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e>EJADCDB36EC>;Õ36;DHJI436EMyAB7*IRHJIRiNH31#ÒDCb 3*Ù<5gJHJ3<OP3<;DH

253<;DHJIK3<E7

x7)56HJIG7)iAB;DH "5

2 ≤ N

x≤ 5

e eDfwO@?¢ y5<EHF3HQ¡k58EJAD1¢AB;DHFO58;>¤ $3=>367OIK;BIK9gJ58EJAHHJ3<76efwO!?5OP5<;>¤.$3:1K5OPADIRHJI+$3=BCS;>ABOPN2EJ3 "5y5<EH,BMB1GC>7QC>;!HJI436E7=DCS;>ABOSNBEJ3 "5¡¢5<EJAD1kMB1GC>7QC>;/e y58EHF5:OP5<;>¤.$31K5OSADIHJI+$3!=DC;>ABOPN2EJ3 "5¡k58EJAD1tyM21RC>7xC>;PHJIK3<E7q=BC;>ABOPN2EJ3 "5:fwO!?SMB1GC>7q=>3CÙke¢¡k58EJAD15OP5<;>¤.$3¿1K3¿O '36OS3¿;>ABOPN2EJ3ÒBCb fwO!?!M21RC>7FC>;@HJIK3<E7F=BC;>ABOPN2EJ3"5y58EHMB1GC>7HEJADIG7<e¡bABOPNBIK3<;P=>3OIG;BI4)g5<EJAHH367#AB;DH**IR1K7FOP5<;>¤.$33<;2793<OPNB1K3Ô eª#;>3xgAD14AB;2;>3OPAHABEIK79$383367*H36;HE5<IG;@=k 5`D58;>g3<E3<;¿H36EE56IK;MB1K5H56C@Et?H*LBOS3x=>3158IR14ABO#"3HEJ367xM258E#L23C>EJ3>e,)65!g*AD1KAB;2;>3!3<7HF14AB;>¤`Cy3!=23

18IR14ABO#"3HEJ32ekª#;gADC>EEI436E3<7H3<;BA6? $3=>3U1t 5<EEI+"36EJ3PB36E7@1t 5`D58;DH:7)C>E¿C>;>3OSAHJA2g?Bg1K3HHJ3EJADC21G58;DH "5PBIH3679793g*AB;27HJ5<;DH32e£1EJ3`HADC>EJ;>3IGOPO#$38=DIK5H36OS36;DH"51G5PO '3<OP3@BIH3679793P3`Hx7,AB;M258EJgADC>E7:5=BC>E%$3

3OIG;BCBH367<e>¥FCy31G1K3 $3H5<IHw7)5¿DIRHJ3<7)7,3Ô

Ð Í e`>EJADCDB3<Ek143QEJ367*HJ3#=>3F1G5=DIDIK7*I4AB;=DC:MyAD1 ?2; 'ABOP3x+x3+x9+x27+x81+x243M>5<E

x2 − 1e

ÐqÐ e $3HJ3<EJOIG;>3<E143qM,$3<EIKO#"3HEJ3¿=¢ C>;@HEIK5<;>¤`143qEJ38gJH58;>¤`1K3¿5`?D58;DHC>;>3L8?BMyAH*L!$36;BC>7,3=>3H3`HC>;>3:5<IGEJ3:=>3

Ae

Ð Ì e¥FC>58;>=:C>;!3<;DHJIK3<En367*H=DIDIK79$3xM>5<E

36143xEJ367*HJ3q3<7H

1eD¥FC>5<;>=

n+13<7H=DIDIK79$3M>5<E

2D1K3EJ3<7H3:3<7H

1e>¥FCy31k3<7H143EJ367*HJ3ÒDC>58;>=

n − 13<7HF=BI BIG79$3M258E

6Ô

©q3*Ù<H3¤ÌB3xH*L2379AD1GCBHJIKAB;27HJAH*L23 h58EIRHJIGOS3@h5H*L23<O56HJIKg7Q¡bAB;DHJ3<7H

&8 eÌÍqÍ

® Å ´ y½Æb½ÇÃ» Å ´ÆÁ¿»#ÇÈ´Æb½*ÄÉ¶¾¿²QÆ/»QÉkÆÐ e@fH¿SAk g1KA2g)>kH*L23SL2ADC>Ex58;>=OIG;BCBH3PLB58;>=B7AB;¬5Sg1KA2g)PiABEJO 5PEI4¤6LHq5<;>¤`1432efiGH36EAk g14A2g)22LB5HQIK7#H*L23!;>3*Ù<HFHJIKOP3!5HwLDI4g)LH*L23g1KA2g)LB58;>=B7xiABEO 5EI4¤6LH58;>¤`1K38Ô ! "#$&%('#*)+,.-'/'#$0 e¢L23=DIK58¤E58O7@56HEI4¤6LH:79L2A8¦H*L23LB5<;>=27!ADiF5g14A2g)#56Hb $ ÜRHABM=BIG5¤E5<O!Ý>58;>=¿58¤5<IG;xH*L23Q;>3)Ù6HyHJIKOP3H*L23,?ÕiABEJO 5EI4¤6LH!58;>¤`1K3ÜN2AHHABO =DIK58¤E58O@Ý9e&¢L23;>3)Ù6HHJIGOS35<M>M>3<5<E7FHJAN>35:1RIRHHJ1K3M2587HF1 $ e

)<3HxN>3PH*L23U;BC>OSN23<EADiFOIG;BCBH367@5<iGHJ3<E:1 H*L23P;>3*Ù<HHJIKOP3!H*L23SLB5<;>=27iABEJO 5SEI4¤6LHq5<;>¤`1432e¢L23L2ADC>ELB58;>=g*ABOM21K3HJ3<7qAB;>3!EJ3,BAD1RCBHJI4AB;SIG;

12L2ADC>E72L23<;>g3BqIG;

xOIG;BCBH367xIRHOSA6B3<7H*LBEJADCy¤6L x

12 × 60EJ3,BAD1GCBHJIKAB;27<e 6IKOIG1G58E1?kS7*IK;>g3H*L23 OIK;BCBHJ3LB58;>=HJ5883<760

OIK;BCBHJ3<7HAhg*ABOM21K3HJ3¬AB;>3EJ3,BAD1GCBHJIKAB;¢IRHOSA6B3<7H*LBEJADCy¤6Lx/60

EJ3,BAD1GCBHJIKAB;27wIK;xOIK;BCBHJ3<76ew£t;H*L23IG;DH36EtD5<1Fi4EJABO $ HJAH*L23EJ38ÒDC2IKEJ3<=HJIKOP3BH*L23OIG;BCBH3ÖLB58;>=&LB5<7¬OSA6B38=H*LBEJADCy¤6L3)Ù<58gJHJ1?LB561Gi5EJ3,BAD1GCBHJIKAB;OSABEJ3qH*LB58;!H*L23L2ADC>EFLB5<;>=/e¢L236EJ36iABEJ3D

x

60=

x

720+

1

2

LDI4g)L!?BIK361K=27

11x = 360kABE

x = 32 811

eQ¢LDC>7`2H*L23EJ38ÒDC2IKEJ3<=HJIGOS3¿IK732 8

11

OIG;BCBH367#56iGH36EF $ e

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6

Ì eDABE5MyAB7*IRHJI B3!;BC>OSN23<E7)Cyg)LU5<73.14

>3gJ5<1R13H*L23 !3'546'7098!$0: 58;>=

0.14H*L23<; 0=>368!$0: eIG;>=5MyAB7*IRHJI B3¿;BC>OPN>36E7)Cyg)LH*LB5HH*L23qi4E5gJHJIKAB;25<1>M258EH,H*L23IK;DHJ38¤636EwM>5<EH,B5<;>=!H*L23:;BC>OPN>36EwIHJ79361Rik58EJ3¿H*LBEJ3<3gAB;27,3<gCBHJIB3qH36EO7Ü45`ÝIK;5<;58EIRH*LBOP3HJIKgF7,3<ÒBCy36;>g*3=2 ÜN8ÝIK;5¤638ABOP3HEI4gF7938ÒDCy3<;>g3>eÜKkL23@7,3<ÒBCy36;>g*3a1a2a3a4. . .

IK7xgJ5<1R143<= $0?3.-1 "'$3 IGi¢H*L236EJ3IG75!;BC>OPN>36Ed7)Cyg)LH*LB5H

a2 − a1 = a3 − a2 = a4 − a3 = . . . = d22IHwIK7xgJ5<1R143<= 4='$ "'$.0?3 IRiH*L23<EJ3IG7Q5;BC>OPN>36E

r 6= 07*Cyg)L!H*LB5H

a2/a1 = a3/a2 = a4/a3 = . . . = re Ý

! "#$&%('#*)+,.-'/'#$0 e)<3H

xN235xM>AB7)IHJIB3#;BC>OPN>36Eè)<3H

nN23#IRH7IG;DH3<¤63<E/M258EH/58;>=

yIRH7i4E5gJHJIKAB;25<1M>5<EHè¢LDC>7`

x = n + y8L23<EJ3

nIG7Q58;IG;DH3<¤63<Ew5<;>=

0 ≤ y < 1eÜ45`Ý 3:5<;DHQHAS>;>=

x79APH*LB56H

yn¢58;>=

x58EJ3gAB;27,3<gCBHJIB3H36EO7qIG;5<;58EIRH*LBOP3HJIKgF7,3<ÒBCy36;>g*32e

IGE7*H7*C>M>M>AB7,3xH*LB5H0 < x < 1

ekL23<;n = 0

58;>=x = y

ekL23¿;BC>OSN23<E7y

n25<;>=

x58EJ3:;>AHH36EO7QIG;58;5<EJIH*LBOS3`HJI4gF7938ÒDCy3<;>g3IK;!H*LDIG7#g5<7,32e

&©qA8587)7)C>OP3

x ≥ 1ekL23<;

n ≥ 15<;>=

y < n ≤ xe!£t;ABEJ=>36E#iABE

yn58;>=

xHAN23gAB;27,3<gCBHJIB3qH36EO7QIG;58;5<EJIH*LBOS3`HJI4gF7938ÒDCy3<;>g3B83EJ38ÒDC2IKEJ3

n − y = x − n = d Ü7&[Ý

iABE#7,ABOP3!EJ365<1b;BC>OSN23<Ede:6IG;>g*3

x = n + y23gJ58;31GIGOIG;256HJ3

xIG;Ü &`ÝHJA¤63`H

n − y = y = dew¢L236;

2y = neF6IK;>g3

nIG7#5@M>AB7)IHJIB3IG;DH3<¤63<EF5<;>=

0 ≤ y < 1H*LDIK738ÒDC>56HJIKAB;¿IG7/795HJIK7*k3<=¿AB;B1?#L23<;

n = 15<;>=

y = 12

ebkLDC>7`H*L23QAB;B1?qMyAB7)7)IKNB1K3D5<1RCy3iABExIG7

x = n + y = 1 + 12

= 32

eÜN8Ý 3U:58;DH:HAhy;>=

x79AH*LB56H

y

n#58;>=

x5<EJ3g*AB;27938gCBHJI B3H36EO7IG;5¤63<ABOS3`HEJIKg:7,3<ÒBCy36;>g*32efw7:IG;hM258EH:Ü45`Ý*/3S7,3<3PH*LB5HqH*LDIK7@=2A23<7;>AH:LB58M2My36;IRi

0 < x < 1e8A3:5<797*C>OS3

x ≥ 1ekL23<;

n ≥ 158;>=

y < n ≤ xe£t;ABEJ=23<E>iABE

yn`58;>=

xHAxN>3Fg*AB;27938gCBHJI B3HJ3<EJOP7¢IG;q5#¤63<ABOS3`HEJIKg¢7938ÒDCy3<;>g3B3:EJ3<ÒBC2IGEJ3

n

y=

x

n= r

Ü ÝiABEw79ABOS3

r 6= 0e6IG;>g*3

x = n + y83:g5<;S31GIGOIG;256HJ3

xIG;PÜ Ý/HA¤63`H

n

y= 1 +

y

n= r

e Ü Ý©qAHJ3H*LB5H

ygJ58;2;>AH¢N>3

0`7)IG;>g*3H*LDIG7yADC214=xOP5883

n/yC>;>=>3y;>3<=/e8A

0 < y < 1e¢L236EJ36iABEJ3D

n/y > n58;>=

1 + (y/n) < 1 + (1/n) ≤ 2Ü47*IK;>g3

n ≥ 1Ý9e¢LDC>7`

n <n

y= 1 +

y

n≤ 2

e>CBH>3Q8;>A8 H*LB5H

nIG7/5QM>AB7)IHJIB3IK;DHJ38¤636E`ebkL23#AB;B1 ?qMyAB7)7)IKNBIR1GIH?x7)56HJIG7)iR?DIK;>¤

n < 2IK7n = 1

e83HHJIG;>¤n = 1

IG;¬Ü Ý*y3P¤63`H1/y = 1 + y

yLDI4g)LgJ58;¬N23EJ3[:EJIHH36;¬5<7y2 + y − 1 = 0

e B?@H*L23¥FC>58=2E5HJI4g DABEOC21K5By =

−1 ±√

12 − 4(1)(−1)

2(1)=

−1 ± √5

2

e6IK;>g3

y > 03:OC>7HQLB5`B3

y =−1 +

√5

2

ekLDC>7DH*L23AB;B1?M>AB797*I4ND143xD5<1RCy3:iABExIK7

x = n + y = 1 +−1 +

√5

2=

1 +√

5

2

e× e¢fEJ38gJH58;>¤`C21G58E/H58;>:LB5`BIG;>¤1K3<;>¤[H*L

60g*O6IK=DH*L

60g*O5<;>=!L236IK¤6LH

40g*OÊIG7>1R143<=@IRH*L:56HJ3<EHJA5@=23<MDH*LSADi

15g*O¦58;>=EJ3<7HJ7AB;P5@L2ABEIRÑ)AB;DH5<12HJ58NB1K3>e )<3H

A

BC5<;>=

DIK;g?Bg1GIKgwABEJ=>36EN>3H*L23qiADC>EN2AHHABOÞg*ABEJ;>3<E7QADi2H*L23HJ5<;>ke6C>M2MyAB793H*LB56H/H*L23q3<=>¤63BC

IG77*14A81 ?E56IK7938=79A¿H*LB56H/H*L23q3<=>¤63AD

EJ36OP56IK;27wAB;H*L23H5ND1432efw7#:5H36E yA8:7qADCBH,>H*L23H58;>IG7xE5<IG7,3<=SC>;DHJIR1¢H*L233<=>¤63AB

O58367q5<;5<;>¤`143!ADi60 IH*LH*L23:H5ND1432ekL233<=>¤63 BC

IK7QH*L23<;S1KA83<EJ3<=PC>;DHJIR1¢H*L23:H58;>AB;>g*3@58¤5<IG;EJ3<7HJ7#AB;!H*L23¿H5ND1432ebfHH*LDIK7QM>ADIK;DH,8LB5HIK7wH*L23:=>36MBH*LADi>:5H36EIG;!H*L23¿H58;>Ô

e¡bAB;27*I4=23<EH*L23 7)IHJC>56HJIKAB;mL236;H*L23hHJ5<;>LB5<7N>3<3<;UE56IK7938=S7,AH*LB5HFH*L23!38=2¤63

ABOP5883<7x5<;U58;>¤`1K3ADi

60 IH*LH*L23UH5ND1432e£iF:5H36E@=BIK=hIK;>=2383<= 7)M2IR1G1i4EJABO H*L23&HJ5<;>LDIR143IHÛ:587N>3IK;>¤E5<IG7,3<=¢H*L23:56HJ3<EQ;>A8ÊEJ3<58g)L23<7QH*L23@1K3,B361bADiEF

BL236EJ3EIK75HABMhgABE;>36EADiH*L23HJ5<;>¬ÜRH*L23Ug*ABEJ;>3<E¿58N>A6B3

AÝ58;>=

FIK7P5MyADIG;DHAB;

AB587P79L2A8:;ÕIK; H*L23h=DIK58¤E58O¬e

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A

B

E F

Ü©xAH3H*LB5H,Q7*IK;>g3H*L23UHJ5<;>hLB587N23836;ÕE56IK7938= N>3,?BAB;>=45 |AF | IG7!1436797H*LB58;

|AE| = 40B58;>=7)IG;>g*3 |AB| = 60

BMyADIG;DHF=>A2367QIG;>=>3<38=!1GIK3:AB;

ABe Ý6IK;>g3

EFIK7QM258E561G1K3612HJA@H*L23¿H5ND143qHABM¢

∠EFA = 60 e 3<;>g3B|AE| = |AF | tan 60 = |AF |

√3ABE |AF | = |AE|/√

3 = 40/√

3e¢L23BAD1RC>OS3UADi:5H36EqIG;H*L23HJ5<;>56HxH*LDIG7M>ADIK;DHqIK73<ÒBC>561HAH*L23S5<EJ3<5UADi

4AEFOC21RHJIGM21RI43<=ÕN<?

60g*O ÜRH*L23=BIGOS36;27)IKAB;ÖADiQH*L23H58;>M>3<EJMy36;>=BIKgC21K5<EHJA

4AEFÝ)e¢LDC>7`H*L23xBAD1GC>OP3:ADi2:56HJ3<EIG7 1

2× 40 × 40√

3× 60 = 48000√

3

g*O 3 ekL23ABEJIK¤`IK;2561BAD1GC>OP3¿ADiB:5H36EIG715 × 60 × 60 = 54000

g*O 3 6LDI4g)LIG7F¤EJ3656HJ3<E/H*LB58;48000√

3

g*O 3 exkL23<EJ3iABEJ3Bk79ABOS3:5H36E#=>A2367IK;>=2383<=U7)M2IR1G1¢i4EJABO¦H*L23:HJ5<;>>y58;>=PH*L23BAD1GC>OP3ADiyH*L23EJ3<O5<IG;BIK;>¤:56HJ3<EIK7 48000√3

g*O 3 e)<3H

dN>3:H*L23y;2561/=>36MBH*LUADik:56HJ3<EFIG;PH*L23:HJ5<;>keq¡bAB;27*I4=23<EIK;>¤¿H*L23:BAD1RC>OS3ADi2:56HJ3<EEJ3<O5<IG;BIK;>¤xIG;@H*L23qH58;>283LB5`B3

d × 60 × 60 = 48000√3

<LDIKg)L@IKOM21RI4367H*LB56Hd = 40

3√

3

ekLDC>7H*L23¿y;2561¢=23<MDH*LPADi>:5H36EIK;!H*L23qHJ5<;>IK7 40

3√

3

g*OeÚ e6C>M2MyAB793xH*LB56HH*L23¿MyAB7*IRHJI B3xIK;DHJ38¤636E7F58EJ3:EIRHHJ3<;@IK;!579MBIKE561<5<7/79L2A8:;/ew«¢31K5HJIB3HJA#H*L23F;BC>OSN23<E

1,L236EJ3Q=>A2367H*L23P;BC>OPN>36E

200658M2My3658EÔÜ DABEq3)Ù<5<OPMB143D

1058M2My3658E7AB;>3C>;BIHwC>M58;>=!HA@C>;BIRH7FHJA@H*L23:EI4¤6LHFADi

1e Ý

& & &9& & e6ee &7 & 6 )+ '5>+ =#1'!6' )4.!#1'7 "#$%!'$# )+".-''$#$330 ¢L23P;BC>OSN23<E7@AB;H*L23S=BIG5¤6AB;2561w¤6ADIG;>¤C>M¬HJAH*L23PEJIK¤6LH5<EJ3

1925. . .

LDI4g)L5<EJ3@H*L237,ÒDC>58EJ367ADiH*L23A2=2=MyAB7*IRHJI B3!IG;DH3<¤63<E76ekL23Pg1KAB7,367*HA2=>=79ÒBC>5<EJ3HA2006

IK7452 = 2025

e6IG;>g*345 = 2 × 22 + 1

¢3P7,3<3H*LB5H2025

1RI436722C>;BIRH7C>M!5<;>=

22C>;BIRH7HA¿H*L23xEI4¤6LHADi

1e>CBH

2006IK71KA2g5H3<=

2025−2006 = 19MyAB7*IRHJIKAB;27#HJAH*L23143iGHADi2025

ex¢L236EJ36iABEJ3D>H*L23@MyAB7*IRHJIKAB;ADi2006

IG722C>;BIHJ7C>M58;>=

3C>;BIRH7FHJA@H*L23:EI4¤6LHFADi

1e

Î ekf $0 ' 8!0 IG75#M>56IKE(x, y)

ADiDM>AB7)IHJIB3FIG;DH3<¤63<E77)Cyg)LxH*LB5Hx+y

58;>=xy58EJ3N>AH*LMy36EJi3<gJH7,ÒDC>58EJ367<eDABE3)Ù<5<OPMB143D

(5, 20)IK75¿79ÒBC>5<EJ3qM>56IKE7*IK;>g3

5+20 = 2558;>=5 × 20 = 100

5<EJ3N>AH*LMy36EJi3<gJHw7,ÒDC>58EJ367<e8L2A8H*LB56Hw;>A7,ÒDC>58EJ3:M25<IGEF3*Ù8IG7*H7IK;@LDIKg)LPAB;>3:ADiyH*L23;BC>OSN23<E7QIK73e

! "#$&%('#*)+,.-'/'#$0 e6C>M>M>AB7,3H*LB56H358;>=

xgAB;27*HJIHJCBH3!57,ÒDC>58EJ3!M>56IKE[e¢L236;

3 + x = a2 5<;>=3x = b2 iABE#79ABOS3@M>AB7)IHJIB3IG;DH3<¤63<E7 a

58;>=be¿6IG;>g*3

3xIG7x5!My36EJi3<gJH79ÒBC>5<EJ3B

x

OC>7HqN23ADiH*L23!iABEJO

3c2 iABE79ABOS3!MyAB7*IRHJI B3!IG;DH3<¤63<E ce!6CyNB7*HJIHJCBHJIK;>¤

x = 3c2IK;DHJA3 + x = a2 <3:A2NHJ56IK; 3(1 + c2) = a2 2<H*LDC>7` 1 + c2 OC>7HFgAB;DHJ56IK; 3

5<7Q5i45gJHJABEè 3xIG1R1y7,L2A8 H*LB56HH*LDIG7FIK7Q;>AHwM>AB797*I4ND1432e L23<;3U=DIDI4=23cN<?

3/H*L23SEJ3<O5<IG;>=>36E¿OC>7HN>3

01ABE

2e£iIRHqIG7

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k2bH*LDIK7:¤ÌB367

1 + c2 = 1 + 9k2 kLDI4g)L¬=2A23<7;>AHQLB5`B335<7#5:i458gJHABEFÜK7)IG;>g*3

3IG7#5:i458gJHABEFADi

9k2 Ý9eQ£iyH*L23:EJ3<O5<IG;>=>36EwIG7 1DH*L23<;

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k26H*LDIK7Q¤Ì B3<7

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35<7Q5i45gJHJABEwÜ47*IK;>g3

3IK7F5i45gJHJABEFADi

9k2 + 6kÝ)eF£iyH*L23EJ3<O5<IG;>=>36EIK7

2H*L23<;

c = 3k +2iABE79ABOS3xIG;DH3<¤63<E

k26H*LDIG7F¤ÌB367

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35<7#5i458gJHABEwÜ47*IK;>g3

3IG7Q5i45gJHJABEFADi

9k2 + 12k + 3Ý)e®

e IG;>=561G1y79AD1GCBHJIKAB;27FIK;EJ365<1y;BC>OPN>36E7x58;>=

yiABEH*L237?27H36OÓADi¢38ÒDC>56HJIKAB;27

2(x + y − 2) = y(x − y + 2)

x2(y − 1) + y2(x − 1) = xy − 1e

e)<3HHJIG;>¤

a = x − 158;>=

b = y − 1H*L23¤ÌB36;P38ÒDC>56HJIKAB;27N238gABOS3

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(a + 1)2b + (b + 1)2a = (a + 1)(b + 1) − 1e

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2H*LDC>7a = 0

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kH*L23!yE7Hx38ÒDC>56HJIKAB;¬N>3<g*ABOP3<72b = (b + 1)(−b + 2)

yLDI4g)LSIK738ÒDC2ID561436;DHHAb2 + b − 2 = 0

e 5gJHJABEJIG;>¤¤Ì B3<7(b + 2)(b − 1) = 0

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28H*LDC>7a = 2

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3Ug5<;h7,3`Ha = −b − 3

IG;H*L23S>E7*H:38ÒDC>56HJIKAB;HJAh¤63`H2(−3) = (b + 1)(−b − 3 − b + 2)

yLDIKg)L7)IGOPMB1GIRk367HA2b2 + 3b − 5 = 0

eD58gJHABEIK;>¤F¤Ì B3<7

(2b+5)(b−1) = 0ebkL23<;

b = −52

ABEb = 1

e6IK;>g3a = −b−3

3A2NHJ56IK;a = −1

2

58;>=a = −4

BEJ3<7)My3<gJHJIB31?be>A7*C>OPO58EIRÑ)3DH*L233<ÒBC>5HJI4AB;27#LB5B3qH*L23iAD1R14A8IG;>¤qDB3:79AD1GCBHJIKAB;27a b x y0 1 1 20 −2 1 −12 0 3 1

−12

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tn = n2 + 20DiABE

n ≥ 1e8L2A8&H*LB5HiABEF561G1

n ≥ 1DH*L23@¤EJ3656HJ3<7HQg*ABOOSAB;P=BI BIG7,ABEwADi

tn58;>=

tn+1OC>7HQN235@=DIDIK79ABEADi

81e

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n+1g*AB;27938gCyHJIB379ÒBC>5<EJ3<7`2N>3<¤ÌK;2;BIG;>¤xIH*L@H*L237,ÒDC>58EJ3:ADi

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ngAB;27,3<gCBHJIB3QIK;DHJ38¤636E7<eÜ7BABE/3)Ù<5<OPMB143D`L23<;

n = 13LB5B3

32 + 42 = 52 D5<;>=:L23<; n = 23LB5`B3

102 + 112 + 122 = 132 + 142 e Ý8L2A8H*LB56HH*LDIG7#M2EJABMy36EHt?PL2AD14=B7FiABEw5<;?n ≥ 1

e

Å × Í Ú e 0 8 $'# )$+ -.+¹ '7!5'57¹0?7µ77¸ $= e

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abcIK7Q5:E56HJIKAB;25<1y;BC>OPN>36E`e

Å × ÍÎ e 0 8 $7'$# )+»$' ¹0 0 '70 $´(6Cµ'0 36 '6? #1'¶$1+7¹0 '51 8! e

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x +√

y +√

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®e 0 8 $7'$# )$+¹056 ' -13² +'0 ' "0?w´(6Cµ'0 3.+* ; ' ² ; !#

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tn+1=2ADIRH'3HEJ3C>;S=DIDIK7936C>Ew=>3

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ABCC>;qHEIK5<;>¤`143Q5`B38g

∠ABC = ∠ACB = 40 e66I P 367*HkC>;¿MyADIG;DH"51 IK;DH%$36EJIK36C>E=BC!HEIK5<;>¤`143=23¿7,ABEH3¿ÒBCy3∠PBC = 20 3H

∠PCB = 30 OSAB;DHEJ36EÒBCy3BP = BA

e

8Å × Í × e 0 8 $' 8!$0 '&µ'7»$0?¼?½40 '$)$7¶0 63' eª#;>3ÖgC>EJIK36C>793ÕEJ361G56HJIKAB;3<;DHEJ3 1K3<7¬g5<EE%$3<7=23Õ;>ABOPN2EJ367;25HJC>EJ361G7¬OPAB;DHEJ3ÒBCy3Û1K5 7,ABOOS3 =>3

n + 1gJ58EJE%$367¬gAB;279$3<gCBHJIGi47`qgABOPOP3<;g*58;DHUM258EP1K3ÕgJ58EJE%$3Õ=>3

n(2n + 1)!3<7H$38¤56143 "5 1K579ABOPOP3=>367ÖgJ58EJE%$367Ö=23<7

n3<;DHJIK3<E7ÕgAB;279$3<gCBHJIGi477)C2I D58;DH7<eFÜ0D5<E!3)Ù836OPMB143Dw7*I

n = 1AB;Õ5

32 + 42 = 52 w3H@7)I n = 2AB;Õ5

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eÅ × Í Ú e 0 8 $' 8!$0 -.+¹ '7!5'57¹0?7µ77¸ $ 6' e8ADIRH

ab3`H

cHEJADIK7!;>ABOPN2EJ367E%$3831K7:H31K7ÒDCy314367!79ABOPOP3<7

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√x +

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© £ ©

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« ................................................................................................................................................................................................................................................................................................................................................................. « Ø

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ABCkEJ3<7)My3<gJHJIB31?kkIG; 4ABC

e Q;mH*L23Û7*I4=23<7ADi 4ABC

3*Ù<HJ3<EJ;25<1R1?k¿58EJ3HEJIG58;>¤`1K3<7DBC

EAC

658;>=FBA

5<7IK;¿H*L23=BIG5¤E5<Oe0>EJA<B3H*LB5HAD = EF

IGi58;>=AB;B1 ?SIGiα = π/2

e

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...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

A

B C

D

E

Fα

βγ

β

β

β γγ

γ

α

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ABC58;>=

DBC5<EJ3g*AB;>¤ECy3<;DH`e DEJABOH*LDIK73gAB;>g1GCy=23H*LB56H

AB = DB58;>=

AC = DCe )<3`H

AG5<;>=

DGN>3H*L235<1HJIRHJCy=23<7@HA

BCIG;hHEJIG58;>¤`1K3<7

ABC58;>=

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AG = AB sin β58;>=

GD = CD sin γ = AC sin γe¡bAB;27938ÒDCy3<;DHJ1 ?¢

AD = AG + GD = AB sin β + AC sin γe Ü &`Ý

£t; 4ACE3qLB5B3

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3FLB5`B3AF = AB sin β/ sin α

e6IK;>g3α + β + γ = π

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AC sin γ

sin α+

AB sin β

sin α

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0 < α < πÝ9e¢LDC>7`

EF = ADIGi¢58;>=AB;B1?!IGi

α = π/2e

!"! #" $ &%' #"(%*),+ " .-/)0 .1324 #" 5768:9; ##2,5=<99> =9?6 A@B 9"+( &6C8D9 #>EF8 &%G%,#9F)7H I)*; .1&5 8J2*5 $K8 # ,L>9/9F)A5 NMO !P" Q)*2 )G"9 1PL$!" H %#>@"(576<)4$ !N)4O=)="9 1 "&H>LIR5 L@B A5= =P) " ) S% TH U@ NV CW 2 YX ,8 Y)=6 F) A%#6IM % > "#

Å ÌqÎqÌ e 0 8 $7'$# )+»$' ¹0 0 '70 $´(6Cµ'0 36 '6? #1'¶$1+7¹0 '51 8!©°

)<3HxyzN23:M>AB7)IHJIB3EJ365<1y;BC>OSN23<E76e#0>EJA6B3¿H*LB5H

(x

y+

z3√

xyz

)2

+

(y

z+

x3√

xyz

)2

+

(z

x+

y3√

xyz

)2

≥ 12e

± '7( .+ .-!'*>$ "'*7?63 )+ -1 "'$#K9 .0?) $04$0! '»' 3µ7# / 3'5 6#1'7( .º& ´ =>µ'70? + !.0 '7&¶ °K 818»-166¹ '7µ'70:36*¸(´ ' G·$ ?<=#1'! $0³ '&µ ¶&'74='7$0³ '&µ$¹ $:= $# /'70 5'74=µ54 -=& .¶$0:.'7+ 0:.-!'70 '7(=·6+1´(6µ'0 3.+ 4-6$# /':4-(&> g´ # 3'Y¸ ' "0 µ7 6#1'7( & &$ $+ =·'7 ' ¶ 3'54' .63'5 0 ($ !# 6$$±° 5>6 >7$4 .-!'7 sº(0 '$' ' efwM>MB1?DIK;>¤#H*L23fTF £t;>38ÒDC>5<1RIRHt?¢<3¿A2NDH5<IG;@H*L23xiAD1G1KA8IK;>¤#IK;>3<ÒBC>561GIHJI4367

x

y+

z3√

xyz≥ 2

√xz

y 3√

xyz

y

z+

x3√

xyz≥ 2

√yx

z 3√

xyz

z

x+

y3√

xyz≥ 2

√zy

x 3√

xyz

e 3gJ58;!H*L236;SgAB;>g1GCy=23¿H*LB56H

(x

y+

z3√

xyz

)2

+

(y

z+

x3√

xyz

)2

+

(z

x+

y3√

xyz

)2

≥ 4

(xz

y 3√

xyz+

yx

z 3√

xyz+

zy

x 3√

xyz

) efwM>MB1?DIK;>¤qH*L23¿fTF £t;>3<ÒBC>561GIH?HA!H*L23:EJIK¤6LHQ7*I4=23@ADikH*LDIG7#1G587HwIK;>3<ÒBC>561GIH?kD3A2NDH5<IG;4

(xz

y 3√

xyz+

yx

z 3√

xyz+

zy

x 3√

xyz

)

≥ 4·3 3

√xz

y 3√

xyz· yx

z 3√

xyz· zy

x 3√

xyz= 12

e¢LDC>7`

(x

y+

z3√

xyz

)2

+

(y

z+

x3√

xyz

)2

+

(z

x+

y3√

xyz

)2

≥ 12e

©qAHJ3¿H*LB5HF38ÒDC>5<1RIRHt?PL2AD1K=27FIGik58;>=PAB;B1 ?IRix = y = z

e !"G *0*5 *G@ L?$A)?+ *5=2C)@ 3 )8" 1 *5 *$*<%&H>6 )>9*E Y)42=)("9 1

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

+ DW " #X > *8 x1

)x2

). . .

)xn

! 3 Q) M % M !

∑

! "

(x1

x2

+x3

n√

x1x2 · · · xn

)α

≥ 2αn

α ∈ (−∞, 0) ∪ (1, ∞)

#A # Y) DM > M n = 3

"α = 2

#

8Å ÌqÎ × e 0 8 $'# )$+ $)o½= &33'6 #$ !$ &$$ ?.+°

¡bAB;27*I4=23<EQH*L237,3`HqADi1K5HHJI4g3MyADIG;DHJ7 (x, y) L23<EJ3 x5<;>=

y5<EJ3!IK;DHJ38¤636E77)Cyg)LUH*LB5H

0 ≤ x ≤ 758;>=

0 ≤ y ≤ 7eDAMyADIG;DHJ75<EJ37,3143<gJH3<=56HxE58;>=2ABOi4EJABOH*LDIK7F793Hè¢f1G1>M>ADIK;DH7QLB5B3qH*L23¿795<OS3¿M2EJA2N258NBIR1GIH?ADi¢N>3IK;>¤q79361K38gJHJ38=!58;>=@H*L23MyADIG;DHJ7;>383<=@;>AHN23¿=DIK7HJIK;>gJHè8IK;>=H*L23qM2EJA2N258NBIR1GIH?H*LB56H/H*L23x58EJ365¿ADiBH*L23HEIK5<;>¤`143Ü4M>AB797*I4ND1?U=238¤636;>3<E5H3[ÝiABEOP38=UN<?PH*L2367,3:HAPMyADIG;DHJ7x5<;>=PH*L23@M>ADIK;DH

(0, 0)IK7x5<;IK;DHJ38¤636EFÜ4M>AB797*I4ND1?

0Ý9e

6 )$+ 3' ¸ '7 "0 3µ"6#1'7( & $+ 6·6' 3' ¶ 3'54'63'70 ($ e

)<3HC>7=23<;>AHJ3¿H*L2358EJ365@ADik5HEIK5<;>¤`143XY Z

N<?[XY Z]

eQ«¢3<g561G12H*LB56HiABE5HEJIG58;>¤`1K3qIH*LB3<EHJI4g3<7A(x1, y1)

B(x2, y2)

B5<;>=C(x3, y3)

[ABC] =1

2

∣∣∣∣∣∣

det

x1 y1 1x2 y2 1x3 y3 1

∣∣∣∣∣∣

=1

2

∣∣x1y2 + x2y3 + x3y1 − x2y1 − x3y2 − x1y3

∣∣e

BABEH*L23PM2C>EJMyAB793UADiwADC>E¿M2EJA2NB1K3<O1K3HH*L23HAM>ADIK;DH7:7,3143<gJH3<=¬56HqE58;>=2ABO N>3P (a, b)

5<;>=Q(c, d)

ex£iOIK7FH*L23@ABEJIK¤ÌK;¢BH*L236;

[PQO] = 12|ad − bc| eq£t;SABEJ=>36EiABE

[PQO]HJAUN>3!5<;SIK;DHJ38¤636E, |ad − bc| OC>7*HN233,B3<;¢k58;>=PH*L236EJ36iABEJ3D ad

5<;>=bcOC>7HFLB5B3qH*L23:795<OS3M258EIRHt?/ekLDIK7IG1R1kA2g*gC>EIK;!H*L23¿iAD1G1KA8IK;>¤HAgJ58793<7

s 5r*ead5<;>=

bc5<EJ3N2AH*LPA2=2=/e¢LDIG7QIG7FHECy3IGi¢5<;>=PAB;B1?!IGi

abc25<;>=

d5<EJ3:5<1R1kA2=>=be6IK;>g3

abcB58;>=

dN>314AB;>¤HJAH*L23¿793H 01234567 8H*L23¿M>EJA2NB5NDIG1RIRHt?H*LB56HH*L23,?5<EJ3¿5<1R1yA2=2=IK7 1

2· 1

2· 1

2· 1

2= 1

16

es 5rre

ad58;>=

bc58EJ3N2AH*L3,B3<;/e¢LDIG7QIG7FHECy3IGi¢5<;>=SAB;B1 ?IRi

a58;>=

d58EJ3:;>AHFN>AH*LA2=2=58;>=

b58;>=

c5<EJ3:;>AHN>AH*LPA2=2=/e¢L23:M2EJA2N258NBIR1GIH?!H*LB5H

a5<;>=

d58EJ3:;>AHFN>AH*LA2=>=IG7 3

4

25<;>=!H*L23:795<OS3IK7QHEJCy3iABEb5<;>=

cekL23<EJ3iABEJ3B>H*L23@M2EJA2N258NBIR1GIH?H*LB5H

ad58;>=

bc5<EJ3!N>AH*LS3,B3<;IK7 3

4· 3

4= 9

16

e6IK;>g3q¡¢5<7,3¿£y58;>=@¡k58793¿££k5<EJ3xOCBHJC>5<1R1?@3*Ù8g1GC>7*IB3D6H*L23M>EJA2NB5NDIG1RIRHt?H*LB5H/H*L2358EJ365@ADiyH*L23qHEJIG58;>¤`1K3IG7Q58;IG;DH3<¤63<EIG7 1

16+ 9

16= 5

8

e !" A97% $ L% (L?54) 8 )6 JV I1,5 8J2*5 $8#7,L>9/9F) 5 M O

P " Q)G2 )!"9 1" $ # 8 O O E/ A,%#9> A%*) + ! >)G 754)!" 9 #7 C "#

Å ÌqÎ Ú e 0 8 $' 8!$0 ±#/² $0 # °QN° 4 ´(6Cµ'0 3 ' & ; 0?# $0?'0 '0?$ °

8D561GCy36EÛ1G579ABOPOP3S2006 =

2006∑

k=1

(−1)k k2 − 3

(k + 1)!

e Q; E58M2My31G1K3ÒDCy3n! = n · (n − 1) · · · 3 · 2 · 1

2M>5<EF3)Ù836OPMB143D6! = 6 · 5 · 4 · 3 · 2 · 1 = 720

e

D6 8!02»'7$ µ53#/ '7 '$6#$$!´(6Cµ'0 3 ' .º& !.0 '!¶ e

0BADC>Ek ≥ 1

B;>ADC>7#7)5BAB;27#ÒBCy3k2 − 3

(k + 1)!=

k(k + 1)

(k + 1)!− k + 1

(k + 1)!− 2

(k + 1)!=

1

(k − 1)!− 1

k!− 2

(k + 1)!

=

(1

(k − 1)!− 1

(k + 1)!

)

−(

1

k!+

1

(k + 1)!

) e0D58EOP5<;BIKMBC21K5HJI4AB;27Q5<1K¤.$38NBEJIKÒBCy3672AB;PA2NHJI436;DH S2006 =

2006∑

k=1

(−1)k k2 − 3

(k + 1)!

=

(2006∑

k=1

(−1)k

(k − 1)!−

2006∑

k=1

(−1)k

(k + 1)!

)

−(

2006∑

k=1

(−1)k

k!+

2006∑

k=1

(−1)k

(k + 1)!

)

=

(

−1 + 1 +

2006∑

k=3

(−1)k

(k − 1)!−

2004∑

k=1

(−1)k

(k + 1)!+

1

2006!− 1

2007!

)

−(

−1 +2006∑

k=2

(−1)k

k!−

2005∑

k=1

(−1)k+1

(k + 1)!+

1

2007!

)

=(

1

2006!− 1

2007!

)

−(

−1 +1

2007!

)

=1

2006!− 2

2007!+ 1 =

2007 − 2

2007!+ 1 =

2005

2007!+ 1

e !5 8J2*5 $8#7,L>9/9F) 5 M O P" Q) 2=)!"9 !1

PL$!" H %#?@" 576<)$ !N)O )&"9 %#?$ Y- #

Å ÌqÎqÎ e 0 8 $7'$# )$+¹056 ' -13² +'0 ' "0?w´(6Cµ'0 3.+* ; ' ² ; !#!# °%»- °

¢L23P1GIG;>3IH*L7*14ABM>3λ > 0

5gJH71RI483S5OIGEEJABEHA5SE5?ADi1RI4¤6LHgABOIG;>¤5614AB;>¤5P1RIK;>3M>5<E5<1R1431bHJAPH*L23x<5`Ù<IK76eQ3`H36EOIK;>3@H*L23!7)1KABMy3ADiyH*L23:EJ3%k3<gJH3<=E5?be -

6

..................................................................................................................................................................................................................................................................................................

6)$+ °# 818»-7x¹ '7µ'70:36 *¸(´ ? e)<3H@H*L23E5?mADix1RI4¤6LHLDIRH@H*L23OIGEEJABE5HMyADIG;DHAq143`HH*L23

x*IG;DH36EJg*36MBHUADiqH*L23ÛOIKEJEJABEN>3

BF58;>=h143`H:H*L23¬EJ3%k3<gJH3<= E5`?ÕADi#1RI4¤6LH!LDIHH*L23

x)5[Ù8IG75H

Ceh£iH*L23S5gCBHJ3U5<;>¤`143PiABEJOS3<=N<?H*L23 OIKEJEJABE58;>=Ê5&L2ABEIRÑ)AB;DH5<1!1GIG;>3IG7

θH*L23<;

λ = tan θe6IG;>g*3qH*L23¿58;>¤`1K3:ADiyIG;>gI4=23<;>g358;>=H*L23F58;>¤`1K3QADi8EJ3(y38gJHJIKAB;¿5<EJ3Q38ÒDC>5<1t,3QLB5`B3 -

6

.................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................

A

BC

.

.

.

.

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.

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.

.

.

..

.

. θ...............

θ

.

.

.

.

.

.

.

.

.

.

.

..

.

. θ...............................

2θ

D &

∠BAC = ∠ABC = θeq¢L236;SH*L233*Ù<HJ3<EI4ABEQ58;>¤`1K3ADi 4ABC

56HwB36EHJ3)ÙCLB587OS36587*C>EJ3

2θB5<;>=!H*L23:7*14ABM>3:ADiyH*L23:EJ3%k3<gJH3<=E5?!IK7wH*LDC>7

tan 2θ =2 tan θ

1 − tan2 θ=

2λ

1 − λ2

eÜ£iH*L23S7)1KABMy3UADiH*L23SOIGEEJABEqIG7

1/H*L23<;¬H*L23SEJ3(y38gJHJ38=E5?LB5<7:C>;>=>3y;>3<=h7*14ABM>37)IG;>g*3¿IRHIK7B36EHJIKg561e Ý

!" 24 G"(576C8D9 ; #!2'5=<9/9 96 =@B =9I)UH I) ; .1 79 &% $ L>% L54)8 )!6JV 1,5 8J2*5 $8#A,L>9/9F) 5 M O P " Q)#2=)!" 9(1(EFL &% Q$ AP 8N$T! #" HCL) " ) @ .; " ! DI) @ )&21 " *HCLFR 5 L>@B 75= P?) " )BS H @ V W '2 YX ,8 Y)=6F) =%#Å ÌqÎ

®e 0 8 $'# )$+,.-' +-!'7 °

IG;>=¬5SÒBC>58=2E5HJI4g¿MyAD1 ?2;>ABOIK561f(x)

7)Cyg)LUH*LB5H,/IRinIG75SM>AB7)IHJIB3!IK;DHJ38¤636Eg*AB;27*IK7HJIK;>¤!ADi/H*L23=BIK¤ÌRH

5EJ3<M>3<5H3<=

kHJIKOP3<7`yH*L23<;

f(n)gAB;27)IG7*H7¿ADiH*L23=DI4¤ÌH

5EJ3<M>3<5H3<=2kHJIGOS367<eÜ DABEF3*Ù<58OM21K3B

f(555) = 555555e Ý

6 )+ 04$ $+!- $# = '7+s° 0 '594='5 'N´(6Cµ'0 3.+ $94='5$ ' $>´ ? e

)<3Hf(x) = ax2 + bx + c

exª#7*IK;>¤¿H*L23!¤Ì B3<;PIK;BiABEJOP5HJI4AB;3@A2NDH5<IG;PH*L23iAD1G1KA8IK;>¤q7?27H36O¦ADik1GIG;>3<5<EF3<ÒBC>5HJI4AB;27f(5) = 25a + 5b + c = 55

f(55) = 3025a + 55b + c = 5555

f(555) = 308025a + 555b + c = 555555

e8AD1DIK;>¤xH*LDIG7#7?B7*HJ3<O83:=>3`H36EOIK;>3¿H*LB5H,

a = 95

b = 2

B58;>=c = 0

ekLDC>7f(x) = 9

5x2 + 2x = x

(95x + 2

) e©qAHJ3¿H*LB5H555 · · · 55︸︷︷︸

k § 3 685 = 5(1 + 101 + 102 + · · · + 10k−1

)= 5

(10k − 1

9

)

= 59

(10k − 1

) e 3gJ58;;>A8 B36EJIRiR?PADC>Ew38ÒDC>56HJIKAB;(f(59(10k − 1)

)=

(59(10k − 1)

) [95

(59(10k − 1)

)+ 2

]

= 59(10k − 1)(10k + 1) = 5

9(102k − 1) = 55 · · · 55

︸︷︷︸

2k § 3 685e

!" ! #" $ &%' #"(%*)#+ " Y-/),0 .124 G"(576C8D9 ; #*2,5=<99> =9?6 A@B 9F)H I)!; .1!5 8N2&5 $ 8 #&,L>9/9F)#5 M O 3P" Q)'2 )U" 9(1 &@B<B' 7HCH>8:%';(L?5 "24 75 5 8 L,@ L?< L?54)>9 M ?@ C9 ," ! !2 *; " I)?@ )("9 1IEIL7% $ =P8 $! #" HCL) " )?@ .; " ! F)?@ ) &21$ # 8 O O#EF 7G%#9 =%*)+ ! C) A54)"9 1?;&" 96=P4 (5 8 @ (L?54)" ! " "G2&L2L/)+ Q) X I1 " *HCLFR 5 L@B A5= =P) " ) S H/@ V W #2 YX #8 Y) 6 F) A%#? C "#

D<TVU*WYX[Z Q1S W ORQK W O

x+x+@ #"%" ¢LDIG7#OPAB;DH*Lk 7QM>EJA2ND1436O IK;BAD1 B3<7QM2EJA2N258NBIR1GIH?be

!)|3(¦Ü $$B Cyg1GIK=P¡¢AB;DH367*HÝ£t;H*L234 × 4

¤EJIK=7,L2A8:;¢kH*LBEJ383Pg*ADIG;275<EJ3E58;>=2ABO1 ?M21G5g38=ÕIG;m=BIRß/36EJ3<;DH7,ÒDC>58EJ367<e Q3`H36EOIK;>3¬H*L23M>EJA2NB5NDIG1RIRHt?H*LB56Hw;>A@HAg*ADIG;27Q1RI43¿IG;!H*L23:7)58OP3EJA8ABEwg*AD1RC>OP;/e.....................................................................................................................................................................

.....................................................................................................................................................................

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j2

j1

6IK;>g3qH*L23qHAH5<1>;BC>OSN23<EFADi2:5`?27QADikM21G5gIG;>¤xH*L23qH*LBEJ3<3gADIK;27QAB;!H*L23:¤EI4=!IG716 ·15 ·14 Ü45<73x795`3<5<EJ1RI436EÝ<H*L23qM2EJA2N258NBIR1GIH?H*LB56H/H*L23H*LBEJ3<3¿g*ADIG;27w5<EJ3qMB1K58g*3<=IK;P=DIGßb3<EJ36;DHFEJA8:7Q5<;>=Pg*AD1RC>OP;27FIK7 16·9·4

16·15·14 = 635

e 3gJ58;U5gJHJC>561G1 ?g*ABOM2CBHJ3!M>EJA2NB5NDIG1RIRHJIK3<7x56H365g)LS7*H5¤63!ADi/H*L23gJ5<1KgC21K5HJI4AB;IK;27H365=ADi:56IRHJIG;>¤QC>;DHJIR1H*L23x36;>=/e/¢L23M>EJA2NB5NDIG1RIRHt?ADi2MB1K58gIK;>¤FH*L23#yE7HbHA:gADIK;27IK;!=BIRß/36EJ3<;DHEJA8:758;>=!g*AD1RC>OP;27IG7 16·916·15 = 3

5

eIB36;H*LB5HH*L23xyE7H/HA@g*ADIG;27w58EJ3IK;=BIRß/36EJ3<;DHEJA8:7x58;>=g*AD1RC>OP;27`>H*L23!M>EJA2NB5NDIG1RIRHt?ADiMB1K58gIK;>¤H*L23@H*LDIGEJ=g*ADIG;UIG;5=BIRß/36EJ3<;DHqEJA8 5<;>=gAD1GC>O;i4EJABO 365g)LADiH*L23yE7HHASIK7 414

= 27

ÜK7)IG;>g*3!H*L236EJ358EJ3458g*g3<MDHJ58NB1K3:7,ÒDC>58EJ367ADCBHFADi

14ABMy36;P79ÒBC>5<EJ3<7JÝ9ekL23:M>EJA2NB5NDIG1RIRHt?PADi¢M21G5gIG;>¤5<1R12H*LBEJ383gADIK;27FIG;P=BIRß/36EJ3<;DHwEJA8:7Q58;>=g*AD1RC>OP;27FIK7wH*L236; 3

5· 2

7= 6

35

e6 kLDIK7wHJIKOP3B83:EJ3<¤58EJ=!H*L23gADIK;27Q587FIK;>=DIK7HJIK;>¤`C2IG7,LB58NB1K3>e 3qIR1G1>>;>=H*L23;BC>OSN23<EFADi¢gAB;Bk¤`C>E5HJI4AB;27FiABEH*L23:g*ADIG;27FIG;@LDIKg)L@H*L23:gADIK;27F58EJ3¿IG;=BIRß/36EJ3<;DHEJA8:7F58;>=g*AD1RC>OP;276ekLDIK7QgADC214=N>3=2AB;>3:N<?@C>7)IG;>¤¿ADC>Eg*ADC>;DHJIG;>¤qOS3`H*L2A2=@iABEH*L23g5<7,3xL23<EJ3qH*L23g*ADIG;27Q58EJ3:=BIG7*HJIG;>¤`C2IK79LB5ND14358;>=!H*L236;S=DIDI4=DIK;>¤N<?

3!2NBCBHFL23<EJ3IG75¿:5`?!HJAgADC>;DHH*L23:g*AB;By¤`C>E56HJIKAB;27=DIKEJ3<gJHJ1?!IK;27H365=be

IGE7*H,BMBI4g)H*L233EJA8:7FIG;@LDI4g)L@H*L23gADIK;27IR1G1kN23:MBCBHè¢L236EJ3:58EJ3 (4

3

)= 4:5?B7¿HJA¬=>AUH*LDIG7<e£t;H*L23HJABM>OPAB7*HxEJA8ÓADiH*L23<793

3EJA8:7`bH*L236EJ3P58EJ3

4MyAB7)7)IKNB1K37,ÒDC>58EJ367@iABE5gADIK;/eÕ£t;H*L23UOIK=>=D143UEJA8ÃADiH*L23<793UEJA8:7H*L23<EJ3U58EJ3

3MyAB7)7)IKNB1K37,ÒDC>58EJ367iABE5g*ADIG;hÜ47*IK;>g3SIH:g5<;¢ H:N>3PIK;¬H*L23U7)58OP3gAD1GC>O;587H*L23gADIK;IG;H*L23HABM2OSAB7HEJA8qÝ)ew£t;:H*L23qN>AHHJABOEJA8 ADiBH*L23<793xEJA8:76H*L23<EJ3x58EJ3

2M>AB797*I4ND143x79ÒBC>5<EJ3<7iABE¿5g*ADIG;hÜK7)IG;>g*3PIH:gJ58;¢ H:N>3PIG;¬H*L23S795<OS3Sg*AD1RC>OP;5<7@3IRH*L236EADiH*L23UAH*L23<ExHAg*ADIG;27Ý)e¢LDC>7`H*L23<EJ358EJ3

4 · 4 · 3 · 2 = 96gAB;Bk¤`C>E5HJI4AB;27FIK;LDI4g)L!;>A@HAgADIK;2758EJ3¿IK;!H*L23:7)58OP3EJA8ABEwg*AD1RC>OP;/e

IG;25<1R1?kH*L23EJ38ÒDC2IKEJ3<=PM2EJA2N258NBIR1GIH?!IK7 96560

= 635

e¢LDIG7¿M2EJA2NB1K3<OÃLB587q58;UIG;DH36EJ3<7HJIK;>¤!LDIG7*HJABEt?/e¿¢L23!IG;BIRHJIG5<1yB3<E7*I4AB;5<7,83<=SH*L23795<OS3QÒBCy367*HJIKAB;qiABE3g*ADIG;27AB;¿5

5×5¤EI4=beÜ4¡¢5<;x?BADC79AD1B3H*LDIG7¢B36E7)IKAB;>ÔÝ wC>EIK;>¤H*L23q=23,B361KABM>OP3<;DHADiH*L23 D Cyg1GIK=¡bAB;DHJ3<7H,6H*L23M>EJA2ND1436O&:5<7g)LB5<;>¤638=HJA¿H*L23iAD1G1KA8IK;>¤qM>EJA2ND1436OÓN236iABEJ3N236IG;>¤g)LB58;>¤63<=SNB5g)HA!IHJ7#ABEI4¤ÌG;25<1>iABEJO ¢LBEJ3<3 #$ '70 '7( ;BC>OSN23<E7#58EJ3:g)L2AB7,36;i4EJABOÞH*L237,3`H

11121314212223243132333441424344 e LB56HIK7:H*L23M>EJA2NB5NDIG1RIRHt?H*LB5H;>A¬HAhADiFH*L2367,3;BC>OPN>36E7LB5B3SH*L23795<OS3¿C>;BIHJ7#=BIK¤ÌRHFABEH*L23:7)58OP3qH36;27=DI4¤ÌH*Ô¢LDIG7!M>EJA2ND1436O 7,3<3<OP38=hÒBC2IH35U14AHLB5<EJ=>36EH*LB5<;H*L23M2EJA2NB1K3<O IH*L¬H*L23gADIK;27`LDI4g)L!IK7Q7HE58;>¤63DB587FIHIK7Q58gJHJC>5<1R1?@H*L23795<OS3M2EJA2NB1K3<O $¡¢5<;@?BADCS79383xL8?2ÔØADC!OIK¤6LH1GIK83HJA:HEt?@79AD1DIK;>¤5¿OPABEJ3¿¤63<;>36E5<12M>EJA2ND1436OL23<EJ3kg*ADIG;27w58EJ3M21G5g38=PAB;5<;

n × n¤EJIK=PÜIRH*L

k ≤ n2ADibgADC>E7,3[Ý9e

|.>eÛkL235<CBH*L2ABEqIG7,L2367:HA58g);>A81K38=2¤63SH*L23g*AB;DHEI4NDCBHJI4AB;27!ADi yEJCyg3¡¢EJADiA2AH,8fw7)7,A2gIG56HJ3 2=DIRHJABE,DIG;!H*L23:M2EJ3<M258E5HJI4AB;PADiyH*LDIK7#gAD1GC>O;/e

D TW[ZMRL#\ TRLU¡LW

y»#´Æ/»QÉkÆ¶¾¿ÇÈÇÃ¾²½¿½É¾ >Éx+x+@ #"%"

ÛAB7*HwADikC>7F143658EJ;>38=5N2ADCBHF¤EJ3656HJ3<7HFg*ABOOSAB;=BI BIG7,ABE7#Ügcd

7Ý/IG;P361K3<OP3<;>HJ5<Et?79g)L2A2AD1DL236;3qyE7H143658EJ;>38=!5N2ADCBHM>EIKOP3¿;BC>OSN23<E7Q5<;>=M2EJIGOS3qi458gJHABEIRÑ*58HJI4AB;276eÜ«¢3<OP3<OPN>36EQH*L2AB793@M>EIKOP3@i45gJHJABEJIÑ*56HJIKAB;HEJ38367,ÔÝ 3@C>7938=U¤EJ3<5H367*HgABOSOSAB;U=BI BIG7,ABE758¤5<IG;L23<;3:143658EJ;>38=HJAP5=2=Pi4E58gJHJI4AB;276ex6IG;>g*3:H*L236;¢kL2A83,B3<E93LB5`B3¿M>EJA2NB5ND1?@iABEJ¤6AHH36;OPAB7*HADiBLB56H/3q1K3<5<E;>3<=$ 3<EJ3qIG7w5EJ3i4EJ3<79L23<EAB;gcd

7#5614AB;>¤#IRH*L79ABOS3EJ31K5H3<=PgJ5<1KgC21K5HJI4AB;27Q5<;>=OP5<;BIKMBC21K5HJI4AB;276es "3 R|<"/e#£i

a58;>=

b58EJ3IK;DHJ38¤636E7QH*LB56HF5<EJ3;>AHQN>AH*L

0BH*L23 40 '7'7 "

#$>µ570 ADia58;>=

b=>36;>AH3<=

gcd(a, b)#IG7H*L23¬1G58EJ¤6367*H@MyAB7*IRHJI B3¬IG;DH3<¤63<EH*LB5H=BI BIK=>3673*Ù<5gJHJ1 ?!IK;DHJAN>AH*L

a58;>=

be£t;UAH*L23<EABEJ=B7

gcd(a, b)IG7QH*L23@¤EJ3<5H367*HQADi/561G1>H*L23@g*ABOOSAB;S=BI BIG7,ABE7ADi

a5<;>=

beÜ QAB;¢ Hw?BADCPIK79LH*LB5H#5<1R1bOP5H*L23<O56HJIKg561=236>;BIRHJIKAB;27xOP58=>3:H*LDIK7xOCyg)L7,36;27,3<ÔÝ2A36OPM>LB587*IRÑ)3D

dIK7w5 #$>µ570 ADi

aIGi

d=BI BIK=>367Q3*Ù<5gJHJ1 ?@IG;DHA

aÜRH*LB56HIG78IRi

a = qdiABEy7,ABOP3wIG;DH3<¤63<E

qÝ9eb2A#HJI4=8?xC>M¿5w14A2AB793F36;>=¢,3w795`?#H*LB56H

gcd(0, 0) = 0eÜ©xAHJI4g3!L236EJ3:H*LB5HwH*L236EJ3IG7;>AH,2IG;Pi458gJH,>51K5<EJ¤63<7HQM>AB7)IHJIB3IK;DHJ38¤636EwH*LB5H#=DIDI4=23<7IK;DHJAN2AH*L

05<;>=

07*IK;>g3U3,B36Et?M>AB7)IHJIB3IK;DHJ38¤636E=BI BIK=>367:IG;DHA

0e¢LDIG7OS3658;27H*LB56Hb3¿3IRH*L236E;>3<38=:HA:IK¤;>ABEJ3H*LDIG7FgJ58793¿36;DHJIKEJ31?kDABEb3x;>383<=:HA7)5?7,ABOP3H*LDIG;>¤79M>38gIG5<1kL236EJ3BB5873:LB5B3:=>AB;>32e Ý

s 6Bn G|%"5Be,IG;>=BIG;>¤H*L23gcd

ADi5QM>56IKE¢ADi8IK;DHJ38¤636E7/IK7/;>AHyHJ3<EJEJIKNB1 ?=DISgC21RH>L23<;H*L23QIK;DHJ38¤636E758EJ3#7)OP561G13gcd(2, −4) = 2

gcd(3, 5) = 1

<5<;>=gcd(−13, 1) = 1

e£4HQIK7QABEH*LS;>AHJIK;>¤H*LB5Hgcd(a, 0) = a

IGiaIG7xMyAB7*IRHJI B3@58;>=

gcd(a, 0) = −aIGi

aIK7#;>38¤5HJIB32e#ÜG¢L2AB793@ADiy?BADCgABOiABEHJ58NB1K3¿IH*LP58N279AD1GCBHJ3¿D561GCy367xgJ58;Sg*AB;>=23<;2793H*LDIK7HA

gcd(a, 0) = |a| e ÝFf1G7,Ak gcd(b, 1) = 1iABE3,B3<Et?@IG;DH3<¤63<E

be¡¢5<;:?BADC7,3<3L8?!H*L23<793¿iABEOC21K5<7#5<EJ3¿HECy3i4EJABOH*L23=>3y;BIHJI4AB;>Ô LB56HwLB5<M>M>3<;27wIGi>H*L23qIG;DH3<¤63<E7F58EJ3q1K5<EJ¤638Ô DABE3)Ù<5<OPMB143DD7)C>M2MyAB793x3:5<;DHHA¿gJ5<1KgC21K5H3

gcd(1977, 2007)e¢ØADC>Ek>E7*HkIG;27*HJIG;>gJH¢OIK¤6LHbN>3wHJAHEt?xHAxi458gJHABE

197758;>=2007

HJA:>;>=:H*L236IGEM>AB7)IHJIB3q=DIDIK79ABE76H*L236;gABOPM258EJ31GIG7*H7HAy;>=:H*L231K5<EJ¤63<7HADi¢5<1R1kg*ABOOSAB;P=DIDIK79ABE7<e)<3H, 7FHEt?@H*LDIK76eIGE7*H,

1977 = 3 × 659efiGHJ3<Ex5SNBIH¿ADiM>56IK;BiKC21/HEJIG5<15<;>=36EEJABE9¢3>;>=H*LB56H

6595<M>M>3<5<E7/HAN23#5M>EIKOP3#;BC>OPN>36E`eÜ A8=>A38;>A8ÕH*LB56H

659IG7M>EIKOP38Ô¢LB5H, 7F57*CyN 1t3<gJHiABE58;>AH*L236E#0D58E58¤6AB;$ Ý¢LDIG7H31G1G7C>7H*LB56HH*L23¿MyAB7*IRHJI B3=DIDIK79ABE7ADi

197758EJ3

13659

B58;>=1977

e©q3*Ù<H,2007 = 3 × 669 = 3 × 3 × 223

eUfF¤56IK;¢56iGH36E¿5NDIRHADi>5<IR1GIG;>¤58EJADC>;>=ky3=DIK79g*A6B3<EFH*LB56H223

IK7qM2EJIGOS3=2bL236;>g*3DyH*L23M>AB7)IHJIB3=DIDIK79ABE7¿ADi200758EJ3

139223

669

25<;>=2007

e¢L236EJ36iABEJ3D8H*L23¿M>AB7)IHJIB3¿gABOPOPAB;=DIDIK79ABE7FADi1977

5<;>=2007

58EJ3158;>=

3LDI4g)L!IKOM21RI4367wH*LB5H

gcd(1977, 2007) = 3e

D<¢LDIG7¿OS3`H*L2A2=PABE*87qEJ3<5<7,AB;258NB1 ?P361R1¢NBCBHIHFADC214=N>3M2EJ3HHt?¬¤ECy3<79ABOS3IGi

197758;>=

200736EJ33<58g)LP3I4¤6LHQ=DI4¤`IHJ7Q1KAB;>¤¿IG;27*HJ3<58=PADi/AB;B1 ?iADC>E,>ABEIRikH*L236EJ33<EJ3@;>APEJ3<58=BIR1?S5<M>M258EJ36;DH#79O5<1R1bM2EJIGOS3i458gJHABE76e 3@g5<;gCBH=2A8:;ADC>EABE5NBIH#N<?14A2A2<IK;>¤:=DIKEJ3<gJHJ1?P5HwH*L23M2EJIGOS3:i458gJHABEIRÑ*5HJI4AB;27QIG;27*HJ3<58=SADib1RIK7HJIK;>¤:=DIDIK79ABE7NBCBHFH*LDIG7x7HJIG1R1¢EJ38ÒDC2IKEJ367qg5614gC21G56HJIG;>¤H*L23@M2EJIGOS3:i45gJHJABEJIÑ*56HJIKAB;27DLDIKg)Lky587#IRHwHJC>EJ;27ADCBH,DIK7Q5qB3<Et?=>36OP5<;>=BIG;>¤M>EJA2ND1436OÓgABOPMBCBHJ5HJI4AB;2561G1 ?/e¢L236EJ3IK7Q5N>3`HH36E:5`?¢8LDIKg)L@HJ5883<7Q5=8D58;DH5¤63ADi>H*L23iAD1R14A8IG;>¤qi458gJH5

8 0:$! => £iabqB5<;>=

r5<EJ3IG;DH3<¤63<E7wIH*L

a = qb + rH*L23<;

gcd(a, b) = gcd(b, r)e

¢LDIG7i45gJH¢IK7;>AHb5<1R1<H*LB5HbIG;DHJC2IRHJI B3ÜK1KAHJ7ADiD;BC>OSN23<EkH*L238ABEt?:N>A2A287gAB;DHJ56IK;:5M2EJA2ADiIGi?BADCb EJ3IG;DH36EJ3<7H3<=ÝNBCBH¢3xg5<;:C>7,3QH*LDIG7i45gJH¢HA=2AqLB5HO56H*L236OP5HJI4gIG58;271KA<B3HA=2A!2HJ5883#5M2EJA2NB1K3<O&58;>=qHJC>EJ;IH¢IK;DHJA¿57)OP561G1K3<EbÜ4ABEb7)IGOPMB1436EÝyAB;>3>e 3<EJ3D 7L2A8,

2007 = 1(1977) + 30 ==⇒ gcd(2007, 1977) = gcd(1977, 30)1977 = 65(30) + 27 ==⇒ gcd(1977, 30) = gcd(30, 27)

30 = 1(27) + 3 ==⇒ gcd(30, 27) = gcd(27, 3)27 = 9(3) + 0 ==⇒ gcd(27, 3) = gcd(3, 0)

BAD1R14A8IG;>¤xH*LBEJADCy¤6LH*LDIG7g)LB5<IG;¢gcd(2007, 1977) = gcd(3, 0)

LDIKg)LP38ÒDC>5<1G73eÜ 3gADC214=:LB5`B3#7*HJABM>M>38=:3<5<EJ1RI436EyL236;q3Q795`m5

gcdH*LB5Hk:5<73<5<7?qHAgJ5<1KgC21K5H3DNBCBHIHF=2A23<7);¢ HFLDC>EHHA83836MS¤6ADIG;>¤qC>;DHJIG1B3:¤63`Hw5

0e Ýq¡¢5<;@?BADCH31G1DLB56H3=BIK=56H/365g)L¿7*HJ3<M>ÔqfH/3<58g)L¿7*HJ3<M¢`3wHA2A2xADCBHb5<7O58;?g*ABMBI4367ADi8H*L23Q79O5<1R1436E¢;BC>OSN23<E5873¿g*ADC21K=!i4EJABOÊH*L23q1G58EJ¤636E;BC>OSN23<E9B58;>==23HJ3<EJOIG;>38=LB5H/:587w1K36iGHA6B3<E[eÜ£t;H3<g)LB;BI4gJ5<1¢H36EO723!My36EJiABEJOS3<=SH*L23 wI BIG7)IKAB;f14¤6ABEIRH*LBO 7,3,B36E5<1¢HJIKOP3<7`¢g5614gCy1K5HJIK;>¤H*L23!EJ3<O5<IG;>=>36E5H3<58g)LU7HJ58¤63>e Ý B36E5<1R1>H*LDIG7qOP3H*L2A2=ADig5614gC21G56HJIG;>¤:H*L23

gcdIG7QgJ5<1R143<=@H*L23 DCyg1RI4=23<5<;f14¤6ABEIRH*LBO¬e>Et?@H*LDIG7F5<1K¤6ABEJIH*LBO AB;

654058;>=

1236eÜ wIK=@?BADCS¤63H

12587?BADC>E58;27J3<EÔÝfiGH36E?BADCP¤63HwgABOiABEHJ58NB1K3xIH*LH*LDIK7FOP3H*L2A2=k8?BADCO5?!;>AHJI4g3qHAHJIGOS3<795`BIG;>¤qi3<5HJC>EJ3<76ekL23>E7*HIK7wH*LB5H?BADCS=>AB;¢ HQLB5`B3¿HJA:EIRHJ3:561G1yADiyH*L233<ÒBC>561GIHJI4367ADi

gcd 7Q=>A8:;@H*L23¿EJIK¤6LH7)IK=>3ÛH*L23<793IG1R1>5<1 :5?B7QN23qHECy3BD7,A3g5<;EJ361G56HJ3qH*L23

gcdADiH*L23UABEI4¤`IG;25<1;BC>OPN>36E7:HJAH*L23

gcdADiH*L23Py;2561;BC>OSN23<E7@=BIGEJ38gJHJ1 ?/ekL237,3<g*AB;>=NDC2IG1K=27FAB;@H*L23xyE7HÛH*L23

gcdIG1R125gJHJC>561G1 ?!561:5`?27FN23xH*L23q>;25<12;>AB;>Ñ)36EJAEJ3<O5<IG;>=>36EIK;!H*L23qf1K¤6ABEJIH*LBOeÜ¡k58;@?BADCP7,3<3qL8?2ÔÝj &"3<Bn G|%"5Be!kL23<793OS3`H*L2A2=27793836OÈHJASABESEJ365<1R1?P361R1iABEx;BC>OPN>36E7k?BADCOP5`?q7)5?k6NDCBHbgJ58;:£C>793H*L236O IK;q5FOSABEJ3w5NB7*HE58gJHk793HHJIG;>¤`1RI483LB56HkOI4¤6LHk5<M>M>3<5<EIK;5g*AB;DHJ3<7HwM>EJA2ND1436OSÔ

C>;2;??BADC7,L2ADC21K=@5879ee6ee:36EJ3xIK7H*L23#B3<Et?yE7HM>EJA2ND1436Oi4EJABOH*L23#B36Et?yE7H#£t;DH36E;25HJI4AB;2561h56H*L236OP5HJI4gJ5<1* F1 ?2OM2IG5=!IK; &< 1 0 1)' eÞ0>EJA6B3H*LB5HH*L23¬i4E58gJHJI4AB; 21n + 4

14n + 3

IK7IGEEJ3<=BCygIKNB1K3¬iABE3,B3<Et?;256HJC>E561k;BC>OSN23<Ene

H36M:AB;>3#L23<EJ3D6587bIK;¿5<;?¿M>EJA2ND1436OIG7¢HAxy¤`C>EJ3ADCBH>LB5H¢IHkIK7/EJ365<1R1?q58798IG;>¤Be¢LDIG7#M>EJA2ND1436OÓg5<;SN>3:EJ367*H56HJ38=5<7<0>EJA6B3¿H*LB56Hgcd(21n + 4, 14n + 3) = 1

iABE

D 3,B3<Et?P;256HJC>E561¢;BC>OSN23<E

n ÜK7)IG;>g*3:5i4E58gJHJI4AB;IK7QIGEEJ3<=BCygIKNB1K3IRi¢IRH7#;BC>OS36E56HJABEQ5<;>==>36;>ABOIG;256HJABEFLB5B3;>Ag*ABOOSAB;i458gJHABE7JÝ9e 3¿HEt?@HA!OSA2=2361kADC>EwOP3H*L2A2=!i4EJABO¦58N>A6B3

21n + 4 = 1(14n + 3) + (7n + 1)

14n + 3 = 2(7n + 1) + 1

7n + 1 = (7n + 1)(1) + 0e

¢LDC>7`gcd(21n +4, 14n +3) = gcd(14n +3, 7n +1) = gcd(7n +1, 1) = 1

5873x:58;DHJ38=be8Ak83:gJ58;5=B58MDHH*LDIK7QOP3H*L2A2= $fw;>AH*L23<Ei45gJHH*LB5HFg5<;PN>3ÒDC2IRHJ3LB5<;>=?! 8 0:$! $6> 7 £i

gcd(c, b) = 1H*L236;

gcd(ac, b) = gcd(a, b)e

¢LDIG7bi45gJHyIK7b5gJHJC>561G1 ?xC>7,3iKC216IK;¿N>AH*L¿=DIKEJ3<gJHJI4AB;27 IRHk561G1KA8:7¢C>7¢HAqg*AB;B36EHgcd(a, b)HA

gcd(ac, b)Ü4561RH*L2ADCy¤6LhIH:IG7;>AH:IGOPOP38=DIK5H31?ÛA2N<BIKADC>7L8?¬3SADC21K= 3,B36E:58;DHbHA=2A¿H*LDIK7JÝ*85<;>=:IRH561G1KA8:7C>7HAgAB;B3<EH

gcd(ac, b)HA

gcd(a, b)ekLDIK7i45gJHIK7QOPABEJ3IG;DHJC2IRHJI B3 ÖgJ58;@?BADCS3)Ù8M21G5<IG;IRHHA?BADC>E79361RiÔ 3:;>A8 HEt?5:7,3<g*AB;>=M>EJA2ND1436O

0 1)' & eÞ0>EJA<B3H*LB56Hgcd(n2, 2n + 1) = 1

iABE@5<;?Õ;25HJC>E5<1;BC>OSN23<Ene

FC>EIG;BIRHJIG5<1DIG;27*HJIG;>gJHIK7HAHEt?:HA:C>793H*L23q58N27HE5gJHbB3<E7*I4AB;!ADiBH*L23 DCyg1RI4=23<5<;:f1K¤6A2EJIH*LBO>NBCBHwIRHIK7wB36Et?S=BI UgC21HHAO5832n + 1

¤6AIG;DHAn2 IH*L2ADCBHwIG;DHEJA2=BCygIG;>¤i4E5gJHJIKAB;27<ekLDIK7FIG7L23<EJ3@£tOPM>ABEH58;DH 5gJH g5<;PN>3¿C>7938=!b7)IG;>g*3

2n + 1IG7#A2=>=kH*L23<;

gcd(2n + 1, 2) = 1e¢LDC>7`

gcd(n2, 2n + 1) = gcd(2n2, 2n + 1)Ü47*IK;>g3

gcd(2n + 1, 2) = 1Ý

= gcd(−n, 2n + 1)Ü47*IK;>g3

2n2 = n(2n + 1) + (−n)Ý

= gcd(n, 2n + 1)Ü47*IK;>g3

gcd(−1, 2n + 1) = 1Ý

= gcd(n, 1)Ü47*IK;>g3

2n + 1 = 2(n) + 1Ý

= 1

587QEJ38ÒDC2IKEJ3<=/e£L2ABMy3x?BADCSLB5`B3:EJ36OS36OSN23<EJ3<=P5NDIRH58;>=!143658EJ;>38=5NBIHw5N2ADCBHgcd

7#L236EJ3>eB?¬;>AOS3658;27¿IK7xLB56H#3PLB5B3S=2AB;>3Sg*ABOM>EJ3<L23<;27*IB3DNBCBHqIHq7,L2ADC21K=¤`I B3!?BADC7,ABOP3¿I4=23<5<7FHJA@H*LDIG;>@58N>ADCBHw5<;>=7,ABOP37*HE5H3<¤`I4367wHA!C>793>e>Et?58M2M21 ?BIG;>¤qH*L236OÞHJAAB;>3ADiyH*LDIG7QOSAB;DH*Lk 7¿h5?2L23<O M>EJA2ND1436OP7 $

£t58;@58;>=23<E>C>EJ¤6L¡b36;DHEJ3iABE B=BCygJ56HJIKAB;IK;h5H*L23<O56HJIKg7Q58;>=P¡¢ABOPMBCBHJIK;>¤ª#;BI B3<E7*IRHt?PADi 56HJ3<E14A2A$$ ª#;BIB36E7)IH?fbB3<;BCy3 367*H 56HJ3<E14A2Ak q©2¡¢5<;25=B5 © ) &[email protected]

D8 Y

@ @"2

3N238¤`IG;&H*LDIK7;BC>OSN23<EADi!H*L23 ¶0 '0 IH*L79361K38gJHJ38=&M>EJA2ND1436OP7i4EJABOH*L23¢LB56I:h5H*L23<O56HJIKg561 F1?BOPMBIK58= eÊkLB58;>87U¤6AÛHJAÖ¡¢LBEJIG7*HJABMyL236E<OP561G1t¡¢5<;25=DIK5<;23<5<O )<3<58=>36EHJA@H*L23@£J ÖIG;@fH*L23<;27`DiABEwg*AD1R143<gJHJIK;>¤xH*L236O iABEC>7<e À

± Å ± À Å

±¶± Å"!

9± ÌÍqÍ ×#%$%&'$)(+*,$)-/.10 2 &'$4365

Ð eÖ>EIK5<;>¤`143ABC

LB587∠A = 70 5<;>= CA + AI = BC

L236EJ3IIG7@H*L23IK;>g3<;DHEJ3ADiyHEIK5<;>¤`143

ABCe 8IK;>=

∠Be

Ì e )<3Hf : 7 → 7 8L23<EJ3 7 IG7wH*L23:793HwADi¢5<1R1yE56HJIKAB;25<1y;BC>OPN>36E727*Cyg)L@H*LB56H

f(x + y) = f(x) + f(y) + 2547iABEw561G1xy ∈ 7 5<;>= f(2004) = 2547

e8IK;>=f(2547)

e× e)<3H

ab5<;>=

cN23qMyAB7*IRHJI B3xEJ3<5612;BC>OPN>36E7w7)Cyg)LH*LB56H

a+ b+ c ≥ 1

a+

1

b+

1

c

e0>EJA<B3qH*LB56Ha3 + b3 + c3 ≥ a + b + c

eÚ e )<3H

ABCN>358;P3<ÒBC2IR1K5H36E5<12HEJIG58;>¤`1K3>e )<3`H

A′ B′ B58;>= C′ N23:M>ADIK;DH7AB;!H*L237,3<¤OS36;DHJ7BC

CA

b58;>=AB

/EJ3679M>38gJHJI B361 ?/e6C>M2MyAB793H*LB5H |AC′| = 2|CB′| |BA′| = 2|AC′| |CB′| = 2|BA| k5<;>= [ABC] = 126

eIG;>=PH*L23@58EJ365ADibH*L23HEJIG58;>¤`1K336;>g14AB7938=N<?!H*L23¿1GIG;>3<7AA′ BB′ D58;>= CC′ e

Î e IG;>=561G1yM25<IGE7(x, y)

LDI4g)L7)56HJIG7)iR?@H*L237?27H36O¦ADib38ÒDC>56HJIKAB;27xx+y = yxy x2y = 1

e®e )<3H

ABCDN23:5g*AB;B3*Ù!ÒDC>5=BEJIR1K5H36E5<1e#0>EJA6B3¿H*LB5H

[ABCD] ≤ 14(AB2 + BC2 + CD2 + DA2)

e eQ36>;>3

fAB;SH*L23!793HADiE56HJIKAB;25<1/;BC>OPN>36E7qIG;SH*L23@IK;DHJ3<EtD561

[0, 1]5<7xiAD1G1KA8:7

f(0) = 0f(1) = 1

B5<;>=f(x) =

f(2x)

4

IGi0 < x <

1

2

3

4+

f(2x − 1)

4

IGi 1

2≤ x < 1

e£i3:EIRHJ3

xIG;¬N25<7,3<

2EJ3<M2EJ3<793<;DH56HJIKAB;587

x = (0.b1b2b3 . . .)2¢y;>=

f(x)IG;N25<7,3<

2EJ3<M2EJ3<793<;DH56HJIKAB;/e

D( e 8IK;>=U5<1R1/M>EIKOP3<7

p7*Cyg)LPH*LB56H

p2 + 2543LB5<7q1K3<7)7H*LB5<;

16=DIK7HJIK;>gJH#MyAB7*IRHJI B3=BI BIG7,ABE76e

e IB36;h5EJIK¤6LH¿HEJIG58;>¤`1K3ABC

IRH*L∠B = 90 143`H P

N235MyADIG;DHAB;H*L2358;>¤`1K3NDIK7938gJHJABE!ADi∠A

IG;27)IK=>3ABC

5<;>= 1K3HM

N>3¬5MyADIG;DHAB;ÛH*L237*I4=23ABÜRIH*L

A 6= M 6= BÝ9e )IK;>367

AP

CPw58;>=

MPIK;DHJ3<E7938gJH

BC

ABF5<;>=

AC5H

D

E58;>=

NEJ3679M>38gJHJI B361 ?/eÛ6C>M>M>AB7,3PH*LB56H

∠MPB = ∠PCN5<;>=

∠NPC = ∠MBPe 8IK;>=

[APC]/[ACDE]e

©q3*Ù<Hb3x14A2A2¿56H/HAHJ3<7HJ7wADi2H*L23 8 §R¨f14NB58;BIG58;Ph56H*L236OP5HJI4gJ5<1. F1 ?2OM2IG5=iABEqI4¤6LÕ8g)L2A2AD1G7<ekLB58;>87P58¤5<IG;Ö¤6AhHA ¡bLBEIK7HABM>L23<E<O5<1R1¡k58;258=BIG58; 23<5<O)<3<58=>36EHAH*L23!£J ÕIK;@fH*L236;27iABEFgAD1G1K38gJHJIG;>¤xH*L236Oe

ÌqÎ±

±·9±· Å ±

À Å±¶± Å"!

,±

¹ À

À ÂQ¶UÀ

6Â

$45,* Ð ex¢L236EJ3!5<EJ3

20M2C>MBIG1G7xIG;U5:BIR1G1G5¤63@79g)L2A2AD1eQfw;?PHASADibH*L236OÃLB5`B3H*L23@795<OS3¤E58;>=Di456H*L236E`e8L2A8H*LB5H/H*L23<EJ3¿3*Ù8IG7*H7w5¿¤E58;>=Di456H*L236E/L2A@LB58756H1K3<5<7*H

14¤E58;>=2g)LDIG1K=2EJ36;/e

Ì e)<3`HMN<5<;>=

PN>3FH*L23EJ3679M>38gJHJI B3OI4=2)M>ADIK;DH7ADiB7*I4=23<7

BCCA

<5<;>=ABADibHEIK5<;>¤`143

ABCy58;>=P143`H

GN23H*L23IK;DHJ3<E7938gJHJIKAB;UM>ADIK;DH#ADi/IHJ7xOP38=DIK5<;27<e0>EJA<B3H*LB56HIGi

BN =√

32

AB5<;>=

BMGPIG7Q5@g?Bg1GIKgQM>AD1?B¤6AB;¢H*L236;HEJIG58;>¤`1K3

ABCIG738ÒDC2IG1G56HJ3<E561e

× e )<3Hxk5<;>=

ykÜKiABE

k = 12. . .

n)N>3MyAB7*IRHJI B3EJ3<561y;BC>OSN23<E7FH*LB5Hw795HJIK7*iR?

kxkyk ≥ 1e

ÜK5Ý0>EJA<B3qH*LB5H n∑

k=1

xk − yk

x2k + y2

k

≤ 14n

√n + 1

eÜN8Ý L23<;P=2A23<7#3<ÒBC>561GIH?L2AD14=!IK;M258EHQÜK5ÝÔÚ e IG;>=M2EJIGOS3:;BC>OPN>36E7

p58;>=

q7)Cyg)L@H*LB5H

p2 − p + 1 = q3 eÎ e#8IK;>=561G1M>56IKE7ADiwMyAB7*IRHJI B3IK;DHJ38¤636E7

(x, n)7*Cyg)LH*LB5H

xn+1 + 2n+1 + 1IG7=BI BIG7)IKNB1K3:N<?

xn + 2n + 1e

$45,*Ð e8ABOS3qM>38ABMB143H583qM258EHIG;@5¿OS3<3HJIG;>¤Be <B3<Et?@M258EHJI4gIGM>5<;DHIG7w58g*ÒDC>5<IG;DH3<=:IH*L56HbOSAB7HkH*LBEJ383#My3<ABM21K3QIK;qH*L23¤EJADC>Mb<5<;>=IRi8HA¿M>5<EHJIKgIKM258;DH758EJ3#;>AHb5gÒBC>56IK;DHJ38=kH*L23<;!H*L23,?PLB5B35@gABOPOPAB;5gÒBC>56IK;DH58;>g3IK;!H*L23:¤EJADC>MeÜK5Ý LB56HIK7wH*L23:O5`Ù<IKO5<1>;BC>OSN23<EFADi¢M258EHJI4gIGM>5<;DHJ7FIK;!H*LDIG7#OP383`HJIK;>¤6ÔÜN8ÝU£i/H*L23<EJ3!58EJ3H*LBEJ383!M>5<EHJIKgIKM258;DH7L2AS58EJ3!OCBHJC>561G1 ?U5gÒBC>56IK;DHJ38=PIH*L3<58g)LAH*L23<E98LB5HIK7wH*L23:O5`Ù<IKO5<1>;BC>OSN23<EFADi¢M258EHJI4gIGM>5<;DHJ7FIK;!H*LDIG7#OP383`HJIK;>¤6Ô

D Ì eQ0>EJA<B3qH*L23IG;>38ÒDC>5<1RIRHt?

1√

a +1

b+ 0.64

+1

√

b +1

c+ 0.64

+1

√

c +1

a+ 0.64

≥ 1.2

L23<EJ3a > 0

b > 0

c > 0

B5<;>=abc = 1

e× e8AD1B3qH*L23iAD1R14A8IG;>¤3<ÒBC>5HJI4AB;IG;IK;DHJ38¤636E7

y2 = 1 + x + x2 + x3 + x4 eÚ eF0>EJA<B3#H*LB5HiABE/58;?:IG;DH3<¤63<E

n ≥ 26H*L23q;BC>OPN>36E

2n −1IK7;>AH=DIDIK7*I4ND143xN<?

ne

Î e£t;q5<;q5gCBHJ38*58;>¤`1K38=#HEJIG58;>¤`1K3ABC

[1K3HHN>3H*L23QABEH*L2A2g*36;DHEJ3B5<;>=x1K3H

dadb58;>=

dcN23H*L23¿=DIK7HJ5<;>g*367i4EJABO

HHJAH*L23q7*I4=23<7

BCCA

858;>=AB

8EJ3<7)My3<gJHJIB31?be0>EJA<B3H*LB56Hda + db +dc ≤ 3r

<L236EJ3rIK7H*L23qE58=BIRC>7FADi2H*L23xIG;>gIKEJg1K3¿ADi2HEIK5<;>¤`143

ABCe

>A!gAB;DHJIK;BCy3#?BADC>EEJ3`HJC>E;HAM>EJA2ND1436OS*7,AD1 BIG;>¤M21K3<5<7)C>EJ36763¿¤`IB3H*L23 & &[§R¨BABEJO ADi¿H*L23 IG;25<1!«¢ADC>;>= ADi¿H*L23 §R¨ ªE56IK;BIG58;h5H*L23<O56HJIKg561 F1 ?2OM2IG5=be¢LB5<;>7#58¤5<IG;P¤6A@HA¡bLBEIK7HABM>L23<EF<O5<1R1yiABEwgAD1G1K38gJHJIG;>¤xH*L23<OÞiABEFADC>EC>7,32e

ÚqÚ ¸¹±·,±· Å ±

À Å±¶± Å"!

9±/

0 3 & -Ð eÜRqe4e )<3IGiKC>E5`Ý8AD1B3qH*L233<ÒBC>5HJI4AB;

arcsinbsin xc = arccosbcos xc L23<EJ3 bac IG7wH*L23¤EJ3<5H367*HIG;DH3<¤63<E;>AHF3)Ù8g383<=BIG;>¤ ae

Ì e Üqe qe)[?BOP5<;27,<I?8Ý¿kL23¬58gCBH3<)5<;>¤`143<=hHEIK5<;>¤`143ABC

IG7¤`IB36;/e )<3HON>3H*L23g3<;DHEJ3ADi#IRH7PgIGEJgC>OSgIGEJg1432e kL23M>3<EJMy36;>=BIKgC21K5<ENDIK7938gJHJABE!ADiQH*L23¬7)IK=>3

ACIK;DHJ3<E7938gJH7H*L237*I4=23AB

5<;>=@H*L23q1RIK;>3BC

5HH*L23¿MyADIG;DHJ7P5<;>=

QDEJ3<7)My3<gJHJIB31?be0>EJA<B3qH*LB56H

∠PQB = ∠PBOe

× eÜRqe fqe6Ø/5<7)IG;27,<I?8Ýk¢L23¿38=2¤63SA

ADiBH*L23HJ3HE58L238=BEJAB;SABC

IK7My36EM>3<;>=DI4gC21G58EHAUH*L23MB1K5<;>3ABC

eDA¬=DIGßb3<EJ36;DHq79M>L23<EJ367σ15<;>=

σ2g*AB;DH5<IG;¬MyADIG;DHJ7

AB58;>=

CekAH*LH*L2367,3x79M>L23<EJ367w58EJ3H58;>¤636;DHIG;DH36E;2561G1 ?:HA5q79M>L23<EJ3

σg3<;DHEJ3<=!5H

Se

)<3Hr158;>=

r2N23xH*L23qE5=DIGI>ADi

σ158;>=

σ2DEJ3679M>38gJHJI B361 ?/e8IK;>=:H*L23¿E58=BIRC>7

RADi

σe

Ú eSÜqe fqeyØ/587*IK;2798I ?Ýq0>EJA<B3!H*LB56H#H*L23<EJ3P=>A2367;>AH¿3*Ù8IG7*Hx58;IK;DHJ38¤636En > 1

7)Cyg)LH*LB56Hn=BI BIK=>367

3n − 2n e

9$Î eÜRqe fqeØ/587*IK;2798I ?ÝbIB36;@58EJ3

2004M>ADIK;DH7IK;:H*L23qMB1K5<;>3>e/¢L23,?@5<EJ3H*L23QB3<EHJI4g3<7ADi5g*AB;B3*ÙMyAD1 ?B¤6AB;5<;>=U;>APiADC>EADi/H*L236OÈ58EJ3!g?Bg1RI4gefÕHEIK5<;>¤`143LB5`BIG;>¤:H*LBEJ3<3ADi¢H*L23@MyADIG;DHJ7587#IHJ7FB36EHJIKg*367IG7qgJ5<1R143<= .-6· IGi¢H*L23!AH*L236E

2001M>ADIK;DH71RI43:IG;27)IK=>3IRH7QgIGEJgC>OSgIGEJg143DD58;>=@IHIK7Fg561G1K38= .-6 IRi>H*L23AH*L236EM>ADIK;DH7w1GIK3¿ADCBHJ7*I4=23qIRH7QgIGEJgC>OSgIKEJg1K3>eÛ0>EJA<B3H*LB56HxH*L23S;BC>OSN23<E:ADi .-=3· HEIK5<;>¤`14367:IK73<ÒBC>561HJAH*L23S;BC>OSN23<EADi

.-6 HEIK5<;>¤`14367<e®eÜ e e¢h561K58DL2A6ÝIG;>=!H*L237)C>OÓADi>H*L23:EJ365<1yEJA2AHJ7#ADiyH*L23:38ÒDC>56HJIKAB;

x +x√

x2 − 1= 2004

e e Üqe4ex«k58=>g)L236;>8A8Ý QA2367@H*L23<EJ3¬3)Ù<IK7H@5iKC>;>gJHJIKAB;

f : → 7)Cyg)LH*LB5H

f(x2y + f(x + y2)

)= x3 + y3 + f(xy)

iABEw561G1xy ∈

e e:ÜRqe fqeØ/587*IK;2798I ?Ý )<3`H

a

b!5<;>=

cN23mMyAB7*IRHJI B3ÖEJ3<561@;BC>OPN>36E7h7*Cyg)L H*LB5H

abc ≥ 1eQ0>EJA<B3qH*LB5H

a3 + b3 + c3 ≥ ab + bc + cae

e2ÜRqe fqe8Ø/5<7)IG;27,<I?8ÝkfÕgAB;B3)Ù2004

)¤6AB;PLB587B36EHJIKg*367A1A2. . .

A2004

ew£t7FIHMyAB7)7)IKNB1K3HA¿gAD14ADC>E¢3<58g)LADi<IRH77*I4=23<7/5<;>=qIRH7=DIK58¤6AB;25<1G7kIRH*L¿AB;>3QADi2003

=BIRß/36EJ3<;DHg*AD1KADC>E7QIG;7)Cyg)L5q:5?@H*LB5HH*L23iAD1R14A8IG;>¤xHAg*AB;>=DIRHJIKAB;27#L2AD14=2ÔÜGIÝ!¢L236EJ3:5<EJ3

10027938¤OP3<;DH7ADib365g)LgAD14ADC>E[e

ÜKIRIÝS£i>58;58ENBIHE58Et?B36EHJ3)Ù¿58;>=HA5<E*NDIRHE5<Et?!gAD14ADC>E75<EJ3¿¤Ì B3<;¢6H*L23<;¢7HJ5<EHJIG;>¤i4EJABOH*LDIG7kB3<EH3*Ù#58;>=xC>7*IK;>¤H*L23F7,3<¤OS36;DHJ7ADi6H*L2367,3HAqgAD14ADC>E7/3)Ù8g1RC>7)I B361 ?¢AB;>3:g5<;!DIK7*IRHw3,B36Et?PAH*L23<EB3<EH3*Ù!AB;B1 ?PAB;>g3>eÐ Í e:Ü£*e0ke©x58¤6361 Ý-)<3`H

ωN23@H*L23@IG;27,g*EI4N238=gIKEJg1K3ADibH*L23@HEIK5<;>¤`143

ABCe)<3`H

L

Ny5<;>=

EN23:H*L23@MyADIG;DHJ7qADi¢H58;>¤636;>g?UADi

ωIH*LH*L23@7*I4=23<7

ABBC

y5<;>=CA

EJ3<7)My3<gJHJIB31?be)IK;>367LE

5<;>=BC

IG;DH36E7,3<gJHF5HH*L23M>ADIK;DHH258;>=1GIG;>3<7

LN5<;>=

ACIK;DHJ3<E7938gJHF5HH*L23:MyADIG;DH

JÜK5<1R1yH*L23:MyADIG;DHJ7

HJNE1GIK3AB;H*L237)58OP3:7)IK=>3ADiH*L231GIG;>3

ABÝ)e/)<3`H

O58;>=

PN>3H*L23POIK=>*MyADIG;DHJ7ADiH*L23P7,3<¤OS36;DHJ7

EJ5<;>=

NHBEJ3679M>38gJHJI B361 ?/e 8IK;>=

S(HJNE)IGi

S(ABOP ) = u2 5<;>= S(COP ) = v2 eÜ 3<EJ3S(F)

IK7wH*L2358EJ365@ADiyk¤`C>EJ3 F Ý)e 3HJC>EJ;¬HAADC>Eq214367ADiwEJ3<58=>36E7 /g*ABOOS36;DHJ7:58;>=7,AD1RCBHJI4AB;27¿HJA¬M>EJA2ND1436OP7¤ÌB36;UIK;SH*L2383<MDH36OSN23<E ;BC>OPN>36ExADi/H*L23 ¶0 !'70 ekL23!>E7*H¤EJADC>M5<EJ3!iABEM>EJA2ND1436OP7ADiDH*L23k31K5<EJC>7Qh5H*L23<O56HJIKg561 F1 ?2OM2IG5= $$= D¤Ì B3<;:IG; 7 e

¶ "'! )+ '0 0 '¹ 0 6 7'5 7 ?3'5 $0! ' e¢L2367,37)I ÙM>EJA2ND1436OP7Q58EJ3¿i4EJABOÞH*L23@£J m8L2ABEH*1GIG7*H $$ &2e8AD1GCBHJIKAB;27#g5<;SN23iADC>;>=5Hhttp://www.mathlinks.ro/Forum/viewtopic.php?t-15624ABEIK; 14Cy<I $g[kqey)58;>8A6BI7$g[£*eh5HJI $g[©ew0B3`HEJA<DI $g[ -' ¶ 8 '#$? <M>EIK;>¤636E,2M/eR < e

9 &× e 8IK;>=@5<1R1BiKC>;>gJHJIKAB;27

fi4EJABOH*L23¿EJ365<12;BC>OSN23<E7HJA:H*L23qEJ3<561>;BC>OPN>36E7F7)Cyg)L:H*LB5H,iABEw5<;?EJ365<1y;BC>OSN23<E7

x5<;>=

y

f(xy)(f(x) − f(y)

)= (x − y)f(x)f(y)

e6Cµ'$#*)+ -!':x¹ &'7x¸ ='6 $0! ' e

¡k1K3<5<EJ1 ?¢H*L23qÑ)3<EJA@iKC>;>gJHJI4AB;IK7Q5:79AD1GCBHJIKAB;/e 3IG1R167,L2A8ÛH*LB5HyH*L23F;>AB;>Ñ)36EJAx7,AD1RCBHJI4AB;27b58EJ3H*L23wiKC>;>gJHJIKAB;27Φa,K

=>3y;>3<=N<?Φa,K(x) =

axIRi

x ∈ K

0IRi

x /∈ K

L23<EJ3KIG75x7)CyN2¤EJADC>MADiDH*L23xOC21HJIKMB1GIKg5HJIB3x¤EJADC>M

∗ = \0 58;>= a ∈ ∗ e¡bAB;27*I4=23<Ea ∈ ∗ 58;>=5P7*CyN>¤EJADC>M K

ADi ∗ e 379L2A8H*LB56H f = Φa,K795HJIK7*k367

f(xy)(f(x) − f(y)

)= (x − y)f(x)f(y)

Ü &`ÝiABEw561G1

xy ∈

e£ixy ∈ K

¢H*L236;xy ∈ K

58;>=¬Ü7&[ÝL2AD14=B77)IG;>g*3IH¿gJ58;N>3EJ3[:EJIHH36;5<7axy(ax − ay) = (x − y)ax · ay

e£ixy /∈ K

H*L23<;Φ(x) = Φ(y) = 0

58;>=Ü7&[ÝIK7wHECy3>e£i27)5?kx ∈ K

y /∈ K

2g3<EHJ56IK;B1 ?xy /∈ K

Ü4AH*L23<EIK793y = xy · 1

x

ADC214=N>3IG;KÝ)e¢LDC>7`

Φ(xy) = Φ(y) = 058;>=Ü7&[ÝL2AD14=B7<e¡bAB;B36E7,31?k61K3H

fN23#5<;?¿iKC>;>gJHJIKAB;i4EJABO

HJA

7)56HJIG7)iR?DIK;>¤FÜ7&[Ý*<5<;>=5<797*C>OS3H*LB56H

fIG7¿;>AHQH*L23!Ñ)36EJAUiKC>;>gJHJIKAB;/e@kL23<;

f(x0) 6= 0iABE79ABOS3

x0 ∈ e@B5<IK;>¤

x = 158;>=

y = 0IK;PÜ &`Ýb?DI4314=B7

f(0) = 0Ü4L23<;>g3B

x0 6= 0Ý)ekL23<;

y = 1¤ÌB367

f(x)(f(x) − ax

)= 0

Ü ÝL23<EJ33793H

a = f(1)e@kLDIK7¿EJ361G56HJIKAB;¬Ü ÝIRH*L

x = x07,L2A8:7H*LB56H

a ∈ ∗58;>=k2OPABEJ3¤636;>3<E561G1 ?¢H*LB5Hf(x) = ax

IGif(x) 6= 0

e©qA8D1K3HK = x ∈ ∗ | f(x) = ax = x ∈ | f(x) 6= 0 e¡k1K3<5<EJ1 ?

1 ∈ KeÊ£i

x1

x2 ∈ KIRH*L

x1 6= x2FH*L23<;

f(x1) = ax15<;>=

f(x2) = ax2e¢L236;¬Ü7&[ÝIRH*L

x = x158;>=

y = x2¤ÌB367

f(x1x2) = ax1x22L23<;>g3B

x1x2 ∈ Keh6IKOIG1G58E1?kÜ7&[ÝQIRH*L

x = x15<;>=

y = 1/x179L2A8:7:H*LB5H

1/x1 ∈ K58;>=k81K5<7*HJ1 ?¢Ü &`ÝyIH*L

x = x21

58;>=y = 1/x1

¤Ì B3<7x2

1 ∈ Ke 3qLB5`B3M>EJA6B38=PH*LB5H

KIK7x5!7*CyN>¤EJADC>MADi

∗ 58;>=H*LB56H f = Φa,KexkLDIK7qg*ABOM21K3HJ3<7#H*L23M>EJA2ADiJe

9¢L23;>3*Ù<HND14A2g)¬ADiQO56HJ3<EIK561IK7iABEqH*L23h0>EJA2NB1K3<O7:HA8361K38gJH!£t;>=DIK5<;Ö£J >3658O ¤`IB36;IK; $$ e

Ð e)<3HA′ B′ C′ N>3#H*L23xOIK=>*MyADIG;DHJ7wADiDH*L23x7*I4=23<7 BC

CA

AB

8EJ3<7)My3<gJHJIB31?kADi58;5gCBHJ3;>AB;>IK79A2g*314367xHEIK5<;>¤`143ABC

b5<;>=1K3HDEFN23!H*L23!i383`HqADiH*L235<1HJIRHJCy=23<7H*LBEJADCy¤6L¬H*L23B36EHJIKg*367

ABCAB;¬H*L23<793U7*I4=23<7`EJ3679M>38gJHJI B361 ?/e¡bAB;>7)IK=>36EH*L23:5<EJg

DA′ ADiyH*L23;BIK;>3<)M>ADIK;DHFgIGEJg143:ADiyHEJIG58;>¤`1K3 ABC1 ?BIG;>¤ADCBHJ7*I4=23¿H*L23HEJIG58;>¤`1K3>e )<3HH*L23PM>ADIK;DH¿ADiHEJIG7,3<gJHJI4AB;ADiH*LDIG7:5<EJg:g1KAB7,36EHJA

A′ N>3 A′′ e#Q3y;>358;25614A2¤6ADC>7*1?H*L23SM>ADIK;DH7B′′ ÜAB;58EJg EB′ Ý58;>= C′′ Ü4AB;58EJg FC′ Ý9e8L2A8¦H*LB5HHEJIG58;>¤`1K3

A′′B′′C′′ IK7#3<ÒBC2IR1K5H36E5<1e6Cµ'$# )+ -!':w¹6& 3'5o¸ 6'7 0?$ '> $# ° » ° $ "''·*½?) "':.-!' '$.-!'70?!#&°0x² ' 4Cµ'¹ &' 7?63©°

)<3HNN23H*L23g*36;DHEJ3ADibH*L23!;BIK;>3<)M>ADIK;DHgIKEJg1K3 N ADibHEIK5<;>¤`143

ABCeqkL23My36EM>3<;>=DI4gC21G58ENDIK7938gJHJABE7wADi

B′C′ 58;>= A′Dg*ADIG;>gI4=23qÜN2AH*LM>5<797H*LBEJADCy¤6L

N5<;>=58EJ3M>3<EJMy36;>=BIKgC21K5<EHJA

BCÝ 2DL236;>g*3D

A′ 5<;>= D5<EJ37?2OOS3`HEJIKg561>IG;@H*L23:=BIG58OP3HJ3<EADi N M>3<EJMy36;>=BIKgC21K5<EqHA

BCeh£4H¿iAD1G1KA8:7¿H*LB56HqH*L23UOI4=2)M>ADIK;DHADiH*L23ÜK79O5<1R1436EÝ58EJg

B′C′ 5<;>=H*L23OIK=>*MyADIG;DH@ADiFH*L2358EJg DA′ 1 ?BIG;>¤ADCBHJ7*I4=23SH*L23UHEIK5<;>¤`14358EJ3=BIG58OP3HEI4gJ5<1R1?ABM2MyAB7*IRHJ3>e......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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..... ........................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................

A

A′

A′′B

B′B′′

C

C′

C′′

D

E

F

N

IRH*L2ADCBH!1KAB797SADi¿¤636;>3<E561GIH?kQ3O5?Õ7)C>M2MyAB793¬H*LB5H N IK7H*L23C>;BIHPgIGE*g143IG;ÕH*L23hgABOPMB143*ÙÛM21G58;>3DQIRH*L 4A′B′C′ M>AB7)IHJIB31?ÖABEJIK3<;DHJ38=mÜK7,3<3k¤`C>EJ3[Ý9e 3POP5`?3,B3<;7*C>M>M>AB7,3H*LB5HxH*L23P58;>¤`1K3<7α = ∠B′A′C′ β = ∠C′B′A′ 5<;>=

γ = ∠A′C′B′ 795HJIK7*iR? π2

> β > α > γ58;>=H*LB5HH*L23@g*ABOM21K3)Ù@5 @ÙADi

B′ IG7 1e£4HIG7wEJ365=DIG1 ?@793836;H*LB5H/H*L23q5 @Ù83<7wADi

C′ 5<;>= A′ 58EJ3 e2iα 58;>= e−2iγ = e2i(π−γ) EJ3<7)My3<gJHJIB31?be/¢L23x5 !Ù¿ADiDH*L23xOIK=>*MyADIG;DHADiDH*L23x7)OP561G1K3<E/58EJgB′C′ IG7 eiα 28L23<;>g3BH*L23:5 !ÙADiyH*L23OI4=2)M>ADIK;DHwADi>H*L2358EJg

DA′ IK7 ei(π+α) e6IK;>g3 π + α < 2π − 2γÜ4;>AHJ3xH*LB56Hα + 2γ < α + β + γ − π

Ý*<3xy;>=DA′′ D5<;>= A′ IG;@H*LB5HABEJ=>36EAB;H*L23gIKEJg1K3MyAB7*IRHJI B361 ?ABEJIK3<;DHJ38=bew£4HiAD1G1KA8:7wH*LB5HH*L23:5 !Ù@ADi

A′′ IK7ei(π+α+(2π−2γ−π−α)/3) = e4πi/3 · e2i(α−γ)/3 e

9£t;5S7*IKOIG1G58EQ:5`?¢k3>;>=H*LB56HH*L23P5 !ÙUADi

B′′ IK7 e2i(α−γ)/3 58;>=H*L23P5 !ÙUADiC′′ IG7 e2πi/3 · e2i(α−γ)/3 e¢LDC>7`H*L23U5 @Ù83<7!ADi B′′ C′′ 5<;>= A′′ 5<EJ3ADiwH*L23iABEO

eiθ eiθ · e2πi/3 B58;>= eiθ · e4πi/3 2<L236;>g*3D 4A′′B′′C′′ IG7#38ÒDC2IG1G56HJ3<E561e ' 17² ' 4>µ' $ "''· 7818!0 =-2°83H

a = BC

b = CA

c = AB

α = ∠CAB

β = ∠ABCw58;>=

γ = ∠BCAe IH*L2ADCBH#1KAB797qADi¤636;>3<E561GIH?k>3!OP5`?5<797*C>OS3H*LB5H

β > α > γe 3>E7*H;>AH3qH*LB56H

BA′ = 12a58;>=

BD = c cos βe¢L236;

DA′ = BA′ − BD = 12a − c cos β = R sin(β − γ)

L23<EJ3

RIK7wH*L23:gIKEJgC>OE5=DIGC>7ADi 4ABC

e6IKOIG1G58E1?kEB′ = R sin(α − γ)

e)<3H

NN23H*L23@g3<;DHEJ3ADikH*L23;BIG;>38*MyADIG;DHQgIKEJg1K3@ADi 4ABC

e¢L236;3LB5`B3∠DNA′ = 2(β − γ)

5<;>=∠ENB′ = 2(α − γ)

e £t; 4NA′B′ 3LB5`B3∠A′NB′ = 2γ

ekL23<EJ3iABEJ3B∠A′′NB′′ = 2γ + 2

3(β − γ) + 2

3(α − γ) = 2

3(α + β + γ) = 120 e£t;hH*L23795<OS3P:5?k

∠B′′NC′′ = ∠C′′NA′′ = 120 emkLDC>7 4A′′B′′C′′ IG738ÒDC2IG1G56HJ3<E561eÌ e IG;>=561G12HEIKMB14367

(a, b, c)ADi¢M>AB7)IHJIB3¿IG;DH3<¤63<E7Q7)Cyg)L@H*LB5HÜGIÝ

a ≤ b ≤ c2

ÜKIRIÝgcd(a, b, c) = 1

258;>=ÜKIRIGI Ý

a3 + b3 + c3 IK7#=DIDIK7*I4ND143N<?P3<58g)LADiyH*L23:;BC>OPN>36E7 a2bb2c

c2a

e6 8!03'70 0 '¹ 0 73':6 &6 ?'7 0?$ '<°8ADIRH

(a, b, c)C>;!H31>HEIKMB143`H`e

Q;hEJ36OP5<EJÒBCy3=¢ 5N2ABEJ=ÛÒBCy3U7*IpM>EJ36OIK3<E:=DIDIK793

a3`H

bIR1F=DIDIK793

a2b3`H=>AB;>g

a3 + b3 + c3 56IK;27*IbÒDCy3 a3 + b3 e/0D58EF7*C2IRHJ3B p =BI BIG7,3 c3 3`HQ=>AB;>g cyg3@ÒDC2Ig*AB;DHEJ3<=BIHQÜGIGI Ý9efIG;27)It

gcd(a, b) = 1e Q3O '36OS3D

gcd(b, c) = gcd(c, a) = 1e

Q;Û3<;Û= $38=DC2IRHÒBCy3a2 b2 3`H c2 7,AB;DH:=>3CÙ0"5=>3CÙ¬M2EJ3<OI436E73<;DHEJ336CÙye02C2IK79ÒBCb IG1G7 =BI BIG7,36;DHg)LB5gC>;

a3 + b3 + c3 @g361G5 5<797*C>EJ3 ÒDCy3 a2b2c2 =DIDIK793a3 + b3 + c3 e Q;55614ABE7 3c3 ≥ a3 + b3 + c3 ≥ a2b2c2 2=¢ A "C

3c ≥ a2b2 e Ü7&[Ýx 56CBHEJ3:M>5<EH,

c2 =BI BIG7,3 a3 + b3 2=2AB;>gc2 ≤ a3 + b3 ≤ 2b3 e Ü Ý

B;Ãg*ABOPNBIG;258;DHÜ7&[ÝÖ3HÜ Ý*SIR1UBIK3<;DH108c3 ≥ 4a6b6 ≥ a6c4 3H=>AB;>g

a ≤ c ≤ 108/a6 23HF36;By; a7 ≤ 108ey¡¢361G53<;DHE56IK;>3

a = 1e6I

b = 1>5<1KABE7

c2 =DIDIK793 a3 + b3 = 2y=k A "C

c = 1e«$38gIGM>EJA2ÒDCy3<OP3<;DH,2143HEJIGM21K3H

(1, 1, 1)367*HFg1G5<IGEJ3<OP3<;DHC>;>37,AD1RCBHJI4AB;/e6I

b > 125<1KABE7

c > b > 1ÜK795<;27ÒBCyADIt>AB;P5<C>E56IRH

c = b>36;Sg*AB;DHE58=BIKgJHJI4AB;5B3<g

gcd(b, c) = 1Ý)e 8HF=>AB;>g

c3 > b3 + 1ek£1BBIK3<;DHw5614ABE7

2c3 > 1 + b3 + c3 = a3 + b3 + c3 ≥ b2c2

9 =¢ A"C

c > 12b2 e%B;g*ABOPNBIG;258;DH#5B3<gxÜ Ý*>IG1yDI436;DH

2b3 > 14b4 kADC3<;>gABEJ3 b < 8

e Q;! $3<EIGy35<1KABE7"5:1G5:OP56IK;PÒDCy31K5:7936C21K37,AD1RCBHJI4AB;3<7H

b = 23H

c = 3e

IG;25<1K3<OP3<;DH,D1K3<7Q79AD1GCBHJIKAB;27Q79AB;DH(1, 1, 1)

3H(1, 2, 3)

e× e IG;>=561G1>iKC>;>gJHJIKAB;27

f : → 7*Cyg)L@H*LB56H,DiABE5<1R1

xyIG;

83:LB5B3

f(x + y) + f(x)f(y) = f(x) + f(y) + f(xy)e Ü7&[Ý

6Cµ'$# )+ 3-'5g¹ &'7N¸ ='6 0?$ '> $# *½- G· ¶ 63.+¶&'74='7U ('0 µ'7 c´ w°, ' 4Cµ' .-'g² 0?'>8 )$+w½- "#$%!'$#,)$+.-!'/'#$0°¡k1K3<5<EJ1 ?¢

f ≡ 0f ≡ 2

D58;>=f(x) = x

5<EJ37,AD1RCBHJI4AB;276e 3¿7,L2A8 H*LB5HH*L23,?58EJ3qH*L23AB;B1?79AD1GCBHJIKAB;27<e83HHJIG;>¤x = 0 = y

IG;Ü &`Ý*D3!¤63`H (f(0)

)2= 2f(0)

e¿£if(0) = 2

2H*L23<;¢N<?143`HHJIK;>¤y = 0

IG;PÜ7&[Ý*837,3<3¿H*LB5Hf(x) = 2

iABEw561G1x ∈

2<H*LB56HIG7f ≡ 2

e6C>M>M>AB7,3!H*L23<EJ3iABEJ3!H*LB56Hf(0) = 0

e )<3`Ha = f(1)

eS83`HHJIK;>¤x = 1

5<;>=y = −1

D3@¤63Haf(−1) = a + 2f(−1)

2BH*LB5HFIK7`f(−1) = a/(a − 2)

e¿©qA87)Cygg*36797*IB31?7*CyN27HJIRHJCBHJIG;>¤(x − 1, 1)

(−x + 1, −1)

25<;>=(−x, 1)

iABE(x, y)

IK;Ü7&[Ý)<3¤63`Hf(x) + (a − 2)f(x − 1) = a

Ü Ýf(−x) +

2

a − 2f(−x + 1) − f(x − 1) =

a

a − 2

Ü Ýf(−x + 1) + (a − 2)f(−x) = a

e Ü 8ÝD1RIKOIK;25HJIK;>¤

f(x − 1)58;>=

f(−x + 1)IG;PÜ Ý*2Ü Ý*B58;>=Ü&<Ý¤ÌB367

f(x) − (a − 2)f(−x) = 0e Ü Ý

«¢3<MB1K58gIK;>¤xN<? −x

IG;PÜ Ý*83:A2NDH5<IG;f(−x) − (a − 2)f(x) = 0

e Ü4Ý£i

a /∈ 13 /H*L23<;N<?361RIKOIK;25HJIK;>¤ f(−x)

IG;Ü Ý5<;>=Ü`Ý*b3P79383H*LB5Hf(x) = 0

iABE5<1R1x ∈

28H*LB5HIK7`f ≡ 0

e£ia = 3

kH*L23<;Ü Ý#¤Ì B3<7f(x) = 3 − f(x − 1)

iABE5<1R1x ∈

e :36;>g*3Df(2) = 3 − f(1) = 0

58;>=f(52

)= 3 − f

(32

)= f

(12

) e Q;H*L23AH*L23<EFLB5<;>=¢>N<?7)CyNB7*HJIHJCBHJIK;>¤ (2, 1

2

) iABE(x, y)

IK;Ü &`Ý*<3¤63Hf(52

)= f

(12

)+ f(1) = f

(12

)+ 3

5@gAB;DHE5=DI4gJHJIKAB;/eIG;25<1R1?kg*AB;27*I4=23<E

a = 1ekL23<;Ü Ý¤ÌB367

f(x) = f(x − 1) + 1iABEq561G1

x ∈ eB?SIK;>=DCygJHJI4AB;¢

f(x + n) = f(x) + nkiABE#5<1R1

x ∈ 58;>=

n ∈ e!£t;M>5<EHJIKgC21K5<E,

f(n) = f(0 + n) = f(0) + n = niABEF5<1R1

n ∈ eQ6CyN27HJIRHJCBHJIG;>¤

niABEyIG;Ü &`Ý*23A2NDH5<IG;

nf(x) = f(nx)iABE#5<1R1

x ∈ 5<;>=

n ∈ e 3<;>g3ByIGi

r = m/n ∈ 7 58;>= x ∈ H*L23<;

mf(x) = f(mx) = f(n · rx) = nf(rx)2

9H*LB56HIK7`

f(rx) = rf(x)eF£t;M>5<EHJIKgC21K5<E,

f(r) = f(r · 1) = rf(1) = re83`HHJIK;>¤

y = rIG;Ü7&[ÝF¤`IB367

f(x + r) = f(x) + re#f1K79Aky793HHJIG;>¤

y = xIG;Ü7&[Ý*23!¤63H

(f(x)

)2= f(x2)

ekLDC>7f(x) ≥ 0

IRix ≥ 0

e6IG;>g*3f(−x) = −f(x)

83:LB5B3f(x) ≤ 0

IRix ≤ 0

e©qA8D1K3Hx ∈

N>3¿DÙ83<=/ew£ir ∈ 7 5<;>= r < x

H*L236;f(x) = f(x − r + r) = f(x − r) + r ≥ r

7)IG;>g*3

f(x − r) ≥ 0e£4HiAD1R14A8:7H*LB5H

f(x) ≥ xe!6IKOIG1G58E1?k

f(x) ≤ riABE561G1

r ∈ 7 7)Cyg)L@H*LB5H r > x8LDIKg)L!IKOM21RI4367wH*LB56H

f(x) ≤ xekLDC>7

f(x) = xe

)<3`HP (x)

N23!5M>AD1?B;>ABOIG5<1yIRH*LPIG;DH3<¤63<EgA23 SgI436;DHJ7q7*Cyg)LPH*LB5HP (n) > niABE5<1R12MyAB7*IRHJI B3qIG;DH3<¤63<E7

ne6C>M>M>AB7,3xH*LB5HiABE3<58g)L!M>AB7)IHJIB3xIG;DH3<¤63<E

m8H*L23<EJ3qIG75@HJ3<EJO IG;PH*L23@7,3<ÒBCy36;>g*3

P (1)P(P (1)

) P(P(P (1)

)) . . .

LDI4g)LPIG7q=BI BIG7)IKNB1K3N<?me8L2A8H*LB56H

P (x) = x + 1e

6Cµ'$# )+ '0 0 'P¹ 0 6 7'5 7 ?3'5 0?$ '> $# ½-! q·¶ $=+1¶ 3'546'5 ('0 3µ'6 ´ ?w°' ' 4Cµ'z½- w² 0?33' :787° 3IG1R1BC>7,3xH*L23¿;>AH56HJIKAB;

P (i)(1)iABEH*L23

i§R¨FH36EOIG;@H*L23¿7938ÒDCy3<;>g3

P (1)

P(P (1)

) . . .

ekL23SgAB;>=BIHJI4AB;P (n) > n

IKOM21RI4367xH*LB56Hdeg(P ) ≥ 1

58;>=UH*L2314365=DIK;>¤g*A23 UgIK3<;DHFADiPIG7QMyAB7*IRHJI B3>e£i

P (x) = x + bH*L23<;

1 + b = P (1) > 1LDIKg)LIGOPMB1GIK3<7wH*LB5H

b ≥ 1eF£4HIK7x3<5<7?HJAP7,3<3H*LB56H

P (x) = x + 1795HJIK7*k367561G1yH*L23!gAB;>=BIHJI4AB;276e¿£i

b ≥ 22H*L23<;

P (1) ≡ 1 (mod b)5<;>=¢2N<?IG;>=BCygJHJIKAB;¢

P (i)(1) ≡ 1 (mod b)iABE5<1R1

i ≥ 1e£i

P (x) = 2x + b2H*L236;

2 + b = P (1) > 1DLDI4g)LIGOPMB1GIK3<7QH*LB56H

b ≥ 0e£i

b = 0H*L236;¢>N<?!IK;>=DCygJHJI4AB;¢

P (i)(1) = 2i iABE5<1R1 i ≥ 1e 3!g*AB;27*I4=23<EwHA2¤63`H*L23<EQ5<1R1kH*L23@EJ3<O5<IG;BIK;>¤@g5<7,367wÜGIÝP (x) = 2x + b

IH*Lb ≥ 1

2QÜGIGI ÝP (x) = ax + b

IH*La ≥ 3

2F5<;>=ÛÜKIRIGI Ýdeg(P ) ≥ 2

em£t;H*L23<793g5<7,367wH*L23<EJ33)Ù<IK7HJ7N ∈ 7*Cyg)LhH*LB56H

P (n) > 2niABE@561G1

n ≥ Ne&6IG;>g*3

1 < P (1) < P (P (1)) < P (P (P (1))) < · · · yH*L236EJ33*Ù8IG7*H7 k ∈ 7)Cyg)LPH*LB5HP (k)(1) ≥ N

e )<3`Hr = P (k)(1)

5<;>=m = P (k+1)(1) − P (k)(1)

e¢L236;r ≥ N58;>=

m = P (r) − r > re BABE

1 ≤ i ≤ k>3LB5`B3

1 < P (i)(1) ≤ r < m2H*LDC>7

m=>A2367:;>AH¿=DIDI4=23P58;?

P (i)(1)iABE

1 ≤ i ≤ ke¬ÛABEJ38A6B3<E9;>AH3H*LB5H

P (k+1)(1) = m + r ≡ r (mod m)ebfw7)7)C>OP3¿587F5<;!IK;>=DCygJHJI4AB;L8?BMyAH*L2367)IG7H*LB56H

P (i)(1) ≡ r (mod m)iABE7,ABOP3

i ≥ k + 1e¢L236;

P (i+1)(1) = P (P (i)(1)) ≡ P (r) = P (P (k)(1))

= P (k+1)(1) ≡ r (mod m)e

:36;>g*3DP (i)(1) ≡ r (mod m)

iABEw561G1i ≥ k + 1

e )<3H

ABCN>3@5HEIK5<;>¤`143Dy58;>=P143`H

rr1r2r3=>36;>AH3IHJ7#IK;2E58=BIRC>7q5<;>=PH*L233)Ù8E5=DIGI>ABM2MyAB7*IRHJ3#H*L23#B3<EHJI4g3<7

ABC8EJ3<7)My3<gJHJIB31?be6C>M>M>AB7,3

a > r1b > r2

c > r3

eQ0>EJA<B3qH*LB56HÜK5Ý@HEJIG58;>¤`1K3

ABCIK7Q58gCBH3D Ü4NÝ

a + b + c > r + r1 + r2 + r3e

9 6)$+ '# ° 0:.+ µ'70 4 ´ ? "#$%!'$#*)+ .-!' '$#$330°Ü45`Ý#)<3H

s=>36;>AH3ÖH*L23m7,36OIGMy36EOI43`H36EADi 4ABC

e DEJABO H*L23;>A8:;iABEOC21K5tan(A/2) = r1/s

58;>=ÕH*L23Û¤ÌB36;IG;>38ÒDC>5<1RIRHt?a > r1

#3ÛA2NDH5<IG;tan(A/2) < a/s < 1

e@6IGOIR1K5<EJ1 ?¢tan(B/2) < 1

5<;>=tan(C/2) < 1

e:kL23<;A < π

2

B < π

2

B5<;>=C < π

2

e :36;>g*3D 4ABCIK7Q58gCBH32eÜN8Ý@6IK;>g3H*L23¬HEIK5<;>¤`143

ABCIK75gCBHJ3BF3LB5`B3H*L23h8;>A8:;ÕIK;>3<ÒBC>561GIH?

s > r + 2R:L236EJ3

RIG7H*L23gIGEJgC>OPE58=BIRC>7hADi 4ABC

e 3Ö5<1G7,ALB5`B3r1 + r2 + r3 = r + 4R

e 3<;>g3Br + r1 + r2 + r3 = 2r + 4R < 2s = a + b + c

e )<3`H

nN235@M>AB7)IHJIB3IG;DH3<¤63<EF5<;>= A

BC 5@M258EHJIRHJIKAB;SADi 1

2. . .

3n7)Cyg)LH*LB5H |A| = |B| = |C| = n

eQ0>EJA6B3xH*LB56HH*L23<EJ33*Ù8IG7*Hx ∈ A

y ∈ B

z ∈ C7)Cyg)L@H*LB56HFAB;>3:ADi

xyzIG7wH*L23:7)C>O¦ADiyH*L23:AH*L23<EHAbe

6 8!03'70 0 '¹ 0 73':6 &6 ?'7 0?$ '<° Q;S=BIGE5!ÒDCy3:143¿HEIKMB143`H

(a, b, c)367*H ) 1KABE7,ÒDCy3

a ∈ Ab ∈ B

c ∈ C

3H1 C>;=23<7w;>ABOPN2EJ367Q367*H1K5¿79ABOPOP3¿=>367F=>3CÙ:56CBHEJ3<76eB<58;27wM>3<EH3¿=>3¿¤.$3<; $36E5<1RIRH%$3DDAB;My3CBHq7*C>M>M>AB7,36EqÒBCy31 ∈ A

b3HqÒDCy31K3M21RC>7My3`HJIRHx;>ABOPN2EJ3D/=DIK79AB;27kbÒBC2I;¢ 3<7HM>5<7=B58;27

A3<7HF=25<;27

Be26C>M>M>AB7,AB;27ÒDCb IR1k;¢ 3)Ù<IK7H3M>5<7=23N2AB;!HEJIGM21K3Hè

Q;g*ABOOS36;>g*3@M258EQOSAB;DHEJ36EÒDCy3!M>ADC>EFHJADCBHx ∈ C

kAB;S5x − 1 ∈ A

e D;36ßb3H,7` IR1w3)Ù<IK7H3x ∈ C

H31wÒBCy3x − 1 /∈ A

5614ABE7`MBC2IK79ÒBCy3(1, x − 1, x)

;>3=>ADIHqM>5<7 '3`HEJ3SN>AB;¢/g[ 367*H¿=2AB;>g:ÒDCy3x − 1 /∈ B

/3`Hq5<IG;27)IÒBCy3x − 1 ∈ C

eD;M>5<EHJIKgC21GIK3<E9x − 1 > k

e¢h5<IG7#g*ABOOS3:56CygC>;S=>367=236CÙ:HEIKMB143`HJ7(x − k, k, x)

3`H(k − 1, x − k, x − 1)

;¢ 367*HQN2AB;¢>g[ 367*HFÒBCy3x − k /∈ A

3`Hx − k /∈ B

e 8HQ=2AB;>gx − k ∈ C

eQ3O '3<OP3B/36;gAB;27)IK= $36E58;DHx1K3<7xHEJIGM21K3H7(x − k − 1, k, x − 1)

3H(1, x − k − 1, x − k)

2AB;P= $38=DC2IRHFÒBCy3x − k − 1 ∈ C

e Q;M>36CBH/5<1KABE7EJ3<g*ABOOS36;>g*36Eg*3E56IK79AB;2;>3<OP3<;DH,<3H/M2EJADCDB3<E/M>5<E/E%$3<gC>EEJ36;>g*3ÒBCy3DM>ADC>EqHJADCBH

i ≥ 0/1K3<7;>ABOPN2EJ367

x − ik3`H

x − ik − 1bHJ5<;DH:ÒBCb IG1G7@79AB;DH7*HEI4gJHJ3<OP3<;DHxMyAB7*IRHJIRi47k58M2M>5<EHJIK3<;2;>36;DHHADC>7q14367:=236CÙ/"5

Ceh56IK7¿MyADC>EC>;

iNBIK3<;g)L2ADIK7*IbC>;h=>3Ug*367@;>ABOPN2EJ367!367*H¿;,$38g3<7)7956IKEJ36OS36;DH¿IK;Bi$36EJIK36C>EADC$38¤561 "5

kg*3UÒDC2IIKOM21RI4ÒDCy3ÒBCb IG1y58M2M>5<EHJIK3<;2;>3 "5

AADC/"5

Be2¡ 3<7HC>;>3g*AB;DHE58=BIKgJHJI4AB;/e

Q;M>AB7,3C = c1

. . .

cn e x 58M2E%"3671G5#M2EJABM>EI+$3`H($3QM>E%$38g($3<=>36;DH3D63H¢M2C2IG7,ÒDCy3

|A| = |C| = n2g[ 3<7HF=2AB;>gQÒDCy3

A = c1 − 1. . .

cn − 1 e¢h5<IG7QM>ADC>EHJADCBH i

AB;5ci > k > 1

=2AB;>gci − 1 > 1

>g3ÒDC2I¢gAB;DHEJ38=DIRHFÒDCy31 ∈ A

e &! />e Q;My3CBHPM>EJADCDB36ESÒBCy3Û7*I

A

B3`H

CiABEJOS36;DHC>;>3ÛM258EHJIRHJIKAB;=>3

12. . .

n 5B3<g |A| |B| |C| > 1

4nB5614ABE7FIR1k3)Ù<IK7H3¿C>;PN>AB;!HEIKMB143`Hè

$!">e & e 6e>Ñ($3<8361 ?¢ ¶('7 -=4-!'70 6.-!'7 ><M2EJIG;>¤63<E92M>EJA2ND1436OÓ¡¢ 8 eÐ Í )<3H

nN>35PMyAB7*IRHJI B3IG;DH3<¤63<Eq¤EJ3<5H36EH*LB5<;

1/58;>=143`H

pN>35PM>EIKOP37)Cyg)LH*LB56H

n=BI BIK=>367

p − 15<;>=

p=DIDI4=23<7

n3 − 1eQ0>EJA<B3qH*LB56H

4p − 3IG7Q579ÒBC>5<EJ3>e

¶ "'! )+ '0 0 '¹ 0 6 7'5 7 ?3'5 $0! '3°¢LDIG7M>EJA2ND1436O IG77)IGOIR1K5<ExHJA¬M>EJA2ND1436O =ADiH*L23 ¡>Ñ)38g)0BAD1RIK79L2961KA<D58OP5H*L23<O56HJIKg561¢gABOPM>3HJIHJI4AB;/ebfÛ7,AD1RCBHJI4AB;58M2My3658E7FIK; $$ D< D e

9D©q3*Ù<H3 HJC>EJ; HA EJ3<58=>36E7 ¬79AD1GCBHJIKAB;27&HJA M2EJA2NB1K3<O7ÊADiÛH*L23 36EO58;h56H*L236OP5HJI4gJ5<1* F1?BOPMBIK58= ¤`IB36;P5H $$ D 9$ e

Ð e Q3HJ3<EJOIG;>3:561G1yM25<IGE7(x, y)

ADi¢EJ3<561y;BC>OSN23<E7xyLDIKg)L795HJIK7*iR?

x3 + y3 = 7

xy(x + y) = −2e

6Cµ'$#)+'0 0 '¹ 0 73':6 7 ?3'5$0! 'º ' 0 '7+w°'$#= $ "#1'7 ¶ :V´ $# ±#/² $0 # °QN° 4 ; 0?3# $0?'0 ´ =>µ'70? + '0?$ °! ' 4>µ'¹ 0 6 7'5 w² 0?'>87°

fw797*C>OS3qH*LB5H(x, y)

IG7Q7)Cyg)L5:M>56IKE[e¢L236;(x + y)3 = x3 + y3 + 3xy(x + y) = 1

LDI4g)L1K3<58=27HA

x + y = 1ekLDC>7

xy = −2e £4HiAD1R14A8:7H*LB5H

x5<;>=

y58EJ3EJA2AHJ7QADi

X2 − X − 2 = 0ekL23<EJ3iABEJ3B

(x, y) = (−1, 2)ABE

(2, −1)8LDIKg)L5<EJ37,AD1RCBHJI4AB;27#ADi>H*L23:M2EJA2NB1K3<O¬e

Ì e£t;H*L23#IK;DHJ3<EI4ABE/ADi25#HEIK5<;>¤`143ABC

8gIGEJg14367K1K2K3¢58;>=

K4ADi/H*L23795<OS3!E5=DIGI/58EJ3=>3y;>3<=7*Cyg)LÕH*LB56H

K1

K2q58;>=

K3HJADCyg)LHAÖ7)IK=>367ADixH*L23HEJIG58;>¤`1K3h58;>=

K4HJADCyg)L23<7

K1K2B58;>=

K3B5<7Q7,L2A8:;IG;!H*L23y¤`C>EJ3>e0>EJA<B3¿H*LB5HH*L23g*36;DHEJ3ADi

K4IG7Q1KA2g5H3<=AB;H*L23 1RIK;>3ÛH*LBEJADCy¤6LH*L23 IK;>g3<;DHEJ3Õ58;>=H*L23gIKEJgC>OPg*36;DHEJ3>e ..................................................................................................................................................................................................................................................................................................................

..

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..

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..

..

..

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..

..

..

..

..

..

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..

..

..

..

.

..

.

..

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..

.

..

.

..

..

...

..

................................................................................................................. .

..

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..

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..

..

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..

..

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..

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........................................................

r r

r

r

A B

C

K1 K2

K3

K4

6Cµ'$# )+ -!':w¹6& 3'5o¸ 6'7 0?$ '> $# ° » ° $ "''·*½?) "':.-!' '$.-!'70?!#&° ' 4Cµ'¹ &' 7?63©°

)<3HOiN>3@H*L23g*36;DHEJ3ADi/H*L23gIGEJg143

KiÜKiABE

i = 1234Ý*¢58;>=S143`H

ρN>3H*L23gABOPOPAB;E58=BIRC>7#ADiyH*L23:gIGEJg14367<e)<3H

γÜg3<;DHEJ3

IBE58=BIRC>7

rÝ58;>=

ΓÜ4g*36;DHEJ3

OE5=DIGC>7

RÝ=23<;>AHJ3¿H*L23IG;>gIKEJg1K3:5<;>=!H*L23:gIKEJgC>OPgIKEJg1K3ADi 4ABC

DEJ3<7)My3<gJHJIB31?be¡kIGEJg143K1

IK7¢H*L23wIKO5¤63QADiγC>;>=23<EyH*L23QL2ABOSAH*L23`H?#IRH*Lqg*36;DHEJ3

A5<;>=¿7,gJ5<1K3i45gJHJABE

k = ρ/r2¢L23<;>g3B −−→

AO1 = k−→AIk7,AH*LB5H −−→

IO1 = (1 − k)−→IAe:6IGOIR1K5<EJ1 ?¢

−−→IO2 = (1−k)

−→IB

5<;>= −−→IO3 = (1−k)

−→ICe/¢L236EJ36iABEJ3D 4O1O2O3

IK7H*L23IKO5¤63ADi 4ABCC>;>=23<EQH*L23L2ABOPAH*L23Ht?PIRH*LUg*36;DHEJ3

I58;>=Ui458gJHABE

1 − ke£4H#iAD1R14A8:7H*LB56HQH*L23gIKEJgC>OPg*36;DHEJ3PADi 4O1O2O3

¢;258OP361 ?O4kIK7H*L23!IGOP58¤63ADi

OH*LBEJADCy¤6LH*LDIK7#L2ABOSAH*L23`H?bebfw7Q5EJ367)C21H,

O4IB58;>=

O58EJ3:g*AD1R1GIG;>3<5<E,D587QEJ38ÒDC2IKEJ3<=/e

13' ehkL23gIKEJgC>OE5=DIGC>72ρ

ADi 4O1O2O37)56HJIG7)y3<7

2ρ =(

1 − ρ

r

)

Re

¢LDC>7`ρ =

rR

2r + R

D5:C>7936iKC21yEJ3<7*C21RHL236;S=BE5IG;>¤xH*L23k¤`C>EJ32e

99× e¢L23g5H36EMBIG1R1K5<E "'0:6? IK7#7*IRHHJIG;>¤¿IG;H*L23OI4=2=B1K379ÒBC>5<EJ3@ADib5<;

N × NN>AB5<EJ=¢/L236EJ3NIG75<;hA2=2=IG;DH3<¤63<EIH*L

N ≥ 3e¢L23UAH*L236E¿7,ÒDC>58EJ367@ADiH*L23N>AB5<EJ=P3<58g)Lg*AB;DH5<IG;P5:MyAB7*IRHJI B3¿IK;DHJ38¤636E,B5<;>=5<1R1kADiyH*L23<793IG;DH3<¤63<E7Q58EJ3:=BIRß/36EJ3<;DH`e©C>OOS36E795HH¢:5<;DHJ7/HAqy;>=:5F:5?:ADß@H*L23xN2AB58EJ=be/kL23xg5H36EMBIG1R1K5<EgJ58;:OPA<B3AB;B1?N>3`H3836;S5=915g3<;DHQ79ÒBC>5<EJ3<7xÜK7,ÒDC>58EJ367qLB5`BIG;>¤:5!g*ABOOSAB;P7)IK=>3[Ý*yABEFADßH*L23@N>AB5<EJ=i4EJABOÃAB;>3!ADi¢H*L23!ADCBH36EOPAB7*HQ79ÒBC>5<EJ3<7`¢LB5`BIG;>¤@AB;>g*3@EJ365g)L23<=U7*Cyg)LS5!79ÒBC>5<EJ3>e- Q;EJ3<58g)LDIK;>¤@5!;>3`79ÒBC>5<EJ3B©#C>OPOP3<E7)56HHLB587#HAU3656HFH*L23@;BC>OSN23<EAB;PH*LB56H#79ÒBC>5<EJ3>e¢L23:;BC>OPN>36E

n3I4¤6LB7 1

n

8¤B5<;>=©C>OOS36E795HHFg5<;2;>AHQ3656HOSABEJ3qH*LB58;28¤Be

Q38gIK=>3L23`H*L23<E;BC>OPN>36E7#g5<;N23=BIG7*HEI4NDCBH3<=AB;H*L23N>AB5<EJ=!7,A:H*LB5HH*L236EJ3IK7Q;>A:5`?PADßSH*L23:N>AB5<EJ=iABE©#C>OPOP3<E7)56HHÜK5Ý!iABE

N = 2003 ÜN8ÝiABEw561G1kA2=2=IG;DH3<¤63<E7

N ≥ 3e

6Cµ'$#*)+ 3'70 0 '¹ 0 6 7'5 &6 ?'7 $0! '3°©qADC>7:561G1KAB;27M>EJADCDB36EÒDCy3B=B58;27¿143SgJ587:¤.$3<; $36E5<1t©#C>OPOP3<E7)56HH¿M>36CBHHADCy

1tADC>E77,ABEHJIKEF=DCHJ58NB1K3<56CU7)58;27#O58;>¤636EFM21RC>7x=230, 9

8¤Be/0BADC>EFg*31K5B>AB;!D5CBHJIR1GIG7,36E1K5:O#$3`H*L2A2=>3M>EJA2NB5NDIG1RIK7H32e>ADCBHq=k 58N>ABEJ=kbAB;EJ36OP5<EJÒBCy3ÒBCy3DbÒBC2IHH3 "556Cy¤OS36;DH36E1K3MyADIK=5`D5<1+$3B¢AB;My3CBHF7*C>M>M>AB7,36EFÒDCy314367#3<;DHJIK3<E7FCBHJIG1RIK79$3<7F7,AB;DH12. . .

N2 e

Q;I4=23<;DHJIRk3g)LB5ÒDCy3@g5<7,3:5B3<gQ7,AB;Sg3<;DHEJ3@=2379ABEHJ3ÒBCy3D>7*IN = 2k + 1

g)LB5ÒDCy3g5<7,3S3<7HqC>;MyADIG;DH:=DChE%$367,365<Ch=>367M>ADIK;DH7@3<;DHJIK3<E7 "5Ug*A2ABEJ=2AB;2;,$38367!=B58;27I = −k

−k + 1. . .

k − 1

k e )<3:g5<EE%$3g3<;DHE561¢367*H (0, 0)

e Q;hD5hgAB;27)IK= $36EJ3<E@1K3<7Pg)L236OIG;27PM258EHJ5<;DH=>3

(0, 0)3`H!7,ABEHJ5<;DH=DCmE%$3<793<56C3<;;>3M2587)795<;DHÛÒDCy3&=¢ C>;ÞMyADIG;DH

(x, y)"5C>;M>ADIK;DH¬BADIG7)IG;

(x′, y′)HJ361PÒDCy3

|x′| + |y′| > |x| + |y| Ü4M258E3)Ù836OPMB143DQ7)I x

y > 0g*31K5h;¢ 5<CBHJABEJIG7,3hÒBCb C>;= $36M21G5g3<OP3<;DH/B36E7

(x +1, y)ADC:B36E7

(x, y +1)Ý)eB;@M>5<EHJIKgC21GIK3<E98C>;HJ3612g)L23<OIK;O#"36;>3Q;,$38g3<7)7956IKEJ36OS36;DH>B3<E7/1G5Q7,ABEHJI43Q36;¿C>;¿;>ABOSNBEJ3Fy;BI8=¢ $3H58M>3<73`H¢795<;27N2ADCyg1432e

)<314AB;>¤=¢ C>;HJ3612g)L236OIG;¢2©#C>OPOP3<E7)56HH793q= $36M21G5g3q3<;@EJ3679M>38gJH58;DH/14367wgAB;>=BIHJI4AB;27=>3w1 $3<;>AB;>g($3Dg*31G1K3wIG;>=BIKÒBC,$3<3#gIK9=23<7)7)C>7/3Hy14367gAB;DHE5<IG;DH367M2EJA2N258NBIR1GIG7*HJ3<7b7)C2I D58;DHJ3<7ÜKIR1¢367*HBI B3<OP3<;DHFg*AB;27936IR1G1+$3:=23i456IKEJ3¿C>;S=23<7)7)IG;Ý • Q3

(0, 0)2IG1bg)L2ADIK7*IRHwC>;=23<7MyADIG;DHJ7

(−1, 0)(1, 0)

(0, 1)

y5<;>=(0, −1)=>3¿i45 g)AB;/$38ÒDC2IKM2EJA2N258NB1K3>e

• B IG1b367*HQ36;(n, 0)

>5`B38gn ≥ 1

2IR1>D5!36;(n, 1)

ADC36;(n, −1)

5B3<gQC>;>3M>EJA2NB5NDIG1RIRH%$3=23 1

2(n + 1)

=25<;27Qg)LB5gC>;=23<7QgJ587`D3H/D5:3<;(n + 1, 0)

5B3<gC>;>3:M>EJA2NB5NDIG1RIRH%$3:=>3 n

n + 1

e• B IG12367*H3<;

(n−p, p)<5B3<g

n > p ≥ 1<IG1<D536;

(n−p+1, p)5B3<gC>;>3M>EJA2NB5NDIG1RIRH%$3¿=>3 2(n − p) + 1

2(n + 1)

3H36;(n − p, p + 1)

5B3<gC>;>3¿M>EJA2NB5NDIG1RIRH%$3=>3 2p + 1

2(n + 1)

e• B IG1b367*HQ36;

(0, n)>5`B38g

n ≥ 12IR1>D5!36;

(1, n)ADC36;

(−1, n)5B3<gQC>;>3M>EJA2NB5NDIG1RIRH%$3=23 1

2(n + 1)

=25<;27Qg)LB5gC>;=23<7QgJ587`D3H/D5:3<;(0, n + 1)

5B3<gC>;>3:M>EJA2NB5NDIG1RIRH%$3:=>3 n

n + 1

e

9 )<3<7:56CBHEJ3<7gJ5877,3!HE5<IH36;DH=>3i45 g)AB;7?BO#$3`HEJIKÒBCy3SÜKM>5<E¿E5<M>M>ABEH "51 C>;=>367:5`Ù8367=>3!g*A2ABEJ=2AB;2;,$38367¿ADCUM>5<E#E5<M>M>ABEH"5

(0, 0)ÝM>5<E#E5<M>M>ABEH "5@1K5@7)IHJC>56HJIKAB;ÒDC2IkDI436;DH=¢ '3HEJ3= $38g*EIRHJ3MyADC>E14367MyADIG;DHJ7

(x, y)5`B38g

x

y ≥ 0e£1x367*H@5<1KABE7i45gIR143=>3 $36EJIRk36E#ÒBCy3@7*Ik1 AB;U;>AHJ3

p(x, y)1K5@M>EJA2NB5NDIG1RIRH%$3@ÒBCy3P©#C>OPOP3<E7)56HHQM>5<79793@M258Ew143MyADIG;DH

(x, y) 6= (0, 0)B5<1KABE7

p(x, y) =1

4(|x| + |y|)e

8ADIRHX1G5D5<EJIG5ND143q561+$3656HJADIKEJ3$3<¤5<1K3q56CM>ADI4=:HAH5<1BOP5<;>¤.$3¿M258EF©#C>OPOP3<E7)56HH5<CSg*ADC>E7#=>37,AB;M,$36EJIGM21K3>e. Q;;>AHJ3

E(X)79AB;P3<7)M,$36E58;>g3:O56H*L!$36OP5HJI4ÒDCy3>e0BADC>E

(x, y) 6= (0, 0)yAB;S;>AH3

w(x, y)1K3MyADIK=2756HHEI4NDC $3@56CUMyADIG;DH

(x, y)Ü47*I1K3S;>ABOSNBEJ3U5gg*EJA2g)L $3"5(x, y)

3<7HnAB;5=2AB;>g

w(x, y) = 1/ne Q3SM21RC>7AB;hgAB;BIK3<;DH,/MyADC>E¿7*IKOM21RIGy3<Ex1K3<7:7,ABOOP5HJI4AB;27:ÒBC2I7*C2IB36;DH,ÒDCy3

w(x, y) = 014ABE79ÒBCy3(x, y)

;>3:i456IRHwM>5<7M258EHJI43=BCSE%$3<793<56CPy;BI¢g*AB;27*I4= $3<E%$3`Ý)e Q;P;>AH3ÒBCy3M>ADC>EC>;PM>ADIK;DH(x, y)

=BCPE%$367,365<CSgAB;27)IK= $36E%$3D2AB;5 |x| + |y| ≤ 2kef14ABE7

E(X) =∑

(x,y)6=(0,0)

p(x, y)w(x, y)

=

2k∑

n=1

∑

|x|+|y|=n

p(x, y)w(x, y)

=

2k∑

n=1

1

4n

∑

|x|+|y|=n

w(x, y)

≤ 1

4

(

1 +1

2+

1

3+

1

4

)

+1

8

(1

5+

1

6+ · · · +

1

12

)

+1

12

(1

13+ · · · +

1

24

)

+ · · ·

=1

4

(

1 +1

2+

1

3+

1

4

)

+1

8

(1

5+

1

6+ · · · +

1

12

)

+ R

A "CR =

2k∑

n=3

[

1

4n

4n∑

i=1

1

2n2 − 2n + i

] e Q;P5 1

4

(1 + 1

2+ 1

3+ 1

4

)+ 1

8

(15

+ 16

+ · · · + 112

)= 143771

221760< 0, 65

3`H

R ≤2k∑

n=3

[

1

4n

4n∑

i=1

1

2n2 − 2n

]

=2k∑

n=3

(1

4n

)(4n

2n2 − 2n

)

=

2k∑

n=3

1

2n(n − 1)=

1

2

2k∑

n=3

(1

n − 1− 1

n

)

=1

2

(1

2− 1

2k

)

<1

4

efIK;27*I

E(X) < 0, 9e

Q;36;= $38=DC2IRHÒDCy3BbM>5<EOI14367g)L236OIG;27g*AB;27*I4= $3<E%$3<7`¢IG13<;3)Ù<IK7H3C>;M>ADC>E143<ÒBCy31y1K3:M>ADI4=B7wHAH5<1yO58;>¤ $3:M258E#©C>OOS36E795HHF;>3:= $36M>5<79793:M25870, 9

8¤Be

Ú e )<3H

A1

B15<;>=

C1N23hH*L23ÛOI4=2MyADIG;DHJ7@ADiH*L23S7*I4=23<7ADiH*L23S5gCBHJ38*58;>¤`1K38=HEJIG58;>¤`1K3

ABCekL23

61RIK;>367 H*LBEJADCy¤6LH*L23<793M>ADIK;DH7M>3<EJMy36;>=BIKgC21K5<EwHJA!H*L23@AH*L236E7)IK=>367/OP383`HyIK;H*L23FM>ADIK;DH7

A2B25<;>=

C2587:79L2A8:;¬IG;H*L23y¤`C>EJ3>e0>EJA<B3H*LB56HH*L2358EJ365ADiH*L23mL23)Ù<58¤6AB;

A1C2B1A2C1B238ÒDC>5<1G7LB561GibADiyH*L2358EJ365@ADi 4ABCe

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................

A

A1

A2

B

B1

B2

C

C1

C2

6Cµ'$# )$+ -!':©¹6& 3'52¸ 6'7 0?$ '>xº ' 0 '7+3w°7!!#=&&G "#1'62¶ :´ $# g° » ° $ "''·½?) "':/.-' '$.-!'70?!#&° '*4>µ'R¹6& 3' ² 0?'>87°

Q3<;>AHJ3FH*L23Q58EJ365xADi 4XY ZN<?

[XY Z]e6IK;>g3 4A1B1C1

IG7bH*L23QIGOP58¤63#ADi4ABC

C>;>=23<EH*L23L2ABOPAH*L23Ht?IRH*Lg*36;DHEJ3:5HH*L23:g3<;DHEJADIK=5<;>=i458gJHABE −12

83LB5B3[A1B1C1] = 1

4[ABC]

e6IGOIR1K5<EJ1 ?¢8C>7)IG;>¤H*L23L2ABOSAH*L23`H?hIH*Lg3<;DHEJ3

A58;>=i458gJHABE 12

83:LB5`B3[AB1C1] = 1

4[ABC]

e©qA8w143`HO58;>=

HN23UH*L23gIGEJgC>OSg3<;DHEJ3¬5<;>=ÕABEH*L2A2g*36;DHEJ3ADi 4ABC

EJ3<7)My3<gJHJIB31?be©qAHJ3wH*LB56HyH*L2367,3QM>ADIK;DH75<EJ3FIK;DHJ3<EI4ABE>HAH*L23Q58gCBH3<)5<;>¤`143<= 4ABCe

)<3HHJIG;>¤haN23¿H*L231K3<;>¤[H*LADiyH*L235<1HJIRHJCy=23i4EJABO

AHA

BCIK; 4ABC

<3:LB5B3[BHC] = 1

2· BC · (ha − AH) = [ABC] − 1

2BC · AH

eª#7)IG;>¤xH*L23q361R14);>A8:;EJ361G56HJIKAB;

AH = 2OA183=238=DCyg*3¿H*LB5H

[BHC] = [ABC] − BC · OA1 = [ABC] − 2[OBC]e

#N2793<EtDIK;>¤H*LB56Hh(H) = A2

ÜRL236EJ3hIG7qH*L23L2ABOSAH*L23`H?¬=236>;>38=58N>A6B3`Ýk3A2NDH5<IG;

[B1A2C1] = 14[BHC] = 1

4[ABC] − 1

2[OBC]

e£t;H*L23P795<OS3@:5?k¢3SgJ58;¬7,L2A8ÞH*LB5H

[C1B2A1] = 14[ABC] − 1

2[OCA]

5<;>=[A1C2B1] = 1

4[ABC] − 1

2[OAB]

ew£4HiAD1G1KA8:7H*LB56H[A1C2B1A2C1B2] = [A1B1C1] + [B1A2C1] + [C1B2A1] + [A1C2B1]

= 14[ABC] + 3

4[ABC]

− 12([OBC] + [OCA] + [OAB])

= [ABC] − 12[ABC] = 1

2[ABC]

587QEJ38ÒDC2IKEJ3<=/eÎ e£i

nIG7:5UM>AB7)IHJIB3IG;DH3<¤63<E9143`H

a(n)N23H*L23P79O5<1R14367*HqMyAB7*IRHJI B3P;BC>OPN>36E¿iABELDI4g)L (

a(n))!IK7#=DIDIK7*I4ND143:N<?

ne Q3`H36EOIK;>35<1R1yMyAB7*IRHJI B3¿IK;DHJ38¤636E7

n7)56HJIG7)iR?DIK;>¤

a(n)

n=

2

3

e

&6Cµ'$# )$+ -!':¹ &'7P¸ ='6 $0! ' 3'70 0 ' ¹ 0 6 7'5 &6 ?'70?$ '> !#[±1#/² 0 # %°QN° $64 & ; 0?# 10?3'70´(6Cµ'0 3.+ 63'70 3$ °! ' 4>µ' 4 ² 0?33' :787°¢L23xAB;B1?7,AD1RCBHJI4AB;IG7

n = 9e©qAHJ3#yE7H¢H*LB56H¢H*L23x¤`IB36;@g*AB;>=DIRHJIKAB;:IG77)IGOPMB1?

3a(n) = 2n¢LDI4g)LIGOPMB1GIK3<7

3 | neS83HHJIG;>¤

n = 3kk3PLB5B3

a(3k) = 2ke¡k1K3<5<EJ1 ?¢

k 6= 17*IK;>g3

a(3) = 3e BABE

k ≥ 2w3LB5B3

3k | (2k)!7*IK;>g3

(2k)! = (2k) · · · 2 · 1gAB;DHJ56IK;27qH*L23P793<M258E5H3i458gJHABE7

k58;>=

33,B36;IGi

k = 3e

:A83,B36E,IGik > 3

8H*L23<;(2k − 1)! = (2k − 1) · · · k · · · 2 · 1 561K79AgAB;DHJ56IK;27H*L23=BIG7*HJIG;>gJHi45gJHJABE7

k58;>=

3B58;>=PL236;>g*3D

a(3k) < 2ke IG;25<1R1?kDiABE

k = 33:LB5`B3

a(9) = 6B7*IK;>g3

9 | 6!NDCBH

9 - 5!e¢LDC>7`

n = 9IG7wH*L23AB;B1 ?79AD1GCBHJIKAB;/e®

ex0>EJA6B3H*LB5HH*L23<EJ3:58EJ3IK;B>;BIRHJ361 ?OP5<;?PM25<IGE7(a, b)

ADibM>AB7)IHJIB3¿IK;DHJ38¤636E7FIH*La > b

LB5DIK;>¤xH*L23iAD1R14A8IG;>¤¿M2EJABMy36EHJIK3<7ÜGIÝ@H*L23¤EJ3656HJ3<7HFg*ABOOSAB;P=DIDIK79ABEwADia58;>=

b3<ÒBC>561K7

12ÜKIRIÝ

aIK7Q5=BI BIG7,ABEwADi

b2 − 5eÜKIRIGI Ý

bIK7Q5=BI BIG7,ABEADi

a2 − 5e

6Cµ'$#)+ -!':¡¹6& 3'5*¸ 6'70?$ '> $# 3'70 0 '¹ 0 73':6 &6 ?'7 $0! '3° ' 4>µ'¹ &' ² 0?33' :787°

)<3H(a, b)

N>3:5@M25<IGEQADibM>AB7)IHJIB3¿IK;DHJ38¤636E7FIRH*La > b

e 3:>E7*Hw7,L2A8&H*LB5H(a, b)

7)56HJIG7)y3<7QÜKI Ý*2ÜGIGI Ý*B5<;>=PÜKIRIGI Ý/IGi¢5<;>=PAB;B1?!IRia2 + b2 − 5

IG7Q5:OC21HJIKMB143:ADiabe6C>M>M>AB7,3xH*LB5HÜKI Ý*DÜKIRIÝ58;>=ÜGIGIRIÝ/L2AD14=bekL23<;

b2 − 5 = λa58;>=

a2 − 5 = µbiABE¿7,ABOP3IK;DHJ38¤636E7λ5<;>=

µe#3<;>g3B

ab2 − 5a = λa2 = 5λ + λµbekLDC>7

b(ab−λµ) = 5(λ+a)e£i

5=DIDI4=23<7

b[H*L23<;¢7)IG;>g*3

a2 = 5+µb`IHkiAD1R14A8:7yH*LB56H

5=BI BIK=>367a<g*AB;DHE58=BIKgJHJIK;>¤#ÜGIÝ)e/kL23<EJ3iABEJ3B

5=DIDI4=23<7

ab−λµe/kL23<;

ab−λµ = 5kiABE7,ABOP3IG;DH3<¤63<Ek85<;>=3xLB5B3

b(5k) = 5(λ+a)2H*LB56H/IK7`

λ = bk −ae/¢L236;

b2 − 5 = (bk − a)a28H*LB5HIK7`

a2 + b2 − 5 = kabe¡bAB;B36E7,31?k¢7)C>M2MyAB793@H*LB56H

(a, b)795HJIK7*k367

a2 + b2 − 5 = kabiABE#7,ABOP3IK;DHJ38¤636E

ke¡bAB;>=DIRHJIKAB;27ÜGIGI Ý>5<;>=ÜKIRIGI Ýyg1K3<5<EJ1 ?L2AD1K=/e£i

d = gcd(a, b)[H*L23<;

a = da′58;>=b = db′ 58;>=P7,A d2a′2 + d2b′2 − 5 = kd2a′b′ ex£4HFiAD1G1KA8:7FH*LB56H d2 =DIDI4=23<7

558;>=

dOC>7*HFN23

1DIGOPMB1?DIK;>¤xH*LB5HFg*AB;>=DIRHJIKAB;PÜKI ÝL2AD1K=27Q58731G1e©qA8B7)C>M2MyAB793¿H*LB5H

(a, b)7)56HJIG7)y3<7

a2 + b2 − 5 = kabiABEw79ABOS3¿IK;DHJ38¤636E

k > 1e EJABO

(ak − b)2 + a2 − 5 = ka(ak − b)B37,3<3:H*LB5H

(ak − b, a)IK758;>AH*L236E/M>56IKEb7)56HJIG7)iR?DIK;>¤H*L23xgAB;>=BIHJI4AB;276eH58EHJIK;>¤IH*LqH*L23M25<IGE

(4, 1)ÜGiABEkLDIKg)L

k = 3Ý58;>=@5<M>MB1?DIK;>¤EJ3<M>3<5H3<=B1 ?:H*L23HE5<;27)iABEJOP5HJI4AB;

(a, b) → (ak − b, a)3A2NDH5<IG;:57,3<ÒBCy36;>g*3xADiB=BIG7*HJIG;>gJH/M25<IGE7H*LB5H/5<EJ379AD1GCBHJIKAB;27<e/<My3<gIGyg561G1 ?¢143`H

a1 = 4

b1 = 158;>=

an+1 = 3an − bnbn+1 = an

iABE5<1R1n ∈ H*L236; an < an+1

iABE5<1R1n58;>=

(an, bn)7)56HJIG7)y3<75<1R1kH*L23gAB;>=BIHJI4AB;276e k367)IK=>367>IHQIG7q3<5<7)IR1?P7,3<3<;PH*LB5H

an = L2n+15<;>=

bn = L2n−1L236EJ3 Ln IK7FH*L23)CygJ5877938ÒDCy3<;>g3@=>3y;>3<=SN<?

L0 = 2L1 = 1

58;>=Ln+1 = Ln + Ln−1

iABEn ∈ e

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LORQWRU*Q1S W UWR\ WgQ1U 4

TFYQT WR\

/- @')-Ö+-/"2 #+ x+ x+ @'9- x"2+

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$D7DE@ ?)F <58( FG,H'= 9. ,IFG' :JD$5 /.2@>5G! =G? 7K! 5 ' 7MLON , (; '*2, QP , R , S , ?)F M/.0+/ ? @ , S9.20T/. ,I=G? 7K! 5 ' 75 ! ' @U( FG, ,0F 3#)#5 5 *, LEVW7X(2DW5G! ?)F D1YJPX.0T/.5 (;D( , &# ';, E1. ' PZ5 [. ' P\PB5G# , 9. , : ?;F = 5 [ , P ,+, 7#)#5 5 *, D7)#X! ' @U( FG, ,0F 3#)#5 5 *,U=G? 7K! 5 ' 72]H() '+* 587):X/. , , ?)F =4' H! ' @U( FG, ,0F 36#;#25 <5 *2,U=G? 7K! 5 ' 725 *, W3 , +<3^Y& ? " <#&%' _ `(; ';'=E=a' H@ , ,0F 36#;#25 <5 *2,>=G? 7K! 5 ' 72 ,+b? 58 , #A=4' <@>5G# FG,,0cT' WLdeDeDe 1a lqp3`vwa lnmGGb i&u1f`l_,abbHm8gGa lnmGh9dih $58@>D43kjDjlidDm)Ln`#fsh9jHeb0o #)#5 5 *,B=G? 7K! 5 ' 72Y! ' @U( FG, ,0F 3#;#25 <5 *2,B=8? 7)!$<5 ' 7;0Y " <#&%' DLfw; =#1#$Cµ' iKC>;>gJHJI4AB;mIG7=236>;>38=5<7U5ÛEJ3<5614D5<1RCy38=m5<EJIH*LBOS3`HJI4giKC>;>gJHJI4AB;

f : →

7)Cyg)LÛH*LB5Hf(nm) = f(n) + f(m)

iABE!5<1R1M>56IKE7SADi¿gABM>EIKOP3IK;DHJ38¤636E7n58;>=

meh£i

f(nm) = f(n) + f(m)iABE¿5<1R1

nm ∈ /H*L23<;3795`?!H*LB5H

fIK7Q5 8 ''5+ =#1#$Cµ' iKC>;>gJHJI4AB;/e£t; &BkL238ABEJ36OqpQ£JBM/e.& 7 DEJ= AB7F7*H56HJ3<7H*LB5HIRi

fIK7F5<;5=2=BIHJIB3qiKC>;>gJHJI4AB;7)Cyg)LH*LB56H

f(n +1) ≥ f(n)iABE5<1R1

n ∈ <H*L236; f(n) = C log niABE5gAB;27*H58;DH

C ∈ e IH*L2ADCBH/H*L23L8?BMyAH*L2367)IG7

f(n +1) ≥ f(n)<H*LDIK7IK7;>AHHECy3xIK;!¤63<;>36E5<1t587IG7x7,L2A8:;¢>iABEFIK;27HJ5<;>g*3D¢N<?H*L23gABOPMB143`H31?S58=>=DIRHJI B3iKC>;>gJHJI4AB;

Ω(n)=>3y;>3<=N<?

Ω(pa1

1 pa2

2 · · · pak

k ) = a1 + a2 + · · · + ak8L23<EJ3

n = pa1

1 pa2

2 · · · pak

k

IK7wH*L23M>EIKOP3@=238gABOPM>AB7)IHJI4AB;PADine DEJ= AB7 kL238ABEJ36O¦IG7#5@=>3<3<MS5<;>=IG;DH36EJ3<7HJIK;>¤EJ367)C21H,NBCBHIHJ7¿M2EJA2ADiIG7¿E56H*L236EqgABOPMB1GIKg5H3<=!2b79383 bMe!& $ 58;>=

§ e 58;>= e kMe 58;>=:ßwe iABE58=>3<ÒBC>5H3qg*ABOOS36;DHJ758;>=5=2=BIHJI4AB;2561BEJ3i3<EJ36;>g*3678IG;>g1GCy=DIK;>¤5M>EJA2ADikÜ=DCy3xHAPÛAB793<E58;>= )65<OSN238Ý¢H*LB5H7*IKOM21RIGy3<7H*L23ABEJIK¤`IK;25612M>EJA2ADikADi%DEJ= AB7<e£t;¿H*LDIK7;>AH3D`3QMyAB793#5F365836EbEJ367)C21H,8NDCBHkIRH*L¿5wB3<Et?:361K3<OP3<;DH58Et?M2EJA2ADiJef1K79Ak8379L2A85:;BI4g3gAB;27,3<ÒBCy36;>g*32en%<|%!srye )<3`H

fN>3w5#gABOPMB143`H31?x5=2=BIHJIB3iKC>;>gJHJI4AB;q7)Cyg)L#H*LB5H

f(n+1) ≥ f(n)iABE5<1R1n ∈ ekL23<;¢<H*L23<EJ3xIG7w5qEJ3<561>g*AB;27HJ5<;DH C ≥ 0

7)Cyg)LH*LB56Hf(n) = C log niABEw561G1

n ∈ e 0 ; 3g1K56IKOÞH*LB5H,2IRi

f5<;>=

g58EJ3HA!iKC>;>gJHJI4AB;277)56HJIG7)iR?DIK;>¤xH*L23@gAB;>=BIHJI4AB;27`H*L23,?OC>7Hw795HJIK7*iR?

f(n)g(2) = f(2)g(n) ∀n ∈ e Ü &`Ý£t;M258EHJI4gC21G58E9>H*LDIK7QADC214=UN23HEJCy3iABE

g(n) = log neqkL23<;¢¢ADC>Eg1G5<IGOÓIKOM21RI4367H*LB56H

f(n) = C log nIRH*L

C = f(2)/ log 258;>=¬7,AUH*L23H*L23<ABEJ3<O IK7M>EJA6B38=be¢LDC>7`3:AB;B1 ?;>3<38=Pg)L23<g)!Ü &`Ý9e

tuJvlw0x y z//|c© 1~J~J t+10lJy 1B6|J/|<y $1+1u0<y <| w

)<3H

n ∈ 58;>=k ∈ e )<3H#C>7HJ5883

l ∈ 7)Cyg)LPH*LB56H2l−1 ≤ nk < 2l e¢L236;

(l − 1)f(2) ≤ kf(n) ≤ lf(2)eh¢L23U7)58OP3SIK;>3<ÒBC>561GIH?IK7HEJCy3SiABEqH*L23iKC>;>gJHJI4AB;

g2[3gJ58;x:EJIH3QIHb587 −lg(2) ≤ −kg(n) ≤ −(l−1)g(2)

ewC21RHJIGM21 ?BIG;>¤H*L23@>E7*H3*ÙM2EJ3<7)7)IKAB;N<?g(2)

>H*L23!7938gAB;>=N<?f(2)

k5=2=BIG;>¤:H*L236Ok5<;>==DIDI4=DIK;>¤N<?k83:¤63H

−1

kf(2)g(2) ≤ f(n)g(2) − f(2)g(n) ≤ 1

kf(2)g(2)

e6IK;>g3PH*LDIK7@LB58M2My36;27iABE:3,B36Et?

k ∈ 38ÒDC>56HJIKAB;hÜ7&[ÝiAD1G1KA8:75<;>=¬H*L23UM>EJA2ADiwIG7g*ABOM21K3HJ3>efw7Q5@gAB;27,3<ÒBCy36;>g*3DB1K3HC>7367*H5ND1GIG7,LH*L23iAD1R14A8IG;>¤¿EJ367)C21H5

n%<|%!2e)<3Hf :

→ ∪ 0 N23@58;PIK;>g*EJ36587*IK;>¤5<;>=Ug*ABOM21K3HJ361 ?S5=2=BIHJIB3iKC>;>gJHJI4AB;/e¢L236;¢fIK7wH*L23qÑ)3<EJA@iKC>;>gJHJI4AB;/e

0 ; )<3H#C>7q7*C>M>M>AB7,3H*LB56HfIK7q;>AHFH*L23@Ñ)36EJASiKC>;>gJHJI4AB;/e B?S¢L23<ABEJ3<O &DH*L23<EJ3UIK7@5EJ365<1QgAB;27*H58;DH

C > 07)Cyg)L¬H*LB56H

f(n) = C log niABE:561G1

n ∈ e£ifH58367!AB;B1 ?IK;DHJ38¤636ExD5<1RCy3<7`H*L23<;

f(n)/f(m) = (log n)/(log m)iABE561G1IK;DHJ38¤636E7

nm ≥ 2

eq¢L236EJ36iABEJ3D(log n)/(log m) = a/b

IRH*Lab ∈ eq¢LDIG7IKOM21RI4367H*LB56H

nb = ma e >CBHQH*LDIK7xIK7xIGOPM>AB797*I4ND143IGi n5<;>=

mLB5`B3;>AgABOPOPAB;M>EIKOP3i45gJHJABEè

¸ '7 $0· e/B?C>7)IG;>¤ BEJ=! AB7` 7kL238ABEJ36Ob3P79383H*LB56HH*L23S795<OS3PM2EJA2ADiw7,36EtB3<7¿HJA3<7HJ58NB1RIK79L@H*L23gABEEJ3679M>AB;>=BIG;>¤EJ367)C21HiABE5=2=BIHJIB3¿iKC>;>gJHJI4AB;276eBABEwgABOPMB143`H36;>3<7)7143`HC>7Q79L2A8H*LB5H,8IRH*L2ADCBHC>7)IG;>¤q¢L23<ABEJ3<O &BD58;>AH*L236EM>EJA2ADibADikkL238ABEJ36O gJ58;PN23¤Ì B3<;/e )<3HC>7#N>3<¤ÌK;@IH*L

u< ye )<3`HfN>3¬58; IG;>g*EJ3<5<7)IG;>¤5<;>=Ög*ABOM21K3HJ361 ? 5=2=BIHJIB3iKC>;>gJHJI4AB;/ekL23<;¢

f(1) = 0e@ÛABEJ38A6B3<E9yIGi

fIK7x;>AHFH*L23Ñ)36EJAPiKC>;>gJHJI4AB;¢yIRH#7)56HJIG7)y3<7

f(n) > 0iABE5<1R1

n > 125<;>=

fIK7Q7HEJIKgJHJ1?@IK;>g*EJ36587*IK;>¤Be

0 ; 3LB5`B3f(n) = f(1 · n) = f(1) + f(n)

>5<;>=H*LDC>7f(1) = 0

e#£ifIK7;>AHFH*L23Ñ)36EJAPiKC>;>gJHJIKAB;¢>H*L23<EJ3!3)Ù<IK7HJ7

a ∈ 7*Cyg)LH*LB56Hf(a) 6= 0

e¿6IG;>g*3a ≥ 1

IRHiAD1G1KA8:7wH*LB5Hf(a) ≥ f(1) = 0

2BL23<;>g3Bf(a) > 0

e#©qA82¤Ì B3<;n > 1

H*L23<EJ33)Ù<IK7HJ7k7)Cyg)LH*LB56H

nk > a2>H*L236;

kf(n) = f(nk) ≥ f(a) > 0k58;>=3LB5`B3

f(n) > 0e

IG;25<1R1?k143`H¿C>77*C>M>M>AB7,3H*LB5Hf(n) = f(m)

IH*Ln < m

ekLDIK7:IG7;>AHMyAB7)7)IKNB1K3@IGin = 1

bN238gJ5<C>793f(1) = 0

58;>=f(m) > 0

2yH*LDC>7>3!O5?U587)7)C>OP3H*LB56H1 < n < m

e )<3HbC>7HJ5883k1K5<EJ¤63q36;>ADCy¤6L:7)Cyg)LqH*LB56H

nk+1 < mk ÜKIH7*C Ug3<7HA@H583k > (log n)/(log m − log n)

Ý9e¢L236;f(nk) = kf(n) = kf(m) = f(mk)

e¡bAB;27938ÒDCy3<;DHJ1 ?¢

f(r) = f(nk) = f(mk)iABEQ3,B36Et?

r7)Cyg)L!H*LB56H

nk ≤ r ≤ mk e£t; M258EHJI4gC21G58E9f(nk+1) = f(nk)

w5<;>=Û7,A(k + 1)f(n) = kf(n)

LDIKg)LIK7IKOMyAB7)7)IKNB1K3BBN>3<g56C>7,3f(n) > 0

eª#7)IG;>¤xH*LDIK7F1436OPO52<3¤63`HH*L23iAD1R14A8IG;>¤Be

86' # 8!0 ; ; -'$0 ' BABEF3,B36Et?

n ∈ DH*L23<EJ33)Ù<IK7HJ7#5<;IK;DHJ38¤636E k7)Cyg)LH*LB56H

2k+1 − 2k = 2k > ne)<3`HFC>7#H583

nIG;DH36EOP38=DIK5H3@;BC>OPN>36E7

riN23H3<3<;

2k 5<;>= 2k+1 28H*LB5HIK7`2k < r1 < r2 < · · · < rn < 2k+1 e

©qA81K3HC>77*C>M>M>AB7,3SH*LB5HfIK7!;>AH¿H*L23UÑ)36EJAiKC>;>gJHJIKAB;/e B?H*L23U1K3<OOP5B

fIG77*HEI4gJHJ1 ?!IK;>g*EJ36587*IK;>¤LDIKg)LIGOPMB1GIK3<7H*LB56H

kf(2) = f(2k) < f(r1) < f(r2) < · · · < f(rn) < f(2k+1)

= (k + 1)f(2) = kf(2) + f(2)e

¢L236;¢kN<?H*L23M2IK¤638AB;>L2AD1K3@M2EJIG;>gIKMB143DD3@LB5`B3f(2) > n

e >CBHwH*LDIG7xLB58M2My36;27iABE3,B3<Et?n ∈ 8LDIKg)L!IK7Q58N27*C>EJ=/e

$H!"&5 & 0keDEJ= AB7 Q;H*L23=DIK7HEJIKNBCBHJIKAB; ADiQ5=2=BIHJIB3UiKC>;>gJHJIKAB;27 91©° ; 6.-2°Ü7&<BÝ &? $ e 0ke DEJ= AB7b58;>=#6e6C>E%$58;?DI 8 33(/.-' -'$0+ ; $ )'70? 6<M2EJIG;>¤63<E9 e £*e BeB«kCBÑ*795B DEJ= AB7/58;>=xH*L23£t;DH3<¤63<E7` » ° )'0-!'0 + Ü7&<8Ý &9& &< e9AB79$3)C2IG7wfw;279ABEJ3<;25 *C>5<; )C2IK758EJAB;25Q3<M258EHJ5<OS36;DHA=>3!h56HJ3<O#$56HJIKg5<7 Q3<M258EHJ5<OS36;DHA=23h5H36O#$5HJI4gJ587?P¡¢ABOPMBCBHJ58gI $AB; ?P¡¢ABOPMBCBHJ58gI $AB;ª#;BIB36E7)IK=258=P=>3)65P«kI4A915 ª#;BIB36E7)IK=258=P=>3)65P«kI4A915¡¢561G1K3 )C2IG7#=>3:ªQ1G1KAB5:7<; ¡k5<1R143)C2IG7#=>3ªQ1R14AB57<; )<A2¤EJA;>AkB<M>56IK; )<A2¤EJA;>Ak2<M>56IK;[email protected] [email protected]

http://www.unirioja.es/dptos/dmc/jvarona/welcome.html

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)<3HABC

N23¿5qHEJIG58;>¤`1K3IRH*L@7)IK=>367a ≥ b ≥ c

ABM>M>AB7)IH3xH*L23¿5<;>¤`14367AB

C¢EJ3679M>38gJHJI B361 ?/e )<3H

AHN23!My36EM>3<;>=DI4gC21G58EQHJAPH*L23!7*I4=23

BCIRH*L

HAB;

BCe83H

m = BH58;>=

n = CHeF0>EJA<B3qH*LB5H

a(bm + cn) − bc(b + c)IK7FMyAB7*IRHJI B3B;>38¤5HJIB3D>ABEÑ)36EJA5gg*ABEJ=DIK;>¤¿587

∠AIG7#A2NDHJC>793BB58gCBH3D2ABEEJIK¤6LH*)5<;>¤`143<=/e

× ÌqÎ Ð e 0 8 $'#*)+ 3-'5k¹ &'77¸ 6'7 $0! ' e)<3H

u1u2B58;>=

u3N23:5<;?EJ365<1y;BC>OSN23<E76e#0>EJA6B3¿H*LB5H

1

6

3∑

i=1

[cos2(ui − ui+1) + cos2(ui + ui+1)

]

≥ (cos u1 cos u2 cos u3)2 + (sin u1 sin u2 sin u3)

2 L23<EJ3¿H*L237)CyNB7,g*EIKMDHJ7FIK;!H*L237)C>OOP5HJI4AB;5<EJ3¿H5836;OSA2=DC214A3e

× ÌqÎqÌ e 0 8 $'#*)+ 3-'5k¹ &'77¸ 6'7 $0! ' e)<3H S N23q5q7,3`HADi>g*ABOM21K3)Ù 2×2

O56HEI4g3<77*Cyg)LH*LB56H,<iABE5<1R1ABC ∈ S 3LB5`B3

ABCAB = Ce

ÜK5Ý8L2A8 H*LB5H(ABC)n = AnBnCn iABE561G1MyAB7*IRHJI B3¬IK;DHJ38¤636E7

n58;>=Ö5<1R1OP5HEJIKg*367

ABC ∈ S eÜN8ÝPI B35<;P3*Ù<58OM21K3:ADi¢7*Cyg)L57,3`H S gAB;DHJ56IK;BIG;>¤¿5H1K3<5<7*HH*LBEJ383OP5HEJIKg*367IH*LHAADi>H*L23<O ;>AB;>9gABOPOCBHJIK;>¤Be

× ÌqÎ × e 0 8 $'#*)+ -.+¹ '5'77¹0?7µ77¸ $= e0>EJA<B3qH*LB56Hloge(e

π − 1) loge(eπ + 1) + logπ(πe − 1) logπ(πe + 1) < e2 + π2 e

× ÌqÎ Ú e 0 8 $'#*)+1ºs° 5 ! ; -!'5>>6q·$7º(0 '$' ' e

)<3H C N>3U5g*AB;B3*Ùk¤`C>EJ3SIK;H*L23UM21G58;>32ef #$ "'.0?? -0 #AB

ADi CM>5<E5<1R1431yHA!H*L23!=DIKEJ3<gJHJI4AB;!B38gJHJABE −→v IG75!g)L2ABEJ=UADi C ADibOP5[Ù8IGOP561k1K3<;>¤[H*LPM258E561G1K361HA@H*L23:=BIGEJ38gJHJIKAB;@B38gJHJABE −→v e0>EJA<B3wH*LB5HkIGiD3,B3<Et?=DIK5<OS3`HEJIKg561g)L2ABEJ=ADi C NBIG7,3<gJHJ7¢H*L23Q5<EJ3<536;>g14AB7938=¿N<? C H*L23<; C OC>7HFN23g*36;DHEJA2)7?BOPOP3HEI4ge× ÌqÎqÎ e 0 8 $'#*)+1ºs° 5 ! ; -!'5>>6q·$7º(0 '$' ' e0>EJA<B3H*LB56H,5<7H*L23¿M>ADIK;DH7ABCOPA<B3¿A6B3<E/H*L23¿7*C>EJi458g*3¿ADiy5<;31G1RIKM27,ADIK=g*36;DHEJ38=P5H

OLDIG1K3¿H*L231RIK;>367

OAOB

OC

7HJ5`?OCBHJC>5<1R1?My36EM>3<;>=DI4gC21G58E9BH*L23M21G58;>3ABC

EJ3<O5<IG;27FH58;>¤636;DHHJA5DÙ83<=7)MyL236EJ3>e× ÌqÎ

®e 0 8 $'#*)+ =3µB !'2¼+k¹14¸(78 3# ´ ef )3 '!.0?3 ÒDC>5=BEJIR1K5H36E5<1>ÜK5<1G7,Ag561G1K38=5 -0 # 4='! ÒDC>5=BEJIR1K5H36E5<1 Ý¢IK75ÒBC>58=2EIG1G56HJ3<E561DH*LB5H/IK77)IGOC21HJ5<;>38ADC>7*1?IK;279g*EJIKN>3<=:IG;!AB;>3qgIGEJg14358;>=@gIGEJgC>OP79g*EJIKN>3<=5N2ADCBHF5<;>AH*L23<E[e)<3H

ABCDN23!5PNDI4g3<;DHEI4gqÒDC>5=BEJIR1K5H36E5<1bIK;PLDIKg)LPH*L23<EJ3!58EJ3!;>AUM258E561G1K3617)IK=>367<ex6C>M>M>AB7,3:H*LB5HFH*L23!gIKEJgC>O7,g*EI4N238=S5<;>=SIG;27,g*EI4N238=UgIGEJg14367qADi

ABCDLB5`B3g*36;DHEJ3<7

O5<;>=

IEJ3679M>38gJHJI B361 ?/e )<3H

ACOP383`H

BD56H

Ee9ADIG;H*L23PM>ADIK;DH7ADiHJ5<;>¤63<;>g?AB;@H*L23ABM2MyAB7*IRHJ3¿7)IK=>367QADi2H*L23ÒBC>58=2EIG1G56HJ3<E5618H*LDC>7QA2NDH5<IG;BIK;>¤#HA@1RIK;>367LDI4g)L!IK;DHJ3<E7938gJHw5Hw5:MyADIG;DH

Te0>EJA<B3H*LB56H

OET>5<;>=

I5<EJ3!gAD1G1RIK;>3658E[e L236;U=>AH*L23MyADIG;DHJ7

E5<;>=

Tg*ADIG;>gI4=238Ô!Ü¡¢ABOPM258EJ3 ( " B= 2 $$D < e Ý× ÌqÎ e 0 8 $'#*)+¹1 & $#>´(6Cµ'0 3.+ ; ¶4$0+7¶&4$0 +7z¹ e

IG;>=PH*L23!;BC>OPN>36ExADiABEJ=23<EJ3<=UM>56IKE7(A, B)

ADi7)CyNB7,3`HJ7qADi 12. . .

137)Cyg)L@H*LB56H |A ∪ B| IG7#3,B3<;/e× ÌqÎ ? e 0 8 $'#*)+ 8 '0¶+´ $ 0?>µ763' =-!&'!$+7'0?& =0·'7+ e

)<3HABC

N235HEJIG58;>¤`1K3IH*L∠ABC = 80 e )<3`H

BDN23H*L235<;>¤`143NBIG7,3<gJHABE:ADi

∠ABCIH*L

DAB;

ACe £i

AD = DB + BC=>3`H36EOIK;>3

∠AC>7)IG;>¤5:MBC>EJ361 ?P¤638ABOP3HEI4gF5<EJ¤`C>OS36;DH`e

× ÌqÎ e 0 8 $'#*)+ '&µ'7»$0?¼?½4$0 ')$7¶0 6 e£t7¿IHxMyAB7)7)IKNB1K3@HASy;>=5PgCyNBIKgqMyAD1 ?2;>ABOIK561P7*Cyg)LUH*LB5H,¢iABEx5<;?MyAB7*IRHJI B3IK;DHJ38¤636E

nDH*L23:M>AD1?B;>ABOIG5<1

P P · · · P︸︷︷︸

n § 3 685LB5873*Ù<5gJHJ1 ?

3n =BIG7*HJIG;>gJHFEJ365<1kEJA2AH7,ÔIG;>=PAB;>3BDIRi¢M>AB797*I4ND143D2ABE79L2A8H*LB5HF;>AB;>3:3)Ù<IK7HJ76e× Ì®Í e 0 8 $'#*)+0?4& :x¹ =-10 '5x¸ 6 e)<3H

abN>3¿=BIG7*HJIG;>gJHM>AB7)IHJIB3xEJ3<5612;BC>OPN>36E7F7)Cyg)L:H*LB5H

(a − 1)(b − 1) ≥ 0e0>EJA<B3qH*LB56H

ab + ba ≥ 1 + ab + (1 − a)(1 − b) · min1ab e

8× Ì® Ð e 0 8 $'# )$+ cµ53#$ 1$0 #&16#1'7(! '73'70 3-=4$R´ =>µ'70? +

! 5$ .*´ e¢L23 8I4N2AB;25ggI;BC>OSN23<E7

Fn58;>= )CygJ587;BC>OSN23<E7

Ln5<EJ3S=236>;>38=N<?H*L23iAD1G1KA8IK;>¤qEJ38gC>EJEJ3<;>g3<7

F0 = 0

F1 = 1 5<;>=

Fn+1 = Fn + Fn−1 iABE

n ≥ 12

L0 = 2

L1 = 1 5<;>=

Ln+1 = Ln + Ln−1 iABE

n ≥ 1e

0>EJA<B3qH*LB56H∞∑

n=1

arctan(

1

L2n

)

arctan

(1

L2n+2

)

arctan

(1

F2n+1

) ≤ 4

πarctan(β)

(

arctan(β) +1

3

)

L23<EJ3β = 1

2

(√5 − 1

) e× Ì®Ì e 0 8 $'# )$+ cµ53#$ 1$0 #&16#1'7(! '73'70 3-=4$R´ =>µ'70? +

! 5$ .*´ e)<3H

mN2358;UIK;DHJ38¤636E,

m ≥ 2b5<;>=U143`H

a1a2. . .

am

N>3M>AB7)IHJIB3!EJ365<1;BC>OSN23<E76e 8D561GC>5H3qH*L231RIKOIRHLm = lim

n→∞

1

nm

∫ e

1

m∏

k=1

ln(1 + akxn

)dxe

e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e× ÌqÌ Ð e¡bABEJEJ38gJHJIKAB;/e 0 8 $'/8!$0*»$¹ $ ¸ "'0 $0 6' ´(6Cµ'0 3 '#1' 6 =#1&# 6 =#1&#7±$ 8!4$' e

8ADIRHABC

C>;HEIK5<;>¤`143=23g('AH($3<7a ≥ b ≥ c

ABM>M>AB79$367#EJ3679M>38gJHJI B3<OP3<;DHw5<CÙ58;>¤`1K3<7AB3H

Cey8ADIRH

AH1K5@M>3<EJMy36;>=BIKgC21K56IKEJ35<Cg('AH%$3

BC5`B38g

H7)C>E

BCe0BAB7,AB;27

m = BH3H

n = CHeÛAB;DHEJ3<E¿ÒDCy3

a(bm + cn) − bc(b + c)3<7HMyAB7*IRHJIRiB; $3<¤56HJIRibADCP;BC21B7)C2I D58;DHFÒDCy31t 5<;>¤`143

A367*HwA2NDHJC>7`B5<IK¤`CSADCS=2EJADIH`e

× ÌqÎ Ð e 0 8 $' 8!0 -!':x¹6& 3'57¸ ='6 $0! ' e8ADIRH

u1u23`H

u3HEJADIK7Q;>ABOPN2EJ367#E%$3<361G7Q58ENBIHE5<IGEJ3<76e¢ÛAB;DHEJ36EFÒBCy3

1

6

3∑

i=1

[cos2(ui − ui+1) + cos2(ui + ui+1)

]

≥ (cos u1 cos u2 cos u3)2 + (sin u1 sin u2 sin u3)

2 A "Cb2=25<;27Q1G579ABOPO56HJIKAB;¢14367FIK;>=DI4g3<7Q7,AB;DHFgJ5<1KgC21+$367QOSA2=DC214A

3e

$$

× ÌqÎqÌ e 0 8 $' 8!0 -!':x¹6& 3'57¸ ='6 $0! ' e8ADIRH S C>;3<;2793<OPNB1K3=>3!OP5HEJIKg*3672 × 2

gABOPMB143*Ù83<7H31G1K3<7qÒBCy3M>ADC>EQHJADCBHAB3H

C ∈ S 2AB;56IRH ABCAB = CeÜK5ÝUÛAB;DHEJ3<EÒBCy3

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AB3`H

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IK7IK79AB7,g361K3<7`3¿LB5`B3∠ABC = ∠BFD

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e

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DA′ = r/√

3e EJABOÓÜ Ý*83:LB5B3

2

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A

A′

B CD

E

F

P

Γ

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√3/2)r = 1

2DA

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o X-pts u!"#N,8O8$6#N,7u(,G570v#% 5M91O7,I.1#')D24H,70.(wY?2x$%0#N5I$-) FC $-#%')O8$-H,/0 . s 5G5I$-E$-Fs 5G,7.COI.1Fy s FK"yzQ|=0 .v&w=0L-,I&~:4?0 .v$#FyCKtR0ODPt.v$ .CO70 F3Pt.4$.vO/0FDR1FK"yz

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iA\fk ≥ 4

jbhX%db

5m + 3m ≡ 0 (mod 16)

D\jb]dbf ` h1bYZ/bX[e`lV<\V<a V<bc%X%ÙdbDfb[NU7W^b[\i5m \1W^1Z7\

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3mXfb3911

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$

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k = 1H*L23C>;BIKÒBCy37,AD1RCBHJI4AB;hiABE

n3,B3<;ÛIG7

m = n = 05<;>=

k = 1e

)<3HnN>3A2=2=/eBABE

m = 0>3!LB5B3HJAS>;>=U7,AD1RCBHJI4AB;27#HJA

1 + 3n = 2k e:A83,B36E,*)<3<A :3<N2EJ3C>7#Ü4ABE )<3,BIyN23<;3<E79AB;¢ &<§R¨xg3<;DHJC>Et?8ÝM2EJA<B3<=@H*LB56HiABE5<1R1n > 2

DH*L23:IK;DHJ38¤636E3n ± 1

LB5<75<;UA2=2=S=BI BIG7,ABE>2>L236;>g*3DBH*L23:C>;BI4ÒDCy3@7,AD1RCBHJI4AB;PADi1 + 3n = 2k iABE m = 0

58;>=nA2=2=IG7

n = 158;>=

k = 2e©qA83!587)7)C>OP3H*LB56H

m > 0eB?g*AB;27*I4=23<EIK;>¤:H*L2338ÒDC>56HJIKAB;OSA2=DC214A

33A2NDH5<IG;

(−1)m ≡ (−1)k (mod 3)2LDI4g)LPIKOM21RI4367H*LB5H

m58;>=

kLB5`B3@H*L23795<OS3M>5<EJIH?be Q;H*L23PAH*L236EqLB5<;>=¢N<?¬3*Ù<58OIK;BIG;>¤@H*L23P3<ÒBC>5HJI4AB;OSA2=DC214A

5k3¤63H

2k ≡ (−2)n ≡ −2n ≡ ±2 (mod 5)

7)IG;>g*3nIK7#A2=2=/e¢LDIG7QIGOPMB1GIK3<7wH*LB5H

kIG7#A2=>=Ü45<;>=!H*L23<;7,A@IK7

mÝ9e©qA87)C>M2MyAB793QH*LB5H

m ≥ 358;>=

n ≥ 3ÜLDI4g)LOS3658;27H*LB5H

k ≥ 7Ý9e83`HHJIK;>¤

A = 22276800 = 26 · 32 · 52 · 7 · 13 · 17 <3g)L238g)83<=N<?5:gABOPMBCBH36EM>EJA2¤E5<OH*LB56H#H*L23<EJ35<EJ3;>A79AD1GCBHJIKAB;27ADi5m + 3n = 2k OSA2=DC214A A

iABEqA2=2=¬3*ÙM>AB;>3<;DH7m ≥ 3

n ≥ 3

D58;>=k > 7

ÜGH*L23g)L238g)<IK;>¤xIK7F5 9>;BIRHJ3M2EJA2NB1K3<OB7*IK;>g3

551 ≡ 53

(mod A)3243 ≡ 33 (mod A)

58;>=2127 ≡ 27 (mod A)

Ý)e¬¢L236EJ36iABEJ3D3OC>7HFLB5B33IRH*L236Em = 1

ABEn = 1

e)<3H

n = 1e 3FOC>7Hk14A2A2FiABEk7,AD1RCBHJI4AB;27/ADi

5m+3 = 2k ÜRH*LDIK7/M>EJA2ND1436O:5<7M>EJABM>AB7,3<=AB;!H*L23>pp#££<M258;BIG7,LSh56H*L236OP5HJI4gJ5<1* F1 ?2OM2IG5=8Ý9eª#7*IK;>¤H*L23OSA2=DC21GC>7B = 65792 = 28 · 257

¢3P5¤56IK;C>7,3<=5UgABOPMBCBH36ExHJA¬793<5<EJg)LiABEq79AD1GCBHJIKAB;27OSA2=DC214ABÜK5¤56IK;¿H*L23xg)L238g)<IK;>¤QIG75 9>;BIRHJ3

M2EJA2NB1K3<O87*IK;>g3

5256 ≡ 1 (mod B)58;>=225 ≡ 29 (mod B)

2<iKC>EH*L236EOPABEJ3B9IG7H*L23q7)OP561G1K3<7HM>A83<EADi

2L236EJ3xH*L23EJ3<O5<IG;>=>36E7qOSA2=DC214A

BN>3<¤ÌK;PHAPEJ3<M>3<5HÝ9ex¢L23gABOPMBCBH36E#M>EJA2¤E5<O¦?BIK361K=>3<=PH*L23iAD1G1KA8IK;>¤xiADC>EFgJ58793<7FiABEn = 1

58;>=m > 0

(m, k) ∈ (1, 3)(3, 7) e6IK;>g3SH*L23PD561GCy367@iABE

k1RI43SIG;H*L23;>AB;>)M>3<EI4A2=DI4g793H:ADiQEJ3<O5<IG;>=>36E7ADiQM>A83<E7ADi

2OPA2=BC21KA

Bb3S79383H*LB56H

k = 1ABE

k = 7e¢LDIG7@¤ÌB367:C>7H*L23S79AD1GCBHJIKAB;27

(m, n, k) = (1, 1, 3)58;>=

(m, n, k) = (3, 1, 7)e 8C>EH*L236EOPABEJ3B¢5<;?AH*LB3<E#79A21GCBHJIKAB;27#OC>7HQLB5`B3

m ≡ 1 (mod B)ABE

m ≡ 3 (mod B)ew6IK;>g3H*L23:79O5<1R14367*HD5<1RCy3<7QiABE

mAH*L236EH*LB5<;

1ABE

35<EJ37)IK¤;BIGyg5<;DHJ1?!HA2A!1K5<EJ¤63HJAPLB5B357,AD1RCBHJI4AB;¢H*L23<793:5<EJ3¿H*L23:AB;B1?7,AD1RCBHJI4AB;27wiABEn = 1

e)6587HJ1?k<3#IG1R1y3*Ù<58OIK;>3

m = 15<;>=

n ≥ 3e/¢LDIG7HJIKOP3B63qC>7,3xH*L23¿OPA2=>C21GC>7

C = 26 ·34 ·17 iABEADC>Eg*ABOM2CBHJ3<Eg)L23<g)ke, Q;>g3xOSABEJ3#H*LDIG7wN238gABOS3675y;BIH3M>EJA2ND1436OÈ7)IG;>g*3321 ≡ 35 (mod C)

58;>=2223 ≡ 27 (mod C)

2>iKC>EH*L23<EJOSABEJ3D558;>=

758EJ3#H*L23x7)OP561G1K3<7HM>A83<E7wADi

35<;>=

2EJ3679M>38gJHJI B361 ?¢L236EJ3#H*L23xEJ3<O5<IG;>=>36E7N>3<¤ÌK;¬HA¬EJ3<M>3<5Hè¬kL23AB;B1 ?7,AD1RCBHJI4AB;OPA2=BC21KA

CH*LB56HxH*L23UM2EJA2¤E58O ¤63<;>36E56HJ38=:587

(n, k) = (3, 5)ekLDIK7w?BIK361K=27wH*L2379AD1GCBHJIKAB;

(m, n, k) = (1, 3, 5)eF6IK;>g3H*L23

$MyA836E7@AB;N>AH*L

258;>=

35<EJ3IK;H*L23P;>AB;>)M>3<EI4A2=DI4g¿7,3`HADiEJ3<O5<IG;>=>36E7@ADiH*L23IKEEJ3<7)My3<gJHJIB3MyA836E7H*L236EJ3:5<EJ3:;>A!iKC>EH*L23<Ew7,AD1RCBHJI4AB;276e£t;@g*AB;>g1RC>7)IKAB;¢H*L23<EJ358EJ3x3*Ù<5gJHJ1 ?BB379AD1GCBHJIKAB;27/HA

5m +3n = 2k <;25<OS31?!(m, n, k) ∈ (0, 0, 1)

(0, 1, 2)

(1, 1, 3)

(3, 1, 7)

(1, 3, 5) e

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

+ "DM M % M N- DM ! &M NM" " C !#

6FM M CX DM DM % " #M !%DM% IW "# S FX W DM " # S % " " "" % M &M NM " " ! #

DH4x q 1N!"#-$&')> ,v$]5S240O/0%,G5_58$F"9$-.1F H0. $

b%`εr bl` h1b=b%ZGZ_U7m1[]b x2

a2+

y2

b2−1 = 0

C j\m<X fHX Z_Z/b%Z`HXV<cbV`[C`\εUGVèbfH[]b `

XY` hUGfW`X V<cb V`lX`M1(x1, y1)

X V<WM2(x2, y2)

h1\j ` hX%`t` h1bdXZG^bD\i(

x21

a2+

y21

b2− 1

)(x2

2

a2+

y22

b2− 1

)

U7[SUGV<W<bmbV<W<bV`=\i<` h1bNh1\[b V`HXV<cbV`[

y 5M91O7,I.') ? z? *!10 #NF#"$v$-.COKC.,IL $%M#N,8OA)F%"$4$%.vO/Ft2x$]57E%,H9 u h1bW<b[NUGfbW fb [ ^1Z_`YU7[Xe\V1[]b n^bV< b\im<f\m<b fMÙ/b[+\im<f\&Ib Ùdb \1\fea

WUGV1X`b [UGVèbfem1fb%`bWU7V` h1b>X!'V<bm1ZGXV<b )( ^<fD \VU7W<b%V<b[3X #M0570g!vN1&49*Or b-`MjbbV(` h1bYm\UGV`[

M1 = (x1, y1)X V<W

M2 = (x2, y2)r

〈M1, M2〉 =x1x2

a2+

y1y2

b2− 1

h1bfbU7[QZ_U,+bjYU7[bX[eXZ7X f m<f\1W^`lW1b<V<bW r ` h1bW^<XZe\VU/ o \ m\[bW\i1` h1b`X V<cb V`H[`\>` h1b \VU7Hp r b%`Mjb b Vm<XU7fH[D\i Z_U7V<b[ U_i

Li = [ui, vi, wi]fb m1fb [b V`

` h1bYZGUGV<b [uix + viy + wi = 0

iA\fi = 1

XV<Wi = 2

` h1b V

[L1, L2] = a2u1u2 + b2v1v2 − w1w2

- m1X UGf3\i=m\UGV`[(\fXm1X UGf3\iZGUGV<b [Xfb \V.&/^c%X%èbUGi=XV<W\VZ U_i` h1b%U7f[eXZ7X fm<f\1W^`SUG[0/Nb f\ 0( V<bb X [NU_Z+[h1\jD[S` hX%`Q` h1bDZGUGV<b1&I\U7VUGV<c

M1`\

M2UG[S`X V<cb V`

è\3` h1bD \VU7lUGi4XV<W\VZ (U_i

〈M1, M1〉〈M2, M2〉 − 〈M1, M2〉2 = 0 o p

$ Ü83<3B¢iABEq3*Ù<58OM21K3B,eR>e4e/¡bA`Ù83`H36E, -!'1¸ '7 0 .³ '>>µ' $' EJ=38=DIRHJIKAB;¢iABEOC21K5 & e @AB;M25¤63 & DiABEH*L23:=>3`HJ56IG1G7<e Ý£t;H*LDIK7;>AH56HJIKAB;¢k35<EJ3PEJ3<ÒBC2IGEJ38=HJA7,L2A8H*LB56H 〈M1, M1〉〈M2, M2〉 IG7IK;>=23<M>3<;>=23<;DHADiBH*L23¿g)L2AB793<;:HJ5<;>¤63<;DH7<e 3#IR1G1B79L2A8H*LB5H,8iABE561G12g)L2ADIKg*367FADiBH*L23H*LBEJ383xHJ5<;>¤63<;DH7 〈M1, M1〉〈M2, M2〉 = 1

e£t;BIK3`&ADikÜ7&[Ý58N>A6B3B<3LB5`B3AB;B1?HA79L2A8H*LB5H 〈M1, M2〉2 = 1e¡bAB;27*I4=23<EhH*L23&M>5<E5<1R14314A2¤E5<O iABEJOS3<=¦N<?H*L23¤ÌB36;¦M258E561G1K361HJ5<;>¤63<;DH7HA2¤63`H*L23<EFIRH*LPH*L23!H*LDIGEJ=UH58;>¤636;DHq5<;>=UH*L23@HJ5<;>¤63<;DHxM258E561G1K361bHJAUIHè k38gJ5<C>793!H*L23H*LBEJ383:=DIK58¤6AB;25<1G7QADik58;?ÒBC>58=2EIG1G56HJ3<E561kgIGEJgC>OP79g*EJIKN>3<=5N2ADCBH5gAB;BI4giABEOÞ5:79361RiMyAD1G58E>HEJIG58;>¤`1K3#ÜRH*LDIK7bIK7¢H*L23Q=BC>561ADi8kL238ABEJ36O>e AB;¿M25¤63 ADi<H*L23#¡¢A`Ù83HJ3<E¢N>A2A2gIRHJ38=!58N>A6B3`Ý8H*L23=DIK58¤6AB;25<1G7QADikADC>EM>5<E5<1R14314A2¤E5<O;258OP361 ?

y1x − x1y = 05<;>=

y2x − x2y = 0B58EJ3:g*AB; 14Cy¤5H32ekLDIK7wH31G1G7FC>7FH*LB5H

x1x2

a2+

y1y2

b2= 0

587#=>367)IGEJ38=be3 Hx Q !"#%$ &') %$-#9 #1,G5_5/0 l91' ,IF=y0 .CO70.4&<$%Fy%!v0%,/.Co X%p b%`

r1 < 0 < r2 < r3

r bY` h1b3fbX Z4f\1\`[\i8x3 − 6x +

√3 = 0

f\ db` hX`

r23 = 4r2

2 − 4r42

XV<Wr21 = 4r2

3 − 4r43

oAr p b%`

s1 < 0 < s2 < s3

r b` h1bgfb XZf\1\`[l\i8x3 −6x+1 = 0

f\db` hX%`r21 + s2

2 = 1

s21 + r2

2 = 1 XV<W

r23 + s2

3 = 1

y 5M91O7,I.')+24H,70.wY?C2x$-0 # 58$-)FtN$#')1OI$%H,70. s 5_58$E$F s 5_,/.vO8.FCy s FK"yzo X-p b%`

x = sin θ h1bc-UdbV bn^<XÙ7\V

8x3 − 6x +√

3 = 0 X% r bfb-jDfHU_`H`b V+X[

√3

2= −4x3 + 3x = −4 sin3 θ + 3 sin θ = sin(3θ)

h1bV

3θ = π3

+ 2πk\f

3θ = 2π3

+ 2πk iA\f=X VU7V`bcbf

k b[ èfHU/ÙGV<c

θ`\

` h1bYU7V`b fIdXZ[0, 2π]

jbY<V<W` hX%`

θ ∈

π

9

2π

9

7π

9

8π

9

13π

9

14π

9

h^<[

r1 = sin13π

9= − sin

4π

9

r2 = sin

π

9

X V<Wr3 = sin

2π

9

=[NUGV<cg` h1bYU/W1b VÙ`8

sin2 2θ = 4 sin2 θ − 4 sin4 θ jbDhX-db

r23 = sin2 2π

9= 4 sin2 π

9− 4 sin4 π

9= 4r2

2 − 4r42

XV<Wr21 = sin2 4π

9= 4 sin2 2π

9− 4 sin4 2π

9= 4r2

3 − 4r43

$DÜN8Ý:83H

x = cos θekL23¤`I B3<;Õ3<ÒBC>5HJI4AB;

8x3 − 6x + 1 = 0O5? N>3EJ3`:EIRHHJ3<;587

−1

2= 4x3 − 3x = 4 cos3 θ − 3 cos θ = cos(3θ)

e¢L236;

3θ = 2π3

+ 2πkABE

3θ = 4π3

+ 2πk2iABEw58;?IG;DH3<¤63<E

keq«¢367*HEI4gJHJIG;>¤

θHJAH*L23IK;DHJ3<EtD561

[0, 2π]83>;>=!H*LB5Hθ ∈

2π

9

4π

9

8π

9

10π

9

14π

9

16π

9

e¢LDC>7`

s1 = cos8π

9= − cos

π

9

s2 = cos

4π

9

58;>=s3 = cos

2π

9

e¢L23¿=>367)IGEJ38=@EJ3<7*C21RHiAD1G1KA8:7i4EJABOH*L23@0D?DH*LB58¤6ABEJ3<5<;@I4=23<;DHJIH?

cos2 θ +sin2 θ = 158M2M21RI43<=!HAH*L23:58N>A6B3qD5<1RCy3<7#ADiri5<;>=

sie

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

cos 3θ#

H+4< - N!"#-$&>')(, 4$5 240O/0%,G5_58$F#"9$-.1F H0. $ b%`

x1

x2

. . .

xn

on ≥ 2

p r b3fb XZxV^ r bfH[[ ^Nh+` hX` n∑

i=1

xi = 0X V<W

n∑

i=1

x2i = 1

TUGV<W` h1b UGVU ^ XV<W XU ^ \i n∑

i=1

|xi|

:#N#%$-.CO7,705_5A)O 4$(#0u$#% 5M91O7,I.')y$6$O zSM#N570.10 E%, HF3Kx.,8L $-H# ,MO/)6Sy 0H0 $L Fy0 0N$L Ft2x#.,700 .v&>@$-$-EL,/.10|v0.4& "$ $-BD0, J 019 F @.1E x.1EF s ,7.<01

bhX-db(

n∑

i=1

|xi|)2

=n∑

i=1

x2i +

n∑

i=1

n∑

j=1j 6=i

|xi||xj| ≥ 1 +

∣∣∣∣∣∣

n∑

i=1

n∑

j=1j 6=i

xixj

∣∣∣∣∣∣

= 1 +

∣∣∣∣∣∣

(n∑

i=1

xi

)2

−n∑

i=1

x2i

∣∣∣∣∣∣

= 2

1n^<X Z_U_`Ih1\Z7W1[3iA\fx1 = 1/

√2 = −x2

X V<Wx3 = x4 = · · · = xn = 0

`liA\Z_Z/\jD[Q` hX%`t` h1b UGVU ^ UG[ √

2

$9

BABEqH*L23O5`Ù<IKOC>O7*C>M>M>AB7,3PH*LB5HmADiwH*L23

xi 7@58EJ3U;>AB;>);>3<¤56HJI B35<;>=

n − m5<EJ3S;>AB;>)M>AB7)IHJIB3DiABE

1 ≤ m ≤ ne IRH*L2ADCBHq14AB7)7ADiQ¤636;>3<E561GIH?k/3587)7)C>OP3H*LB56H

x1. . .

xm

5<EJ3F;>AB;>*;>38¤5HJIB3w5<;>=xm+1

. . .

xn58EJ3w;>AB;>)M>AB7)IHJIB32e

B?P8g)L8:5<EÑ[ 7q£t;>3<ÒBC>561GIH?km∑

i=1

x2i ≥ 1

m

(m∑

i=1

xi

)2

58;>=n∑

i=m+1

x2i ≥ 1

n − m

n∑

i=m+1

xi

2

=1

n − m

(m∑

i=1

xi

)2 e£4HwiAD1R14A8:7H*LB5H

1 =n∑

i=1

x2i ≥

(1

m+

1

n − m

)( m∑

i=1

xi

)2

=n

m(n − m)

(m∑

i=1

xi

)2 e:36;>g*3D n∑

i=1

|xi| = 2m∑

i=1

xi ≤ 2√n

√

m(n − m)e

BABE5<;?@EJ365<1B;BC>OSN23<Er<H*L23qÒBC>58=2E5HJI4giKC>;>gJHJIKAB;

x(r −x)LB5875qOP5[Ù8IGOC>O

r2/456H

x = r/2ex¢LDC>7`yiABE

n = 2k2L236EJ3

kIK7x5!MyAB7*IRHJI B3IK;DHJ38¤636E,23!LB5`B3

m(n − m) ≤ k2 5<;>=n∑

i=1

|xi| ≤ 2k√n

=√

ne

2ÒDC>5<1RIRHt?PL2AD1K=27FiABEx1 = x2 = · · · = xk = −xk+1 = · · · = −xn = 1/

√ne

BABEn = 2k + 1

8L23<EJ3kIK7Q5:M>AB7)IHJIB3¿IG;DH3<¤63<E9

m(n − m) ≤ maxk(2k + 1 − k)(k + 1)(2k + 1 − k − 1) = k(k + 1)

e:36;>g*3D

n∑

i=1

|xi| ≤ 2√

k(k + 1)√n

=

√

n − 1

n

e2ÒDC>5<1RIRHt?PL2AD1K=27FiABE

x1 = x2 = · · · = xk =

√

k + 1

k(2k + 1)58;>=xk+1 = · · · = xn = −

√

k

(k + 1)(2k + 1)

e 3gAB;>g1GCy=23¿H*LB5HH*L23OP5[Ù8IGOC>OÞIG7 √

n +(−1)n − 1

2n

e !"U!5= A< + *5 + &5=)(H/ " ! DI)0 N)(H / 1(OG" H!+5*2 L>%*)

" ! '6N-Q),L " XI)6 RA)!"9 1#2'8 O 2 " 5768:9I) @ &9 M 9 " ! DI)E C) @B ()K"9 1S*HC6C,L?5E &%' #"9I)K" CX )T8 NV/) 1TE/ AL?H92,H 9>+L5=;) + " Y)(%'<)&"9 1(O L 6L?5< #(SU 7 () + " ! F)(H!@ " F)=2=)*" 9(1 "DM #*6IM % > #

$ × Ð Î × e $$ , $ k B7 0 8 $7'$#*)$+ 3-'5x¹6& 3'57¸ 6'7 0?$ ' e

BABELDI4g)L!IG;DH3<¤63<E7n=>A2367FH*L23:38ÒDC>56HJIKAB;

3xy − 1

x + y= n

LB5B3:5:7,AD1RCBHJI4AB;!IG;IK;DHJ38¤636E7x5<;>=

yÔ

± '7( .+".-'/>$ "' )$+¸ +¹0:)$0? =')!'5'R´(6Cµ'0 3.+$$$0=')! ¹0?$ ° ¹ '7 3'7+ 0 '5)$+'0?$¶ 3'546'5 ¶= !36 ¶6´ ? ''? /4 64o¶-=4c»':=-6$)'70?4g¹+ #1'7 N´ ?9$# ''0 2°! $x¹1[´ =>µ'70? + 0=#=7¶ ´ ? e

)<3HkN2358;?hIG;DH3<¤63<E[e £i

n = 3kH*L23<;H*L23¬¤ÌB36;Õ3<ÒBC>5HJI4AB; N>3<g*ABOP3<7

3xy − 3k(x + y) = 12LDI4g)LULB587x;>AS7,AD1RCBHJI4AB;276e:£i

n = 3k + 1>H*L236;

x = k58;>=y = −(3k2 + k + 1)

IG7579AD1GCBHJIKAB;/eh£in = 3k − 1

H*L236;x = k

5<;>=y = 3k2 − k +1

IG75¿79AD1GCBHJIKAB;/e3<;>g3B<H*L23¿3<ÒBC>5HJI4AB;!LB5<7w5q7,AD1RCBHJI4AB;:IGi>5<;>=!AB;B1?IGinIG7Q;>AHw5OC21RHJIGM21K3:ADi

3e

!" 2,8NO 2 "(576C8D9F) @ &9 DM 9 " ! DI)AE >)G@B ()"9 12&5 H>LC95#=$ 8:@ 8 %G%,8 L) >X 39 " ! DI)79CX F),6 RA)," 9(1&5 8J2*5 $ 8#=,L>9/9F)5 NM O ,P" Q)=2=) " 9(19&H>L>@ @ &H>8 82C) " )(9 ! 2 YX Y)(9 !F) + " W YX ! I1@ 8N2&L?HO&5(@ L%6L?54)@ A" ! % " "/) 9 #FE/M Q)%,HI1(O7%' =9!L # 6/9 #"9/9 7;(H #"4) M Q) ; .1" DM #

~Q % N!"#-$& ') s 105_5/0 Q?Gy<?/R3?/ ?=y 0. 0 F=zS.4&*<H0 !v0&<$-# 1FI.4&,/01o X%p i

β > 1U7[XfbX Zg \V1[e`X V` W<b-èbf U7V<b` h1bV^ r bf+\im\[N[NU r Z7bfbX Z

[]\Z_^Ù/\V1[\i` h1bb n^<X%Ù/\V

x − β log2 x = β − β ln β

oAr p iα1 < α2

Xfb3`Mj\m<\[NUÙdbfb XZ"[]\Z_^Ù/\V1[\ix` h1bb n^<X%Ù/\VU7V o X-p X V<WUGi

x1X V<W

x2XfbX V(`Mj\fbX Z4V^ r b fH[g[NXÙG[Ni_U7V<c

α1 ≤ x1 < x2 ≤ α2

m<f\db` hX%` iA\flXZGZ

λ[N^Nh3` hX`

0 < λ < 1

λ log2 x1 + (1 − λ) log2 x2 ≥ ln(λx1 + (1 − λ)x2

) kb%`b f UGV<bjh1bVbn^<X Z_U_Ì\1ee^<fH[

y 5M91O7,I. ') y$6$OCzSM#N570.10 E%, HFKx.,8L $-H# ,MO/)6ty0 0N$L Fly 0H0 $L F=2x#].,/0(0.4&@$%$EL-,7.<0FCu1&,C$ &(0 .v&$ !v0 .v&<$ &')+O 4$$&,8O8

o X-p \V1[NU7W<bfQ` h1bYi7^<V<Ù/\Vf(x) = x − β log2 x − β + β ln β

iA\fx > 0

b3hX-db

f ′(x) = 1 − β/(x ln 2) q jh1b V<eb x = β/(x ln 2)

UG[S` h1b\VZ fUÙ/X ZdX Z_^b U7V<eb

f ′(x) < 0iA\f

0 < x < β/ ln 2X V<W

f ′(x) > 0iA\f

x > β/ ln 2

jbD[bb` hX%`f

UG[=W<b fb X [NUGV<cY\V(0, β/ ln 2)

XV<WU7V< fbX[ U7V<cY\V(β/ ln 2, ∞)

h^<[

fhX [=XDfbZGXÙ db UGVU ^ X`

x = β/ ln 2

&

©qAHJ3¿H*LB5Hlim

x→0+f(x) = ∞ 58;>=@H*LB56H

limx→∞

f(x) = −β + β ln β + limx→∞

x(

1 − β log2 x

x

)

= ∞ 7)IG;>g*3

limx→∞

β log2 x

x= 0

e 3:;>A8&7,L2A8 H*LB56Hf(β/ ln 2) < 0

e6IK;>g3f(β/ ln 2) = β/ ln 2 − β log2(β/ ln 2) − β + β ln β

IRH7*C Ug3<7HAS79L2A8H*LB56H1/ ln 2 − log2(β/ ln 2) − 1 + ln β < 0

2LDIKg)LSIG7q38ÒDC2ID561436;DH#IG;7)CyggC>797*I4AB;!HJA1 − (ln 2)

(log2(β/ ln 2)

)− ln 2 + (ln 2)(ln β) < 0

1 − ln 2 − ln(β/ ln 2) + (ln 2)(ln β) < 0

1 − ln 2 − (ln β − ln(ln 2)

)+ (ln 2)(ln β) < 0

1 − ln 2 + ln(ln 2) − (1 − ln 2)(ln β) < 0

LDI4g)LqIK7bHECy3#7*IK;>g3(1−ln 2)(ln β) > 0

5<;>=1−ln 2+ln(ln 2) < 0

eb¢L236EJ36iABEJ3DfLB587 '$6>+ HAEJ365<12EJA2AH7

α158;>=

α27*Cyg)L:H*LB56H

0 < α1 < α2e/¢LB5HIG7<H*L23¤`IB36;S3<ÒBC>5HJI4AB;PLB5<73*Ù<5gJHJ1 ?@HAEJ365<1y7,AD1RCBHJI4AB;276eÜN8ÝEJABO M258EH@ÜK5Ý#3U7,3<3SH*LB5H

f(x) ≤ 0iABE:5<1R1

x ∈ [α1, α2]L236EJ3

0 < α1 < β/ ln 2 < α2e¢LDC>7`

log2 x ≥ (x/β) − 1 + ln βe6IK;>g3

0 < λ < 1DIHiAD1R14A8:7H*LB56H

λ log2 x1 + (1 − λ) log2 x2

≥ λ

(x1

β− 1 + ln β

)

+ (1 − λ)

(x2

β− 1 + ln β

)

=1

β

(λx1 + (1 − λ)x2

)− 1 + ln βe Ü7&[Ý

©q3*Ù<H37,L2A8 H*LB56H,DiABE5<1R1t > 0

t − β + β ln β ≥ β ln t

e Ü Ý)<3H

g(t) = t − β + β ln β − β ln tekL23<;

g′(t) = 1 − β/t7,L2A8IG;>¤H*LB56H

t = βIK7wH*L23@AB;B1 ?Pg*EIRHJIKg5612D561GCy32ew6IK;>g3g′′(t) = β/t2 > 0

83:7,3<3¿H*LB5Hg(β) = 0

IG7#5EJ361G56HJI B3:5<7361R1y5<7FH*L235NB7,AD1RCBH3OIK;BIGOC>O¦ADige :36;>g*3D

g(t) ≥ 0iABE561G1

t > 058;>=PÜ ÝiAD1R14A8:76e£t;PM258EHJI4gC21G58E9DiABEt = λx1 + (1 − λ)x2

83:A2NDH5<IG;λx1 + (1 − λ)x2 − β + ln β ≥ β ln

(λx1 + (1 − λ)x2

) e Ü ÝDEJABO¦Ü7&[Ý58;>=PÜ Ý*83qH*L236;SLB5`B3

λ log2 x1 + (1 − λ) log2 x2 ≥ ln(λx1 + (1 − λ)x2

) e2ÒDC>5<1RIRHt?PA2g*gC>E7FIGi¢5<;>=PAB;B1?!IGi

x1 = α1x2 = α2

25<;>=λα1 + (1 − λ)α2 = β

LDI4g)L@?BIK361K=27λ =

α2 − β

α2 − α1

eQ6IG;>g*3f(β) = β(ln β − log2 β) < 0

D3:7,3<3H*LB5Hα1 < β <

β

ln 2< α2

8LDI4g)L!IG7#g*AB;27*IK7H36;DHIRH*LH*L23:5<797*C>OPMDHJI4AB;0 < λ < 1

e !"UDM #

&9&

× Ð ÎqÎ e $$ , $ k B7 0 8 $7'$#*)$+0?4& 3:x¹ 6-$0 '77¸ $= e£t; 4ABC¢143`H

DEFN>3!H*L23IG;DH36E7,3<gJHJI4AB;27ADiH*L235<1HJIRHJCy=23<7qi4EJABO

A

BCHAH*L237)IK=>367

BCCA

AB

>EJ3<7)My3<gJHJIB31?be )<3HHN>3H*L23!ABEH*L2A2g3<;DHEJ3@ADi

4ABC`1K3H

LN23H*L23wIK;DHJ3<E7938gJHJIKAB;¿ADi

AT5<;>=xH*L231GIG;>3H*LBEJADCy¤6L

BMy36EM>3<;>=DI4gC21G58EHA

BCD58;>=!143`H

TN>3qH*L23IG;DH36E7,3<gJHJI4AB;PADi

BE58;>=

DFe8L2A8H*LB56H

BL = BCIGi¢5<;>=PAB;B1 ?IRi

∠ACB = 45 e6)$+ 3-'5x¹6& 3'57¸ 6'7 0?$ ' e 3:OPA2=BIRiR?@H*L23:EJ38ÒDC2IKEJ36OS36;DHQADiyH*L23M>EJA2ND1436OÞHAN>36

$-² .-BL = BC

; !# 6+/ ;∠ACB = 45 0

∠ACB = 135 $# .-

AT&<.-' 8 $0 ;

C²9.-0 '7 8 '$>

γe©qAHJ3@H*LB5H

BFH¢5<;>=

D1RI43AB;SH*L23gIGEJg143

γIRH*LU=BIG58OP3HJ3<E

BH5<;>=H*LB56H

ATIG7#H*L23!MyAD1G58EIRH*LSEJ3679M>38gJHFHJA

γe 3!gJ5<1R1

L′ H*L23!MyADIG;DHADi/IK;DHJ3<E7938gJHJIKAB;ADiyH*L23:M>3<EJMy36;>=BIKgC21K5<E7QHJABC

56HB58;>=!HJA

AC5H

Ce

IGE7*H,k7)C>M2MyAB793H*LB56H∠ACB = 45 ABE

∠ACB = 135 ekL23<;UC>7*IK;>¤:H*L23i45gJH7QH*LB56HBE ‖ CL′ 58;>=

∠BCL′ = 45 D379383¿H*LB56H 4CBL′ 5<;>= 4BDH58EJ3IK79AB7,g361K3<7QEI4¤6LHHEJIG58;>¤`1K3<7`IH*LEI4¤6LHF58;>¤`1K3<7#5HB5<;>=

D2EJ3<7)My3<gJHJIB31?be)<3`H

UN>3FH*L23OI4=2)M>ADIK;DHADi

CL′ e/kL23<;¢∠UBH = 90 <7,AxH*LB56H UB

IK7/H58;>¤636;DHbHJAγ56H

B58;>=H*L23gIGEJg143

γ′ = (BCL′)IG7QABEH*L2A2¤6AB;25<1DHA

γe6IK;>g3

CL′ IG7w5:=BIG58OP3HJ3<EADiγ′ >H*L23@MyADIG;DHJ7 C

58;>=L′ 5<EJ3gAB; 14Cy¤56HJ3:IRH*LPEJ3<7)My3<gJHFHA γ

e3<;>g3BL′ IG7qAB;H*L23:MyAD1G58E

ATADi

CD58;>=!H*LDC>7`

L = L′ 5<;>= BL = BCe

...............................................

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A

B CD

EF

H

L L′

T

Uq

∠ACB = 45

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A

B

C

D

E

F

H

L L′

Uq

∠ABC = 135

!" &O =9?6 7HC8:9 #$ L>@ 8:9I) P !FV XFM9 M ) M Q); .1S &H>6'L5E7%' G" 9F)!" CX ),8 NV/) 1 "P *2H=P ( =%,L 2?% <),+ X5 " Q)@ 8 )&" 9 #

6FM >X ! " "DM ∠ACB = 45 9>L0 (L 6%&59>H &%*;(8 2>)

" ! !9 !F) 9 !F)3+ " W YX ! I1 0>5&%'28D924 +LH>H> &6 5= =9 *$# )8# + #/L *0 )?P " "/)C9 142,8NO2 " 5768:9I)>@ /9 M #9 #" ! F) E >)@B (),"9 1*;(L? 70>05 L?< %#7%,$ &H>HI)&% " >)#246.)," 9(179 *HCL@ @ *HC8(82C) " )79 !2 YX Y)39 !F)+ " W YX ! I1$ # E #G9C@ LL% ) ) M % DM " J1O L 6L?5 < #S & )(+ " ! DI)H!@ " F)(2=)&" 9(1 " M #

& × Ð Î

®e $$ , $ B7 0 8 $7'$#*)$+0?4& 3:x¹ 6-$0 '77¸ $= e)<3H

ΓN>3H*L23UgIKEJgC>OPgIKEJg1K3UADi 4ABC

e)<3HM

N>3S5<;IK;DHJ3<EI4ABE¿M>ADIK;DHAB;H*L237*I4=23AB

b5<;>=1K3HNN235<;IG;DH36EJIKABEMyADIG;DHqAB;SH*L237*I4=23

ACe )<3`H

DN235<;IK;DHJ3<E7938gJHJIKAB;MyADIG;DHFADi

MNIH*L

ΓeQ0>EJA<B3qH*LB56H

∣∣∣∣

MB

MA· AC

DB− NC

NA· AB

DC

∣∣∣∣

=BC

DA

e6)$+ '$3'70 2°! $x¹13´ =>µ'70? + 0?6#=s¶ *´ e

)<3HP = AD∩BC

5<;>=Q = MD∩BC

e ±1#$330 "'! y A2Aq=23<561K73)Ù8M21RI4gIHJ1?¿IH*LH*L23qgJ58793QL23<EJ3DIK7wABM2MyAB7*IRHJ3

BAB;@58EJg

ACADiDH*L23¿gIGEJgC>OSgIGEJg143D58;>=

B1RI4367N23H3<3<;

Q58;>=

Ce IH*L¿H*L23C>793xADi>=BIGEJ38gJHJ38=@=DIK7HJ5<;>g*36758;>=@=DIKEJ3<gJH3<=58;>¤`1K3<7`¢3PgJ58;¬5`BADI4=79M>38gIG5<1g5<7,367:3)Ù8g3<MDH#L23<;

PABE

QIK7¿56HxIG;By;BIH?2bH*L2367,3MyAB7)7)IKNBIR1GIHJI4367q5<EJ336587*IG1 ?LB58;>=D143<=UC>7)IG;>¤gAB;DHJIK;BC2IH?U58EJ¤`C>OP3<;DH7<e B?Û36;>361G5<C>7` ¢L23<ABEJ3<O¦5<M>MB1GIK38=@HAH*L23¿HE5<;27B3<E7)5<1

QMDADi 4BAP

5<;>=!HJA@H*L23¿HE5<;27B3<E7)5<1QND

ADi 4CAP

BM

MA= −BQ

QP· PD

DA

5<;>= CN

NA= −CQ

QP· PD

DA

ek38gJ5<C>793 4PCA ∼ 4PDB

83:LB5`B3AC

BD=

PC

PD

2

7)IGOIR1K5<EJ1 ?¢ 4PBA ∼ 4PDCIGOPMB1GIK3<7wH*LB5HAB

CD=

PB

PD

e£4HwiAD1R14A8:7H*LB5H

BM

MA· AC

BD− CN

NA· AB

CD= −BQ

QP· PD

DA· AC

BD+

CQ

QP· PD

DA· AB

CD

= −BQ

QP· PD

DA· PC

PD+

CQ

QP· PD

DA· PB

PD

=−BQ · PC + CQ · PB

QP · DA

=−BQ · PC + (CB + BQ) · (PC + CB)

QP · DA

=CB · (PC + CB + BQ)

QP · DA

=CB · PQ

QP · DA=

BC

DA

587#=>367)IGEJ38=be

&

!" @ 8N2'LH + &6*8 H>HCL)*5 >)#0 .1*O 96 AH>8D9 #A$ L>@ 8:9I) P !FV X/M 9 M ) M Q),; .1S*HC6C,L?5 E &%' #"9I)!" >X/ )G8 V) 19&H>L>@ @ *HC8(82C) " ) 9 !2 YXY) 9 !F)T+ " W YX! 1 E/ AL?H92,H 9>+L5=;) +/" Y) %'<) "9 13$ # E#!9@ L?L>% 4) ) DM % M " J1 " DM #

% %!" " " 4M

•

8 BN ∩ CM = G M "N) M ∣∣

∣∣

b

DB− c

DC

∣∣∣∣=

a

DA

#

•

8 BN ∩ CM = I M N) DM ∣∣

∣∣

1

DB− 1

DC

∣∣∣∣=

1

DA

#

•

8 BN ∩ CM = H DM M N) M ∣∣

∣∣

cos B

DB− cos C

DC

∣∣∣∣=

| cos A|DA

#

H+< . N!"#-$&>')(,,0%5A)+2x$-. $Ft24H0 #-L-F#"u0.,/01 b%`

pr bgX b W\1W<WDm<fHU bV^ r b f b%`

α(n)W<bV<\èb` h1bZGXfcb[ `CU7V`bcbf

k[N^Nh3` hX%`

pk UG[XV(UGVèb c%fHX Z4WU dUG[]\fl\i11 · 22 · 33 · · · nn

f\ db` hX`

limn→∞

α(n)

n2=

1

2(p − 1)

y 5M91O7,I.')BD0%5MO 4$% ]0.49 #NFKC# 9%5G,7.v$-.1E)u.10 # ,H9 u+FA.<.#'9* F1z9#OAH,70< V>\fW1b f è\(cb%`lX V(U/W1b X\i` h1b r b hX%dU/\fl\i

α(n) jbYm<f\1ebb W3dbfI ^Nh

ZGU +b bcbV<W1fbU7VhUG[gfb X []\VUGV<cDiA\fQ` h1bbm\V<bV`\ip

U7V+` h1b3m<fHU bW<b \ m\1a[NUÙ/\V+\i

n!" h1bDfb[N^1Z`t` h1b V(UG[

α(n) = p

(

1 + 2 + · · · +

⌊n

p

⌋)

+ p2

(

1 + 2 + · · · +

⌊n

p2

⌋)

+ p3

(

1 + 2 + · · · +

⌊n

p3

⌋)

+ · · ·

VX \fbD \V<eUG[]biA\f

α(n) =

N(n)∑

j=1

pj

2

⌊n

pj

⌋(⌊n

pj

⌋

+ 1

)

jh1b fbN(n) = bln n/ ln pc

" bDhX%db(

n

pj− 1

)(n

pj

)

<

⌊n

pj

⌋(⌊n

pj

⌋

+ 1

)

≤(

n

pj

)(n

pj+ 1

)

^1Z_ÙGm1Z UGV<c r pj U7bZ7W1[

n

(n

pj− 1

)

< pj

⌊n

pj

⌋(⌊n

pj

⌋

+ 1

)

≤ n

(n

pj+ 1

)

&7

¢LDC>7`1

2

N(n)∑

j=1

1

pj

− N(n)

n

<α(n)

n2≤ 1

2

N(n)∑

j=1

1

pj

+N(n)

n

e

)<3HHJIG;>¤n → ∞ 5<;>=;>AHJIG;>¤xH*LB56H

limn→∞

N(n)

n= 0

83:¤63Hlim

n→∞

α(n)

n2=

1

2

∞∑

j=1

1

pj=

1

2(p − 1)

e !" @ 8N2'LH+ &6*8 H>HCL)*5 >) 0 .1,2,8NO 2 " 5768:9I)&@ A9 M

9 " ! F) E C)&@B ()," 9(1&5 8J2*5 $ 8# 'LC99I)&5 M O P " Q)*2=),"9 14E/ AL?H92,H 9>+L5=;) + " Y)(%'<)&"9 1(O L 6L?5< #(SU 7 () + " ! F)(H!@ " F)=2=)*" 9(1 "DM #

DH+< . N!"#-$&>')(,,0%5A)+2x$-. $Ft24H0 #-L-F#"u0.,/01 b%`

E = (x, y) ∈ × | x + yUG[Xm<b fHiAb`l[n^<X fb

X V<WZ7b%`N(n)

r b` h1bD[NU /NbD\i` h1bY[]b-` (x, y) ∈ E | x ≤ n

XV<Wy ≤ n

iA\fn ∈

f\db` hX%`

limn→∞

N(n)

n√

n=

4

3(√

2 − 1)

y 5M91O7,I.') %1$]5Cy58#%'1$%ME F 240) # ,8&<$F R1F=Kxy-zlF4u1&, v$&>')(O v$$ &,MOIT\fSb XNh

n ∈ XV<W

k ∈ Z/b-`

ϕ(n, k)r bY` h1bV^ r bf\ivU7V`bcbfSm<XUGfH[

(x, y)jYU` h

1 ≤ xy ≤ n

XV<Wx + y = k

h1b VN(n) =

∞∑

i=1

ϕ(n, i2)

\ b]dXZG^<X%èbϕ(n, k)

jbfH[e`\ r [b fIdb` hX`(U_i1 ≤ x

y ≤ n

` h1b V2 ≤ x + y ≤ 2n

`DiA\ZGZ7\jD[D` hX`ϕ(n, k) = 0

^<VZ7b [N[2 ≤ k ≤ 2n

i2 ≤ k ≤ n+1

` h1b V` h1bb n^<X%Ù/\Vx+y = k

U7["[NXÙG[NbW r ` h1b=m1X UGfH[(1, k−1)

(2, k − 2)

. . .

(k − 1, 1) q h1bV< b

ϕ(n, k) = k − 1

l in + 1 ≤ k ≤ 2n

` h1b V` h1b(b n^<X%Ù/\V

x + y = kU7[g[NXÙG[NbW r

(k − n, n)(k − n + 1, n − 1)

. . .

(n, k − n) q h1bV< b

ϕ(n, k) = 2n + 1 − k

h^<[

ϕ(n, k) =

k − 1U_i

2 ≤ k ≤ n + 1

2n + 1 − kU_i

n + 1 ≤ k ≤ 2n

0\` h1bfMjYUG[]b

h1bfbiA\fb iA\flb XNhi ∈

ϕ(n, i2) =

i2 − 1UGi

2 ≤ i ≤ I1

2n + 1 − i2

UGiI1 ≤ k ≤ I2

0

\` h1b f8jYU7[b

jh1b fbI1 =

⌊√n + 1

⌋ X V<WI2 =

⌊√2n⌋

& ©qA8

N(n) =∞∑

i=1

ϕ(n, i2) =

I1∑

i=2

(i2 − 1) +

I2∑

i=I1+1

(2n + 1 − i2)

=

I1∑

i=1

i2 − I1 + (2n + 1)(I2 − I1) −I2∑

i=I1+1

i2

= 2

I1∑

i=1

i2 −I2∑

i=1

i2 + 2n(I2 − I1) + I2 − 2I1

=I1(I1 + 1)(2I1 + 1)

3− I2(I2 + 1)(2I2 + 1)

6+2n(I2 − I1) + I2 − 2I1

=

√n · √

n · 2√

n

3−

√2n · √

2n · 2√

2n

6+ 2n

(√2n − √

n)+ O(n)

= n√

n

(2

3− 2

√2

3+ 2

√2 − 2

)

+ O(n)

=4

3n

√n(√

2 − 1)+ O(n)

e¢LDC>7`

N(n)

n√

n=

4

3

(√2 − 1

)

+ O

(1√n

) 58;>=!H*L236EJ36iABEJ3D

limn→∞

N(n)

n√

n=

4

3

(√2 − 1

) 587#g1K56IKOP38=be

!" 9CL?0 L46 *59CH7% ;(8 2>) " ! D !9 !F) 9 !F) + " W YX! 1B5= A<+ *5 + &5=)3H / " ! F) 0 N) H / 1 @ 8J2,L?H+ &6*8 H>HCL)5 C)40 .1 2'8 O 2 "(576C8D9F)?@ C9 M ,9 ," ! F)IE C)@B ) " 9(15 8N2&5 $8#?,L>9/9F) 5 M !O P" Q)2 )7" 9(1 S &H>6'L5E7%' G" 9F)A" >X/ ) 8 NV/) I1(O L46CL5 < #SU 7 ()+ " ! DI)H'@ " F) 2 )&" 9(1 " DM #

DH+< . N!"#-$&>')(,,0%5A)+2x$-. $Ft24H0 #-L-F#"u0.,/01 b%`

nr bDXDm\[ U_`U dbU7V`bcbf X V<W(Z/b-`

γr b ^1Z7b f [= \V1[e`X V` f\ db` hX`

γ − 1

48n3< 1 +

1

2+ · · · +

1

n− ln

(

n +1

2+

1

24n

)

< γ − 1

48(n + 1)3

y 5M91O7,I.')+O 4$!vN!"#-$%T\flb XNh(m\[ U_`U dbU7V`bcbf

n Z7b%`

xn = 1 +1

2+ · · · +

1

n− ln

(

n +1

2+

1

24n

)

+1

48n3

&<¢L236;

xn+1 − xn = f(n)L236EJ3

f(x) =1

x + 1− ln

(

x +3

2+

1

24(x + 1)

)

+ ln(

x +1

2+

1

24x

)

+1

48(x + 1)3− 1

48x3

e 3qLB5B3

f ′(x) > 0iABE

x > 0e ±# ª#7)IG;>¤5¿g*ABOM2CBHJ3<E5614¤63<N2E5q7?B7*HJ3<O63q¤63`H

f ′(x) = 2656x6+10096x5+15008x4+10836x3+3870x2+652x+3716x4(x+1)4(24x2+12x+1)(24x2+60x+37)

e 8C>EH*L236EOPABEJ3Blim

x→∞f(x) = 0

e:¢L236EJ36iABEJ3Df(x) < 0

iABE#5<1R1x > 0

>LDI4g)LSIGOPMB1GIK3<7H*LB5H#H*L237,3<ÒBCy36;>g*3 xn∞n=1

IK77HEJIKgJHJ1?P=>3<g*EJ3<5<7)IG;>¤Be6IK;>g3lim

n→∞xn = γ

D3OC>7H#LB5`B3xn > γ

iABE5<1R1ne¢LDIG7#M2EJA<B367FH*L23¿143iGHIK;>3<ÒBC>561GIH?be

BABEw3<58g)LMyAB7*IRHJI B3¿IK;DHJ38¤636EnD1K3H

yn = 1 +1

2+ · · · +

1

n− ln

(

n +1

2+

1

24n

)

+1

48(n + 1)3e

¢L236;yn+1 − yn = g(n)

8L23<EJ3g(x) =

1

x + 1− ln

(

x +3

2+

1

24(x + 1)

)

+ ln(

x +1

2+

1

24x

)

+1

48(x + 2)3− 1

48(x + 1)3e

3¿LB5B3g′(x) < 0

iABEx > 0

e ±# ª#7)IG;>¤x5¿g*ABOM2CBHJ3<E5<1K¤638NBE57?B7*HJ3<O63¿¤63`Hg′(x)=−8864x7+72336x6+247520x5+456204x4+483110x3+288492x2+86997x+9472

16x(x+1)4(x+2)4(24x2+12x+1)(24x2+60x+37)

e C>EH*L23<EJOSABEJ3D

limx→∞

g(x) = 0e kL23<EJ3iABEJ3B

g(x) > 0iABE:561G1

x > 0LDIKg)L7,L2A8:7FH*LB56HH*L237,3<ÒBCy36;>g*3 yn∞

n=1

IK7#7*HEI4gJHJ1 ?IG;>g*EJ3<5<7)IG;>¤BeQ6IK;>g3lim

n→∞yn = γ

3:OC>7*HFLB5`B3yn < γ

iABEw561G1ne¢LDIG7QM>EJA6B3<7FH*L23EJIK¤6LHIK;>3<ÒBC>561GIH?be

!"# OG"(H*+5 2 (L% " %#F%*$ L G"4)" ! D46N-Q)?L " XI) 6 RA) "9 1"S*HC6C,L?5 E &%' #"9I)*" >X/ ) 8 NV/) #,6IM % #

D 3 . N!"#-$&')DwY? ?y u$$-. F 0%5MOH'1uu$]57FCO 4$RD$O $%H570.4&#

b%` 4ABChX%db3X ZÙ_`^W1b

ADX V<W\fM` h1\1 bVèfb

H b-`

Er bD` h1b U7W<a

m\UGV`S\iAD

X V<WM

` h1b U/W1aNm<\U7V`S\iBC

o X%p iAD = BC

m1f\ db` hX%`HM = HE

oAr p [l` h1b \VdbfH[]bD\i o X%pxèfH^b

& r*e 8 $ 33' ; 0/")$+V¸ $+V¹$0:)0 =')!'5'´ =>µ'70? +10='$)$=!$#º '$ 0 '7+ w°#$#= ' $ "#1'77¶ >´ e

Q3<561GIG;>¤IH*LhM258EHJ7Ü45`Ý¿5<;>= Ü4NÝxHJA2¤63H*L236E,3M2EJA<B3SH*LB56HHM = HEIGi58;>=AB;B1?UIRi

AD = BCe 3!IG;DHEJA2=BCyg3PgA2ABEJ=BIG;256HJ3<7IH*L

D(0, 0)E(0, 1)

58;>=A(0, 2)

AB;¬H*L23y)5[Ù8IG7/LDIR143

B(b, 0)58;>=

C(c, 0)=236>;>3PH*L23

x*5`Ù<IK7:iABEEJ3<561;BC>OPN>36E7

c > be£4HqiAD1G1KA8:7¿H*LB5H

MIG7¿H*L23SMyADIG;DH (1

2(b + c), 0

) 58;>=H*L231GIG;>3CH

#M>5<797*IK;>¤UH*LBEJADCy¤6LC5<;>=ÕMy36EM>3<;>=DI4gC21G58EHA

ABLB587!H*L2338ÒDC>56HJIKAB;

y = 12bx − 1

2bcekL23

yIK;DHJ3<EJg3<MDHQADi

CHIK7

H(0, −12bc)e 3¿H*LDC>7#LB5B3

HM2 − HE2 =(

b + c

2

)2

+b2c2

4−(

1 +bc

2

)2

=b2 + c2

4− bc

2− 1

=1

4

((b − c)2 − 4

)=

1

4(BC2 − AD2)

e¢LDC>7`

HM = HEIRi¢58;>=AB;B1?!IGi

BC = ADerr*e 8 $ 33' ; &$0 7?636 )+ ¼6' ; ·6' 90? 14 ?c´ =>µ'70? + ;

$0³ '&µ$ $0?³ '&µ$¹ $>6,!#T/'70 5'74=µ54 !# 7½3µ$05¶ $'&¸ 6 e

k38gJ5<C>793 4ABD ∼ 4CHDÜ47*IK;>g3

∠BAD = 90−∠ABD = ∠HCDÝ*3=238=DCyg*3UH*LB56H AD

CD=

BD

HD

e&£t;hH36EO7ADi7*I4¤;>3<= 7938¤OP3<;DH7H*LDIK73<ÒBC>561GIH?N>3<g*ABOP3<7DA · HD = DB · DC

e 3¿H*L236EJ36iABEJ3LB5`B3HM2 − HE2 = HD2 + DM2 − HE2

= HD2 + (DB + BM)2 − (HD + DE)2

= HD2 +(DB + 1

2BC

)2 − (HD + 1

2DA

)2

= 14(BC2 − AD2) − DA · HD + DB(DB + BC)

= 14(BC2 − AD2) − DA · HD + DB · DC

= 14(BC2 − AD2)

e¢LDC>7`

HM = HEIRi¢58;>=AB;B1?!IGi

AD = BCe

!" 9CL?0 L46 *59CH7% ;(8 2>) " ! D !9 !F) 9 !F) + " W YX!

( "

)1 @ 8N2'LH+ *6&8NHCH>L/) 5 C) 0 .1 2,8NO 2 " 5768:9I)

@ 9 M K9 " ! DI) E >) @B )"9 1 *O 96 AH>8D9 #U$ L@ 8D9F) P !FV X/M 9 M ) M Q),; .1S*HC6C,L?5 E &%' #"9I)!" >X/ )G8 V) 1P 2H AP ( =%,L 2?% <?)'+ X 5 " Q),@ 8 )U"9

(

)1 EF 7LH 9>2'H> =9C+L?5=;)#+ " Y),%'<)

"9 1A+( 7+ 9CL?5(L<)AH F)A%AE.)#"9 1AO L46CL5B< #=SU 7 ()7+ " ! DI)AH @ " F)&2=)#"9 1" M #

" " M CX ABC

DM0/- "3 BC

&M MAD = BC

#G6FM % DM H

A

! >X M

BC DM /- " "

AD = BC % C &M M "

M

BC)

BC

) " " - DM M " E

AD

#&6IM D HM"

HE DM ! D '0 C I)DM

" DM " - M " DM #

&

× Ð ® Ð e $ 0 8 $7'$# )$+ g° » ° "'$'7/·3½?) "':(.-' '$.-'0?$# e

)<3HDN235M>ADIK;DHwAB;!H*L237)IK=>3

BCADi 4ABC

D58;>=!143`HPN2358;5<E*NDIRHE5<Et?MyADIG;DHFAB;!H*L23:7938¤OP3<;DH

ADe )<3`H

BPOS3<3H

AC56H

E58;>=

CPOP383`H

AB5H

Fe

ÜK5ÝS£iAD ⊥ BC

BM2EJA<B3¿H*LB5H∠BDF = ∠CDE

eÜN8ÝU£t7wH*L23g*AB;B36E7,3:ADibÜK5Ý/HECy38Ô6)$+1º ' 0 '7+ w°#$#= ' $ "#1'77¶ >´ ! "#$%!'$#*)$+,.-'/'$#$330 e

)<3H`N>3H*L23Û1RIK;>3H*LBEJADCy¤6LÖB3<EH3*Ù

AM>5<E5<1R1431HJAH*L23Ö7*I4=23

BCe )<3`H

DE58;>=DF

OS3<3H`56HMyADIG;DHJ7

G5<;>=

HEJ3<7)My3<gJHJIB31?be )<3`H

1234558;>=

6=>36;>AH3H*L23@58;>¤`1K3<7BDF

FDA

EDA

CDE

AHD

b58;>=AGD

/EJ3<7)My3<gJHJIB31?be¡k1K3<5<EJ1 ?¢∠5 = ∠1

58;>=∠6 = ∠4

eIGE7*H:379L2A8 H*LB5H

HA = AGe>EJIG58;>¤`1K3<7

AHF5<;>=

BFD5<EJ3U7)IGOIR1K5<E,LDI4g)L¬IGOPMB1GIK3<7¿H*LB56H AF

F B=

HA

BD

e )IK838

.............................

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A

B CD

EF

GH

P

`

1..............................................2

...............................

3...................................

4.........................................................

5..............................................

6.........................................................

IK793B>HEIK5<;>¤`14367AGE

5<;>=CDE

5<EJ3!7)IGOIR1K5<E,B?BIK361K=BIG;>¤ CE

EA=

DC

AG

efwM>MB1?DIK;>¤¡b3,D5B 7QkL238ABEJ36OÞHJA 4ABC83:¤63`H

1 =AF

FB· BD

DC· CE

EA=

HA

BD· BD

DC· DC

AG=

HA

AG

e¢LDC>7`

HA = AGeÜ45`ÝU£i

AD ⊥ BC¿H*L236;HEIK5<;>¤`14367

DAH58;>=

DAG5<EJ3ÖgAB;>¤EJCy36;DH,IKOM21 ?BIG;>¤qH*LB5H

∠1 = ∠4e

ÜN8ÝF£i∠1 = ∠4

H*L23<;∠5 = ∠6

LDIKg)LIGOPMB1GIK3<7wH*LB56H 4DGHIG7QIG7,AB79g*314367ÜRIH*L

DG = DHÝ)e/¢L23xOS3<=BIG58;

DAIG; 4DGH

IG7H*L236EJ36iABEJ3x5<1G7,A¿H*L23x561RHJIHJCy=>3i4EJABODHA

GHekLDIK7QOP5883<7

AD ⊥ `58;>=!H*LDC>7Q5<1G7,A

AD ⊥ BCe

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

"'@ V " M DM > V 1 M " "/) " ) % M M > M '2 " 3@ DM ( " M C M " 2 Y- # "

( ! #"$&%'"(#)*+ )( "UL!Q),- F) "U+C) # ./)#

&<× Ð ® Ì e B $9 0 8 $'# )$+[±1·$$0 # 78 '$-! ?3 #µº 6'70?3·6'´ =>µ'70? + 46#1'$) $047º '0 + e

Q3HJ3<EJOIG;>3:561G1>IG;DH3<¤63<Ew79AD1GCBHJIKAB;27(x, y)

ADiyH*L233<ÒBC>5HJI4AB;x5 + y7 = 20041007 e

6)$+ '$' '/64 47¶-61 e 3h79L2A8 H*LB56H!H*L23Û3<ÒBC>5HJI4AB;LB5<7U;>AÕIK;DHJ38¤636E7,AD1RCBHJI4AB;27SN<?g*AB;27*I4=23<EIK;>¤EJ3<7*I4=DCy3<7QOSA2=DC214A71e6IK;>g3

71IG7#5M2EJIGOS3:5<;>=

(16, 71) = 13LB5`B3

(1614)70 ≡ 1 (mod 71)N<? B36EO56H, 7 )IHHJ143¿kL238ABEJ36Oe :36;>g*3D20041007 ≡ 161007 = (1614)70(1627) ≡ 1627 = 40969

≡ 499 ≡ 1176493 ≡ 23 = 8 ( (mod 71)e

IRH*L¿H*L23qL2361GMADi25¿g*ABOM2CBHJ3<E963>;>=:H*LB5H/H*L23qÒDC2IK;DHJIKgEJ3<7*I4=DCy3<7wÜKOSA2=71Ý58EJ3

0120232630323437394145485125<;>=

7058;>=!H*L237,36MBHJIKgFEJ3<7*I4=DCy3<7#ÜKOSA2=71Ý5<EJ3

01514172546545766B58;>=

70e

£4HiAD1G1KA8:7QN<?@HJ38=DI4ADC>7QNDCBHw7HE5<IK¤6LHJiABE:5<EJ=g5614gC21G56HJIKAB;27H*LB5HH*L23¿EJ3<7*I4=DCy3<7ADix5 + y7 (mod 71)

5<EJ3M2EJ38gIG7,31?¬H*L2AB7,3kL236EJ3

0 ≤ k ≤ 707*Cyg)LH*LB5H

k /∈ 81011606163 e6IK;>g3xH*L23¿EJ3<7*I4=DCy38IK7wOIG797*IK;>¤63gAB;>g1GCy=23xH*LB56HH*L23¤`IB36;3<ÒBC>5HJI4AB;LB587;>A!IK;DHJ38¤636Ew7,AD1RCBHJI4AB;276e

!" CX DM% X &U2,8NO 2 "(576C8D9F)@ 49 M 9 " ! DI)CE C)@B )&" 9(1 " M #&6FM #

6FM JV " DM 71

M &M M M %JV #

Y( <4v !"#%$ &') %# $(J19,G#w 0 240 eN$-NFKx.,8L $-H# ,MO701O

5G,8O$., M0+&<$ s 01O705M9.<)0 Ft240 e$5I.<0Fxy%!v0%,/.C? XZ/e^1ZGX`b

limn→∞

ln

(n∏

k=1

(k2 + n2

n2

)k)1

n2

s u!"#N,8O8$6SL,/]O9 05_5A)(,I&<$%.vO7, M05 #- 5H9O7,8.#')O 4#%$#% 5IL $%M#,8&<$-.CO7,C$ &')0.0 #%O8$-H,7#>'1$-# ,8&<$3O 4$,7l.10u$#,/.O 4$g5G,G#Ov01OtO v$D$%.4&4

b%`Sn = ln

(n∏

k=1

(k2 + n2

n2

)k) 1

n2 " h1b V

Sn =1

n2

n∑

k=1

k ln

(k2 + n2

n2

)

=1

n

n∑

k=1

k

nln

(

1 +k2

n2

)

5!«kI436OP5<;2;@7)C>OÊ587)7,A2gIG56HJ38=qIRH*LH*L23qg*AB;DHJIG;BCyADC>7iKC>;>gJHJIKAB;

f(x) = x ln(1+x2)58;>=!H*L23EJ38¤`C21G58EwM258EHJIRHJIKAB; 1

n

2

n

. . .

n

n

ADiyH*L23¿IK;DHJ3<EtD561[0, 1]

e:36;>g*3D

limn→∞

Sn =

∫ 1

0

x ln(1 + x2) dx =1

2

∫ 2

1

ln u du =1

2(u ln u − u)

∣∣∣∣

2

1

=1

2

((2 ln 2 − 2) − (−1)

)=

1

2(−1 + 2 ln 2)

e9 !"3N@ 8N2'LH3+ &6*8 H>HCL)A5 >)70 .1J$ 8J =%G%,L+ &8NHCL<)ALH9C8NL 2&@ O +LH>HI)

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

!

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acbCde ( *+%'-,/.0*%1#32+f bCd .g9=OEXhH8;: hHij7X8;: <P=;?DG@IKJ@LB&89MOND<PBQEH?DGkR>lm=;hX89MU<>BUEX?FGUTnYHLB&=;oH=Op 8K[&MUE9qrLBstg;?FB]8XBU?F: NSuwvZ VnZHxM&GkuOBQp p798;: <>=;?FGyBUAzBU?F: <>:HIKJ@LB&89MOND<PBQEH?>RSB&ACB&?D: <F:DT3vZ +Z;[&M]hX89GU|J@Z 7ZO~=9=989?>=]^|VX?FEHNPBKYXZ J@Z9[]\;MU^_9BU?

( *+%'-,/.0*%1#32* 4( * 6'-,/.g9=OEXhH8;: hHij7X8;: <P=;?DG@IKJyLB]89M]NF<>BUEX?FGkR>l =;hH89MQ<PBQEH?DGUTn;MU<>?F: NSu[UEH?F?_Cs JMQq;:9rMOu&: p7X8;: <P=;?DGyB&ACB&?D: <F:IJyLB]89M]NF<>BUEX?FGkRSBUAzBU?F: <>:TWH\O: p : owS=;hXi;|&SBU1`: iU\9M&A|&Z 3Z9vn?F=;GSGSACM&h|&hX89?>B3\;M]hXi;|XM]=9u]:[&MQ<P=|;9_9?DEHGjGP: M9|;[]\9MQ^hCvW=98;: h

wholeissue 33 5 · Ú ebª#; Òbc>58=2e ig1g56h %" 36ej3 g igejg*ab;279g*eji h#c>; g 3ey...

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wholeissue 33 5 · Ú ebª#; Òbc>58=2e ig1g56h %" 36ej3 g igejgab;279geji h#c>; g 3ey...