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!#"$&%!')(*',+-/.' 02143658793:7,36;=A@CB=DE7,@C1FBAGIH4@A;27EGJ@?G*K23:L=DI@2MC1...

Author: Canadian Mathematical Society

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4

(4 + 1)(4 + 1)(8 + 1) = 225

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2004

2

×3

× 167

= 2004

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√

2

cot β −

√

2− 4

√

6

b

3KA5A3 e 561 e B21S56GI38< ◦ 5;=< ◦ M>!;=@GIHQ;=G*KA5G ◦ ◦ ◦ = >C 60 58;=<5L=LA1>CHQ;=oG*K23P58<=
=

=

√ √ √ 4 2+ 2− 6 √ √ 5 2− 6 = √ √ 4 √ √ 6+ 2 6+ 2 4 √ √ √ √ √ √ (5 2 − 6)( 6 − 2) 12 3 − 16 = = 3 3−4 4 4

b

√ K236DI36g]@ADI3 ~ KCH e KTHS758L2L=DI@[ßHQN56GI361 > −1 ◦ QHS;27*GI358<@Cg β = cot (3 3 − 4) b c e 58;M=3!= S1 1pG*K23TDI3N5HS;AHQ;=P58;=[1Q37 39.896 < 3[GJ36DJNHQ;=3< ◦ 5877,K2@8~P;OHQ;G*K23!
8

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5[A3E>A@CB:3,A3DGJDIHQ38A@CB:5;=< HQN KA5[A3 × 1SHQ8361 >OKA58<7*HQNHS1S58Dv3*ßL=3DH436; e 367 b DI3561S1 > 36; |@>G*K23Ý3;CGI3D¬GI56HQ;AHS;= L2DI@2MA1Q3Nx7JG>C143Þg4DI@AN G*K2367,3Ý e @GGIHQ79K § 56G*K236× NO5GIH e 51o¡¨KA561S1Q3;=6367 b KCHS7:L258D¬GIH e B21S58D!L=DI@2MC1436N KA57M23836; )GJD5;27)1S56GI38< 5 MAHGg4DI@AN HGI7@ADIHQ[HQ;2561g]@ADINO5G ~HFG*KUL=@CB=;=<27DI3LA1Q5 e 3<ÛM>W<2@C1S1S58D7!58;=< ®MCB=7 DI3LA1Q5 e HQ;= e @A5 e K b KCHS7¨L=DI@2MC1436N¢HQ7u7)HSNHF1Q5D2GI@oL2DI@2MA1Q3N7uG*KA5G2~3v51F16795[~ ~K23;#~3Å=D7*G7JGI5DGI38< 143658DI;AHQ;=51Q638MAD5 MCBAG¨HGGIB=DJ;27@CBAGGI@M=3#5g45HSD/MAHG¨N:@ADI3 e @ANL21FH e 56GI38

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DI@AN G*K23[H A3;HQ;Ag]@ADINO5GIH4@A; ~3KA5[A3

«8G*KA5GÅHQ7 T T +3= N N −2

T (N − 2) + 3N (N − 2) 3N 2 − 6N

= TN b = 2T

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T = 6n − 6n

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6n2 − 6n 3n2 − 3n T = = N +4 2n + 4 n+2

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= 3n − 9 +

n+2

K23DI3P<2@P~3P6@Tg4DI@ANÜK236DI3 G*KA56G

3n − 9 +

18 n+2

n+2

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3n − 9

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(7a + 3) + (7b + 5) = 7(a + b + 1) + 1

6 HQNHS1S58DG*KCHQ;=7ÅKA5L=L=3;E~K23;i>A@CB1Q@2@2o56G/7)BpMGJD5 e GIHQ@A;5;=A@CB=D7,31Sg b § 56G*K236NO5GIH e HS58;27 M236HS;=3)ß6GJDI3NO 361 >1S56>OM=36587JGI7 5DI351 ~P5>A714@2@2HQ;=Eg]@AD5 7,K2@AD¬G~P5[>@Cg¨~PDHFGIHS;=G* KCHQ;=7 b @79K2@8~ G*KA56GG*K23;AB=NO M=36D7 5;=< 5DI3THQ;:G*K23 33 5 795N:3P38CB2HC561436; e 3 e 1S587)7 G*K23,>~PDIHGJ3 b 33 ≡ 5 (mod 7)

b KA5GG*KCHS7NO35;27HS7iG*KA5G 3 DI358<G*KCHS757 HQ7 e @A;=DBp3;CGGJ@ N:@2 HSgp~3T ~363GÅG*K2537)58NO3DI36NO 7 @C g b 33 58;=< 5PNB251FGIHSL21Q3 5 7 3KA5[A3M23836;iHS;A37JGIH45GIHQ;=7,@ANO3@Cg6G*K23MA587*H e L=DI@AL=3D¬GIH4367@Cg e @A;=DIBp36; e 367 b 3 e 58;T~PDHFGI3EG*K236N <2@8~P;587g]@C1F14@8~P7 b

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l y G*K23;

a + c ≡ b + d (mod m) a − c ≡ b − d (mod m) a × c ≡ b × d (mod m) ak ≡ bk (mod m)

¶

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45 + 20 ≡ 3 + 5 ≡ 8 ≡ 1 (mod 7)

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23 495 × 103 621 = 2 433 505 395

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b

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β

γ

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β

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=

a

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sin γ

=

sin ∠B 0 LM

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(B 0 C 0 )2

= (LB 0 )2 + (LC 0 )2 − 2(LB 0 )(LC 0 ) cos(β + γ) = λ2 (b2 + c2 + 2bc cos α) = 4λ2 21 (b2 + c2 ) − 41 a2

= 4λ2 m2a

«

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2

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P (x)P (x + 1) = P (x )

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b

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C

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A

h(t − k)

E

D

B

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k(h2 − kt) hk(k − t) , 2 h2 − k 2 h − k2

b

G58L2Lp3658D7G*KA5GÅ~3NB=7*G3)ß e 1FBp<=3G*K23 e 57,3E~K236; «CG*KA5GHS7 ~ K23; 5ID 351F1 h@Cg =G*K2±k b e 8 @ ~ , 3 A 3 D S H ; * G C K Q H 7 5 , 7 3 * G 2 K E 3 * G A K I D 8 3 P 3 S 1 S H = ; 3 7 3 2 L I D 2 @ A M Q 1 3 N ◦ ∠ACB Lp36DJL=3;= b

×

BC

Ñ b 3[G M23G*K23E793G@Cgp561S12MAHS;258D|>T7,3ABp36; e 367@Cg21436;=®G*K 5;= ¨3DHS;3)ß5 e GI1> GI3DINO7ÅG*K23;!G*K23 b k e 367 = 0 5;=< 7,3ABpf 36; (0) ¨3D/561Q79@Pa3)ß5 e GI1> HS; GI3DINO7 b 0=DI@6A3 G*KA56G/HFg 5;=< k G*K23; a b e 37g4DI@AN f (b)7*B e KEG*KA5G b 58DI37938CBp3f; (a) c A a+b+c = 0 f (a) + f (b) + f (c) = 0 (# !!0 @ ! ) "* 3)+ ) 0."> (!" ,$<

H½ ´ » #¼ ¹

K23793G HQ75iA3 e GJ@ADÅ7)L=5 e 3@A; fFG*K23P7*B=NHS7vG*K23P@A;=3<=3p;=3<HS;!G*K23 /2 7*G56GI3NO3;CG 58;=A <1Q3G 5;=< ~HFG*KO
n P

i=1

b/ K236;

HS77*HQNL21 >G*K23;AB=NOM=36D@Cg=GI3DINO7

á d(a, b) = |ai − bi | d(a, b) G*KA56GP ¨3DiHS;RG*K23:7938i=1 CBp3; e 37 58;=< b DI@ANG*K23:7JGI5GJ36N:36;CGP@CgG*K23:L=DI@2MC1436N b GÅg]@Ca1S1Q@8~P7vGJbDHCHQ561S1 >TG*KA5G HS7HQ; |3 e GIHA3 b d f (a), f (b) = d(a, b) × b f 58;=< ~3?<238

HS75TLp36DJNBAGI5GIH4@A;@Cg

58;=<

G*K23DI3g]@ADI3

HQ751S7,@5

B (fD 1 , .b . . , fn ) (e1 , . . . , en ) M2527)HS= 7g]@A A b: K23 ; b 3G ~HFG*K á a = (a , a , . . . , a ) d(0, a) = k d 0, f (a) = k 1 2 n B 1 b 3G á i ∈ {1 . . . n} b KCB=7 g G*K23; «6G*KA56GHQ7 G*K23 d fi , f (a) = k − 1 • × ai = 1 d(ei , a) = k − 1 F e @2@ADI
n n n P P P bP KCB=7 Gg]@C1F14@8~P7#G*KA56G FH g G*K23; a= a i ei f (a) = a i fi = ai f (ei ) i=1 i=1 i=1 HS71SHS;=35D b f K236; HFg ~3PKA5A3 a + b + c = 0 b

×

f (a) + f (b) + f (c) = f (a + b + c) = f (0) = 0

£i3*ßGi~3:GIB=DI; GI@ 7,@C1FBAGIH4@A;27g4DI@AN @CB=DPDI358<=36D7GJ@ 2 L DI@2MA1Q3N7@CgvG*K23 8 b Ù361S58DB=7)HS58; § 5G*K23N56GIH e 561 1>ANOLAHQ58G*K23g]@C1F14@8~HS;=58NO3 bc GoG*K23:M238[HS;2;AHQ;= 0A3GI3 e K2@2@A7,367 5;AB=NOM=36D G*K236;fÙ=HF1S1 e K2@2@A7,367P5:;AB=N:M23D 58;=<G*K23;Û0A3GI3 e K2 @2@A79375;AB=NO a b M=36D b ¡58; 0A3GI3 e K2@2@A7,3TKCHQ7o;AB=NOM=36D7oHS;7*B e K:5P~P5[>G*KA5GG*K23PG*KADI333AB=5GIH4@A;27 c 58;=< 3 3 2 3 2 x + +N: bx x + cx2 + ax + b = 0 KA5A3Pax 5 e @AN @A; + c = 0 x + bx + cx + a = 0

45 4M

DI3651pDI@2@G

;=356GIH A3PDI@2@G

MH·½ }pµ´ L»¸ A½ ¼ /¹

´ ½ ¹ À A½ %·

(# ( !!0 @ ! ) - "+ ) ) ."*> 3 ! "( 6 " () "!"* $#$" ( 3) .$ (+ -"* !0( ! ( /" ,# # "!0# 7(< 1 < "( -! #."(! +! $ ! ) - "+! (8;:=< + " ) ) < e K2@A793;RM> 45 367 b 0A3[GJ37*HQNL21 > e K2@2@A7937 58;=<G*K236; g]@ADo5;> a3= b b¨ K23G*KADI383 Ù=HS1F1 0A3[GJ 3 e K2@2@A7937 AB=5−1 GIH4@A ;27M23 e @AN:3 3 2 c = −b 5;=< 3 ~KCH e xK e −x 1Q35DI1 +bx−b >KA5[A3 = 0 3 2 2 x +e bx − bx − 1b = 0 x − bx − x + b = 0 x=1 5875 @ANN:@A;DI@2@G 58;=<7*B=L=L=@A7,3 HS7i5;=356GIH A3!DI@2@G ]M £o@ b 6B=L=L=@A7,3Ù=HF1S1 e K2@2@A7,367 GI@51F12G*KADI383P38CB=56GIHQ@A;27 b K236b; =HS;0L= 5DGIH e B21Q5D ~3PKAr5[A3 e @ANN:@A;! r 3 + cr + a = 0 b 58;=< 3 2 r + cr + ar = 0 b DI@ANÐ ~363G 2 @AD 6 p B 2 M J 7 I G F H I G A B I G S H = ; Q H C ; I G @ 2 * ~3EG*K236;: 63[G 3 r 2+ cr + ~KCaH e = K>C0H43 14

n o ~K23DI3 Ó b @8~ NO5;>UL25HSD7 795GIHQ7*gF> HS7T5Lp@A7*HFGIH A3 n! (n, q) {q 2 } = n 2000 HQ;CGI38636DP5;=< HS7T5;=@A;=JHQ;CGI38636DD56GIHQ@A;251v;AB=NOM=36DP7*B e KRG*KA5G #( ! ) q + , N:3658;27vG*K23 9g4D5 e GIH4@A;2561L258D¬G i@Cg b 0 < q < 2000

9¶1¶ H½(# f ¹¶"#(!"* ,( # $#$ "%5(· 3 ´ )4, +.$!(! 2#( F ·( »@*! < "!£ )#µ¼! £ ¸ !¹ ("+ ) < ) ."*> " !0d !»@ "(! 6 )# ¼ " ,)=# ) ´"3 ½ ( )

{r}

r

2 K 36DI3P58DI3 7)B e KL25HSD7 b 2400 M=35O L=@A7)HGIHA3!HS;CGJ363Do5;=< 5;=@A;=*HS;CGJ363DoD5GIH4@A;2561;AB=N:M23Do7)B e K 3G n b 3G G*KA56G á 58;=< M=3L=@Aq7)HGIHA3HQ;CGI38636D7o7*B e KG*KA56G 5;=< 0 < q < b 2000 a 47*HQ; e b3 K23; á HQ7;=@GÅ5;!HQ;CGI38636D b 3E7)5>G*KAq56G = a/b HS75 gcd(a, b) = 1 b>1o q (n, q) n # ,"* ! HSg b K23TL=DI@2MC1436N HQ7GJ@p;=<K2@8~ÝN58;>:L25HSD7 n! 2 {q } = (n, q) 2000 58DI3P6@2@2<L25HSD7 b b K23; @AD5;>!DI36512;AB=NOM=36D 1Q3G FG*K23iHS;CGJ363DL=5DG@Cg = x − {x} x HQ756@2@2<L25HSDÅHFgu5;=!HFbxc g

(n, q)

a2

G*KA56GÅHS7

b2

−

a2 b2

=

n! 2000

−

n! 2000

«

n! b 2000a2 = b2 2000bq 2 c + n! − 2000 2000

dP

gG*KCHS7 e @A;=

a

2

n!˜ b2 n! b 2 2 2 ˜ ˜ = 4b bq c + − 4b 500 2000

Gg]@C1F14@8~P7#G*KA56G n!˜b2 HQ7i5;?@2<=<HS;CGJ363D bP KCHQ7 e 5;2;=@GiM23TGJDBp3!HSg HS7E «8G*K236DI36g]@ADI3 n≤3 8 KCB=7 ~3NB=7*GKA5A3 b500 K236; HS7;=@G b DI@AN ~3 * b=5 a 5 KA5A3 n! n! b 2 2

a

= 25bq c +

80

− 25

2000

b Gvg]@C1F14@8~P7ÅG*KA5G n! HS758;HS;CGJ363Dv;=@Gv «G*KA56GÅHQ7 × n ∈ {6 7 8 9} 5 80 3EKA5[A3iDI3 b§ @ADI538@6A3D ;=@GJ3iG*KA56GÅg]@ADÅ58;>Lp@A7*HFGIH A3EHQ;CGI38636D 5 a (

25 ± a 5

2 )

( ) a2 a 2 b = 25 ± 2a + = 25 5

K236DI36g]@ADI3 HQ7

HSg

25 − a 5 L=56HQD7 a n,

n,

C 5 r

HSg

5

a 5

n,

HS7i56@2@2<L25HSD

b P36; e 3

G*K236;

58;=<7,@ ~3E~HS1F1¨M23T58MA1Q3GJ@O<=3[GJ36DJNHQ;=351F1¨6@2@2< n,

25 + a 5

HQ7o56@2@2<L=56HQD

M>=;=
b n=6 K236; n n! o b

9 25

58;=<

6,

a 5

HS7¨5#6@2@2HSg

a2 25

=

9 b 25

13 = 25

58;=<

a 7, 5

HS7¨5#6@2@2HSg

a2 25

=

13 b 25

=

e K23 e OG*KA56G b GiHS7P357>GJ@ 2000 HS7iG*K23:@A;A1>?79@C1SBAGIHQ@A;R7)56GIHS7)gF>CHQ;= × GTg]@C1S1Q@8~P7TG*KA5GG*K23 6@2@2<W a L2= ≤g]@A12 5HSD37O@Cg#G*K23?g]@ADIN 58DI3R561S1L=56HQD1 7O@C≤gG*aK23? DJN (6, q) × g]@AD b# K23DI3T5DI3 7)B e K 3 + 25k 5;=< 22 + 25k 6, 6, k = 0 1 . . . 399 800 5 5 6@2@2<L25HSD7 b

C 5 r

b

b

n=7 K236; n n! o 2000

gG*KCHS7#38CB=56GIHQ@A;HQ77)56GIHS7)p38< ×

G*K236;

a2 = 25

a2 25

+ 13 ≡ 3 (mod 5)

b

b KCB=7 G*K23DI3!HS7i;=@6@2@2<L=56HQD Ù=BAG579AB=5DI3!HS7i;=3,A3D e @A;=DIBp36;CG#GI@ 3 (mod 5) HQ;!G*KCHS7 e 58793 b

C 5

b } b n= 8 o n K236; n!

r

P36DI3

2000

=

4 25

58;=<

8,

a 5

HS7¨5#6@2@2HSg

a2 25

=

4 b 25

=

11 b 25

b P QH 7G*K23?@A;A1>U7,@C1FBAGIH4@A;U7)56GIHS7)gF>CHQ;= G g]@C1S1Q@8~P7G*KA5G 1 ≤ a ≤ 12 × G*K236@2@2<L=56HQD7E@CgG*K23Tg]@ADJN 5DI3561S1/L=56HQD7@Cg/G*K23!g]@ADIN 2 + 25k 5;=< g]@AD 23 + 25k

C 5

r

a = 2

8,

b

5

b

(8, q)

k = 0 1 . . . 399

n=9 K236; n n! o 2000

=

11 25

58;=<

9,

a 5

b K236DI3P58DI3

8,

800

5

7)B e K6@2@2<L=56HQD7 b

HS7¨5#6@2@2HSg

a2 25

b £ @8~ i HQ7G*K23R@A;A1>U7,@C1FBAGIH4@A;U795GIHQ7*gF>AHS;= G g]@C1F14@8~P7G*KA5G a = 6 1 ≤ a ≤ 12 × G*K236@2@2<L=56HQD7E@CgG*K23Tg]@ADJN 5DI3561S1/L=56HQD7@Cg/G*K23!g]@ADIN 6 + 25k 5;=< 19 + 25k 9, 58DI3P<=@A;=53 b

P

g]@AD

(9, q)

k = 0 1 . . . 399

b K23DI3i5DI3

9,

5

7*B e KT6@2@2<L=56HQD7 5 ;=<~3 800

Õ b H A3;?5ß838<HS;CGJ363D 58;=7,3ABp36; e 3 ~K236DI3 N ≥ 5 N58 ;>RAA3 g]@AD 5N:@6A3:HS7NO58<=e31 M > e2e K2.@[email protected]* HQ;=e e 7)HQ;27 bv ~@7)B e K7,3ABp36; e 36758DI37)5HQ<GI@ e @Ai ;2∈ 793 e {1 BAGI H A−1} 3GJ36DJN758i ;== < e 1KA 5;=2[ HQ.;=.E. G* K2N 36HSD M=3o7*HQNHS1S58DuHSg=@A;=3o@CgCG*K23N e 5;!M23i@2MGI56HQ;=3

8

¹ H½

f%· ´ M· » 7µ #¼ ¸ ¹

´ 3½

H½ » #¼

(# +!! @ ! ) "+ ) ) ."*> ! "( 6 " ,# " () "!"* $#$" ( 3) 9.$ (+ -"* !0( ! ( < * @ ! ) ) )+ @< K23PN5[ßHQN51=;AB=N:M23D@Cgu7938CBp3; e 37 ;=@TG¬~@@Cg=~KCH e K5DI3P7*HQNHS1S58D HQ7 b ;=3TN:@6A3T;AB=N:M23D GI@:M=3PG*K23T NO16 ~3!<236= @AD @A3 27)HQ .;2.7T. @C g N − 4} bc 7938CBp3; e 3@Cg7)B e iK NO@A367E~HS1F1vM=3 G*KA56G e KA5;=6i 3∈7PG*{1 K23: e . . . e i i+4 b DI3L2DI3793;CGI38T5i7,3ABp36; e 3i@Cg=N:@6A3o;AB=N:M23D7 <23;=@GI38 561Q79@B=793PG*K23U;=@=GI5(U Up ) K231Q3;=®G*K:@CguG*K237,3ABp36; e 3HS7 b 3 GIH4@A1; , . . .g]@A, Dv G*K23 ) NO =@pA36756Gv561S1p5;=W5?=;AHFGI3R7938CBp3(e ; e 13 , [email protected] NO@A367 @ADT35 e K (e01 , . . . , e0N ) G*K23PC51FBp3@Cg U = (U ,U ) @A;A1>:HFg/G*K23 1g , . . .HSg p< C < > H ¨ 3 D o 7 4 g I D A @ N * G 2 K P 3 C 5 F 1 p B 3 C @ 8 5 = ; 0 iL=5= ei HS;G*K237938CBp3; e 3 DIHG1> @C.g.G*.K2 3N GJ@ G51;AB=N:M23D@Cg=e@ ieJe B=DJDI36; e 367Å@Cg i − 4 i − 3 . . . i HQ7#@2<2< b GÅg]@C1S1Q@8~P7vG*KA5G®Ø × U 45 G*K23P@ADI<=36D@CgpG*K23N:@6A37#KA587;=@!HSNOL=@ADG58; e 3A« 4M HFgpG*K23P7)58NO3N:@6A3HQ7B=7938 M23@ANHFGGI38<~HG*K2@CBAGv5& ¨3 e GIHS;=oG*K2 3=;251p7*G56GI3A«

e

~3N5>7)B=L2Lp@A793EG*KA5G 3 e 1Q56HQN

U

HQ7#@ADI<23DI3<7,@G*KA56G

U1 < · · · < Up

b

G*KA56G~K23;=3,A36Di5O7,3ABp36; e 3O@CgÅNO@A367 e KA5;=63758;HQ;AHGIHQ5617JGI5GJ3 IG @5=;2517*G56GI3 G*K236DI3#HQ75#B=;AH4CBp3i@ADI<23DI3<7938CBp3; e 3 0 0 (e . . .e , e58NM=@6) A3 G*KA56GvL2DI@2 1e 58;2;=@GEM2 0 0 (e , . . . , e ) U V 1 N b ;AB21S1 b 35797*B=N:3 b U6HS6= 3G ; e ∅3 g]@AD G*K23oC51FBp3P@Cg HS7 e KA5;=638<@A;A1>M> M> e 3a G*K23#pDá 7JG/NOa@A=3#U HS;E1G*K237938CBpU3i; > DI3L2DI379i3≥ ;CGI382< g]@AD¨5;> ~KCHF143G*eK2a3C561SBp3@Cg HS7;=@G e KA5;=638<M>E5;>EN:@6A3HS;oG*K237,3ABp36;Ue 3 b/ KCB=7 0 5;=e< i 0 ig]@A< eaG*KA6= eb i 6= b ;@G*K23Dp~@ADIAHS;=iiGJ@ i < a G*K2 e = 6 e i 3P5DIi[B=N:36;CG G*K23P795N: 0 e = 6 e V V a a e b G*KA56GKA57 4B=7JGM=33;58L2L21FH43 C @ g Q H N 2 L F 1 4 H 6 3 7 * G KA56G ˜ ) < `(U ) `(U `(U ) V˜ = (V2 , . . . , Vp ) e b e e e ~ C K H K A @ C ; J G D 5 C < H G 7 C @ = B D 8 K 2 > = L @ * G 2 K 3 * 7 HQ7 b 5 = ; < 2 K 6 3 ; 3 C K = B 7 ˜ ˜ U V B== D e ∅1Q56HQNÜKA 57#;=@8~Þ M=33=;∅L=DI@6A38< b U = V bÐ K23; 3G <=36;=@GJ3 G*K23W793G:@CgP561S1E7JGI5GJ367 Nb . . . , eN ) 3 <236á =;=3 E58;f38CB2HC561436; e 3 DI31Q5GIH4@A; @A; (e1M, >Û 7JGI5GIHQ;=G*KA5GTG¬~|E| @7*G56= GI37258DI3 38CB2HC561436;CGHSg25;=PHFgCG*K23,>58DI3o7*HQN∼ HS1S58D b¨ E K23oN5[ßHQN51C;AB=N:M23D@Cg27,3ABp36; e 367 ;=@iG¬~@E@Cg~KCH e KE58DI37)HSNHF1Q5D HQ7¨G*K23#;AB=NOM=36D@CgA3AB2H C51Q3; e 3 e 1Q5797937u~HG*KEDI37)Lp3 e G GJ@TG*KCHS7#38CB2HC561436; e 3PDI31Q5GIH4@A; b

K236DI3 58DI3 3*ß5 e GI1 > e HQ; e 1FBpf7)B e KÛ7,3ABp362; e 3U3HFG*K236DNO58837B=7,3U@CgiNO@A3 ;AB=N:M2 3D @AD<2@23∅7 i b ÙA>W@CB=DPB=;AHQABp36;=37)7 e 1S5HSN ;=@GTN583B=793R@CgHG g]@AD e @A;C4G 5HS;27 N −4 361Q3NO3;CG7g4DI@AN b 5M2@A3 HGug]@C1S1Q@8~P7¨G*KA56 G/365 e iK38=CB21HC 5621436 ; .e .3 . e 1S58N7)7 − 2 E N K236DI36g]@ADI3 G*K23P;AB=NOM=36D@Cg e 1Q5797937vHQ7 2 b = 16 N −4 2

° b

bc 1FHQ;=3 HS;CGJ36D7,3 e GI7 M23#5iCB=5
H½ ²»

x¶ ¹ ¶

H½ M¶

· À H½

`

AB

(#' 6! $ 4%" +! ! "* +!)0 %: ,#( "( 6# %< + 4 )0 :'. ,"( 6# 7() 4 5$ 5 *"* " "*) " ( " ," @< 2* 5 -*" ) )+ @< 3 e 5;2;=@G e @A; e 1FBp<=3G*KA5G HQ7ÅL258D561S1Q361GI@ b P36DI3oHS7Å5 e @CB=;CGI3DI3*ß58NL21Q3 b AB 58;=< b 3[G 3G M23O5GJD5Lp3[)@ClH4<:~HFG*K á G*KA5PG 7)Lp3 e GI= HA32CD 1> 7)B e K? 58;=< á M23OABCD HQ;CGI3DH4@ADEL=@CHQ;CG7T@CgG*K23:7)HQ<=367 AB 5k;=< CD DI3AB S AD BC b 3G ~KCH e KHSNOLA1SHQ37vG*KA56G HS7#;=@GL=5D51F1431=GJ@ 58;=< AP CS PS AB á Q R = 6= 1 P D SB e e b M=3G*K23!HS;CGJ36D7,3 GIH4@A;27E@Cg ~HG*K 5;=< DI37)Lp3 GIHA31> 3!79KA51F1/L=DI@6A3 P S AC BD b G*KA56G

P Q = QR = RS

D..............................................................................................T C ............................................................

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

X

Y

O

S

R

Q

P

A

á

3G

T

B

M23EG*K23Lp@CHS;CG@A;!G*K23P7*H4<23

CD

7*B e KTG*KA56G

CT AP CS = = TD PD SB

b

b 6HS; e 3 CT b 6HQ; e 3 CT AP ~3iKA5A3 CS ~3EKA5[A3 = = P T k AC T S k DB T3D T~D G 5;=P< D M=3vG*K23HQ;CGI3D793 e GIHQ@A;27@Cg HFG*K SB5;=< DI37)Lp3 e GIHA31> PT AC 58;=<1Q3á G XM23EG*K23EOHQ;CGI3D793 e GIHQ@A;O@Cg ~HG*K BD b Y T S AC 6HQ; e 3 5;=< ~3P@2MCG5HS; XR k T S P T k AC AB k CD 6HQ; e 3

PR PX AO AB 2 = = = = RS XT OC CD 1

QY k P T T S k DB

5;=<

AB k CD

×

PQ TY DO CD 1 = = = = QS YS OB AB 2

Gvg]@C1F14@8~P7vg4DI@ANÜ

58;=

G*KA5G

«¦K23; e 3

b

P R = 2RS

~3PKA5A3

«¦K236; e 3

P Q = QR = RS

b

QS = 2P Q

b

b Q5 HQ;= ! "( 6 <

c 1S1=Lp@A7*HFGIH A3EHQ;CGI38636D7 3)ß e 3LCG 58DI3P79@C1SBAGIHQ@A;27 b n b n=2 4B=7JG e K2@2@A7,3 × g n = 1 e K2@2@A793 a=b≥2 bi 2 5;=< K 36; × n ≥ 3 a = (n − 1)n−1 b = (n − 1)n a b ≥ 2

45 58;=<

R´ » ¼ ¹

g

(aa )n = aan = (n − 1)n(n−1)

n

(n − 1)n

=

£i@8~Þ7*B=L=L=@A7,3iG*KA5GG*K23DI33*ß8HS7*GHQ;CGI38636D7

a b ≥ 2

b

(aa )2 = bb

(n−1)n

= bb

b

7)B e KTG*KA56G

3 e 5 ;2;=@GEKA5[A3 b M=3 e 56B=7,3!G*KCHS7P[HA367 b a a 2 7)HS; e 3 3 e 5 ;2;=@GKA5[A3 b ≤ a M23 e 5B=793G*K23; b b ≤ a2a < (a2a) a 2 a >a 12 b KCB=7 b GÅg]@C1S1Q@8b~P7Å≥ ~ 3NB=7*GKA5[2a A3 ≤ b G*KA56(2a) G <=@236= 7#;=2@Gvb (a ) 3G M=3P5L=DHQNOa3< b < 2a × a ;AB=NOM=36D~KCH e KO b DI@AN& ~3KA5A3 b J α<β 2aα = bβ = <1 β 2a 6HQ; e 3G*KCHS7!HS7GJDIBp3g]@ADT365 e KUL2DIHSN:3

n=2

b

] M ¡1Q35DI1 > HQ75T79@C1SBAGIHQ@A; b £o@8~Ý7*B=L=L=@A7,3G*KA56G 5;=< (1, a>1 5DI3PL=@A 7)HGIHA3EHS;CGJ 3(a, 63Db) 77)= B e KT G*KA1) 5G (aa )5 = bb

c 7¨HQ;E45

b

b ~3;=@8~k<238 ~3O@2MGI56HQ; 587iHS;R45 g4DI@AN ~KCH e KO~3556HQ;R<238 ~3TKA5A3793836;OHS; a = 27 ~KCH e K14365

3 *G

© £i@8~ ~3UGIB=DJ; GJ@ ,7 @C1FBAGIH4@A;27?M>@CB=D:DI358<=36D7GI@dL2DI@2MA1Q3N7 @CgEG*K23 8 65 H~P5;=3793 § 5G*K23N56GIH e 561 1 >2NL2HS5<[HA36; © Ø b Ö b HS;=<561S1pL25HSD7 @Cu g L=@A7)HGIHA3EHS;CGJ363D7#7*B e KTG*KA56G x2 y+2 b

R´ » ¼ ¹ (x, y)

y

=x

?( +!! @ ! ) - "+ ) ) ."*> ! "( 6 < £i@GI3G*KA5G 58;=< 5DI3!7,@C1FBAGIH4@A;27 bo 3!L2DI@A3G*KA5GG*K2367,3T58DI3G*K23 (2, 2) @A;A1>7,@C1FBAGIH4@A;27 b (1, 1) 3G 58;=< M23PL=@A7)HGIHA3EHS;CGJ363D77)B e KTG*KA5G á x y x2 y+2 b

y = x b £o@8~Ü5797*B=N:3!G*KA5G gv36HG*K23D @AD G*K236; 5;=< × b DIx@AN = 1 ~y3P7,= 33E1G*KA56G 58x;=< = yKA5[= A3T13*ß5 e GI1 >!G*K237)58NO3PL=DHQNOx3T> 1 x y Q αi 58;=< Q βi M23G*K23#L2DIHSN:3#<=3 e @ANLp@A7*HFGIHQ@A;27@Cg 5;=< b/ K23; 3G x y g4á DI@AN x = g]@ApDvi 35 e K y = pi * j b x2 βj = (y + 2)αj £i@8~&7)B=L2Lp@A793EG*KA5G@A;=3P@CgpG*K23PL2DIHSN:3P HS7#@2<=< b b 6HQ; e 3 αi HS;=

9 nKC> B=7 nG*2 K 3B=;AH4CBp3iL2DIHSN:3i
5;=< HQ7 b 3[G 58;=< y 2 á x = 2 α b g y = 2β b 2CB=56GIHQ@A;? DI38 5;U36587>RHS;=K23Pn38CB=56GIHQ@A;DI38 nb g]@ADi35 e KHQ;CGI38636D 4M>?58;=@G*K236Di357>HS;= BC M=3G*K23E56M 1FGIHGIBp<=3og4DI@AN NO383[GIHQ;= AB AP b 3[G#G*K231FHQ;=367 b 0=DI@6A3TG*KA56G#G*K23!B1SHS;=37 56á G AP 58;=< A á N:BQ 33Go5G 5;=< H 58DIá 3Lp36DJL=3;=
H½ 1¸ ¹ %Á %¶1¶ M· 7µ µ M· 7¸ ¸

CM

%¶ M¶ µ À p¹Q´· ½ » ¹ H½ »

L¾ %· »

(#?( 4" #=;")0)0 -"*9 ! ")+ (!I(!" , 4, 6" "* 3 /

! "( 6 9( ,!0 < " ,#$"+33 1" #% 7 " () "!"* $#$" ( 3) .$ (+ -"* !0( !0( 7*() 4$ 5 -*" " "*) " ( " ,"( "( 6# 6" ) +!* 4%"+ ( #$" ! ( < %! )+,* 6" "* 3 ) ) < 6HQ; e 3 ~HFG*K N:33G56G HS7@A;:G*K23L=@C1Q5Do@Cg Γ «=~K23; e 3 HQ7QP G*K23Lp@C1S58AB D@Cg b 6HQ; e 3 R HS7G*K23 e 36;C GJH b HGÅg]@C1S1Q@8~P7G*KA5GC DI3T@Cg HS7#Lp36DJ L=CR 3;=

A½ x¶ ½

"*3)+'+ $ 5 *" ) ! ) < C ............................. ........... ............. .................. ........ ...... .... ... ... .... ....... .... .... ... ... .... . . . . . . .... 2 . ... .... . ... . . ... . . . .. .. .. . ... .. . . . . . ... . .. .. . . .. . . . . .. .. ... .. ... ... . . .... . . . . . . .. . .. . . . . .. .. ... .... .... ....... .. ... ... . . . .. ... ... ... .. . . . . .. ................................................... . . . . . . .. . .......... ....................... . .............. . . . . . . . . . . . . . . . . ....... . . ....... .. .. ................. .. ..... .... ... .... .... ..... .................................. ..... .. ... .... ........ .......... 1........ ....... .... .......... ........... .... ......... . . .......... ..... ........ ..... .. ....... ... ...... . . . ........ ... .... ... ... ...... .......................................................................... ................................................. ............................. .... ....... .............. .. .. .. ... ......... .. ................. .......................................................... ................... .. ................. ............. ..................... ........... .................. . . . . . . . . . . . . . . . .... ................. ................ .......... ............. .... ........ .......................... ........... .... ............ ............................... .... .......................... ........ ................. .. .. ......... ........................................................................................................................................................................................................................... ............ . .. .... . . . . . . . ... ... . .. ... ........ . . . . .. ... ... . .. ..... .. . . . . .. . . .... .. .. .. .... .... .. .. .. ... ... .... ... .. .. .. ...... .. .... .... ... ... ..... .. .... ..... . . ... .... .. ..... . . . .. . ... .. . .. ..... . ... . . . . .. ...... .. ...... . ... . ... . . . . . . ....... .. . .. .......... ... . . . . . . . . . ... . . . ........... .. .. ............ ... . .. . . . . . . . . . . . . . . . . .. .. . ................................. . . . . . . .. .. . .. .. ... ... .... .. .. .. .. .... .. .. .. ... .... .. .. .. .. .. .. ... ... . . . . . . . . ... . .. ... ... ... .. .... ... ... ... .... ... .... 3 .... .... .. .... ..... ... .. .. .... .... . . . . ... . ... .... . .... ...... . . . ...... . ........ .... .......... . . . . ........ ........ ... ........ ........... .......... ............... ................................................

Γ

Q

Γ

P

T

A

H

B

M

R

Γ

D

e HQD e 1Q3

6HQ; e 3 5;=< 1SHQ3@A; 5 ◦ G*K23:Lp@CHS;CGI7 b 3[∠AP b Q6HS; e 3 M=Bi 3 G*= K23∠AQB g]@2@G@Cg=G*= K23P90 L=3DILp 36;=
b ~3P79383iG*KA5G 58;=< 561S1=1FH43P@A;5 e HSD e 143 C PL2 ID Q ΓM223 G*K23 Lp@CHS;CG 7*B e K G*KA5G ; @2A@A;=< 143[G CM b D HQ7Û5ÞL258D561S1Q361Q@2D58N b K236;ÜCB=5 b ;=< ~3KA5[A3 AD ⊥ AP BD BQ ⊥ AC BQ ⊥ BD KCB=7 k CA ◦ b ∠HBD = ∠HAD = ∠HT D = 90

K236DI36g]@ADI3 b 561S1=1FH43P@A;5 e HQD e 1Q3 5;=< T @A; e K2@ADI@Cg¨561S1p7*BpM2793G7 P~31S5M2361CHFGu~HFG*K5i;AB=N:M23D e K2@A793;T5TG e K7,3[ G @Cg á ~HFG*SK = {1 2 3b [email protected]. 3 5 100} HQ; b 0=T D58;=S<2@ANg4DI|T @AN | = 49 DI@6A3EP G*KA 5GÅG*K236DI33*ß8HS7*G757*BpM2793G @Cg ~HG*K {1 2 . . . 100} 7)B e KTG*KA56GÅg]@ADv35 e K HQ7;=@GÅ1S5M2361F143<~M HFG*K b S |M | = 50 x ∈ M M \ {x} x ?( # "(! # 7< 1 < "( ; 3 ! # ."(! +! +!)0 * "! ( 8;:=<

À H½

;63;=36D51 ]g @AD5;>:L=@A7)HGIHA3HQ;CGI38636D 1Q3G 58;=<143[G bo 31Q58M=n3 1365 e SKO= ,7 3[G {1 2 . . .~ HFG*2n} K:5! ;AB=NOM=36D P = e {T |= T ∈P b K2@A7,|36T ;56⊂ GÅDS 58;= <2|T @AN g4DI@AnN − 1} ×

`(T )

S

b @AD¨35 e K 3G G*K23793G 793G7 T M=∈ T ∪{`(T e K QH 7 e @A;CáG5HS;=F 38 HS|M g | = n} b K23T;AB=NO 36DP@Cg/ 7)B )} F `(T ) ∈ 6 T M ∈ F G*KA56G g]@AD7,@ANO3 HS7 e 3D¬GI56HQ;A1 >W;=@DI3656GI3DG*KA58;UG*K23 M = T ∪ {`(T )}

T ∈ P b ;*G K23i@G*K236DKA5;=< G*K23#GI@GI561C;AB=N:M23D 2n P n − 1 @Cg2793G7HS; HQ7 2n b/ K23DI3g]@ADI3 G*K236DI3oNB=7*GM=35G/1Q357*G 2n 793G7 2n F − n n n−1 b @AD5;>!7*B e K 7)B e KPG*KA56G g]@AD5;> HSgAG*K23DI3 M ∈ HFP M e KG*6= HQ779@A∈N:F3 7)B M KA5TG ∪ {`(T )}HQ7v1S5M2361F143T
~KCH e KHQ7

;AB=N:M23D@Cg2793G7HQ;

° b

á

3G

<23;=@GI3WG*K23Û;AB=NOM=36D?@CgTL=@A7)HGIHA3HQ;CGI38636D7

7)B e K G*KA5G

φ(k) b 6 ng]@ADO ≤7,@Ak B=L=L=@A7,3 G*KA5G NO3ULp@A7*HFGIH A3 m n gcd(n, k) = 1 φ(5 − 1) = 5 − 1 b = b HQ;CGI38636D7 0 DI@A3iG*KA56G 5;=< m n gcd(m, n) > 1

R´ » ¼ ¹

?( +!! @ ! ) - "+ ) ) ."*> ! "( 6 b b 3[G 6B=L=L=@A7,3 g]@ADoG*K23:L2B=DILp@A793@Cg e @A;CGJD5
− 1) = 2

p1

· · · pk

(p1 − 1) · · · (pk − 1)

£i@GI3iG*KA56G SH g m HQ75PL=@8~3Dv@Cg G*K236; 4@G*K23D~HQ793 ~3<=@!;=@G 214365 3 5 −1 e KA5A3 ~ C K H ? K 5;=< m n J ϕ(5 − 1) ≡ 0 (mod 4) 5 − 1 ≡ 0 (mod 8) e @A;CGJD58
C 5

C 5

b

d=4

HS7#@2<=< b KCm b MAHQ@CB=7)1 > ;=@A;=3E@Cg B=7 = 2k +581;=< g4DI@AN 5M2@A3 m · pk G*K23 HS7v3A B=a561=GJ@ 2 b 3; e 3 g]@AD365 e K 5 ~−3i1KA5[= A3 4p1 · · 2k m p 5 i 5 × 5 = 5 ≡ 1 (mod i b DI@AN pG*iK2)3 g4DI@AN ~KCH e KR~3<=3 5[~ HG/g]@C1F14@8~P7/G*KA56G b HS75ECB=5

K236DI36g]@ADI3 g4DI@AN n k 1 = 2(p1@AD − 1) · · · (pk − 1) 5) ~KCH e K~37,33iG*KA 565G − 5 e≡ @A;CJG 2(−2) D5
µ69¶1¶+ ,<

gcd(m, n) > 1

Ù @ADJ;27JIGJ3HQ; 561Q79@kL=@CHQ;CG7?@CBAGG*KA5GG*KCHS7:HQ7L=DI@2MC1436N 8 C @CgiG*K23 c N:36DIH e 5; § 5G*K23N56GIH e 561 § @A;CG*KC1 > buc 7,@C1FBAGIH4@A;58L2Lp3658D7ÅHS;PG*K23£i@6A3NOM=36D 8 HQ7)7)Bp3P@CgpG*K23 § @A;CG*KC1> L b 8 b Ñ b 3[G M23O5gQB=; e GIH4@A;?g4DI@ANÐG*K23O793GE@CgÅL=@A7)HGIHA3HS;CGJ363D7EGI@G*K23O793GE@CgÅ;=@A;= f HS;CGJ363D77)B e KTG*KA5G ;=385á GIHA3 5;=<

f (1) = 0

g]@ADv561S1

H½ À ²¹ A½ » n≥2

f (n) = max{f (j) + f (n − j) + j}

b Ú3GI3DINHS;=3

O¾

f (2000)

b

A½ »

(#= 46 6"-"+ 3 / H ! "( 6 " ,#=( # "!0# 7< 1 < " * -! # ."%! ! $ ! ) -; "+! (8;:=< + 6"-"+ 3 ) ! < ;:~KA56G#g]@C1F14@8~P7 G*K23N5[ßHQNB=N HQ;:G*K23<=3p;AHGIH4@A;?@Cg HQ7 e @A;27*H4<23DI3< @A36DÅ56× 1S1 7)B e KG*KA5G FG*KA56GHS7 51F1 g]@AD/~KCH e K f (n) 5;=< j 1 ≤ j ≤ n−1 j f (j) f (n − j) b 58DI3i<236=;=38< b 3~HS1F1CL=DI@6A3iM>HS;= e K23 e 838< G*KA56G G*KA56G × f (2) = f (1) + f (1)5;=+ b
f (k) =

f (n − 1) + f (1) + n − 1 =

58;=<g]@AD

2

1≤k
n(n − 1) (n − 2)(n − 1) +0+n−1 = 2 2

1 ≤ j ≤ n − 2

f (j) + f (n − j) + j

= =

j(j − 1) (n − j)(n − j − 1) + +j 2 2 n(n − 1) − 2j(n − 1 − j) n(n − 1) < 2 2

b

Gug]@C1S1Q@8~P7uG*KA56G K HS7 e @A; e 1SBp<237 n(n − 1) b¨ C × f (n) = max{f (j)+f (n−j)+j} = 2 G*K23HQ;=
¾ ;¶ ·

f (2000) = 1999000

9 "(! 6< 5;=o;=@GJ367ÅG*KA5G n+1 ~KCH e KTHS7~361F1=;=@8~P;GI@ f KC (n +G*1) M 3G*K23;AB=N:M23D@CgADI38[HQ@A;27HQ;C1GI+ = @i~ H e KE K23= LA1Q51;=+ 3#HQ7 1FHQ;=367HS;T636;=3D561 Lp@A7*HFGIHQ@A;O43,A3D|>L25HSD@Cg1FHQ;=367HQ;CGI3D793 e Gv5;=<;=@TG*KADI331SHS;=375DI3 ne @A; e B=DJDI36;CG b

µ

KA5G e @ANOLA143[GJ367G*K23 6 ! ,! g]@ADÅG*KCHS7#HS797*Bp3 b 83;=A@CB=D;AH e 379@C1SBAGIHQ@A;27 58;=ANOLAHQ58< e @A;CGJ367*G7

A¹ ¾ &½

/

}

+ " o+E$ ()'9+

· 7»

7µ !! + ¶ S

" !$# À $ A½! ) -'"79» &Á "+(L¶

7+! " 4$ H6 ! @ 6# 46 M> *Bp58;=<0C5B21 b 51F1> 2 79@CgSG e @A36D 88 2Ù/£ ×9 (# /

46

)D b L2BpMC1SHS7,K23

c L256367 ¤ ©8 b b

0A3[GJ36D7

©

À

4 ! )+-/: , ib ;Ý35 e KÞ@Cg!G*K23fGI3; e KA58LCGJ36D7@CgTG*KCHQ7UM=@2@2 G*K23 56BAG*K2@AD7 HS;CGJDI@2&3*ßLA1Q5;256GIHQ@A;5;=<¢HQ;27JGJDIB e GIHQ@A; G*K23,>&D58&MCB2HS1Q< GJ@8~P5DIA@CB=D@8~P; o56gSGJ36D365 e KABp367*GIHQ@A; @AD e KA561S1Q3;=63 b 8@C1FBAGIH4@A;27GI@PG*K23EABp367*GIHQ@A;27v58;=< e KA51F1436;=637v58DI3E[H A3;THQ;PG*K23oGI3)ß6G b 7+! " 4$ HQ7ANONO3GJD|> 5;=<63@AN:3[GJD|> b ;OG*K23 e KA5LAGI3D 56GGIH e 3O0A@C1>A6@A;27 × á 7*GIBp<23;CG7M238[HS;M>T56GGI3NLAGIHS;=#GJ@NO5883#C58DH4@CB=7G|>2L=37@Cg=L=@C1>A6@A;27@A;T5i1Q5GGIH e 3 b K23,>O~@ADJGI@8~P58DI<[Bp36797*HQ;=T58;=?GJD|>RGI@ NO588379KA58L=37E~HG*K 79L=3 e HF3< 5DI357 b? K23:HS;A37JGIH45GIH4@A; G*K23;g]@ e B= 7937i@A;~KCH e K:5DI357 e 58;M=3!5DI357E@Cg/1S56GGIH e 3T79AB=5DI37 bi KCHS7o1Q358<27#GJ@:5 63;=36D51FHF*5GIH4@A;O5;=<p;2561S1 >5TL2DI@2@Cg b ;G*K23 35[> HFgSGIHQ;= i793 e GIHQ@A;O5GÅG*K23T36;=< @CgG*K23 e KA5LAGI3D 7*GIBp<23;CG7Å3*ßLA14@ADI338CB2× HS1S56GI3D5618GJDHQ5;=[1436á 758;=<L=DI@6A3G*KA56G¨;=@A;=3 e 58; HQ;51Q5GGIH e 3 b 8@AN:3P@CgpG*K23@G*K236D3)ß8L21Q@AD56GIHQ@A;2758DI3L=561SHS;=<2DI@ANO37 M=3 e @A;27*GJDB e GJ3< G*K23g]@CB=Dv;AB=NOM=36D758NO3 e HSD e 143L=5 e 8HS;=iHQ;5PL21S58;=3 58;=HS7 e 1Q3,A3D MCBAG=G*K235G*KC143[GIH e H e @A;27uB=7938<#G*KADI@CBp6K2 @CBAGvG*K23!M2@2@2 e @A;AgQB=7938<:NO3T5G=D7*G b K23T793 e GIHQ@A;GIHGI14367o@Cg/DB=; 7I~HQN 58;=<MCH483 <=@O;=@G e @A;A3,>NB e K:58M=@CBAGvG*K23 e @A;CGJ36;CG@CguG*K23T7,3 e GIH4@A;27 b CgQB21/GJ@A;=3@CgG*K23OM2@2@2 b KA5<× G*K23g]3831SHS;=TG*KA5GG*K23 36DI3iGI5614HQ;=oGJ@! N:358;=fDI3 e @ANN:36;=<ÛHG!g]@AD *>A@CB=;= 7JGIBp<=36;CGI7 b&c G!=D7*G[1Q5; e 3 G*K23 L=DI@2MC1436NO7E~@CB214< 7,33NGI@ DI38CB2HQDI379@AN:3:561463M2D56H e 7,@AL=KCHQ7JGIH e 56GIHQ@A; b @8~3, A3D G*K23,>!L=DI@6AHQ<=3i5EN:3658;AHS;=[gQB21 e @A;CGI3)ß6Gg]@AD143658DI;AHQ;=o561463M2D5 b K23oGI35 e K23D e 5;K2361SL =1F1=HS;!G*K23P58L2751Q@A;=oG*K23o~P5[> b K23L=DI@2MC1436N e @A;CGJ3*ßG75DI31FH4831>GJ@M23PNO@GIHC58 GIH4@A;25612g]@AD7*GIBp<23;CG7HQ;KCH46K!7 e K2@2@C1=@AD e @C1F14363ENO5G*K23N56GIH e 7 e 1S587)7,367 @ADÅ3,A3;!g]@AD L=DI@A7)Lp3 e GIHA3ENO5G*K23N56GIH e 7ÅGI35 e K23D7 b ~@CB21Q<3; |@6>TGI35 e KCHQ;=og4DI@ANÝG* KCHS7M=@2@2P57 × 5 e @C1S1Q3863P3143 e GIHA3 b e B21FG|>~HG*K e KA58LCGJ36D HQ; ~KCH e K7*GIBp<23;CG7¢<2@ 638@ANO3× GJDH e e @A;27JGJDIB e GIHQ@A;27 b ;ÞN!> 3*ßL=3DH436; e 3 3,A3; e @C1F14363 7JGIBp<=36;CGI7KA5[A3 × GJDI@CBpMA1Q3~HFG*KG*K23N:@A7JGoMA587*H eoe @A;27JGJDIB e GIHQ@A;27 b G NH46KGoM=3M=3[GGJ36DGI@5OHS;A@C1A3PG*K23TDI365<23D b GL2BAG7G*K23!@A;AB=7 g]@ADv<2@CHQ;=G*K23NO5G*K23N56GIH e 7@A;TG*K23DI365<23D b/c 7*GIBp<23;CG~K2@P~@A× D*87ÅG*KADI@CBp6KTG*KCHQ7 M=@2@2E~HS1F1=B=;=<23D7JGI5;=N:3658;27vGJ@<=@N56G*K236NO5GIH e 7 b

A¹

X[Z &U ¬ ` [URc ]b` e` U YVc Ub^pU c&Z

"A')-p$= T#"

:%"

P(,

;?~KA5Gig]@C1S1Q@8~P7 HQ7P5 e > e 1SH e AB=58<2D5;=[143OHS;~KCH e KG*K236DI3:58DI3O;=@ G*K23

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

A

B

E

D

C

F

r

M

G

H4[B=DI3

;UG*KCHQ7L258L=3DE~3?58DI3?;=@G e @A; e 36DJ;=3< ~HFG*K G*K23RKA58DIN:@A;AH e L=DI@AL=3D¬GIH4367@Cg G*K23CB=× 5M=@2@287 7)B e K:57 58;=< b# 3 <=@ K2@8~3,A36D 793 GG*K23E7*G563og]@AD/~KA56Gg]@C1S1Q@8~P7M>CBp@ GIHQ;= ~HG*K2@CBAGÅL2DI@2@Cg 7,@ANO3 ~36 1F14);=@8~P;63@AN:3[GJDIH e L2DI@ALp36DGIHQ37@Cg e > e 1FH e CB=5

b 7*4, - ,! %"+"( * 3)+ !) 6 4, " + )" b

¶

b

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B

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

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

A

B

B

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!

35. %" A½ ¶1¶

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A

F = ∠AF G G = ∠AGF b 5;=< 0 F G0 = ∠CGF K236= ; ∠CF G 5;=< F + G = C = 2(π − Ab 0 6 ) B=NO 0 0 0 F ) @2MCG5HS; NHS;=+G 58;= − C~3 2 0 b (A = 2π − (A0 + C5;= )< K236+ DI36C) g]@ADI3 = π/2 A0 +b C 0 = 3π/2 B 0 + D 0 = π/2

D

D

G

B

C C

F

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35. %"A½ ( ¶

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; 4 "( 6# 0 "(! 0 0 0 0 0 B ,"(! F"* AC ; D 4 G 0 <0 3)+( B G = D F 0 0 "(! BD , F!A!0 C -"*G) "(! F C = GA AC F G A C 0 0 < * - "* * ) 46=+ , !0' !0) 0 0 " ,# 0 0 < ]833 BD E ABF CDG HQ[B=DI3 F Gb B D

¶

B0

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

A

B

D

C

E

D

F

C

H

G

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k2 + k + 1 r (k + 1)2

´ ! /()+# ¹

3G

~K23DI3

a>0

»

r

7)B e K

b

5DI3EG*K237)HQ<=367#@Cgu5EGJDIHS58;=[1Q3 b

M23G*K23 e HSD e B=NOD58
<23;=@GI37vG*K23PL=3DHQNO3GI3D@Cg=GJDIHS58;=[1Q3

[Tk ] =

b

= k > 0

AA1 BB1 CC1

3G á

"( $ -"2b

<23;=@GI37vG*K23P5DI35@CgpGJDIHS58;=[1Q3

HS7vG*K23HQ;2D58
7¾

4ABC

T

T

«

«

b

46 6" "* 3 / (!" ,Cb b 0=DI@A3iG*KA56G

a2 + 1 3a2 − 1 5a2 + 1 7a2 − 1 π b + + + + ··· < ea 3e3a 5e5a 7e7a 4

ÕÔÓ

§´ !0 /()(# (1¸9%Á ! <

%¶ 7µ À

(" 6#$"* 91" #% 7 ib b M23T58;>CB=5 b AD 3[G M=3G*K23NH4<2)L=@CHQ;CG@CCBE g 58 ;= < M23G*K23ADG NHQ<= N EF P á b Lp@CHS;CG@Cg G 0=DI@FA3i G*KA56G HS7#38CB2HS1S56GI3D561 b

b

4M N P

µ

7¾ ´ 9¶

>¹"(!(4$ #$" @*¬µ6 * *': " %"+ < <µ ;¼0"0

ÕÔÓ b ! /()+# ! 6 !" -"* / (" $ -"2b

b 3G M=3Lp@A7*HFGIH A3vDI3561;AB=N:M23D7 5;=
1

1+

(a1 a22 ) S2 a1 + 2a2

µ

+

7¾ ´ 9¶

1

(a1 a22 a33 ) S3 a1 + 2a2 + 3a3

+···+

a1 + 2a2 + · · · + nan

≤

b

2n n+1

>¹"(!(4$ #$" @*¬µ6 * *': " %"+ < <µ ;¼0"0

ÕÔÓ b ! /()+# ! 6 !" -"* / (" $ -"2b

Sn (a1 a22 · · · an n)

58;=<1Q3G 3G M=3PDI3561u;AB=NOM=36D7oDI35GJ36DÅG*KA5; M=3P5;> .M=.36.D b a0=nDI@6A3EG*KA5GÅHSg −1 G*K236; α Lp@A7*HFGIH Aá 3DIa 35611 ;AaB=2NO a1 + a2 + · · · + an ≤ αn

µ

1 1 1 n b + + ··· + ≥ a1 + 1 a2 + 1 an + 1 α+1

7¾ ´ 9¶

>¹"(!(4$ #$" @*¬µ6 * *': " %"+ < <µ ;¼0"0

ÕÔÕÔ b ! /()+# ! 6 !" -"* / (" $ -"2b

58DI3Lp@A7*HFGIH A3DI3561p;AB=N:M23D7 G*K23; 8K2@8~G*KA56G FH g a1 a2 . . . a n 1 a1

~K23DI3

+

2 (a2 )

´ ! /()+#

1 2

+

3 (a3 )

&½

Sn = 1 + 2 + · · · + n

ÕÔÕ Ö b

1 3

b

+ ··· +

n (an )

C¿

1 n

≥

Sn 1

(a1 a2 · · · an ) Sn

7µ

5: (! "*(! ( !0 " -"2b

c AB=58<2DB=L21Q3 @CgALp@A7*HFGIH A3HQ;CGI38636D7HS77956H4¨Ø

(3, 5, 16, 1008) (3, 21, 40, 10208) (3, 56, 85, 57408)

(3, 8, 21, 2080) (3, 33, 56, 22360) (3, 65, 96, 75208)

(3, 16, 33, 6440) (3, 40, 65, 31416) b (3, 85, 120, 122816)

HS;= 58;=
§´ !0 /()(#

ÕÔÕiÓ b

Wµ ! " À $ A½! ) -57´ )* 3¾99¶

("*)0

" "=b

b 3G M=3T;=@A;=*;=385GIHA3TDI3561¨;AB=NOM=36D7i7*B e KG*KA5G 2 a + b2 + c2 = 1 0=DI@A3iá G*KA56aG b c 1 1 1 9 b 1 − ab

+

1 − bc

+

1 − ca

≤

2

¹

´ 7¸ ¶

ÕÔÕiÕ b ! / ()+# "*#%( (! 9+ ! " 2b

·

"!0#

½ .¸

/(4,-

· À $ A½! ) -

8 $+!

3G M=3!G*K23HS; e 36;CGJDI3O@Cg 5;=<143[G 58;=< M23HFG7 e HSD e B=NOD58 b 0=DI@4ABC A3iG*KA56G p b 2 2

¹

6r ≤ AI + BI + CI ≤

´ 7¸ ¶

ÕÔÕv° b ! / ()+# "*#%( (! 9+ ! " 2b á

3G

a b c x y z

·

"!0#

12(R − Rr + r )

½ .¸

/(4,-

· À $ A½! ) -

8 $+!

M23PL=@A7)HGIHA3DI3651p;AB=N:M23D7 b 0=DI@6A3EG*KA5G

(bc + ca + ab)(yz + zx + xy) ≥ bcyz + cazx + abxy + 2

58;=@ e*e B=D7 b

b´

p abcxyz(a + b + c)(x + y + z)

,¼¹ 4"%¶1¶

ÕÔÕ b ! /()+# 3 7 ! ( 8;:Tb

"$#( )*$#%+ ,-

À H½+!)0 7 ! (

c DI3iG*K236DI3HS;Ap;AHGJ31>!NO5;>L=DHQNO3;AB=NOM=36D7G*KA5G e 58;2;=@GvM=3~PDIHGGJ36;57vG*K23 7)B=NÜ@Cgu5L2DIHSN:3;AB=N:M23Dv5;=<5L=@8~3Dv@Cg

§´ !0 /()(#

»

7¾ 9¶

2

( ! + 3@: +-"* 94"(! )+- / "( $ -"2b 3G M23RG*KADI383ULpA@C HS;CB GÅ;= g b K23iLp36DJL=3;=
Q5 ]M

MC

0=DI@A3iG*KA5GG*K231SHS;=3

?á

3G

Y

XY

L2587)7,367vG*KADI@CBp6K5Cß83<Lp@CHS;CG

M23TG*K23L=DI@ |3 e GIH4@A;?@CgG*K23L=@CHQ;CG

Z b ∠BZD = ∠CZD

b´

,¼¹ 4"%¶1¶

ÕÔÕ b ! /()+# 3 7 ! ( 8;:Tb

A

@A;CGJ@OG*K23!1SHS;=3

"$#( )*$#%+ ,-

D

b

XY

b 0=DI@A3TG*KA5G

À H½+!)0 7 ! (

K236DI3T5DI3 7,36;256GI@AD7#HQ;O5!793;25GJ3 b A5 e KO7,36;256GI@ADKA587o36;=3NH4367~HFG*KCHS; G*K237,36;256GI3 b 0=DI@6A2005 3G*KA5GÅG*K23DI3HQ7#5;=@A;=936NOLCG>O7*BpM2793G @Cg¨793;25GJ@AD7#7*B e KG*KA5G g]@AD3,A3D|>T793;25GJ@AD/HQ;PG*K23i793;25GJ3 G*K23i;AB=NOM=36DÅ@Cgp36;=3NH4K 367@CgAG*KA5G7,36;256GI@ADHS;G*K23 7,3[G HQ75;:3,A36;;AB=N:M23D b

§´ !0 /()(#

K

»

7¾ 9¶

( ! + 3@: +-"* 94"(! )+- / "( $ -"2b b 0=DI@A3G*KA5GG*K23 ¡¨@A;27*H4<23D#5GJDIHS58;=[1Q3 HQ;~KCH e K b c} a = max{a √ABC√ √ 5;=< √ √ 3)ß8L=DI36797*H4@A;27 (aKA5+A3E b G*+ c)795N: 23 −7*H4; b a + b + a − b · a+c+ a−c 2 K 3 2 2 2 ÕÔÕ

b

b +c −a

bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

ÕÔiÔ

d´ !0 /()l º ,"!¹ 4"*º -R»@ ,A¼ 7»6!"*)+3½+¾9 ¶

b ¡¨@ADIDI3 e GIHQ@A; b

8@CHFG

ABp3

B=;GJDIHS58;=[1Q3O3Go79@CHFG

ABC

"( $ b

BA1

CB1

=

A1 C

A1 ∈ BC B1 ∈ CA =

B1 A

AC1

= k > 0

C1 B

3[G

C1 ∈ AB

GI361S7

b

b§ @A;CGJDI3DRCBp3k1Q37 7,3N:36;CGI7 3[G 7,@A;CG1Q37 e @G 37 < B=; AA1 BB1 CC1 GJDIHS58;=[1Q3 b

e 3GJDIHS58;=[1Q3 L=5D Ú 37*H4;=@A;27L258D 3G 143RD5[>A@A;f<237 e 3D e 1Q37HS;27 e DHFG!3[G rk Rk e QH D e @A;27 e DHFG<23 Tbuk § A @ ;CGJDI3DvABp3 Tk

b

P (Tk ) < P (ABC)

© b

@ B

P (T )

@ B

k2 + k + 1 [ABC] (k + 1)2 √ k k P (ABC) Rk ≥ (k + 1)(k2 + k + 1)

[Tk ] =

b

b

rk >

k2 + k + 1 r (k + 1)2

§´ !0 /()l º ,"!¹

ÕÔÓÑ b

8@CHFG

a>0

+

ea

< 37*H4;=3E1 5HSDI3
T

T

«

«

«

r < 3 7*H4;=3E143D5[>A@A;:
»

e 3D e 1Q3HS;27 e DHFGv<=3

7¾

4ABC

b

4, 6"-"+ 3 9 $+ (!" ,Cb

3a2 − 1 3e3a

+

5a2 + 1

§´ !0 /()l º ,"!¸9(%Á,!0 < b

[T ]

bu§ @A;CGJDI36DABp3

a2 + 1

ÕÔÓ

@ B

< 367)HQ;=3143L 3DHQN 3GJDI3

5e5a

+

7a2 − 1 7e7a

+ ··· <

%¶ 7µ 7 À

"( 6#$"* 91" #%

π b 4

ib

8@CHFG B=; AB=58<2DHS1S56G 36DI3ABp31 e @A;=ABp3 3[G79@CHFG 1Q3 Lp@CHS;CGNHS1FH43B b 6B=ABCD < 3 = DP1Q37 e @G 37 58;=< @A; e @A;27JGJDIB2HGTM 3)ß6G 3DH43B=DI3NO3;CGP14367 GJDIHS58AB ;=[1Q37 3AB2HF1Q5G 3D56Bß CB DC 3[G AD b 8@CHFG 143Lp@CHS;CG#NHS1FH43B<=3 CBE DCF ADG N EF b 3G 143L=@CHQ;CGvNHS1FH43B:<23 P FG § @A;CGJDI3DCBp3143iGJDHQ5;=[143 37JG 3AB2HF1Q5G 3D561 b

µ

7¾ ´ %¶ lº

¹

MNP

ÕÔÓ b ! / () ,"(! 2 ! - 6 !" -"* / " Cb

Rµ

Wµ ;¼0"0

"! 4 #$"( 6 * * : " %"+ < <

8@CHFG 3[Go79@CHFG <237E;=@AN:MADI37ED 3361S7 1 A+ a 1 a2 . . . a n Lp@A7*HFGIHFg47 b§ S@AkC ; GJ= DI3Dv Bp3 2 + · · · + k 1

1+

(a1 a22 ) S2 a1 + 2a2

1

+

(a1 a22 a33 ) S3 a1 + 2a2 + 3a3

1

+···+

Sn (a1 a22 · · · an n)

a1 + 2a2 + · · · + nan

≤

2n n+1

b

µ

7¾ ´ %¶ lº

¹

Rµ

Wµ ;¼0"0

"! 4 #$"( 6 * * : " %"+ < <

ÕÔÓ b ! / () ,"(! 2 ! - 6 !" -"* / " Cb

8@CHFG <=367v;=@ANOM2DI367D 3831Q7ÅLA1SB=7D5;=<27ABp3 3G79@CHFG B=; −1NO A ;=@AN:MADI3PD 3a3611p Lpa@A27*HFGI.HF.g¨.C Bpa36n1 e @A;=CBp3 b 6H @ ;CGJDI3Dα CBp3 a1 + a2 + · · · + an ≤ αn

1 1 1 n b + + ··· + ≥ a1 + 1 a2 + 1 an + 1 α+1

7¾ ´ %¶ lº

µ

ÕÔÕÔ b ! / () ,"(! 2 ! - 6 !" -"* / " Cb

@ B

§ @A;CGJDI3DCBp3

7)H

¹

Rµ

Wµ ;¼0"0

"! 4 #$"( 6 * * : " %"+ < <

a1 a2 . . . a n

79@A;CG<237;=@AN:MADI37D 3361S7Lp@A7*HFGIHFg47 5 1Q@AD7

2 1 3 n Sn + ≥ 1 1 + 1 + ··· + 1 a1 (a2 ) 2 (a3 ) 3 (an ) n (a1 a2 · · · an ) Sn

b

l´ !0 /()l º ,"! : &½+ (! C¿"*(! (7µ! "

Sn = 1 + 2 + · · · + n

ÕÔÕ Ö b

º º

Cb

;U

(3, 5, 16, 1008) (3, 21, 40, 10208) (3, 56, 85, 57408)

(3, 8, 21, 2080) (3, 33, 56, 22360) (3, 65, 96, 75208)

(3, 16, 33, 6440) (3, 40, 65, 31416) b (3, 85, 120, 122816)

ID @CBCA3DuB=;=33)ß8L=DI36797*H4@A; 3; 36D51Q3#Lp@CB=Du1S5#7)B2HGJ3#<=3#AB=58<2DB=L21Q3G7 bn3*,ß8c3n AB2H!L=@A797 38<23;CG?1Q5 L2DI@AL=DH 3[G 3d

§´ !0 /()l º ,"!") 3Wµ ! " 0 À H½+!)0 'º #%´ )* 3¾9 ¶

ÕÔÕiÓ b

"( $ b

b 8@CHFG 3[G <=367;=@ANOM2DI367D 3831Q7v;=@A;!; 356GIHFg47GJ31Q7CBp3 2 a + b2 + c2 = 1 § @A;CGJDI3DCaBp 3 b c

¹

´ l¶ º

1 1 1 9 b + + ≤ 1 − ab 1 − bc 1 − ca 2

·

ÕÔÕiÕ b ! /() ,"(! ("(! # "*#%( (!3 "*( 6Cb

R

/4 -

À $ A½! ) & º

½ .¸

·

8 $+!

8@CHFG 143 e 3;CGJDI33G 143PD5[>A@A;:A@A;: 3 D e 143 e HQD e r@A2 ; 7 e DHFG bu§ @A;CGJDI36DCBp3 6r ≤ AI + BI + CI ≤

p

12(R2 − Rr + r 2 )

b

¹

´ l¶ º

·

ÕÔÕv° b ! /() ,"(! ("(! # "*#%( (!3 "*( 6Cb 8@CHFG

3G

a b c x y

z

/4 -

À $ A½! ) & º

½ .¸

·

8 $+!

<=367;=@AN:MADI37#D 3831Q7L=@A7)HGIHSg47 b§ @A;CGJDI3DCBp3

(bc + ca + ab)(yz + zx + xy) ≥ bcyz + cazx + abxy + 2

3GGJDI@CBCA36DAB=5;=<1 38561SHG 3P51FH43B b

1´

lº

p abcxyz(a + b + c)(x + y + z)

v¼¹ 4%" 1¶ ¶ "#( 3 º $#( -" - À $ A½! ) & º #%=7 ! (

ÕÔÕ b ! /() ,"(! 3 7 ! ( 8;:Tb

)58JG HS1B=;T;=@AN:MADI3oHQ;A=;AH2<=3o;=@AN:MADI37ÅL=DI36NHQ3D7vAB @A;!;=3oL=36BAGL=57 3 e DHQDI3 e @ANN:3EB=;=3P7,@ANN:3P< B=;O;=@ANOM2DI3PL2DI3NH436D3[G< B=;=3LAB2HQ7)795; e 3P<=3

§´ !0 /()l º ,"! -!* @: +-"*»9"(! )+-7¾9 ¶

ÕÔÕÑ b

2

"( $ b b 8@CHFG ;U<2@A;2;=3GJDI@CHQ7PLp@CHS;CGI7P51FH4; 367T

MC

Y

b Q 5 § @A;CGJDI3DvABp31S5<2DI@CHGJ3 L2587)7,3L258DB=;OL=@CHQ;CGÅß83 XY D b8§ @A;CJG DI3DuCBp3v14367¨58;=[1Q37 ]M 8 @CHFG 1Q5L=DI@ |3 e GIH4@A;E

1´

lº

BZD

CZD

v¼¹ 4%" 1¶ ¶ "#( 3 º $#( -" - À $ A½! ) & º #%=7 ! (

ÕÔÕ b ! /() ,"(! 3 7 ! ( 8;:Tb

1>R5 7 36;256GI36B=D7<25;27PB=; 7 36;256G b ¡¨KA58ABp3:7 3;25GJ3B=DE5<=367T36;2;=3NHQ7 5O1 HS;CG × 3DH43B=D2005 R5OB=; 79@CB=7,)3;2793NOMA1Q3O;=@A;AHQ<=3 <=3 7 36;256GI36B=D7iGJ31CBp3 L=@CB=D e KA5CBp3ONO3NOM2DI3O

§´ !0 /()l º ,"! -!* @: +-"*»9"(! )+-7¾9 ¶

K

ÕÔÕ

b

"( $ b

b¨§ @A;CGJDI3DCBp3P14367 8 @CHFG B=;GJDHQ5;=[143<25;27#143ABp31 b c} ABC a = max{a √ 3[G √ √ √ √ 3)ß8L=DI36797*H4@A;27 (a@A;C+ b + c) 2 − a + b + a − b a + c + a − c · GÅ143N 36N:37)HQ;=3 b 2 2 2

b +c −a

d

;¶ &½

¶

6

¶

: ,! %

) !/+! " , - ()(#6< 746 #( ! 3) *" " *) ")(# )0 #%+! ! " 6 )+ ) ! 6 ) 4 3) , "*)* ,!0% )*<

&½ ?

&½

7 »

´ N¼ 3½ M·

.º À $ A½! ) -?

Ó Ó b ! /()+# $ (/;!)0 "( "+ Ø 8 ("!" 0 (@"(! " ( @() $ -" "( 6# ' 1 +! % "2b

6B=L=L=@A7,3EG*KA5G a3

b2

À

dµ

− bc +

c2

a b

+

5;=<

c

5DI3PL=@A7)HGIHA3DI3651p;AB=NOM=36D7 b 0=DI@6A3EG*KA5G

b3

W¾ M¶ c2

− ca +

a2

+

c3 a2

− ab +

≥

À H½

b2

3(ab + bc + ca) b a+b+c

4%"! ) < 5 % $ @; " +!)0 * (" ; (27 ib G/HS7¨~361F1C;=@8~P;45;=iGI@L=DI@6A3 G*KA56G 2 bc79KA+58DIca) bÅ KCB=7 H(a e 36c) ~HFG*KO3× AB=561SHG>HSgu58;=HSg G7)+ B1 b + 7GJ@≥ L2DI3(ab @A3EG*+ K23 Lp36D a = b = c HQ;=3AB=561SHG> X

a3 b2

− bc +

c2

≥ a+b+c

~KCH e K~3 e 5;ODI3®~PDIHGJ3EHS;!G*K23g]@C1S1Q@8~HQ;=E3AB2H C51Q3;CGÅg]@ADJN76Ø

X

X a3 (b + c) b3 + c3

a3 (b + c)(c3 + a3 )(a3 + b3 )

≥ a+b+c ≥ (a + b + c)

3KA5[A3 X

a3 (b + c)(c3 + a3 )(a3 + b3 ) − (a + b + c) = = =

X

X

X

a7 b(a2 − b2 ) +

X

Y

Y

(b3 + c3 )

b

(b3 + c3 )

ab7 (b2 − a2 )

(a7 b − ab7 )(a2 − b2 ) =

X

ab(a6 − b6 )(a2 − b2 )

ab(a2 − b2 )2 (a4 + a2 b2 + b4 ) ≥ 0

b KCHS7 e @ANOLA143[GJ367vG*K23PL2DI@2@Cg b ~HFG*K3AB=561SHG>!HSgu5;=HFg a=b=c 9I3

#F $ 7$8$/0 ..0 .0/$4)$07D7%#$ 4 ) $ 7 @A8 .FC3@ $ )"#. $ '(:=A ;.@ 8@ F0/ $&#6 '( 6 - ! )96 - 6 $ JI#6 " 0 # @ $ $@ # %& . 0' $ &#6 '( 6 - .- / 69- ( =>.69=>I)K8.#- ?>" ?$ /'C= 79)?$ " )/6

8 0 G4 @A045@CF $ "#=(B.'C- $ JI#6 ?$//F#@ F$%#;/014 :$G8/J # /- 6 $ JI#6 " 3#F 4 0$ &'A?"? 6 :#%JI) 6>"JK$@ ).'7".= $/H"-9'(6 ,E @ 0 ,4 )043$ G#' - .$0 $ 7 /8 /8 KF $01-(- & &C".'(69= .$:.69'( 69-9?$ ?%.) "'>%$'AJI " #4 ;G@ .@ 8## $#= / H6 %B.=>B:. '( $'( $ GC.6 : '77%##6 )" &F E I0 0$ 6 &#6 '( 6 - $ 6 '(#$ 8$ 7 F@CF %#;/014 :$,8/J # /- 6 $ JI#6 " )@% H0G$3-(".)/- $ H 6 8/ 97%6 .- K@ )- 6 -(".- 7$3F'C:- $ 014G: .@ % 0/$/ 8/JIJ "#6 - 8/ 97%.7$ ,6 :- '##$ ?$

7E

)K9'(5"#=(B:"'?2 5'' "?6 'AB/.:2 5@ : " 6 - E - '( )/'C#-9B/,CB)' )G' 6 / "? 69-9 E,8 ' -> ,B.. )". 6 CB/ -9B#6 6 / "? 69-956 - B: JI#6 )"'A.:7- .- J & J<:- 6 / - B)#-- B:K'(6 %?6 .)/'C. J D)/'C)G .- 6 = $* ++ $)E)E %)'7"I/ @(#6 - K 6 ('( # #)-> ! :#76 = ! 'CJ "%# #6 ".-:.-3" #--9B/ / #-9B/JI.- 6 = 0G J ) 6 # * +#E3FB/,%?6 H,'C !>'C:=A,->I:7- (.- JI.- 6 = $ *)* E K. )/'C#K-9B/3%./'( 6 #- 69

(

!>' /'C?

X ak (b + c) ≥ a+b+c bk + ck

k≥1

E /)" ?"") ) 69 =(B.6 < /! %./'( 6 #- 69)E 6 '(-:B/)/'C#I- B)#X a3 (b + c) X a3 ≥ 2 ≥ a+b+c 3 + c3 2 + c2 b b

K- B: )- :)- B:,CB)' )G' 6 : " 6 -

)

P aλ+1 $&DB:'C E 6 λ≥0 bλ + cλ

P a3 a+b+c ≥ 2 b + c2 2

'C) .= 6 #%- B: >! -G 6 )

)? $/B:H?"?%%#->5,%./'( 6 #- 69<-

n

'(6 # E

!#"%$&''(*),+.-/&,01+2-3547698:;<:=?>@BACEDGFG8HI=5J:;KFL>58NM/OQPL8R,@S>?C/TUPL8H=J:/SITWVUXY" Z[1\1\1],^`_bac,da ^ d/fg_/h,[ d5ak_ld akl d/f1m _nd/f1o*ac,da ^ dn\9] f.a f ABC P "Nr cG_pe \1_/l\9_f1o eji [ d/lUqtli ],u e acG_p\ df1_ ].q aki ] uB_/_5a^ e da e 0 acGe _ eIs i i 4ABC ak] BC AB X 0 P \9_lk\1_/f1o [ dlUqtl],u df1ovacG_p\1_/l\9_f1o [ d/lUqtl],u uE__?a^ d5a P CA BC Y P t e s i I e s i " ak] uE__?a^ d5a AB CA Z +G"#w q 0 \1l]y,_pac,d5a " ^ fbacG_ f.a_/l ],lx].q P e e 4ABC [XY Z] ≤ [ABC] e e &G"#w q 0 \Gl]/y,_|ac,d5a 0 0 d/f1o _/^B],fzacG_ l [1u l _{].q dl_ ske s ske ski P f1i_/e d/l " ABC X Y Z ] s ijie "} 9~.:/S,JIH:2A2CB/:=F,H:~.>HIHCRTUXLR,WR=`R/HIT,M`R5;KR/N"

0 0 +1" ? _ a _acG_g\1] f.a^B].q D E F ac e 0 f.a_/l^_ a ],f{].q ac P X BC P Y e e 0 s e 0 " ac l_^\1_ a y,_ Z fe _ s e i CA P Z e dAB 0e s _ f1o ◦ XD ⊥ BC ∠ABC = 60 ^ 0 | " r c,d5y,_ c.[1 ∠P XF = ∠DXB = 30◦ ac,d5a ^ 0 ◦ e P F = P X sin = 1P X " Z 30 u d/l 0 2 P X = 2P F P Y = 2P D e j e i i " df1o PZ Zf =_ 2P E 0 e s ∠DP Z = ∠XP F =0 60◦ df1o ∠EP X = ∠Y P D = 0 60◦_c,d5y,_ ◦ ∠F P Y = ∠ZP E = 60 "x _f _ 0 ◦ ∠XP Y = ∠F P D = 120 s

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

X

E

F

P

B Y

D

C

Z

[XP Y ] [F P D]

=

K236DI36g]@ADI3

PX · PY

PF · PD

[XP Y ] b= KC4[F B=7 P D] [ZP X] = 4[EP F ]

2P F · 2P D

=

PF · PD

b 6HSNHF1Q5DI1 >

~3KA5A3

= 4

b

[Y P Z] = 4[DP E]

58;=<

A

[XY Z] = [XP Y ] + [Y P Z] + [ZP X] = 4 [F P D] + [DP E] + [EP F ] b = 4[DEF ]

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

k

58;=< £ @8~143[G i P@CE 58;=<143[G M=3G*K2P3D 51GI= HFGIBpx<2 3T g = y bc P7#FHS7= ~36z1F1 k 4ABC b 6HQ; e 3 ;=@8~P; ~3PKA5A3 x+y+z =k

F

∠F P D = ∠DP E = ∠EP F = 120◦

B

E

z P x

y

D

C

~3KA5[A3 [DEF ] = [DP E] + [EP F ] + [F P D] = 21 xy sin 120◦ + 12 yz sin 120◦ + 12 zx sin 120◦ =

6HQ; e 3

√

3 (xy 4

+ yz + zx)

b

(x + y + z)2 − 3(xy + yz + zx) = x2 + y 2 + z 2 − (xy + yz + zx) ≥ 0 xy + yz + zx ≤

[DEF ] ≤

K236DI36g]@ADI3

√

3 4

· 13 k2 =

1 2 b k 3 √

3 k 12

b [XY Z] ≤ [ABC] b HFG*K2@CBAGo1Q@A797@Cgv636;=3D561SHG> ~3EN5>T5797*B=N:3G*KA5G HQ7Å5ELp@CHS;CGÅ@A; P b 6HQ; e 3 G*K23NHQ;=@ADp5D e BC XD ⊥ BC b ~3!KA5A3 ∠P XB = ∠DXB = 30◦ 6HQNHS1S58D1> ◦ ∠P Y B = ∠EY C b =6HQ30 58;=< ; e3 ◦ ∠P ZC = ∠F ZA =793830 ~3O 3!G*KA56G ∠P XB =5DI3 ∠P Y B B e @A; e > e 1FH eb KCB=7 P Y X b ∠P Y X = ∠P BF 6HQ; e 3 ~37,33vG*KA5G ∠P Y B58DI= 3 e ∠P @A; e ZC > e 1SH eb KCB=7 P Z C Y b ∠P Y Z = ∠P CZ

KCB=7

·k =

√

3 k 12

·

√ 3 BC 2

=

1 [ABC] 4

b

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

X

Y

B F

D

E

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

~3P79383iG*KA5G

C

P

Z

6HQ; e 3

58DI3 e @A; e > e 1SH e ~3KA5A3

A B P C b ¤#7)HS;= ~ 3 63[G ∠P Y X = ∠P CA )

∠P BF = ∠P CA

∠P Y X + ∠P Y Z = ∠P CA + ∠P CZ = 180◦

5DI3 e @C1S1FHQ;=3658D b X Y Z q¬q b ?( .+ 46 6

K236DI36g]@ADI3

b P36; e 3

/¿ ´ x· µ 9¶1¶

1µ

g4DI@AN

*

b

3½

À

( $ - 6 +!1" + .* ib bÛc 7)7)B=NO3G*KA56G 143[G KA5877*H4<238J1436;=®G*K M=3G*K23 e 36;CGJDI@CH4<W@Cg 4ABC 2 Gq b 3~HS1F1=L=DI@6A3 58;=!HSg P ∈√Ω e @2@ADI27JGJ36N7,@PG*KA5G B =DI@C(0, 3)GJ@! b 6B=L2Lp@A793oG*KA56G b K23;!HA 58;=< × GHS7 BAGIHS;=3o <23= DHA(1, 3oG*KA0) 5G C = (−1, 0)

KCB=7

P = (u, v) √ √ X = u, 3(1 − u) Y = (u + 3v, 0) b √ √ √ Z = − 21 (u − 3v + 3), − 23 (u − 3v + 1)

[XY Z] =

=

6HQ; e 3

√ 3(1 − u) 0 √ √ 3 − 2 (u − 3v + 1)

 u√  1 u + 3v det  2 √ 1 − 2 (u − 3v + 3) 2 √ 3 3 2 1 4 b √ − u + v − 4 3 3

FG*K23!M2@CB=;=<25D|>@Cg

KA587G*K23!38CB=56GIHQ@A;

x2 + y −

2

= 38 √ buc 1S7,@ 3AB=561SHG> ~3i79383#G*KA56G HSg=5;=HSg P ∈Ω K2@C14!Z] HSg ≤ 3 = @AD [ABC] b P =G P ∈Ω 2 b g b 1FH4367@A;!G*K23 e HSD e B=N e HSD e 143@Cg G*K23; 2 × P 4ABC u + v − √13 = 34 P36; e 3 «8G*KA5GÅHQ7 5DI3 e @C1S1FHQ;=3658D b X Y Z [XY Z] = 0 ∂Ω

Ω

 1 1   1

1 √ 3

! @(8 .FC3@ $ )"#.$''(:=A58 5@ 8 F@ ?$ 6 C)"'(6#)".- B:'A .- .- ,8/ 7%#7$ ) 6 #$ 0G$ 3#F 4 0$ :'A?" 6 /.%JI. 6 "JK$&@ ).'7".= $#3"- '(6 " ,E @ 0 ,4 )043$ G#' - .$0 $:F E I0 0$ 69 &#6 '( 6 - $ 6 '(#$8$ 7 F#@ F %#;/014 :$8/J # :`- 6 $ JI#6 "&:I- B:)/'C)G'E

#" $&/'2'5(7)p+2-&.0E+2-354 698:;<:=?>@ A2C{M/R/ VN.H />?8=H@.R.@B@1> ,RSjStR.@1:/SH@2T ,RSjStR.@1:/SH@2TN~?;KRHIN"

PN:= : :2>?8:R/ 8 L.> T

" _5a Z[1\1\1],^`_nac,da _EacG_ c,d/^ df1o 4ABC ∠A = 90"B◦r cG_E\9∠B > ∠C_^v],f H df1o ^ q ]G]2a ].qUacG_E\9_lk\1_/f1o [ dl%qtl],u a] ] f.a 0 ets i e A BC " Z[1\G\9],^_neac,da B ije ^pacG_EqBC acG_*u ll],l uEdm_*].q f|acG_ f1_ ]G]2a ].qYacG_ e e e j i e e B AH D 0 ac,d5a \9_lk\1_/f1o [ dlNqtl],u ak] ^

©

&¿

¹ pµ

= *4%"$( )*$#%+ ,- 2 3 ) 63+ -" )* 7* !0 ,(68;:Tb 6HQ; e 3 e @ADIDI37)Lp@A;= 5DI∠BB 3 e @ADIDIA 37)Lp=@A;=∠B
+

FG AH

=

B0C BC

B0B

+

BC

= 1

58;=)JIJ<7 $H-9B/ 45-9B/'( : #4 ;G@ .@ 8## $ #= /36 %B.=>B:. '( $'( $ GC#6 5:5' 77%.#76 . &F E K00G$ 69 &#6 '( 6 - $ 6 '(#$ 8' $ 7 #@ % 0 /$/ 8/JIJ ".69-98/ 97%.7$ 6 /->'##$ ' $ 7 F#@ F %#;/014 :$8/J # :` - 6 $ JI#6 " &:I- B: )/'C )G'E

a b

0 d/f1o

R PLR/8k8>?8:T%VN.H />?8=HJIRGJ

d/l_#\1],^ a y,_ l_d f,[1u _ l^ " l]y,_pac,d5a e e i c 1 1 1 ab bc ca −2 + 2 + 2 − 27 + + a2 b c c a b " 2 # 1 1 1 1 1 2 1 1 2 " ≥ − + − + − 3 a b b c c a ~,:/SGJIH:2 A2C 9 >`H F1R.:T2=J.@1>5KJtT %:2 HSjSI=xDL:/SS> 5HtR,J> G=5JIHJ,J>TQ/2 : 8:2QJ:T Z[1\1\1],^`_pac,d5a

0

#"$&/'2' *)1+.- ,0K+.-/34 698:;<:=5>/@ AC M5:= > GH= 69:/SHJ > .H Rn@1> DGRGJIRS1CR/TUPLR/8 >S:21RTN~5;KR5HtQ"

\ d/f1o 1 1 f mpacG_vl mc2a ^ o1_vd/f1oE^ uE\ q f1m 0 _v^__ ac,d5axacG_ f1_h.[1d a ^ e e e e ie ,e e ije Ee _h.[ y.d _f.aYa] e i 1 1 1 1 2 ab bc ac −2 + + ≥ 27 + + 3 a b c c a b ac,da ^ 0 e 1 1 1 2  ab bc ac 2 0 + + + + b c  c a b  ≥ 1 a 3 3 w c c ^ _/d/l akl[9_ acG_ f1_/h,[1d a " ets e ski i / ie

O

"!$#% &(' )+*)+*,'.-,/102435&678&9:%&;=A@,B1CEDF*1)+/G>H1&>JI@+&KM=L@ NPOP7N<J&6$@=;G&QR&TSU=H8V >WIC=XY-1/+Z[)\XY]Y3%*,^ @1B,Z[B`_\/,-,*,>A@,B1Z[B?_\/,-1*,>Ra$*1)?C8X BbB,C0cJ/+ZedG/Gf=*8-GX C=BFK9U[gAQhQJNi<4j$a=&TUe;=N:P>AN;1@=UeNkjH1&Rkmla$N;8;,>B,C0 Hnc&6;=N8@6Aj$9UekmU.QhQJU[N1>&AC8f=/G'Y*@8] B8]e/IC8X.-,/+Z[)\X.] 3,>@,B1CE&(C=f8/G'.*,>pI@+&K(kmU[HncWNP;a=& 6*Pq=/+C8>O1Z[B1Cnr\/`KkmUec TV& ;1:a$NPQWH8s4N1>$anZ[B1)+*8-G>$6*1tEB,C=X BFKHncJ6U.@<4gRlcJN6bD4ja$6&9;=N:P>$anZ[X )+]e*,' >RI7nK HncWUelHI6R<4U.@F>kiX )?)+*Pq,Z[XR@8*Pq=] u/+ZvC@8] B8]e/Hn*,' './Gf8/G>(D,*Fw' X C8>pkxg>5I$@G&KD,gRNic5gTy&69A>pln*1Z`] BF'./+)\> Qhki>TI@+&KAy&;=<4cWNP6Dv&QWgpI$@F>IZv)FqF' X C/+C8f43,tECPB,)\Xeq,tb>UYC=CP)+24ZGq8r[z1>&q,)+]YZ[X BFK9hjn7Uell5D,gRchQp@Pg(QT>

G.' ->#$10 $ 7 4 0 :EF 7H0 )0 )0 :$ -9B/) $:'C= A 4 ;G@ .@ 8## $ #= / 36 %B .=>B:. '( $ '( $ GC#6 : 5'77%.#76 .&F E K00G$ 69 :.69'( 69-9?$ 6 '(#$&8$ #@% 0 :$ / 8/JIJ "#6 - 58/ 7%#7$ 6 /->' ##$ ?$GF@CF %#;:04 /$ 8/J # / - 6 $ JI#6 "&:I- B:)/'C)G'E G: = 7 )GH- B:

!> #D 6 .%%./'( 6 #- 69GE@ ! !>' $ $ $ $-9B/

xk > 0

n X

k=1

1 x2p k

!m−2p

m X

− nm

k=1

1−2p

x1 · · · xk−1 xk

k = 1 2 ... n !−2p

xk+1 · · · xn ≥

" $&''( )W+.-,0v+2-354 698:;<:=5>/@AC

#>2JFL>58SIRL@2=/" 0 _?a 0 0 # e y,_/f h.[10 d2o,l eji d50 ak_ld i 0 ABCD i P Q 0

1 n

X

1≤i<j≤n

1 1 − xi xj

2p

m

O M/O%~>/>5G2T RSJAG:2E>ST*JFL>

0 0 _BacG_*u o d/f1o R S \G\9],^_gac,e d5a 0 " NZ[1 \9] f.a^*].q df1o l_^\1_ Ma y,_ e s e i AB BCd/f1CD DA AC a d5a BD "7 _?a acG_ o dm],fGd ^ o f.ak_l^`_ b _acG_|\1] f.a ^[ c{ac,d5a e i AC BD e s E O e s " h,[1doGl da_/ld ^ dv\Gdld _ G ] m5ldu ei l]/y,_{i ac,Nda EM O e ii i cG_/l_ [OP AS] = [OQBP ] = [ORCQ] " = [OSDR] l_\1l_^`_f.a^ acG_ dl_d ].qKh,[1doGl da_/ld [W XY Z] ei i W XY Z ~,:/SGJIH:2 AC M5:G>SK~ F.S:=5AG>58 /T PLR2C=H@1>?T GTWV< ~ %"U$ o ") 9>58C#=HIHSIR8U=5:/SGJIH:2G= %>58>v=/,A2HJJ>@|AC pOUXLH ;1;*M5:2FGG=?:2GT PN>?R />?8`J:2,T QTpVN ~ W/:=F,H:B~,>`HtHtCR/T XKR.WR/=RHTYM`R5;KR/ #69>J>?8 N O b:G:TpP1H:/SIR VQ,H />58=HJCT /R {HI8[email protected]/T D xT V< ~ 5H /FL:9/T<69:/StBDL:2E|.HJtCnDL:/SjS> ,> T pHtKJ>? 8 pR />5, T T V< ~ LR/K@EJFK>W;K8:;<:=5>?8" 4

G

Jx

P

R

M

da ],f ac _/f.akl_ df1o qtd ak],l e i e e s s A A .......... ],f.a] df1o ],f.a] ...... ..................................S ... ... ........ 2 M P AS ... .. ..................................... D .... ... " PS w aq] ] BD . . . .. ... ... ..... ^*ac,da d/f1o ..... ... ... ... . . ............ ... .. . . . ......... ... . . . . . . . . CBAD iji ......... .. . ... . . " Z Pf S_ k BD . . . . . . . . . . . . . . . . .. .. . . .. .......... .. .... .. . ... .E.. ........ ... .. . .. e s MO [CBAD] = 4[M AS] 0 P "n k_/BD .. P............................. . N ................................................ df1o _nm_?a f _ 0 .. . . . . . ......... .. ... . .. . . ... . ...... . ................... .... .. s BD k P S d/f1o MO k P S ............ ..... ... ... . .... ... .... .. .................... ........ ....... ......... ... .. ........ ...... .. .. ......... M . . . . . . . . . . . . . . [OP S] = [M P S] ......... ........ .. .. ......... ............................................................ .... ........ .. ... .. R .... ........ .. ... ......... ... .. .. ............ .. " ... ... O ................... . ... .. .. ........... . 1 . . ... ........... . B .. .. ........... .. [OP AS] = [M P AS] = [ABCD] ... ... ........... ... ........... .... .. 4 .. ............ .. .. ........... . . . . ........ . . r cG_#^duE_ l_/d/^`],f f1mp^`cG] ^xac,d5a Q .......................................... ................. e ............... .. C 0 1 [OQBP ] = [N QBP ] = [ABCD] 4 0 1 [ORCQ] = [M RCQ] = [ABCD] 4 0 1 [OSDR] = [N SDR] = [ABCD] 4 m y f1m acG_voG_/^ l_/onl_^[ a " e e e i ad_/^

o

&(' )+* )+*,'Y-1/10 243 kmU[HncWNP; a=&6*Pq=/+C8>O4Z[B,Crv/`K HncW6UY@n<4gAlcWNP6 D4ja$6&T9;8Nn:P> anZ[X )+].*,' >TI7KyT&T;8<8cJN6%Dv&QWgpI$@F>IZv)Fq,' X Cn/+C=f131tEC=B1)\X.q1tb>UYC=CP)+24ZGq8r[z1>&Tq1)+] Z[X B+KA9pgpI(SQJNyWkM&QT> ;1B,Cr.B1)+]./+Z[>TH1&(>hI@+&K:AIOPNPUs4cR&g$>()+][q=0P/+Cn] >R9*1CkiX ' ' )WHn*,' 'Y/Gf,X B=]./EUYC=)+] XY]vq8]e/G><,*1Z?*1C].*,>g(QKB,C0 <8U\<Is ngAQR&6TI>Hn*1t B,V C/ )+ ] X >$6*1tEB1C8X BPj

Ó

ÕiÓ b

88 Ø ©

´ !0 /()(#

²¿9½ %"! »9$4%"!0 )*-¾99¶

(7

" "=b

; 7)B=L2Lp@A793G*KA5GG*K23iL=@CHQ;CG7 1SHQ3E@A;PG*K23o1FHQ;=3o7,3N:36;CG M5 ;=N <G*K23OL=@CHQ;CG G*K23OLp@C× HS;C4ABC G 1SHQ37P @A;G*K231SHS;=3O7938NO3;CG 1FH4367P@A;G*K23BC 1FHQ;=3 P CA Q 7*B e KTG*KA56G 7,3N:36;CG HS7579AB=5DI3 b 6B=L=L=@A7,3EgQB=DG*K236DÅG*KA5G AB MNP Q √ AM AC + 2AB b = √ AN AB + 2AC

¡¨KA5D5 e GI3DHF)3

§¹ ½ »

µ

4ABC

b

À ¾

( 4, 6"-"+ 3 9 $+ (!" , 63$; - ,!1" .* ib 3P~HS1F1¨L=DI@6A3TG*KA5G AM AN

b

=

¿ ´ M· µ69¶1¶

" ,# .+ *4, *

√ AC + 2AB √ AB + 2AC

HSg58;=<@A;A1>OHSg

∠BAC = 3π b 58;=< 3G 4 á a = BC b = CA c = AB ¡¨@A;27*H4<23DG*K23WK2@ANO@G*K23G|> [3 G AB b h A, U á B

sin A − cos A =

~3@2MGI56HQ;

AU 2

√ π 2 sin A −

AB = AC

@AD

A .......... Q ........................................................................... P

AQ

58;=< M23¢G*K23¢HQN56367f@CgL=@CHQ;CG7 5;=< M563@Cg¨G*N DI37)Lp3 Ve GIHA31> b K23; HS7G*K23THQN K23 BU V C 7,CB=58DI3 58;=

4

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

M

U

N

C

V

π = c2 + a2 − 2ac cos B + 2

= c2 + a2 + 2ac sin B = c2 + a2 + 2bc sin A

= c2 + (b2 + c2 − 2bc cos A) + 2bc sin A

= 2c2 + b2 + 2bc(sin A − cos A) b √ 2 √ π = b + 2c + 2 2bc sin A − −1 4

KCB=7

b √ 2 √ ~K23DI3 π AU 2 = b + 2c + m m = 2 2bc sin A − −1 4 √ 2 b 6HS; e 3 G*K23K2@AN:@G*K23[G>fHQNL21FH4367OG*KA5G 6HQNHS1S58D1> 2 AV = c + 2b + m AM AU G*K23[HA36; e @A;=
AV

√ 2 √ 2 b + 2c b + 2c + m √ 2 = √ 2 c + 2b c + 2b + m

KCHS7P7)HSNOLA1SHF367iGJ@

b

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q " a 5 ije sin(40◦ ) < 12 q 0 %]2a_9l^ka ac,d5a 5 ^ _/h,[ y.d _/f.a a] 1 5 e e i sin(40◦ ) < 12 1 − cos(80◦ ) < 12 2 0 c c ^NacG_^d/uB_Wd^ "x _5a "Nr cG_/f ],l 1 cos(80◦ )" > sin(10◦ ) > 16 c = sin(10◦ ) 0 l],6u ets e _ 3 1 ◦ ◦ 3 0 0 s −6c + 1 < 0 d/f1o _ dl_#o1],f1_ c > 16

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R

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∠P F1 F2 = ∠EF A = ∠C = ∠DF B = ∠P F2 F1 " _ " _f _ 0 x d/f1o ^`_ 0 s P F1 = P F2 e e P D2 = P D3 P E3 = P E 1

E1 F1 cot A + F2 D2 cot B + D3 E3 cot C = (P F1 + P E1 ) cot A + (P F2 + P D2 ) cot B + (P E3 + P D3 ) cot C = (P F1 + P E3 ) cot A + (P F1 + P D2 ) cot B + (P E3 + P D2 ) cot C = P F1 (cot A + cot B) + P D2 (cot B + cot C) + P E3 (cot C + cot A) P F1 · AB P D2 · BC P E3 · CA = + + CF AD BE P F1 · sin C P D2 · sin A P E3 · sin B = 2R + + CF AD BE PN PL PM [AP B] [BP C] [CP A] = 2R + + = 2R + + AD BE [ABC] [ABC] [ABC] " CF = 2R

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x

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A

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k=1

n X

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