2Mm T` HH;Q`Bi?K2 /m THmb T`Q+?2 pQBbBMX
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lM2 TQTmHiBQM 2bi `2T`ûb2Miû2 T` mM2 HBbi2 /BM/BpB/mb- +?[m2 BM/BpB/m ûiMi 
`2T`ûb2Miû T` mM +QmTH2 UHQM@
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+Q``2bTQM/Mi mt BM/B+2b
/2b TQBMib /BMiû`i /Mb H2 i#H2m /2b /BbiM+2b T`Q/mBi T` H 7QM+iBQM 
+H+mH2`n/BbiM+2b c
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H TQbBiBQM /2 /ûT`i /m
`Q#QiX
ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 +`ö2`nTQTmHiBQMUK,BMi- /,MTXM/``vV @= HBbi,
[mB +`û2 mM2 TQTmHiBQM /2  BM/BpB/mb HûiQB`2bX *2ii2 7QM+iBQM T`2M/ 2M T`Ki`2 
H2 MQK#`2 /BM/BpB/mb ¨
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/m `Q#QiV i2H [m2 T`Q/mBi T`
H 7QM+iBQM +H+mH2`n/BbiM+2bX 1HH2 `2MpQB2 mM2 HBbi2 /BM/BpB/mb- +2bi@¨@/B`2 /2 
+QmTH2b UHQM;m2m`- +?2KBMVX
AAAX"  aûH2+iBQM
ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 `ö/mB`2UT,HBbiV @= LQM2,
[mB `û/mBi mM2 TQTmHiBQM /2 KQBiBû 2M M2 +QMb2`pMi [m2 H2b BM/BpB/mb 
+Q``2bTQM/Mi mt +?2KBMb H2b THmb
+Qm`ibX PM `TT2HH2 [m2 T 2bi mM2 HBbi2 /2 +QmTH2b UHQM;m2m`- +?2KBMVX G 
7QM+iBQM `ö/mB`2 M2 `2MpQB2 Tb /2
`ûbmHii KBb KQ/B}2 H HBbi2 Tbbû2 2M T`Ki`2X

kyRd@yj@kd ky,Rj,98

S;2 ef3

AAAX*  JmiiBQM
AAAX*XRV ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 Kmi2`n+?2KBMU+,HBbiV @= LQM2,
[mB T`2M/ 2M T`Ki`2 mM +?2KBM 2i H2 i`Mb7Q`K2 2M BMp2`bMi HûiQB`2K2Mi /2mt /2 
b2b ûHûK2MibX
AAAX*XkV ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 Kmi2`nTQTmHiBQMUT,HBbi- T`Q#,7HQi- /,MTXM/``vV @= LQM2,
[mB T`2M/ 2M T`Ki`2 mM2 TQTmHiBQM /QMi 2HH2 7Bi Kmi2` mM +2`iBM MQK#`2 
/BM/BpB/mbX G2 T`Ki`2 T`Q#
U+QKT`Bb 2Mi`2 y 2i RV /ûbB;M2 H T`Q
/ 2bi H Ki`B+2 /2b
/BbiM+2b 2Mi`2 TQBMib /BMiû`iX
AAAX.  *`QBb2K2Mi
AAAX.XRV ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 +`QBb2`U+R,HBbi- +k,HBbiV @= HBbi,
[mB +`û2 mM MQmp2m +?2KBM ¨ T`iB` /2 /2mt +?2KBMb Tbbûb 2M T`Ki`2X *2 MQmp2m 
+?2KBM b2` T`Q/mBi 2M
T`2MMi H T`2KB`2 KQBiBû /m T`2KB2` +?2KBM bmBpB /2 H /2mtBK2 KQBiBû /m /2mtBK2 
TmBb 2M  MQ`KHBbMi 
H2 +?2KBM BMbB Q#i2MmX
AAAX.XkV ú+`B`2 mM2 7QM+iBQM /2Mii2
/27 MQmp2HH2n;öMö`iBQMUT,HBbi- /,MTXM/``vV @= LQM2,
[mB 7Bi ;`QbbB` mM2 TQTmHiBQM 2M +`QBbMi b2b K2K#`2b TQm` 2M /Qm#H2` H2z2+iB7X 
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T(BYR)V 2i UT(K@R)- T(y)V
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/2 /ûT`iX
AAAX1  H;Q`Bi?K2 +QKTH2i
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/27 H;Qn;öMöiB[m2USA,MTXM/``v- K,BMi- T`Q#,7HQi- ;,BMiV @= 7HQi- HBbi,
[mB T`2M/ 2M T`Ki`2 mM i#H2m /2 TQBMib /BMiû`ib U};m`2 kV- H iBHH2 K /2 H 
TQTmHiBQM- H T`Q
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+Qm`i +?2KBM /2tTHQ`iBQM 2i H2
+?2KBM HmB@KK2 Q#i2Mmb m #Qmi /2 ; ;ûMû`iBQMbX
AAAX1XkV 1bi@BH TQbbB#H2 p2+ HBKTHMiiBQM `ûHBbû2- [mmM2 Biû`iBQM /2 HH;Q`Bi?K2 
/û;`/2 H2 `ûbmHii , H2
K2BHH2m` +?2KBM Q#i2Mm ¨ H ;ûMû`iBQM   2bi THmb HQM; [m2 +2HmB /2 H ;ûMû`iBQM  \
.Mb H{`KiBp2- +QKK2Mi KQ/B}2` H2 T`Q;`KK2 TQm` [m2 +2ii2 bBimiBQM M2 TmBbb2 
THmb ``Bp2` \
AAAX1XjV Zm2HH2b mi`2b +QM/BiBQMb /``i T2mi@QM BK;BM2` \ úi#HB` mM +QKT`iB7 
T`ûb2MiMi H2b pMi;2b
2i BM+QMpûMB2Mib /2 +?[m2 +QM/BiBQM /``i 2MpBb;û2X

PTû`iBQMb 2i 7QM+iBQMb Svi?QM /BbTQMB#H2b
6QM+iBQMb
 `M;2UMV `2MpQB2 H bû[m2M+2 /2b M T`2KB2`b 2MiB2`b U    V
 HBbiU`M;2UMVV `2MpQB2 mM2 HBbi2 +QMi2MMi H2b M T`2KB2`b 2MiB2`b /Mb HQ`/`2 
+`QBbbMi ,
HBbiU`M;2U8VV  (y- R- k- j- 9)
 `M/QKX`M/`M;2U- #V `2MpQB2 mM 2MiB2` HûiQB`2 +QKT`Bb 2Mi`2  2i #@R BM+Hmb U 2i 
# 2MiB2`bV
 `M/QKX`M/QKUV `2MpQB2 mM MQK#`2 ~QiiMi iB`û HûiQB`2K2Mi /Mb < < bmBpMi mM2 /Bbi`B#miBQM mMB7Q`K2 `M/QKXb?m77H2UmV T2`Kmi2 HûiQB`2K2Mi H2b ûHûK2Mib /2 H HBbi2 m UKQ/B}2 mV `M/QKXbKTH2Um- MV `2MpQB2 mM2 HBbi2 /2 M ûHûK2Mib /BbiBM+ib /2 H HBbi2 m +?QBbBb HûiQB`2K2Mi- bB M = H2MUmV- /û+H2M+?2 H2t+2TiBQM oHm21``Q` Ki?Xb[`iUtV +H+mH2 H `+BM2 +``û2 /m MQK#`2 Ki?X+2BHUtV `2MpQB2 H2 THmb T2iBi 2MiB2` bmTû`B2m` Qm û;H ¨ t kyRd@yj@kd ky,Rj,98 S;2 df3 Ki?X7HQQ`UtV `2MpQB2 H2 THmb ;`M/ 2MiB2` BM7û`B2m` Qm û;H ¨ t bQ`i2/UmV `2MpQB2 mM2 MQmp2HH2 HBbi2 +QMi2MMi H2b ûHûK2Mib /2 H HBbi2 m i`Bûb /Mb HQ`/`2  Mim`2H  /2 b2b ûHûK2Mib UbB H2b ûHûK2Mib /2 m bQMi /2b HBbi2b Qm /2b imTH2b- HQ`/`2 miBHBbû 2bi HQ`/`2 H2tB+Q;`T?B[m2V PTû`iBQMb bm` H2b HBbi2b H2MUmV /QMM2 H2 MQK#`2 /ûHûK2Mib /2 H HBbi2 m , H2MU(R- k- j)V  j c H2MU((R-k)- (j-9))V  k m Y p +QMbi`mBi mM2 HBbi2 +QMbiBimû2 /2 H +QM+iûMiBQM /2b HBbi2b m 2i p , (R- k) Y (j- 9- 8)  (R- k- j- 9- 8) M m +QMbi`mBi mM2 HBbi2 +QMbiBimû /2 H HBbi2 m +QM+iûMû2 M 7QBb p2+ 2HH2@KK2 , j (R- k)  (R- k- R- k- R- k) 2 BM m 2i 2 MQi BM m /ûi2`KBM2Mi bB HQ#D2i 2 };m`2 /Mb H HBbi2 m- +2ii2 QTû`iBQM  mM2 +QKTH2tBiû i2KTQ`2HH2 2M k BM (R- k- j)  h`m2 c k MQi BM (R- k- j)  6Hb2 mXTT2M/U2V DQmi2 HûHûK2Mi 2 ¨ H }M /2 H HBbi2 m UbBKBHB`2 ¨ m 4 m Y (2)V /2H m(B) bmTT`BK2 /2 H HBbi2 m bQM ûHûK2Mi /BM/B+2 B /2H m(B,D) bmTT`BK2 /2 H HBbi2 m iQmb b2b ûHûK2Mib /QMi H2b BM/B+2b bQMi +QKT`Bb /Mb HBMi2`pHH2 < < mX`2KQp2U2V bmTT`BK2 /2 H HBbi2 m H2 T`2KB2` ûHûK2Mi [mB  TQm` pH2m` 2- /û+H2M+?2 H2t+2TiBQM oHm21``Q` bB 2 M2 };m`2 Tb /Mb m- +2ii2 QTû`iBQM  mM2 +QKTH2tBiû i2KTQ`2HH2 2M mXBMb2`iUB- 2V BMb`2 HûHûK2Mi 2 ¨ H TQbBiBQM /BM/B+2 B /Mb H HBbi2 m U2M /û+HMi H2b ûHûK2Mib bmBpMibV c bB B =4 H2MUmV- 2 2bi DQmiû 2M }M /2 HBbi2 m(B)- m(D) 4 m(D)- m(B) T2`Kmi2 H2b ûHûK2Mib /BM/B+2 B 2i D /Mb H HBbi2 m mXbQ`iUV i`B2 H HBbi2 m 2M TH+2- /Mb HQ`/`2  Mim`2H  /2 b2b ûHûK2Mib UbB H2b ûHûK2Mib /2 m bQMi /2b HBbi2b Qm /2b imTH2b- HQ`/`2 miBHBbû 2bi HQ`/`2 H2tB+Q;`T?B[m2V PTû`iBQMb bm` H2b i#H2mt UMTXM/``vV MTX``vUmV +`û2 mM MQmp2m i#H2m +QMi2MMi H2b ûHûK2Mib /2 H HBbi2 mX G iBHH2 2i H2 ivT2 /2b ûHûK2Mib /2 +2 i#H2m bQMi /û/mBib /m +QMi2Mm /2 m MTX2KTivUM- /ivT2V- MTX2KTivUUM- KV- /ivT2V +`û2 `2bT2+iBp2K2Mi mM p2+i2m` ¨ M ûHûK2Mib Qm mM2 K@ i`B+2 ¨ M HB;M2b 2i K +QHQMM2b /QMi H2b ûHûK2Mib- /2 pH2m`b BM/ûi2`KBMû2b- bQMi /2 ivT2 /ivT2 [mB T2mi i`2 mM ivT2 biM/`/ U#QQH- BMi- 7HQi- wV Qm mM ivT2 bTû+B}[m2 MmKTv UMTXBMiRe- MTX7HQijk- wVX aB H2 T`Ki`2b /ivT2 M2bi Tb T`û+Bbû- BH T`2M/ H pH2m` 7HQi T` /û7mi MTXx2`QbUM- /ivT2V- MTXx2`QbUUM- KV- /ivT2V 7QM+iBQMM2 +QKK2 MTX2KTiv 2M BMBiBHBbMi +?[m2 ûHû@ K2Mi ¨ H pH2m` xû`Q TQm` H2b ivT2b MmKû`B[m2b Qm 6Hb2 TQm` H2b ivT2b #QQHû2Mb XM/BK MQK#`2 /2 /BK2MbBQMb /m i#H2m  UR TQm` mM p2+i2m`- k TQm` mM2 Ki`B+2- 2i+XV Xb?T2 imTH2 /QMMMi H iBHH2 /m i#H2m  TQm` +?+mM2 /2 b2b /BK2MbBQMb H2MUV iBHH2 /m i#H2m  /Mb b T`2KB`2 /BK2MbBQM UMQK#`2 /ûHûK2Mib /mM p2+i2m`- MQK#`2 /2 HB;M2b /mM2 Ki`B+2- 2i+XV û[mBpH2Mi ¨ Xb?T2(y) XbBx2 MQK#`2 iQiH /ûHûK2Mib /m i#H2m X7Hi Biû`i2m` bm` iQmb H2b ûHûK2Mib /m i#H2m XKBMUV- XKtUV `2MpQB2 H pH2m` /m THmb T2iBi U`2bT2+iBp2K2Mi THmb ;`M/V ûHûK2Mi /m i#H2m  c +2b QTû`iBQMb QMi mM2 +QKTH2tBiû i2KTQ`2HH2 2M # BM  /ûi2`KBM2 bB # 2bi mM ûHûK2Mi /m i#H2m  c bB # 2bi mM b+HB`2- pû`B}2 bB # 2bi mM ûHûK2Mi /2  c bB # 2bi mM p2+i2m` Qm mM2 HBbi2 2i  mM2 Ki`B+2- /ûi2`KBM2 bB # 2bi mM2 HB;M2 /2 MTX+QM+i2Mi2UUR- kVV +QMbi`mBi mM MQmp2m i#H2m 2M +QM+iûMMi /2mt i#H2mt c R 2i k /QBp2Mi pQB` H2 KK2 MQK#`2 /2 /BK2MbBQMb 2i H KK2 iBHH2 ¨ H2t+2TiBQM /2 H2m` iBHH2 /Mb H T`2KB`2 /BK2MbBQM U/2mt Ki`B+2b /QBp2Mi pQB` H2 KK2 MQK#`2 /2 +QHQMM2b TQm` TQmpQB` i`2 +QM+iûMû2bV r r r 6AL r r r kyRd@yj@kd ky,Rj,98 S;2 3f3