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!#"%$'&)(+*," -/.103245064,03798:=9?@4,;>.A=...

Author: Canadian Mathematical Society

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!#"%$'&)(+*," -/.103245064,03798:<;>=9?@4,;>.A=CN4P0L8gWhjikBE0L8OBE;;<790)H9?D0L =97.N=.NCN7909bF/0M890 d CJ4GC1;<7 ;>gnBGF/060L8B!45;>.N=B[4CJ7M?[28/0 ;=/CQ>23.1037>B ;=n0645;>.N=CN4KO;<7>BGF 46>=n0L4EBDC1;<7/4+2?D0~BD20L7g1?D;2R d ;BDCN22BGF'L0L2=n0 d F<2KMH/CJ;<7/4,F>CNHfF/0.18MCN72L.ACNg12¡ ~;<72: ¢ L£ bF<2794@BE;¤YC d F<2?D8f+;<4,F>CN79; ;>gu¥23.1F/;>=94GC10¦s7B! d L;>=n3F>.ACJ76;>g>BGF/0¦s7gP0¡} ©n?D=97/4D}+C d gq;CJ79BGF/0L450<=n034GBDCJ;<7/4Lb ªj«¬P¯®°_«@±u²E´³_±µ·¶¸@¹!º6¸#»s¼¸ ½~¾»f¿«º6ÀÂÁTº)¸@Ã±²G«Ä+Ã Å/ÆnÇÈZA\N\ÉÆL\ÊLËOÌqÍË6ÎÆL\N\{Æ/ÈZ1Ï<ÐfÌ{ËÏOVYXÆ9ÑL\{ËÒMÉ)ÌÆÐ>ËÌqÍËXÓLÉÓfÌ{ËÓLÒÔUÌ@ÌqÍË ËÏÕÆÎ9Ön`ÒZ1ÏuÇ/Ì{ËÉ]ÈPËÈZN\A\u×EÆL\N\{Ë×5ÌYØÆnÇLXWÓÏ<ÉÈPËX_É,ÍËËÌDÔÅ/ÆnÇXÌËÓÒÙÈZA\N\9XËL×GË,ZÊLË Ú VÆLZ1ÏYÌJÉ@ÎÆXsËÓ>×DÍf×EÆXEXËL×5ÌÓLÏ/ÉÈPËX¡Ô Û b c \1Ó<Ì[ÌJZ×EËVÆLZJÏuÌCN4@26H9;>CJ7>B }6F/0L?D0BGF/0 d ;/;CJ7/2BE034s2L?D06I/;BGFCN7>BE0L30L?[43b .N23B[BDC d 0+H9;>CJ7>B[4 I>=gBGF/0 ¥@0BD0L?DKM CN(3, 790−4) BGF/0T7<=9KRI/ 0L?;>g.N23B[BDC d 0H9;>CJ7>B[4R;<7'BGF/0 (2, d CJ? d .J0 2 b 2 x + y = 25

Þ T b L0B 2L42 22 − 1 32 − 1 42 − 1 20042 − 1 bß3 H/?D0L44 S = × × × · · · × S 2 2 2 2 2 3 4 2004 g1?[2 d BDCJ;<7 ?D0 8>= d 0L8BD;~.1;}6034GB_BD0L?DKO43b à b c VYÓ3\AZ1ÏÕ>XÆÒfËPCN42)7<=9KOI903?!BGF<2B@?D0328<4@BGF/0)42KO0gq;=9?G8C1CQB B H923.NCN798/?D;.10>2L.A=n0L4sC ;>g b A + B = C C ½ bF/0 d CJ? d .J0}+CABGF0>=923BDCJ;<7 CN7>BE03?[4,0 d BD4BGF/0s.NCN790 23BuB}6; 2 2 x + y = 1 5 0L2L4=9?D0 d d 8CN7>BD4 2L798 bL0B I90sBGF/0 0L7>BE?D0;>gg b ∠AOB

å b@¥@0BD0L?DKMCN79023.N.9CN7>BE0L30L?[4

x

4= d F)BGF<2B

(x2 − 3x + 1)x+1 = 1

b

æ bL0B 8/0L79;BD0@BGF/0s4G=9Kç;>gBGF/0@CN7>BE0L30L?[4WI90B}60037 2L798 CN7 d .N=94GCQ<0/bWÜ<;+23.N2=n+ 0;>3g + 4 + 5 = 15 b f (133333, 533333)

b c F/0G L23;<7O2798M27R0>=/CN.N23BD0L?[23.9BE?DCN279.J0)F<2<0)0>=92L.Hn03?DCNKR0BE03?[4LbsãqgBGF/062L?D0L2 ;>gnBGF/06F/0G L23;<7CJ4 √ 45<=92L?D0+=97gnBGF/0BE?[CJ2L79.10 6 3 b@¥@0BD0L?DKMCN79023.N.<>23.N=n034;>g gq;C d F x

(1999x − 99)3 = (1234x − 56)3 + (765x − 43)3

b!ÜCN798M23.N.n45;>.N=
(x, y)

b

CJ7M?D032L.n7<=9KRI/0L?[4BD; 1 1 + x y

x2 y + xy 2

=

5 6

= 30

b

Û b!ÜCN798~BGF/0+7<=9KRI/0L?;>gY?D032L.n4,;>.A==923BDCJ;<7 P0G LBÝ}60 C <0BGF/0 ; d CN2L.O4,;>.A=¢£ b

L

sin(x) =

x b 315

2?[CABDCNKR0 2BGF/0LKM23BDC d 4

à³s¿n²q±²E¼¸¯³_±µ+¸@¼´_±²G°ÃM«+¼´À¸_±u²q±²G«Ä Þ < !#"%$'&(*!, ) +-$.&/102!,3!,4"%$5/6&/7+8!,9Y:

Û b F/0372)g123BGF/03?8>CJ4EBE?DCJI<=g d 2L798CN4 d F>CN.J8/?D037 0L2 d F d F>CN.J8M?D0 d 03C <0L4 ¢6d 2798>C1034s2798~BGF/03?D0+CJ4s;<790+.J03gNB;L<03?bãqg F/;}60,<0L? B}6;)g1?D CJ0L798<4 ?D0 d 0CQ<034 ;{;>CJ7~BGF/06?D;>=9HO2798~BGF/0 d 2L798=CN.J8M d 2L798g d 2L798_L0B I90@BGF/07<=9KOI903?W;>g d 2798>C1034ä2L798.10B I90@BGF/07<=9KOI903?W;>g d F>CA.18
x = 11(y + 2) + 3

YF>=94 }6F>C d F:g 2L798=9? d F<2<=n06037>BDC103?!4EBE?DC d BE03KR037>B!Hn;<4GCABDCAg H ;<45;<7/4 n n b 2 2 2 f (n) =

4(1) − 1 × 4(2) − 1 × · · · × (4n − 1)

->2?!0 L2LKOH<.10 0B b 9?D;>=><0L ?9fBD;>(1) =2Lf.J03(2) =9?[4= 890 3 × Hn;>15 =9?9= .1034,45 >=n03.A.1034

= ÆL\Ç/ÌJZ{ÆÏ >n-/=/CN4,>=n0 f (n)

n

f (n)

034GB9=97 d 2L?E?0 @ H/2?[g12LCQBb ;<7M2

4k2 − 1 = (2k)2 − 1 = (2k − 1)(2k + 1) = 2(1) − 1 2(1) + 1 × 2(2) − 1 2(2) + 1 × · · · × (2n − 1)(2n + 1) = (1)(3) × (3)(5) × (5)(7) × · · · × (2n − 3)(2n − 1) × (2n − 1)(2n + 1) 2 2 = 3 5 · · · (2n − 1)2 (2n + 1) 2 b = (3)(5) · · · (2n − 1) (2n + 1)

-<;>=9?W<=n0 4,;>CQB=97 d 2?D?0 @ H92L?Dg123CAB CA.>g12L==n0 45;>CAB=97 d 2L?E?0 @ 2n + 1 H92L?Dg123CABb f (n) 3C.10f79;CJ45;<7/4 2 2 d CNH9?D;/>=n0LKO0L7>B 4GC 2n+1 = (2a+1) 89;<7 d b<¤ 0 @ 2a+1 H9;>=9?=9= 7 d 4a 03?[B[2L+4a+1 CN7037>BDC103? n = 2a(a+1) n = 2a(a+1) d 2L?E?0P Hn;<4GCABDCAg 2L.J;2L.J03=9?[4!8/0 H9;>=9? 2 034GB=97 a 2n + 1 = (2a + 1) n d .1034,>=n03.A.1034 034GB@=97 2?D?0~ @ H92L?Dg123CABs45;<7>B8/;<7 d 8/;<7/70@ 034PH92L? f (n) n = 2a(a + 1) b a = 1 2 3 . . . à b c GKR0BE?D0f.N28/8903??D0L4EBD4~22LCN7/4GB2R<03?[BDC d 23.W}23.N.b'YF/0fKC18/H9;>CJ7>B6;>gBGF/0 .J2898/0L?ä10 CJ4!B}+C d 02L4@g12?äg1?D;=9798O2L4@CQBäCJ4@g1?D;L0BBGF/0Hn;>CN7>BD4P2B}6F>C d FOBGF/0~.N28/8903?@BD;>= d F/0L4 ..... ... BGF/0!}23.N.L2798BGF/0?D;>=9798I/0 2798 ?D0L4Hn0 d BDCQ<0.Q: 2L798 A ................... A B ... .... .10B I/0BGF/0sH9;>CJ7>Bu23B9}6F>C d FBGF/0!}23.N.KO00BD4uBGF/0?D;>=9798b ... .... ... ... C ... ... ... L0B I90BGF/0KMCJ89GHn;>CN7>Bä;>gCJ7>B ... ... ... ... ... ... P D ... . . ... . I90M;<7RBGF/06}2L.A.45;OBGF<2B CN4H/2?[23.N.J03.YBE;OBGF/0M?D;>=9798b ... ... ... ... ... P ... P D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................ ÜCN7/2L.A.Q: .10B I90RBGF/0Hn;>CN7>B6;<7ÝBGF/0f?D;>=9798'4G= d FTBGF<2B D ..... ........... ... E .... ..... ... .... ..... ... CJ4@H/2?[23.N.J03./BE;6BGF/0P}23.N.b .. ... .... ... ... P E L0B ... ... ... ... .... . I906BGF/0~.J0L79¡BGFf;>g b+YF/037BGF/0).10379¡BGF;>g ... ... . ... . .... .. ... . x d DP . ... ... . . . ... . . CJ4 b_3CN7 0sBGF/0s.J2898/0L?uCJ4 KO0BE?D034W.J;<79 }60F<2<0 ... . ... . . . .. . EP 2x 10 ............................................................................................................ b E B C |AP | = |BP | = 5 P;} 2L798 CJ4 }6F/0L7 d 0 4DAP b =b6∠EP ∠DP A = ∠EBP d ;<79?[=n0L7>BBD∠DAP ; YF>=94 B 2798 = |DP | = x c H9H<.Q:>CJ79BGF/04EP ->:BGF<23B;B[2L|EB| CN7 2 2 2 4DAP (2x) + √ bW3CJ7 d 0 }6F>C d F6:
|AC| = 4x = 4 5 √ KR0 BE?D0L4s;<7~BGF/0P}23.N.b 4 5

BGF/0+.N28/8903?!?D032 d F/034s26F/03CJ3FB;>g

½ b 9?D;>=><03?O=97 79;Cu?D03489;<7>BM.10'H/?D0LKC103? d F>CY?D0á034GB 0B>=/C .{ 23==C1037>BBE?D;>CJ4Pgq;>CJ4H<.N=94?[2L7984GC.J0MH/?D0LKC103? d F>Cu?D0O034GBP80L@ H<.J2 d 0 @ 2M Hn;>=9?8/0,<0L7C u?D068/0L4=97;>CQB .1079;=n0L4EBDC1;<7b@72 ; = 5 1abcde = 10 . 2L==CY?D060L4EB803@ H/.N2 d 0 @ 2 ;<7O+ ;/IBDxC1037>B x = abcde b 0T79;>=><032L=j79;CN4gq;>CJ4MH/.A=94R?[ 2L798j>=n0.10 abcde1 = 10x + 1 H9?D03KMCJ0L? d 06>=/CY8/;<7/790 10x + 1 7x x

L079;23.N=92BE0 q 3

= 3(105 + x) = 3 × 105 − 1 b = 42857

142857

b

√ 5 + 2 13 +

q 3

√ b 5 − 2 13

£ p p √ √ = ÆL\Ç/ÌJZ{ÆÏ?>L0B 2798 2798.J0B b 3 3 x = 5 + 2 13 y = 5 − 2 13 s = x+y YF>=94 CN4!BGF/0?D0L<=/CN?D08M4=9Kb 06F<2<0 s b 3 3 3 2 2 3 3 3 = (x + y) = x + 3x y + 3xy + y = x + y + 3xys √ 2L798 3 √ 2798 x3 = 5 + 2 13 y = 5 − 2 13 q p √ √ √ b 3 xy = (5 + 2 13)(5 − 2 13) = 3 25 − 4(13) = 3 −27 = −3

P;}

s

037 d 0

√ √ s3 = 5 + 2 13 + 5 − 2 13 + 3(−3)s

}6F>C d F4GCJKMH/.ACNn0L4äBD; 3 b@©<:~CJ7/4Hn0 d BDC1;<7 CN4@2?D;/;B!;>g9BGF>CJ4 s + 9sg12 − d BE10 CN4!2P ;= g 30 b@©<:6.J;< 79sP= 8>CQ>1CJ4GC1;<7 }60;/I>B[2LCN7 0>=923BDCJ;<7bF>=94 24!BG F/s0−;1 BGF/0L?Wg12 d BD;
;} 2 P F<2L4W79;?D032L.?D;/;BD43bW037 d 0 BGF/0;<7<.Q:+4,;>.A==923BDsCJ;<7M s=1 q 3

√ 5 + 2 13 +

q 3

√ b 5 − 2 13 = 1

æ b<9?D;>=><0L?WBD;>=BDC103?[4sHn;<4GCABDCAg14

(x, y)

x2 − 11y! = 2003

BE0.N.J0L4s>=n0

b

0B d b 1! = 1 2! = 1 · 2 = 2 3! = 1 · 2 · 3 = 6 = ÆL\Ç/ÌJZ{ÆÏ >_3C 034GB6H/2LCN? 8>CJ45;<7/4 23.1;CJ= = 4k2 2L.J;2L?80@ n7
2 2 2 2 x H/.A=9= 4W=9(2k 7b @+ 7 d 1) ;<7 d = .N=<_ B 4k >=n0+Hn;>4k =9?+ =97 1d = 03?[[B 4(k 2LCN7)0L7>+BDCJk) 0L? + 1 2 ;>= 2 b d ;<7>BDC1037>B!.J0+g12 d m ¥@0H/.A=94 .1;CABP=97TKM=/.QBDCJH<.108/0<=92BE?D0 2 4,;>CQB=97RK=/.ABDCNH/.J0~8/0~>=923BE?D0)x H<.N=9− 4s=911y! 7b->2 ? d ;<7>BE?D0 .106KO0LKOI/?D0~8/0~8CABD0 d 0L4EB 2

8 CN?D0 < 0L4EBs=97fKM=/.QBDCJH<.10890<=92BE?D0MH<.N=94BE?D;>CJ43 bW>70@ >=923BDCJ;<7f7Y 2O8/;<7 d H/24890 2003 4,;>.A==92798 b @7)03 @ ?DCA0=9y7O≥ H/24?W=97M.J0L4 d 24 b y = 1 2 3 8/;<7 d 3C ;<7á2 2 b @703@ ?DCA0 y >= 11(1!) x289;<= 2LCN4 03@ KR037>B =n0 1 7Y 0Lx4EB+H/−24 =97 d 2?D?=0R @ H/2003 2?[g12LCQ Bb!ãq.ä7Y 2f 7 d 2014 H/246890R45;>.N=CN46;<7 x − 11(2!) = 2003 x = 2025 BE?D;>=><0 y = 2b ß<7<x9= 7 45 4C ;<7O2 2 89;<7 d 2 b @7~03@ ?DCA0 3 =97 d 2L?Ex?0T x8/0Ý =45;>2069 <=n0 7Y 0L4EyB= H/24 @ − H92L11(3!) ?Dg123CABbãq= .7Y2003 289;< 7 d H92L4R .N=70L@ <=92BDC1;<7MHn;<44 0L 8908/;<7 d .J24,;>.A=
¢ P0G LBP}60CQ<0OBGF/0f; d CN2L.ä4,;>.A=BD0L4EB L L£/ 3£/ b µ¸

Û bW;>.Q<0

¸@Ä@±u²G¸_±µ [º6Ä«Ä ³_±µ+¸@¼´_±²G°ÃM«Ä@±¸@Ã±

log2 (9 − 2x ) = 3 − x

b

= ÆL\Ç/ÌJZ{ÆÏ?>¦s4GCJ79BGF/068/0397gnBGF/0.1;/2L?DCQBGF
F/0 d CN79@BGF/034,0sCJ7+BGF/0;=923BDCJ;<7

Þ bWF/;}BGF<23B

= = = = =

3−x 23−x 8 · 2−x 8 0

=

0 2x = 8 b x = 3

;

}606F<2<0

}60sn798BGF<23BuBGF/0,:62L?D0PI/;BGF4,;>.A=
√ √ 2L7989798CAB[4ä>2L.A=n09b 1 1 CN4@?[23BDCJ;<7/2L. ( 5 + 2) 3 − ( 5 − 2) 3 √ √ = ÆL\Ç/ÌJZ{ÆÏ?>äL0B 1 1 bYF/037 x = ( 5 + 2) 3 − ( 5 − 2) 3 √ √ 2 √ 1 x3 = ( 5 + 2) − 3( 5 + 2) 3 ( 5 − 2) 3 √ √ √ 1 2 + 3( 5 + 2) 3 ( 5 − 2) 3 − ( 5 − 2) √ √ √ 1 1 √ 1 1 = 4 − 3( 5 + 2) 3 ( 5 − 2) 3 ( 5 + 2) 3 − ( 5 − 2) 3 b = 4 − 3(1)(x) √ 037 d 0 √ 1 CJ4@20L?D;;>g 1 ( 5 + 2) 3 − ( 5 − 2) 3

P (x) = x3 + 3x − 4 = (x − 1)(x2 + x + 4) √ }6F>C d FF<24 2L4CAB[4W;<7<.Q:+?D0L23.03?D;ubuYF/03?D03gq;
/ÕÔuYF>CN4H/?D;/I<.J0LK CJ44GCJKCN.N2?9BD;PH/?D;/I<.J0LK ¢ ;<7BGF/0 > 2L?DCQBDCJKO0P2BGF/0LKM23BDC d 4 < ; KMHn0BDCABDCJ;<7 gq;C d F24,;>.A=CJ79H92309b à b!ãqg 2798 }6F/0L?D0 2L798 2?D0+?D0L23. 9798 b a3 + b 3 = 4 ab = 23 a b a+b = ÆL\Ç/ÌJZ{ÆÏ?>W;BE0n?[4EBWBGF<23B (a + b)3 = a3 + 3a2 b + 3ab2 + b3 = a3 + b3 + 3ab(a + b) }6F>C d F6:>1C 0.18<4 L0B bYF/037 3 x=a+b x = 4 + 3 32 x

.J0L2L?D. :

x=2

x3 − 2x − 4 = 0 b (x − 2)(x2 + 2x + 2) = 0

CJ4!BGF/0;<7<.Q:?D0L23.?D;/;Bbä037 d 0

a+b=2

b

b

½ ! b ÜCN798 2L798 4= d F6BGF<23B}6F/037M27:;<790;>g9BGF/0LKCN4@2898/08)BE;6BGF/0+H/?D;/8<= d B y B }6; BGzF/ ;>gnBGF/06;BGxF/03 ?W 0 ?D034=/.QBäCJ4 b 2 = ÆL\Ç/ÌJZ{ÆÏ?>©<:4[:=923BDCJ;<7 F<2B+CN4 x = y =;=94 x = 1 x = −2 x = y = z = 1 x = y = z = −2 å bäãqg 2L798 2?D0sBGF/0sBGFg n798 P (x) = x3 − x2 + x − 2 a+b+c 2798 a2 b 2 c 2 b a +b +c

= ÆL\Ç/ÌJZ{ÆÏ?>_ãqg

a b

2L798

x3 − x 2 + x − 2

c

2L?D0BGF/0+?D;/;BD4s;>gnBGF/0CQ<037 d =nI
= (x − a)(x − b)(x − c) b = x3 − (a + b + c)x2 + (ab + ac + bc)x − abc

ß/>=923BDCN79 d ;/0 d CJ0L7>B[4P;>gYBGF/0 2 BD0L?DKO4 }60)4500+BGF<23B b <=92L?DCN79 x a©9=< +Bsb0>+=923cBDCN= 1d / BGF>CJ4 }60)30B 2 b 9 7 ; 0 d CJ0L7>B[4 a4s+CQgnBGF/ 0 BD0L?DKO 0L7 d 0 = 1 x ab + ac + bc = 1 a2 + b2 + c2 + 2(1) b a2 + b2 + c2 = −1

æ b!ãqg

sin x + cos x =

r

√ 2+ 3 2

}+CQBGF

= 1

0<x<

= ÆL\Ç/ÌJZ{ÆÏ?>>=92?[CJ79PBGF/0CQ<037R0L<=92BDC1;<7

}6030B

π 2

sin2 x + 2 sin x cos x + cos2 x

=

1 + 2 sin x cos x

=

sin 2x

=

n798

x

b

√ 2+ 3 2 √ 3 1+ 2 √ 3 2

b~YF/03?D03gq;
b-9?D;3<0+BGF<2BäB}6; d ;<7/4,0 d =BE0L30L?[4 d 2L7/79;B@F<2<02 d ;
= ÆL\Ç/ÌJZ{ÆÏ?>n>}6; d ;<7/450 d =BD0303?[4 d 2L7~I/0@}?DCQB[BE037)2L4 2L798 2k gq;BD0303? bW3=9H9H9;<4,0 CN4s2 d ;g − 1 2k k p 2k − 1 2798 + 1 b_F/0L7 2k + 1

2k − 1 = mp

2L798

2k + 1 = np

gq;BE0L30L?[4 2L798 bs3=nIBE?[2 d DB CN79BGF/0n?[4EB@0>=923BDCJ;<7Og1?D;=94 n 4,0 d ;<798OC <0L4 ;
b69?DCN279.J0 2798 F<24<03?[BDC d 034 bMÜCJ798f2L.A. 45; ABC C(1, y) y BGF<23B!2L79.10 CJ4@2?[C13FB!279.J09b A(3, 1) B(5, 7) C = ÆL\Ç/ÌJZ{ÆÏ?> 6 B(5, 7) BC ⊥ AC .......

= = =

;
−8 0 0 b y = 3

A(3, 1)

b

ã{7jBGF/0'8
= ÆL\Ç/ÌJZ{ÆÏ?>äL0B (P T )2

c .J45;

QR = x

P

bYF/037

= (P R)2 + (RT )2 = (64 + x2 ) + 144 + (20 − x)2 = 2x2 − 40x + 608

(P T )2 = 202 + 42 = 416 2x2 − 40x + 608 x2 − 20x + 96 (x − 8)(x − 12) x = 8

= = =

;
-

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

Q

T

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

7 − 8y + y 2 y 2 − 8y + 15 (y − 5)(y − 3) y = 5

R

S

b

b037 d 0

416 0 0 b x = 12

P

T 4 U

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

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

=

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

C(1, y).................................

20

8

Q

x

R 20 − x

8

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

7−y 4

−1 mAC −2 1−y

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

=

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

mBC

S

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d ;<7/4CN4GB[4P;>g4C 7<=9KRI/0L?[43b F/0L7M}606B[20~23.N.uHn;<44CJI<.J0)4=nI<4,0BD4P;>g c 4,0B d ;<7>BD23CJ7BD4 BGF/0~4=9KM4;>guBGF/00.103KR037>BD4;>guBGF/034,0)4=nI<4,0BD42?D0 ?D0L4Hn0 d BDCQ<0.Q:ub@¥@0BD0L?DKMCN790BGF/0+4C 67<=9KRI/0L?[4@CN7 S 5 2L798 b 87 92 98 99 104 110 S = ÆL\Ç/ÌJZ{ÆÏÑØUä\N\{ËÏ ÓX[ÓL]ÐLXDÓ>Õ9Ë S`ÉÌDÇ<Õ9ËÏuÌ11 ] W Ó 0>X[Z{ÕÐ<Ë = Ë×EÆÏYÕ>ÓLXEØ = ×DÍYÆ/ÆL\N] >ÆÏYÕ9ÆÏ< ] )b

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L0B b Ü?D;C18/08 }60 d 27 {a d ?D0L2BE0BGF/0Sgq;>= .N.J;}+ CJ79 Pb45 0cB ;>dgY 0e>=9 23fBD}CJ;<7/4 a+b+c+d+e a+b+c+d+f a+b+c+e+f a+b+d+e+f a+c+d+e+f b+c+d+e+f

= = = = = =

87 92 98 99 104 b 110

£ ¢

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}6F>C d F d 2L7OI90+4CNKOH<.NCA0L8)BE;

5a + 70 = 110 }6F>C d Ff}6030B bYF/037 2L798 a = 8 b = 14 b c = 19 d = 20 e = 26 b_F>=94 f = 31 S = {8 14 19 20 26 31}

g1?D;
23 $+-&587 9;<5= >? @ A )!< BC A)D)E393! , *+ A '<-4 !<- A3>;!<'<587 <!# #-8'7 #< A>;!<'<587 A?>;E#'-

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£

Å

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oGb = ÆL\[Ç<ÌJZÆÏ)VYÓX Æ9Ñ/ËX,Ì 9ZA\NZJÏ/É#0ZJ] sÇ/ÌqXGËÒfÆÏYÌ1] ub -<;<4,;<7/4 2L.J;=n0 4,;>CQB=97 x(x + 60) d 2L?E?0 @ H/2?[g12LCQBW?Dx0,<= CJ0L7>n B 2 −d F/30 03? d F/0L? BE0.9<x=n> 0 0 2 −<30)(n +?D030) d F/03= c CJ900 0 d 2?D?0 @ H92L?D(n g123CABä =n0.{ ;<7~ ? d F/n09b − 7/4GC 4,;>CQB_=97 d 2?D?0 @ H92L?Dg123CABbnP;d 30 k > ; R = > I 1 C 3 0 7 b n2 − 900 = k2 n2 − k2 = (n − k)(n + k) = 900 = 22 32 52 P;>=94890,<;<7/4PKO23CJ7>BD0L7/2L7>B4 03@ H92L?D0L? 037890= Mg12 d BE0=9?[4 0B BE0.J4>=n0 900 a b 0B b @7'23=9?[22LCN7/4C@>=n0 0 B a+b b−ab n−k = a n+k = b n = k = -/=/CJ45<=n079;>=94ä<;>=/.1;<7/4s<=n0 0B 4,;>CJ0L7>B!89034037>BDC103?[4 CA.9g12L=<2B<=n0 0B 4,;>2CJ0L7>B 8/;>CAB 0BE?D0H9a23CJ? .1b0348/03= 890PK 0LKO0PH/2?[CAB0/@ bn-/=/CJ45<=n0=9n 78/0L4!k890= g12 d BE0=9?[4890 900 BE?D;>=><0L?W890= g12 d BD03=9?[4 .10P450L?D;<7>BbnL0PH/?D;/I<. 0L KO0P?D0,BW89;<7 d 2 0B 8/ 0 >=/C r 0B b @?.1034!H/2LCN?D0L4!8BD0L4225 8/0g12 d 89;<7/7903?D;<7>B n 45= BE0=9?[4s890 ;<7>rB + s > 30 k = r − s > 0 0B bnß>.A.1034@89;<7/79037>B 225 (1, 225) (3, 75) (5, 45) (9, 25) ;>=RI>C1037 b (15, 15) n ∈ {226 78 50 34} x ∈ {196 48 20 4} ooGb = ÆL\[Ç<ÌJZÆÏ·ÑØ äÇLÉÌ1Ó>ÊLÆ uX[ZJÒ10/ËXD] uÏÓ<Õ U ] _Ç>ËÏÆÉ UäZ1XGËÉ] UX[Ð>ËÏYÌJZ1Ï/Ó9b

È

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L0B I/0@2sHn;<4GCABDC <0CJ7>BD0303?buF/00L<=92BDC1;<7 2 F<2L4Hn;<4GCABDC <0 2 x +60x−k =7O0}60)K=94GB@9798 CJ7>BD0303?s4,;>k.A= 7 D B 0 3 3 0 ? b Y / F 3 0 2 2L.A.P->:BGF<23;gäBGF/0gq;CJ46CJ4~0L<=/C >2L.J0L7>B 2 (30, BE;R9798:BGF<23;;!<'<587 A;>;EF.

587

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2002!

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ÓØÍËÒ

<!#

<-8' 7

< < A

= ÌJÓWb

d ¥@0BD0L?DKMCN790+BGF/0+.J2L?D30L4EBH9;<4CQBDCQ<0CJ7>BD0303? gq;C d F n CN4s2g12 BD;g 2002 b F<2BF<2LH9H90L7/4@CAg CJ4@?D03H/.N2 d 08OIL: n ;
ÆÂ Å Ä

2002

Æ

2003

2004

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L0B

bÝ3CN7 d 0

CN7

BGF/03?D0M}+CA.N.äI90

F = 2002! = 2002! KM=/.QBDCJH<.1034)2002 ;>g 2798 2 · 7 · 11 · 13 KM=/.QBDCJH<.1034);>g 2 2 b2002/13c = 154 13 b2002/13 c = 11 2798 M K / = Q . D B J C < H 1 . 3 0 4 > ; g b Y / F 3 0 D ? 6 0 + } A C N . . 9 I ~ 0 R K < ; D ? ~ 0 K=/.ABDCNH/.J0L4P;>g 13 3 3 11 2L79b2002/13 8 bYF/03?D03cgq;<=?D0 0 BGF/0~.N2?D3034GBsCJ7>13 BD0303? 4G= d FRBGF<2B CN4 n 8>CQ>C18/0L4 7 2 b_F>=94 n 2002 2002! gq;g b_YF>=94 ;<7<. : + } A C N . . > 8 Q C > 1 C / 8 0 { _ b F>=94 gq;g 2798)2004 4CN7 d 0 = 2 2· 3 · 167 BGF/2004! 0L?D0P2?D0 KM=/.QBDCJb2004/167c H<.1034;>g b / F L 0 D ? 0 gq;BD0303? 4167 = d F)BG> F<22004 B < 8 C < J C 9 8 3 0 4 J C 4 b n 2004n 2004! n = 12 23 $+- 587 9;<5= >? @ A4 )!< BC A D(E (' -&23 ,

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L0B I90f2gq;>=9?E58>C1CQB+CJ7>BD0303?b ¥@0BE03?EKCJ7902L.A.!Hn;<44CJI<.J0M>2L.A=n0L4~;>g gq;C d F 2003 0L798<4CJ7OBGF/0)gq;>=9?8
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L;/;/CN792B!BGF/0)H/?D;/I<.J0LKÙKR;/8>=/.1; }6064,0L0+BGF<23B 2003 b 10 k ≡ 3 (mod 10) b k ≡ 7 (mod 10) 0f790 3B679;BDC d 0RBGF<2B 2 2L798 79;<79 7 ≡ 49 (mod 100) 73}6≡ 43 (mod 100) ) b L 0 B / F 3 0 D ? 0 J C 4 ~ 2 4 N C 9 7 1 . ) 0 4 7 10a 790≡ 2BDCQ1<06(mod 8>C1CQBb100) YF/037 =94GCJ79k≡ BGF/0 ©9CJ79+ ;=94

k2003

YF/037

4GCJ7 d 0

k2003

≡ (10a + 7)2003 ≡ 72003 + 2003 · 10a · 72002 b ≡ 43 + 3 · 10a · 49 ≡ 43 + 70a (mod 100)

03798/4@CN7~BGF/068>C1CQBD4 43 + 70a 70a 7a a

037 d 0

≡ ≡ ≡ ≡

2003

}60F<2<0

3 (mod 100) 60 (mod 100) 6 (mod 10) b 8 (mod 10)

b CQBGF62.ACAB[BD.J0PF/0.JH6g1?D;=9?E k ≡ 87 (mod 2798 100) bÝF>=94 BGF/0f k ≡ 587 (mod 1000) k ≡ 587 (mod 10000) b 8=94 BGF/2004 03?D02 ?D0+79;4,;>.A=
+¼

qË

c KM2C d 45<=92L?D0 d ;<7/4CN4GB[4s;>g723.N=n034 4= d F6BGF<23B032 d F;>g d ;>3.N=9KM7/4 27988BW4=9Kb@©0 .1;} NC 42L7M0 L2LKOH<.10 BGF/0+?D;}34 × KM2C d 4,> =92?D0/b ;>gY2 3×3

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T = 3E

2 9 4 7 5 3 6 1 8

ÆÂ Å Ä

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L0BWBGF/0+KO2C d 45<=92L?D06I/0 a1 a4 a7

YF/037 b 3CJ7 d 0fBGF/04G=9K =CN4sE 8=92L./BE; BGF/037 T 3T

2798OF/037 d 0

a2 a5 a8

a3 a6 a9

;>gBGF/003.J0LKO0L7>B[4OF/;B[2L.A.Q:

<03?[BDC d 23.N. :

= (a1 + a5 + a9 ) + (a2 + a5 + a8 ) + (a3 + a5 + a7 ) = (a1 + a2 + a3 ) + 3a5 + (a9 + a8 + a7 ) = T + 3E + T = 3E + 2T

T = 3E

b

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= = =

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BY

= = =

s s

q

−1 + 2

√

3

−1

2

+

Y CBE

√

3−1 −0 2

2

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√ √ 4−2 3 = 3−1

"

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3−1 √ 2

2

=

√ √ 4−2 3 = 2− 3 2

b

√ b YF>=94 GB F/0+27/4D}60L?_BE;6BGF/06<=n034GBDCJ;<7CN4 1 : (2 − 3) YF<2B, 423;/;/8)4GB[2?B }60< <023B_.J0L2L4GBä3;BW2L72L7/4E}603?BE;BGF/0H/?D;/I<.J0LKb@©9=B_}2: 034

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= (2r) 2

= 4r b = 2r 2 2

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

D

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2

n

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

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H7

A7

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

Â

A1 H 1 + A 2 H 2 + · · · + A 7 H 7 = H 1 A2 + H 2 A3 + · · · + H 7 A1

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

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A4

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YF>=94

7 X

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= Ai Hi2 − Hi A2i+1 = (Ai Hi + Hi Ai+1 )(Ai Hi − Hi Ai+1 ) b = a(Ai Hi − Hi Ai+1 )

(OA2i − OA2i+1 ) = a

7 X

i=1

(Ai Hi − Hi Ai+1 )

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i=1

}6F>C d F~CJ4!BGF/089034CN?D08?D0L4G=/.ABb

Ai H i =

7 X

b

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Hi Ai+1

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8[Æ ¼ ]Æ Ä

(x + y) + 3x + y

Æ

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E"6$ p

b 2 2b ãqg a + a ≤ n < (a BGF/+ 0L7 1) x + y ≥ a + 1

n = (x + y)2 + 3x + y ≥ (a + 1)2 > n

2 d ;<7>BE?[28>C d BDCJ;<7b!ãqg n

x + y ≤ a − 1

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= (x + y)2 + 3x + y ≤ (a − 1)2 + 3(a − 1) = a2 + a − 2 < a 2 + a ≤ n

£ 2 d ;<7>BE?[28>C d BDCJ;<7bYF/03?D03gq;
x+y =a

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n = (x + y)2 + 3x + y = a2 + a + 2x

b

037 d 0

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9b

2 2 ãqg a ≤ n ≤ a + BGaF/0L−7 2 x + y ≤ a − 2

b

n = (x + y)2 + 3x + y

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x + y ≥ a

≤ (a − 2)2 + 3(a − 2) = a2 − a − 2 < a 2 ≤ n

BGF/037

n = (x + y)2 + 3x + y ≥ a2 + a + 2x ≥ a2 + a > n

2 d ;<7>BE?[28>C d BDCJ;<7bYF/03?D03gq;
x+y =a−1

b_F/0L7

n = (x + y)2 + 3x + y = (a − 1)2 + a − 1 + 2x = a2 − a + 2x

b

037 d 0 6 } 0F<2<0 2798 b 1 1 2 2 −8/a a)IL:BGF/y P ;}.J0B x279=8 2 (n I/0O 03979+08 034,= 0M0a H9−?D03145− 4GC1;<x7/43=b c 2 (a 4PCJ7 + 2a45− 0 n − I/;2) BGF 2798 2L?D0fCJ7>xBD0303?[4Lybe3CN7 d 0 2 2L798 }60F<2<0 be 3CJ7 d 0 x y a ≤ n a > 0 x > 0 Ngq;CN4 d 2450 }602L.N4,;fF<2<0 b~YF>=94 BGF/0MH/2LCN? 2 n (x, y) CQ<≤ 034sa2+=9+ 7a=n− 0?D20LH/?D0L450L7>B[23BDCJ;<7R;>Eg CJ7~BGF/068/0L4GCJ?D0Ly8> gq;
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ABC

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I

AI ID

= =

AB AC = BD CD AB + AC AB + AC = = 2 BD + CD BC

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

b

D

B

C

M

¢L 037 d 0

AB = 2BD

2798

AI = 2ID

b ;<7/450>=n0L7>BD. : b

AD = 3 ID

L0B

YF/037

I90WBGF/0!4,0 d ;<798CJ7>BD0L?[450 d BDCJ;<7;>g

AD

}+CABGF@BGF/0 d CN? d =9K d CN? d .10;>g

∠M BD = ∠M BC = ∠M AC = ∠M AB

3CJ7 d 0

YF>=94

M

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∠BM D = ∠BM A

4M BD ∼ 4M AB

DM BM BD 1 = = = BM AM AB 2

037 d 0

DM DM BM = · = AM BM AM

AM = 4DM

g1?D;C d F6}60;/I>B[2LCN7

2 1 2

b

4ABC

b!ã1Bägq;>.N.J;}4!BGF<2B

b

1 4

=

b

b

Ü ?D; bF>=94 b = ID + DM = IM YF/03?D03gq;CJ4= CJKMDM H/.AC1034!BGF<23B AI = 2ID b OI ⊥ AM AI ⊥ OI

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Y (t = t2 ) ....... ... ...... .... ... .... .. . .... ... .... .. .... . 1 .. .... . .... .. . 2 4 .... .. . .... .. .... . .............. .... . 1 ... ........................... .... ................. ... .................. ...... 4 ..................... ... .................. .... ................... ... ................. .. 4 .... . 4 1 ................. 6 .... .. .................. . .... ................. ... 1 .................. .... 4 5 .. .................. . .... .................. .. .... ............... .. ......................................................................................................................................................................................................................................................................................................................................

(t2 − t )u P (t = t )

A

-

B

|t − t |v

Q(t = t ) |t − t |w |t − t |w t u |t − t |v

R(t = 0)

t3 x

6 S(t = t3 ) D

C

? (t = t6 )

|t5 − t3 |x

X (t = t5 )

¥ 0L79;BD0BGF/0+Hn0L89034GBE?[CJ2L7/4sIL: @ b CABGF/;>=gY30L7903?[2L.ACAB{: }60 AKR B 2L798RBGF<2B@BGF/0,:KR0 B2B KO2:244=9KO06BGF<23BBGF/0~9?[4GBsH/2LCN?@BD;R 0L0 B!C} 2D 4 2798 A D BDCJKO0 b6P0G LBs4G=9H9H9;<4,0BGF<2B 2798 KR0Bs2B 2798MBGF<23B 2L798 KO0Bs2B A B }6F/0L?D0 0 bä3=9H/Hn;<450 gJ=9?[BGF/03C?_BGF<23B 2t791 8 KR0B!2A B 2798) BGF<23B 2Lt7928 t > t > 0 B D t B 2 1 3 KO0B!2B b 0}2L7>BWBE;~4,F/;}BGF<2B 2L798 2L.N4,;~KR0Bb C YF/0ft?D4;<28<4~2L.J;<79M}6F>C d F 2798 C }60LD ?D0RBE?[2<03.A.NCN79KO00B62B62H9;>CJ7>B b D d 0L4 3=9H9H9;<4,0sBGF<23B ?D0L2 d F/08 23BuC BDCNKR0 2L798 ?D0L2 d F/08 23BuBDCNKR0 b¥!CN4GB[27 X D

X

t5

C

X

t6

b

¢ 2?D0)2445F/;}7RCJ7MBGF/0)n=9?D0 2L454G=9KMCN79+BGF<23B CJ4PI/0B}60L0L7 2L798 b 0~F<2<0 t3 2L.N4,;890379;BE0L8OIL: 2L 798 BGF/0?D0L4Hn0 d BDtCQ4< 0+45H900L8/4s;>g t2 2 798 b ¦s4CN79'037903u.N 2Lv=94 3wY F/0L;
XS

QY

·

PR

= −1 =

¦s4CN79)'0L790.J23=94F/0;
4P RX

t3 − t 5

·

t2 − t 4

}+CQBGF)BE?[27/4[<03?[4523.

·

−t1

Y QS

}60F<2<0

t2 − t 1 t3 t4 − t 6 b P Y RS XQ · · = −1 = · · Y R SX QP −t2 t5 − t 3 t1 − t 4

ß/>=923BDCJ;<7 CQ<034 27980L<=92BDC1;<7 CQ<034 t1 t2 t3 + t1 t3 t5 + t2 t4 t5 = t1 t2 t5 + t1 t3 t4 + t2 t4 t5 b t1 t2 t3 + t1 t3 t6 + t2 t4b t5©9=<=B t1 t2 t5 +}6tF/ 1t 3=nI>BE?[2 d BDCJ79:

C D - !"#$ %&

$!'! %&( ) " *+ (%,

+ .-) /. -0 $ +! %&( ! 1# 2)!t( $3 2%&4) / + ,76 8 + + -9 . ):( $0 &x! $$y &%;4( ) . =< 8 %>4 + ?6; xy + 5 0 $3 xyt + 5 , " $( ! ( # 8 .!( $3 . " $3 + + @ xyt 3 5 %& 2 (2 7%;)$ %&) 8 !B+ 6= 8 ( # ( 2 5 $' $3 73 A -0!( $xy $"*2+ xyt, 5 , , ! 1$ ! C D. E 8 @! $$ BB 4(F 8 $%B ( G,G F ( " 8 (xyt 7=5+ ( " ( 7 (,-' LF1L02 L 3 *+ 0LH4($ IJ (6

! A 0$;+ , , ! %0< $*F$+ D, $& K ( ) %&0% L 3 . B L(4 7 2:( %B + + 6GL , 1 , L2 , ! L3 $/+ A ($ 8 , + + 8 <2+ + ' 0 L L L 4)*B (+ + (+?,G* < 7= "-<+ !F %;+ 6& %;B+ ;L)-9-B$= 1 2! !( $3 ! -/($/-' + M L=3N L (+4+ (+ ! )O (@= %;0 !P,-/ & ). + + -! 8 %QF 3*+ (%R4 $$ (%&((JO S@;- 6K 7; / T$ !)F UV3$ 0 8

! $;$$$! F+ / $ P A 8 $,

!

$3 73L3A < (, L xyt5 L3 L4 4!( $3 ! %;D W

T

3

4

Û b c B CJ79 c ?BGF>=9?, 4 ;>=9?B 7=9798O/2I<.J09bPß<2 d F 2n F<24s2BKO;<4GB 03790LKC1034s2 KO ;<79PBGF/06;BGF/03?[4Lb-9?D;3<0BGF<23B'0L?[.NCN7BGF/0P}+CQG2?D8 d 2L7f890,>CJ450~2~n450L− 2BD1CJ792?D?[279303KR037>B4= d FMBGF<2Bs79;7g F>CJ4s0L7903KMCJ0L43b

u¼ ¡Æ

Ë

= ÆL\Ç/ÌJZ{ÆÏÑØ /Z{ËXEXGË ÆXEÏ/É ÌË,Z1Ï<]

ÓZJÉÆÏ/É L ÓÌ[ÌË¡] X[ÓÏ×GËb

0244=9KO0)BGF<2BsBGF/0?D03.N23BDCJ;<7 CJ4P79;B2L703790LK):;>g +CJ4P4[:/KMKR0BE?DC d b ;<7/4GC18/0L?!BGF/0)4CNKOH<.10~?[2LHnF}6F/;<450+<0LA ?BDC d 0L42L?D0BGF/07gY}6F>C d FR2?D0 032 d F;BGF/0L?¡b F/0 ;{;>CJ790L8 IL:2L7 0L8930CNg@2798';<7<. :ÝCNg!BGF/0,:2?D0f79;B60L7903KMCJ0L4~;>g H9?D;/I>.103KCJ4!BGF/037BD;H/?D;L<0BGF<2BWBGF>CJ4s?[2LHnF d ;<7>BD23CJ7/4@2~+2LKMCA.ABD;<7g 2 . . b Ü>?D;B ;>gBGF/ 0H/?D;/iI<= .J0LK 1 2}6 0 F<2<0 K d ≥ n i i gq;=94 b min{di } ≥ n i i BD; 10 03? d CJ450 2 ©9=CJ79+ Hb 0,<0L?{:R4GCJKMH/.J0~?[2LHnF}+CQBGF G k <0L?BDC d 0L4423BDCN4gA:>CJ79 k ABGF>CJ4+CN4679;}724¥!CJ?[2 d 4 d ;<798.N.J;}4Lb

&P#$

Â

¼

"

/b9L;L2@ 4E ÆÒÑLZ1Ï/Ó/Ì{ÆX[Z1Ó\ nXÆ9ÑL\{ËÒMÉäÓÏÕ <ËX×ZNÉË¡É ;.A.J2L798

b

¢ Û Û b ;<7/4EBE?D= d BP2R450B;>g d CN? d .1034P}+CQBGF79;<79E03?D;?[28>CNCW4G= d FfBGF<2B+0G L2 d BD. :T;<790O;>g P BGF/0LK#H/244,034!BGF=n3FO032 d F H9;>CJ7>B;>gnBGF
Â

u¼ ¡Æ +Ë

Ë

ÓZJÉÆÏ/É L ÓÌ[ÌË¡] XDÓLÏY×EË>b

ÆÒOÒfËÏYÌWÑØ /Z{ËXEXGË ÆXEÏ<É Ì{Ë¡ZJÏ/]

¼

" :

YF>CN4sH9?D;/I>.103K }+CQBGF~CAB[4s45;>.N=BD0303? 4= d F)BGF<23B n

ÆÉ ¼ 8Ä Æ ¿ n

Ë Á

n

27 + 84 + 110 + 133n = 144n

ÆÆ

n

u¾

Â

¿

= LÆ \{ÊLËÕ ÑØ /ZËXEXË ÆXEÏ<É Ì{Ë¡ZJÏ/] ÓZJÉÆÏ/É LÓÌ[ÌË¡] XDÓLÏY×EË /Í/X[ZJÉÌÆ3VYÍËX LÔ XDÓ>Õ\ËØ3] X[ZNÉ3Ì{ÆL\J] +Ô LÔ = ÒfËLËÏ 0] LÓ\Ì[Ñ/ÆÒOÒfË,\J]@ÌqÍYË ËÌqÍËX[\JÓÏÕÉLÔ ~Ë6Ð3Z{ÊLË ÆXEÏ/É ÌË,Z1Ï ÉsÉÆL\[Ç<ÌJZÆÏub

L0B

f (x) =

27 144

x

+

84 144

x

+

110 144

x

+

133 144

x

890n790L8f;<7 bã1BCJ4P0324[:MBD;R4500BGF<2BBGF/0)gJ=97 d BDCJ;<7 CJ4P8/0 d ?D0324GCJ79.N==.A.1;1}4äBGF<23B CN4!BGF/06;<7<. :M45;>.N=gnBGF/0H/?D;/I<.J0LKb f (5) = 1 n=5 Û ½ b@-9?D;L<0PBGF<23B

u¼ ¡Æ

∞ ∞ X X ai aj b ≤ π a2k i+j i,j=1 k=1

Ë

= ÆL\Ç/ÌJZ{ÆÏÑØ /Z{ËXEXGË ÆXEÏ/É ÌË,Z1Ï<]

YF>CN4sCN4s2)4Hn0 d CJ23. d 2L4,0 p>1

2798

p0 =

p p−1

¢ CJ7 6 } F>C d FM4GB[23BD0L4 L0B ∞ P BGF/0L7 0 bpn ≤ B

;>gYF/0;
(p = 2) ∞ b!ãqg P 2L798 apn ≤ A n=1

∞ X

∞ X

m=1 n=1

';B

&P#$

ÓZJÉÆÏ/É L ÓÌ[ÌË¡] X[ÓÏ×GËb

n=1

1 1 a m bn π ≤ A p B p0 m+n sin(π/p)

π sin(π/p)

CJ4!BGF/0I9034GB!H9;<454GC1I>.10/b

Ç Ê

¨bJb +2L?D8: §3bJßub CQB[BD.10¡}6;/;/8 ¨sb - ;>@ .Q:>2 23BGF/03KO2BDC d 2L .C1I b Û æ b@¥@0 d ;BD;)B}6;~g12 BE;
¿ Ë 3Æ

Â

235 + 972

N¾Æ

b

M8Â Ä

Â

{ÏË ÇLÓ3\AZÌJZ{ËÉ

Ë

2KOI/?[C18/30

= LÆ \{ÊLËÕ ÑØ ZNÐ9Ç>Ë¡\UÒfËÏ/Ð/ÇÓ\ Æ>ÊÓLÉ] /Ó3\JÓ LZNÐ9Ç>ËX[ÓL] Ó\N\{ÆX×ÓL] = VYÓZ1Ï /Í9X[ZNÉ3Ì{Æ3VYÍYËX LÔ XDÓ>Õ\ËØ3] X[ZJÉÌÆL\N] ÓLÏYÕ /Õ9ÈsÓXGÕ LÔ ~Ô ÓÏ<ÐL]ZA\JÎ1X[ZÕ L9 Ó ÇLX[ZËX Ï>ZÊLËX[ÉGZÌ1ØL] Ó/Ì{Ë[X \{Æ/Æ] )Ô~ËPÐZÊLË)ÌqÍË6ÈsX[ZÌË ,ÇVÝÆÎ/UÒfËÏ/Ð/ÇÓ\ Æ>ÊÓLÉ3Ô

3CJ7 d 0

8Â

2352 + 9722 = 55225 + 944784 = 1000009 = 10002 + 32

¢ }602LH9H<.Q:)BGF/0+CJ89037>BDCAB{:

}+CABGF

(ab + cd)2 + (ad − bc)2 = (a2 + c2 )(b2 + d2 ) a = 17 b = 58 c = 2 d = 7

}6F/0L7 d 0

BD;M;/I>B[2LCN7

10002 + 32 = (172 + 22 )(582 + 72 ) = 293 · 3413

2352 + 9722 = 293 · 3413

b

Û b!BD=n89037>BD4@CJ7M264 d F/;/;>.3;gq;=9H94;>gu2B!.J0L2L4GBWB}6;ubs;~B}6; ?D;>=9H/4)F<2<03;<790 4GBD=n8/0L7>B[4}+CA.N.W3;BE;/30BGF/0L?PKR;BD4sF<2<063;<790PBE;/30BGF/0L?!0G L2 d BD. :M;<7 d k0/b@>-9?D1;L<0PBGF<2BWBGF/0+7<=9KRI/0L? ;>gY4GBD=n8/0L7>B[4@CN7~BGF/04 d F/;/;>./CJ4@2B!KR;<4EB b

u¼ ¡Æ

Ë

k

ÓZJÉÆÏ/É L ÓÌ[ÌË¡] X[ÓÏ×GËb

= ÆL\Ç/ÌJZ{ÆÏÑØ /Z{ËXEXGË ÆXEÏ/É ÌË,Z1Ï<]

ÜCN?[4GB!79;BE0+BGF<2BäBGF/06H<=nI<.ACJ45F/08O4EBD2BE03KR037>B@;>gBGF>CJ4sH/?D;/I<.J0LK}2L4;/IL=94. : }?D;<79 4CN7 d 0CABä24508~gq;g9BGF<23B BGF/0+7<=9KRI/0L?;>g4GBD=n8/0L7>B[4@CN7~BGF/0+4 d F/;/;>.9CN4 Ó/Ì\ËÓLÉ Ì b 0+H9?D;3<0BGF/0 d ;.A.1;}+CN79 d ;<7/4,0L<=n037 d 034 c 7:MB}6;?D;>=9H/4F<2<0~23B@KR;<4EB;<790)4GBD=n8/0L7>B@CJ7 d ;BD4f}6;>=/.18 I90 BE;/30BGF/0L?CJ7 B}6; 8>Cu0L?D037>B?D;>=9H94 d ;<7>BE?[2L?{: BE; BGF/0 F:=9H d ;<7>BD23CJ7/423.N.sBGF/04GBD=n8/0L7>B[4Lb¯ !BGF/03?}+CN4,0 }60 d ;>=/.J8j79;B F<2<0R2L79;BGF/0L??D;>=9H;>g!2BP.J0L2L4GBB}6;4GBD=n8/0L7>B[4 2L4?D0>=/CJ?D0L8 }6F>C d F}6;>=/.18 d ;<7>BE?[28>C d BJ2 b d ß>2 d F4GBD=n8/0L7>B+I/03.J;<794+BD;2BP.J0L2L4GBB}6;?D;>=9H943b{ãqg45;B}603?D0OCN7 ;<7<. :T;<790?D;>=9H BGF/0370CABGF/03?BGF<2B+?D;>=9H}6;>=/.J8 d ;<7>B[2LCN7Ý23.N.BGF/0O4GBD=n8/0L7>B[4 d ;<7>BE?[28>C d BDCN79M1I ;B@}6;>=/.1879;BPI/0~BE;/30BGF/0L?@}+CABGF0,<03?{: ;BGF/03?!4EBD=n89037>B0 LE2 d BD.Q:M;<7 d 0 d ;<7>BE?[28g23.N.BGF/04EBD=n89037>BD4 2L798T.J0B bãqg }60 L0B x ∈ S 890379;BE06IL: S BGF/07<=9KOI903?;>gu?D;>=9H94 d ;<7>B[ 2LCN7Bb)©<: d 2I9;3<0 BGF/03?D0~2L?D0xB}6;R8=9H/4 2L798 bO3CN7 d 0 a G0G CN4GB d FRBGF<23B 2L798 2L798 BGF/03?D0R 0 4= 0 0 G a 2 ∈798 G a 4G∈ |G|2L79≥ 4GBD=n8/0L7>B[4 = dG F6BGF<2B 8 2 b|G ';<|?D0L≥;L<203 ? IL:12 }60 b∈G c ∈ G0 b 6= a c 6=d ad F<2<0 bYF/03?D0)CJ42?D;>=9H ;<7>B[2LCN7/4 2L 798 JG4CN7 d 0 0 2798 00 }6F>C F b ∈ / G c ∈ / G G b c 2798 KM=94EBsI90+BD;/30BGF/03?s;<7 d 0 bs¦s4CN79612 }60)F<2<0 bã1Bgq;>.A.1;}4BGF<2B G b c b a∈ / G00 d(a) < k 06F<2<0 12

X

(x,G) x∈G

  XX  X b 1 = 1 = d(x)  x

G x∈G

x

¢£ @7BGF/0;BGF/0L?!F<2798

X

X

1 =

G

(x,G) x∈G

YF/03?D03gq;
X

 

X

x x∈G

d(x) =

x



X

1 = X G

|G|

G

|G|

b

b

L0B I90O2O4GBD=n8/0L7>B 2L798.J0B I90O2R?D;>=9HT79;B d ;<7>B[2LCN7B x BGF/0L?D00G CN 4GB[42M?D;>=9GH }6F>C d F d ;<7>BD23CJ7/4 2798 xb~;BE06BGF<2B a4∈ G G a a CN7 d 0 b!';=9H9x4 2798 G = 6 G x ∈ / G b ∈ G b = 6 a Gã1aBWgq;>.A.1;G a 2?D0+8=n0?D;>=9H d ;<7>BD23CJ7BE?[28>C d BDCJ;<7 BGF<23B G n>k H923CJ? 4= d F)BGF<23 B }606F<2<0 (x, G) x 6∈ G b 0 < k − d(x) ≤ k − |G| < n − |G|

YF/037

d(x) b |G| < n − |G| k − d(x)

4= d F 3=9KOKCJ79)2L.A.YBGF/0)CJ790L<=923.NCQBDC1034 ;/IBD23CJ790L8Rgq;
BGF<23B

X

|G|

(x,G) x6∈G

n − |G|

X

<

(x,G) x6∈G

d(x)

b

k − d(x)

£

L0B I/0@2?D;>=9HbF/0@7<=9KRI/0L?u;>g4GBD=n8/0L7>B[479;BuI90.1;<79CN79äBE; CJ4 b G n−|G| YF/03?D03gq;
(x,G) x6∈G

|G|

n − |G|

=

x x6∈G

G

=

c .J45;

X X X G

|G|

n − |G|

n − |G|

X

(x,G) x6∈G

d(x) k − d(x)

=

X X x

=

X x

G x6∈G

X |G| b = |G| n − |G| G

d(x) k − d(x)

k − d(x)

!

!

X d(x) b = d(x) k − d(x) x

¢¢ ã1B!gq;>.A.1;}4äBGF<2B £

KO2:OI/0?D0¡}?DCQB[BE037O2L4 X G

|G| <

X

d(x)

x

2L4 d .J23CJKO08ub }6F>C d F d ;<7>BE?[28=92L.ACAB{:'CJ4)BGF/0TI9034GB)Hn;<44CJI<.J0 CJ7BGF/0gq;>.N.J;}+CJ79f450L7/4509b L0B b ãqg gq;.1x n }2L ?D0452BDG i 0L= C1;<17/4s;>gnBGF/0+ 03K# CJ4G 82L{x 798 1 xi } b i = 2 . . . n k=n Û b 0 d F/;/;<4,0 03.J0LKO0L7>B[44= d FBGF<23BW27:6BGFBb@-9?D;L<0PBGF<23BWBGF/0,:2L.A.pF<2<02 d ;Bb

'#

u¼ ¡Æ

Ë

= ÆL\Ç/ÌJZ{ÆÏÑØ /Z{ËXEXGË ÆXEÏ/É ÌË,Z1Ï<]

:¡ ¼

ÓZJÉÆÏ/É L ÓÌ[ÌË¡] X[ÓÏ×GËb

I9027CN7>BE0L30L? L2 798.10B I9024,0BY}+CQBGF 0.103KR037>BD43bL0B I n0P≥ / 8>CJ4E1BDCJ7 d B4=nI<4,0BD4ä;> g 4G= d FS BGF<23B gq;BD0L?[450 d BDA CJ;<17 A2 . . . A k S i j CJ4@79;B03KOH>B:ub_F/0L7 n−1 b r

nb!L0B

Ai ∩ A j

k≤2

nXÆ/ÆÎ!ÆÎ\ËÒOÒOÓ9b~Ü>;g i =b1qã g . . . k gqB ;C F ;<7>BE?[28.A.1;}4YBGF<2B Ai ∩Aj = Ai ∩(S\Ai ) = ∅ 2L?D0)8g b!F>=94 2k ≤ 2n A S k b B1 B2 . . . B k BGF<123 BäA CN4 2 . . . A n−1 k≤2 0~79;} H9?D;3<06BGF/089034CN?D08R?D0L4G=/.ABIL:OCJ798>= d BDC1;<7Rgq;B p CJ4= >2 d 2=n ;>=94G.Q:BE?[=n0 4GCJ7 d 0+}60 d 2L7 d F/;/;<4,0);<7<. : p−1 4G=nI/450B[4Lb@F/0)?D0L4G=/.ABCN4 2 <3 g12L.N4,0gq;=9?450B[4 p = 3 S = {x1 x2 x3 } A 1 A2 A3 A4 2KO;<79 ∅ {x } {x } {x } {x x } {x x } {x x } 1 2 3 1 2 1 3 2 3 .1031 2 ?[x.Q: 2 79x;<379}0 4= d F~BGF<23BäBGF/0CN7>BE03?[4,0 d BDC1;<7R;>gY27:BGFgBGF/03K#CJ4s79;<790LKMHguBGF/06gq;>=9? d 27fI/0 b c .N4,; BGF/0)450B[4 ∅ {x x } {x x } {x x } 1 2 1 3 2 3 I90 d F/;<450L7 I90 d 23=94,0OBGF/0CJ?CN7>BE03?[4,0 d BDC1;<7TCJ4)0LKMH.A.1;}4+BGF<2B;<790f;>g!BGF/0 03.J0LKO0L7>B 4,0BD4WK=94GB_I/0 d F/;<4,037 452: bYF/037~032 d F6;>g>BGF/0 d F/;<4,037)450B[4WKM=94EB 1 d ;<7>B[2LCN7 bYF>=94 BGF/0 d ;<7 d .A=94CJ;< 7OF/;>.J{x 8/41gq};BD0303?bR3=9H9H9;.18<4+gq;B[4Lb@L0B }+CABGF .J0LKO 2798~.10B p − 1 A1 A2 . . . A2p−1 I9068>CJ4EBDCJ7 d Bä4=nI<4,0BD4s;>g 4G= Sd F)=BGF<{x 23B 1 x2 . . . xp } S

Ai ∩ Aj ∩ Ak 6= 0

gq;
i<j
b

L0B I/06BGF/07<=9KOI903?;>g4,0BD4 BGF<23B d ;<7>B[2LCN7 b6ãqg BGF/0L7 xp np < 2p−2 BGF/07<=9KOI9n 03?_p ;>g>4,0BD4 79;B d ;<7>BD23CJ7 28.A.10 d BDC1;<7 ;>geKR N C G 4 D B N C 7

B G 4 n = / I ;>g p−2 d ;<7>BE?[28=94 p−2 b n ≥ 2 p CABGF79;Ý.J;<454;>g303790L?[23.NCQB: }60KM2:79;}4=9H/Hn;<450BGF<23B I/03.J;<794)BD; xp b A1 A2 . . . A n p

¢

E"6$ p

b p−2 b L0B np 0> 2 gq;.A.10 d BDC1;<7)i;>g9KR;g b_¦s4CN79 } 1 BGx 2L798 4G{x = d FR F<223B . . .0 xp−1 bRã1B BGF/0.103KOKM2 }60M890L8<= d 0BGF<22BsBGF/0L?D0O0G CN4GB 0 r s A ∩ A r s = ∅ gq;>.N.J;}4WBGF<23B b!Ü>; }60 890L8<= d 0BGFB;>gY23.N./BGF/0 4Lb

E"6 p

Ai

9b

p−2 b

L 0B np I/=042= d FsBGF<23B 27988/039790 buYF/037 b x Bk 6=gq;.N.J0 d BDCJ;<7);>g Bp−1 B_4G=nI/450B[4!;>g 279A 8+1BGF/ 0CN27> BE.03.?[.4,0 d A BDC12; g/B 27k:6H923CJ? 2 + 1 S ;>gu8CN4 d ;>.N.J0 d BDCJ;<7~CJ4@79;<790LKMHBE?[28g xp 6∈ 4s}6A F>C kd F89;R79;B d ;<7>BD23jCJ7 CN4BGF/0 Ak ∪ {xp } = Aj 452LKR0+2L4!BGF/0+7<=9KRI/0L?;>g 4!}6F>C d F d ;<7>BD23A CJ7 i }60890L8<= d 0BGF<2BWBGF/0 d x;>p.A.10 d BDC1;<7 Ai xp ;>gnBGF/0 4CJ4 Ai

A1 A2 . . . Anp A1 ∪ {xp } A2 ∪ {xp } . . . Anp ∪ {xp }

}6F/0L?D0

gq;CJ4EBDCJ7 d B!4G=nI/450B[4;>g x;p} 6∈ Ai P p−2 {x , xBG2F/,06 . .CJ79. 8> ,x 1 23A 4= d FMBGF<23B gqA ;B 452: b .J0L2L?D. : gq;iCN∪ } BGF/0 0L2 d F 2L798M BGF/0L?D0gq;Bs;>ag23.N.nBGF/0 a 43∈b@Y 4P03{x 798/p4s Ai CJ798>= d iBDC1;<7M4GBD0LHM2798)BGF/ 0aH9?D;/;>gDb Û b@L0B

a b

2L798

c

b@-9?D;3<0BGF<2B

I/0?D032L.n7<=9KRI/0L?[4!}+CQBGF4=9K

a7 + b7 + c 7 = 7

a5 + b5 + c 5 5

Ä ¼ ¼ Ë Æ Æ d¡Ë Æ §Å É

Â

a2 + b2 + c 2 2

b

+¿ Æ ¾ §Æ Ë

= ÆL\{ÊLËÕ'ÑØ Z{×DÍYË,\ Ó/ÌJÓ3ZA\N\{Ë] _ÆnÇ>ËÏ<] X[ÓÏ×GË /Í/X[ZJÉÌÆ3VYÍËX LÔ X[Ó<Õ\{ËØL] X[ZJÉÌÆL\N] /Z{ËXEXGË ÆXEÏ/É ÌË,Z1Ï<] ÓZJÉÆÏ/É LÓÌ[ÌË¡] X[ÓÏ×GË ÓÏÕ nËLÕÇ3\JÓ )Ô eÇ X,Ì1ØL] PÆ>ÊLËXD] 3U!] = UPÔ~ËPÐZÊLË X[Ó<Õ\{ËØ ÉsÉÆL\[Ç<ÌJZÆÏub

YF/037

L0B

I/0BGF/0+?D;/;B[4@;>g

a b c 2798 S1 = a + b + c = 0 S2

P;} }60F<2<0

YF/037

x3 + qx − r = 0

2L798~.J0B

S k = ak + b k + c k

= a2 + b 2 + c 2 = (a + b + c)2 − 2(ab + bc + ca) = −2q

S3 + qS1 − 3r S4 + qS2 − rS1 S5 + qS3 − rS2 S7 + qS5 − rS4

= = = =

0 0 0 0

S7 S S = q2 r = 5 · 2 7 5 2

b

==⇒ ==⇒ ==⇒ ==⇒

S3 S4 S5 S7

= = = =

3r 2q 2 −3qr − 2qr = −5qr b 5q 2 r + 2q 2 r = 7q 2 r

b

¢ Þ Û bç0LKR037>BD4 ;<7 2L798 CJ7>BD0L?[450 d B~2B~Hn;>CN7>B b-<;>CJ7>B[4 2798 EBGF/04,0LKR03K M ,4 0LKR037>BD4 279AC 8 ?D03BD 45H90 d BDC <03. : 2LD? 04= d F)BGF<23BW 7>B H/2 44,034 KM BGF=n3F AB b@-9?D;L<0PBGCD F<23B b E KM ≤ max{AC BD} = ÆL\{ÊLËÕÑØ /Í/X[ZJÉÌÆ3VYÍËX LÔ XDÓ>Õ\ËØ3] X[ZJÉÌÆL\N ] ÓÏÕ 3ÆÉ,Í>ZÆ = Ë¡ZJÒZqØ3Ó3] Ó<ÈsÓ3É,ÓF0ZJ] ,ÓVYÓLÏÔ ~ËPÐZÊLË = Ë,Z1ÒZ1Ø3Ó ÉsÉÆL\[Ç<ÌJZÆÏub

Â

¿

¿ 4É Æ

Æ

Ä

Ä

YF/0+gq;>.A.1;}+CN79.103KOKM2CN4ä}603.A.79;}7

:¡ E"6$ r

nb

ãqg

CN4 2

P b AP ≤ max{AB, AC}

p

b

H9;>CJ7>B

;<7

2L798CAB[4@H9?D;/;>g9}+CA.N.979;BI90CQ<037b BGF/0´4CJ890 ;>g BE?DCN279.J0 BGF/0L7 BC ABC

b_q0L0BGF/068>CJ2?[2KÙ;<7)BGF/0+.10gNBI/03.J;}Ob

AB k CD I90H9;>CJ7>B[4~;<7 4G= d FBGF<23B 2798 b'3CJ7 d 0 X Y2L798 CD KX k AC 2?D0I9;BGFH/2?[23.N.J03.J;/?[2KM4 }606F<2 <0 KY k BD AKXC BKY D

L0B

KX = AC

3CJ7 d 0

M

CJ4@2Hn;>CN7>B;<7~BGF/0450KO0L7>B

YF/03?D03gq;
2L798

XY

KY = BD

}60F<2<0

IL:)BGF/0+.J0LKMKO2

KM ≤ max{AC, BD}

A

E"6 p

B

C X

b

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

T

E

Q

K

P

B

M

D

A

E

M

Y

Y

D

K

b

KM ≤ max{KX, KY }

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

b

S

C

9b

X

bWq0L0BGF/068>CJ2?[2K#;<7~BGF/0?[C13FB!2I/;L<0/b ã{7AB BGF>CN4 ∦d L2 CD 4,0 }6F/037f}60 d ;<7/4GC18/0L?<=928/?[CN.N23BD0L?[23. }60RF<2<003CQBGF/0L? ;=g ∠A + ∠D > 180 ∠B + ∠C > 180 30L7903?[2L.ACAB{:)BGF<23B H/2?[23.N.J03.=n3F ∠A + ∠D 2798 2B 2798 > ?D180 0345H90 d BDC <03. :bÝF/0L7 2LD 798 2?D0H9;>CJ7>AB B[4~;<7TBGF/0 EC Q ?D03 45H90 d BDC <03. :b©<:''037903P .N2L=94YF/Q0L;BD4 EM 2L798 P EC EM 4DP C }606F<2<0 P Q DM CE · · = 1 QD M C EP

b

¢ YF/03?D03gq;
BGF<23BäCN4

CM P Q bW3CN7 > MD QD

d0

DP k AB

}6030B P Q

CM AK > MD KB

QD

=

AK b_F>=94 KB

b

I/0R2RHn;>CN7>B4= d FfBGF<23B 2798 bL0B I/0R2 L0B XBGF<2B CX k AK H ;>CN7>Bä4= d F6 n 2L798 bL0B I9KX 0BGF/0kCN7>BEAC 03?[4,0 d BDC1;<7Y;>g DY k BK KY k BD S XY }+CABGF b CD YF/037 2798 bO3CN7 d 0 CX = }6AK = AC Y D = KB KY = BD 0F<2
b

;<7/450>=n0L7>BD. : =94GCJ796 }60)F<2<0 CS CJ42~Hn;>CN7>B;<7OBGF/0 CM b++0L7 d 0 < M G SD MD 4,0LKR037>B b DS L0B I90BGF/06CN7>BE03?[4,0 d BDC1;<7;>g }+CQBGF bsYF/037 CN42~Hn;>CN7>Bs;<7OBGF/0 KM d 4,0LKR037>B T b6F>=94 b)3CJ7 0 CNSY 42MH9;>CJ7>B;<M 7BGF/04,0LKR037>B ≤ KT T XY }606F<2<0 KT IL:~BGF/0.103KO KMKM 2 KT ≤ max{KX, KY } = max{AC, BD}

YF/03?D03gq;
KM ≤ max{AC, BD}

b

b

ÞPÞ b@-9?D;L<0PBGF<23B n X n

k

k=0

(a + k)k−1 (b + n − k)n−k−1 = (a + b + n)n−1

Ë

Æ

8Å

1 1 + a b

b

= ÆL\Ç/ÌJZ{ÆÏÑØ ÝZ×DÍË¡\ Ó<Ì1ÓZN\A\Ë¡] ÆnÇ<ËÏ/] X[ÓÏ×GËb

¦s4CN79

L0B n k

8/0L79;BD06BGF/06.10gNBs4GC18/0~;>guBGF/0)2I9;3<00L<=92BDC1;<7 2798O.10B b N = n−1 }6F/0L?D 0 N N }60}?DCQBE0 + k−1 S = S 1 + S2 k

S =

2L798

S1

=

S2

=

N X N

k k=0 N +1 X k=1

(a + k)k−1 (b + N + 1 − k)N −k

N

k−1

(a + k)k−1 (b + N + 1 − k)N −k

b

"

¢ ;
N

S1

n?[4EB 6 } 0+79;BE0PBGF<23B gq;

(a + k)k−1 (b + N + 1 − k)N −k N −k N = (a + k)k−1 a + b + N + 1 − (a + k) k N −k X N −k−j N N −k = (a + k)k−1 (a + b + N + 1)j −(a + k) k j j=0

k

N −k X

=

j=0

N −k X

=

j=0

k

N −k

N j

N −j (−1)N −k−j (a + b + N + 1)j (a + k)N −1−j k

N

3=9KOKCJ79;3<0L?

S1 =

k

N X N j

j=0

j

(−1)N −k−j (a + b + N + 1)j (a + k)N −1−j

2L798CJ7>BD0L? d F<2L79CJ79PBGF/0;gY4G=9KOKM23BDCJ;<7

(a + b + N + 1)

j

N −j X k=0

N −j k

}6030B

(−1)N −k−j (a + k)N −1−j

P;}j}60KM20=9450;>g>BGF/0gq;>.N.J;}+CJ79303790L?[23./?D0L4G=/.AB gq;. :/79;
(−1)m−k

k=0

m b P (k) = 0 k

JF/04G=9K ;<7ÝBGF/0.J03gNB4CJ890RCJ4 }6F/03?D0 CJ4BGF/08.N.J;∆ }4WBGF<2BBGF/0PCJ7/7903?W4=9K CJ76BGF/0 890n790L8MIL: 2I/;L<060G H/?D0L∆P 44CJ;<(x) 7gq;<=? P (x CJ4 + 1) 0 d − 0LH>PBW(x) }6F/03 7 bYF>=94 S1 0 j=N S1 =

Ü<;
b 0 1 (a + b + N + 1)N 0 (−1)0 (a + 0)−1 = (a + b + n)n−1 a N N

S2

S2

}60F<2<0

=

=

N +1 X

N (a + k)k−1 (b + N + 1 − k)N −k N − k + 1 k=1 N X N (a + N + 1 − j)N −j (b + j)j−1 j j=0

c 9 H H<.Q:>CJ79BGF/0452LKR02?D=9KO0L7>B@24@gq;.A.1;}43b

b

Â

u¼ ¡Æ ¼

Ë

ÆÒOÒfËÏYÌWÑØ /Z{ËXEXGË ÆXEÏ<É Ì{Ë¡ZJÏ/]

Â

ÓZJÉÆÏ/É L ÓÌ[ÌË¡] XDÓLÏY×EË>b

"

YF>CN4sH/?D;/I<.J0LKCN4sH/?D;.A=.A.J2L798 b ÞPà bYF/0H<.J2L790+CJ4s8>CQ>C18/08CN7>BE;?D0LC1;<7/4sIL: .NCN790L4CN7R303790L?[23.H9;<4CQBDC1;<7/43b@-9?D;L<0 BGF<23B!2Bä.10324EB ;>gnBGF/0?D0LC1;<7/4@2L?D0BE?[CJ2L79n.1034Lb

u¼ ¡Æ n−2

Ë

= ÆL\Ç/ÌJZ{ÆÏÑØ /Z{ËXEXGË ÆXEÏ/É ÌË,Z1Ï<]

ÓZJÉÆÏ/É L ÓÌ[ÌË¡] X[ÓÏ×GËb

KO00BD4PBGF/0CN7>BE03?DCJ;gBGF/0M79;<7958/0303790L?[2BE0 ÜCN?[4GB79;BD0~BGF<23B CAgBGF/0.NCN790 BE?DCN279.J0 BGF/037 K=94GB d ?D;<4`4B}6;f4GC18/0L4+;>gBGF/0~BE?[CJ2L79.10 452: 2B ABC 2798 2B b!ã1BWgq;>.N`.J;}4äBGF<2B CJ42+79;<7958/0303790L?[2BE0PBE?[CJ2L79[AB] .10/bYF>=94 MCAg 2.ACJ79[AC] 0KO00BD4BGN F/0sCJ7>BD0L?[C1;g/2sBE?[CJ2LAM 79.10 NCQB_8>CQ>C18/0L4BGF/0@BE?DCN279.J0CN7>BE;PB}6;?D0LC1;<7/ 4 23Bä.J0L2L4GB!;<7906;>g/}6F>C d F~CN4@2+BE?[CJ2L79.10/b L0B I90O2L7CJ7>BD0303?bL0BsBGF/0OH/.N2790OI90O8BD;?D0LC1;<7/4IL: n ≥ 3 .NCN790L4 CN7f303790L?[23.H9;<4CQBDC1;<7b CABGF/;>=g_303790L?[23.NCQB: }60)KM2n: ` ` . . . ` 1 2 n 4=9H/Hn;<450+BGF<2B gq;CN7>BD4 . BG F>M 30L7903?[2L.H9;<4M CQBDC11;< 7 M2L279 8. CJ.7~ CN4sn−1 ;g `i b)`' i+1 2798 ?D06nB}6 ;T gYiBG F/0 4PM ;<7 i M T M i+1 i i BGF/0.ACJ790 27:)B}6;~;>g9BGF/0BE?DCN279.J0L4 F<2<0+79;6CJ7>BD0L?[C1;CN7>B CJ7 d ;=94 }60F<2<00 L2 d BD.Q: T1 T2BE?[ CJ.2L.79. .103T4Ln−2 b I/0> 0L8b!ãqg79;<79n0− L0B ;>g92BGF/0P.NCN790L4 KO00BD4äBGF/0CN7>BE03?DCJ;g k ≤ n − 2 CJ4;<790~;>gYBGF/06?D0CJ;<7/4Lb !BGF/03?}+CN4,0 26.NCN790 `i KR0L0B[4sBGF/06CN7>BE03?DCJ;g Tk b BGF/0L7 `iCN7>BE03?DCJ;g k 0PBGF/0.J0L2L4GBWCN7>BE0L30L?ä4G= d F)BGF<23B L0B TI9 KO 00BD4äBGF/0+ bYF/037 g1?DT;BE;B}6;T?D0CJ;<7/4 2B+ ` T i k 1 ;<790);>gn}6F>C d FCJ4s2+BE?DCN279.J0 42: b;BE0BGF<23B79;<790);>gnBGF/0.NCN790L4 KR0L0B[4@BGF/0 T k1 `i CJ7>BD0L?[C1;g gq;g9BGF/0+.ACJ79034 ãqg gq;BD0L?[C1;g BGF/037 CN4 i > i1 0+gq;>.N.J;T 1 ;<790);>gBGF/06?D0CJ;<7/4Lbsã{7MBGF/0);BGF/03? d 2L4,0 `i4=nI<4GBDCQBD=Tg9k.N CN790L4!?D0LKM2LCN7CN4H/?D; d 03454@}+ CA.N.u0,<037>BD=92L.A.Q:R4EBE;C d F~CJ4@2BE?DCN279.J09b CABGFRBGF>CJ4H9?D; d 0L44 032 d F;>gBGF/0 4.10328<4PBD;2BE?[CJ2L79=/.J2L??D0CJ;<7bO3CJ7 d 0 27:OB}6;f;>guBGF/0?D0LC1;<7/4 F<2<079;RCN7>TBEi03?DCJ;CN7>BsCN7 d ;CJ4EBDCJ7 d B 2L798)

}Â

YF<2B d ;CN4sCN454G=n09b@0L798OKR0:<;>=9?7.N=I/0KM2LCJ79@2760 ;g<2H/?D;/I<.J0LK 4,0B2L798H<=nI<.AC d 23BDCJ;<7 ;>gnBGF/045;>.N=

V 2\ ^ b]` W}^ V º`+`NWYX e'W2^

6

Ã

1

â"~(

T

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T

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¼

H(X) ∈ Z[X]

G(X)

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u

Z

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) (uA)(u(2S − B)) = (u

αβ)

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YF/03?D03gq;

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A(2S − B) = T

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2

2(1 − cos x) x2

2 + cos x 3

r

>

1 + cos x 2

1 + cos x 2(1 − cos x) < 2 x2

Á

Æ

b

8Å

Þ Pæ å b nXÆ3VÆÉ ËVYÓLX ZÌDÇ ÊLÆÏ9ÓLX¡ÇL] _Ç<×ÓXGËÉÌ1] ÆnÇÒOÓLÏb

;>CAB =97~H92L?[2L.A. 0 @ .1;/?[2LKOKO09b c .{ 23C18/08Y =9790? 0 .J0P452L7/4?[28<=92BDC1;<7 ABCD BE?D;>=><0L?ä=97M Hn;>CN7>B 4=9? EB 0.Y>=n0 b 1 M

AB

AM =

5

AB

¼ ¿ ½ ¾ }Ë

¾

zÄ

'd½ Ë

Þ PæPæ b nXGÆ3VÆÉ R Ë VYÓX ÝZ0>Í/Ó<Ë,\ Æ9Ì×DÍËÌ{Æ<Ê] Ï>ZÊLËX[ÉGZÌ Ë jËÒfÆX[Z1Ó\Õ9Ë LËXEXË Ë>Ç>ÊLË] = ÌDÔ ËÓLÏ/] /b

d ;<7/4CJ8 03 ?D0 8/03= d 0L? d .J0L4f8CN7>BD4f0BO79;<79 d ;<79?[=n0L7>B[4 @7 0B 890 d 30 7>BE?D0L4ä?D0345H90 d BDCAg14 0B b;>CQB .J0H9;>CJ7>B_8 CN7>BE03?[4,0 d BDC1;<7)8/0L4ä890Γ=1 B[279Γ3032 7> BE034 d ;BH92L4M.10T450KO0L7>B 0B O1 >=/OC2790 d ;>=9Q bs¦s790BD2L7930L7>BD0 t 1 1O d ; =9Hn0O.J0R4,0LKR037>B CJ7>BD0L?[450 d BD0O.1O 034+ B[22793037>BE034 0B O 1 O2 t1 t2 0L7 0B tc ?D0L4Hn0 d BDCQ<03KR037>Bb E1;>CAB E2 .J06Hn;>CN7>B@KCN.AC10=f8>=4,0LKR037>B b';<7>BE?D03?s>=n0 0B P K 03KR0 d 03? d .10/b O 1 O2 P Q E 1 E2 4,;<7>B!4G=9?ä=97M

È

¤¼

Þ Pæ

½

YÂ

"½ ¡¼

;>CAB

a1 a2 . . . a n

8/0L4@79;
n n−1 X X

i=1

3C

Å

b nXÆ3VÆÉ ËVYÓLX <ÓLÉGZN\{Ë X,ÌÆÓË¡]uÏZËb

r=

aji

j=0

−1

b

n−1 X −1 H9;>=9? KO;<7>BE?D0L?!<=n0 √ n a 1 a2 · · · a n ≥ 1 En ≥ n rj

qI d b j=0 n = 2 n = 3 ?n≥4 Þ Pæ b nXÆ3VÆÉ ËVYÓLX <ÓLÉGZN\{Ë X,ÌÆÓË¡]uÏZËb

12

¤¼

;>CAB

½

YÂ

a1 a2 . . . a n En =

;>CAB 12 1I

';<7>BE?D03?>=n0

1 + a 1 a2 1 + a 2 a3 1 + a n a1 b + + ··· + 1 + a1 1 + a2 1 + an

En ≥

n(1 + r 2 ) 1+r

? ';<7>BE?D03?<=n0;>=O?0@ gJ=
¤¼

Å

8/0L4@79;
b √ n a 1 a2 · · · a n ≥ 1

r=

"½ ¡¼

½

H9;>=9?

En ≥

YÂ

n=3

5(1 + r 2 ) b 1+r

0B

n=4

b

"½ ¡¼

Å

Þ Pæ b nXÆ3VÆÉ ËVYÓLX <ÓLÉGZN\{Ë X,ÌÆÓË¡]uÏZËb √ ;>CAB 0B 8/0L479;BE?D0L?>a=n 0 b c d r 1 (1 +

Þ

¤¼

a)2

½

+

1 (1 +

b)2

Á

+

1 (1 +

Æ

c)2

+

1 (1 +

8Å

d)2

≥

b

4 (1 +

r)2

b nXÆ3VÆÉ ËVYÓLX ZÌDÇ ÊLÆÏ9ÓLX¡ÇL] _Ç<×ÓXGËÉÌ1] ÆnÇÒOÓLÏb

3C 0B 45;<7>B89034+037>BDC103?[4Hn;<4GCABDCAg14BE0.J4<=n0 0B4C m ≥ n a b c > 0 KR;<7>BE?D03?m <=n0 n am bm cm an bn cn b + + ≥ + + m m m m m m n n n n n n b +c c +a a +b b +c c +a a +b

¤¼

½ Ë

Æ

¤¼

½

YÂ

8Å

Û b nXÆ3VÆÉ ËVYÓLX Z{×DÍYË,\ Ó<Ì1ÓZN\A\Ë¡] _ÆnÇ>ËÏ<] XDÓLÏY×EË>b -<;>=9? BE?D;>=><03?!.N26H/.A=94sHn0BDCABD06I9;B d ;BE08/0.N2 d ;<7>BE?[23CJ7>BD0 a + b + c + abc b Þ

ab + bc + ca = 1

Þ

12

1I

"½ ¡¼

Å

Þ b nXÆ3VÆÉ ËVYÓLX <ÓLÉGZN\{Ë X,ÌÆÓË¡]uÏZËb

';<7>BE?D03?<=n04GC y z t

3C

0 ≤ λ ≤ 4

2L.J;=9?WBE;>=94.1034s79;
(t2 + 1)(x3 + y 3 + z 3 ) + 3(1 − t2 )xyz b ≥ (1 + λt)(x2 y + y 2 z + z 2 x) + (1 − λt)(xy 2 + yz 2 + zx2 ) t=

1 4

0B

λ = 4

. CJ70L@ 2L.ACAB0 @ d CJ58/0L44=94s8/0,B

17(x3 + y 3 + z 3 ) + 45xyz ≥ 32(x2 y + y 2 z + z 2 x)

9?D;>=><03?WBE;>=23.10=9?[4sHn;<4GCABDC <0L4@890

δ

b

BD03.A.1034s<=n0. CJ70L@ 2L.ACAB0 @

x3 + y 3 + z 3 + 3δxyz ≥ (1 + δ)(x2 y + y 2 z + z 2 x)

'È ½ 0¿ Æ ½ zÅ

45;>CABs>2L.N2I>.10OH9;>=9?BE;>=94 <=/Cä45;<7>B NC 8/0L479;Ë G]Ï>ZÊLËX[ÉGZÌ ËÕ9Ë <Ó\N\JÓ<Õ9ÆL\AZÕ] <Ó3\A\1Ó>Õ9ÆL\NZ{Õ] ÉGVYÓ3ÐÏË ×EÆÏ/É5Ó<×DX Ë Ó 3ÆÉ,Í>ZÆ = Ë¡ZJÒZqØ3Ó b ( ) ;>CAB =97MBE?[CJ2L79.10?D0 d B[279.J0679;<79GCN4,; d 0 .10 8Y 279.J0P8/?D;>ABC CQB_037 0BuBD03.><=n0 b;>CAB .J0 E...............................................................................................................................................G .. . ..... .......... H/CJ08O890+.J26F<2L=4=9AB ?ä.10 d ;B 0 @ D b ....... .... .. ... ... ......... ........ ... ... ........ ... ... ........ ... ... A BC ........ ... .. ....... ...... ..... . . . . . . . . . . . . . . ........ ;>CAB .10)Hn;>CN7>B8 CN7>BE03?[4,0 d BDC1;<78/0).J2M8CABD0 0B ... ... ........ ............... .... ..... ..... .... .... ....... .... ... ....... .............. ...... .. .... ........ ... ..O2............... ... 890~.J2MGH/2?[23.N. 03 .J0 2 H92L45427>BH92L? bu;>CQB AD 0B .. ...... . . .. . . .... ........... .. ............ ... . . AB C E F . . .. ........ ... ............... .. . . . . . . . . . . . . . ........ ..... ... ....... .. ........ 89034H9;>CJ7>B[4BE0.J4<=n0 0B gq;B ... .......... .............................................................. .....H . . ............................................................................ .. . ....... . . . . ACGE BF GE . . . . . B ..... ............ ..... .......................... ..................... .... F ?D0L4Hn0 d BDCQ<03KR037>B89034P?D0 d BD2L79.1034LbY;>CAB .{ CN7>BE03?[4,0 d ... ....... .. .. . .. ..... ....... .. .... .......... ......... .... H ....... .. ..... ..................... .... ... .......... ............... ...O ... .... BDC1;<7890 0B bu;>CABsn7/23.103KR037>B 0B .J0L4 ................ ... 1 ... . . .... ... .. .......... ... . AG BF O O . . D . ... ....... ... 1 2 . ... . . . . . . .... d d . . CJ7>BD0L?[450 BDCJ;<7/4w?D0L4Hn0 BDCQ<034 8/0L4 8>CJ23;<7/2L.J0L4w89034 ... ....... ... . . . .. . . . .... . ....... ... .. ... . . . . . . .... . ....... ... ... .. . . . . . . . . <=928/?[CN.N23B 03?D0L4 0B b ....... ... ... .... .. . . . . . . ....... ... ... .... .... CDHF BDGE . . . . ....... ... .... ... .... .......... ... ... .... ';<7>BE?D0L?O<=n0Ý.1034OBE?[CJ2L79.1034 0B .. .. .................................................................................................................................. ABC DF E 4,;<7>B!450LKOI<.N2I>.1034Lb A C

¼

qË ½ Ê

'½

'È

¼

DO1 O2

½ 0¿ Æ zÅ

qË ½ Ê

Þ ½ b nXÆ3VÆÉ ËäVYÓLX LÇLÓÏ ÆÉ×GÆ _ÆÒfËXÆ <Ó3\A\1Ó>Õ9ÆL\NZ{Õ] ÉGVYÓ3ÐÏË>b

'½

ÓLX Ç>Ë G]Ï>ZÊLËX[ÉGZÌ ËÕ9Ë <Ó\N\JÓ<Õ9ÆL\AZÕ]

È

@789;<7/790=976BE?DCN279.J0 0BW45;>CAB =97~H9;>CJ7>Bä>=n03. d ;<79>=n04=9?.N2+8/?D;>CQBE0 P 0@ 0PHn0=CQBE0H92L? b ;>CAB > .{ CN7>BE03?[4,0 d BDC1;<7~8/ABC 0 JH9?D;>.J;<79 BC H92L?[2L.A. 0.10 2 A1 0B_45;>CAB . CJ7>BDAP 0L?[450 d BDCJ;<7~8/0 JH9?D;>.J;<79 0@ 0Hn0=CQBE0+H92L? ';<7>BEC?D0L?<=n0).{ 23CJ?D0AB 8>=BE?[CJ2L79.10 [email protected]KO;L:<037/790M.0L@ ;
A1 BC

A2 BC

È

¼

½ 0¿ Æ zÅ

qË ½ Ê

Þ På b nXÆ3VÆÉ ËäVYÓLX LÇLÓÏ ÆÉ×GÆ _ÆÒfËXÆ <Ó3\A\1Ó>Õ9ÆL\NZ{Õ] ÉGVYÓ3ÐÏË>b

'½

ÓLX Ç>Ë G]Ï>ZÊLËX[ÉGZÌ ËÕ9Ë <Ó\N\JÓ<Õ9ÆL\AZÕ]

'È

?D0~CJ7/4 d ?DCNHB.1034P.1;<79=n0=9?[48/0L4 @789;<7/790~=97T>=928B H9?[CJ4)8/2L7/4 d 0B;CJ23;<7/2L.J0L46KO0L4G=9?D0L7>B 0B >n=n 0 p√ q d ?D03? 0B 0 KR;<7>BEm d mp + nq ≤ 12 (d + d0 )

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

¼

Þ Pæ+à b

Ë 8½ Æ

8Æ

8Å

nXGÆ3VÆÉËLÕRÑØ ÝZ1Í Ó\qØ ËÏ× ¡Ë] X[ÓLÉÆ<Ê] ÆÒOÓÏ>Z1Ó/b

I/0s2L7:+2 d =CN=94_2L798 L0B d CJ? d =9KM?[28>ABC CN=94 ?D0L4Hn0 d BDCQ<0.Q: 2L798·.10B I/0´BGF/0 r 4,03KMR CNHn03?DCNKR0BE03? BGF<2B CN4 bL0B I90RBGF/0f.J0Ls79¡BGF';>gBGF/0fKR0L8gBGF/0CJ7>BD0L?D7/2L.WI>CJ450 d BD;g g1?D;
X ma 3s2 − r 2 − 4Rr s2 − r 2 − 4Rr ≤ ≤ 8sRr 7sRr awa

1I

3

4

≤

X

¼

m2a b2

+

≤

c2

¿

4R + r b 4R

8¼

¾qË " :

Þ Pæ@½ b nXGÆ3VÆÉËLÕRÑØ Æ/Ë 6Æ/ÈsÓXGÕ] nÆX,ÌJÓ3\{ËÉ] ] = UPb ã{7/4H/CN?D08OIL:-9?D;/I<.J0LK qb ¥ 2 BGFubn¨_230B[BE0 £ Hb L0B π bWF/;}BGF<2B

12 1I

h

x ∈ 0, i

2 + cos x 3

2 + cos x < 3

Þ Pæ å b

¼

b

2

2(1 − cos x) x2

r

1 + cos x 2

>

1 + cos x 2(1 − cos x) < 2 x2

§Á

b

8Æ

Å

nXGÆ3VÆÉËLÕRÑØ ZÌDÇ >ÊLÆÏ/ÓX¡Ç3] _Ç>×DÍ9ÓLXË¡É3Ì1] _ÆÒOÓLÏ
L0B I902H/2?[23.N.J03.J;/?[2KbW¦s4GCJ79;<7<.Q:~2L7=97/KM2?E084GBE?[23C13FBD08/30 n798M2Hn;>CNABCD 7>B ;<7 4= d F)BGF<2B b 1

¾

¼

M

¿ É Ë¾ AB

Ä

AM =

Þ PæPæ b nXÆ3VÆÉËÕ ÑØ ÝZ0>Í/Ó<Ë,\ ËÈ@ÎÆnÇÏÕ\1ÓLÏYÕ] = ÌDÔ ÆÍ/Ï É] /b

5

AB

Æ9Ì×DÍËÌ{Æ<Ê]

Ë

âËÒfÆX[ZJÓ3\

Ï>ZÊLËX[ÉGZÌ1ØÆÎ

;<7/4GC18/0L?679;<79GCN7>BE03?[4,0 d BDCJ79f2L798 79;<79 d ;<79?[=n0L7>B d CN? d .1034 2L798 }+CQBGF Γ2;>g_BGF/0 d 037>BE?D0L4 ?D0345H90 d BDC <03. :bL0B 2798 I90~BGF/0OHn;>CN7>B;>gWCJ7>BDΓ 0L?[1450 d BDCJ;<7Ý O320L 7>B[4 d FÝ89;T79;B+CN7>BE03?[4,0 d BPBGF/0R.NCN7904,0LKR037>B B}6; d ;C Q t t 1 2 }6F>C d F6CJ7>BD0L?[450 d B[4WBGF/0P4,0LKR037>B b cd ;B KO00BD4WBGF/0 O O 1 O2 2798 2B 279 8tc ?D0L4Hn0 d BDCQ<0.Q:ub BD2L179O 30L27>B[4 E;>2g BGF/0.NCN790M4,0LKR037>B L0B t1 I/0~BGtF/20MKME CJ891GHn;>CN7>B b-9?D;3<0~BGF<23B P O 1 O2 P Q 2798 2L?D0 d ;<7 d : d .AC d b E1 E2

Þ Pæ

¼

¡Â

¼

L0B

I/0H9;<4CQBDCQ<0+?D032L.n7<=9KOI903?[4 L2 798.J0B

a1 a2 . . . a n

n n−1 X X

En =

i=1

ãqg

r=

Å

nXGÆ3VÆÉËLÕRÑØ <Ó3ÉZA\Ë X,ÌÆÓË¡]Ï>ZÊLËX[ÉGZÌ1ØÆÎ /\ÆLZ{ËÉÌJZJ] ÆÒOÓÏ>Z1Ó/b b

√ n

a1 a2 · · · a n ≥ 1

H9?D;3<0BGF<23B

aji

j=0

b

−1

En ≥ n

n−1 X

r

j

−1

gq;
j=0 qI d b n = 2 n = 3 ?n≥4 Þ Pæ b nXGÆ3VÆÉËLÕRÑØ <Ó3É ZA\Ë X,ÌÆÓË¡]Ï>ZÊLËX[ÉGZÌ1ØÆÎ /\ÆLZ{ËÉÌJZJ] ÆÒOÓÏ>Z1Ó/b

12

L0B

¼

¡Â

a1 a2 . . . a n

L0B 12

r=

1I

+

1 + a1

1 + a 2 a3 1 + a2

+ ··· +

Å

1 + a n a1 b 1 + an

b √ n a 1 a2 · · · a n ≥ 1

-9?D;3<0BGF<2B

I/0H9;<4CQBDCQ<0+?D032L.n7<=9KOI903?[4 L2 798.J0B

1 + a 1 a2

En =

¼

En ≥

n(1 + r 2 ) 1+r

? -9?D;3<06;CJ4H9?D;3<0BGF<23B

¼

Þ Pæ b

gq;
En ≥

¡Â

n=3

2L798

n(1 + r 2 ) 1+r

n=4

gq;
b

n=5

b

¼

Å

nXGÆ3VÆÉËLÕRÑØ <Ó3ÉZA\Ë X,ÌÆÓË¡]Ï>ZÊLËX[ÉGZÌ1ØÆÎ /\ÆLZ{ËÉÌJZJ] ÆÒOÓÏ>Z1Ó/b

√ L0B 2L798 I/0sH9;<4CQBDCQ<0@?D0L23.7<=9KOI903?[4_4= d FBGF<23B b 4 r r = abcd ≥ 1 -9?D;L<0PBGF<23aB b c d 1 1 1 1 4 b + + + ≥ 2 2 2 2 (1 + a) (1 + b) (1 + c) (1 + d) (1 + r)2

Þ

¼

b

§Á

8Æ

Å

nXGÆ3VÆÉËLÕRÑØ ZÌDÇ >ÊLÆÏ/ÓX¡Ç3] _Ç>×DÍ9ÓLXË¡É3Ì1] _ÆÒOÓLÏ
ãqg 2L798 2?D0~H9;<4CQBDCQ<06CN7>BE0L30L?[4P4G= d FOBGF<2B 2L798RCNg n m ≥ n a b c > 0 H9?D;3<0BGF<m 23B am bm

Þ

+

Û b

cm

¼

+

bm cm

+

am

Ë

+

cm am

Æ

+

bm

≥

8Å

an bn

+

cn

+

bn cn

+

an

+

b

cn an

+

bn

nXGÆ3VÆÉËLÕRÑØ ÝZ×DÍË¡\ Ó/ÌJÓ3ZA\N\{Ë] ÆnÇ<ËÏ/] XDÓLÏY×EË>b

Ü<;=9798R2L798BGF/0)?D0L2BE034GB!.1;}603? 9 I9;>=9798O;>g a b c ∈ (0, 1) G4 =nI;{0 d BWBE;)BGF/0 d ;<7/4EBE?[2LCN7>B b a + b + c + abc ab + bc + ca = 1

Þ 12

1I

¼

Þ b

¡Â

¼

Å

nXGÆ3VÆÉËLÕRÑØ <Ó3ÉZA\Ë X,ÌÆÓË¡]Ï>ZÊLËX[ÉGZÌ1ØÆÎ /\ÆLZ{ËÉÌJZJ] ÆÒOÓÏ>Z1Ó/b

-9?D;3<0BGF<2BäCNg

0 ≤ λ ≤ 4

BGF/0L7

gq;
x y z t

(t2 + 1)(x3 + y 3 + z 3 ) + 3(1 − t2 )xyz b ≥ (1 + λt)(x2 y + y 2 z + z 2 x) + (1 − λt)(xy 2 + yz 2 + zx2 )

Ü>;
t=

1 4

2L798

λ = 4

BGF/02I/;L<0CJ790L<=923.NCQB:MI90 d ;
17(x3 + y 3 + z 3 ) + 45xyz ≥ 32(x2 y + y 2 z + z 2 x)

ÜCJ7982L.A.nHn;<4GCABDC <0>2L.A=n0L4s;>g

δ

b

4= d F)BGF<2BWBGF/0+CN790>=92L.ACAB{:

x3 + y 3 + z 3 + 3δxyz ≥ (1 + δ)(x2 y + y 2 z + z 2 x)

F/;>.J8/4gq;C d F2L?D0 N C H9;<4CQBDCQ<0+?D032L.n7<=9KRI/0L?[4 J CAC 4GC18/0+.J0L79¡BGF<4 ;>g2+BE?[CJ2L79.10/b x y z

M¼

¿ Æ hÅ

Ë ½ %Ê

Þ à b nXGÆ3VÆÉËLÕ ÑØ LÇÓLÏ ÆÉ×EÆ _ÆÒfËXÆ ÓX Ç<Ë G] uÏÓ<Õ Õ9Ë <Ó3\A\1Ó>Õ9ÆL\NZ{Õ] <Ó\N\JÓ<Õ9ÆL\AZÕ] = VYÓZ1Ï Õ9ËÕZ{×[Ó<ÌËLÕÌ{Æ 3ÆÉ,Í>ZÆ = Ë¡ZJÒZqØ3Ó b ( ) L0B I90279;<79ECJ45;<4 d 03.J0L4~?[C13FB+BE?DCN279.J0 }+CABGFf?[C13FBABC 279.J0M2B 2798 bfL0B I/0 E..............................................................................................................................................G >BD; AB . ..... .......... BGF/0Rgq;/;B;>gWBGF/023.ABDCQBDA =n890Og1?D;CJ7>B@;>gYCN7>BE03?[4,0 d BDC1;<7O;>gBGF/0.ACJ790 ........ ............... .... ... .. ... ... .... ... ....... .... G AD ... ....... .............. ...... .. . . . . . . .... . ........ ... ..O2....... ... q0G LBD0L798/08 }+CQBGFOBGF/0~.ACJ7906BGF=n3F }6F>C d FRCJ4PH/2?E .. ....... . . ... . . ... .......... .. ........... . . . .. . . . . C ... ........ ............... .. . . . . . . . . . . . . . . ........ ..... ... ....... .. ........ 2L.A.10.CN7>Bä4= d F6BGF<2B .......... ... ............................................................. .....H . ............................................................................. .. . ....... . . . . . . . . AB E ACGE B ..... ............ ..... .......................... ..................... .... F CJ4~2?D0 d B[279.J0 2798.J0B I/0BGF/0H9;>CJ7>B64G= d FÝBGF<23B ....... .. ... ..... ... ............. ........... ....... .. ... ....... .. ...... ..................... .... F ... .......... ............... ...O ... .... CN4P2?D0 d BD2L79.10/bML0B I/0)BGF/0)CN7>BE03?[4,0 d BDC1;<7 ................ ... 1 ... . . ... ... ... .. .......... . . . .... BF GE H D . ... ....... ... . . . . . . . . .... ....... ... ... ;>g 2798 bsL0B I90PBGF/0+CJ7>BD0L?[450 d BDCJ;<7O;>gnBGF/0 .. . . . . . ... . .... . ....... ... .. . . . . . . AG BF O . . .... . ....... ... ... 1 .. . . . . . . . . ....... 8g=928BD0L?[450 d BDCJ;<7~;>gCJ23;<7/2L.N4;>gÓ<Õ Õ9Ë <Ó3\A\1Ó>Õ9ÆL\NZ{Õ] <Ó\N\JÓ<Õ9ÆL\AZÕ] = VYÓZ1Ïub

M¼

¿ Æ hÅ

Ë ½ %Ê

L0B I/0+27:~Hn;>CN7>B!;<7)BGF/0.NCN790 CJ7 bL0B I90PBGF/0PCJ7>BD0L?[450 d A BDC1;<7;>g P JHn;<44CJI<. :0 3BE0379890L8 }+CQBGFBC BGF/0).ACJ794ABC 0BGF=n3F }61F>C d FOCN4H92L?[2L.A.10.BD; 279AP 8M.10B I/0BGF/0CJ7>BD0L?[450 d BDCJ;<7R;>g JHn;<44CJI<. :R0 3BBE0379890L8 }+CQBGFBGF/0.ACJ790 AC A AP 2 BGF=n3F }6F>C d F~CJ4@H/2?[23.N.J03./BE; b C AB -9?D;L<0BGF<2BBGF/0O2L?D0L2;>g CN4BGF/0R30L;g_BGF/0O2?D0324;>g 4ABC 2L798 b

M¼

4A1 BC

4A2 BC

¿ Æ hÅ

Þ På b nXGÆ3VÆÉËLÕ ÑØ LÇÓLÏ ÆÉ×EÆ <Ó3\A\1Ó>Õ9ÆL\NZ{Õ] <Ó\N\JÓ<Õ9ÆL\AZÕ] = VYÓ1Z Ïub

_ÆÒfËXÆ

Ë ½ %Ê

ÓX Ç<Ë G]

uÏÓ<Õ

Õ9Ë

¨_C <0L72L7CJ7/4 d ?DCJI90L8 d ;<7<0G f>=928g_.10379¡BGF DB 20L7~CJ7M;=9798~BGF/0>=928CJ23;<7/2L.N4s;>gn.10379¡BGF m2 9 7 n8 p0 q d d H9?D;3<0BGF<23B √ 1 0 b mp + nq ≤

2

(d + d )

¾

ÆVYXGÆ9ÑL\ËÒ·ZJÉË3ÊLËX+VËXEÒOÓLÏYËÏuÌJ\1Ø'×\ÆÉËÕÔ LÍËTËLÕZÌ{ÆXZNÉMÓ3\{ÈsÓØ3É~Vn\ËÓLÉËÕâÌ{Æ ×GÆÏ<ÉZ{Õ9ËXäÎÆXVWÇ/ÑL\AZ×Ó/ÌJZ{ÆÏOÏËÈâÉÆL\Ç/ÌJZ{ÆÏ/ÉPÆX!ÏËÈáZJÏ/ÉGZJÐLÍuÌNÉÆÏ)VYÓ3É3ÌnVYXÆ9ÑL\{ËÒMÉ3Ô

"

0 F<2<08>CJ4 d ;3<0L?D0L8RBGF<2BH/?D;/I<.J0LK CN4CJ89037>BDC d 2L.YBD;RH9?D;/I>.103K }6F/;<450 4,;>.A=BE0L8 > >¢ b ;<7/450>=n0L7>BD. : H9 ?D;/I>.103K CJ4 d .J;<4,0L8 2L798~79;)45;>.N=BE0L8b 0P2H9;>.1;/CQ 0gq;CJ4 ;L<03?[4CJ3FBb

¼

Þ Pæ@½ b > <

u¼

nXÆ3VÆÉËÕRÑØ <ÓÏÆÉnÔ ¡É5Ó<ÆnÇ3É5ÉÆÐ3\{ÆnÇL]/UÌqÍYËÏ/É]uXËLË×EË>b

2L?D0BGF/0+4CJ89034s;>gY27M2 d =
ãqg

a b c Xp

uÂ

a2 + b2 − c2

3¿ RÆ

H/?D;L<0BGF<2B

p b a2 − b2 + c2 ≤ ab + bc + ca

8Æ

8Ä

oGb = ÆL\[Ç<ÌJZÆÏfÑØ /Í9X[ZNÉ3Ì{Æ3VYÍYËX LÔ XDÓ>Õ\ËØ3] X[ZNÉ3Ì{ÆL\J] äb L0B

2798

2 2 2 2 2 2 2 2 x2 = b2 + cb¯ −F/a + b2 −2Lc7928 0L7 y2 = c1 +2a − 2b 2 z 1= a 2 x y z > 0 a = 2 (y + z ) b = 2 (z + x2 ) 1 2 2 2 bYF/0CQ<037MCN790>=92L.ACAB{:OI/0 d ;
}6F/0L?D0

;
X

(yz + zx)

≤

Xp

y2 + z2

p z 2 + x2

}6F>C d F~CJ4!BE?[=n0 4GCJ7 d 0 b 2 2 2 2 2 2 2 F/;>.18<4(yCN7T zCNgW)(z ß/>=92L.ACAB{: + 2798+ ;<7<x. :f)CNg − 2(yz + zx) 2 = (z −2L79xy) 8 2≥ 0 b z = xy xd = yz YF/034,0 d ;<798>CABDCJ;<7/4s2?D060L<=/C >2L.J0L7>BäBD; bs+0L7 0 0L<=923.NCQB:OF/;>y.18<4@= CJ7zx BGF/0 x=y=z H9?D;=92L.ACAB{:+CAg<2L7986;<7<. :CNg b BGF<23BYCJ4 CAg<2L7986;<7<. :CNg x = y = z a=b=c ooEb = ÆL\Ç/ÌJZ{ÆÏfÑØRÅnÇ3ÎË,Z LÍ/Ó<Æ]ÉÌDÇ<Õ9ËÏuÌ1] ÆÏ ZA\N\NÉ ÆL\N\{ËÐZ1Ó<ÌË qÏ/ÉÌJZÌDÇ<ÌË¡] LÆXGÆÏuÌ{Æ] )b

N¾

037 d 0

0Á

¢Ë Â

©<: 2L= d F: 4Pã{790L<=923.NCQB:

¡Ç

gq;
}606F<2<0

x y z w ≥ 0 p √ b √ xy + zw ≤ (x + z)(y + w)

Xp

a2 + b2 − c2 =

p a2 − b2 + c2

p 1 X p 2 a + b2 − c2 a2 − b2 + c2 2

p p c 2 + a2 − b 2 c 2 − a2 + b 2 X b 1 Xp (2a2 )(2c2 ) = ac +

≤

2

£

EÁ ¼ ¢Â

uÂ

oooGb = ÆL\[Ç<ÌJZÆÏfÑØ Z 3ÍYÆnÇ3] nÆL\0 ÆÒOÒÇLÏË] Z1ÏYÌËX Ó<ÊLËÏ/] 3 ] = UPb ©<:~BGF/0 d ;<7<0G CQB:M;>g √

x

}60F<2<0

p 1 p 2 a + b2 − c2 + a2 − b2 + c2 2 ≤

s

037 d 0

(a2 + b2 − c2 ) + (a2 − b2 + c2 )

P √

a2 + b2 − c2 ≤ a+b+c =9Hn;<7M4,>=92?[CJ79
2

= a

b

}6F>C d F+I90 d ;
23 $+- 5872(9 #2 2 A ' *A&E2A *2 C29 2 )2

E< A <$+ $$7 ' ' + A ' ' + A&> < ' ' - * ##+ ' E ><22 A:9;!<#A ' E E 9 A ! <!# # '#"E $ <*A ? #A ;A *2 !" 9# AC !<- A $ ,% $# ' F#,+ , 7 A ' 'B6'&*< A A *2 ' : $ 2(9# A= ' ,A ( A *2 $ 2 ) 9 2 A ! 7B6<',!B A% $<58*!# @A12 ! $ ' * 99?2! ?. 2 # A F#,+ , 7: 2,5= ' A -8BC, ,+ $2 2 A4 AE # <' # 2 :.- 9A #+ . A 23A 2 E 829# E /A )"" + + + !# ,"@ <$+ $$7 A0 ' #-5?! A ;B6' ?7 ><21 > - 9

A4E ' @,-8' A 2 93;.4$6 ) ;A >?$ ' F#,+ , 7 A '. '#-8' A:E2A *2 5 6 75 )2( 9 A;>;!# <' A;9;B6' < ' '-6 . 25!# ' & !#58B ,""- "!# $C! - &' BC& ' -8' - !<58 , !< ,< $+ < ,!< , '<5#+ *A;, < )%)$ & ) 2 + 18 $: 9?!' $$7 '<- ;?E<'!# 7 4 : 9?! ' , 7?. " 5< 23$B6' @- <'<? % (B ! , ? 74 $+4 : 9?!' $$74587 ' - '<-- 2 2 a + b2 + c2 ,-A % 4<5 '

%

#% ,

K

'-

X p a 2 + b2 − c2 ≤ a + b + c

s

-<" '' '-6 B BC" ; 4 ' , *A C!

a2 + b2 − c2 = 2ab cos C = 4

1 ab sin C 2

cot C = 4K cot C

'- ,% 4 B ' $- , ,AF :9?!' $$7 '#5#+4,'C5 %( ,"":'

*'

s K :9?! ' , 7 % <$ ) #(<'& -6:=

√ √ √ cot A + cot B + cot C ≤

! <5 ' -6 " $#%) X

a(b + c) cos A ≤ ab + bc + ca

F !#58 $!# $ B '3, < ,!< ,/ ?'#5#+*A" $#% - 587 <E ' :9?! ' , 7 A ' B6@8 : '#5 - 3BC < ' ! $ '#5"F' #+$--6 9?!

X p p , n an + bn − cn n an − bn + cn ≤ ab + bc + ca

' $ $ ! -( '<-$"' )' ' ? + <' ,+ #.

n≥2

¢

" ¼

¾

Æ

¾

Þ Pæ å b > 3 nXGÆ3VÆÉËLÕ)ÑØ _ËLÆX[Ð>Ë Ó\ÆÐ\ÆnÇ3] = ~Å _ÉÈPË¡Ð<Æ] _ÉÈPË¡Ð<Æ] ~Å/] = P U b 3=9H9H9;<4,0sBGF<23B 2?D0sBGF/0H9;>CJ7>B[4ä2Bu}6F>C d F+BGF/0 d ;<7 d =9?E?D037>B_.ACJ79034 F;>gs2TCQ<037BE?DCN279.J0 D4CJ 89E034M AD KR0L0B+BGF/0 b L0B 2L798 I/0f BGF/0 BE CF ABC p p 1 Hn03?DCNKR0BE03?[4M2L798 2L798 BGF/02L?D0L2L4M;>g 2798 ?D0L4Hn20 d BDCQ<0.Q:ub δ δ 4ABC 4DEF 1 2 -9?D;L<0PBGF<23B J2 qI

2p2 ≤ p1

CNg CNg

AD BE

2L798

CF

2L798

2?D0+279.J06I>CJ450 d BD;

AD BE CF gq;
d

2p2 ≤ p1

0Á ¼ ¢Â

2Â

= ÆL\Ç/ÌJZ{ÆÏÑØ Z LÍÆnÇL] nÆL\ 0 ÆÒOÒÇÏ>ZÌ1Ø ÆL\A\Ë¡Ð<Ë¡] ZJÏuÌ{ËX Ó>ÊLËÏ<] 3 ] = UPb

> >¢>¢£ b CN462 d =CJ79O;>gäBGF/0RH/?D;/I<.J0LK 4G=n330L4EBD4+BGF<2B 4ABC ?D0L4G=/.ABM89 ;/034R79;BMF/;>.18 gq;CJ4O H9?D;/I>.103K CN4R79;}7j2L4 Ü>27/279; 4-9?D;/I>.103K ;>gY2L.A.9BGF/0BE?DCN279.J0L4@CN7 d 4 d ?DCJI90L8MCJ7 BGF/06;C d 4ABC BE?DCN279.J0s2B[BD23CJ7/4uBGF/0sKCJ7C d BE ?[CJ2L79.10sF<24 Hn03?DCNKR0BE03?!79;M?D0L2BE03?WBGF<2L7 1 b 12

YF>CN4CJ4 ¤¦ H9?D;/I>.103K

2

2502

3£9¢

p1

d ãqg CJ4+79;BP0>=/CN.N23BD0L?[23. 2798 CN4PBGF/0M.J;<7930L4EBP4GC18/0 BGF/0L7}60 d 2L7 d F/;/ ;<4,0 4ABC BE;fI90)GB F/0KMCJ89GHn;>CN7>B;>g bOBC L0B 2798 BD;30B D BC E → C F → B b 2 p2 → 2BC ;<7<03?[> 4,0.Q3: p14G=9H9H9;<4,0 CN4ä0L<=/CA.J2BE03?[2L.3}+CABGF bL0B u = AF 2798 bF/4ABC 0L7 2L798 AB = 1 v = BD w = CE 0 < u v w < 1 Xp b 2 2 p2 = f (u, v, w) =

u + (1 − w) − u(1 − w)

YF/037

∂f ∂u

=

2

0B[BDCN79 ∂f ∂u

p

u2

= 0

v − 2(1 − u) b + p 2 2 2 v + (1 − u) − v(1 − u) − u(1 − w)

2u − (1 − w)

+ (1 −

}60630B

w)2

[2u − (1 − w)]2 · [v 2 + (1 − u)2 − v(1 − u)] = [v − 2(1 − u)]2 · [u2 + (1 − w)2 − u(1 − w)]

}6F>C d F?D08>= d 034BD;

u(u + w − 1)v 2 = (1 − u)(1 − u − v)(1 − w)2

b

YF>CN4 d 27gJ=9?[BGF/03?I90+KO2L7BD;

uv(1 − u)(1 − w)(1 − v − w) = 0

IL:=94CN79 0,>2< 4s?D0.J2BDC1;<7 uvw = (1 − u)(1 − v)(1 − w)

b

/Õb_00@BGF/079;BE0sgq;>.N.J;}+CJ79!BGF/0.NCN4GB;>g/45;>.Q<03?[4!I/03.J;}Ob +0L7 d 0 BGF<23B)CN4 2798 b YF>=94 b ©<: 14[:/− KMvKR−0BEw ?{: =}600 v = 1 − w u = 21 w = 1−v d ;<7 d .A=n890 1 1 1 CJ4sBGF/0;<7<. : d ?DCQBDC d 2L.uHn;>CN7>B;>g b6P;BD06BGF<2B 3b 1 1 1 , 2gJ=9, 7 2d BDCJ= f0O;>g f BGF/20M c .J45; 24PBGF/20 , d 2;<,7 2d =9?D?D0L7>BH9;>CJ7>BP2H/H9?D;<2 d F/0L42O4CJ89 ;<7 2 4ABC f d .1032?[.Q:T2H/H9?D;<2 d F/0L4CAB[4)KM2 LCJK=9K >23.N=n0;>g b 037 d 0 1 gq;
q8

&P#$ '6 : T

4DEF

YF>CN4sH/2?Bä[email protected];6}60.N.79;}7 b

¼

"

Ë

M b¥ ; > b

Ç

8Ë

¢

e bA9b .J2LKRLCJ7 qÏuÌ{ËXEÏ/Ó/ÌJZ{ÆÏ9Ó\ Ó<ÌqÍYËÒOÓ/ÌJZ{×[Ó\ \1ØÒ~VnZ{×É6SSa c_c H b 2L79 8MH/Hb 1H/?D;/I<.J0LKÙ¨sbJãGb £ 2798~CAB[4@4,;>.A=
23 $+- 587 E ><22 A69;!<#A ' 9?2 E, E > 93 *2 A B $ ' A<' '#- $-A ' ' = O P,O *P,O !P E 9# ,8 908.> 9?2 A(> " A ' = 0 O *P,O !P E E 9 A4 ! <!# < '<,E $ <*A ?
3. # 9A ' -? ' $*A82(> , 99?2! ?. 2 # A F#,+ , 7 82,5= ' A -8BC"#A#2> *2 ::. 9 -A #+ ,A 2A 2 '= O *P,O !P, 2 . 2 < < A 2(A '= @ O *P,O !P 9 ;" . $6 ) ;A;>?$ ' F#,+ , 7 A ' ' '#-8' AE2A *2 '-6 .

6+#,6 ) '#- A)% ,+ 66%('7C )<5 ' # C:9?!'# $O P B O DP ! /O DP6=

. >.

uv 2 (1 − u − w) + (1 − u)(1 − u − v)(1 − w)2

=

0

+ (1 − u)(1 − u − v + uv)(1 − w)2 − uv(1 − u)(1 − w)2

=

0

+ (1 − u)2 (1 − v)(1 − w)2 − uv(1 − u)(1 − w)2

=

0

+ uvw(1 − u)(1 − w) − uv(1 − u)(1 − w)2 uv(1 − u)(1 − w) v − (1 − v) + w − (1 − w)

=

0

=

0

=

0

uv 2 (1 − u − w + uw) − u2 v 2 w uv 2 (1 − u)(1 − w) − u2 v 2 w

uv 2 (1 − u)(1 − w) − uv(1 − u)(1 − v)(1 − w)

2uv(1 − u)(1 − w)(v + w − 1)

, , , ,

,

"!$#&%'(%)*+-,.0/ %)2134%65*739839.: ;=<>:?@: )2:A3B;DC7:'>:EF GHIFJLKMHONP QR S6TUPVIWTUXVR Y S\[ J_^DR`XaPVS\RY S\[ IWRbR&cJURIUS X\TUHdedeHS Γ2 ] lZ J \ S NR SJY gf ^DReXaPVS\R JUHh$T&^DΓR1Y Z S\[ J S\[ m IWZ RY2i\RTWJWP QRX Z j S\[kJ_^DR lS\[ m]l h\i\i\HY7RnJ_^ J ]=Z PVYmJ_^DR XBPBZ S\R JWHh$T&^DR Y J Γ1 S\Z [ Γ2IWRZ Y2i\RATWJWZ P QRX j B Γ1 Z doPV[\p_m i$HPBSJ(HqJ_^DR@ Y7R Nrd-RSΓJ 2 Z C Z S\[bJ_^ D J ] S\[ IWRsJ_^DR@Y7R T_HS\[bPVSZ JWR M IY2RTWJWPVHSDY AB P Q ] Z Z Z Z Hq S\[ KnPaJ_^ S\[ IWRY2i\RTWJWP QRX j MC Z MD Γ1 Z Γ2 ] t\IWH QRJ_^ J S\[ IWRT_HS\TujTUXBPVT QZ Z A ] B ] P ]Z

Á

N¾

Ë ¡Â

uÇ

oGb = ÆL\[Ç<ÌJZÆÏTÑØÅnÇ3ÎË,Z LÍ/Ó<Æ]ÉÌDÇ<Õ9ËÏuÌ1] PÆÏ ÝZN\A\JÉ ÆL\A\Ë¡Ð3ZJÓ/Ì{Ë {Ï<É3ÌJZÌDÇ/Ì{Ë] LÆXGÆÏuÌ{Æ] )b

3CJ7 d 0 CN4!;<7+BGF/0?[28>C d 2L.<2¡ CN4ä;>g 2798 b 2 2 }604,0L0sBGF<2B M 4,;fBGF<2B ã1Bgq;>.A.1;}4BGM F<2B A = M B CJ4 d : d Γ.AC 1d F/0L7 Γd 20 M Pb@L· 0M ?D0345H90 d B C 2L=798 M Q 89· 03M 79;D BE0P BGF/0 d 037>BE?DCDQP 0L4s;>g 2798 ∠M P Q = ∠CDQ O O Γ Γ 1 2 1 2 BDCQ<0.Q:ubYF/037BGF=n3FM45;
2798~BGF/0+?D0L4G=/.ABägq;>.A.1;}43b

Å

∠AP M + ∠M P Q + ∠QBA 180◦ − ∠CP A + ∠CDQ + ∠QBA 1 ∠AO1 C + ∠DBQ + ∠QBA 2 ∠AO1 O2 + ∠DBA ∠AO1 O2 + ∠O1 O2 B = 180◦

ooEb = ÆL\Ç/ÌJZ{ÆÏÑØ @ÓÏ>ZË,\ Ë¡ZNÉ G] /ZJÏY×EË¡\A\Ë¡É5] X[ÓÏ×GËb

É

/ÕZÌ{ÆX É×EÆÒOÒfËÏYÌDÔWF/0+gq;>.N.J;}+CJ79PH9?D;/;>g=9450L4ä<0L?{:MK= d F~BGF/042KO02L?D=9KR037>B 2 4CJ7;>.A=CJ?D0 d BE0L8b'ã1BCJ4CN7>BE03?D0L4EBDCJ79 BE; d ;g!=9798CJ?D0 d BE0L8Ý279.J0L43b Ü>;=9?!Hn;>CN7>BD4 2 d CN= ? d .1(Y 0/b X, Y Z) 0 X Y Y Z _ËÒOÓX Ç<Ë¡ÉLÔ

Å %Ê b

2 2 MP · MC = 4,;<> 7M B dA ; d : d= .NCJ<M = 034LBb = M Q · M D n C P Q D

89;<7 d .1034)>=923BE?D0RHn;>CN7>BD4

b)-<;>=9?_8/0L4W?[2LCN4,;<7/4ä8904[:/K40@ BE?[C10J.N28/?D;>CQBE089034 d 0L7>BE?D034 034GBK40@ 8>CJ2BE?DC d 0 d ;B!H/2?[23.N. 03 .J0L43b O1 O2 AC BD b @7M79;BE03?[2 J . 2 O K L 0 G 4 9 = D ? 0 R K ;/8>=/.1; 890. 279.J068/0L4s8/?D;>CQBE034 0B 0 b 0 (d, d )

π

d

ß<7)<03?[BD=R890348>C 0L@ ?[2L7>BE034 d ; d : d .AC d CQB 0L@ 4@;<7MHn0=
d

(P A, P Q) = (P A, P C) + (P C, P Q) = (CA, CD) + (DC, DQ) = (CA, DQ) = (DB, DQ) = (BA, BQ)

.N2 d ; d : d .AC d CQB 0 8Y ; =M @ 8/0L4@Hn;>CN7>BD4

oooGb = ÆL\Ç/ÌJZ{ÆÏ _ËXEÒOÓLÏnØ/b

b

A B P Q Ñ Ø 9 × 0ÓXGÕ = VË×DÍYÌ1] @ Ì[ÌÆ GÊLÆÏ ä Ç<ËX[Z× <0 Ë

Ë

uÏ
Ó3Ð<Õ9ËÑnÇXÐL]

}+CABGF d 0L7>BE?D0 2L798 L0B I90)BGF/0CJ7<03?[4CJ;<7R}+CABGF?D0L4Hn0 d BBD;RBGF/0 d CN? d .10 i Γ [? 28>CN=94 b~YF/037 2L798 bf©<:RBGF/0OM 0 d 27>B i(B) 2L= /2L7930L7>rB=F/0M ;= d 0~BGF<2i(A) B 2= A 798 B 2 b r = MP · MC r = MQ · MD

YF/03?D03gq;CJ= 86}+ bYF>CN4 d CN? d .10 ABCD AC k BD AB = CD d 2L7/79;BH92L454BGF=n3FfBGF/0OKMCJ89GHn;>CN7>B ;>g CABCJ4PBGF/0L?D0gq;BD;2 d CN? d .10PBGF=n3F 2L798 M24@?D0LAB <=/CN?D 08ub A B P Q

23 ,+ - 5870 &2 2 E # 2 A E<' ' !# ' A( ' ,,' A ' 29 2 )2

E< A <$+ $$7 ' ' + A ' ' + A3> < ':' - 4* <#+* <' # E ><22 A 9;!<#A ' 9?2 E, E 1> < &93 2 ;A . >. B , ' A3' -29 2 2 <E 5 7CE )2 993 ;A . >. < $-E<' A' '<- ,-A ' > * ,<3' !#58B ,""- '6 "!# $ 1E 9# ,8 9%. > 9?2 A > , A #1 ( ;9## A& !<- A$ , , < ' ) F#,+ , 7 A ' ' B6<' *< A A 2 3. ! 9A '-'? ' $*A2> $2 5 9 2 A ! 7B6<',!B ( A $<58*!# @A(2! ' <E < #> 9 A( !<- A 4 % ?@ F#,+ , 7 A ' A *2 ;2 9 5

A)!#- A4E< <$0" 2(-+' BC <,, '- 2 = A 4 % ?@A 'A 2 . . A 7 ' ,"5B6BC* A( 4 '-8 9 ;. $ ( A >?, ' F#,+ , 7 A / ' '<-8' A)E23 A 2 17 ! A @ EB6B ! <$$7&E , #* A $ " '<+
b`O $ ! #_%'(%)r* + ,.M!$#&%'(%)*+,.sE #_*r:)`! b: U3'(% 9: A3V) :E + ! : 3 ; 4%#3u+: 9: EV3FE3 r* #L)&3 .; % L:"!>: 4%ED;#;$(5%

& P QR SJLKMHoi$HPBSJWY B S\[ C S\[oJ_^DRnX9HDTUh\YHq J_^DRMi$HPBSJ A Y_h$T&^J_^ J J_^DR Z T_RSJUIWRMHq$J_^DRSPBS\Rp_i$HPBSJs TUPVZ IWTUXVRMHq ](' XaP9RY@HS6J_^DRnXaPVS\R 4ABC

BC

) $5% (V3u%Eg,.Cr%D*% F5*W1( 4%),D*#+; ),- +\*rE.9;0/*?1D%#UA2FE3 *r#L)&3 .;0/3D;$(5% 44SJUIWHD[h$TUR T_HDHIW[PVS JURoYjY_JWR d KnPJ_^ B = − 1 a, 0 S\[ C = 1 a, 0

] KM^DR IWR PVY`J_^DRoXVZ R S\N J_^kHq Y2Z RNrdeR SJ S\[gXVRrJ ^ 2 QR-TUHDZ HIW[PBS JWR Y 2 a BC ]FZ A Z (X, Y ) Z 4 q D E S\[ F IWRnJ_^DRMdP9[Dp&i\HPVSJYHqFY&PV[\RY BC AC S\[ AB IWR Y&i$R TWJWPQRrXj Z ]DZ ] S\[ ] ef ^DR ] J_^DR S ] ]DZ D = (0, 0) ] E = 14 a + 21 X, 21 Y ] Z F = − 41 a + 21 X, 12 Y SPVS\R p&i\HPVSJbTUPVIWTUXVRPVYJ_^DReTUPVIWTUh\deTUPVIWTUXVR-Hq J_^h\Y PJWYTUR SJUIWR doh\YUJ76\R N HS6J_^DRi\R IWi$RS\[PVTUhDX 8 I 6PBY7R TWJUHI Hq 4 q PVY XBY7H KM4DEF ^P9T&^6PBY J_5^DRbXBPBS\R ] 1 HSoJ_^DRMXaPVS\R KM^Z P9T&^-^ YR :h JWEF P9HS ] J_^DRS dxh\Y_=J@^ 2 X QR6TUHDHNIW[PBS Z JWR Y Z Z (f ^DBC RS Y_9 PVS\TUR IWRyI = [PaP>0H; ] q\J_^DRSN PVS\R p&i\HPVSJsZ TUPBIWTUX9R KMR^ Z QR 1 X, 0 N D = N E ] Z Z ] Z 2 N D2 =

<RS\T_R

] XZ

S\[

Y

Y WJ PVY_qaj Z

2 1 X 2

= N E2 =

x2 − y 2 =

1 2 a 4

2 1 a 4

+

1 Y 2

2

= HSQRIY7RrXj q HI Sji\HY&PJWP9HSHq\QRIJWR&c A KM^DHY2RTUHDHIW[PBS JWR YsY JWPVY_qajMJ_^PVY R>:h WJ PVHS @?D+ R_c] T_RiJ Z HI KM^DR IWRnJ_^DRnJUIP S\NX9RZ PBY`[DRNZ RS\R I JUR A = B A = C Z ] PVYnJ_^DR TUR SJUIWRgHq J_^DR0SPVS\R p&i\HPVSJTUPVIWTUXVR Y&ZPBS\T_R-PJnPBY6R:hDPV[PBY_Z J SJ ] N = 12 X, 0 f ^DR IWRrq HIWR J_^DR X9HDTUh\YeHqmJ_^DRgQR] IJWR&c q9IWHd S\[ PBYeNPQRSA6 j J_Z ^DR D E F A ] m ] Z ] >R :h JWPVHS 6\HQR KM^P9T&^n[DR Y2T_IWP6\RY IWRTWJ S\NhDX I ^jDi\R IB6\HX KnPaJ_^bd HI c PVY BC Z ZC Z @ ?D+ Z KnPaJ_^OZ J_^DRni$] HPBSJWY S\[ IWRZ d-HQRZ [ Z BZ C

0Á ¼ ¢Â

2Â

ooEb = ÆL\Ç/ÌJZ{ÆÏÑØ Z LÍÆnÇL] nÆL\ 0 ÆÒOÒÇÏ>ZÌ1Ø ÆL\A\Ë¡Ð<Ë¡] ZJÏuÌ{ËX Ó>ÊLËÏ<] 3 ] = UPb

©<:fBGF/0M4,;>.A=BBE;H/?D;/I<.J0LK >¢ > Y£ ¢ BGF/0 7CJ7>B d 037>BE?D0.NCJ0L4f;<7 CNg+2L798 ;<7<.Q:âCNg b ;<7/4G C18/0L? BC |B − C| = π/2 2L798 }+CABGF bW3=9H9H9;<4,0BGF<23B C = Bb+ A = (x, y) L0π/2 B 2L798 B = (−1, 0) Cbs= F/(1, 0L7 0) x>1 P = (x, 0) Q = (2x − 1, 0) B = ∠CAP = ∠P AQ BGF>=94 4GCJ7 d 0 b!037 d 0 y x − 1 BGF<23BäCN4 A + 2B = π/2 AB ⊥ AQ = x+1

x2 − y 2 = 1

y

b

0@30B9BGF/0@;BGF/0L?I/?[2L7 d F ;>gLBGF/0@F:.J2ä}6F/037 b (xBGF/< {ã 7 d ;<7 d .A=94CJ;<7 BGF/0.J; d =94ä;>g CN4 0?D−1) 0 d BD2L79=/.J2L?_F:.J2 d 037>BE?D0B8= 23BYC+π/2 BGF/0KMCJ89 }+CQBGF)BGF/0P<03?[BDA Hn;>CN7>B;>g C d 034 0G d .N=n8/08ub BC B C

23 $+-587 E ><22 A 9;!<#A ' E 9# ,8 9 8.C> 9?2 A > , A E ' E ;95 A ! <!# < '<,E , #*A ?
3. # 9A ' -? ' ,*A 2> $2 5 9(2 A F ! 87 B6' "! B A- $<58*!# @A2! ' . 8. A7 ' $ 5B6BC* A 4 ' -8 E 829# 5 E 2=A )"" + + + !# ,"@ <$+ $$7 A ' #-5?! A ;B6'7 9 ;. $6 ) ;A)>?, ' F#,+ , 7 A ' '#-8' AE23A *2 #7 )2 A !#- 4 A 6E , '#" , !<,*A' , A (5 6 75 )2(9A>;!#<' A 9;B6' < ' '- . > '<-$7 B6'@<- <'<4'&' :9?!# < <5,B A6% <5 ' 1 7 =5 , '<' $ !# 6 0 'B ' B '#5!<:' $ " ,5- B ,, ,,&= $ - ' BC" )'&B ,, ,, ' # A % 6B6' BC ' 6 BC '4 ' ! ' ?7 =5 ': ' # AA B + C = π/2 . A

/* 1 *r# V:E +) 44 S 4ABC

k

|B − C| = π/2

!$#&%'(%)*+

,. C 5 8g**E\A; : ,D%8e8g* V ;

1 *

KMR^ Q R 4 c = a4 + b 4 Z l 9 Z ; ^DHK J_^ Z J 4ABC PBY Z TUhJWR Z S\NX9R [ 9 6 ; RJURIUdPVS\RJ_^DRI Z S\NRMHq ∠ACB 9 T ; ?
5% V34%Eg, . D13a ' # V3V)& ; 3B)2) % #3 5 % ( 1>*#UEe( 5 9:(u* % 4* +*;Cr%' a39E;=;$ 5 RbUT X PBd J_^ J l h\i\i\HY7RJ_^ J IWR X n n n KnPJ_^ n>29n Z ; Z Z 4ABC PVY TUhJUR p S\NX9R [ Z S\[6cJ_^ =J a + b Z Z Z Z 2 2 − 2n π ]

cos−1

4q

!$#&%D% l RrJ

n > m > 0]

J_^DRS

≤ C <

2

1

2

1

(1 + λn ) n < (1 + λm ) m 1

F (λ, n) = (1 + λn ) n

(f ^DRS

q HI

Z

∂F λn ln(λn ) − (1 + λn ) ln(1 + λn ) = 1 ∂n n2 (1 + λn )1− n

XaX

λ>0

P;}

450B

CN4s890 d ?D0L2L4CN79;<7

b_F/0L7

2798~CJ7 d ?D0324GCJ79+;<7

f (t) t ln (0, e−1 ) }+CABG= FR2~ KMCNt7<=9KM=9K´f23(t) B −1 b6ãqg −1 BGF/037 (e f (t) (1~+ 2798CN,g ∞) }60RF<2<0 t = e 2L79t8 ≥ e NC KO< H<.Q:>fCJ79 BGF<2t)B 0 < t < e−1 f (t) < 0 f (1 + t) > 0 −1

b!037 d 0 ∂F bYF>=94 CJ4@8/0 d ?D0324GCJ79P242gJ=97 d BDC1;<7 223CJ7 F f (t) < f (1 + t) <0 ;>g b_F/04EBD2BE03KR037>BäCJ7~BGF/0.103KOKM∂n 2679;}gq;>.N.J;}4Lb n

Ü>?D;
}6064,0L0BGF<2B

CN4@BGF/06.N2?D3034GB@2L79.10/b0B

b

cn = an + b n C λ = b/a 1 1 YF/037 IL:~BGF/0+.103KOKM2nb+0L7 d 0 1 n n n n n 2 2 c = (a + b ) = a (1 + λ ) < a(1 + λ ) 2798BGF>=94 CN4 2 2 2 2 bF/0L?D0gq;
©<:~BGF/032}

;>g ;<4GCJ79034 6 } 0F<2<0

cos C =

2

a2 + b 2 − c 2

a2 + b2 − (an + bn ) n

=

2ab

2ab 2 1 + λ2 − (1 + λn ) n

=

2λ

¥@039790

2

1 + λ2 − (1 + λn ) n b 2λ c H9H<.Q:>CJ79sBGF/0.J0LKMKO2A}+CQBGF }60P4500sBGF<23B gq;0 λ>0 gq;2 lim g(λ, n) − g(1, n) ≤ 0 λ>0 g(λ, n) =

E"6$

ÜCN?[4GBä79;BE0PBGF<23B

p

n→∞

g(1, n) =

b b 0<λ<1 YF/037

E"6

9b

b

λ>1

YF/037

1

9b b YF/λ 037 = 1

b+0L7 d 0

n→∞

lim g(λ, n) − g(1, n)

lim (1 + λn ) n = λ

n

F/037 d 0

lim g(1, n) = n→∞

lim g(λ, n) =

n→∞

n→∞

E"6 p

F/0L7 d 0

n→∞

p

YF>=94

1

lim (1 + λn ) n = 1

YF>=94

gq;
2

2 − 2n 2

=

n→∞

lim g(λ, n) − g(1, n)

n→∞

b lim g(λ, n) − g(1, n) = 0

=

(1 + λ2 ) − 1 λb = 2λ 2

b

λ−1 < 0 2

lim g(λ, n) =

1b 2

(1 + λ2 ) − λ2 1 b = 2λ 2λ

1−λ < 0 2λ

b

n→∞

YF>CN4 d ;gu;>gY;>=9? d .N2LCNK#2I9;3<09b P;} }60)}+CN.A._4,F/;}BGF<2B gq;;

=94CN79PBGF/0.103KOKM2nbF/0L7 g(λ, m) − g(1, m) ≤

lim g(λ, n) − g(1, n)

n→∞

≤ 0

gq;=94 b g(λ, m) ≤ g(1, m) m bWã1Bgq;>.A.1;}4nBGF<23B9BGF/0?[27930@;>g ÜCN7/2L.A.Q: 79;BD0äBGF<23B lim g(λ, n) = 0 g(λ, n) λ→∞ 2 24@2+gJ=97 d BDC1;<7O;>g CN4 2 − 2 n bYF>=94 λ 0, g(1, n) = 0, 2

cos

−1

2

2 − 2n 2

≤ C <

π 2

b

23 $+- 587 E C><22 A)9;!<#A ' ' = 'O (P)'-FO *P E 9# ,8 9 8.3> 9?2 A> " A ('= =O P'-O *P'- ' '= ' 3( "!# $ " '=@O )3P )4E 2 9 93 )> ;93 8A )' < F#,+ , 7 -?!#"'# $ E <' #
;9## A !<- A $:" $# ' F#,+ , 7 A ' ' B6'<*< A2 A *2 9# E (2(9# . A 9'< ' $'-A)E2A 2 ' = )O P '-&O *P $ 2 92 A ! 7B6<',!B A $ <58*!# @A2 ! $ ' > 99?2! (. 2 # A F#),+ , 7 )2,5= ' A -8BC, 2 E 2! ; E A >? 9' ?,-8A A 2 -+ $ 2 2 A A E # <' 2 :. 9A #+ A( 2A *2 ' = =O (P '-KO *P (2 9 5 A)!#- A4E< <$0" 2(-+' + BC #"$' (- 2 = A 4 % ? @A) ' A 2 ' = O (P ' -/O *P 3." 2 5 2 A ' ,# @ A

(' = @O P;'-=O *P;'- ' ' = ' ? "!# $C( " ' =O 3P ) ) .1$2 A @8' A A *2 '= @O P;'-0O *P 1 9; .1 $ ( A;>?, ' F#,+ , 7 A '> '<-8' AE2A *2 ,7 : A1 @ EB6B ( !#, 76E $ <*A!$ ) , '<+
"!$#&%'(%)*+-,.0/ %)2134%65*739839.: ;=<>:?@: )2:A3B;DC7:'>:EF

&

P QR S IWR TWJ S\NX9R KnPaJ_^ IWR X9RJ S\[ 6DR-i\HPVSJYMHS Y&PV[\RY E Z KMF^DR IWR S [ \ IWR Z Y&i$R TWJWPABCD QRrXj Y&h$T&^-J_^ Z J Z S ] [DR S\HJWR Y AB Z [P QR] ] Z [CEF ] = 31 S ] J_^DR IWR HAD q ] QR Z t\IWZ H QR4P J_^ J π Z ∠ECF ≤ 6 5% V34%E , . 5 * A* s#L@) V:ED: +r3 ;7 E3 *#)_3 9.0% 5: #W:&*, %; 5 :#:_*, %;$ %)7E39::E + M*r# %* +*% 3VE\:b:E " + #r3 1> % ; !$% 9 A %8e8 E3 9. % 4* +*; b3VE4*r#8b:( *ED;0 #;8 5 RrJ

a = AB ] b = BC ] α = ∠BCE ] θ = ∠ECF Z

S\[

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

D.................................................................................................................a C ................................................................................................................................. ... ... .............. .... ... ... . ........ .. .. β ... ............ ............ ..... . . . . . . . . . .... . . .... ... ............ θ............... ........ ... ............ . ............ ... ... ................ ... ............ ... ... ......... ............ ... ... .... ............ ... ............ .... ... ............ ... ... . . . . . . . . . . . .... . . . ... . ....... . . . . . . . . . . . . . ... .... . . . ....... . . . . . . . ... . . . . . . .... . . ........ .. . ... . . . . . . . . . . . ... . . . .... .. . . ........................ ... . ... .. ................ . .... . . ............ ... . . . ... . ............ ... . . . . . . . . . . .... . . . ............ ... ............ . ..... . .... . ............ . . ............. . . . .. ...... . ...................................................................................................................................................................................................................................................................

α

F

b

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

A

E

B

β = ∠F CD

YF/037

cos(α + β) = cos( π2 − θ) = sin θ ab = S

037 d 0

sin θb = π

2L798)}60F<2<0

= 3[CEF ] = 3 21 · CE · CF · sin θ 3 b a = · · · sin θ 2 cos α cos β 3ba sin θ 3ba sin θ b = = cos(α + β) + cos(α − β) sin θ + cos(α − β) 1 2

cos(α − β) ≤

1 2

bF/0L?D0gq;
θ≤

π 6

b+ß<<=923.NCQB:F/;>.J8/4}6F/0L7

α=β= 6 23 & ,+ - 587 2 2 E ! 2 A E<' ' !# ' A ' $"' A ' 4 E >< 2 2 A 9;!# 93 2 ;A/ . >. B $ ' A' '<- ,-A ' E # 9 ,8 9 . > 9?2 A&> , A CE E 9 A ! #!< # '#" E $ <*( A ? #A ;A 2 9# E )29# . A 9' <' $-ACE23A *2 $2 5 9 2 A ! 7B6<',!B > A $<58*!# @A 2 ! $ ' 9 2 . 82 2 A ' BC- ;99?2 ?. *2 A <$+ $$7 23$5= ' A -8BC"#A 2> ,2! 9# 9A F#,+ ,-8'#-E2 E A >;!# 2 A 23 < ' 2 9 5 AC !<- A E< F#, " 2-+*'BC #"$ ' -123= A 4 % @8A 'A 2 0 . 8. 5 A7 ' ,"5B6BC* A 4 '-8 E 82# 9 E /A )"" + + + !# ,"@ F#,+ , 7 A ' #-5?! A ;B6' ?7 >" 2(9 9 ;1 > A ' '- A ' 1 . 9 ;" . $6 ) ;A >?$ ' F#,+ , 7 A '' '#-8' A;E2A *2 ,7 (2 ;A !<- 1A E , '#" , !<,*!A " A '-6 . $ < B6BC"- <'<; <5$B A') '<,-A -<)< <,"!#- "' % E ' - F , ' ' 4 -&" BC '. 4 <--&, + '<,& AB $5AD ,&,'( . 2 )< $+ ,"!#-AB AD) B ? $"$ 7 ' ! BC) <'< ' $4& BC 'A( ,+ * 7. E

F

AB

AD

O!$#&%'(%)*+-,.0/ %)2134%65*739839.: ;=<>:?@: )2:A3B;DC7:r'>: E

& P QR SoJUIWP S\NXVR ABC KnPaJ_^oPBS\T_RSJUIWR I TUPBIWTUh\d-TUR SJUIWR O S\[0T_RSJUIWHP9[ G ] ]$Z ] t\IWHQRbJ_^ J Y&h\iDi$HY2RbJ_^ J Z Z ∠AIO = 90◦ Z IGkBC 5% V34%E ," . 1 *'>#&%'(%)*r# [\RS\HJUR J_^DR TUPBIWTUh\d-TUPBIWTUX9R Hq RrJ j 6DRkJ_^DR Y2RTUHS\[ 4ABC Γ KnPJ_E^ 6DR PVSJWR IY2RTWJWPVHSnHq S\[`X9RJ Γ ]rZ D RrJ J_^DR-PVSJWR IY2RTWJWPVHAI S Hq S\[ l Z PBS\T_R BCPVYMJ_^DR d-R RrJ J AE AG BC M Z T_RSJUIWHP9[ S\[ PVY d-R [P G S Z AM Z Z ] AG : GM = 2 ^ : 1QR RM S\[ ∠ABI = ∠DBI Z Z

R

6

∠BAE

f ^DRS

= ∠BAI = ∠IAC = ∠EAC = ∠EBC = ∠EBD

A

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

Γ

I

·

B

·

O G

DM E

C

S\[e^DR S\TUR

∠BIE = + ∠ABI = ∠EBD ∠DBI ]Z Y&PBdoPaX I`JUH l PBS\∠BAI T_R PJbq HXB+ XVHK YnJ_^ J = ∠EBIPBYM BE = EI ∠BAE = ∠EBD ] 4ABE Z Z S\[6J_^DRIWRq HIWR 4BDE ]Z AB : BD = AE : BE = AE : EI

02?D0TC <0L7'BGF<2B BGF<23B)CN4 b YF>CN4CJKMH/.AC1034 = 90◦ d0 BGF<23B 2L798F/037 ∠AIO b OI F/0L7 ⊥ AE b AE AB : BD =bá2YF>: =914 3CJ7 d 0 AI = IE }60 30B : EI = 2 : 1 F>C d FCNKOH=94 = 2 : 1 b IG k BC AI : ID = AG : GM IG k DM

23 & ,+ - 587 2 2 E ! 2 A E<' ' !# ' A ' $"' A ' 4 E ><22 A 9;!# 93 2 ;A/ . >. B $ ' A' '<- ,-A ' E 9# ,8 9 . > 9?2 A&> , A CE E 9 A ! #!< # '#" E $ <*A ? 2 9 5 A4 !<- A E< <$ " 2(-+' BC& <,, '- 23= A 4 %8?@A 'A *2 . 8. A 7 ' $ 5B6BC* A 4 '-8 E #2(9# E /A )"" + + + !# ,"@ F#,+ , 7 A ' #-5?! A

;B6'7 9 ;. $ ( A)>?$ ' <$+ $$7 A ' '#-8' A(E2A 2 7 : A @EB6B ! <$$7 E $ <*! A $ , '#+#A A 2 ' - 75 )29A;>;!< ' A 9;B6'# '. < + ' ,+ 3<58+- '# + ' $*!#: =? A . ◦ #+ ( # . # %(' ' IG + %&kMBC

< M B 2a <=. (b<+ c 2 (aWAFb) (c ,-:+ ,4 & W<. ) "AF% <'<@ . 2 BC ! ' (E#+*':" ' B6): <'< A "4 ' , % A< * '< ,< < ' $+: 5 W ' -6 " ,#%) "& = 2a = b + c (W . 3. + . A H $ ( .! %D 3 A C-$ $<1 A <' A . <.

(W . 3. + . A H $ ( .!"3 (% $ 1A <' A . A #.

(W . <' ! #"A ! *+ (%; .!%&D#$ %!+ %&D#$ A -,, ' A =A . 8.

'&O(D "!$#&%'(%)*+-,. 3V1 :, 9." *E. * ;

#W:)r%*r;!=%8e: E3V:\

44S [DR S\HJWRJ_^DROY_P9[DR Y6 j J_^DROYjdedeR[P SDY Z t\IWH QRJ_^ J a ] b ] c ] S\[6J_^DRTU4ABC PVIWTUh\doI ] [PBh\Y 6 j R Z Z Z bc sa

5% V34%E ,. 8 3u+3 <>: 9: 8e: %D%; *) ;(5%

+

ca sb

+

ab sc

≤ 6R

Rsi\IWHQR d-HIWRmNRS\R I X PBS\R>:h XaPaJ4j Z Z Z +-, .-/ RrJ ABC 6\R JUIP S\NX9R S\[eX9RJ 6DR IB6PJUI I4ji$Z HPBSJWZ Y@HS6J_Z ^DRnXaPVS\R M] N] P] Z Z Y7R Nrd-RSJWY IWR Y&i$R TWJWPQRrXj BC ] CA ] BA ] f ^DRS !$#&%D%10

s a ] sb ] sc ]

3W13 +: E

E3

*#)_3 9 . ;

A

≤ 6R

j

#&+r3V; ) -+\*E9; 6* ),4*r#UE

bc ca ab + + AM BN CP

6

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

β

γ

P

N

α

B

M

C

RJ S\[6X9RJ 6DRJ_^DR WI R Hq B ] β = ∠BN A ] γ = ∠AP C ]Z ∆ Z Z αR= ^ Q∠AM R 4ABC Z

1 abc a · AM sin α = ∆ = 2 4R

]

£

037 d 0 b3CNKMCA.J2L?D. : 2798 ab b bc ca = 2R sin α = 2R sin β = 2R sin γ AM BN CP YF>=94 b bc ca ab + + = 2R(sin α + sin β + sin γ) ≤ 6R AM BN CP 23 ,+ - 587* - 29 2 )2

E< A <$+ $$7 ' ' + A ' ' + A&> < ' ' * ##+ ' E &><22 A 9;!<#A ' 9?2 E, <E > 93 2 ;A . >. B ,

' A <' '#- $-A ' E 49# -8 9 . > 9?2 A E# ""#A > , A $2 5 9 2 A ! 7B6<',!B A <58*!<"@8A 2 ! $ ' 8 23A '#%(' '@ A0*' ' 2 9 A !<- A;E< F#,>" 2(-+' BC; <,,' - 23= A 4 % ?@A 'A * 2 # 2 . 2 < < A2 <A ( ,% "!# $ ) 5 "7 : A# @CEB6B ! <$$7 E $ <*A!$ , '#+#A A 2 ' -6 . 23 3< $+ #+ -C <' '- BC4<,- '#$ %(' <3' 7&,C ' @ 7B6BC- '< . >'# ' ,<,-6< '<; <'# !' 7 $+)' $ <3 ! ,= max

$--A$

,,' # B

A

A

AM BN CP

bc , ca , ab sa sb sc

'#5#+ <; '# max

%

bc , ca , ab AM BN CP

' ' ?7 +* '

≤ 2R

4ABC

.

≤ 2R

,

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