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"!$#&%('*)+! ,.-0/21435/635/87:9;=:?@3+-A=BDC0CABDCNCN3 BE?CN187:¡-0/*143M1*aN=:...

Author: Canadian Mathematical Society

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n!

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

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(m + 1)!

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∗ _ =jI:/23 z*CBGF3WCN9:/23<>a-N/837:UBGF _ J \¿ a 9./87:a 3 × 3 × 3 = 27 _J _J _ J _ 1=jI:/23 BGFa1BM18?D/7:BD/49K1B18-A35JO18-A- _ =jI./83LBGF1BCL3 1 <7{<×7:/( 1 3WCN9:/ × 1BGF./7=:JKI:/2 ?oH.18CL7>BE/89<7OB½z<¦3GC09./83 187:9 BGF./(7=:J¦I./8?oH.18CL7>BE/8 9 b BGF./7=:JKI:/2? H:12CN7>BD/49K<7BGF?D/8/3WCN9: /2c3 9./BD/8?DJOCL7:/ \ d a−b−c+d

\ ñC _ F~>12-L=j/18-A=j/83o<>a

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x2 + k

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D .......... ........... . ....... ........... .... ............. ........... ...... . ........... . . . . . . . . . . . . . ............. ... .. ........... .... ........... .................... .................. . ..... . ... ........ .......... . . . . . . . . . . . . ... . . ....... . ....... ..................... .... ..... ........ ..... . . . . . . . . . . . . .... . ....... . .. . . . . . . . . .... . ....... . . ....... . . . . ....... . . . . . . . . . . . ....... .. .. ............... ....... ... ............. ..........

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c

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

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î mì

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\ <4zçJK187;(Hj<>CL7>BD3 187:9 (A/6, (0, 5) z*CABGFCL7>BE/82/8? _ <.=:? 187.3Ez/2? CN7KI.<¡ò(² <>ajBGF./*?D/2-L1;147.3Dz/8?°35F./4/¡B \ B È Ë \ óM/BD/8?DJOCL7:/ CLa c

(B + 18)c + 7d = 4 d + e = 20

?DCBE/cBGF./c~>12-L=j/<>a

e+f f + 5c

c

= 36 \ = 15

CN7KI.<¡ò(²j³<>ajBGF./*?D/2-L1;147.3Dz/8?°35F./4/¡B \

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r

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r B

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È

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Ô* Ø ÝjÙ×*Ð" É â *ÝjãDØ Í *Ë ÕÖcãGÔ*Ý ÛÐ *Ð É \ F./-N/87:UBGF3<>ajBGF./3WCN9:/23<>aÄ1*BE?DCL147:¡-N/ 18?D/ r b c = b + 4r/ 4 -0/27:UBGF3<>a@BGF. zF./8?D/ CL36BGF./K?149CA=:3<>aBGF./OCN7.3 _ ?DCNI:/89 _ CNABC ? _ -N/ \ óM/BD/8a?DJO=CL7:b/O− BGF./ 3WCN9:/23o<>rajBGF>CN3 BE?CN187:¡-0/*143M1*aN=:7 _ BDC0<7K<>a <7-; \ r

\ ¿7 @ ! GB F./8?D/ z 183187/2?E?DBg<>a4BGF./H.?D<.I-N/8J \ F./2?D/CBcz143(3GÈ B12BD/49|BGF12B r \& F>CN3F143I./4/27 _ -0<.¡CW/6am
"$#&%('*),+ -,. +(/01'2.3g\

\ ñ?Da16BE?CN187:¡-0/

·8/B

2/B

s = 1 (a + b + c) p 2 \ ¿ B°am<>-L-N<4z3 BGF1B sr = s(s − a)(s − b)(s − c)

sr 2 = (s − a)(s − b)(s − c)

º;CL7.3+/2?BDCL7:BGF.//GòH.?D/83W3WCN<7.3am
zF>C _ F6CLJKH-LCN/83ÄBGF12B

\ ¿a

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b 1 + r 2 4

c b 2

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15 \ r 4

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|x + a| − |y − a| < a \

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−a < |x + a| − |y − a| < a

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y−a≥0

4³ \ l 187:9 y−a<0 µc<4xzç+za/< F1¡~0/

−a < y − x − 2a < a

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x + a < y < x + 3a

\

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y
\ l 187:9 y−a<0 µc<4xzç+za/≥ F1¡~0/

−a < x + y < a

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−x − a < y < −x + a

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\ l 187:9 y−a≥0 µc<4xzç+za/< F1¡~0/

−a < −x − y < a

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\

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y≥a

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.. . . .. a.. . + . . x .. . . − .. .. = .. . . y . .. .. . . . . .. . . .. . . .. . . . .. . . . .. . . . . .. . . . . . .. . . ... . . .. . .. . . .. . . .. . ................ ........................ . . .. . . . ..... .. . .. . . .. . .. . .... . . . .. . . . .. +

a

x

=

y

y

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. .. .

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y

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¡ Ë \ ñCL7:9O12-L-:CL7>BE/82/8? 35<>-L=BDCN<7.3 BD<(BGF.//8=:1BDC0<7 √ \ √ √ 4x + y + 4 xy − 28 x − 14 y + 48 = 0

. +(/01'2.3 <> / > / 3?7.3 '*).5 0 :!) % ? :,A 5 . +B/ 6 < '*) 2370 :!A 3 ) 01' .3)+ . +9+2::!5 )6' +(02.35 "(\ ñ>?D=:12BDCN<7

z/2/B

√ 2 √ √ √ √ 2 √ 2 x + 2(2 x)( y) + ( y) − 28 x − 14 y + 48 √ √ √ 2 √ 2 x + y − 14(2 x + y) + 48 √ √ √ √ 2 x+ y−6 2 x+ y−8

= 0 = 0 = 0

\

√ > \{ F./Oj?3EB*<>aBGF./835/¦/4>=:12BDCN<7.3CL3 BE= /82/86?cH:12CN2?3 x + y = (x, y) ∈ (0, 36) (1, 16) (4, 4) (9, 0) \ BGF./(3+/ _ <7:9KCL3351BDCN3G/89{I8; F>=:3 BGF./ _ -L=BDCN<(x, 7O35y) /B@CN∈3 (0, 64) (1, 36) (4, 16) (9, 4) (16, 0)

(0, 36) (1, 16) (4, 4) (9, 0) (0, 64) (1, 36) (4, 16) (9, 4) (16, 0)

\

Í \ <4zJO147;am<>=:?E}59>C0¡CBcH:<3WCBDC~/CL7>BE/82/8?39>C~>CN3GC0I>-0/OI8; F1~/BGF./OH.?DBE/2? _ F147:2/89 7BG. F /p?D/23W=.-BCN3¦1&07:=:?E}59>C0¡CB Hj<3GCABDC ~/6CN7>BD/42/2?9>C~>CN3GC0I>-0/I8; 7 . +(/01'2.3 <> :!37: ' : : )+2.3 : 5 ) --,: > 5" (\

¸2=:H:H:<3+/ CN3°1cam<>=:?G}W9CN¡CAB7=:J¦I./8?BGF1BCN3 100b ¿ ajz/*=LC 7>1000a BE/2? _ F147:+2/* BGF./*+ j?10c 3EB187:+9Od-L143EB9CN¡CAB3 z/2/¡B°BGF./7=:J¦I./8? 9C ~CL3WCNI-N/I8; \oabcd 7 \ F>CN37:/Uz»7=:JKI:/2?ÄCL33G=:H:H: <3+/89cBE<*I:/9>C~>CN3GC0I>-0/ dbca = 1000d+100b+10c+a I8; \o F./2?D/2amC999a =/K999(a d)/ _ / − dbca JO=:73EB612-N35<|I./¦9>C~> CN3GC0I>-0/KI8; \ ¸2CL7 abcd CL37:CN3GC0I>-0− /KI8999d ; z J=:3GB6− F1¡~ 7 999 7 \ BGF1B°CN3 7 | (a − d) 4 a ≡ d (mod 7) ¸2CN7 _ / z/J=:3GBF1¡~/ 7 | abcd 1000a + 100b + 10c + d ≡ 0 (mod 7) −a + 10(10b + c) + d ≡ 0 (mod 7)

F./{7=:JKI:/2?

10b + c ≡ 0 (mod 7)

_ 187|~>18?x;a0?D
BD<

¡C ~CL7:

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H:12CN?3

3G= _ F|BGF1B

10b + c \r] _ FBGF1B b c 7:9(BGF./2?D/*14?D/ 0 H:1299 CN?3 3W= 15 10b + c ≡ 0 (mod 7) 14 a d z*CABGF 147:9 N3WCL7 _ /BGF./2?D/(14?D/ H.18CL?3Mz*CBGF a ≡ d (mod H:12CN7) ?3 1 ≤ a ≤ 9 0 ≤ d ≤ 9 9 a = d 3 \M F./8?D/ama z*CABGF 187:9 H.18CL?3z*CABGF a = d + 7 7=:J¦I.2/8?3 z*CABGFBGF./d9./8 =3GCNa ?D/8+ 9OH:7?D

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"$#&%('*),+ -,. +(/01'2.3g\

º; /8?D<7 3ñ=:14?D/<>agBGF./14?D/21K<>aÄBGF./(BE?DCL147:¡-N/z*CBGF{3WCN9:/23 \¿ aÄz/-0/¡B zF./2?D/ CL3 a b c A2 = BGs(s − a)(s s = 12 (a + b + c) F>CN3 _ 187KI:/z−?DCb)(s BBE/27O− 143 c) p = a + b + c \ 2 16A

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= p(p − 2a)(p − 2b)(p − 2c)

p

JO=:3EBI://+~/27 \ F./8?D/18?D/6B½z<Hj<3W3WCNICA-LCBDC0/23

l DC ] -L-BGF./7=:J¦I./8?3 18?D//+~/27 \ c187:9OH:?CNJK/ z/*JO=:3EBF1~/ \ º:=B º/ _ 18=:35/ 18?D/a/+~ /2b7K a√ b czF>C _ FCL3M7:BE/82/8? \ a=b=c=2 BGF>CN3o¡C~/23 A = 3 ¯ 7:/<>a:BGF./7=:JKI:/2?3 l CL3o/+~/87 147:9(BGF./=B-N<353<>a2/27:/8?12-La CB; b zc/6JO1;(143W 3W=:JK/BGF1B 147:9BGF1B 147:9 b c\ \Ä F./87 14?D/<.9:9 \@¿ a z/ _ 147B14/ B4 GFa12B:= CN3 2 b = 6 c b < c c−b ≥ 2 c ≥ b+2 = b+a \ µ<4z º:=BBGF>CN36CL3*1 _ <7>BE?19>C _ BDCN<7<>aBGF./ ?DCL147:¡-N/ ¿ 7:/8=:12-LCB; \ /87 _ / b = c 187:9(z/F1¡~ / p = 2 + 2b 16A2 = p(p − 4)(p − 2b)2 = (2 + 2b)(2b − 2) · 4

zF>C _ F;>C0/-093 2 2 4 BGF12BrCL3 (b−A)(b+A) = 1 \ º:=BgI./ _ 18=:35/ b+A ≥ 2 BGF>CN3CN3M7:-0/ \ = 1

\ F./H.?D<.9= _ B<>ajBGF./*Hj<3GCABDC ~/*?D/812-7=:JKI:/2?3

x y

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z

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1

\ ¿a

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x+y+z 0 0 \ 0

m

¸2CNJCL-L14?-; amBE/82/8? BGF./CN7:/8=:12-LCB;BGF12BCN3@BD<I./6H.?D<8~/89 k CN3o/8=.C ~>18-N/87>B@BE< \ ÈU (1 − xk )(1 − y k )(1 − z k ) ≥ 0 /JO=:3EB3+F.<4zÓBGF12BMx CLJKH-LCN/83 \M F>CN3MCN3<.I8~CN<>=:3oCLa a 147:9 147:1912B-N/8183G B<7:/ J 3GB1I: / xyz = 1 x y z 1 JO=:3EBI:/?D/81BE/2?BGF147 \ CABGF.<>=BM-N<353<>a2/27:/8?12-LCB; 143W3W=:JK/BGF12B 187:9 x<1 \ F./87(z/*JO=:3EB1F1~/ BE<351BDCN3GaA;Ox \ y>1 z>1 ¸2CN7 _ / /4>=.C~>12-0/27>BD-;143 1 z/ _ 187(z?DCBE/ x= ÈU yz

1 1− k k y z

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L

/ _ -N12CNJÚBGF12BcCLa CL3187;Hj<3GCABDC ~/¦?D/218-°7=:JKI:/2? BGF./27 1\ t t−1 ≥ 1− t <H.?D<8~/6BGF>CL3 7:
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. +(/01'2.3 <> :!37: ': : ) +2.3 : 5 ) --,: > 5"(\

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

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202000

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20

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187:9

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A

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

................... ....... .... ........ ........ ..... ..... .............. ........ ... ... ........ ........ ........ . . ... ... . . . . . ........ ... ........ ....... ..... ... ........ . . . . . . . ... ... ◦ ..................... .... . . . . . . . . . ... .... ...... . .............. . . . . . . ... ......... ................... ..... ... ..... ... ........ ... .. ................. ... .... ........ ... ........ ... . .... ........ . . ◦ . ........ . . . . ... . . ... . ... .. ... .... ........ . . . . . . . . . . . . . . . . . . . . ...... ............ ... ............................... ........ ..... .... ....... . ....................... ............... .. . . . . . . . . . . . . . . . ... ... ◦ . .................................. .. .............. ........ ......... ........ ..... ...... ........ ...... ............................................................................................................................................................................................................................................................................................................................................................................

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2 5·8

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2 1997 · 2000

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µ <4z -0/¡B 3cH:?D<2~/ /+~>1 3 F./8-~/23618-A-4CL7:9.3*<>a?12BDCN<3 c <7:/JC02FBI:/BE/2JKH>BE/896BE<612BB1 _ oBGF./H.?D<.I-N/8Jz*CBGF~/ _ BD=JK1¡;9.=z*CN35F am=:H-0/*<>a.-0/2JKJO143 \ F./835/614?D/ /83W3+/27>BDCN12-L- ;BGF./4=B z/ _ 12-L-BGF. /2J-0/2JKJO143°BE<(CL7:9C _ 1BE/cBGF1BBGF./+;14?D/ 0=:3EB 3GBD/8H.3<7BGF./z1;(BD =:? JO18CL7BGF./4z?DCBE/ \ BDar16BE?DCL147:¡-N/ [ABC]

ABC

n ¨(¨ C¿ aB½za>BGF./CN?g14?D/2143CN3°/4>=:18 BE<(BGF./*?12BDCNajBGF./CN?°?D/83WHj/ _ BDC~/F./2CN2FBD3 \ A.. <7.3GC09./8?*BGF./{B½z<BE?CN187:¡-0/23 AXY B /F1~/ 147:9 CL7 BGF./9>CN14?14J \ ..... ....... .... ...... ........ . BXY . . . . . ... .. .... . . . \ . 8 1 : 7 9 . . ... ... ..... 1 A ... ... ...... [AXY [BXY ] = 21 bhb ....... ... ... ........ ....... . . F>=:3 ] = 2 bha . ... .. . ........ .................. . ..... ......... . ... ... .... . . . . . .......... ....... .. ... . h . . . . ... b .... ..... ....... ......... .. ..

¡

h

[BXY ]

=

ha \ = hb

.......... .. .. . .. a.............. .......... .. ... ... .. .. .......... .. .. .. .... .. .......... .. .. ...... .. ......... .... ... .... ............ . ............ ... ............. ........... .... ... ............ .. ............ .......... ...................................................................................................................................................... ....... .........

[AXY ]

1 bha 2 1 bhb 2

X

b

Y

n ¨(¨ |¿ a°B½z<BE?DCL147:¡-N/83(F1¡~/KBGF./{3W14JK/{F./2CN2FB BGF./¦?12BDCN<¾<>a@BGF./2CL?*18?D/8183CL3 /4>=:18-:BD<(BGF./*?12BDCNajBGF./CN?°?D/83WHj/ _ BDC~/I.183+/23 \ F./H.?D<.<>aÄ<>ajBGF>CL3MCL3-0/aLB 183oF.
U µc<4zz/cz*CL-A-H.?D<8~/ /+~>1 3 F./8CN7>B6<>a _ <7 _ =:?D?D/87 _ ;p<>aBGF./BGF?D/4/ _ /+~>CN187.3I:/ 143¦CN7`BGF./¶9CL1?18J <7eBGF./ P ?DCN2FB \

A ...... ..... ... ... ... ..... ... .. ....... .... ... ... . . .... .. ... .... ... ... ... ... ... ... ......... . . . ....... ....... . .. . .. . . . . . . . .... . .... . .. . . . . . .... . . . . . ....... ........... .... .... ... ..................................... ... .... ......................... . ... . . .... . . . . . ........... .... ..... ... . .. . . . . . . . . . . . . ........... .... . .... . .. . . . . . . . . . . .... . . . . . ........... .. .. ........... .... . . . . . . . . . . ........... . . .. .......... . . . . . . . . . . ........... ....... .. ............ . ........... .... . . . . .. .. . .......................................................................................................................................................................................................

Y

-N/818?D- ;

81 7:9 F1~/1 _
BX \ XC

B

P

C

X

147:9 F1¡~/ 1 _ a:BGF./9C g/2?D/87 _ /83CL3 [P BX] BX = XC 18[P-L3+XC] <BGF./3W14JK / 4 BGF12B°CL3 ¸2CNJCL-L14?-;

3GCN7 _ /

=

Z

BX \ [ABX] − [P BX] = [AXC] − [P XC] XC

. F /87z/*-0<.<.12B°BGF./9CL1?18J z /12-N35
=

BX \ XC

x

º;O3GCNJCL-L14?@14?D¡=:JK/87>B3 z /2/¡B

¼o3WCL7:*x

147:9K

È¡E

[AP C] AZ \ = [BCP ] ZB

187:9

[BCP ] CY = [ABP ] YA

ÈU

z/F1~/

BX CY AZ [ABP ] [BCP ] [AP C] \ · · = · · = 1 XC Y A ZB [AP C] [ABP ] [BCP ]

F./ _ <7~/2?3+/{<>a /+~>1 3 F./8
·

CY YA

CN318-L3+<{BE?D=j/

·

AZ ZB

4

BGF12BcCN3

CLa@BGF?D/8/ _ /+~CL147.3

= 1

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+ ,.-0/213

1,5 5z,

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& &

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\ ·8/B

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t =

FG

2 cos 18◦ (cos 36◦ )2 cos 6◦

147:9BGF1B@BGF./35=:18?D/F143M3WCN9:/6-N/87:UBGF

√ t 3 \ √ 2+ 3

Ë \ ·8/B \ ñ CL7:9¦BGF./-0/2143EBHj<3GCABDC ~/CN7>BD/42/2? 3W= _ F 13 5 f9(x) a BGF12B C ~CN9:/2= 3 5x ma + 9ax BE/82/8? \ Í \

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41 7:9 amaa-N1/87:< {a1 a2 a3 } UBGF a2\ < a3 0I

3

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10

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\ ·8/B

Cp,q

x ≥ 0 y ≥ 0

I:/*?D/812-7=:JKI:/2?3 z*CABGF

x+y =2

x2 y 2 (x2 + y 2 ) ≤ 2

\ ·8/¡B

\ ,:?D<8~/cBGF12B

\

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óM/49>= _ /6BGF1B

=

3

16Q2

R≥

(abcd) 4 √ Q 2

z*CBGFO/8=:12-LCB;CLar187:9K<7-;CAa

ABCD

CN3M13+>=:14?D/ \

\ ñBE/82/8?

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+

x2

+

x3

+ ··· +

xn

=m

Ê \ ,:?D<8~/BGF1BCN7/21 _ F35/Bc<>aBE/27 _ <7.35/ _ =BDC ~/OCL7>BE/82/8?3cBGF./8?D/CN3<7:/(zF>C _ FCL3 _ ajBGF./BE/82/8?3 \ ñBD/42/2?3 18?D/ _
É \ ·8/B n I./1Hj<>- ;.7:B = ?D/812- _ <./ p(x) 3 \ a¸2=: F12B· · · + an x 187:9 =8 \ p(8) ≤ 4 147:9:7:9 z*CABGF: H ?D<.<>a 12-L-j3G= _ FHjp(4) <>- ;.7:<=J2CN12-N3°z*p(16) CBGF p(8) = 4

)

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f ¤¨ u É \ ·8/B \ ¸2=:H.Hj<35/ \ ·8/B I./BGF./ _ CN? _ =:J _ /27>BE?D/6<>a 4ABC_ _ \ ·8/> I:/(1H:<>O CN7>Bo<7OBGF./(JCN7:arBGF./ CN? =:J _ CN? _ -N/ AB B AC 187:> 9 BC I./(Hj<>CL7>D BD3 BC \ ·8/B <7 3G= _ F¦BGF12B 147:9 I./(E BGF./CN7>BDF /8?35/ _ BDCN<7<>a AD AC ⊥ OF P \ ¿ a AB ⊥ OE \ 147:9 H:?D<2~/6BGF12B BE CF P B = P C + P O ∠BAC = 30◦

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p

187:9

q

3W= _ F6BGF1B (7p − 2p )(7q − 2q ) CN3147*CN7>BD/42/2? \

¢

pq

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C

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

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

B

6 3 s

C

A

¤ *¥

. +(/01'2.3 <> . < :,A0 '9+ '*3 - '95 " / 0@A:!6 .3;0*5

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50 + 50 + 22 = 122

\

> F CL3o?D/2H:?D/23+/27>BD3 BGF./*7=:J¦I./8?<>aH:1BGF3Ma0?DCN9CL7: C

A

BD<

<> 0@?7: : ' 0 .A

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

17

6

1

50

11

5

1

3

A

22

C

B

50

11

5

1

17

6

1

B

¦

2 Ë \ mBGFK?149:/

,:?D<8~/cBGF12B@BGF./7=:JKI:/2?

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9999999 + 1999000

¯ I.35/8?x~/BGF1B \ 187:9 9999999 = 107 − 1 1999000 = 2 · 106 − 103 F./2?D/2am
= 107 + 2 · 106 − 103 − 1 = 12 · 106 − 103 − 1 \ = (3 · 103 − 1)(4 · 103 + 1) = 2999 · 4001

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an = n

n

. + : <> ' :!A A: .A 3 - !02: '135 ) '9-,.3 - ) # 0B02:!5A ) 37% : )3 / 6'*3 ?7:!3)3 ) A N ) 35 ' + *AB' )/ AB'2:,A 3 ' :!AB- '(0 > 5 )02:,AB+2..5 " : ,' :=0@? : AB'(02: / . ?7:!3 ) 3 ) 3

¦

¦

\ 2¸ CN7 _ / /*H:?D< _ /4/89KI8;CN7:9>= _ BDC0<7 \ ñBD/42/2? z/(J=:3GBF1¡~/
a` + ` = aan+1 + an+1 = 2(n + 1)

` ≤ n

BGF./87

a` = `

0I8;BGF./*CL7:9= _ BDCN<7¦F4;Hj
E

147:9KF./27 _ /

2(n + 1) = a` + ` = ` + ` ≤ 2n < 2(n + 1) 1 _ <7>BE?19>C _ BDCN<7 \ ¿a \ BGF./27 3WCL7 _ / z/6JO=:3EB F1~/ 2(n a` < n + 1 F>C _ + FCN1) JOH.-AC0/23 BGF12B º;BGF./6CN`7:9>>= _ nBDC0+ <7K1F4 ;.H:BE?19>C _ DB CN<7O112CN7 \ F>=:3 \ F12B°CN3 _ = _ BDC0<7 \ `=n+1 an+1 = n + 1

\

I / 1Hj<>- ;.7:BD/42/2? _ <./ _ C0/27>BD3 \ : m4BGFÚ?149:/ ·8/B 35/4>=j/87 _ /Ã<>a CL7>BE/82/8P?3 (x) 351BDCN3G/23¶BGF./ _ <7:9CBDC0<7.3 x1 x2 . . . x\ nñ CL7:.9 . .BGF./~>18-A=j/<>a x1 = x2000 = 1999 xn+1 = P (xn ) n ≥ 1 x1 x2 x1999 \ F./

¤

x2

+

x3

+ ··· +

x2000

. + : <> ' :!A A: .A 3- !0 :'*3 5 )'9-,.3 - ) # B0 02:!5 AB)3 % : ) 3 . / < '137.. + " 6 )A :: 5 ) AB'1-5A ) 37% : <> -'16'9+1)A6 :0@? . - : ' : .A 3- !0 :'*3 - BA '(02: /

¡¡

¦

t¤

D¢ ¦

\ ñBE/82/8? ¸2=jI.3 _ ?DCLHB314?D/ _ <7.3WCN9:/2?D/49JK<.9=.-N< -0/¡B 1999 n \ F.2/ 7

yn = xn − xn−1

1999 X i=1

yi =

1999 X i=1

(xi − xi−1 ) = x1999 − x0 = 0

\

x

4 \ ¸2CL7 _ / ¸2=:H:H:<3+/¾BGF12BKamBD/42/2? BGF12B amBD0/42/2?3 P (x) BGF./¶CN7>BD/42/2? _ <./ _ NC /87>B3 CAB¦CN3{z/-L-47:<4z \é¿ BOam<> -L-N<4z3OBGF12B 9C ~CN9: /23 9>C~>C0a9./83 6= b am=j/87 _ n/ \ a − b P (a) − P (b) y y18n+1 n7:<7:}W9:/ _ ?D/8 F./27KBGF./(7=:JKI:/2?3 am= _ /BGF1B |y1 | = |x1 − x1999 |\ = =JK|y 2000 ·8/B |x2000 − I./6xBG1999 F>CN3 _ |18-A|=j / \ |y1 | =·8/|yB 2 | = 6=140 I:/c·BG·F.·/=7=:|yJ¦2000 I./8?| <>ajBD/8?Da JK3M JK<7: zF>C _ FOF1~/6BGF./ k y1 y2 . . . y1999 \ \ /87 _ / ~>18-A=j/ F./87BGF./?D/2JK12CN7CL7: BD/8?DJK3oF1~/cBGF./c~>18-A=j/ a 1999 − k −a 1999 X i=1

yi = a(2k − 1999) 6= 0

_ <7>BE?149C _ BDCN7:*x \ ¿ BMam<>-A-0<4z3 BGF12B BGF./2?D/CN33+C _ F 4 GB F1BMCL3 xn = xn−1 \ n ] 7K/8183;CL7:9= _ BDCN<7-0/2193BD< \ 0 F.2/ 7 am
n

x1 x2 x1999 \ + + ··· + = 1 + 1 + · · · + 1 = 1999 x2 x3 x2000

\ x82BGF(?149:/ ñ>7=:J¦I./8?3

x31 x52 + x53 + x54 + x55 + x56 + 5

+ +

§ ¦ ¤ ¡ u£ ¤ D¢ ¦

H.?D<8~/oBGF./CN7:/8=:12-LCB;

x1 x2 . . . x6 ∈ [0, 1] x32

x51 + x53 + x54 + x55 + x56 + 5 x36 x51

+

x52

+

x53

+

x54

+

x55

+5

+ ... 3 \ 5

≤

£

. + : <> '2:,A A : . A 3- !0 :'*3 5 ),'1-!.3- )# 0B0 : 5 AB)3 % : +B/ < : 6 ) 0@? : 6 ) 01' / : - /+ > % : : ,: 3AB' 5 )AB'9- 5 A ) 37% :) 3 . / < '*3 .. +" 6 )A :: 5 )AB'9- 5 AB)3 % : .A 3 - !02: '13 ) 3 " 6 ) A : : / -!: - '*6' +*) A 6 :0@?7. - : ,' : " 6 ) A : : AB'(02: / ¸2CN7 _ /

x1 , . . . , x 6

¡

¡

¦

14?D/6CN7BGF./6CN7>BD/8?x~>12-

[0, 1]

x52 + x53 + x54 + x55 + x56 + 5 ≥ x51 + x52 + x53 + x54 + x55 + x56 + 4

\

º;Hj/2?EJ=BDCN7:MBGF./3W=jI3 _ ?CNH>BD3 z/3+/8/oBGF12BÄBGF./-0/aLB3GC09./c<>aBGF./CL7:/4>=:18-ACABx;CN7BGF./ H:?D<.I>-0/2JCN3M1B JK<3GB 6 X

x3i

x5 i=1 1

ñ
y ≥ 0

+

x52

+ ··· +

x56

+4

=

6 P

i=1 6 P

i=1

BGF./ ] ¹|´4±M¹

x3i

\

x5i + 4

¿ 7:/8=:12-LCB;K¡C ~/83=:3

p y5 + y5 + y5 + 1 + 1 ≥ 5 y5 · y5 · y5 = y3 5

4

BGF12B°CL3

3y 5 + 2 ≥ 5y 3 5

BGF12B°CL3

6 X

x3i

i=1

\ F>=:3 ≤

6 X

(3x5i

+ 2) = 3

i=1

6 X

i=1

6 P

i=1 6 P

x3i

x5i + 4

i=1

≤

x5i

+4

!

4

3 \ 5

F>CL3 DB <.2/BGF./2?z*CBGFO<>=:?°CL7CABDCL18-<.I3+/2?x~>12BDCN<7.3 _ a \ \ m¡BGF(?149:/ ·8/¡B I./oBGF./12-ABDCBD=j9:/23°<>a1 _ =BD/MBE?DCL147:¡-N/ AA \ ·8/B=:3M9./87: CABE?18?x; H:<>1CN7> BB BCN7.3G1C0 9.CC /BGF.1/BE?DCL147:¡-N/ I8; O A B C M 1 1 1 147:9 BGF./|I.183+/23O<>aH:/8?DHj/27:9C _ =.-N18?3K9.?1¡z7¶a0?DB \ BGF./*-LCL7:/83 M N P Q RS . +(/01'2.3 <> N .-?' . :'*6' > ) 5 7) ) - ) '95 ) 7)3

M¥ ¢ ¦

A ........ ..... ... .. .. .... .. ... ... .. ..... ...... . .... ... .. .... .. .. .... .. ..... .... .. ... .... .. . .... ..... .. . .... ... .. . .... .. .. .... . . ... . .... ... ... .... . . . ... .... .. . . .... .. ..... . .... . .. ... . . .... .. ........... . .. .... .... . . . . .. .... ... .. . . .... .... .... .. .... ... .. 1 ..... .. ... .. .. . . . .............. . . . . . ... .............................. ...... . . . . . .. . . . . . . . . . . . . . . . . .... .. ... ... .......... . . . . . . . .. . . . . . . . . . . . . . . . . . . . ... ... ............... .... ...... .. .. .. .... ....... .. .................. .. .... ...... ....... ..... ................. .. ..... .... ............... .... ... ....... .. .. .................................. .... . .... ...... . .... . . . . . . . . . . . . . . .... ..... ....... 1 .................. . .... .... . . . . . . . . . . . . . . . . . .... . .... . . ..... .. ..... . . . . .... . . ... ...... ................... .. ........... ......... . ... . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . .............. ... . .. ...... ... . . .... .. . . . . . . . . . . . . . . . . . . . ... .................... .. ........ . .... . .. . . . . . . . . . . . . ... .... .. ... ... ...................... . . .. . . . . . . .... . . . . ... .. .......... ... ............. . .. . . .... . . . . . . . . . . .. ....... ... ... .......... ... ... .... .. . . . . . . . . . . . . . . . . . .......... ... ........ ... .. .. ... .... .. . . . . . . . . . . . . . . . .......... .......... ... .... ..... . .... . .. . . . . . . . . . . . . . . . . .... .......... . .. ..... .... .. ..... ...... . . . . . . . .... . . . . ... . .......... ... .. ..... ..... . ... . . . . .... . . . . . . . . . . . . . . . .......... .... .. ... ... ... . .... . .... . . .. . . . . . 0 . . . . . . . . . . . .......... ... .. ... .... ... .... . ..... . .. . . . . . . . . . . . . . . . .......... .... ..... .. ... .. ... ... . . .. . . . . . . . . . . . .... . . . . . . .......... .. ...... ... ... .. ... .... . . . .. . . .... . . . . . . . . . . . . .......... . .... ... .... .. ... ..... .... . .. . . . . . . . . . . . . . . . .......... ..... ... ........ ... . .. ............ . . . . . . . . . . . .......... ...... .............. . .. ......... . . ......... ... . . . . . . . . ..... . .. .......................................................................................................................................................................................................................................................................................................................................................................

Q

B

C

O

S

?

R

M

P O

B

N A1

C

2¸ CN7 _ / 18?D/ _ <7 _ ; _ L- C _\ ◦ BGF./Hj<>CL7>BD3 B C 14B?D/ 1 _ C \ 8/ 7 _ / 81 ?D/ _ 1
∠B1 A·81/A \g F./87 B =0 : I ∠B /BG1F.BA /oCL3+<.= 2<∠ACC 7.12- _ <7 10=j= 1∠AA BE/<>a 1 Cz*1CBGF*?D/235H:/ _ BrBD< O O 4A1 B1 C1 ∠AA1 O

= ∠AA1 B1 − ∠OA1 B1 \ = ∠C1 A1 A − ∠C1 A1 O 0 = ∠O 0 A1 A

x

È

¸2CN7 _ /

OM A1 N

CL3M1?D/ _ BD187:¡-0/

z/2/¡B°a0?D
m

∠A1 M N = ∠OA1 M = ∠OA1 A = ∠O 0 A1 A

F./2?D/2amBD1/8O?35/ _ BDCN<7<>a L\ F./87 147:9 CN3OBGF./pJC09.}WH:OA <>CN7>1B M N L \ ·8/¡B <>a I:/BGF./OCN7>BD/8?35/ _ BDCN<7<>a OA1 T\ * z C G B F ¸2CL7 _ / 0 OO M N k A1 O 0 z /F1¡~/ M N OT : T O 0 = OL : LA1 = 1 : 1

\

A ..... ............ ... ... .... .. ..... ...... . . .... .. ... .... ... .... .... .... ... ..... .. .... . . .... ..... .. . .... ... .. . .... .. .. . .... . ... .. .... . . .. .... .. . . ... 1 .... .. . . ..... .. ........... . .. ....... ...... . . .. . . . . . . .... ... .... . . . .. . . . . . .... .. ... .... 1 .............. ....... ........ ....... ....... .... .. ................. ........... ....... ........... .... .. ........... ..... ....... .... .. ..................... . . .... . .. ...................... .. . .... ............... ... ............. .. .... . . . . . . . . .......... ....... .............. . . .... .. . . . . . . . . . . . .......... .... ..... .. ..... ...... . .. . . . . . . . . . . .......... .... . ..... .... ... . . .. . . . . . . . . . . . . . .......... .. .. .. .. .. .... ..... . 0 .. . . . . . . . . .... . . . . . . . .......... .. ... .... ....... .. ........... .... . . . . . . .......... . .......... .. .......... . . . . . . . . .......... ...... . . ........... ........ ...... .. . . . . . . ..............................................................................................................................................................................................................................

\

B

F>=:3 H:183535/83 0 147:9 BGF?D<>=j 2OT FBGF./=JC0T9.}WOH:<> CN7>B<>a M N0 \ ¸2CNJCL-L14?-; 187:9 OO H:18353 P Q RS \0 /27 _ / BGF?D<>=j2FBGF./MJC09.}WH:<>CN7>BÄ<>a 18?D/ _ OO <7 _ =:?E?D/27>B612B 187:9 M N P Q RS BGF./JOCN9:}GHj<>CL7>B<>a 0\ OO

C

O

M

L

T

O

NA1

B

C

µc<4z$z/pBD=:?E7»BE<»3+<>-A=BDC0<7.3am-0/2JK3|<>a*BGF./ · ¿¿¿ ¹p12BGF./2JK1BDC _ 18¯ -;JKHCN149K<>a¹&<>-N9:<2~>1 8 ñ³ , \ È È É \ ·8/BBGF./KaN=:7 _ BDCN<7 3 I./ _ <7.3WCN9:} 2 2 → F>C _ (FBGfF.(x) /*CN7:/8= =:x 12-LCB− ; 2ax − a − CN3 BE4?D=j/*am12-L=j/23 f :am
¤ F¥

x ∈ [0, 1]

¤

. + : <> $. < :!A0 ' +9'13- '15 " / 0@A:!6 .3;0*5 '2:,A A : .A 3 - !02: '135 ) '9-,.3 - 0B02:!5 A ) 37% : )3 )A N ) 35 '9+ @AB' ) / AB' :!A 3' !: AB- '(0 > 5 )02:,AB+2..5" : ,' : ) 3 - -!. +B/ 01' .3 576 . '187: =<> 0@? :: ' 0 .A(-

) #

¦

¡

¦ .¦

¦

/6z*CL-A-rH.?D<8~/*BGF12B GB F./6~>18-A=j/83<>a amC _ FBGF./CN7:/8=:12-LCB; |f (x)| ≤ 1 CN3 BE?=j/*am
− 21 ≤ a ≤

2

/cz*CA-L-:7:/4/89BGF./*am<>-A-0<4z*CL7:caN=:7 _ BDCN<7(~>18-A=j/83 f (0) = −a2 −

3 4

1 √

\

f (1) = −a2 − 2a +

¿a BGF./27 |a| > 21 |f (0)| = a2 + 43 > \ 7:
3 4

> 2

1 2 2

2

1 √

x

2

2

1 4

+

2

f (a) = −2a2 − 3 4

+

= 1

3 4

3 4

\

187:9KBGF./2?D/2am
= 1.

F>=:3 \ < 7 _ /1418CL7 CAB@CN3M7:

³ F./<7-;~>18-A=j/83@<>a ?D/8JO18CL7CN7: BD<*I:/ _ <7.3GC09./8?D/89*14?D/BGF.<35/o351BDCN3GaA;CL7:Mx \ /cz*CA-L-:H:?D<2~/6BGF12B ama µc-N1 2 f JK1Uò4CLJO=:J"187:9 <(x \o¿ B am<>-L-N<4z3 BGF12B°BGF./ z*CABGF(~/2?BD/Wò(1B f (x) = Hj/27−CN7:a) 6=:H+ z14f?D9 (a) a,<>fa (a) <7cBGF./CN7>BD/8?x~>12- _ 147*< _G_ =:?Ä<7-;ozF./8?D/ JOCL7CNJ=:Jç~>12-L=j/23 18-A=j/o<>x a = 0 CL7>BD3 \ <7.3GC09./8?°147; 3W12BDCL3WaA;>CN7:6x \ F./87 3WCL7 _ / z/F1¡~/ a |a| ≤ 12 \ 3 1 2 3 2 ¸2CN7 _ /

|f (0)| = a +

a ≥ − 21

z/F1~/

1 − f (1) = a2 + 2a +

147:9BGF./2?D/2am
f (1) ≤ 1

\Ä] -N35<

3WCL7 _ /

≤

4

3 4

+

2

=

a+

− 52 < a <

f (1) − (−1) = −a2 − 2a +

5 4

5 2

=

= 1

4

3 2

1 2

a+

z/2/¡B

/ _ <7 _ -L=j9./6BGF1B

\ ·8/B

|f (x)| ≤ 1

3 4

am
≤ 2

2

1 √

2

x ∈ [0, 1]

\

2

+

1 2

3 4

= 1

+a

\ > \ 147:9BGF./2?D/2am −1147:9 3W 12|f BDCL3W(1)| j/83M≤ x 1 BGF./87 a ∈ [0, 1] G a |f (a)| = 2a2 +

1 2

≥ 0

−a

> 0

\

I./|1|7.1BD=:?18-M7=:J¦I./8?3W= _ F¾BGF1BBGF./7=:J¦I./8? F 183 9CL3GBDCL7 _ B 2 2n 28 _ \ 9 C ~CL3+ 9 N C E 3 D B N C 7 B > 9 C > ~ N C 5 3 < ? 3 4 < z J K 8 1 7 p ; 9 CL3GBDCL7 _ B 2 30 9C ~CL3+CN3M6nH.?D<.I-N/8J"F143M7:<3+<>-A=BDC0<7 \

. +(/01'2.3 <> ¡ '2:,A A :t¤ .A 3- !0 :'*3 5 ,) '1-!.3- )# 0B0 : 5AB)3 % :

¦

/Kz*CA-L-°H:?D<2~/¦BGF12B6BGF./3EBD1BE/2J¦/27>B<>a BGF./H.?D<.I-N/8J$CL3CN7 _ aMBGF./ 4/ 9>CABDaH:<3WCBDC~/9C ~CL3+BD/42/2? \ 01 /6z*CL-A-Äk35F.<4z BGF12BBGF./2?D/CN37: F N C ° 3 * z L C A 5 3 . F 4 < z G B F 2 1 @ B G B . F / : H D ? <.I>-0/2JCN3CN7 _ a _ <7>BE?149C _ BDC0<7 BGF1BMBGF./7.12BD=:?12-7=:J¦I./8? n \ 351BDCN3G/23 147:9 2 2 ·8/B

d(2n ) = 28 d(3n ) = 30 k Q : I / G B F./H.?DCLJ¦/9./ _ a zF./2?D/BGF./(CN7>BD/42/2?3 i n = pα n i i=1

pi

14?D/CL7KCN7 _ ?D/2143GCN7:
BGF12B°CL3

n

CN3<.9:9

BGF./87

d(2n2 ) = 2

k Q

i=1

k Y

i=1

(2αi + 1) = 14

\

(2αi + 1) = 28

4

F>CL3MCL3CNJOHj<3W3WCNI-N/

3WCL7 _ /cBGF./6-0/aLB 3WCN9:/6CL3o<.9:9O187:9BGF./*?DCN2FB3GC09./6CN3o/+~/27 \

k Q J=:3GBÄI./o/+~/27 \Ä F./27 n d(2n2 ) = (2α1 +2) (2αi +1) = 28 B4 GF1B°CN3 i=2 01H.?D<.9= _ B°CN7:9./Wò4/89KI8;BGF.//8JOHBx;O3+/¡B°CL3o/4>=:18-.BE< 1

F./2?D/2am
(α1 + 1)

k Y

i=2

(2αi + 1) = 14 = 2 × 7

\

¼o3WCL7:H:18?DCB; z/9./49>= _ /BGF1B/CABGF./2? BE= ?1493C _ BDC0<7 \ d(3n ) = d(2 × 3 ) = 3 × 8 6= 30 /6z*CA-L-r35F.<4z BGF12B BGF./2?D/CN37: F N C ° 3 * z L C A 5 3 . F 4 < z G B F 2 1 @ B G B . F / : H D ? <.I>-0/2JCN3CN7 _ =.CN?D/89BDa _ <7>BE?149C _ BDC0<7 BGF1BMBGF./7.12BD=:?12-7=:J¦I./8? \»¿ a CN3<.9.9 BGF./27¾BGF./ _ <7:9>CABDCN× BE?179>C 6= 7 \ 2 d(3n ) = 8 6= 15 F./2?D/2amCABDCN<7 -0/2193BD< n d(2n2 ) = 14 k Q \ µc<4z BGF12BMCN3 (α1 + 1) (2αi + 1) = 7 4 n = 26 d(3n2 ) = 2 × 13 6= 15 i=2 zF>C _ FCN3M1 _ <7>BE?19>C _ BDCN<7 \ mI

Ë \K] -A-BGF./¦7.1BD=:?18-@7=:J¦I./8?3a0?DCABE?18?DCA-;|18-N<7:O1 _ NC ? _ -N/ \g F./63W=:J<>a/+~/8?x;BGF?D/4/ _ <7.351/ _ =BDC 100 ~/2- ;14?D?147:2/897=:JKI:/2?3CN3 _ 12- _ =.-L12BD/49 \ ,:?D<8~/°BGF12BjBGF./8?D/M/Wò8CN3EBjB½z<3W= _ Fc3W=:JO3 z*CBGFBGF./o9>C Ä/8?D/27 _ /oI./B½z/8/87BGF./8JÓI:/CN7: ?D/81BE/2?@BGF147 \

¡ t¤

. +(/01'2.3 <> '2:,A A : .A 3- !0 :'*3 5 ,) '1-!.3- )# 0B0 : 5AB)3 % : 2

¦

ñCL?3GB C~>C09./18-A- BGF./?D/8JO18CL7CN7:7=:JKI:/2?3CL7>BE< H:12CN?z*CN35/ 9CL3 x<>CL7>B ?D<> =:H:3o<>a:1BGF?D/8/ _ <7.3+/ _ =BDC~/*7=:JKI:/2?3M18-N<7:BGF./ _ CL? _ -0/ \ F.33 /BE < B18-:3W=:J <>a BGF./23+/

33

P ?D<>=:H:3CN3 100

i = 5049

187:9BGF./{1¡~/8?142/{3W=:J

CL3 5049 33

\

= 153

¿ B am<>-A-0<4z3°BGF1B@BGF./8?D/*CL3Mi=2 \ 12B°-N/8183GB <7:/*3W=:J 351¡; 3W= _ F(BGF12B µc<4z BE153 < H:12CN?z*CN35/ 9CL3 x<>CL7>B ?D<> =:H:3o<>a:100 BGF?D/8/ _ <7.3+/ _ =BDC~/*7=:JKI:/2?3M18-N<7:BGF./ _ CL? _ -0/ \ F.33 /BEa BGF./23+/

33

99 ?D<>=:H:3CN3 P

i = 4950

187:9BGF./{1¡~/8?142/{3W=:J

CL3 4950 33

¿ B am<>-A-0<4z3°BGF1B@BGF./8?D/*CL3Mi=1 \ 12B°-N/8183GB <7:/*3W=:J 351¡; 0 3W= _ F(BGF1B 0 \ S S ≤ 150 F./27 4 1 : 7 9 z / 8 1 D ? / . 9 < : 7 / S − S0 ≥ 3

= 150

\

\ ñCL7:9O12-L-.BGF./*aN=:7 _ BDCN<7.3

zF>C _ F3W12BDCL3WaA;(BGF./*?D/2-L12BDCN<7

→ ( x · f (x) = bxc · f {x} + {x} · f bxc , f :

∀ x ∈ e zF./8?D/ 147:9 9./87:BE/8?18-H:18?BO147:9`a0?1 _ BDC0<7.12-H:18?BaN=:7 _ BDC0<7.3 ?D/83WHj/ _ BDb·c C~/- ; \ {·}

z ¤ ¦ U¤

~ ¤ ¡ ¦

. + : &<> '2% ?7:+ ) 0*),'9+ + : 5 . / :!35 AB)3 % : ) 3 ' :!A A: .A 3- !0 :'*3 5 ) # 0B02:!5A ) 37% : : ,' : )01)' +9+2: - -!. B+ / 01' .3

)'9-,.3 -

F./35<>-L=BDCN<7.3M18?D/6BGF./ _ <7.3EBD187>B°aN=:7 _ BDC0<7.3 \ ] 7; _ <7.3EBD187>BaN=:7 _ BDCN<7(351BDCN3G/23BGF./c¡C~/27(?D/2-L12BDCN<7 3WCL7 _ / \ <7~/2?3+/-; _ <7.3WCN9:/2?16aN=:7 _ BDCN<7 amC _ FBGF./BGF./*¡C ~ /87?D/-N1xBDC0<=7bxc F.<>-09+3 {x} 187:9 \ ¸2=jI.3EBDCABD=BDCL7:c14714?EfICBE?14?x;7:<7:}DW/8?D<CN7>BD/42/2? am=:3 \ x ∈ (0, 1) µc f (x) = C I:/e187 18?GI>CABE?18?x;Ó7:<7:}EW/2?D<Ó?D/218-7=:J¦I./8? \ ¯ I.35/8?x~>CN7:¶BGF1B <4z -N/B x 187:9 z/2/B bxc ∈ {x} ∈ [0, 1)

= C·x am
xf (x) = bxc · C + {x} · C = C · bxc + {x}

\ f (x) = C x \ ñ4CN7:91`Hj<>- ;.7:a9:/8?D/4/ z*CBGFç?D/812- _ . < / _ C0/27>BD3|3W= _ F»BGF12B/81 _ F <>a^CAB3 ?D<.=:18-`BE<ÀBGF./ 35=:318?D/$<>a<7:/ D? <.a»BGF./ j H <>- ;.7:
147:9

f (x) = C

am<>-A-0<4z3 \°¿ 7 _ <7 _ -L=:3GC0<7

P (X) = X + 9X + 9X + 9

y¤

¤ ¤

. + : <> ,:!)3 +1) / :*H3 AB' :/5 :,)/3 :!5 AB)3 % : ' % ? : + )01)' +9+2:!5 . / :!35 : . < :,A0 ' +9'13- '15" / 0@A:!6 .3;0*5 )3 ' :!A A: . A 3- !0 :'*3 5 )9' -,.3 - ) # 0B02:!5A ) 37% : : ,' : )01)' +9+2: - -!. +B/ 01' .3 -

¦ ¤ F¥ U¤

AB)3 %

¦ ¡

\Ä / _ 187(<.IBD12CN7cBGF./?D/8=.CL?D/49 ·8/B I./MBGF./ _ a w P (X) Hj<>- ;.7:
. +(/01'2.3

k \ ¸2CN7 _ /

u + v + w = −9 uv + vw + wu = 9

187:9

uvw = −9

u + v + w = (u + v + w) − 2(uv + vw + wu) = 63 u v + v 2 w 2 + w 2 u2 = (uv + vw + wu)2 − 2uvw(u + v + w) = −81 \ u2 v 2 w 2 = (uvw)2 = 81 2

2

2

2

2 2

F>=:3 GB F./*?D/4>=.CN?D/89KH:<>-;7:
Q(X) = (X − u2 )(X − v 2 )(X − w 2 ) = X 3 − 63X 2 − 81X − 81

. +(/01'2.3

Q(X 2 )

/27 _ /

kk \ F./H:<>-;7:
Q(X) = (X − u2 )(X − v 2 )(X − w 2 )

= (X 2 − u2 )(X 2 − v 2 )(X 2 − w 2 ) = (X − u)(X − v)(X − w)(X + u)(X + v)(X + w) = P (X) · −P (−X)

\

3W12BDCL3Wj/83

= (X 3 + 9X)2 − (9X 2 + 9)2 = X 6 − 63X 4 − 81X 2 − 81

Q(X) = X 3 − 63X 2 − 81X − 81

\

\

\ ,:?D<8~/cBGF12B°am
a b

147:9

c

BGF./*CL7:/4>=:18-ACABx;

(a + b + x)−1 + (b + c + x)−1 + (c + a + x)−1 ≤ x−1

√ \ F.<>-093 z F./2?D/ 3 x = abc

m¤

. +(/01'2.3 <>

' % ? : + ) 0*),'9+ + : 5 . / :!35 AB)3 % :

¦

\Ä] 7K/2143; ·8/B −1 −1 −1 _ 12- _ =.-L12BDCN<(a 7(;+ CN/2-Nb9.+ 3 x) + (b + c + x) + (c + a + x) = N/D N D

= 3x2 + 4x(a + b + c) + a2 + b2 + c2 + 3(ab + bc + ca) = x3 + 2x2 (a + b + c) + x a2 + b2 + c2 + 3(ab + bc + ca) + a2 b + ab2 + b2 c + bc2 + c2 a + ca2 + 2abc

\

¯ I.35/8?x~>CN7:BGF12B a2 b + ab2 + b2 c + bc2 + c2 a + ca2 = (a + b + c)(ab + bc + ca) − 3abc

147:9

x3 = abc D

z/F1~/

= 2x2 (a + b + c) + x a2 + b2 + c2 + 3(ab + bc + ca) + (a + b + c)(ab + bc + ca)

\

/27 _ /

\ D − xN = (a + b + c) (ab + bc + ca) − 2x2 − 3x3 √ 187:9 º;BGF./ ] ¹|´±¹ ¿ 7:/8=:12-LCB; z/{F1¡~/ 3 a + b + c ≥ 3 abc = 3x √ \ > F : = 3 3 ab + bc + ca ≥ 3 a2 b2 c2 = 3x2 \ 2 2 3 D − xN ≥ 3x(3x − 2x ) − 3x

¿ B am<>-A-0<4z3°BGF1B N

.6

¡ t¤ Y' ≤

1 x

= 0

143M?D/8=.CL?D/49 \

6 :,370 <> '2:,A A : .A 3 - !02: '135 ,) '1-!.3- )# 0B0 : 5A ) 37% : D

¦

¦

·8/BBDCL7: 147:9 BGF>CL3H:?D<.I>-0/2JCL33GCNJCL-L14?gBE<(H:?D<.I>-0/2J 3 b33+<>-A=BDC0γ =<>ca3z F>C _ F¶18H:H:/818?3CN7 )0@? ) ) '*3 : ² CN7&BGF./¼¸ ]α¹ =¯ a 8β =1 <7Â @<>- \ 4 7:< \ ³ G=:7:/(8 H \ ³ \ E È

\`¯ 7BGF./3GC09./83

147:9 <>aBGF.//8=.CA-N1BE/2?18-@BE?DCL147:¡-N/ BGF./Hj<>CL7>BD3 AB ABC ?D/235H:/ _ BDBC C ~/2- ; 18?D/ B14/27`3G= _ FpBGF12B D√ E \ F./`3GBE?12 0C 2FB{-LCL7:/83 187:9 CDCL7>:BE/2DB ?3+/ _ B{=CN7^BE BGF./`:Hj<>EA CL7>B =\ ( 5 + 1)/2 AD CE O F./¶Hj<>CL7>BD3 147:9 14?D/&CL7>BE/2?DCNCL7>BD3<>a*BGF./¶35/4JK/87>B3 187:9 M N OD OC \ F./OH:18?18-A-0/-gBD<{BGF./ ?D/83WHj/ _ BDC~/-; 3W= _ F¦BGF12B 147:9 M N k BC AN = 2 OM 3GBE?12C02FB-LCL7:/ 9.?1¡z7(BGF?D<>=j2FBGF./6Hj<>CL7>B CN7>BD/8?35/ _ B3°BGF./635/4JK/87>B CL7 BGF./cH:<>CN7>B \ AC ,:?D<2 ~/oBGF12BÄBGF./cF18-Aam}G-ACN7:/ CL3OBGF. /cICL3+/ _ BE?DC ò*<>a>BGF./147:¡-N/ M C \ 187:9

P

AP

M AN

.6

=¥ D¢ ¦

6 :,3701- <> N .-?' . : '16' > )5 ) )-) '15 ) 7) 3

F./ _ <7:9>CABDCN<7.3<>aBGF./KH:?D<.I>-0/2J 3EBD1BE/2J¦/27>B _ 147.7:CABDCN<7 < 7:9CBDC0<7.3 AN = 2OM 14?D/cBE?D=j/ \ √ z/ ¸2CN7 _ / CD : DB + 1) : 2 \ º; √ = ( 5√ 2/B B 3)F1~/ ...... ... .. z/ ¹&/87:CD /-N12=::3 CB

F./4=
4

BGF12B°CL3

BE AO DC · · = 1 EA OD CB √ √ \ 5 + 1 AO 5+1 · ·√ =1 2 OD 5+3 √ AO 2( 5 + 3) = √ = 1 OD ( 5 + 1)2

4

\

zF./87 _ /

F>=:3

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

D

M

E

O

P

K

N

A

C

AO = OD / 14?D/(¡C ~/87OBGF1B 187:9 18?D/Hj<>CL7>BD3<7BGF./(35/4JK/87>B3 187:9 N \ ·8/¡B ?D/83WHj/ _ BDC~/-; 3W= _ FBGF1B M I./OBGF./KHj<>CL7>B*<7 OD 3G= _ FOC BGF1B \M F./27 M N k DC K OA \ ¸2CL7 _ / KN OK : OA : OC = OM : OD OA = OD \ ¸2=CL7 ON _/ z /3+/8k/AC BGF12B ◦ OK = OM ∠KM N = ∠ADC > ∠ABC \( F./8= 147:9 ?D/am<60 ?D/ ◦ z/OF1~/ ∠KN M = ∠ACD ∠KM N > ∠KN M \ ¸2CL7 =_ / 60 KN > KM = 2OM

∠AKN = 180◦ − ∠DAC > 180◦ − ∠BAC = 120◦

\ F./2?D/2am-A-0<4z3°BGF12B AN AN\ > 2OM F>=:3 CB°CL3M 7: BE?D=jKN /cBGF12B AN = 2OM

µc<4zÁz/ÂBD=:?E7BE<Ó35<>-L=BDCN<7.3|BD<ÓH:?D<.I>-0/2JK3p<>a(BGF./ · ¿¿¿ ¹p12BGF./2JK1BDC _ 18¯ -;JKHCN149K<>a¹&<>-N9:<2~>1 4 ´ ñ>³ ¡´³ \ È È È É \ ±?147:9>a012BGF./2?o9>CN3EBE?DCNI=BD/83 3Ez/8/B3o14JK<7: ?187:9 _ F>CA-09?D/87¦18?E?187:2/49K18-N<7: 1 _ CN? _ -N / O:?3GB(<>ao18-A-oF./|¡C~/2n 3O<7:/3Dz/4/¡B*BECA-09 BGF./87ÂF./|¡C ~/83 <7:/O3Ez/8/BoBD<¦BGF./O7:/Wò2Bc?147:9 _ F>CL-N9 BGF./87|<7:/O3Ez/8/B354CLH:HCN7:< 7:/K?187:9 _ F>CA-09 9 _ F>CA-09?D/87¾147:93+<|<7 \ F./{9>CN3EBE?DCNI=BDCN<7CL3 BGF./87p<7:/K3Ez/8/B6354CLH:HCN7:B½z<?187: /Wò4/ _ =BE/89KCN7OBGF./3518J¦/(9>CN?D/ _ BDC0<7 \ ñ>12-L=j/23c<>a 9:<./23c/+~/2?x;{?187:9 _ F>CA-09 n 2/B 13Ez/8/B

.6

¡ t¤

6 :,370 <> '2:,A A : .A 3 - !02: '135 ,) '1-!.3- )# 0B0 : 5A ) 37% :>\

F>CL3OH.?D<.I-N/8JÀCN3O/8=.C ~>18-N/87>BBE-0/2JÁ² <>aMBGF./ <>=:?D7.14JK/87>B<>aMBGF./ <4z7.38 ¶0¸8H:?CN7: ¹p18CL7 @/2?3WCN<7 \ ] 3+<>-A=BDC0<È 7`14H.Hj/214?3¦CL7 @3;02:,A 3)01'2.3 ),+ ) 0@?7:!6 )01'2%(- N . /A 3) 6 :!3;0 . N . 3 - 5 /:!-,01' .3 - ) 3 -,. +(/01'2.3- /49>CABD/49I8; , \ \ 1;>-0-LCL3+F./89I8; ] =:3GBE?12-LCL147 ¿ 7>BD/8?D7.12BDCN<7.18- /27>BE?D/am7.?DC _ FJ¦/2 7>B ] =:3EBE?18-ACN1 \

'

§

\

·8/¡BBGF./¾7=:JKI:/2? óM/87:a18-A-?D/218∗ I./¾¡C~/27 \ M 7=:J¦I./8?3 amC _ Fn BGF./8∈?D/(/Wò8CN3EBD3o1(:7CABD/3+/8=j/27 _ / z*CBGF x (ap ) p = 1 . . . n 3G= _ F(BGF12B ap ∈ {0 1} p = 1 . . . n \ −1 −2 −n x = 2

· a1 + 2

· a2 + · · · + 2

· an

187:96H.?D<8~/°BGF1Bjam=j/87 M _/ 1=: z*CABGF(BGF./*J¦/27>xBDC0∈ <7:M /89OH.?D
¤

. +(/01'2.3 <> ' :!A A: .A 3 - !02: '135 )'9-,.3 - ) # 0B02:!5 AB)3 % : 56 . '98;: <> 0@? : : ' 0 .A(-8\

¡

n \@ F./27 _ <7.3WCL3GB3<>ar18-A-?D/812-r7=:JKI:/2?3 01 ·8/B 0 I:/*?D/8M :x M0 y BGF12B _ 41 7K H.?D/8= 35/87>BD2/49xCN7 BGF.∈ /6amM
\ F./{14I:<2~/{?D/8H.?D/835/87>B12BDCN<7¾am2a .\O . . F>n p BG∈ CN3*3WCLJKHa-;{ F./¦{0 I>CN7. ?x;/GòH.p147.= 3GC0<7 =:3 187:y9 0 n y M = {0 1 . . . 2 − 1} −n BGF./8?D/am=j/87:/2353<>aBGF./O?D/2H:?D/23+/27>BD1BDC0<7 y a0?D: -A2-0<4z3@ BGF.M /c=:7C0>=j/87:/2353M<>a.BGF./*I>CN7.18?x;O/GòH.147.3GC0<7<>a/81 _ F x∈M 0\ y∈M

F./27

mI

·8/B

2−1 a1 + 2−2 a2 + · · · + 2−n an ∈ M Px = −p \ ¿ B°am<>-A-0<4z3°BGF12B x= 2

\ ·8/B

I = {p | ap 6= 0}

p∈I

X

2−p =

p=1

p6∈I

/9./49>= _ /6BGF1B

f (x)

n X

=

X 2000

p∈I

2p

2−p −

+

X

p∈I

X 1

p6∈I

\ ñCL7:9O12-L-.BGF./*CL7>BE/82/8?~>18-A=j/83o<>a

F143

1999

m

b·c

\

amC _ F(BGF.//4>=:12BDCN<7

m2 x − 13 1999

9CL3GBDCL7 _ B ?D/218-j3+<>-A=BDC0<7.3M

\

= 2000x + 1 − 2−n − x

2p

= 1999x + 1 − 2−n

2−p = 1 − 2−n − x

=

x − 12 2000

9:/27:BD/4?12-H:18?B°aN=:7 _ BDCN<7 \

\

4

¡ t¤

. +(/01'2.3 <> '2:,A A : .A 3- !0 :'*3 5 ,) '1-!.3- )# 0B0 : 5AB)3 % :\

¿ a CN3135<>-L=BDCN<7O<>a:BGF./¡C~/27K/4>=:12BDCN<7 BGF./27 amBD/42/2? x \ ¹&=:12BDCN<7|JK 1¡;|I:/Ox?D/U= z?D2000k CBBE/27|18+ 3 12 k k = m2 k + 2 zF./8?D/ m k + 12m2 − 13 4 BGF12B°CL3 f (k) = 0 1999

2

f (k) = (m − 1)k +

m2 k + 12m2 − 13 1999

\

\ ¸2CL7 _ / CN3°BGF./63W=:J<>aj147CL7 _ ?D/8183WCL7:aN=:7 _ BDC0<7187:9 2¸ =:H:H:<3+/BGF12B 2 \ ¿ B@am<>-A-0<4z3 17:<7:}W9:/ _ ?D/8183WCL7:caN=:m 7 _ BDCN> <7K1<7 BGF.f/6aN=:7 _ BDCN<7 CN3 CL7 _ ?D/8183WCL7:6<7 BGF12B°BGF.//4>=:12BDCN<7 F143o1BJ¦<3EB<7:/35f<>-L=BDCN<7 \@ F./8?D/am-A=BDC0<7O<>ajBGF./H.?D<.fI(k) -N/8J = ¿a BGF./27OBGF./(¡C~/27{/8=:1BDC0<7¦I:/ _ C _ F¦9.<./83 m = 0 −1 = 3+<>-A=BDC0<7.3 \ F>=:3 CL3M7:-A=BDC0<7O<>aj2000 BGF./H.?D<.I-N/8J \ 7:=:12BDCN<7 F183o35<>-L=BDCN<7.3 F>=:3f (k) \ BGF./=9:/203WCL?D/49~>18-A=j/83o<>a k =18?D1/ 2 . . . 141999 7:9 m m=1 m = −1

\

,:?D<2~/

BGF1B¶BGF./

¡ t¤

am + a n ≥ m m + n

.6

7=:J¦I./8? mn+1 + nn+1 W3 12BDCL3Wj/83ÂBGF./ a = m m + nn \ 41 7:9 n am
?D/2-L12BDCN<7

n

6 :,370 <> '2:,A A : .A 3 - !02: '135 ,) '1-!.3- )# 0B0 : 5A ) 37% :>\

F>CL3KH:?D<.I>-0/2JÁCN3O²Â<>aoBGF./¾¼¸ ] ¹ ¯ 44> \Ó] 35<>-L=BDCN<7Â18H:H:/818?3KCL7¶BGF./ I:<.<.8-0/¡B<>a@ó? \ 18-BE/2?o¹¾C0/27>BG81 H.=jI>-LCL3+F./89KI8;BGF./¹ ]]c\ \¯ 7*BGF./3GC09./83 187:9 <>aBGF.//4>=.CL-L12BD/8?12-BE?CN187:¡-0/ BGF./Hj<>CL7>BD3 ABC 147:9 ?DBC /83WHj/ _ AC BDC~/-; 14AB ?D/ _ <7.3WCN9:/2?D/49¾3G= _ FBGF12B M N P \ ñ4CN 7:9{12-L-BGF./~>18-A=j/83 _ CN? B_ -N/=z*CBM amC _AP FOBGF.:/ P BGFKBGF./ : M C = CN : N A = λ λ _ <2~/8?3BGF./¦BE?CN187:¡-0/I:<>=:7:9./49¶I8;BGF./3EBE?18CN2FB-ACN7:/23 9CL14JK/BD/8? \ AC 147:9 ¿ 7BGF./ _ 183+/{<>a _ <7 _ =:?D?D/87>B3EBE?18CN2FB6-LCL7:/83 BGF./{JK/87>BDCN<7:/4AM 9|BE?CN 18BN 7:¡-0/ CP 9:/82/87:/2?12BD/83MCL7>BE<1Hj<>CL7>B \ . +(/01'2.3 <> N .- ? '2. :'*6' > ) 5 ) )-) '15 ) 7) 3g\

=¥ D¢

8· /BBGF./|CL7>BE/2?3+/ _ BDC0<7.3K<>a 187:9 AM BN 147:9 147:9 I./ 187:9 187:9 BN?D/83WHj/ _ CP CP AM X Y BDC~/-; 187:9-0/¡B I./oBGF./ _ CL? _ =:J _ /87>BE?D/ Z O \ <>a 4ABC ¸2CN7 _ / z/K3+/8/OBGF1B AP = BM = CN a0?DC _ F 4P z /FAC 1¡~/ ≡ 4M BA ≡ 4N CB ∠P CA = ∠M AB = ∠N BC

¿ B am<>-A-0<4z3°BGF1B

\

B

.... ............ .... .......... ..... .......... ..... .... ............. . .. . . .. ... ... ... .. ... ... ... .. ... .. .. .. .. .. .. ... ... ... ... .... ..... ........ ...... . . .. .... .... ....... ............. ... .. ......... ... ... ... ... 2.... ...................... ... ... ... ....... .. .. .. ... . ...... ... .. ... . ... . . . . . . . ... ... ... ... ... ... . ... . . . . ... ... ......... ... . . .. . . . . ... ...... ... . . . ................... ........ . . . . . ... . . . . . . . . ............ ................ .................... . ... . . . . . . . .. ... ... ......... ........................................ ....... ................. . ... ............... ... ....... .. ...... ........... ..... . ... ........ ............. . . ..... ......... . . . . . . . . . . . . . . . . . . . . ... ........................ .......... ..... .......................... .............. . . . . . . . ... .................................. ... ................... ................. 1 . . . . . . . . . . . .... ....... .. .........................................................................................................................................................................................

M

Γ

P

Z

O

Γ

A

∠AXB = ∠BY C = ∠CZA = 120◦

X

Y

\

N

C

.4 /&9./87:a 187:9 I8; 147:9 4OAB z/OF 4OBC Γ1 Γ2 ?D/83WHj/ _ BDC~/-; \ ¸2CN7 _ / 1¡~/ ◦ ∠AOB = ∠BOC = 120 ∠AXB = ∠AOB \° F./8?D/ama 187:9 =JO∠BOC AOB Γ1 \ -AC0/2∠BY 3o<7C BGF./* CL7:a X Y BOC Γ2 ¿a ~>18?DCN/83(<7BGF./¦ 3WCN9:/ a0?D18?DCN/83<7BGBF./K3GC0C 9./ a0?DX CL7 _ C09./6z*CBGF \¿ a BGF./27 187:9 BGF./87 -AC0/23<7 λ < 1 \°¿ BGF./JCN7:<λ?g= 14? _ 1 <>a X 14Y7: 9 -LCNZ /83°<76BGF./JCN7:a a X BGF./87 Y<>a CO -LCN/83o<7BGF./*BO JOCL7:<>a 1 \ X AO Γ Y BO Γ2 1 µc<4zz/ _ <7.3GC09./8?cBGF./ _ 183+/ 1 \Â¿ 7BGF>CL3 λ = 2 _ 183+/Kz/9./87:BD/8?35/ _ BDCN<7<>a ............... ............ ........................ ......... ........ .. ......... ........ ...... . . . . T BN . . . 0 ..... .. .... .... ..... .... \g F./27 z*CABGF(BGF./ _ CN? _ =:J _ CN? _ -N/<>a .. ... ... ... . .... . . . . .. .. . .. 4ABC

187:9 F>=:3

X

∠AT N0 = ∠AT B = ∠ACB = 60◦ \ ∠CT N0 = ∠CT B = ∠CAB = 60◦

∠AT N0 = ∠CT N0 4

zF./27 _ /

M

. ... ... .. ... 0 ....... ... ...... ... ... ... . ... .. .. ... .. 0....................... ...... ... .. . .. . . . ... .. .... .. ... .. ... .. ... ..... .... ...... ..... . . . . . ... ... . ... .... ... .... .. . ... . . .... . . . .... .. ... . .. ... . . . ... . . . . .... ... ... .. ... ... ........ . . . ... .... ... ... ... . .. ....... . . . . . . .. ....... . . ..... .. . . . . . . . .. .. ..... .. ....... .. ... ..... . ...... .. .. ........ .. . 0 ... ............................................................................................................................................... .... ............ ... ........ . . .... .............. . ... .......... ............ .... . . . . . .. ............ .. ..... ............ . ......... ....... ............ .... ...................... ........ ... .... ........... ...................................................

N

A

C

T AT : T C = AN0 : N0 C = 1 : λ = 2 : 1 \ zF>C _ FCNJOH.-AC0/23 BGF1B AT = 2CT ¸2CN7 _ / CL3 ◦ 147:9 ◦ z/|3+/8/BGF12B ∠AX T ∠AT X0 \ = 4AT X 0z3 _/ /4>=.CL-L12BD/8?12- \ /87 0_ / = 60 ¸2CN7 60 A C B m a > < A 0 4 < ◦ \ X 0/2T7 _ = BGF12B / AT = 2CT ◦ \ F./8∠X ?D/am<0?DT/ C = 60 ∠X0 CT = 90◦ ∠T X0 C = 30 ◦ ◦ ◦ \ ∠AX0 C = ∠AX0 T + ∠T X0 C = 60 + 30

= 90

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x

<7.3GC09./8?BGF./>=:147>BDCB;=:7:9./8?@BGF./?149C _ 12- \ º;(BGF./*CL7:9= _ BDCN<7KF4;.H:

uk+1 uk+2 + uk+1 uk+3 + uk+2 uk+3 = uk+1 uk+2 + (uk+1 + uk+2 )(uk + uk+1 + uk+2 + 2m) = uk+1 uk+2 + uk uk+1 + uk uk+2 + (uk+1 + uk+2 )2 + 2m(uk+1 + uk+2 ) = m2 + 2m(uk+1 + uk+2 ) + (uk+1 + uk+2 )2 \ = (m + uk+1 + uk+2 )2

F./2?D/2am
BGF./>=:147>BDCB;O<7BGF./?DCN2FB 3WCN9:/<>aÄm

CL3o/4>=:18-.BE<

uk+1 +uk+2 +uk+3 +2(m+uk+1 +uk+2 ) = uk+3 +3uk+2 +3uk+1 +2m

\

º:=B I8;BGF./CN7:9>= _ BDC0<7F4;.H:4 C _ FKCN3oBGF./ BGF./ ?C02FBo3GC09./<>ax CN3/4>=:18-BE< 2m = uk+3 − uk+2 − uk+1 − z <>a 2uk+3 + 2uk+2 + 2uk+1 − uk \ ?D/ _ =:?3GC~/9./2:7CABDCN<7K u k+4 F>CL3°35F.<4z3BGF1BÄBGF./35/4>=j/87 _ /3W12BDCL3Wj/83gBGF./12-ABD/8?D7.12BDC ~/?D/ _ =:?3GC~/?D/2-L12BDCN<7 <>aH:?D<.I>-0/2J 4 am
9 I G

I, # " +" 0 ,0 F F , + , #0 9 , *6 , 00 +

7 , 9 I +

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147:9

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0 ≤ c ≤ 1 b sin x a \ c + cos x <

m¤

c+1

' % ? : + ) 0*),'9+ + : 5 . / :!35 AB)3 % :\

x

/(19.9OBGF./F4;.H: 0 f (x) = tb _ \ _ CN7 ?D/2143GCN7:*<7 ¸2CN7 / z/F1~/ c + cos x < 1 (0, ∞) 0< c+1

¸2=:H:H:<3+/7:<4zÓBGF12B I:/ _
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_

c + cos x c+1

b

< 1 =

0
<

sin x \ x

\ ·8/¡BBDCN7:

¸2CN7 _ /

c + cos x c+1

r

r ≥ 1 ≤

sin x x

r=

0 \

BGF./H:?D
b a

z/*35/4/cBGF12B

c + cos x c+1

\

sin x \ º:=B°am=:3 N C 3 N 8 / 8 1 D ? O ; L C 7 D ? 8 / 8 1 W 3 L C : 7 < 7 t+α 1−α f (t) = [0, 1] =1− t+1 t+1 /83 BE<3+F.<4zBGF1B 1 + cos x sin x zF>C _ FCL3o/4>=.C~>12-0/27>B°BD< < 2 x \
_ /23BE<H.?D<8~/6BGF1B c + cos x

. F /(-L143EBMCL7:/4>=:18-ACABx;¦CN3Mz/-L-g7:<4z7KBD<{I:/BE?=j/(am=:?oH.?D<.<>aCL3 x ∈ (0, π) _ '.E '#, 2,

" , " , L$$ , "3 "1: ' - 8A?
9 I G 7 2,0 7 , F + 0 # , ! = 0 M +"K + 9 I A, / , , #D/ , ,0 +#". , , / 2 /, G@5 /, 9/<= + G 9 # F0 + 0

G /+a9/b<= C +" G1, bK6=

900

a = b = 0 a > 0 a < b +%/ % + K+$ I + 9 , /0 .F +" 7 " /, 0 "G

&

U Q f5 LO! " $ +b L6 4(4) 4-!#"4%$ '$694' )97;:( )*) L)]+ $+ 4Q4 ')7 4Q4 ')781/5)94W4-!-,W /. ! ) )4( 0H(\(\(D5?BAUCLY5NOC NJ(VZCLY5NOC x + y + z = 1 i IQDR5ACLY5NC x T y T z ∈ [0, 1) r r r xy yz zx 3 z + xy

+

x + yz

+

y + zx

≤

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BGF./ ] ¹|´4±M¹ ¸2CN7 _ / ¿ 7:/4>=:18-ACABx;(;zCN/2+ -N9.3 xy = z(x + y + z) + xy = (x + z)(y + z)

/27 _ /

Xr

$

r

xy = z + xy

xy z + xy

r

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1 X 2

=

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=

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$

xy 1 ≤ (x + z)(y + z) 2

x y + x+z y+z

x y + x+z y+z

\

x y y z z x + + + + + x+z y+z y+x z+x z+y x+y x+y y+z z+x + + x+y y+z z+x

=

\

3 2

?451 $:D-'E4G 7 4 E 7 , 0 I'.B3%/ $"1 "E?1F 4: /E%'%",0 2,0 ; -0, , I4.
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9 I 2 F" , G ( ) ++=, I, 9AF9+ +D + / , J+0 K, .# = J, 1 D ;# ) + 9 I 2 , F, A<= # ,0x = ) y+", 2= z,0 = + 3 7 ! 0 , "K$$ I + J

, " +# " J# ;+ 9/ L E 9 !G /+" !
= +# 2 /, 5 !<= x y z>0 x y z A +# ;, / , J !<= a + b + c = abc ;+, +, L 8"

1 1 1 3 p + p + p ≤ 2 1 + a2 1 + b2 1 + c2 ;C# I 9 , = +# B.9 & " + :I < ,

>

" G

[

W O !"$)_B691/']6,44! 7%3('4 57[ &4 3 0H(\(\(D5?BA CLY5NOC NIQA CLYA ?LP VA? D FN J(D5J(KdDh CQH(?BA CXIQP*NJ(M `9A P R5A N MA"D !&AOCXI P ab\IQDD F-NJ(V&aYT AbJ(T aLA cT NMAD"!#A CXIQP9aEP9J CQAIQ\(IQA CQNOCQP D5J&D FCLYAUP9J(Ac5H(N`*PSC_^ Xp 2 2 2 a+b+c ≥

6/$+9), "

# $#% & #

a +b −c

$ ' $]B) & '(4 7 +$ !( -+ 7)[ * )*6j69E%36*6,O) 45+_,+ 5+9)_+$+, 7.- -+_7)/0 ( NJ(V NIQAfC:9WD \D P*J CQ?[D H5C ?P9V(AWNZaXP*IQaX` AaXAJ CXIQAV#NC O 132546487 0H(\\D5?dA CLY5NOC ?HaY CLY5NC AB P9J CQAI ?dAaQC A?CLYAaXB P*IQaX` A ; F X NJ(V Y NIQAWC:9WD#\(D P9J C ?UD5J&CLYAaXP*IQaX` A ?HaYCLY5NC AX NJ(V BY NIQACQNJ(MAJ C ? T CLYAJ AB ≥ AX + BY

¡

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p a2 + b2 − c2 + a2 − b2 + c2 √ √ √ √ 2ab cos C + 2ac cos B = 2 · CB · CD + 2 · BC · BD = √ √ = 2 CM · CD + 2 BD · BM = 2 · CZ + 2 · BY ≤ 2a

zF./8?D/6BGF./6CN7:/8=:12-LCB;am<>-L-N<4z3a0?D=:?°-N/8JOJK1 \ F>=:3 ¸2CNJCL-L14?-;

p

147:9

p \ a2 + b2 − c2 + a2 − b2 + c2 ≤ 2a p

p b 2 − c 2 + a2 p p c 2 + a2 − b 2 + c 2 − a2 + b 2 b 2 + c 2 − a2 +

≤ 2b ≤ 2c

\

. F /ÃH.?D=:18-ACABx;"am<>-A-0<4z3»I8; 19.9CL7:ÓBGF./ -L143EB¶BGF?D/8/ CN7:/8=:12-LCBDC0/23 \ .>=:18-ACABx;KF.<>-N9.3CLar187:9K<7- ;CAajBGF./6BE?CN187:¡-0/6CL3o/4>=.CL-L12BD/8?12- \ /7:C _ FCN3 BE?=j/ 3GCN7 _ / 2 2 a + b ≥ 2ab > ( 2 − 1)ab ] 2/4BD/8?DH:?D/¡BD1BDC0<7CN3@143am<>-L-N<4z3 .BGF./H:/8?CNJK/BD/8?<>a.1c7:<7:}5<.IBD=:3+/ BE?DCL147:¡-N/CN3612-z1¡;.3*?D/212BD/8?oBGF187aBGF./-0/27:UBGF3<>a@18-A-3WC ò BD187:2/87>B3Äa0?Da2BGF./@BE?DCL147:¡-N/°BD<CAB3g7CL7:/4}GHj<>CL7>B _ CN? _ -N/ z*CABGFc/8=:12-LCB; CLar187:9K<7-;CAajBGF./6BE?CN187:¡-0/6CN3o/8=.CA-N1BE/2?18- \ + 9<= <= , , #0 - 9 " , ; F &*1 ? E 7 9K1 "E 4. " ! 63J# +# 9 I G -%' E E4 .G C: :. 7 ,0 '#, 2, / & , /?A'%$ F# ) , 9 <= , , #0 - 9 " , ,0 , 0 " +, # 9 /, 9 , , / 2 9 ,0!3 +#2 L = , F0 ,0 +# 9 G*E( F+* + = +9 C + /" <,0FF 9 + L, / , G + F , "J) ;+ J 9" +# L, # ,0 E $ E B.'E ;4I$ [email protected] ? '%",0 2,0 $ 2 ;$ 2 7 9, "", 3?#: (;& :D45E -%4: 7 E& @.4: '.- + ", 9+ '#, 2, ( ME 0 ,0 ? I +"

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A .. ........ .. ...... .. .. .... .. ... ...... .. ... S .. . .. ..... . .. ... ... .. .. ... ... .. .... .. 0 .... . .... . C ..............T ........ . .... . . .. .. .................... .... . . . .......... ... ...................... . . ... . ................ .... . ...... . . . . .... . . . ..... ................. . . .. ..... . .... . . . . . . .... .... . .. ... ........ O .... .................. ....... .. ............ ................... . ........ ....... . . .. .... . ....................... . . . .................. .. .. . .................................................................................................................................................

A0

B

C

=

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¸2CN7 _ CH /

=

CG

187:9

B

D

C

AC B|am<>-A-0<4z3|BGF12B ∠GBE = ∠ACH ∠EBC > ∠HCB \¦ F./87 \ G B F 2 1 B N C 3 2 / 7 _ / BE BG ∠GBC > ∠GCB BG < CG 4 < 1 < 1 CG CH \ 147:9BGF>=:3 BE < CH < CF ¸2CN7 _ / 147:9 CABam<>-A-0<4z3BGF12B BE CF 4 BGF1BCL3 AB < AC BE < CF < AC AB BE CF \ F>CL3 _ <7>BE?19>C _ B3 BE CF \ < = b ¿a c \ z/F1¡~/13WCLJOCA-N18c? _ <7>BE?14c9C _ BDC0<7 \ F./8?D/am
2¸ CNJCL-L14?-; a0?DC _ F3+F.<4z3°BGF12B

\ AD BE CBam<>-A-0<4z3MBGF1B = a=b a CN3o/8b=.CA-N1BE/2?18- \

/27 _ /

a = b = c

4ABC

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/27 _ /

BC

AD < BC

2 tan θ sin 2θ =

z/F1¡~/ ·8/B

=

\ ¸2CN7 _ /

2DC

< 1

2 tan θ

2 sin θ × 2 sin θ cos θ = 4 sin2 θ < 1 cos θ

sin 2θ <

H

=

AD 1 = < 1 2 tan θ BC

I:/cBGF./*am<.ajBGF./H:/8?DHj/27:9C _ =.-N18?°a0?D
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AC

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\

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=

AB

=

AC

8· /B I:/6BGF./*Hj<>CL7>B<>aCN7>BD/8?35/ _ BDCN<7K<>a 147:9 I:/cBGF./Hj<>CL7>B<>a 147:9-N/B \ ¸2CL7 _ / BECL3@BGF./cAD CN7>BD/8P ?35/ _ BDCN<7<>a z*CABGF H:/8?DHj /27:9C _ =.-N18F?°ICL3+/ _ BEa CP AB AD BC \ F>=:3 AD z/F1¡~/ BE CF 4 BGF12B°CL3 CF = BE = = BC

AC

AB

AD BE CF = = a b c

\

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4ABC

¨2Wª F./6am<>-L-N<4z*CN7:BGF./4
_ 147K/8183WCA-;I:/H.?D<8~/89 \

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9 I G (

)

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abc = 1

ab2 + bc2 + ca2 ≥ ab + bc + ca

v¢

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¢

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abc = 1

p 187:9 ab2 + 2bc2 ≥ 3 (ab2 )(bc2 )2 = bc 3 \ ] 9.9CL7:ÂBGF./ÂBGF?D/8/ ca2 + 2ab2 ≥ ab 3

°\

BGF./*CN7:/8=:12-LCB;CN3o/8=.C ~>18-N/87>B@BD< x

b c a 1 1 1 + + ≥ + + c a b c a b

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a b

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147:9

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x y z

3G= _ F*BGF1B

F./¡C~/27OCL7:/4>=:18-ACABx;CL3 BGF./87K/8=.C ~>18-N/87>B@BE< xy yz zx + 2 + 2 z2 x y

≥

z x y + + z x y

x y

a=

\

UÈ ¿ 7:/4>=:18-ACABx;p am<>-L-N<4z3*a0?D-093oCLag187:9 ~/ _ BD=.-09OI: /*14H.H.-AC0/89BDaB18= CL7 b = c = 1 x3 y 3 + y 3 z 3 + z 3 x3

x3 y 3 + 2y 3 z 3 ≥ 3

p 3

≥ x3 y 2 z + xy 3 z 2 + x2 yz 3

(x3 y 3 )(y 3 z 3 )2 = 3xy 3 z 2

\

¸2CNJCL-L14?-; 147:9 3 3 3 3 3 3 3 2 \ 2z 3 x3 ≥ 3x2 yz 3 ] 9:9> CNy7:zBGF./8+35/6 BGF?D/8/*CL7:/4>=:18-ACABDCN/83 z ma x <>-A-0<4+z2x 3 \ y ≥ 3x y z È¡

8

(0 &*1!?E 7 )9;1 E)4." 2 63-D$1 E ? 1 4:'E*'#, 2, - ,0 , 4.
='%$.3JE?B";4 $-%E? 9:D0%29@5 , 8 '#, 2, / & " F/ @. !
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U

5 aXD5IXIQAaQCXAV / !@"$! $O6 4 0 * $B+ $7 )OQ7 E7;: ; F NIQA;CLYA?LP VA?ED F P*J(9WY P aYWNOC !&D5?LC D5J(A[NJ(M `9AUA aLAAV5? π T 3 NJ(VPjF RaP*T ?bbPjT C ?;c aXP9IQaXH ! I N V P*H(? T \IQ4ABC DR5AUCLY5NOC X 2 2 2 2 ≤ 6R

a +b +c

cos A

# $#% & #

" $[.O ,W ;4! 7 B+ 4O6_ d7 * ;7 :

AfH(?dACLYAfFGD `*`9D 9fP9J(M,9WAO`*`9KJ(D 9 JF NaQCQ? ?dAA OL

6/$+9),

Y

# $#% & #

cos A = X

# $#% & #

X

# $#% & #

J(V(AI8CLYA M

A \(NJ(V?bCQD

8

Y

# $#% & #

s2 − 4R2 − 4Rr − r 2 4R2

R+r

cos A =

cos B cos C =

R

8

4R2

# $#% & #

cos A + 2

s2 − 4R2 − 4Rr − r 2 4R2

s2

T

a2 + b2 + c2 = 2s2 −; 2r 2 − 8Rr 2 T

H 9 ` A I ? ( J A 5 c ( H N * ` S P _ C ^ R ≥ 2r P R5AJ&aXD5J(V5PSCQP D5J T 9WA(!H(?LCEY5NOR5A P (2 cos A − 1) ≤ 0 T 9WY P9aY

WY P aY?P !&\5`*Pjg3A?ECQD 9

T

r 2 + s2 − 4R2

X

# $#% & #

cos A ≤ 1 + 4

9 A Dh C NP*J ?P*J(MfAc H(NCQP9D5J? T NJ(V T W

T

+2

R+r R

X

# $#% & #

cos B cos C

≤ 1+4

r 2 + s2 − 4R2 4R2

≤ 3r 2 + 3R2 + 6Rr = r 2 + 3R2 + 7rR + r(2r − R) ≤ r 2 + 3R2 + 7rR T

T

È

=:3WCL7:6 \ F>CL3 -L143EBCN7:/8=:12-LCB;CN3/8183WCA-;(35/4/27BE<I./*/8=.C ~>18-N/87>BBE<BGF./6H.?D=:18-ACABx;KF.<>-N9.3CLaÈU 147:9K<7- 0; CLajBGF./cBE?DCL147:¡-N/*CL3o/4>=.CL-L12BD/8?12- \ U¥½W£.U

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X

X

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>

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X

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>

, F " C . +# , , /, # ,0 G 6D" 9 #2, <, ;0 ;9 9 3+# K !< <= #0 +; , " L/ F+MJ +# C# 22,0 9 C, + 9 # 9 0 < 9 J9 F !9 =0 , ( D 8I /, " 7 / $ I 11!G

f

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# $#% & #

a b +c −a

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6 $5+9)_ " $ %'5)S : hA[CLYAf A \IQA??P9D5JDh C NP*J(AVZh^Z?Hh CXI N aQCQP*J(MCLYA`9AF*C8?LP VAUD F AOC CLYA \(IQD5\(D5F?BA(a, VP*J(b,Ac) c H(N`jPjC,^F IQ"D ! CLYA I P VA ?LP VA e YAJ

È

F (a, b, c)

=

X

$

a6 + 3abc X

−

$

X

$

(a2 b + ab2 ) − 3abc

4 2

2 4

X

2 2 2

(a b + a b ) − 6a b c

$

a3

\

/7:/8/49BD<35F.<4zBGF1B \ 0 CABGF.<>=B-0<3W3<>a2F/27:(a, \¦ F./87 /8?12b,-LCBc) ; ≥ z/K JK1¡;|143W3W=:JK/BGF1B C /87 14H.H.- ;CL7:BGF./O·21¡z <>aA <≤3GCN7:B/23 ≤ \ . F D B < 1 8 : 7 ¡-0/ 1 cos B z /FC1¡~≤/ 2 ≤ cos B ≤ cos A \ x c2 + a2 − b2 − ca ≥ 0 /7:<4zJK144/(BGF./ _ F147:2/O<>ag~>18?DCL1I>-0/23 \ 187:9 z = c−b \ F./87Kx CN3o/8=.C ~>18-N/8x7>B@=BE< a y = b − a -N/818?D- ; x y z ≥ 0 (x + y + z)2 + x2 − (x + y)2 − x(x + y + z) ≥ 0

zF>C _ FCN7BD=:?D7OCL3o/4>=.C~>12-0/27>B@BE<

z(x + 2y + z) ≥ xy

CABGFaN=:?½BGF./8?F./-NHOa0?D

z/<.I>B18CL7

ÈU

= F (x, x + y, x + y + z) = z 6 + 6xz 5 + 6yz 5 + 25xyz 4 + 14y 2 z 4 + x4 yz + 10x2 z 4 + 6x3 z 3 + 16y 3 z 3 + 39xy 2 z 3 + 29x2 yz 3 + x4 z 2 + 8y 4 z 2 + 26xy 3 z 2 + 11x3 yz 2 + 27x2 y 2 z 2 + x4 y 2 \ − (xy)2 (2y 2 + 2xy + 3xz) − (xy)(4xy 2 z)

F./*CL7:/4>=:18-ACABx;K 147:9

\

È¡

CLJKH-LCN/83 BGF1B

−(xy)2 (2y 2 + 2xy + 3xz) ≥ −z 2 (x + 2y + z)2 (2y 2 + 2xy + 3xz)

\ z/<.I>B18CL7

−(xy)(4xy 2 z) ≥ −z(x + 2y + z)(4xy 2 z)

¸2=jI.3EBDCABD=BDCL7:6CL7>BE<(BGF.//Wò4H:?D/2353GC0<7am
5

5

14I:<2~/

≥ z + 6xz + 6yz + 25xyz + 14y 2 z 4 + x4 yz + 10x2 z 4 + 6x3 z 3 + 16y 3 z 3 + 39xy 2 z 3 + 29x2 yz 3 + x4 z 2 + 8y 4 z 2 + 26xy 3 z 2 + 11x3 yz 2 + 27x2 y 2 z 2 + x4 y 2 − z 2 (x + 2y + z)2 (2y 2 + 2xy + 3xz) \ − z(x + 2y + z)(4xy 2 z)

8ò4H:187:9CL7:*187:9

6

F (a, b, c)

4

_ C0/-093

F (a, b, c) ≥ z 6 + 3(x + 2y)z 5 + (4x2 + 11xy + 12y 2 )z 4 + (3x3 + 8y 3 )z 3 + xy(13x + 11y)z 3 + x(x3 + 2y 3 )z 2 + x2 y(9x + 13y)z 2 + x4 yz + x4 y 2 ≥ 0

143o9:/23WCL?D/49 \

È4È k½k \

¡ R¥

. +(/01'2.3 <> ' ? . / 5 . + .6 6 /3'(0 > . + + : : 5 1' 3;02:,A ) :,3 5 5

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¼ 3WCL7: o CN7:/8=:121-LCB+ ;KI.cos / _
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X

$

cos A

\

A + sin2 B + sin2 C)

4(sin2 A + sinB + sin2 C) ≤ 6

X

$

cos A

BGF./p1I.<8~/

\

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a b2 + c2 − a2

&

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8a2 b2 c2

+ F

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R

"$8* ) ! '3 6 1345+ 4O)*6j6_ 7 $57" 4+ ! A C NJ(V h(AfCLYAWP*J(aLAJ CXIQANJ(V D5I]CLYDaLAJ CXIQAZD F O

A J(D 9 ICLY5NC H 6/$+9),

4ABC T

IQA?\AaQCQPSR5AO`S^

IH = 2r − 4R cos A cos B cos C ?dAA \(IQDh ` A ! 3

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A(z) =

n Q

(z − zk ) T

NC T FGD5I i IQDR5ACLY5k=1

P*N`

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aQNJ h(A F NaQCXD5IQAV k=0 NIQAf\D5?LPjCQP R5AfIQAN`3J5H!&h(AI ?

A(z) = z n + zk

n−1 P

ak z k

k = 1T 2T . . . T n − 1T 1 1 k+1 an−k k ≥ an−k−1 T C(n, k) C(n, k + 1) CQA?WCLYA h P9J(D"!P9N`;aXDA #aXP AJ C n

WYAIQA C(n, k) VAJ(D 4

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{¦ : '9- 5

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. +B/ 01' .3 <> 7) 0@? +2::!3

"$- : .5 "$- : .5

5

F./CN7am=B@7=:JKI:/2?3 147:9 1 _ BD=:12-L- ;BE/-L-L3°=:3 N <7-;(BGF12B@BGF./+;O18?D/7:<7:}G7:/41BDC~/6CN7>BD/42/2?3 \ B C P F./ 35=:18?D/ <>apBGF./"-0/aLB3WCN9:/ <>apBGF./ H:?D=B CN3 1*3G=:J<>aj3WC ò8BD/4/27 (B \g F./2?D/2amCL3 3W=:J 7:<7:+ }W7:P/8)(C 12BDC ~+ /BDN /8?D)(B JK3 + B½zC)(P <(<>az+ F>C _ N F()18 ?D/ 2 2 187:9 2 2 B N C P CN3?D/212BD/8?oBGF147aBGF./O?DCN2FBc3WCN9:/CL3 Bz N + C P F>C _ FKCN3-0/2353oBGF187=:14+ ?D/(C <>arP BGF./ ?DCN2FBM3WCN9:/CL3o-N/83W3MBGF187{=:18-jB BE<BGNF./+ 3+>=:C14?DP /(<>a BGF./K-0/aLB \ ¸2 CL7 _ /OB18CN7:BGF./¦Hj<3GCABDC ~/¦35=:18?D/¦?D<.CNH BGF./*CN7:/8=:12-LCB;OF.<>-093 \ k½k \

¢ ¢

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(B + P )(C + N )(B + C)(P + N )

. F /H:?D=:18-ACABx;¾CN3CNJOJ¦/89CL12BD/2- ;&<.IBD12CN7:/89&I8;¶<.I3+/2?x~CL7:KBGF./7:<4z7 \ CN7:/8=:12-LCB; −1 ≤ rx,y ≤ 1 C&*1? E 7 9K1 "E4." 2 63 ? 451 $ :D'- E4 7 : J E K L/ # , @#9 2 63? 451 $ :D'- E 4 G 7 4) "5E 7 , 0 '5B3451!? K4 12G E"$$/4# + - 9 2 ?K'I$ 53K 00E@ 4 ! :L'I$/K'%!// , # "
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\(IQDD F

È k \. +B/ 01' .302. 01 <> 7' % ?)A -+ ' ?;01+ > <> 0@? :: ' 0 .A )

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AB =

a b c d

¦ 187:9

¡

: - -5 ;) 37% ? . )+2.- :,A :!-5 BA =

P = αI + βAB + γBA

F./27¦I8;K9>CN?D/ _ BmF>=:JO147

k½k \

187:9

e g

f h

IH 5

H

(

6 . '187:

\ ¸4/¡B

Q = αI + γAB + βBA

\

_
det(P ) − det(Q) = (α + βa + γe)(α + βd + γh) − (βb + γf )(βc + γg) − (α + γa + βe)(α + γd + βh) + (γb + βf )(γc + βg) = α(β − γ)(a + d − e − h) + (β 2 − γ 2 )(ad − bc + f g − eh) = α(β − γ) tr(AB) − tr(BA) + (β 2 − γ 2 ) det(AB) − det(BA) = 0

\

. +(/01'2.3 <> 01

·8/¡B

m¤

' % ? : + ) 0*),'9+ + : 5 . / :!35 AB)3 % : 147:9

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C = βAB βBA \ ñ>+ - ;.7:
BGF./27 _ -0/214?-;

γ = 0

P (x) = det(xAB + γBA) − det(xBA + γAB)

\

/?D/219>CL- ;*3+/8/MBGF1B \ ¸2CL7 _ / F183@9:/8?D/4/1BgJ¦<3EB P=:(−γ) B½zP-;7:-A-0<4z3 \ BGF12B amò 147:9 \ µc-;7:= B147:9*BGF./3518J¦/MBE?1 _ / \g F12BgCN3 det(xI − C) = det(xI − D) CABGF z/<.I>B18CL7BGF./?D/8=.CL?D/49O?D/83G=.-AB \ x = −α \ mI µc< BGF./6C09./87>BDCB;CN7KN1 9.<./83M7:-N9CAa 9./87:BDCABx;OJO12BE?C ò 18n ≥-0/¡3B ·8/ B 7:9 In n×n A3



 1 1 −1 1  =  −1 0 2 1 1

147:9

B3



1 1 =  −1 1 1 1

 −1 \ 0  1

F./27*3EBE?18CN2FBDam 3 W/8⊕ ?D<A JO312BE?C ò \@B F./8= 7 O a0?Dn−3 aÄ9CL?D/ _ B 3G=:J (n<>ar−JO3) 12BE?× C _ /8(n 3 z /F1~/ \ 147:9

147:9

/27 _ /

An Bn = On−3 ⊕ A3 B3

Bn An = On−3 ⊕ B3 A3

det(In + 2An Bn + Bn An ) = det(In−3 ⊕ (I3 + 2A3 B3 + B3 A3 )) = det(In−3 ) det(I3 + 2A3 B3 + B3 A3 ) = 352

È

147:9 det(In + An Bn + 2Bn An ) = det(In−3 ⊕ (I3 + A3 B3 + 2B3 A3 )) = det(In−3 ) det(I3 + A3 B3 + 2B3 A3 ) \ = 348

;0 6E 1 !$8%" 9 "1!:G"@;'.E4I$"E )/K1@5'#, 2, ?;E?E 7 , 9 2F , 35?(451 $1- E & ): 41!?1 / +, ='%",0 2,0 F" , 0 I4.
=

α2 + α(β + γ)tr(AB) + (β 2 + γ 2 ) det(AB) & + βγ (tr(AB))2 − tr(A2 B 2 )

U

!#"$ ! $O6 480)* $B+G$7 [) Q78E7;: ; J KO 9WAUg(J(VZCLY5IQAAf\(IQDD F ?;D FCLYAWaX`*N??P9aXN`(P*J(Ac H(N`jPjC,^ T X 3 1≤

# $#% & #

cos(A) ≤

2

J Q K " T 9WA gJ(V Q `9N ! P9J ?P*`j`*H(?XCXI NCQP9D5J? D F;CLYA :@N,D5IQPLNCQP9D5J GD5I Q NI N !&NOCQN 0; J(Ac5H(N`*PSC_^ i IQDR5A CLYA Nh(DR5A = aX`*N??P9aXN` P9J(Ac5H(N`*PSC_^T> H(?LP9J(M CLYA :@N,D5I PLNOCQP D5J ; J(Ac H(N`jPjC,^ 1 !3 I5D5I[CLYA aLD5J R5AJ5P AJ(aLA D FECLYA#IQANV(AI 9WA#IQABR5P9A9 CLYA.:@N,D5I PLNOCQP D5J ; J(Ac H(N`jPjC,^ NJ(V TT = (b , b , . . . , b ) T 9WYAIQA AOC S = (a1 , a2 , . . . , an ) n 0H(\(\(D5?BAU1 CLY52NC P n n P N ( J V a1 ≥ a 2 ≥ · · · ≥ a n b1 ≥ b 2 ≥ · · · ≥ b n aj = bj j=1 j=1 NJ(V Pk a ≥ Pk b FGD5IWAN aY k = 1 T 2 T . . . T n − 1 e YAJ 9WA ?NO^ CLY5NC S j j & 4 'L j=1 ) . NJ(j=1 ;F V 9W, A 9 I PjCQA NJ(V F P9?ENfaXD5J R5A UF9H(J(aQCQP D5J T CLYAJ TT ST ST ;

n X

j=1

F (aj ) ≥

n X

j=1

F (bn )

N J(V F P*?=NUaLD5J(aQNOR5AF9H(J(aQCQP D5J T CLYAJ CLYA[N hDR5AP9J(Ac5H(N`*PSC_^ P9?8IQABR5AI ?dAV 6/$+9), "$6 4O6]+G'3O/.B4+3 $7:2Q$O6*)- $ 4L) $ 7 + 5" %$ ! 7 $+G )94( PSCLYD H5C3`9D5?d?/D F e YA NJ(M ` A?2P9J[CLYAb\(IQDh ` A ! NIQAbNJ(M `9A?2P*JUNbCXI P9NJ(M ` A MAJ(AI N`jPjC,^ T 9WAWN?d?LH !#AUCLY5NOC A ≥ B ≥ C AWV5P*?LCQP*J(M HP9ABC ?dYZCLYAaQN?dA 9WYAIQAfCLYA CXIQP*NJ(M `9AbP*?Dh CQH(?dA A > π/2 F IQD"! CLYA;aQN?dA 9WYAIQAbPSC3P*?/J(D5J(KdDh CQH(?BA A ≤ π/2 ; J#h(D CLY aXN?BA? 9WA 9fP*`j`H(?dAUCLYAfF NaQC=CLY5NOC=CLYAfF9H(J(aQCQP9D5J P9?;aXD5J(aXN R5A f (x) = cos x D5JZCLYAfP9J CQAI,R NT ` [0, π/2] ;

F

ST

È \ F./27

ñCL?3GB°3W=:H.Hj<35/6BGF1B

A ≤ π/2 π π π π π \ , , 0 (A, B, C) , , 2 2

] H:H-;>CN7:BGF./¹p1 x
π 2

z/F1¡~/

3 3 3

+ cos 0 ≤ cos A + cos B + cos C ≤ 3 cos

¸2CNJOH.-ACLaA;>CN7: z /<.I>B18CL7BGF./?D/4>=.CN?D/89CN7:/8=:12-LCBDC0/23 \ \ F./27 µc<4z 3G =:: H H:<3+/cBGF1B

\ π 3

A > π/2

(π − A, 0) (B, C)

] H:H-;>CN7:BGF./¹p1 x

\

π−A π−A , 2 2

z/F1¡~/

π−A cos(π − A) + cos 0 ≤ cos B + cos C ≤ 2 cos 2
F./27

\ A 1 ≤ cos A + cos B + cos C ≤ cos A + 2 sin 2 ñCL7.18-A-; z/*7:
9 I G)/ +# 9 ; K, 2< F KF" , IG

3O !#"$ ! $O6 4 0 * $B+G$7 [) Q78E7;: P9?&N aXH5CQAKLNJ(M `9AV PjLC Y CLYA@?XCQNJ(VNIQV J(D C N CQP9D5J T 0H(\(\(D5?BA CLY5NC 4ABC \(IQDR5AUCLY5NC 9 2 2 2 4abc <

6/$+9),

a +b +c

a cos A + b cos B + c cos C

"$[.O W , ;4! 7 B+ 4O6_ d7

*

7;:

≤

2

abc

AOC K = (b2 + c2 − a2 )(c2 + a2 − b2 )(a2 + b2 − c2 ) 0P*J(aLA 4ABC 9P ? N aXH5CQA T 9WA !H(?LCbY5NOR5A K > 0 A C S hAUCLYA NIQANWNJ(V R CLYAWaXP*IQaXH!&I NV5PjH(?D F A 9fP*`j`(H(?BACLYAfFGD `j` D 9fP*J(M,9WA`j` K J(D 9 J D5IbAN? ^KXCQDK\IQDR5AUF N aQC ?

4ABC

(a2 + b2 + c2 )(16S 2 ) = K + 8(abc)2 X 2S T a cos A = R # $#% & # 4SR = abc T

a2 + b2 + c2 ≤ 9R2

T

È

F./-L143EB°CN7:/8=:12-LCB;am<>-L-N<4z3a0?D9R − (a2\ + b2 + c2 ) O _ /27>BE?D/ BGF./ _ CN? _ =:J _ /27>BE?D/147:9 BGF./=:12BDCN<7.3H m 187:9Km³ 147:9(BG4ABC F./*CL7:/4>=: 18-ACABx; z/<.IBD12CN7 E È¡E E K > 0 (a2 + b2 + c2 )

X

$

a cos A

= (a2 + b2 + c2 ) · = =

2S R

(a2 + b2 + c2 )(16S 2 ) 8SR K + 8(abc)2 2abc

=

K

+ 4abc > 4abc

2abc

zF>C _ FH:?D<2~/83 BGF./*-N/2aLBG}WF147:9CL7:/4>=:18-ACABx; \ ¼o3WCL7:cCN7:/8=:12-LCB;OL 147:9K/8=:1BDC0<7.3o 147:9Ox³ z/2/¡B ÈU G (a2 + b2 + c2 )

X

$

a cos A ≤ 9R2 ·

2S 9 9 = (4SR) = (abc) R 2 2

zF>C _ FB14/23 _ 14?D/¦<>a@BGF./K?DCN2FBG}5F187:9|CN7:/8=:12-LCB; \ .>=:18-ACABx;¾F.<>-N9.3*CL7BGF./¦?C02FBG} F147:9CN7:/8=:12-LCB;CLa147:9K<7- ;CAajBGF./cBE?DCL147:¡-N/*CL3o/4>=.CL-L12BD/8?12- \ 0 K45B)) E9)$ / ?5 '%$.3 &*1@;'5E ;&*EE @;' ?#: D$ ? , " .& / F/I $ /, 3 $" E B5E ;4I$ [email protected] "? K'#, 2, J I$ 2 I$ 2 7 9, " , /3&*1? E 7 )9;1 EL4." 2 63J? 451 $ :D'- E 4 G 7 4 E 7 , F '5B3 $?# : 7 4) : K " ! '#, 2, / &* "<= ! '%$.3(451!? K4 1!G E$$ 4# +)-/0 9 2?5'%$.3': @G EI'%E4#K@# -/, , . 7 3 EE4 2 : '%$ '%!// , # "L'56E )'#, 2, / ,0 / ,0 $ /, 3 &MK4?E :O4K' 1 :O : "1%E1F4 7 < 7 !, 3 :0$ 1: $"E10&*1 /5JB ;98, $ 3 G GK$&*EEE B# //0F
9 I G 7 F+9 .4.
U

3O !#"$ ! $O6 4 0 * $B+G$7 [) Q78E7;: 0H(\(\(D5?BA CLY5NC NJ(V B NIQA C:9WD ABR5AJ CQ? 9fPjCLY \(IQDh5N h P*`jPjCQP9A? P (A) NJ(V A L ? H a Z Y L C 5 Y O N C NJ(V 0 < P (B) < 1 AOC P (B) 0 < P (A) < 1 K =

P (A) + P (B) − 2P (A)P (B)

CLY5NC |K| < 1 T NJ(VP*J CXAIX\IQAOC8CLYAR N`*HA K = 0 6/$+9), "$8* ) ! '3 6 1345+ 4O)*6j6_ 7 $57" 4+ !

A \IQDR5ACLY5NC |K| ≤ 1 I P*I ?LC T J(D CXACLY5NC

0YD

2[P (A ∩ B) − P (A)P (B)]

9

P (A) + P (B) − 2P (A)P (B)

= P (A) 1 − P (B) + P (B) 1 − P (A) > 0

È ¸2CN7 _ /

P (A ∩ B) ≤ P (A)

147:9

P (A ∩ B) ≤ P (B)

z/F1¡~/

2P (A ∩ B) ≤ P (A) + P (B)

ma <>-L-N<4z3 \ K ≤ \ /?D1/2JK18?GBGF12B F.<>-N9.3CAa147:9<7-;6CLa F>CL3< _E_ =:?3 amC _ F18J¦<>=:A7>B∩3BDN < =∅ K ≥ −1 \ x P (A) + P (B) + 2P (A ∩ B) ≥ 4P (A)P (B) .>=:12BDCN<7Kx _ /2?B18CL7-;OF.<>-093MCAa P (A) + P (B) ≤ 1 3GCN7 _ / CL7BGF12B _ 1435/ 147:9

P (A) + P (B) + 2P (A ∩ B) ≥ P (A) + P (B) ≥ \ ≥ 4P (A)P (B)

P (A) + P (B)

2

\ F./87K12B -N/8183GB<7:/<>a µc<4z 3G=:H:H:<3+/6BGF12B 147:9 > 1 P (A) \ 1¡; (A) + P (B) CL3?D/81BE/2?jBGF187 1 35P 8 · ¡ / B 8 1 : 7 9 P (B) P (A) > 21 P (A) = 21 +h P (B) = 12 +k 2 zF./8?D/ 1 187:9 1 \ F./87 0
2

|k| <

2

P (A ∩ B) = P (A) + P (B) − P (A ∪ B) = 1 + h + k − P (A ∪ B) ≥ h+k

147:9

\ F>=:3 m z*CL-A-.am<>-A-0<4zçCAa \ F>CN3 CL3 _ /2?B18CL7-;BE?=j/6CAa 3WCL7 _ / h + k − 4hk ≥ 0 k ≤ 0 √ ° \ ¿ a 187:9 BGF./27 187:9 −4hk ≥ 0 h + k = P (A) + P (B) − 1 > 0 k > 0 hk ≤ 12 √ √ √ \ 4P (A)P (B) = 1 + 2(h + k) + 4hk

h + k − 4hk ≥ 2 hk − 4hk = 2 hk(1 − 2 hk) ≥ 0

zF./27 147:9 µ 0B P (A ∩ B) = P (A)P (B) A B /+~/87>B3 \ L0 ?451 $:D-'E4 7 :)EE K' , I@#9 ! 63I?4510$ : -'E4 G 7 4) "E5 7 , 0 I'5B3B5E1 *EB 7 / / K%'I$ 530 )9 1:JG@;'5E4I$EE );1!@I'#, 2, ?KE?E 7 %5, 9 2DF" # , /3#451? ;;4 *1!EG E"$$/4# F+#- 9 2#?5'I$ 53 K 00E4 2 : '%$ '%!// , # "
:. L ;:. L

L'I$ 535&*1!? KE -K4.&*EE "E4#D&* <=, '%",0 2,0 F # $!9G /+ 2 3 " 1 : '#- 8J?
9 I G

3O !#"$8*@)9' 4 6 $ /1 ! . 7 13Q4O )O7= 3&45)94( 0H(\(\(D5?BAUCLY5NOC NJ(VZCLY5NOC T GF D5I n = 1 T 2 T 3 T . . . T 0<x <x 0

1

p √ √ √ 1 + xn 1 + xn−1 xn+1 = 1 + xn−1 1 + xn xn+1

N i IQDR5ACLY5NC8CLYAf?BAc5HAJ(aLA Gh IP9J(V lim xn

n→∞

{xn }

P*?;aLD5J R5AIQMAJ C

³8

. +(/01'2.3 <> / (:'? ).5-0 / :,370*5 . 3

'9+ +1- . +9+2:,'*) 0 : 2 3 -01' 0 /02:!5LN.A.370 .5 "(\

F./¡C ~/87{?D/ _ =:?3WC ~/(amaÄBGF./(3+/8=j/27 _ / BD< }F1B n BG _ F -0/¡B I:/c7:<7:}W7:/812BDC ~/ \ ñ>
xn = tan an

sec an (1 + tan an−1 tan an+1 ) = sec an−1 (1 + tan an tan an+1 )

¹¾=.-ABDCLH.- ;CL7:I.
cos an cos an−1 cos an+1

\

z/<.I>B18CL7

cos(an−1 − an+1 ) = cos(an − an+1 )

\

\ F./87 ¸2=:H:H:<3+/BGF1B amBD/4[0, 2/2π/2) ? 3G= _ F(nBG F1B

1B18-A- \¾i ≥ F>=:13 7:<|B½z< _ aajBGF./*3+/8=j/27 _ / 18?D//8=:12- \ {aBGnF.}/O?D/-N1BDC0<7 ¸2CN7 _ / −BGF a1n+1 \ ¿ B°− CNJOH.-AC0/23 BGF12B an−1 6= an ama -L-Nn+1 <4z3M)I8= ;CN7:cos(a 9>= _ BDC0n<7 B )

a 147:i−1 9BG= F./2?Da/2i+1 amBE/82/8?

2an+1 = an + an−1 an =

n

a0 + 2a1 a0 − a 1 − 3 3(−2)n−1

zF>C _ F<.I8~>C0<>=:3G-; _ <7~/2?D2/83BE< a0 + 2a1 \6 F./8?D/am=j/87 _ / 18-L3+< {xn } 3 _ <7~/2?D2/83 187:9(z/F1~/ lim xn = tan

n→∞

2

a0 + 2a1 3

= tan

2

tan−1

√ √ ! x0 + 2 tan−1 x1 \ 3

0 $" E B5E MK4I$ @.1 ? '#, 2, * J$2 A$2 7 9, # "", 3 &*1!?E 7 )9;1 ED4. " ! 63K?4510$ : -'E4 7 :JE K , ;@#9 ! 63 ? 451 $:D-'E4G 7 4 E 7 , F L'.B3 )9 1: G.@;'.E4I$"E )/K1@ '#, 2, =?;E?E 7 M, 9 2 F , 3 451?;;4 12G E$$=4#+ -/0 9 2 ?5 '%$.3"K 00E4 2 : '%$ '%!// , # "
9 I G

U

!#"$ 4 ! 694)< 3 ! $ 7 1() f-4O),! d7* +G7;: N 0YD 9CLY5NOC;PjC P9?\(D5?d?LP h ` A CXD V5P R5P9V(AZN aXP9IQaXH`*NIV P9?daP9J CQD#FGD H(I;\(NI C ?,9fPjCLY&CLYA ?dN !#AfNIQANh8 ^ !#ANJ?;D FCLY5IQAAU`*P*J(A ?dAM5!&AJ C ?D F(CLYA ?dN !#AU`9AJ(M CLY Gh DA?/CLYAIQA[A P9?XC/N?XCXI NP9MY C AV(MANJ(VWaLD"! \(N?d?8aLD5J?XCXIQHaQCQP9D5J 9P*JfCLYA[aX`9N?d?LP aQN` ?BAJ?BA CLY5NOC=P9? T 9fPSCLYN g(J5PjCQAfJ5H !#hAIbD F-?LCQA\? 5 7 ,U469+4 3! )97 2L ! /6 $+9), "$&%' )9O+_-'3O1/ N PSCLYD H5C ` D5??bD F-MAJ(AI N`*PSC_^ T 9WAUN?d?LH !#AUNH(J5PjC=I NV5PjH(?b?BD CLY5NOC CLYAUNIQAN D F3CLYAWaXP*IQaX` AfP*? π AOC AB hA NaYD5IQVCLY5NOC V PSR P VA?ECLYAWNIQAN P*JCLYA I NOCQP D 1 : 3

³4 ó?1z`1MH.14?12-L-N/2- _ F.a>/8=:12-8-N/87:UBGF \] 7;-LCL7:/ BGF?D<>=j2FBGF./ _ /27>BE/2? CD z*CABGF <7 187:9 <7 z*CL-A-.9C ~CN9:/oBGF./14?D/21*<>EF a>BGF./ _ CL? _ -0/cI./B½z/8/87BGF./2O 3+/ E AB F CD B½z< _ F.BD=:18-H:18?B3 187:9 BGF./8?D/amBD3 <>ag/4>=:18-r18?D/81 \¿ BM?D/2JK12CN7.AB 147:9 9>C~>C09./*BGF./ _ CN? _ -N/CL7>BE<am<> =:??D/8¡C0<7.3 3MBD< CD EF \ CN7.3G=:?D/6BGF12B \ ¸2CL7 _ /¦BGF./18?D/81|<>a BGF./3WJK12-L-N/8?*3+/8J¦/27>B(0<>aBGF./ óM/2:7:/ EF = AB = ∠BOA _ CN? _ -N/ 9:/¡BE/2?EJ2θ CN7:/89¦I8; CN3 z/F1~/ AB (2θ − sin 2θ)/2 = π/4 \ x 2θ − sin 2θ = π/2 > F CL3M35F.<4z3°BGF1B CL3°BGF./cam<.a.BGF./*H:/8?DHj/27:} sin 2θ π π \ /27 _ / AC a θ= + > H 4 9C _ =.-N18?@BE< a0?D
tan θ

< BH

_ 187.7:=B BGF.BH /-0/27:< UBGF OB _ <7>BDCN< \ F>OE z*CL-A-~>14?x; OH 7=jBH <>=:3G-;a BE<1JOCL7CNJO =:J <>a =:3 OB OH 3+ajBGF>CL3M?Da.BGF./835/63+/8J¦/27>BD3JO=:3EB@CL7~<>-~/61B@-N/8183GB°<7:/ _ F.a-0/27:UBG F 3W= _ F¶143 3WCL7 _ / CLa187:Bz/8?D/BE<&CL7>BE/2?3+/ _ B BGF>CN3 _ F.CN7>BCN7>BD/8AB ?C0BD3 BGF./B½z<¶35/4JK/87>B3(z<>=.-09 F12BCN3{9./83GCN?D/89 \¯ 7:/ _ 147 I8;ÂBGF./2JK35/2- ~/83 _ ?D/212BD/am<>=:?O?D/8¡C0<7.3 _ <7>BE?18?x;ÂBDa:9:/8?D/4/ n <8~/2?ÄBGF./?1BDC0<7.12-N3amBD/42/2? \ ¸2CN7 _ / 2

z/<.IBD12CN7

n

cos k = cos(sin 2θ) = cos 2θ −

π 2

= sin 2θ = k

k − cos k = ke0 − 21 eik − 21 e−ik = 0

\

º;¦BGF./ /8?DJOCBE/8}·CL7:9:/2JK187.7 F./8C _ F{H:?D<2~CN9:/89{BGF./:?3GBH.?D<.<>a@<>aBGF./ BE?147.3 _ /27:9:/27 _ /¶<>a CAa z/8?D/p12-02/8I.?12C _ BGF./27`BGF./&14I:<2~/p18-N2/4I?18C _ -ACN7:/214? _ a./GòH:π<E7: /87>BDk CL18-L3 _ <>=.-N97:BD12- zF>C _ F{CLJKH-LCN/83BGF12B _ <7.3EBE?D= _ BDCN<7|I8;?D=.-N/8? CNJOHj<3W3WCNI-N/ \ 451!? K4 12G E"$$/A4# +(-/0 # 9 2M?)'I$ 53 G G$&*EEE B# /
+ # !3-%'E E4 .G C: :. 7 ,0 C'#, 2, / L& , " %?55'I$ 53 E1 : ' - 8A?
9 I G

!#"$8*@)9' 4 6 $ 1/ ! . 7813Q4O O) 7= 3&45)94( P9?;V PSR P9?LP h ` Afh^ Y i IQDR5ACLY5NC (2 )! n

n

1!2!4! · · ·

(2n−1 )!

2k−1 + 1

k=1

³ È

.6.¢ .- '(02: . :!--,:!3;01'1)+ + > 0@? : -) 6 : -!. +B/ 01' .3 <> : #¥:0 HAB- +*) 3 )'*%B£ 5 3' :!AB- '(0 > . )AB) : .5) A ) : .5 ¤ .- 3 '1) )3 :,A : . !'13 ) '2% ?7:+8¤ ) 0*),'9+ + : 5 $. /:,3 5 A ) 37% : ?AB'1-,0 .¢7? :,A ¤ . :!35 )+1)3 AB'15 A :: % : )01)+ ' . .¦ / :!AB-!:,3 ! )' 5 3' :!AB- ' ) IH 5¤$/ :!37.-*H'1A: - 5=HA :!3;01'13 ) )A N ¦ .¦ )3 ) 3 )'16'*3 ? ).5 '9+ @AB' )/AB'2:,A 3' :!AB- '(0 > 5 )02:,AB+2..5 " ) 3 / (:' ?) .5 -0 / :,370*5 .3 ' +9+9- . +9+2:, '*) 0 : @ 3-,01'(0 / 0 : 5CN .A .3;02.5" ¦ k\

¸2CN7 _ /

(2n )!

=

1!2!4! · · · (2n−1 )!

_ /83 BE<3+F.<4zBGF1B

CAB 3G=

¸2CN7 _ / CN7>BD/42/2?3 9C ~CN9:/23

2k + 1 2k−1 + 1

2k + ! 1 2k

187:9

\

!

n Y

(2k )!

CN3o9>C~>CN3GC0I>-0/I8;

2k 2k−1

=

2k + 1 2k k−1 2k−1 + 1 2

2k−1 + 1

/ _ 18-A-:BGF12B@BGF./ am
m = 1 2 3 . . .

·8/B

An =

¸4<>-L=BDCN<7 ¿ 1I.<8~/

/27 _ /

m

<>

k=1

2k−1 + 1

!

am

2k−1 k∈

\

187:9 3GCN7 _ /`BGF./ CBMam<>-L-N<4z3oBGF1B k−1

CN3147ÓCL7>BE/82/8?

18?D/?D/2-L12BDC ~/2- ;¦H:?CNJK/

.6.¢ .- '(02: . - '*6' +*) A -,. +(/01'2.3 233- < A / %¥5H /-0@AB'1) ) 3 ' ?7. /5 5 H ¦

k½k \

2k−1

=

(2k−1 )!(2k−1 )!

k=1

n Y 2k

2

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