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!#"$&%!')(*',+-.' /1032546728672:9=@?BAF*K12ML1CH?1N@0O2:PQ6RGS9F*KBGO62:;@GT...

Author: Canadian Mathematical Society

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A
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3 sin θ + 4 cos θ = 5

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3¢

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d

600

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625

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650

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−1

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70

c ¤#9<20EG3[9<2 c ¤#9<2X672 f ?@9<;<2 4A<9<2XLp259BFJ2#;12 2§FvL<4:67672L<4:C032L
a

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100 231

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115 231

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CH?BAB@2:C~0O2PQ4:GOPA
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118 231

f (x) = ( 21 )x

2

−2x c

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

A

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¥25æ 6,?BAp;@CH2450312:æ N1CG3BAp2:PV2:9BFL
log10 x2

2

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ØÙ c K12>9@A
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1

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d

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600

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4

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675

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f (x) = ( 21 )x

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−2x c

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

A

C

C

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Ø c

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GS6X4L
n!

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n

h34 f FJ?@CGO450+*

n · (n − 1) · (n − 2) · (n − 3) · · · 3 · 2 · 1

GO6#;125<9<2;FJ?QN12 c

?@CR2 :4:PVL@032 c 4:9<; 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 f 4! = 4 · 3 · 2 · 1 = 24 f ÞX2[FH2:CHPQGS9<2oF*K128LF*K@45FF*K1289@A
(m + 1)!

,.-0/2143657- 8)9:;3=<>?>@5+3+- A c K12X9@A4R=@254CGS6 c 7 3 3 S = 365×24×3600 f FJ?@C6#?Bh FH?CH2:672:PVN@ 0O24>h34 f FJ?@CGO450 252§F = 2 ·3 ·5 ·73 S

hp2?@CH[4:9@GE)28F*K12>h34

S

= 2 · 3 · 4 · 5 · 6 · 8 · 3 · 52 · 73 5 · 73 365 c = 10! · = 10! · 2·3·7

42

?2,@25C c KBA<6 KBG f KQGSPVL@0SGO2:6F*K@45F 10! < S < 11! <365 < 11 m = 10 426722: c GO6RF*K12>GS9BFJ2: 512:C 2 :@2:9VGSh2K@4;4:676*ACH2:PQ4:GS9<2;!F*K12>674:P2 c ! "#$ % &! '% )(+* ,-/. 10' 2/. &% $354'&67% .2!892& % .2%:%7/ (<;&=>8(, ?A@;' <$=>B3+. 36::<6<%3 &% . C8DE. %7%7'? ß c d PV?:BG326,K1?>GS9<84645FFH2:9<;12;N:= c d ;@A10TFH6L<45G3;GF L<2?@L@032 2:4 f K 10 c? ? 49<; ∗f KBGE03;@CH2:9HF 2:4 f K c8 K12FH?FH450Ä4PV?BA<9BFXF4500 259h_?@CRFHG f §2 FH6X46IF 4,160 PV4:9= f KBGS0O;1CH259V44[FFJ259<;<2:; ,.-0/2143657- 89 :KJ /7>8LGM 5N >PO>Q0A8R@ >TSASKU 3 1@ >8 3:O-V>'R3U 57@ >TW 57Q< ,N <-4-0/ OEX - A - 8 3+-AOZY\[^] d 9t4;BA10EFL<4§=16_F P?@CH2VF*K@49t4 f KBGE03;+`~F*KBA<6 F*K125CH24:CH2 4(500) ; ?B0E0O4:C68GO9674503256ML@0SA<6M4692 5FJC4aF < GO9674503256 K125CH2 GO6F*K12Q9@A< PVN<25C8?B= hD42000 ;@A10TFH6 6a a câ ?B0TBGO9<h_?@C GO9F*K12V[CH?BA8(, ?

c d 6)6)AF*K@45F~F*K12!254CÝF*KQGS646)LpK125CH2>>GTF*K f GSC f A03259< F*KQ?BhpF*K12N1250TFN<2 ?BA10O;F*KBGO6#;@GEå25CH2:9 f 2N12 34 3N f 3;

2π

PV2[FJCH256

40,250

PV2[FJCH256

P2§FJCH2:6

40,250π :GS0O?@P2§FJCH2:6

40,250 _ 2 9
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Ú 2[F

4:9<;

N<2!GS9BFJ2:52:C6 cIb ?BA4:CH2Q§GT@259F*K@4[F

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(a ? b) ? c − a ? (b ? c)

c

a?b

GS6M;<2[p9<2:;FH?N<2

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1BA1GTB4503259BFH0T=

a?b

cI K12:9

GO6#;125<9<2;Q4:6

(a − 2)(b − 2) + 2 (a ? b) ? c = (a ? b) − 2 (c − 2) + 2 = (a − 2)(b − 2) + 2 − 2 (c − 2) + 2 c = (a − 2)(b − 2)(c − 2) + 2

cI KBA<6 F*K128;@GEå25CH2:9 f 2oGS6I)2:CH? c a ? (b ? c) = (a − 2)(b − 2)(c − 2) + 2 - N R.365+- 8)R / qÝq c,-0/ 1.365+- 8 9 :\J / R8 1.-AOQ0A8R@ >S\U3 1.@ >8$3:OY [ > 5 / / -0/ /7>PQ5:R.3+>R8@ ) 8U 365+3 1437>PO YZU <)R 5R3OEY\[^]

â GOPGS0S4C0T=

(a ? b) ? c − a ? (b ? c)

= (ab − 2a − 2b + 6) ? c − a ? (bc − 2b − 2c + 6) = (ab − 2a − 2b + 6)c − 2(ab − 2a − 2b + 6) − 2c + 6 − a(bc − 2b − 2c + 6) − 2a − 2(bc − 2b − 2c + 6) + 6 = (abc − 2ac − 2bc + 6c − 2ab + 4a + 4b − 12 − 2c + 6) − (abc − 2ab − 2ac + 6a − 2a − 2bc + 4b + 4c − 12 + 6) c = 0

c

d

fP

f ApN12>GEF*Kt6*G3;12:6?BhX0O2:9< F*K

GS6L14:GS9BFJ2:;CH2;4:9<;çF*K12:9 f A@FGO9BFH?

∗ f ApN<256R>GTF*K6)GO;<256?Bh0O2:39< F*K f P c h ;12:99@A
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.......................................................................... .... .. .. .. K12#;@GS4[C4:P 4[FÄCHGO5KFÄ6,K1?6?@9<2Xh34 f 2#?BhF*K12XL<45GO9BFH2; f ApN<2 .... ... ... ... .... d ..... c ..... d ..... .................................................................................. K12:CH2oF*K128032§FFJ25C~?@9!4>67@A<4:CH2M6,K1?6DFJ?>KBG f K f ?BA<9BFIF*K@4[F~6)PV450S0 .... .... .... .... ... .. .. .. ... c .... b .... c .... f ApN<2 f ?@9BFJCG3NBA@FJ256 c Ý¥v2:PV2:PVN<25CMF*K@45F254 f K6,BA<4CH2?@9F*K12h34 f 2 ................................................................................ .... .... .... .... .... ... ... ... CH2:L1CH2:672:9BF6#46)PV450S0 f ApN12 c ... d .... c .... d .... ............................................................................... . ¢0O2:4:CH0 = 6)GS9 f 2?@9@0T=F*K1267PQ4:0E0 f ApN12GO9F*K12 f ?@CH2?Bh~F*K12N@GO f ApN12 CH2 f 25G @2:69GO6 f ?BA<9BFH2;FÝ>G f c2!K12592Q6)AF*K12M;BGO4[C4P 4:9<;!254 f K2?BhBF*K12 f ?@CHCH2:6)Lp?@9<;BGO9<6)PV450S0 f ApN<256 d c ¢Ä?@91672BAp2:9BFH0 = GO6 f ?BA<9BFJ2:;F*K@CH22MFHGOPV2:6RK1259!26*A

a − b − c + d = 1 − 6 − 12 + 8 = −9

ä c ?@CKBG f KB450SAp2?@CB4:0EAp2:6#?Bh

k

GEhv4:9=

GO6

x2 + k

x4 − 3x3 + 6x2 − 3kx + 8

4h34 f FJ?@CR?Bh

A 5N <

,.-0/2143657- 8)9:

â A
h_?@CJP

x2 + ax + b

R08)Q3O+QA R.@ >

11

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GO6o4h34 f FH?@C co K12:9VF*K12?F*K12:Ch34 f FH?@C#PA<6*FN12?BhÄF*K12 2 :4 x9<;+k2K@4[@2

x4 − 3x3 + 6x2 − 3kx + 8 = (x2 + k)(x2 + ax + b) c = x4 + ax3 + (b + k)x2 + akx + kb

1BA<45FHGS9< f ?12 2BA<45FHGO?@916

f G3259BFH6V?@9F*K12032[hSF!4:9<;CHGO5KF!6)GO;<256

a b+k kb

2?1NFH45GO9F*K126=16JFJ25P

?Bh

= −3 = 6 c = 8

CH?@P 252[F `:6*ApN16JFHGEFHA@FHGS9<F*KBGS6IGS9BFJ? 2o?1NFH45GO9 ` b = 6 − k c K125CH25h_?@CH2 k(6 − k) = 8 c F*K@45F~GS6 * ?@C * (k − 4)(k − 2) = 0 k=2 k=4 - C+2 . 6:/' !-"" % &! '% & . ! B. ; 6 ;'& )@<3% (, ?

Ù cvd 949BFv>GS6,K1256FH?MFJC4§@250h3CH?@P FH? ?@9MF*K12 A>GTF*K D;@GSP259< 6)A
GS9<;!F*K12 cE ;BGO6JFH4:9 f 2

D

2

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

F

B

4

BE

E

8

A

C G

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FD

AF

BE =

2

AB 2 8 f P c × FD = ×8 = AF 6 3

E

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

A

8

G

- C+2 . 6:/' !-"" % &! '% & . ! B. ;6 ;'& )@<3% (, ?

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,.-0/2143657- 8 9 : 3=< >?>0@5237- !A ]

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Ú 2[FIF*K12>L14:GSC c Ú 2[F CH25LF*K12 f 4503?@CG32L<45GOCR < h_?@C4!K@4PVN@A
259 f 2

GEh

M

= (427, 48% × 427) = (427, 204.96) 31 × 9 = , 31 × 9 ≈ (526.42, 279) 53% = CB − HB ≈ (99.42, 74.04) = CB + C ≈ (625.83, 353.04) = (220, 12 × 9) = (220, 108) c = (360, 28% × 360) = (360, 100.8)

CH2:L1CH2:6)9BFH6F*K12 f 4503?@CG32>L14:GSC~h_?@CIF*K12PV2:450
c

F*K1259

K125CH25h_?@CH2

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cÄd N@4 f ?@9BFH45GO916 CH2:; f 4N1N1452:6R4:9<; [CH2:2:9 f 4NF*K128N@4 49<;4:0S6,?L@0O4 f 2568F*K@45FoGS9F*K12QL<4:9 cQ K@4[FoGS68F*K12VL<2:C f 2:9BF452Q0EG32[0SGOK1?1?1;F*K@4[F ?Bh F*K128FÝ? f 4N1N1452:6~F*K@45F~4CH2M9L<4:9 ?@9<28GO6CH2:;49<;F*K12?F*K12:CIGO6X[CH2:2: 9 - -AU 5+3+> - U-0/2143657- 8)U 9 :J /+> L M 5N >PO)QA R.@ >ISASU 3 1@ >8 3:O$V>PRU 5+@ >IW 5 Q0< ,N < -4-0/7O X0- A - 8$37-AOEY\ [ BR08 @ A 5 OZV5 UQAR0A-0/ /+>!Q 5 R437>PO Y 3 3:R 5ROEY\[^] 11 !#"%$'&)(* ( 2 f 49 4 f KBG32,@2kF*K12¡;12:6*GOCH2:; ?BA@F* ,+.-!/ 10324 5+ f ?@PV2>N:=67250O2 f FHGS9<825GTF*K12:CDCH2:;!h_?B0E03?2:;N:= 1 . . . . . . . P = 25 .................................................. P = 4 [CH2259ç?@C8[CH2:2:9h_?B0E03?2:;N:=CH2;36722!F*K12 ....... . . . . . . . . . . . . . . . .. .................................... ;@GS4[C4:P cR K12
........ ................ ................ ......... ................ ......... ........ ......... . . . . . . . P = 34 LFH?L@G f ?@9<2?BhIF*K12QFÝ?CH2:; f 4N1N1452:6 KBG f K f 49çN<2 5 = 10 2 ;GS9 31 4§=16 c KBA<6 F*K12L1CH?1N14N@GE0SGTF=Q?Bhv6)A fJf 25676GS6 2·3 ?@C = 3 60% 1 10

Ø à c K@CH2:2V67PQ4:0E0 f GSC f 032562:4 f K?Bh~C4;@GEA<6 f P 49<; ?@9<20S4CH525C f GSC f 0324:CH203? f 45FH2;4:6!GO9<;BG f 45FH21; ?@9tF*K12 CHGO5KF c ÞX2§FJ25CJPGO9<2MF*K12>4CH254?Bh0S4CH525C f GOC f 0O2 c b ?BA
√ a+b 3 π c

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

OZV5 UQAR0A-0/ /+>!Q 5 R437>PO Y 3 3:R 5ROEY\[^]

Ú 2[F N<2F*K12 f 259BFJCH2:6ç?BhF*K12F*K@CH22 GO9BFH2:CG3?@C f GOC Af 0O 2:6 B0O 2[F C N12#F*K12oL0S4CH52 f GOC f 0O24:6X6,K1? 2o;12:9
A
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p

AC 2 − DC 2 =

259 f 2

2 3

r = √ +1 πr

A

F

252[F AO AE

AC 2 × AE = √ AD 3

E

O

B

√ c √ 4−1 = 3

4AEO ∼ 4ADC

AO =

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

=

D

C

AC `F*K@4[F~GO6 AD

c

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2

= π

2 √ +1 3

('+. '6 :<6 %3 7% .2=> <6:/. '?

2

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√ 7+4 3 3

c

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,.-0/2143657- 89 :KJ /7>8LGM 5N >PO>Q0A8R@ >TSASKU 3 1@ >8 3:O-V>'R3U 57@ >TW 57Q< ,N <-4-0/ OEX - A - 8 3+-AOZY\[^] K129@A9@AGO6 .. ..... .. .... ∆ = 10 . . ... ?Bh0O4[FFHG f 2>Lp?BGS9BFH6GO916*G3;128F*K12MFJCGO4:9<§032MGO6 1 ... ... .... ... ... ..... ... ............. ∆ − E/2 + 1 = 10 − 5 + 1 = 6

.. ..

c

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c

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= = = =

4 20 36 c 15

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B

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67PQ4:0E0p67@A<4:CH2 c KBA<6 2 4:0E0 K@4[@2 6)PV450S0B67@A<4:CH2:6D4:6F*K12o67K@4;12;4CH254 cIâ AGS6 c K1259 =@GO250O;@GS9< x 3x2 = c2 = 1 x = √13 K@4[FN1CGO9<[6XA<6RFJ?F*K12259<;V?Bhv49

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f (x)

RA 1U3]

N124>hOA<9 f FHG3?@9KBG f K6)45FHGS6)p2:6

f (29 + x) = f (29 − x) _h ?@C450S0:B450SAp256R?Bh c h K@4:6R2 : 4 f FH0 =>F*K@CH2:2oCH254:0BCH?1?FH6 49<; ;12[FH2:CHPQGS9<2 x f (x) α β γ c F*K128B4:0EAp2?Bh

S

α+β+γ

ß c 7?1K@9032[hSFoFJ?9 LGEF*K!F*K12K1?BA

S

A

cXÚ 2[F

x1 =

B

?@C

1 c 2002

n ≥ 1

;<2[p9<2

nxn+1 = (n + 1)xn + 1

c GO9<;

x2002

c

ã c ?@CGO9BFH2525C6 c 032§F 49<; 2 dn = gcd(an , an+1 ) ÞX2[FH2:CHPQGS9<2MF*K12>0S4CHn 5256*≥ FB4:10E Ap2?Bh an c = n + 500 dn

c FÄGS6D§G @2:9MF*K@45FF*K12LGO9BFH2525CCH?1?FH6#49<; p(2002) q(x) 2002 K@4689
ä c GS9<;oF*K120S4CH5256*FvLGTF*KM2 : 4 f FH0T=FÝ2:9BF|=< N N! @@2 )( )2:CH?*8;BG3§GTFH6 c

Ù codef GSC f 032!L146)6,256F*K@CH?BAp5KVF*K12@2:CÝFJ2 ?Bh4CH2 f F49<§0O2 4:9<;FH?BA f K12:6 EG F6D6*G3;12:6 4:9<; 45FL4CH254N ?Bh GS9 f P 2 c C

MN

2

ABCD

0 cXÚ 2[F

m = 144sin

c B450SA<4[FJ2

2002 P k=1

2

x

+ 144cos

2

x c

PV4:9=Q6*A f K

?

m

6#4CH2MGO9BFH2525C6

c

2002 P k! k · k! 2003! − − 2002 k k 2 2 2 k=1

Ø à c ? PQ49=Q4[=@684CH2F*K125CH2FH?4:CJC4:9<52 G3;12:9BFHG f 450CH2; G3;12:9BFHG f 450N@0EAp2 5 52,@25C|= PQ4CJN@0O2GS6 49<; OG ;<259BFHG f 4:0#[CH2259 PV4:C*NB03256!GO9 46JFJC4:GO5KF 0EGO9<26*A f KçF*K@45F! 4; _4 f 52:9BD F FJ?4[F~0O2:4:6*FR?@9<2PQ4CJN@0O2?BhpF*K12>674:P2 f ?B03?BA
¾

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R0A 1U]",0< -

3=< > U3+> ? U 5 8": - 10A

Ø c 9F*K12oL10S49<2 GS6~4 f GSC f 032>GEF*K f 259BFJCH2 49<;C4;BGSA<6 49<; 4:CH28;@GS6*FHGS9 f F Lp?BGS9BFH6X?@9 GS 6Γ4MLp?BGS9BF~?BA@FH6*G3;12 4O 9<; 4CH2F*K12MrPQ P GO;<*Lp?BGS9BQ FH6?Bh 4:9<; Γ M 49<; N CH2567L<2 Γf FH G A @250 = c8â A
¾

ß cÚ 2[F 2002

a1 a2 . . . a n

4:9<;

GO9 f 0EA<6)G @2 cDâ A
N<28CH254:0@9@A
b1 b2 . . . b n n P c /
n P

i=1

1001

4:9<;

i=1

n n X a3i 17 X 2 c ≥ a b 10 i=1 i i=1 i

ÞX2[FH2:CHPQGS9<28K12:9V2:@A<450SGTF=QK1?B03;@6 c

cXÚ 2[F

n

N124Lp?@6*GEFHG @2MGO9BFH2525C c ÞX2[FH2:CHPQGS9<2MF*K12>67PQ4:0E03256*F~L6*A

49<; 4CH2CH2:4:CJC4:9<52:PV2:9BF6?BhF*K12 K12:CH2 b1 b2 . . . b2n+2 N@GS9
S

2n + 1 0

*A<6*FHGEhE==@?BA

2n + 1 1

2n + 1 . . . 2n + 1

ã c GS9<; 4:0E0XCH2:4503B4:0EAp2;thOA<9 f FHG3?@916 C45FHGO?@914:0p9@A

S

x y

GO9

49<;

f (1) = 2002

c

;<2[p9<2:;?@9tF*K12672[F?Bh4:0E0

f (x + y) = f (x) + f (y) + 2xy

h_?@CR450S0

c *A<6*FHGEhE==@?BA
1 f ?@CJ9<25CvGO6F*K12 F4:0EGO4:9 K12Xp91450LFJ?QÜ

+

D²

Ø c GS9<;4:0E0

7²kµ

&+$#+'*)&%+%+)

² 5«² ¾ Ë²) <Ë É :! Í|Ò ¾ Î : É Ï i

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3

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7²

FHGSP256DF*K1286*A
ß c K12L@0O4:9ç?BhR4K1?BA<6,2K@4:6>F*K12V6,K@4:Lp2?BhR4 f 4L@GEF4:0 ?1NBF4:GS9<2;N:=6*A1GEF4NB0T= LPV 2[FJCH256D03?@9< c K1282 5FJ25C* L10S4 f GS9<6*G3;12)N:=<*6)GO;<2#h_?BAFJ?@L?BhvF*K126)G 2 5FJ25CJ91450p4:0E0O6 4:9<;2:4 f KVh34 f 2h_?@CJPQ6o4:94:9<§032?Bh >GEF*K!CH2:6)Lp2 f FDFJ?4K1?@CHGT)?@9BFH450pL10S49<2 c 30◦ ÞX2[FH2:CHPQGS9<2F*K12@?B0SA

¾

cÚ §2 F 4:9<; N<2>FÝ?VLp?BGS9BFH6o?BhvF*K12L@0O4:9<2 4:9<;V032§F N<2>F*K12PQGO;<*Lp?BGS9BF#?Bh L1CH? |2 f FHGO?@M cÚ 2[A F N12B 4>0SGS9<2 49<;032§F 4:9<; N12MF*K12 916?Bh 4:9<; ?@9BFH? c 49<; R dAB 676*A9C4;BGSA<6X46RF*K12 f GSC f A

BSM

ã c GS9<;450S0[B4:0EAp2:6I?Bh h_?@CpKBG f KM4:0E067?B0SA@FHGO?@916D?BhF*K122BA<45FHGO?@9 n x3 −3x+n = 0 4CH2MGO9BFH2525C6 c c /
1

ä c 2>4:CH2§GT@259Q4 f K1256767NGEF*K f ?B0EAFJ?N12 R@ 8RN >8$3 100 GSh@F*K1,2 =!K@4[@284 100 P ?@9!6*G3;12 c 9@GTFHGO450S0 = 4:0E0 6,BA<4CH256#4CH2oKBGEFH2 c 34

6~GTFDLp?@6)6)GON@0O2FH? f ?B0O?BA
3N

6GTF#L4:9<;4:0E0ÄF*K12CH2:PQ4:GS9@GO9< 42 :4 f FH0T= f ?B03?BA
f

¾

¾

2

;¾

4

;¾

6GTF#L4:9<;4:0E0ÄF*K12CH2:PQ4:GS9@GO9< 22 :4 f FH0T= f ?B03?BA
¾

£82 : F~2§G @24 f ?@PVPV2:9BFX?@94L

I4C4V£45JGO?@914:0O2;@G ©

ß cvd 914[FHA@ GEhBF*K129@AN1450O4:9 f 2:; c 9@A

½

>8 39 : ,.3:R08 M R3Q.- 8O

º ½ [ O¨Á PJ]

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2kFHA67?B0SA@FHGO?@916çFH? L
½

n

- < R

,.-0/2143657- 8)9:

>@ J\RUU 5 /:R3O$,43=A RU94- 1A/QO

2X>GE0S0L
0 ≤

n P

n P

o(k) −

A8R08BN >] K12:CH2

e(k) ≤ n

o(k)

4:9<;

e(k)

; 259N:= 4:P?@9< d 1 2 . . . n j k GO6 n c 2:9 f 2 d

n X

k=1 n X

4:9<;

o(k)

=

d

49<;

≥

j

e(k)

=

n X

k=1 n X k=1

d

k 2K@4§@2 n

d+1

$$ X

d

k=1

â GO9 f 2 j n k

X jnk

o(k) − o(k) −

n X

k=1 n X k=1

e(k)

=

#

=

n d

j k

∞ j X

i=1

e(k)

d

n 1

j k

=

n 2i − 1

−

∞ j X

=

i=1 ∞ X

i=1

k

−

i=1

2i

j

j

∞ j k X n

n 2i − 1 n 2i

n 2i

−

k c

k

j

k

≥ 0

n 2i + 1

k

≤ n

c

: ß c ¢v?@916*FJCA f F8F*K12L
½

P

,.-0/2143657- 8)9:

5N <> /+R.3:R5 / /+>!O - 1.>8)O

A R8+N >1]

Ú 2[FF*K12>§G @2:9FJCGO4:9<§032MN<2 49<;0O2[F 49<; ;12:96)GO;< 256 CA AB

GSC6*F 6*A
Ý£o?FJ2F*K@45F 6)GS9 f 2 4:0E00EG32?@9F*K12 f GSC f 032>GTF*K ◦ A P E F49<; c ;@GS4PV2[FH2:C ∠EP 49
=

PD

sin B sin A

=

b a

4:9<;

PF PE

sin C

=

sin B

=

c b

c 4:6A<6)A<450 a = BC b = CA c = AB KBA<6 P D GS6XF*K12Lp?BGS9BFGO916*G3;12 6*A f KQF*K@4[F PE P F 49<; P 4ABC = = b c F*K12;@GS6*F49 f 2:6 a 4:CH2L1CH?@Lp?@CÝFHG3?@91450FH? CA) ac b c Ú 25P?BGS9<2L
K12:CH2

a

b

c

[P EF ] = P E · P F · sin A = λ2 bc

a 2R

=

λ2 2R

abc

cÄ KBA<6 EK125CH2 GO6ÄF*K12 f GOC f A
c K12>0O4[FFJ25C~h_?B0E03?6XN<2 f 45A<6,2 [A0 AB] = [A0 AC] KBA<6 GSh f GO6XF*K12CH2 2 FHG3?@9?Bh GS9VF*K12GO9BFH2:CH914:0NBGO672 f FH?@C#?Bh SA AA0 ∠BAC F*K12:9 h_?@C#4:9=Lp?BGS9BF 2K@4§@2 d(M 0 , AB) d(M 0 , AC) c8 K120EGO9<2 0 ?Bh M SA = c ¢v?@916*FJCA f FHGO96)GSPQGE0O4:CH0 =!F*K12b6=@PVPV2;BGO4:9 GS6XF*K126=1PQP2:;@GS49F*K@CH?BAp5K S SB c F*K@ACH?BAp5K 2?1NBF4:GS9 4[FDF*K12MAGO9BFH2:C672 f FHGO?@9V?Bh 49<; B P SA SB

£ ? 203?1?145F67?B0SA@FHGO?@916h3CH?@P ?BA

ß cÄ CGO4:9<§03256 4:9<; 4:CH2 f ?@916JFJCHA f FH2;8?BA@FH6*G3;12DFJCHGS49<§0O2 A3 A1 O24:9<; A1 A2 O3 A1 A26)AAf 3K cvd LGEF*K GO6D?BA@F6)GO;<2 O A = O A O A = O A O A A 2 3 2 1 3 1 3 2 1 1 F*K@45F 49<; 4:9<; 2 A3GS68F*K12 1 1 ∠A O A ∠A O A ∠O A A = ∠O A A = T 1 3 2 1 2 3 1 3 2 1 2 3 c /Lp25CJL<2:9<;B2G f A10S4C~h3CH?@P FH? O1

34 3N

A 1 O1

GS6XLp25CJL<2:9<;BG f A10S4CDFH?

A 1 O1 O T =2 1 O2 O3 A2 A3

,.-0/2143657- 8)9: >0-

O 2 O3

`

c

A >:IJ]+R8@R / / OZWIR

A2 A3

¨Á !J]

@ >8)O XO

,

Ú 2[F 0 N<2gF*K12è0EGO9<254C A1 7→F*K@45X .................................... FJC4916*h_?@CJPQ45XFHGO?@9 FCH?F45FH2:6#4M@2 f FH?@C ........................... .... .......................... .... ........................... ... @ A . .....................O . .. ... 2 f ?BA<9BFJ25C f 03? f [>GO672 N:= c Ú § 2 F . . ... . .. ◦ ... ... . . A @ X 4:9<; 90 ... ..... .. θ . . . . ............... N12F*K12 PQGO;<*Lp?BGS9BFH6?Bh ... . .. . .. A @ . . . 0 M θ ........ .... a ..... ϕ 49<; N CH2567L<2 f FHG @250 = cÚ 2§F . . ... . .... .. 0 A @ ... O3 ....X .... ......... A1 A3 A1 A4:29< ; ... ... .b .. ........X c A @ . .. ..... .. z R @ X ... . . . . . . ... ......... θ = ∠O A A ϕ = ∠O A A 1 2 3 1 3 2 ... . A M@ .. ϕ K1259 N ... ... ... . .. ∠A O M = ∠A O M = θ ... A 3 2 1 2 . @ ... ... c 49<; .. .. . ... A ... ∠A O N = ∠A O N = ϕ @ 2 3 1 3 ... . .. Ú 2[F 49<; . .. A . . @ ......... .. a = cot θ b = cot ϕ

.. .. 0 0 ... cVÚ 2§F − −− → − −− → AUA T b .@ .. a R @...... .... . .......c = a+b = A1 N = N A 2 .... .... ...... ...A . . A . . . . 2 ...... θ −−−→ −−−→ −−→ ϕ ................. 3 .... .... = T O1 = A1 M = M A 3 .... ... .... m −−→ 4:9<;!w −−→ c ... .... ... .... = A1 T = NM .. .... . .... ... −−→ £8? −−−→ .... ... .... .. M O 2 = a O3 N = b .... .. . .... .

49<; −−→ 3c £o?FJ2 −−→ .... ... .... ... .... A .. .... .. 3 c F*K@245TF = a w5T`7A F*K@34[F=GO6 bw . . . .... ... .... = 2c 2 .... ... d 0O67? (a+b) 6)GS9 f 2 = 2 .... .. .. .... . . .... A2 T : T A 3 = a : b ... .... K@4[@2 .. ...... ..........

.. .. .

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

... ...

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

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

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

!

!

m KBA<6

!

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

?

!

a b 2 c = (2 )+ (2 ) = (a +b ) a+b a+b c

O1

!

m c a +b = 2c £8? −−−→ m 49<; A 1 O1 = +

−−−→ O3 O2 = b

!

0

+

w

+a

0

!

= (a + b )0 +

w

=

cm 2

0

+

c 2

0

=

c −−−→0 c A 1 O1 2

KBGS6#L1CH?:@256#4:67672:CÝFHG3?@9Q34 c d 0O67? 49<; 2O1 T : A2 A3 = 2| | : c| | = 2 : c KBG f KL
c â A
!a

n 1X 2 x n i=1 i

·

!b

·

n 1 X 3 x n i=1 i

!c

≥ 1

c

f 49tN12CH2:L1CH2:672:9BFH2;ç4:649
½

(−2, 1, 0)

4º

½

(1, −2, 1)

,.-0/ 0>@ 9: - < R >@ \ J R3UU 5 / RO ,.3=A8R3U 94- 10A Q3O A8R08BN > - 94>A 3 Y 143=A > - 8 3:O CR 8@ 457>A A > - A 8U37> 568)O R5 U- 8)U VR 3 3+>!O A80R 8BN >] J\RUU 5 /:R U A 5+3+> 1 ]

/

5 / 568)U 5 O M _ > Q 5 0>

c K1259 a+2b+3c c

GSC6*F~6,2§F h_?@C~450S0 3 c x1 49<; = (x1 )a (x221PQ )bA<(x x1 > 0 c â OG 9 f 2F*KBGS6DGS6IFJCApn2= h_?@CI1N 1 x1 < 1 a + 2b + 3c = 0 £8? c (b + 2c)(−2, 1, 0) + c (1, −2, 1) = (−2b − 3c, b, c) = (a, b, c)

FCH25PV45GO916RFH?6,K1?¡F*K@45F

c 49<; b + 2c ≥6702[F c≥0 ?L 0 §GT@259QGS9<2BA<4:0EGEF|=VN12 f b?@ P +252c 6 ≥ 0

Ú 2[FFHGS9<

t → 0

1+t 2

2?1NBF4:GS9

a

1 + t2 2

b

c 1 ≥1 2a+b+c

1 + t3 2

259 f 2

c

≥ 1

c

a+b+c≤0

b + 2c = (a + 2b + 3c) − (a + b + c) ≥ 0

49<;

?L
xk+1 = 1 + kt n X

i=1 n X

i=1 n X i=1

c≥ K125CH02

49<; c

6,2§F n Lp= +2M1GO9B FHx21525= c x k K1= 2:9 1 − t GO6X4 ?@6*GEkFHG @ CR4:x 9<;2 = · · · = I

k

t ∈ (0, 1)

xi

= k+1

x2i

= (k + 1)(1 + kt2 )

x3i

= (k + 1) 1 + 3kt2 + (k2 − k)t3

K12§G @2:9GS9<2BA<4:0EGEF|=VN<2 f ?@PV2:6

c

c c 1a (1 + kt2 )b 1 + 3kt2 + k(k − 1)t3 ≥ 1 c √ cX K12:9 b câ GS9 f 2>F*KBGO6#GS9<2BA<4:0EGEF|=K1?B0O;16 £8? FH42 √ t = 1/ k 2 4 + k−1 ≥1 k 4:9<; F*K1259 PQA<6JFN<2 h_?@C450S0RLp?@6*GEFHG @2GS9BFJ2:52:C6 √ k lim 4 + k−1 = +∞ c k

9

49<;!F*K12>L
ä c /GEF*KGS9BFJ2:52:C f ?12 f GO2:9BF686)A f KVF*K@4[F 4:CH2Q;BGO6JFHGO9 f F#Lp? 25C6 p(1) p(2) . . . p(n) ?Bh c

uº

2

,.-0/2143657- 8)9: 457>A A >

½

- A 8)U 3+> 5 8O

R5 U- 8)UVR 3 3+>!O

A R8+N >1]

2#>GE0S0@LGEF*KGO9BFH2525C f ?12 f GO2:9BF6X49<;V;12[CH2:24[FRP?@6Jn F 6)A f K p(x) 4CH2;@GS6*FHGS9 f FRLp?25C6#?Bh c n F*K@45F p(0) p(1) . . . p(n) 2 ÞX25<9<24:9 PQ45FJCG 46h_?B0E03?6

(n + 1) × (n + 1)  1 0  1 1   M =  1 2  .. ..  . . 1 n

M

0 1 22 .. .

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

0 1 2n .. .

n2

...

nn



  c    

?@C f ?@9@2:9@GO2:9 f 2 2V>GE0S0DGO9<;12 F*K12CH?649<; f ?B0EAPQ45FJCG f 2:64:9<; GTF*K @2 f FH?@C686*F4CÝFHGO9<M> C4[F*K12:CF*K@4:9 cM K12:9F*K12!2:9BFJCG3256M?Bh 4:CH2 04:9<; 1 M mij = ij c h_?@C T>GEF*K 0 i= 2 ?B.h . . n Ú 2[0F 1 ;122: 9<.?.FH.2 F*nK12;12[jFH2:= c&â GO9 f 2 0 =GS6Q14 49;<25C ¨ ?@9<;12 CHPQ0GS91 419B F d M Q PV4[FJCHG 2K@4[@2 N:=F*K12!250E03)M 9
   M  

2PA<6*FK@4§@2

M

K125CH25h_?@CH2

−1

0

      

n X tij

j=0

1 0 0 .. .

d





       =      1 1 1 .. . 1

=



1 1 1 .. . 1



       =      (

1 0



      1 0 0 .. . 0



  c    

GEh i = 0 c EG h i>0

£8?FH2>GS9QL<4:CFHG f A10O4:CDF*K@4[FDF*K124N1?:@2>6)A?1i;1; cVÚ 2[F N<2F*K12 a d = 2 b a b ≥ 0 b c 49<ω ; ?@CH;<25C?Bh GO98F*K12#CGO9<X?BhGS9BFJ2:52:C6IPV?1;@A10O? c K12:9 ω 2 GO9BFH2525C 2 ≡1 ω2 ≥

?@CX2,@2:C|=V 2!K@4§@2 kω b 49<(mod ;K12:9 f b) 2F*K12:CH2! GS6*1F6 2 c ≡ 1 (mod b) 49GO9BFH2525C 6*A f K!kF*K@≥4[F 0 kω ck

2

= 1 + ck b

Ú 2[F

K1259

a0 a1 . . . a n 

h_?@C~4:0E0

     

i

n X tij

ai =

j=0

d

N126*A f KF*K@45F 

a0 a1 a2 .. .

2a

  2a+ω   a+2ω    = M −1  2 ..     . 2a+nω

an

2a+jω =



n X tij

d

j=0



  c    

2a (1 + cj b) = 2a

n X tij

j=0

d

+

n X

tij cj

c

j=0

n â GO9 f 2 P 2>67228F*K@45F GO6X4:9QGS9BFJ2:52:CDh_?@C254 f K c tij GS6X49GS9BFJ2:52:C ai i j=0 d K125CH25h_?@CH2 F*K12MLGO6X4[=FRPV 6*F + a 2:4:CH0 x= cR¨ ?@+CH2·?5·@· 2:+C a6)1GSx 9 f+ 2 a0 n



2K@4§@2

   M  

p(i) = 2a+iω

a0 a1 a2 .. . an

h_?@CR2:4 f K



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2a

  2a+ω   a+2ω    =  2 ..     . 2a+nω

i = 0 1 . . . n



     

4:9<;2>4CH2;
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` c

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=

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≡ p (mod (b + 1))

c

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c

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p = nr

c

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(x + y + z)3 = n2 xyz

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(k + 1)u − (7k − 2)u + (k + 1) = 0

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c

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2

,

2

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E /7} E /7 }.

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= ≤

25CH2

m(S1 ) + m(S2 ) + · · · + m(Sk+1 ) k−3 k (k + 1)2 4 k+1 c = 2 4 k−3

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(k + 1)

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B E H

C F I

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P

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6 ......... .. ......... .... ......... .... ......... 5 ......... . . 1 . . . ... ......... ......... ......... ......... .... ......... ......... .. ......... ......... ......... ..... ......... ........... ......... ... ......... .... ................. ......... . . ......... 2 ......... ......... 3 ......... ......... ......... ......... ......... ......... ......... ..

.................P ....r..................P ...r ... P ...r...................r ...P ... .r P4

?L<4[F*K@6M4:CH26)4:GO;VFJ?N<22BA1GTB4503259BFGEhD?@9<2GO6M?1NFH45GO9<2:;h3CH?@P¿F*K12?F*K12:C c ?@C N:=4 f GOC f A10S4CD67KBGShSF~?Bh1F*K12 f ?1;BGO9<o6,2:@Ap259 f 2 F*K12 D67p+q 1 25D 2BA1GTB4503259 f 2 f 0S46)6 f ?@9BF4:GS9@GO9< K@4:6 2[03D 25P 9BFH26 .N1.2 .f 4: A< 2 4;@PQGTFHγ6IL1∈CH2 Γf GS 6,2[0T= γ f = f 0SG f 67KBGShSF6M49<;4:0E0I?BhγF*K12:P p+q 4CH2Q;BGO6JFHGO9 f F c h p + q γ = D 032§F N<2V6)A f KF*K@4[F GO68F*K12V0O?2:6JF84PV?@9<!F*K12 6 câ GO9 1f 2Q D F*K122Q 0S.GS.9<.2: 6 Dp+q 4CH2 m ` ` ` mA<9@GO@Ap2 cI K12:9!F*K12L<4[F*K i i ;@GS6*FHGS9 f F 6)A f K4:9 GO6 m Dm . . . Dp+q D1 . . . Dm−1 c :@2:C|=?F*K125C f = f 0SG f 6,KBGEhSFo§GT@256MCGO672FH?4QL145F*K GO684N1?:@2F*K120SGS9<2 +@qy ck KBA<6 2:4 f K ?BhKBG f Kt45F0O2:4:6*Fpx ?@9<2 25CFH2=0GO6QN1250O?ÃF*K120EGO9<2 +L10 qy =F*K@045F~F*K129@A< PN12:C 2BA1GTB4503259 f 2 f 0O4:676 f ?@9BF4:GS916o2 :4 f FH0T=V?@9<2G3;12:450vL<4[F*K pxGOPQ =@GS9<M p+q c ?BhG3;12:450pL<4[F*K@62:@A<450O6 1 p+q

p

ä c A R8+N >

?@C4L
,.-0/2143657- 8 U 1

d(n) = 4n

/ 5+>0@ 9 : >- A/Q.>

PR3Q.> /+

- 1/+-AUO J 3=< >8)UO $A >0>AN >]

Ú 2[F c ?@C2:4 f K L1CHGSP2 N124L254 f K GO6R4M9L
d(n) = (2 + 3β2 )

Y

(1 + 3βp )

c

p≥3

KBGS6CH2[0O4[FHG3?@96,K1?6DF*K@4[F ;
c

?@C c F~h_?B0E03?6RF*K@4[F 2K@4[@2 βp βp 4)K1β?Bp03;@≥6X1 F*K12CG35KFR6)pGO;<≥ 2?B5hÄ GO6X45F~03p 2546JF ≥ 5:4 9<;V= 2B(1 A<4:+ 0EGEF|=V GShv+ 49<4β ;V?@p9@0 =GSh h_?@C 1 βp = 0 c 2;12;BA f 2MF*K@4[F c f 4:0E0 : 2 9 2 p≥5 2 + 3β2 ≥ 21+β2

2 + 3β2 ≥ 2(1 + 1)

KBGS6§G @2:6

β2

β2 ≥ 2 1+ 1

β2 + 2

= 2 + β2 + β22

c

c β2 ∈ {0 1 2} h B = 0 ?@C B = 2 F*K1259 2 + 3β2 = 1 4:9<;N1?F*KQ6)GO;<256?BhÄ2BA<45FHGO?@9 2 2 1+β 4CH22:@A<450
n=2

GSPVL@0T=BGO9<oF*K@45F

n = 27 = 128

β2 ≤ 2

c KBA<6

:

h

F*K12:9 2 + 3β2 = 1+β CHGO5KFX6)GO;<2?Bh2:@A<4[FHG3?@9 2 `F*K@245FGS6 F*KBGO6?BA103; §GT@2 5 β5 5 cè β2 = 1

5 4

49<;F*K12:CH2PA<6*F8N124h34 f FH?@Co?Bh

β5 > 0

c 2 f 4919
β5 > 1

5

?@9F*K12

N12 f 4:A<672

ce K12:9 5 β5 5c β5 = 1 = 4 â ApN16JFHGEFHA@FHGS9<GO9 1 + c# KBA<6 2?1NBF4:GS19Q+F*K13β §GT@3β 256 5 h_?@CX450S0 256,?B0EA@FHG3?@49 c βp = 0 p≥7 >

K125CH25h_?@CH2

n = 2 49 · f 5?@39 f=0SA<2000 *6 G3?@9 F*K126,?B0EA@FHG3?@9164CH2

c 4:9<; 2 128 2000 Ù c 3 R / : K128;@GS4[C4:P 67K1?6~486*F4:GSC f 4:6,2:7N@CHG f >GEF*K 6JFJ25L<6R?Bh>G3;F*K PV4;<2 ?Bh A<9@GE F f ApN<256 c ÞX2§FJ25CJPGO9<2M450S01L6*A f KQN1CG f 6 c N@A112 GE03;Q4:9 n×n×n

,.-0/2143657- 8 U 1

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

PR3Q.> /+- - 1/+-AUO J 3=< >8)UO $A >0>AN >] c KBA<6 489<2 f 2:6)674:C|= f ?@9<;BGEFHGO?@9 K12X@?B0SAL14C450S0O250O2:L@GOL<2;!?Bhp6*GE)2 4:9<;!6*A f K!L14C450S0O250O2:L@GOL<2;@6 f 4:9N<286*F4 f 2:;FH? 24:6~ ×325× c 2 f 0S4:GSP F*K@45FIF*KBGO6R0O4:6*F h_?@CJP4 f ApN12?Bh6*G3;12 0E0p4:46X 4:9=PQA10TFHGOL@032?Bh f ?@9<;BGEFHGO?@9GS6X450O67?!9<2 f 12 f 4:9VN<2 f ?@916*FJCA f FJ2:; 2:6) 674:C|=B`:F*K@45FDGO6 4 f ApN<2?Bh6)GO;<2 12 n = 6` ?@9@0T=!GSh GS6#2,@2:9 c f £8`?6)A /12 49<;QF*K@45FXGTFGO6oLGTF*K GEF*KV= 4@18` 2:CÝFJ2 Q4[FF*K12 c ¢v?B03?BAGEF*K?@9<2>?Bhv2[G35KF f ?B0O?BA?@9F*K12>[i, L14iCGE+ F|=Q1]?Bh × [j, j4:9<+; 1] GS× 9[k, F*K128kFJC+ GOL@1] 032 i j k c 92:4 f K>N1CG f 4:0E05F*K122[G35KF f ?B0O?BAA<9@GTF f ApN<256 c ¢ÄK1?1?@672Q?@9<2?BhF*K12Q25GO5KF f ?B03?BA4:LF*K12:9 c KBA<6 F*KBGO6 f ?B03?BA
/ 5+>0@ 9 : >- A/Q.>

K@4[F f ?@PVL@032§FJ256F*K12 - A 8>A h_?@C>F*KBGO6!GO6)6)Ap2 c >GO67K=@?BAg5?1?1;L?:@25CDF*K12>6)A

½

½

+ " o+M$ >()'7+

R.3=< > R4365PU Ü@= â @2§FJ?@6*0O4§ â 4§ f K12,t49<; GTFHA d 9<;1CH2:2:6 d 6767? f GO4[FHG3?@9Q?Bh d P25CHG f 4 B â Ü£ gOL< 4:Lp25C J : L<452:6 > 5+> >@9 :

O R N R/+>!U 37>A

%½

fA

L@ApN@0EGO67K12;N:=F*K12 ¨ 45F*K125PV4[FHG f 4:0

º ½ [ O¨Á PJ

¤ â F c : c -0/ /+>!Q.>!O ,.38] R 1/ O

?¾

, 8c

KBGS6!N4:CH2GO9 f 0EAp;<2:; c¨ ?@6*F GOPQLp?@CÝFH4:9BFH0T= F*K12Lh_?@CvF*K12[GOCN1CH?@4;4L1Lp254:049<;4:CH2o[CH?BAFJ2 f K@9@G3BAp2?@CF*K12:PV2*49<;1HB4:CHGS45FHGO?@916 c K12M6JF=B032oGO6R9@G f 2[0T= 2;BA f 4[FHG3?@91450 4:9<; h_?BA<9<;!F*K12N1?1?14L10O2:4:6)AGTF*K!48FÝ?17;BGOPV2:916*G3?@91450BB4CGO4[FHG3?@9 GO6o9GS6F*K@45FvF*K125CH24:CH2#GS9@p9@GTFJ2[0T=>PQ49=MGO9BFH2525C6 6*A f K8F*K@45FvF*K12XGO9BFH2525C6Ih3CH?@P FH? 2 f 4:9N<2!4:CJC4:9<52;GO9467@A<4:CH2!4CHC4[=6,?F*K@n45FF*K12!4[@25C452?Bh49=VFÝ?VGS6 1 9CH254;BGO9< c 24: 0E09h3CH?@P486JFHAp;<259BFI?@C |?BA67?@CF 6x + 8 1 − x2 = 5 1 + x + 1−x ?Bh1F*KBGO9<8K@4§@2>N<2:2:99;1?@9<2 GO9g49k2,=@2:N@0EGO9< N:=g4 f ?@PVL@A@FJ25CQ450352:N1C4ç 6=16JFJ25P c â ?@P2ç?BhF*K12:672ç45035?@CGEF*K@PQ6 4CH2 @2:C|= 67?@LpKBGS6*FHG f 4[FJ2:; Oh_?@C¡2 :4:PVL@032 A<6*GO9< f =@0EGO9<;@CHG f 4:0O52N@C46¡FH?á@2:CGShE= GO9<2:@A<450SGTFHG3256 c 9032:;<52 ?BhF*K12:P3 4:9<;F*K12[GOC 0EGOPGEF45FHGO?@916 GS6 f CGEFHG f 450FH? 4L1L67225PV66)4; F*K@45FDF*K12L1CH?1N@0O2:P f ?@PQPQA<9@GTF=GO[9GE0S0o2:9 |?5= 6*FHAp;=@GS9<8F*K1 2 f ?@9BFH2:9BF6#49<;6,K@4:CHGS9<8F*K125P >GTF* KQKBGO6#?@CRK12:C~6*FHAp;12:9BF6 c

U

¾

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¾

gcd(Fm , Fn )

gcd(Lm , Ln )

gcd(Fm , Ln )

F*K@45Fv4CH2 f ?@9BFH45GO9<2:;8GO9 â 2 f FHG3?@9 cd CHP2:;>GTF*KF*K12:672X6*F45FH2:PV2:9BF6 GO9 â 2 f FHG3?@9o#2 ?1NBF4:GS9 f ?@PVL@032§FJ2 f K@4:C4 f FH2:CGE*4[FHG3?@916D?BhBF*K12CH2:@A1GSCH2;;BGTBGO6*G3NBGS0EGEF|=>CH 2:6*A10EF6 c FÄFHA8 >A8R/ 6*GEFHA<4[FHG3?@9 c KBA<6 h_?@ C2 :4:PVL@032 GO9 K12:?@CH2:P 24:916J25CF*K12BAp2:6JFHG3?@9 K12:9GO6 4 K12M 4916H2:CIGS6~F*K@4[FIF*KBGO6X? fJf ALp?@6)6)GON@GE0SGTF=?Bh ?@mC N<2[GOn9<M9<2[4[FHGT@2_?@CI)2:CH? GO90EG35KF?BhvS c m n â 2 f FHGO?@9 f ?B0E032 f FH6XFH?152[F*K125CRF*K12!250O2:PV2:9BF4C|=L1CH?@Lp25CFHGO2:68 ?Bh G3N1?@914 fJf GÄ4: 9<; Ú A f 4:6#9@A0S46JFDFÝ?!6,2 f FHG3?@916 c Õ Ï|ËÊgË ÉÔ DÔ DËÔ :Í|Ë

!

K122[0325P259BFH4:C|=0EGO9<254C~L
Ln = Fn−1 + Fn+1

49<; £82:[45FHG @28B450SAp256?Bh

"!$# %'&'(

5Fn = Ln−1 + Ln+1 n

4:CH2>GS9 f 0SAp;12;!F*K@CH?BAp5KF*K12CA103256

F−n = (−1)n−1 Fn

c

c

L−n = (−1)n Ln

)+**, -+."- /# -+.10-2(2&3'4-2(5# 67-+89+"65# 35($

S

Ü
KBG f K!h_?B0S0O?GOPQP2:;@GS45FH250 =!h3CH?@P Fn =

K12:CH2

αn − β n α−β

αβ = −1

2>GSPVPV2;BGO4[FJ2[0T=Q;12;BA f 2

*

Ln = α n + β n

α+β = 1

CH?@P

c

c

*6 GOPQL10O2QGO9<;BA f FHG3?@9 N@4672;?@9F*K12VCH2 f AGTF*K N1?F*KMCH2:4; 2 m = 0 d

F2n = Fn Ln

0 1 1 0 1 1 ...

6,?F*K@45F

l

CH?@P

! &! º :y

Fm

49<;V

* GEF*K125C

2,@2:9

⇐=⇒

2MGS9@h_2:CDF*K@4[F

Lm

gcd(Fm , Lm ) = 1

?@C

2,@259

⇐=⇒

3|m

gcd(Fm , Lm ) = 2

c

c

c N:@GO?BA<6)0 = K12:CH2V49<;6*ApN1672BAp2:9BFH0 = 2K@4[@2 GEh gcd(Fm , Lm ) = 2 c 49<;V?@9@0 =GEh

3|m

A -4-16

gcd(Fm , Lm )

GS914:0E0T=

= = = =

gcd(Fm , Fm−1 + Fm+1 ) gcd(Fm , Fm + 2Fm−1 ) gcd(Fm , 2Fm−1 ) ?@C N:=V 49<;Q c 1 2

:N =V N:=F*K12CH2 f A
ÜGEF*K PV42>GTFRL10S4:GS9!F*K@45F * GSh

Fm | F n

49<;

Lm | Ln

GEh

:

m|n

m|n

?1;<;B0T= c

¥v2 f 450S0

gcd

9F*KBGO6Q672 f FHGO?@9ç2CH2 f 450S0F*K12F*K@CH2:2ç2,= gcd F*K12:?@CH2:PQ6QCH250S45FHGS9<FH?tF*K12 c> K@CH?BAp5K1?BA@F#F*K12!CH2:6JFo?BhÄF*KBGO68L14L<2:C ,6 2:@Ap259 f 256 49<; 49<; 4CH2 {Ln } m c 2 n >GS0E0 49<; Lp?@6*GEFHG @2!GO9BFH{F 25n25} C6 µ n) |m| 4:0E03? ?BAGEF*2K= 9<2[max{µ 4[FHGT@23?@:CD2)25CH? | m} B4:0EAp2:6#?Bh m N@A@FR9<2,@2:C~GS9!F*K126JFH4[FJ25P259BFH6#F?Bm hÄ?BA

! ) ! # l

:y

l

:y

! l

:y

}

gcd(Fm , Fn ) = Fd

gcd(Lm , Ln ) =

(

gcd(Fm , Ln ) =

(

c

Ld ?@C 1 2

Ld ?@C 1 2

GSh |m|2 = |n|2 ?F*K12:C>GO672 c GEh |m|2 > |n|2 ?F*K125CÝ>GS6,2 c

d 9!250O2:PV2:9BF4C|=L1CH?1?Bh?Bh K12?@CH25P PQ4[=!N12oh_?BA<9<;GO9 K12?@CH25P c c M 2 ;h_?BA<9<;GO9 4:9<; c Ð & $Ë Í Í=@Í|ÓÍ3Ï3Í DÔ DËÔ :Í|Ë

!

l

º

! :y

Fm | Fn

GSh49<;V?@9@0 =GEh

m=2

?@C

m|n

c

A -4- 16 K12M;@G @GS6)GON@GE0SGTF=6JFH4[FJ25P259BFF*K@4[F GEh GO6R : cDâ GS9 f 2 m | Fn c £8F c# 2M>GE0S0vF4:2676*= ? 6)AGEF*K Fm `F*K@4[F~GO6 m c GSh `F*KBA<6 GEh Fm = 4:9<; Fd Fm | FF*K1n2:9 m≥3 m|n m≥2 F m | Fn m=d l

º

! :y

Lm | Ln

GSh49<;V?@9@0 =GEh

m=1

?@C

m|n

?1;<;B0T= c

A -4- 16~ K126*F45FH2:PV2:9BFRF*K@45F GSh ?1;<;B0T=VGS68 c8â GO9 f 2 GTF m|n GO6o?1N:BG3?BA<6F*K@45F GEh Lm | Lc n£8? 6)AF*K@45F 4:9GTF*K Oh_?@C L
! :y

OG SGSG SGSGEG SGT

Fm | L n

GEhv4:9<;V?@9@0T=!GShv?@9<2?BhpF*K12Mh_?B0S0O?>GO9<MK1?B0O;16

m = 1 m = 2 m=3 m=4

49<; 49<;

3 | n

?@C

n ≡ 2 (mod 4)

c

º

A -4- 16 2#F*KBGO6 26JFH4:CFXN:=V :GO9<≥ 8F*3K126*FJCHF?@m 9<525CXK=@Lp?F*K1256)GS6XLF*nK@4[ F Fcç m ≥ ct K12:9 h_?@ C>6,?@PV2 KBA<36 N:= F = L n gcd(F , L ) = F = L ≥ 3 m2:?@CH2:P¿n m K1 câ nGS9 f 2 2K@4§@2 4:9<; m n GS686*FJCG f FH0 = |m| > |n| F = L = L L 2 L@0T=BGO9<Mm GO9 f CH2546*GO9< 26722>F*K@4[F 2 GSPV F*K@4[F n c Ü |n|2 , c ?@9<2K@4:6 KBA<6 2n | m c Ln | F2n | Fm

c K1259 Ü A@F < `BK1259 f 2 GOPQL10 =@GS9<MF*K@45F ?@C c co K125CH25h_?@CH2 n = 1 4:9<2; ÜGTF*K £8? 46)6)A 2:6XF*|n| K@4[2F Fm c = L d Fm = L d ≥ 3 Fm = 3 KBA<6 F*K12 f ?@9<;BGEFHGO?@9 c h GOPQL10EG3256XF*K@4[F `@F*K@4[FGS6 Fm | Lcnd f ?BA
GS91 c>â GO9 f 2 4:0E0T= 6*AGTF*FK m | Ln CH254;@6 3 | Ln 3 n = 1 1 0 1 1 2 0 2 2 1 0 ...

K125CH25h_?@CH2 c GEhv49<;Q?@9@0T=!GSh | Ln n ≡ 2 (mod 4) KBGS6 f 3?@PV c L@032§FJ256RF*K12L1CH?1?BhÄ?Bh K12:?@CH2:P A
!

l

º

:y

Ln | Fm

GEhv4:9<;V?@9@0T=!GSh

n=1

?@C

2n | m

c

A -4-16 h c h F*K1259 49<; F*K1259 n = 2n : 1c ¢Ä?@9@L cL nK125|9 F2n | Fm :N =ç 49<; 25Cn6,2[= 0T= 1 6* AhOA<9 f FHG3?@9?Bh 2K@4§@2 ` |m| n n=d 2 GS> c F*K@45F~ 6 |n|2 c Ü |n|2 , 2n | m K12:?@CH2:PQ6 49<; K@4[@2MF*K12Mh_?B0S0O?>GO9<8C4[F*K12:CR6JFJCHGOGS9< f ?@91672BAp2:9 f 2 c

;! . &!o+

ry )yzTz

º

Ln | Fm

GShv4:9<;V?@9@0 =GEh

F2n | Fm

c

A -4-16 Ü?F*KQ;@G @GS6)GON@GE0SGTF= f ?@9<;BGEFHGO?@916#4:CH22:@A1G B4:0O2:9BFDFH?

n=1

?@C

2n | m

c

h f ?BAF*K12 f ?@9@25C6,2GSPVL@0SG f 4[FHG3?@9KBG f KçGO6!67?@P2 K@45F6*A
#0!) %

/ )##! 1" ! 2 ( ! 2|n

3

3 = L2 n ≡ 2 (mod 4)

"#%$'&$ ()"!* ,+(-) .

L2 | L n

/

0 Ë)ËÔ5ËDÒBË < Ü A c

>8 3:R0A : [ 1

94>ATX < >0- A :

S À » ½ » H¼ `½ ½ S

F*K2;BGEFHGO?@9

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£8?FH2MF*K@4[F

cos

C−B 4

6)GS9 f 2 C − B

6= 0

4

π−A 4 π−A sin 4

2 sin

π π c 259 f 2 ∈ − , 4 4

π + 3A 4 π − 3A π−A sin − sin 4 4 A π−A c 2 sin sin 2 4

= sin = =

6*GO9 f 2 π − A π−A 6= 0 ∈ 49<;!F*K12>CH2:6*A10EF~h_?B0E034?6RGOPQP2:;@GS45FH250 = c 4 £8?FH2>F*K@45F

sin

º /

qÝq c ,.-0/2143657- 8)9: V5 0<- 1O

-0/

0,

π 4

c 2:9 f 2

1 = 2 sin

18523 : -0/ /+>!Q.>!OM 568$37>AKWIR 0>8O

Á

1 A 2

V3O P, J8c c K152 9GTF

d 6IA<6)A<450 O0 2[F 4 9<; a = BC b = CA c = AB s = 21 (a + b + c) p GO6~2[0S09
s

ac(s − b)

49<;

s

ab(s − c) s

4:9<;

BI =

â GOPGS0S4C0T=

£8?

259 f 2

b

b a+c

ac(s − b)

c IE = a+b

s

ab(s − c) c s

ID =

s

DC

s

s

CI =

1 BI 2

ID

0O2[F

θ = ∠BIP

c K12:9

· CI sin(2θ) = [BIC] = [BIP ] + [P IC] =

1 P I(BI 2

+ CI) sin θ

c

2BI · CI cos θ cVÚ G32 >GO672 2ID · IE cos θ câ ApN@6*FHGTFHA@FHGO9< PI = IQ = BI + CI F*K12!4N1?:@2!2 L
1 p

c(s − b) +

√ 2 bc

p p p = b(s − c) (a + b) b(s − b) + (a + c) c(s − c)

KBG f K f 4:9VN<28FJC4916*h_?@CJPV2;GS9BFJ?

259 f 2

p p √ p c b(s − c) + c(s − b) = 0 2 (s − b)(s − c) − bc

c K1259 2(s − b)(s − c) 1 67?F*K@4[F 4(s − b)(s − c) = bc cos A = 1 − = bc 2 c A = 60◦ - >8 53 6d 6X4 f ?@916,2:@Ap259 f 2 GO6#2:@A1GE0O4[FJ25C4:0@>GEF*K 46GEF6 f 259BFJCH?BG3; c 4DEP I 2 P>,Z0'9 !03 ( !(>@( / (A$9 ? ?DE. . & : . 4& ! .2 '. 0 9 '@ ( ? P

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ßXãà c

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∠ACB = 120

TÁ º

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Á

qA½ ½

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2

>GE0S0L
sin ∠EAC

=

sin ∠ECA

sin ∠A

=

1 ∠C 2

∠AEC

PV22§F

C

1 ∠C 2

DC c 1 ∠C 2 DA

AC

D D

B

= 2 cos

sin BC = 2 cos 21 ∠C = 2 cos BA

Ú 2[FDF*K12>49<§0O2NBGO672 f FH?@CR?Bh

E

4[F

D0

sin ∠A sin ∠C

c K1259

DC c DA â A 1 D0 C EC = = 2 cos 0 DA AE

1 ∠C 2

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AC

259 f 2

0EG32MGS9!F*K@4[F?@CH;12:CR?@9

∠AED > ∠AED 0 = ∠CED 0 > ∠CED

c

∠ADE − ∠AED < ∠ADE − ∠CED = ∠DCE =

1 ∠C 2

c

< 60◦

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ßXã Ø c

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∠AED = ∠CIF

-

J

A -

/ (À

º

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AC 4:9<;QAB 032§F N
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.... ............ ... ... ...... .... .... ........ .... .. ... .... ... ... .... .. .... ... ... .... .... .... .. .... .. ... .... .. .... ... ... .... .. ... .... ... ... .............................................................. ... .... ........ .... . . . .... ........................................ ..... . .... .... . . ... . . ... ................ . .... .... .......... . .... .......... .... ........... .. ..... .............. ... .... .............. . . . . ... .... .... .... ............... .... . . . . ... .......... .... ... .. . . . . . . .... . ... . . . .......... .. .... ..... . . . . . . . . . ... . . . .......... .... . .... . . . . . . . . .... . . ... . . .......... . .. . .......... ... ........... ..... ........... ........ ... ........ ... ........... .... ...... ............... ........... . ................................................................................................................................................................................................................

αα

D

E

β

I

γ γ

β

B

C

F

Ú 2[F c8 K12:9 c 4:9<; γ = ∠C/2 α + β + γ = 90◦ £8?FH2MF*K@4[F α = ∠A/2 β = ∠B/2 c ∠ADE = ∠CED + ∠DCE = ∠CED + γ d 0O67? A<6*GO9<oF*K12K=1L
F#h_?B0S0O?6#F*K@45F

Lp?BGS9BFH6

∠ADE = ∠BIF = 90 − β = α + γ

cM KBA<6

∠CED c ∠IED K12:9 = ∠IAD 4:9<; 4:= CH2 αf ?@9 f = f 0EG f[ A E I D

cM KBGO6GOPQL10EG3256F*K@4[FXF*K12

∠AED = ∠AID = ∠BAI + ∠ABI = α + β = 90◦ − γ = ∠CIF

c

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DI = EI

'

∠A = 60◦

?

1 !" #%$ &(' )* #,+.-/10$,-324-3)568795.:;5#<5 = ->6?,5%!95 @A { M_HKJ F CDDEF
=

AB

−

AC

<5 N < R0A @ Z]

,.-0/2143657- 89 : Ú 2[F

¯½ R :R?Á

4@ >8OJ 3+>8 >0- @ > cè K1259

08.5 /

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∠ACB =c 2θ ∠ABC = 4θ ∠BAC = 180◦ − 6θ 6,?! F*K@4[F £o? ◦ ∠ADC = 90 − 3θ = 7θ −â 90◦ Ú ∠DAB = 4θ − (90◦ −c 3θ) ¤#6*GO9<!F*K12 GS9<2 4§ h_?@C ◦ ◦ ∠DAC = θ + 90◦ 2?1N−FH456θ) GO9 FJCHGS49<§0O2:6 = (7θ − 49<90 ; ) + (180 DAC BAC 1 CD

+

1 AC

= = = =

KBG f K f ?@PQL10O2[FH2:6RF*K12>L
sin(90◦ − 3θ)

1 + AC sin(90◦ + θ) AC 1 cos 3θ +1 AC cos θ sin 2θ cos 3θ + cos θ

AB sin 4θ cos θ 1 2 cos 2θ cos θ 1 · = 2AB cos 2θ cos θ AB

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CB

1 !" #%$ &(' )* #,+.-/10$,-324-3)568795.:;5#<5 = ->6?,5%!95 @A W3G%H \G HKJG DE.MON.H8EN DRSET.CPKGT1\G,XENT FKCP JHKJLH 8M VGN 4ABC ABDRSET.CPKGT \G,XENT B FKCP J]HKJLH L NT WOGH D \ G K H J ( G D . E > M . N H E N BD = BC Y BE LNT CD LNTjF C CDDEFN.HQGR F,GPSHSM3ENjE.AC CE = BC a RSEVGHKJLH ∠BAC DP EP + = 2 sin ∠BAC = 90◦ BE CD 2 0d[efO-g@]' )4ij-/kS+l$md8nl5f35->dod/$m6 p"e$%@68r 5 @lkK$ GH AB = c BC = a LNT CA = b BMONPQG^HQRSM>L N|%WOGF DBC L NT ECB L RSG4M>F,EF
2

¤#6)GS9<oF*K12 â GO9<2 Ú 4[h_?@CDFJCGO4:9<§03256 BE = 2a cos

1 C 2

¤#6)GS9<oF*K12 â GO9<2 Ú 4[h_?@CDFJCGO4:9<§03256 DP =

a 1 A 2

cos

BCE 4:9<;

sin A + 21 C

4:9<;

CBD

2?1NBF4:GS9

CD = 2a cos

49<;

DBP 49<;

ECP

1 A 2

sin A

=

1

sin A +

2

cos

CD

1 C 2

1 C 2

1 = sin A + cos A 2 = sin A +

FRh_?B0E03?6~F*K@4[F

1 2

cos A

+

sin A +

2

1 B 2

cos 21 B sin 12 C cos

1 B + sin 12 B 2 cos 12 B cos 12 C

1 A 2

cos cos

1 B 2

!

sin A + 21 B

cos f 49VN128CHGTFFJ259Q46

KBA<6 *F K12K=1L

cos

1 C 2

c

28F*K12:9V?1NBF4:GS9 a

EP =

1 B 2

!

cos

1 C 2

c

!

c

c 49<; F*K125CH25h_?@CH2 ∠BAC = A = 90◦ 0 -9 0 @9 !I. < ! . &!%2;' : % %7%3&.2 .2%288# P . &'! (1 / 9:0 ? !,+Z+% 3 !& ! &. 3&.2 .2%28T@;' P;. . P. 3 / B &@ ,Z(' :88 . 3 DC! .7DEa92&! /6 )^B %2 . # '( B? ,* 1 8 D5! &G03@0 / $? ? !-!,*' . 8 %7!DCD5/ 3% ;3+,-% ;3 '/ 3@ 1? C( (1 P.2 $ 3&.2 .2%28 '!I. &8A0 / 9 . Z(''' )C0'DCDK&.2%2 0' // $. 3% B& & 8 / @9<@ . 43(, 'A0'& D 2 ' 3 % . 1DC. 3C%2;' '<# A ?

cos A = 0

4vS " ~ !" #%$ & ' )5k[dO5 7q@9$kS @) 6n- jp95!-g&#<6i 36

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¾

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0d[efO-g@I')e[$,- +5. 6#fSe.&$@9f36 @^ij->dodO# ld>dg$" -35f/$@#fO-[fSef/$m6A*K@Afg 6

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GTi = Ai Ti

EHSM>N|HKJLH

sin ∠GAi Ti

LNT

sin ∠Ai GTi

GUi = Ai Ui

sin ∠GAi Ui sin ∠A GUi

{ G^|G%H i Ai T i = A i U i ∠Ai GTi + ∠Ai GUi = π GTi sin ∠GAi Ti sin ∠GAi Ai+1 L NT

GUi

GNPKG

3 Y GTi

i=1

GUi

=

=

=

sin ∠GAi Ui

=

sin ∠GAi Ai+2

3 Y sin ∠GAi Ai+1

i=1 3 Y

sin ∠GAi Ai+2

sin ∠GAi Ai+1

sin ∠GAi+1 Ai

w JGRSGF CW_HYE.W>WOE { ~F M GTM>LHSGW X U U i=1

=

3 Y GAi+1

i=1

GAi

= 1

:

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x

y

z

1 1 1 + + x y z

√ + (x + y + z) ≥ 4 3

c 0.d efO-/@Z') Q=5.&) d/f36A0 5@^? #%$%6 ~6 0 L EHSG R FKH9HKJLH YERqLNX HKJG;MONGCLW>M_H/X u{ > 0 MOF;GCM V.LWOGN.H HQE4GL P JhE.YHKJGYE.WoW3E M>N|

√ √ 4 3 1 −u ≥ − 2 3u2 u 3

√ √ 6 3u3 − 3u2 − 4 3u + 3 ≥ 0 √ √ √ 2( 3u)3 − ( 3u)2 − 4( 3u) + 3 ≥ 0 √ √ ( 3u − 1)2 (2 3u + 3) ≥ 0

w JGW>L FQHM>NG .CLWoMoHgX4M>F;PQW3GL R W_XIHQR CG ~ GNPKG

1 1 1 + + x y z

− (x + y + z) = ≥

\ ERL NX HKJGM>NG .CLWoMoHgX u{ > 0 HQEGLP JE.YHKJGYE.W>WOE MON|

X 1

−x

x

X 4√ 3

3

− 2 3x

+ (x + y + z)

= ≥

X 1

≥ 0

x

+x

X 5√ 3

3

√ = 2 3

M>FGCM V.LWOGN.H

≥ 0 ≥ 0

w JGW>L FQHM>NG .CLWoMoHgX4M>F;PQW3GL R W_XIHQR CG ~ GNPKG 1 1 1 + + x y z

2

√ √ 1 5 3 +u ≥ − 3u2 u 3

√ √ 3 3u3 + 3u2 − 5 3u + 3 √ √ √ ( 3u)3 + ( 3u)2 − 5( 3u) + 3 √ √ ( 3u − 1)2 ( 3u + 3)

√

√ − 3x2

√ = 4 3

/

N ARA @",

qÝq c ,-0/ 1.365+- 8 9: >A R08 :)] 34

9<2BA<4:0EGEF|=

_N

/ Á

1>A 5

q½

85 0>A U 523 :O

2K@4§@2

√ c 3

x+y+z ≤

2K@4§@2

√ 1 1 9 1 + + ≥ ≥ 3 3 x y z x+y+z

49<;V

R3Q.@ >9 1A/QO

x2 + y 2 + z 2 1 = √ 3 3

Ü@=!F*K12 d¨ ¨

r

x+y+z ≤ 3

9<2BA<4:0EGEF|=

Ü@=F*K12Q¥v?1?F ¨ 2:4:9 â BA<4CH2

49<;VK1259 f 2

CH?@P

>AN < 3:O Y 3 37- 0- 8

F*K12 f O0 45GOPh_?B0E03?6 c * 24:LG f 2

c

1 1 1 + + + (x + y + z) x y z

1/4

≥ 4

x+y+z xyz

= 4

y z y2 z x z2 x y x2 + + + + + + + + yz z y zx x z xy y x

≥ 4 9

r 9

x4 y 4 z 4 x4 y 4 z 4

= 4

!1/4

= 4

(x + y + z)(x2 + y 2 + z 2 ) xyz

√ 4

1/4

1/4

√ c 9 = 4 3

T 3@ , B19 0&2 3&.2 .2%2^ A<&. T3 Z // . ! !^9:0 * &+@<>9 1!& '6 !^9 2 , !,Z0 . -' 1DC&. # 0 9 ' @( $?P ! ' . % + 197, 1,ZI0 A,Z'3 %:! 3% 3&.2 .2%2 )@<3% @<'%7)(),)0 9 I0 @9 8!I. < ! .&!%2;' : % %7 . . % # P . ! (1 /0 > )? B9!^9 ,$,-9 )&/2 % %7 . . % )&/2\@) '9:0 ? !,+)>%7 ' ! ! . 3&.2 .2%28B@8;3 P;. . P. 3 / + @ * :>,Z( 8'3:8 . ' &DC& . DE 92&! /6 ) B %2 . #$9 0 > 9 ? 8 *3367;3 &8 K4' <! E0$/ ( ( B )'% 2 Z,'!I / \4 1 / , ?*! @,! B \ '( @ 19 B9 . . 3<6 I9 >, 4 ,* + B 3&.2 .2%2I $ &% & $ &% &C(,*@, 1? C ( (1*P. 3&.2 .2%28 !I. &8 0 / &( . 1,Z( 9 )&/ & % &% . . % )&/ @) )? ! ? ? !9 . > +(1> % &! '% * B !I. 0' /8. %7K9 ! % . % &% /E@<'%7Z(, & 9 . Z ('' ' ) 0'DCDK. % 0' // $ . 3% B & & 8 * B@9 @ . 43(, '+0'D 2 3 3 % . 1DC. >3K%2;' '<# A ?T+!!%-%7= % ;&. <$%2;'8DE. %7%7 7% .2!5 3 ZD5< $%2;' 7 = . & !! 92'P8 .2%28$! ' ! 9 3P . % ! ! ^ 9 3P . % \0&!6 ;! 6 ;P=> 92'P8 .2%2!? @;'K< % T678 6< C6:3 . % ' 8 ! A* 592'P8 .2%2!? (13K 2 T%2;'KD5%2;'&^ 8 &&* ! 8 % . P . ? .

! 2 >< DC/)&\% ;!%)P . % Z;32>. K/.2%2;' +% ;3Z%7=Z. 3P . % . . '& √ ? x=y A =36 /z5=&!1/% 8. '3I%2;'E& ' . % .2T%2;&&% . xk > 0 (7 k = 1 2 . . . n) 8&67;T% ;!% n P 1% ;3 78 2

k=1

!

xk = 1

a b>0

a

n X 1

k=1

xk

!

±b

n X

k=1

xk

!

≥ (an ± b)

√ n

3

0&;&Z3/ 3! - . DE. >6 DCD5 '%1=>;&.26 ;E. % ;3B/ A67. 67 -=>;3 ?>9 3! /3!> 3 8 . /C %2;' A. λ ≥ µ > 0 'Z. x > 0 (7 k = 1 a2 =. .b. =n)1 8&67;K% ;!% n P

k=1

xλ k = 1

k

A%2;'

n X 1

xµ k=1 k

g 5 @A ?

!

±

n X

k=1

xµ k

!

≥ n(λ+µ)/λ ± n(λ−µ)/λ

3

; m `K!" #$&' ) ' '5# it$%+ 5'-O5 @6#fSe.&$@9f36A*$+S5@6

w JGYGLHSCRSG T F,E.WoCHSM3EN`HQEjDRSE\WOG % u 8 MOF U M>FWOE { MON|`DRSE\WOG U GH A0 B 0 C 0D0 \G1L4CL TR M>W>LHSGR LW { M_HKJ(L N(MONFNF MOTG HSE4\GHKJGPQENV G ^CL TR M>W>LHSGR LW { JEF,GFKM3TGPFDLFL R LH HKJGVGR[HQG IHSE4HKJG^WoMONG gE.MONM>N|MoH~HSE P a RSEVGHKJLH P MOF HKJGMON.HSGR FN.H E.YHKJG^T.MOL |ENLW>F AC L NT BD MoY9L NTENW_X1M>Y P = O 0d[efO-g@I' ) e [$,- +5. 6#fSe.&$@9f36 @^ij->dodO# ld>dg"$ -35f/$ g@#fO-/fSefg$%6A* @9f/ 6

I GHACF"WOL \GWHKJGDE.M>N.HSF8F,E HKJLH A0 B 0 C 0 D0 WoM3GEN DA AB BC CD RSGF DGPSHSM_VGW_X

∠A0 OB 0 = ∠A0 D 0 O + ∠OC 0 B 0 s]GJLVG CFKMON|HKJGY3L PSH8HKJLH8HKJGMONPQGN.HQRSGWoM3GF ENIHKJGLN|%W3G\M>F,GPSHQER F ∠A0 D 0 O + ∠OC 0 B 0 − ∠A0 OB 0 = ∠A0 D 0 O + ∠OC 0 B 0 + ∠OB 0 A0 + ∠B 0 A0 O − 180◦ =

1 2

(∠C 0 D 0 A0 + ∠B 0 C 0 D 0 + ∠A0 B 0 C 0 + ∠D 0 A0 B 0 ) − 180◦

= 180◦ − 180◦ = 0◦

BCDDEF,G NE { HKJLH P = O F M>N| OW LM U LWOEN| { M_HKJ HKJG]Y3LPSH^HKJLH L NT OB 0 BC 0 L RSGP[XPQWoM3P { GJ L%VG OD 0 DA0 ∠DOA0 + ∠A0 OB 0 + ∠B 0 OB = 90◦ − ∠A0 DO + ∠A0 OB 0 + 90◦ − ∠OBB 0 = ∠A0 OB 0 − ∠A0 D 0 O − ∠OC 0 B 0 + 180◦ = ∠A0 OB 0 − ∠A0 D 0 O − ∠D 0 C 0 O + 180◦ = 180◦

GNPKG L NT LRSG4PKE.WoW>M>NGLR BM M>W>L R W_X LNT L RSG4PQE.W>WoMONGL R } H U O D Y E.W>WOE { F~B L ANT OBD C EKH NJVLGH R OF,GW_=X PFKCM>DF~DHKEJF,GGhMON.HKJHSGLR H FF MONhER~EN1HKJGHQR MOLN|%W3G C 0D0 O

} & )

r
º

c

∠A0 P B 0 < ∠A0 OB 0

c

A -4-16 ¢Ä?@916*G3;12:CMF*K12FH4:9<52:9BF>FJ?F*K12 f GSC f 032 c CH?@P ¢0S4:GSP 4[F 2 A0 OB`50 GTFh_?BO K@4[@2 E 0 3 0 ? 6 * F @ K 5 4 Ä F * F 1 K # 2 H F : 4 < 9 5 : 2 9BF 0 0 0 0 0 0 +h ∠OC 0 D 0 > ∠A B O 5?12:6XF*∠A K@CH?BOD Ap5KVF*= K12∠A GS9BFJ25B CHGO?@O C#?B 4 < 9 ; * 6 O G P S G S 0 4 C T 0 = * F @ K H C B ? p A 5 Q K * F 1 K 2 O G B 9 H F : 2 C 3 G @ ? X C ?Bh 0 0 GS49<§0O2 ∠A OD c f f f f c 5 2 9 2 S G C 3 0 2 S G B 9 J F 5 2 C , 6 2 H F v 6 J F H C @ ? @ 9 0 8 = 5 4 < F * F 1 K X 2 < L B ? O G B 9 F 0 0 0 0 ∠B C 0 D 0 O 46 f 0S4:GSP2:; c O KBA<6 OC GS6#?BA@FH6*G3;1 2 f GSC f 032 A OB 4 < 9 ; P A0 OB 0 ∠A0 P B 0 < ∠A0 OB 0 d 676*A
259 f 2 49<; 4CH2898 53 6 £o25GTF*K12:C?BhçF*K12á6*ApN1PGEFFH2; 6,?B0EA@FHG3?@916&4;<;@CH2:6)6,2:; F*K12 ?@C PG35KFX9 90 f 45FHBGO?@91=6 90 4CH§AGTF*K?@9@0T= PQGS9F*K12o6*A

F*K12032[hSFIN:= O49<§0O2:6 0 0 KBG K2:@A<450O6 ∠B P B −∠BP B −∠BC 0 B 0 46DGS9 GO916 f CHGON<2:;GS94 f GSC f 032o4:CH2M2:@A<450 KBG f KGS9FHAF*KBGSCH;Q0EGO9<2?BhpF*K12<914:0672[FJ?B h2:@A<4[FHG3?@916XGS9F*K122:4:CH−(∠P 0EG325CR4CHC §A
D

- C1@,1?E(>(A P . 3&.2 .2%28 1!I. &80 B?

u . K!" #$& ')(?e5 @-g&6 5dod35.&d>-g&6q0%!95-O@q

@.- $%a ? RSEVGER T.MOF DRSEVGHKJLH HKJGCNMOCGF,E.WoCHSM3ENE.YHKJGF XFQHQG U E.YAG .CLHSMOENF bb0 + cc0 = aa0 − rr 0 2 2 2 a2 = b 2 + c 2 a0 = b 0 + c 0 0 0 0 0 2r = b + c − a

2r = b + c − a

L U EN| G RSENZRSMO|JH QH RSM>L N|%WOGF { J GRSG r LNT r0 LRSG^HKJGM>RLFF %| M_VGN(\X L NT a0 = 13 b0 = 12 c0 = 5 r0 = 2 a=5 b=4 c=3 r=1

132> 5 < R-AO1U 3 1@ >8 3:O

,.-0/2143657- 8 9 :

½

- 8

5 / / U -0/ /7>PQ5:R.3+> 8)U36523 1.3+>!OX0- A - 8$37-AOBY\[^]

K126=16JFJ25P ;GE0S0GEF*KQ;BGSåÄ2:CH259BFL14CGEF|= 6*A f KF*K@45F 4:9<; 2 2 m > n {b c} = {m − n 2mn} KBA<6 0O2[F K125CH2 4:9<; 4CH2FÝ?CH250S45FHG @250 = 2 2 {bGO9B FHc} {m n c â GOPGS0S4C|= L n0 =QL 2 Ý F ? H F G @ 5 2 {b0 c0 } = {x2 − y 2 2xy} x co y GO9BFH2525C6M?BhI;@GEå25CH2:9BF#L14CGEF|=49<; K12:9 x > y a = m2 + n2 r = mn − n2 c 2 2 0 2 a0 = x + y r = xy − y K125CH24CH28FÝ?Q2567672:9BFHGS4:0E0T=;@GEå25CH2:9BF f 4:6,256RF*K@45FI2>9<2:2;!FH? f ?@916)GO;<25C c

E /7

c 0 0 2 − n2 , 2mn, x2 − y 2 , 2xy) K1(b, 259c, F*K1b2,2cBA<)45= FHGO?@(m 9 S G # 6 2 B 1 A T G B4503259BF~FH? 0 0 0 0

r

bb + cc = aa − rr

c

(m2 −n2 )(x2 −y 2 )+4mnxy = (m2 +n2 )(x2 −y 2 )−(mn−n2 )(xy −y 2 )

K12:9h34 f FH?@CH2;

F*K124N
(nx + ny − 2my)(my + ny − 2nx) = 0

c

259 f 2 25GTF*K12:C cMâ GO9 f 2F*K1256,2FÝ? f 4672:6 ?@C P2§FJCHG f nx 2nx 4CH2>6=1PQ 8 2 +>ny GE0S0<4= 6)6)A<2my PV2MF*K@4[FDmy F*K12>+ h_?@ny CHP25= C~GS6R FJCHAp2 cI KBA<6 c n(x + y) = 2my

c â GO9 f 2 4:9<; 2PA<6*FÄK@4§@2 n) = gcd(x+y, 2y) = 1 m = x+y K125CH25gcd(m, h_?@CH2 39 n gcd(m, n) = 1 c 49<; m + n ≡ 1 (mod 2) c r 0 0 2 2 2 2 K1(b, 259 c, b0 , c ) 0= (m 0 − n 0, 2mn, GO6#2:@A12xy, G B4:0O2:9BxFDFH− ? y )

E /7)

bb + cc = aa − rr

2(m2 −n2 )xy+2mn(x2 −y 2 ) = (m2 +n2 )(x2 −y 2 )−(mn−n2 )(xy−y 2 )

KBG f K!h34 f FH?@C6GO9BFH?

(3ny + my + nx − mx)(mx − my − nx) = 0

c

h F*K1259 KBG f K8GO6ÄGOPQLp?@6)6)GON@0O2 f 2DF*+my )6 GS9 3ny K12R0O25hSF+nx−mx 6*G3;12~GO62,@2598=49<0; F*K12RCHGO4ny 5KF6)= GO;<2~(m−n)(x−y) GO6?1;1; c 259 f 2 mx−my−nx = 0 KBG f K!GO6#2:@A1G B4:0O2:9BFDFH? d [4:GS9

(m − n)(x − y) = ny

c

6*GO9 f 2 2K@4[@2 4:9<; − n, n) = gcd(x − y, y) = 139< ?FH28F*K@4[F mPQ− c gcd(m KBA<6 A 6Jn < FRN<= 2>2,y@2:9 c (m, n, x, y) = (x, x − y, x, y) x−y = n x

d 0S06,?B0EA@FHG3?@916FJ?!F*K12?@CHGO§GO91450vLGEF*KFJCGTBGO450~Lp25CJPA@FH4[FHG3?@91649<;t;BGS0S45FHGO?@916 c 4@+]%6 ?@CM4:9= c GS9<; LA<6*GO9<oF*K12>CH2: 6*A10EF6 cÄd 0E0p6)A f KL14:GSC6#4:CH2MF*KBA<6 f x?@916JFJCHA y

0 -9 0 @9 !I. < ! . &!%2;' : % %7%3&.2 .2%288# P . &'! (1 / 9:? 8 3'6 ;' &8 \4' <! 0/)' &9 . ( 3 ' ) 0'DCDK&.2%20' 2/&/ $. 3% B?

9:0

^ j 9 8 K!" #$&j' ) 5d/f+9$% ?,5 @le# 6 S#edo-3@l$@ )2 @5# -[e 2? 6@@#'me.kS=6%e#f -O5 GH n ≥ 3 \GL N && NLHSCR LWlNC U \GR GHSGR U M>NGHKJGF U LWoW3GFKHNC U \GR FKCP JIHKJLH HKJGGN.HQR M3GFE.Y9LNXRSE { L NThE.Y9LNXhPQE.W>C U NhE.YHKJG U LHQR M µ = µ(n)     

1 2 .. .

a1,2 a2,2 .. .

··· ··· .. .

a1,µ a2,µ .. .

n an,2

···

an,µ

    

L RSG^TM>FKHSM>NPSH~NC U \GR F;Y3RSE U HKJGF,G%H {1 2 . . . n − 1 n} L NT1HKJGNC U \GR F;M>N GL P JRSE { F C U HQEIHKJGFCG & -/f/ # l2h2Z$@9f E F<E.W>CHSMOENF { GRSGvRSGPKGM_VGT YERZHKJ.MOF DRSE\WOG U E { G,VGR I{ G { GRSG MONYER U GT\ X T { L RST wAO4 sZLN|~LN T 9LM U MON| JL E sM>WoY3RSMOT LCRSMOGR NM_VGR F M_H/X

sZLHSGR W3EE HKJLHHKJG,X JL%VG`E\HSLMONGT F<E U GZDLR HSM>LW RSGF CW_HSF BDG PQM lPSLWoW_X HKJG,XhJLVGDRSEVG T1HKJLH YERLWoWlETT n ≥ 3 µ(n) ≥ 3

µ(3) = µ(7) = µ(9) = µ(11) = 3 µ(5) = 5

A YERLWoWlETT n { J.M3P J4LRSG^T.M_V.MOFKM3\.W3G\X µ(n) = 3 a RSEE.Y3F { M>WoWlNEH \GI|%M VGN1XG%H F M>NPKGsZL N|L NT JL EL RSG^PKEN.HSM>NCMON|HKJGMOR { ER EN1HKJGDRSE\.W3G U w JG;DRSEDEFNJ.MOF"F C\ M>FWqETT n w JGIU DLR HSM>LWARSGF CW_HSF L RSG1GNU E.C|JHQEhF,JE { HKJLH HKJ.M>F M>F µ(n) = n Y3LW>F,G sZL N|LNT JLE\GWoM3G,VGHKJLH YERLWoWlETT G PQGD.H µ(n) = 3

n

n=5

0

º

ß à c Q : 8 R3U 521 O 84) 8 U9 A 1 N 1 O J 1U3=A 5:R1c

/

?

Ú 2[F

A

-AU>0@

9:

?À

M R /+3=< >A &R8- 1UO

N124MFJCHGS49<§0O2oK1?@6,2M0S4CH5256*FR4:9<§032;1?12:6X9
ABC f ?@916*G3;12:CDGO9<2:@A<450SGTFHG3256#?BhpF*K12Mh_?@CJP

Á 8U

A 1 / 5 8>8

2π/3

c ?@C

λ µ ∈ c A B C A B C cos · cos · cos ≥ λ + µ · sin · sin · sin 2

O4

_N

2

/
2

λ ≥

2

2

2

√

2 3−1c 8

/
√ 2 3−1

49<;

µ = 1+

8

√ 3

f 4919
0/

@5+3+- A U A > RA 6~ K125CH225CH2!9

"!$#&%

'(!)#+*-,.'$!/10$' 23%

' 4#+*-,

57698 :;9<=57>@?"@?"
'(!)#+*-,.'$!/10(d `@b ,

e7;LBUfE698 fEgh5U698 :M;9y?R@?"
'(!)#+*-,.'$!/10$' 2%

'4#+*-,

e7;LBUfE698 fEgh5U698 :M;9@?"y?R{ "X 1X7ul@J"f}{"TfE67

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