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"!#$!&%(' *) +-,/.103254687/9:,/,/6;03@?;AB.C6$AB6 DFEG,DFEIHJ.CK8AIL M...

Author: Canadian Mathematical Society

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4

kV/,jD ?;,zDFEG,=HJ6 N ,/6ID1.C,gAILD1.aHJ4/6;]jZq, k >5>5y;93,zDFEB4PD`DFEG,=H\6 N HJ. N ZJ,TDCAIy N EG,/> I ABC N 46;K 4PD 46;K QB.C,P>oe;, DCH XB,PZ ukEG,TZ[HJ6;,VDFEB.CAIy]PE 46;K 9:,/,bDa>|DFEG, AB Y X I kVBAIH\6ID N kz/X ,jD 46;K HJ6 N H\. N Zq,@4bAC D 0 ?G,TDFEG,@eAIH\6IDOAILpHJ6IDC,/.a>o, DCHJAB6:AIL AB HJ>dAB6DFEG,@>RM ,/]b9:,P6ID X0 >5y N EsDFEB4PD kz;.CA/XB,T DFEB4bD Q CM Q4/6;K 4.CL, 0 X C X L = CM A L Y N AIZ[Z\H\6;,/4/.H\Lp4/6;K3AB6BZuxH[L k

¬o

AB = AC

/,bD Q Q46;K ?;,eAB>FH[DCH XB,.C,/4PZB6By;93?;,P.a>>Fy N ETDFEB4bD 2 k c a + b2 + c2 + abc = 4 ;.CA/XB,VDFEBa4PD b k a+b+c≤3

E~@¦} ( 6|,} 9P +* 2 × 4.5 1H %$

O

kH\6;K4PZ\Z.C,P4/ZeAIZ uG6;AB9sHJ4PZJ>

P (x)

>Fy N EgDFEB4PD

P (a) ∈

H\93eBZ\HJ,/>DFEB4bD

a∈

k

k^/,jD ?G,:43I,/K8,bZ\Z[HJeG>R,Gk$/,bD ?;,346n4/.F?IH[D1.a4/.vuneAIH\6IDTAIyBDC>FHqKG, k{EG, B1 DC4/6;]P,/6IDE Lq.CAB9 D1A DCAIy N EG,/> 4PD 4ze;AIHJ6ID k|/,bD ?G,z4ze;AIHJ6IDÀB6=E DFEG,VZ[HJ6;, E kTBAB.O,P4 N C >5y N EBDFEB1 4bD E E:1e;AB>5HDCHXBB,=2HJ6IDC,]P,P. QpK;,b6;, AIL B C B C = C B iHJ>d?GAIy;6;K;B 1 1 1 1 1 2 i+1 HJ6DC,/.C93>dAIL HJ6DFEIH\>934/6G6;,/.rk=;.CA/XB,TDFEB4PDDFEG,@>R, By,P6 N , ,/K: H\6 B {B } i i DFEG,=eGZ\46;,Gk

¨

r

§

k~v6x4VD1.aHJ4/6;]jZq, QBKG,P;6;, D1Ag?;,dDFEG, N H\. N Zq,Da46;]P,P6IDD1A Q/D1A Q4/6;K ABC AC D1A3DFEG,gHJ6 N H\. N Zq,sAILw DFEG,gD1.aHJ4/6;]jZq, Ca Qp4/6;K{ZJ,bD ?G,gDFEG,x.a4KIH\y;AB >zAIL k+-,b6;, ra 46;K HJ6{DFEG,3>5493,gcAILDFEG,zD1.aHJ4/6;]jZq, k

ª

ABC

k/,jD 4/6;K ?;,:HJ6IDC,]P,P.a>g>5y N EYDFEB4bD k/,jD ?G,t4t>Fy?G>o,bD=AIL kFDa>d4/6 Q nQ u nQ P ({1 . . . n}) HJ6IDC,]P,P. >5y N E@DFEB4PD 46;K Q Q Q F G∈F k-;.CAPXB, t 1 ≤ t ≤ n {t t + 1 . . . t + k − 1} ⊆ F ∩ G DFEB4PD k n−k

«

|F | ≤ 2

k$IAB.T,RXB,P.vu8.C,P4/Z`6By;93?;,P. KG,P;6;, QdDFEG,xLq.a4 N DCHJAB6G4/ZeG4.DVxAIL k:;.CAPhxi ; XB,gDF= EB4PDRmin({x}, QLAB.d,RXB,/.vu{ 4bDCx}) HqAB6G4PZ6By;93?;,P{x} . 46;K:,RXB,P.vu3e;AB>5HDCHXB,=.C,P4/Z6By;93?;,P. x QIDFEG,/.C,@,5/HJ>1DC>O4@e;AB>5HDCHXB,TH\6ID1,/]P,/. >5y N ExDFEB4bαD ε n k 2

hn αi < ε

9(

,{6;,F/D=]jHXB,:4684/ZD1,P.16G4bDCHXB,:>RAIZ[yBDCHqAB6YDCAne;.CAG?IZq,P98AIL`DFEG,YlTAB6;] wAB6;] wAB6IDC,/>1DRQ`LAB.VRAIZ[yBDCHqAB68H\68DFEG,$+-, N ,/93?;,P. ª/ 6By;9:?G,/.|AILDFEG, k

#pEIHJ6G4 &

ª k ª/ 5 NM

* 4 8 ª/ ªª ª,« *4&U *4> µ+124 d² , ¶, 1 k

+-,bDC,/.C9H\6;,=4PZ\Z;LJy;6 N DCHJAB6G>

f :

f x + yf (x)

→

>Fy N ExDFEB4bDRQILAB.`4/Z[Z

k

= f (x) + xf (y)

Q

x y∈

Q

,./38 4 0/38 * , -+/21;*4 84 +/ 7µ , ¤ 4 1 4¤34 /21;/ -0/38 *¯ )+7.1384 7 8 4 ,;* 8`¤ 41 pEG,zLJy;6 N DCHJAB6gHqKG,/6IDCH N 4PZ\Z u f x + yf (x)

N ZJ,/4/.CZ us>54PDCH\>5,/>`DFEG,=]jH XB,/6 N AB6;KIH[DCHJAB6pQ

0

= f (x) + xf (y)

#vS &

k

hVA54PDCH\>5L[uIHJ6;]ODFEIHJ> N AB6;KBHDCHqAB6k ,oEGA1D?;f,T?GADFEx4KGKBHDCHXB,z4/6;Ks9yGZDCHJeBZ\H N 4bDCHXB,IQ/`DFEB4bD LAB. f f (x) = x 4/Z[Z k x∈ 4/6;K H\6 S QoAB9:, G40H\6;] x=1 y=0 f (0) = 0 f (x) = 0 QDFEG,/6

# &

x

0 = f (x) = f x + yf (x)

= f (x) + xf (y) = xf (y)

k

wEGAGAB>FHJ6;] >5y N ExDFEB4PD QIo,,zDFEB4PD kpEIy;>jQ HJ9eGZ[Hq,P> y f (y) 6= 0 x=0 f (x) = 0 k x=0 GyBDaDCHJ6;] Qpz]jH XB,/> f (1) 6= 1 y = 1/ 1 − f (1) 1 + yf (1) = y EG,/6 N ,BQb f (y) = f 1 + yf (1) = f (1) + f (y) Zq,P4KB>D1A@DFEG,T4?G>Fy;.CKBHDcugDFEB4PD kl=,P6 N ,IQ kG40H\6;] f (1) = HJ6 0vS QOXI4/Z[HqK{LAB.

f pEI(xy) y;>jQ = f (x)f (y) LAB.w4/Z[Z Q f (xy) = f (x)f (y) x y∈

y

QDFEIH\>O>FHJ9eGZ[H\,/>`D1A

xf (y) =

kxpEIH\>VZJ4/>FDd,

k-p,PeGZ\4 N H\6;]

1 + yf (x) f (x)

x∈

y f (x)

1 1 + f (yx) x f (x)

=

46;K 4]b4/H\6@XI4PZ\HJKgLAB.4/Z[Z x 6= 0 y∈ 46;Ksy;>FHJ6;] QxH\6 =ª 0jQ bQ H\>V?GADFE34KGKBHDCHXB,g4/6;K:9syGZ[DCH\eGZ[H N 4PDCH XB,;k7PHJ6 N , H\>6;AD y6 ∈ f N DCqH AB6pQIH[DLAIZ\ZJAfDFEB4bD DFEG, 5,/.Cx AgLJy; LAB.4PZ\Z k f (x) = x

x∈

,|6;A< DCy;.C6dD1Ad>oAIZ\yBDCHJAB6G>wLq.CAB9iAIy;..C,P4KG,/.a>pD1Ade;.CAG?IZq,P93>AILPDFEG, ª 5, N EG AIZ\H\>REG57PZqAPXI40$4PDFEG,P934bDCH N 4/ZppAB93e;,bDCHDCHqAB6pQ Q4/6;K 1 p KBH XBHJK;,P> 3 kn7G E >ApDFEG,=> By;4/.C,@AILp4/6snHJ6IDC,]P,P.jk p − 1

¨

)+* ,.-0/213*4 0138 8 * 4 /;8 1 4 1L *4 °6 / /;8 `¢ 4 7 8 n −1

4p − 3

7PHJ6 N , KIHXIHqKG,/> Q-e;.aHJ93,BQ5> 2 pEG,/AB.C,/9&DFEBp4PD KBH XBHJK;n,P> −21 = (n − k 1)(n /,bD +2 n + 1) Qw@4 n +n+1 n + n + 1 = kp eAB>FH[DCH XB,zHJ6IDC,]P,Pp .jkpEG,P6 k|ÂB.C,APXB,/.okQ 2 k ≡ kp = n + n + 1 ≡ 1 (mod n)

n2 + n + 1 n2 + n + 1 k ≤ < n+1 p n+1

k =

pEG,P.C,PLAB.C,IQ

k=1

4/6;K

p = n2 + n + 1

kpEG,P6

4p − 3 = 4(n2 + n + 1) − 3 = (2n + 1)2

Q

46;Kg 46;K kg~v6t46t4 N B y D1,/54/6;]jZq,/K:D1.aHJ4/6;]jZq, O P Q ZuIHJ6;]z.C,P>oe;, N CD H XB,PZ u3AB6>FHqKG,/> 4ABC 6;K 4 .C,T>5y N EgDFEB4bD AC

BC

BC AP = PQ AB

7EGA

Q

Q

QB4/6;K

4/6;K

AC k BQ = PQ AB

4.C, N AB6 N u N Z\H N k

)+* ,.-0/213*4 !1;7µ 89, +/ 1 ,:,;8 * -+8 4 r¢ 4 7 8

AIL

O P Q

C

,go,>FDa46;KB4.CK 6;ADa4PDCHJAB6tLAB.ODFEG,>5HJK;,P>bQw4/6;]jZq,P>bQw46;K N H\. N y;93.a4KBH[y;> k-+-,b6;, kÒ>5H\6;]dDFEG,@]jH XB,/63, By;4bDCHqAB6G>jQ-]jHXB,P> k 2 c2

= P Q2 = (a − kb)2 + (b − ka)2 − 2(a − kb)(b − ka) cos C = (1 + k2 )(a2 + b2 − 2ab cos C) − 4kab + 2k(a2 + b2 ) cos C Q = (1 + k2 )c2 − 2k 2ab − (a2 + b2 ) cos C

46;K3EG,P6 N ,IQ c2 = 2k 2ab − (a2 + b2 ) cos C

};yBDRQIy;>5H\6;]VDFEG,sP4bbQo4/9:, ApEIy;>jQ;BZJ,bDaDCH\6;] eATAIL Qo, = N DCABU.`B LAIZ\ZJADFEBΓ 4PD Q46;K H\>-AB6@ KIH A AIL − ρkEIHJ> .C,/94.10de;.CAPXBHJU K;,PA>4=6=U ,P4B >au N AB6G>1D1U.Cy N DCHJAB6=AILIeAIH\6IDC> Q >54PDCH\>5L[uIHJ6;]DFEG, N AB AB6;KBHDCHqAB6G> 4/6;K QrAIQ L 4PD kOEG,/6DFEG, N HJ. N ZJ,To>FHJ6;]zDFEB.CAIy]PAB E 93,,jOC DC>-DFEG,g>5HJK;,P> U U C AC 46;K 4]b4PHJ64bD 46;K QB.C,P>oe;, N DCH XB,PZ uk

* /;8

BC

P

Q

* 4 8

hV,F/DT >RAIZ[yBDCHqAB6G>|DCAe;.CAG?IZq,P93>AILDFEG,g7PH\6;]b4e;AB.C,$4PDFEG,P934bDCH N 4/Z M Z uB93eBHJ4K ª/ª Q M e;,/6 7, N DCHJAB6pQpI4.D QG]jH XB,/6sHJ6 ªI ª S ª S/ k

4

k-/,bD

f (x)

¤

?G,=4TLJy;6 N DCHqAB6g54PDCH\>5,/> Q

f (29 + x) = f (29 − x)

LAB.4PZ\Z/XI4PZ\y,P>AIL k`~L EB4/>,F/4 N DCZ uTDFEB.C,/,d.C,P4/ZI.CAGADC> Q Q/46;K QKG,bDC,/.C9H\6;, x f (x) α β γ DFEG,VXI4/Z[y,@AIL k

)+* , 8 * 08 / 1 ,:1 4 1 * ,:,8 +8 *4 /N¶5*8 4 7 °, P 0138 8 * 4 /;8 1 4 1 *4 ° 6 / /38u¢ 4 7 8 8 * 8 SU 4, , u« ¶ 84 0 © ) - / * U 1L¶ 08 41 8T 1 d¦0 ¦ -+8 4 * 1 8O +8 4/2124 ,3* P +* - 1 01 8P 8 87 8 4¦ ¨ © N« ¬ 4 ¬ 1 ,¯ 1 ° - 1;8 41 8T 1./ s ¬ 0/38 ,3*0*o²

`¬ 8x1 8 4 U 1B¶ +8?³ * , -+/21;*4 α+β+γ

EB4/>,F/4 N DCZ ugDFEB.C,,T.C,/4PZ.CAGADa>-4/6;K EB4>DFEG,T>o4/9:,dXI4PZ\y,T4PDèAIH\6IDC> 7PHJ6 N , a> uG99:,jD1.CH N 4f?;I A yBD QAB6;,3AILDFEG,.CAGADa>T9sy;>FDV?;f, k /,bD ksEG,3ADFEG,/. DbQ 4/; 6 K 29Q9y;>1Dd?G,s>auB9393,bD1.aH N 4?GAIyBD 29wEG,/6 N ,BγQ = 29 493K αLAB.>oABβ α = 29 + x 9:,T.C,P4/Z6By;93?;,P. kEG,/.C,b29 LAB.C,BQ

M

β = 29 − x

x 6= 0

α + β + γ = (29 + x) + (29 − x) + 29 = 87

O

k

0

k:2oAGEB6YZq,bL\DdD1A@e;4/>FDT e;9 4/6;Kn.C,/4 N EG,K D1A5493,xKB4buwkTlT,g6;ADCH N ,KDFEB4bDO4bDO?;ADFEsDFEG,x?;,/]jB 9H\6ByBD1,P>de;4/>FDO e;A 9 DFEG,g HJ6G6BH\6;y] 46;KzDFEG,V,P6;KgAILDFEG,OD1.aHJepQPDFEG,9H\6ByBD1,dEB46;K=934K;,-DFEG,>o4/9:,4/6;]jZq,dAIL K;,/]b.C,,P> 110 |HJ9eGZ[H N HDDFEB4bD k`7PH\6 N ,=4/646;]jZJ,@AIL jQPk pEIH\>-H\>DF12 EG,zxDCH\= 9:,= AIL−DFEG,z

§

k-/,bD

y 55 k y − 30 + = 12 3 55 QGLq.CAB95,/>DFEG,z.C, N y;.1.C,P6 N ,z.C,PZ\4PDCHJAB6sAIL pEG,z>o, Iy,/6 N , DFEG,=e;.CAG?IZq,P9nk~v6;KG,,/x Kpn Q = n/2002 +Q;n4/6;−K 1 nxn+1

pEG,P6pQ

ª

x1 = 1/2002 n+1 n(n + 1) = n +n = + n2 2002 2002 n k = (n + 1) + n − 1 + 1 = (n + 1)xn + 1 2002

x2002 = 2002

k

k*BAB.gHJ6IDC,]P,P.a> QOZq,jD 46;K k 2 dn = gcd(an , an+1 ) +-,bDC,/.C9H\6;,zDFEG,TZ\4.Cn]P,P>F≥ DXI4/1Z[y,@AIL an k = n + 500

)+* , 8 01;8 8 * 4 /;8 1 4 1L *4 °6 / /;8 z¢ 4 7 8 9,;* P+* -1 01 8P 88 7 8 4R¦ ¨ ~© N« z¬ 4r¬ 1 ,¯ 1 R°0- 138 41 8T 1./ y¬ 0/38 ,3*0* ²

` ¬ 8x1 8/Nµ 8d * ,.-0/213*4 ¬ 4¶5* 1 8 /Nµ 8 81./3* dn

pEG,=4/6G>1 QG4bDaDC4PHJ6;,/KgO?GADFE 4/6;K QnH[D`=L AI1000 Z[ZqA`DFEB4PD dn

pEG,P6

an

an+1

dn

KBH XBHJK;,P>

an+1 − an = (n + 1)2 + 500 − (n2 + 500) = 2n + 1

¨

Qw4/6;K N AB6G>o, Iy,/6IDCZ upQ

KBH XBHJK;,P>

k

4/Z\>RAtKIHXIHqKG,/>

2 dn n(2n + 1)k= dn Q/R, kEG,/6 H\>|KIHXIHJ>FHq?IZq,V?/u Q Q4/6;K k dn = 2001 = 3 · 23 · 29 dn 29 7PHJ6 N , Q/bQ 2n + 1 ≡ 0 (mod 3) 2n ≡ −1 ≡ 2 (mod 3)

M

7PHJ9sH\Z\4.aZuQB>FHJ6 N ,

n ≡ 1 (mod 3) 2n + 1 ≡ 0 (mod 23)

k

QFHJ6 N ,

2n + 1 ≡ 0 (mod 29)

Qo, p,P934PHJ6;KG,/.EG,AB.C,P9&4/6;Kty;>FHJ6;]@DFEG,s>FDa46;KB4.CKt9:,jDFEGAGKpQ oAI23 46;Kg1= D1,P9 3 ·AIL29N AB= 6;]b.a87 y,/6 N ,/> M3 = 3 · 23 = 69

667x 87x 69x

Q

≡ 1 (mod 3) Q ≡ 1 (mod 23) k ≡ 1 (mod 29)

$# & # &

#\ & # & # # &

}Bu .CAIyBDCH\6;,s9:,jDFEGAGKG>jQFDe;AB>5HDCHXB,s>oAIZ\yBDCHJAB6G>TAIL \ Q Qw46;K D1At?;, Q Qw4/6;K Qw.C,/>5e, N DCHXB,bZuwkl=,P6 N ,IQw4>RAIZ[yBDCHqAB6:DCA:DFEG, = 3 ?/ >auG>1D1,P9 x1S =Q 1ª QBx4/26;K = 9 HJ>O]jH XBx ,/63 u 8

# & 0# &

# &

& k

n = 1 · 667 · 1 + 11 · 87 · 9 + 14 · 69 · 8 = 17008 ≡ 1000 (mod 2001)

wAB6XB,P.a>R,bZuQFHJ6 N , dn KBH XBHJK;,P> k n = 1000

«

dn

dn

&

2001

k`~qDwH\>]jH XB,/6zDFEB4PDDFEG,e;AIZuB6;AB9H\4/Z EB4>DFEB.C,,KBH\>FDCH\6 N D p(x) = x3 +ax2 +bx+c e AB>FH[DCH XB,THJ6IDC,]P,P.|.CAGADC>O46;K kV/,bD k~qD 2 p(2002) q(x) 2002 EB4>V6;A: HJ>d4/Z\>RA:]jHXB,P6:DFEB4PD|DFEG,xe;AIZuB6; AB9H\4/Z = 2001 .C,P4/Z= .CAGx ADa>/− kg+-2x ,bDC,/+.C9 H\6;,gDFEG, p q(x) XI4/Z[y,@AIL k

)+* , 8 01;8 8 * 4 /389124 1L *4 °6 / /;8 ¢ 4 7 8 - / * U 1L¶ 08 41 8T 1 d¦0 ¦ -+8 4 * 1 8O T84 /21 4 4 ,3* P+* -1 01 8P 8 87 8 ¬ 8x1 8 /Nµ 8x * , -+/21;*4VU 1B¶ +8z¶5* 1 8 /Nµ 8 81;/;* a

pEG,eAIZ uG6;AB9sHJ4PZ DC40,/>=AB6Y,RXB,/.vu:XI4/Z[y,sHJ6tDFEG, 2 HJ6IDC,/.vXI4PZ kx7PHJq(x) 6 N , = (x −EB1) 4>z6;+ A{.C2001 ,/4PZ.CAGADC>jQp4/Z[ZwDFEB.C,/,.CAGADa>TAIL 9y;>1D|?;,T[2001, ZJ,/>5>DFEB∞) 4/6 kO+-p,P6;q(x) ADCHJ6;]dDFEG,T.CAGADC>OAIL ?/u Q Q Q1D|2001 7PHJ6 N ,@p(2002) ,/4 N ExLq4 N DC= AB. (2002 − x1HJ)(2002 >-4=eAB>FH[− DCH XBx,z2 )(2002 HJ6IDC,]P,P.RQ−FDpe;AB>5HDCHXB,H\6ID1,/]P,/. >Fy N EDFEB4PD ,P6;KG>pV4bD-DFEG,,/6;K AIL k=EG,/6 f (N ) , Iy;4/Z\>pDFEG,|f6B(N y;93)?;,P.wAIL/Lq4 N D1AB.a>AIL DFEB4bD934juz?;,`LAB.193,KdH\6 kN 7PHJ!6 N , 10 N ! 10 = 5×2 46;K:DFEG,/.C,s4/.C, N ZJ,/4/.CZ u{L,rdDFEB4/6 >bQHDOLAIZ\ZJADFEB4bD H\>T, By;4PZwD1A3 DFEG, 5 2 f (N ) 6By;9:?G,/.|AILLq4 N DCAB.a>OAIL H\6 kEG,/6 ?/u4V5>R,P>DFEB.CAIy]PE3DFEG,=XB,/.D1,F f

¨

#

N

HJ>

109

AIL4s.C, N Da46;]jZJ, 4/6;K:DCAIy N EG,/> H[Da>>FHqKG,/> 4/6;K 4PDe;AIHJ6IDa> 46;K C Q.C,P>oe;, N DCH XB,PZ uk7PABCD y;eGeAB>o,ODFEG,dKIHJ>1DC4/6 N , Lq.CAB9 DCAB A HJ>ADN Y 9 kH\6;KxDFM EG,T4.C,P4gN AIL H\6 N 9 2 k

)+* , 8 9-7 8 *¯;*0* / ©Pµ *'¶,z *4 1 8T 41 8T 1./ U¶,;*0*Pz 8* 8 G U 4, , d« ¶ 8 4 © ) 4 9,;* * - 1L 01 8P 8 87 8 ¬ 8t 8O 84 / 7 *'¶, * .1 /38 *¯ /Nµ 8d * ,.-0/213*4 V *¯;*0* / 4U 4 ,:, MN

2

ABCD

P

M

∠BM C = ∠M N C = ∠P N C

¨

BC d

=

u v

¨

4/6;K

DC d

=

v k

u

k

l=,P6 N ,IQ H\>W>5H\9H[ZJ4/.8DCA k C HJ>V>5H\9H[ZJ4/.-DCA 4P N C k 7PHJ9sH\Z\4.aZu4BM Q 4P M C wAB6G>o, Iy,/6I4DN DCZ upQ C

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

AB

A...............................................................................................M ..................................................................................................B ....... ........

N9 k pEG,=4/6G>1 4 KIHJ>1DC24/6 N , ÂB.C,d]P,P6;,/.a4PZ\Z up[ABCD] QPZJ,bD ?;,|= DFEG,d Lq.CAB9 DCA kO ,@HJ6ID1.Cd AGKIy N ,g4KGKBHDCHqAB6G4PZ C M N 6;ADC4bDCHqAB6n4/>@>oEGADC4/6;]P,/6IDD1A=DFEG, N HJ. N ZJ,z4bD QPo,zDbQbQ k BC · DC = d2 [ABCD] = d2

k

ª k-/,bD

x klTAFy

N E

¤ >O4.C,zHJ6IDC,]P,P.a>%¢

+) * , 8 * 08 / 1 ,:1 4 1 * ,:,8 +8 *4 /N¶5*8 4 7 °, P 0 138 8 * 4 /;8 1 4 1 *4 ° 6 / /38j¢ 4 7 8 ,3* P+* -1 01 8P 8 87 8 4 ¦ ¨ © N« ¬ 4¬ 1:,L¯ 1 9°0- 138 4+1 8 1;/ ¬ +/;8 ,;*0*t²

¬ F8 1 8 N/ µ x8 * , -+/21;*4¬ 4 m = 144sin

2

x

+ 144cos

2

m

k3~v6;KG,,/KpQVLq.CAB9 DCA pEG,s46G>C 24 H\6 N Z\y;>FHXB,T4/.C,=4bDaDC122 4PHJ6G4?BZJ,;k 145 wAB6G>FHqKG,/.-DFEG,xLJy;6 N DCHJAB6 0 < t < 1 f 2 ,/6 N ,BQ/DFEG,z4?G>oAIZ\yBDC,V9H\6BHJ9sy;9 N K;, N .C,/4/>5H\6;]VAB6 / 4 ; 6 g K \ H 6 C . / , / 4 5 > \ H ; 6 V ] B A 6 ` k T l 1 0, 12 ,1 2 H\> J4PDaDa4/H\6;,K{RAIZ[yBD1, f f = 24 x = 934rH\9y;9 2AIL HJ> q4bDaDC4P4HJ6;,/Kg4 N AB6IDCHJ6ByAIy;>`LJy;6 N DCHqAB6pQDFEG,=~v6ID1,P.193,KIHJ4bD1, 4PZ\y,dpEG,/AB.C,/9DFEG,/6 ]jy;4.a4/6ID1,/,/>fDFEB4bD,RXB,/.vu=HJ6IDC,]P,P.?G,bD`4PZJ>oA=4PDaDa4/H\6G4?IZq,IQ46;KxAIy;. 24 145 N ZJ4PHJ9³LAIZ\ZJA/k

#

kXI4PZ\y;4bD1,

2002 P

#

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

&

&

k

)+* , 8 01;8 8 * 4 /389124 1L *4 °6 / /;8 ¢ 4 7 8 - / * U 1L¶ 08 ¬ 41 , 1 ¯ 81 V T °1 0- 138 d¦04+ 1 ¦ 8 1;-/ 84 ¬* +/; 81 ,;*08O * o²

+ `8 ¬4/28d124 1 8 U 4 1B¶ ¦ + 8¨?³ x ¨ 1;©/; N8« -O 7 ¬ 4 k=1

pEG,@,FeG.C,/>5>5HJAB6g5= HDCHXB2002 ,zH\6ID1,/]P,/. k n hVADC,xDFEB4PDODFEG, N Z\4/H\9 H\>dD1.Cy,sLAB. k7Py;e;e;AB>R,6;AXI4/Z[HqK@LAB. k n ,OREGA`LAB. n k`≥~v6;2K;,/,KQ

¨

n X (k − 1) · k! 2k k=1

n−1

=

n−1 X k=1

= = =

n

(k − 1) · k! n!(n − 1) + k 2 2n

n!

n!(n − 1) 2n 2n! + n!(n − 1) n!(2 + n − 1) −1 = −1 2n 2n n!(n + 1) (n + 1)! k −1 = −1 n n 2 2 2n−1

−1+

pEIy;>jQIDFEG, N ZJ4PHJ9³H\>D1.Cy,TLAB.`4PZ\Ze;AB>5HDCHXB,zH\6ID1,/]P,/.a>

n

k

kgl=AxHJ6 4Y>1D1.a4/HJ]PED@Z[HJ6;, >Fy N E8DFEB4PDx,RXB,P.vu 94.1?BZJ, H\> 4K¡4 N 5,/6IDD1As4bD`ZJ,/4/>FDAB6;,=94.1?BZJ,@AILDFEG,T>o4/9:, N AIZqAIy;.`4>|HDC>o,PZ[L@¢ )+* ,.-0/213*4 ,3* P+* -1 01 8P 8 87 8

k

pEG,P.C,=4.C, /k B4 N Ed>R,jDpAIL 42694.1?BZJ,/>AILPDFEG,|>5493, N AIZJAIy;.9sy;>FD.C,P934PHJ6DCAG]P,bDFEG,P.pAB.,PZ\>R,|?;, 5As]b.CAIy;eG>bQ/R,Pe;4/.a4PDC,KxH\6ID1A@D|46;K 4K 4 N ,P6ID94.1?BZJ,/>Pk 3 pEIy;>jQ

¨

#&

¡

¡

y;6BHD1,/K¤Jk

DFEG,/9Yk

JH EG,zDFEB.C,, IyGHJ6IDCy;eBZq,jDC>AIL;DFEG,=>5493, N AIZqAIy;.`.C,P934PHJ6 pEG,P.C,s4/.C,@DFEG,/6 ]b.CAIy;e;>zAIL934/.F?IZq,P>/kV ,xEB4bXB, 3

\¤ N¤

# &

3! = 6

ODCA{4/.1.a4/6;]P,

JH[H , N EGAGAB>o, AB6;, N AIZqAIy;.=4/6;K ?B.C,/40 HD=H\6ID1AYDFEG,:D5>5HJ?BZJ,teG4.DC>jQ PkiEG, D3AILODFEG, ?B.CAG0,/6 N AIZJAIy;.>REGAIyGZJK?G,z6;AB6;544K 4 N ,P6ID4/9:AB6;]ODFEG, N EGAIH N ,P> ]b.CAIy;eG> /DFEIy;>bQ/o,VD-46;K ?B.C,/40 DFEG,/9Yk pEG,P.C,x4/.C,=DFEG,/6 ]b.CAIy;eG>zAIL94.1?BZJ,/>Pk ,geBZJ4 N ,=DFEG,=DVAILpDFEG,sS ?B.CAG0,/6 N AIZJAIy;.OHJ6:Doe;ADC>jQw4/6;K:o4/9:,xLAB.-DFEG, ª N EGAIH N ,P>/k ?B.CAG0,/6 N AIZJAIy;.jk ,=EB4bXB,

¤ ¤

,@EB4jXB,

¡

6

S

N AIZqAIy;. #S Q & #S Q & #S Q & #S Q & #S Q &

# ª Q &

# &

A¤

6 · 3 · 2 · 2 · 2 = 144

ª N AIZqAIy;. ª Q ª Q ª Q Q ª Q Q

#

&

# # #

N AIZqAIy;. #S Q & #S Q & #S Q & #S Q & #S Q & S

#

Pk

ª r N AIZqAIy;. ª Q ª Q ª Q ª Q ª Q

#

&

# #

&

#

4

&

#

,@EB4jXB, /k 2,= · EB 2·4jXB=, 240 Z[DCAG5]P·,bDF3EG·,P2.RQ·Pk pEIH\>]jH XB,/> ]b.CAIy;eG>AILp94.1?BZJ,/>Pk ,=EB4bXB, 6

&

&

&

&

&

&

5 jN AIZqAIy;.

N EGAIH N ,/>

#\ Q & # Q & # Q # Q && # Q &

6 + 36 + 144 + 240 = 426

/k

HJ6

O

hVA< |?;,jD5H XB,;k7Py;eGeAB>o, P k ;.CAPXB,zDFEB4bD 2 2a1 a2 . . . a n

i=1

ai =

i=1

bi

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

¨

+-,bDC,/.C9H\6;,V/k

)+* ,.-0/213*4 01;8 8 * 4 /389124 1 *4 ° 6 / /38V¢ 4 7 8 ~¶5* 1 8 /Nµ 8 81;/;*

¨

pEG,P.C,gH\>d4g9H\>oeG.CH\6IDjkpEG,x]jH XB,/63HJ6;, By;4PZ\HDcu3HJ>OLq4PZJ>o,@H[L LAB.O,/4 N E k ai = b i i pEG, N AB.1.C, N D`HJ6;, By;4PZ\HDcuxHJ>

¨

¨

n X a3i

i=1

n 17 X

≤

bi

10

,VjQIH[L bi = 2ai ai =k 2bi AB.

(ai , bi ) = (1001, 2002)

(ai , bi ) = (2002, 1001)

7Py;939sHJ6;]=APXB,/.

i

# &

H\6 U

n 46;Ks.C, N 4/Z[Z\H\6;]dDFEB4PD P

i=1

A¨

(ai , bi ) = (2002, 1001)

¨ § k-/,bD

i=1

a2i

QOK;,P>5H\.C,Kwk Iy;4/Z[H[Dvu3A N1N y;.a>-H\Lp4/6;K3AB6BZuxH\LQILAB.,/4 N E AB.

n P

b2i =

kpEG, N AB6;KBHDCHqAB6

i n P

i=1

QG,PHDFEG,/. a2i

(ai , bi ) = (1001, 2002) n P N 46g?G,d.C,r = b2i

Q

kEIy;>bQ

i=1

2 2 2 2 1001 p = 21LABn.EB4/Z[LAIL N y;.a−>`H\p)2002 , Iy;4/Z[H[pDvus+A NF(n L4/6;KsAB6BZ= u@H\2002 L H\>|,RpXB+ ,/6x1001 46;K (n − p) n (ai , bi ) = (1001, 2002) DFEG,=>5y?B> N .aHJeIDC> FH[DCH XB,zHJ6IDC,]P,P.jk-+-,bDC,/.C9H\6;,zDFEG,T>o94/Z[Zq,P>FDè;AB>o>FHq?IZq,T>Fy;9 a1 b1 + a2 b2 + · · · + a2n+2 b2n+2

Q

Q Q Q 46;K Q Q Q 4.C,s.C,/4/.1.a4/6;]P,/93,/6IDa>@AILDFEG, 2n+2

2n + 1 0

2Fy;>FDCH[L[uguBAIy;.4/6G>15y;9H\>9sHJ6BH\9H 5,KzOH\6 N .C,/4/>5H\6;] A 6;,xAILpDFEG,@>R, By,P6 N ,P> B b1HJ4PZ b2N AG, . . . N HJ,/b6I 2n+2 46;KnDFEG,tADFEG,P.TH\>xK;, N .Ca ,/4/1>5H\a 6;2]Bk . 7P. H\.6 N a,:2n+2 DFEG,t?IHJ6;AB9s Da>x4/.C,{H\6 N .C,/4/>5H\6;] Lq.CAB9

¨

2n + 1 0

D1A

DFEG,=9H\6BHJ94/Z>Fy;9*H\>

2n + 1 n+1

46;KYKG, N .C,P4>FHJ6;]gLq.CAB9

2n + 1 n+1

D1A

2n + 1 Q 2n + 1

n n X X 2n + 1 2n + 1 2n + 1 2n + 1 Q 2 = 2 k n+1+k k n−k k=0 k=0

FD`>FDC,/esy;>R,P>DFEG,V=46;K H\.CZ\/> − k:j ,xR,bZq, N D

2n + 1

n

2n + 1 4n + 2 TD1A8KGA n

e;,/.a>oAB6G>@Lq.CAB9DFEIH\>x]b.CAIy;ek8EG,/.C,{4/.C, N Zq,P4.aZu

DFEB4PDjk M 6YDFEG,tADFEG,/.=EB4/6;KpQZq,jDaDCHJ6;]

k

?;,3DFEG,{6By;93?;,P.@AIL-?;APuG>@H\6nDFEG,{>o,PZJ, N DC,K

n ]b.CAIy;ewQPo,,ODFEB4bDDFEG,OD1ADa4/ZB6By;93?;,P.AILIHJ>`4PZJ>oA P

pEG,P.C,PLAB.C,IQIDFEG,=9sHJ6BH\934PZ>5y;9³HJ>

4n + 2 k 2 n

k=0

2n + 1 k

2n + 1 k n−k

ª

kiH\6;K 4/Z[Z-.C,/4PZqaXI4/Z[y,K$LJy;6 N DCHqAB6G> .a4PDCHJAB6G4/Z6By;9:?G,/.a> >o4bDCHJ>FL[uBH\6;]dDFEG, N ABf6;I K : H[DCHJAB6G−→ >

LAB.4PZ\Z

Q

K;,b6;,/K(AB6$DFEG,Y>o,bDgAIL4/Z[Z

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

HJ6 46;K

Q

k2Fy;>FDCH[L[uguBAIy;.`46G>CPk

)+* , 8 "137µ 8 , 0 / 1:, ,38 * -84 4¢ 4 7 8 0 1;8 8 * 4 /389124 1 *4 ° 06 / /1:,/3,388 ³ ¢ * , 4 -+7 /281;*4 4 ,3* P+* -1 0 1 8P 8 87 8 ¬ 08 8 8 4/ x y

f (1) = 2002

~qDOH\>V.C,P4KIH\Z u N EG, N 0,/K:DFEB4bD-DFEG,gLJy;6 N DCHJAB6 H\>V4x>RAIZ[yBDCHqAB6k ,zREGA-y;6BH Iy,;kz/,bD ?G,@4634r.1?B7→ HD1.ar(r 4.vu3>o+AIZ\2001) yBDCHJAB6kV+-,/6;ADC,g?/u f DFEG,d]jHXB,P6 N AB6;KIH[DCHJAB6 Q4/6;KT zQ k`~qDwHJC> f (x + y) = f (x) + f (y) + 2xy z ∈ z > 0 ,/4/>5H[Z[email protected]/XB,/K@DFEB4bD q?/u@HJ6;KIy N DCHqAB6pQ 2 LAB.4PZ\Z f (nz) = n f (z) + (n − 1)z Q y;>5H\6;] 1D1,Pe knEG∈,/6p QILAB.4PZ\Z

¨

C

#

&

x = nz

y=z n∈ Q 1 1 1 2002 = f (1) = f n × = n f + (n − 1) · 2 n

46;KxDFEIy;>jQ f

m n

f

1 n

=

1 n

1 n

2001 +

n

n

kEG,/6pQILAB.4PZ\Ze;AB>5HDCHXB,zH\6ID1,/]P,/.a>

m

46;K

n

Q

k 1 1 1 m m = f m× = m f + (m − 1) · 2 = + 2001 n

n

n

n

n

pEIy;>jQ EGAIZJKG>@LAB.T4/Z[Ze;AB>5HDCHXB, @k^7PHJ6 N , r ∈ =HDF0 N AB6;KBHDCfHqAB(r) N AB6;KIH[DCHJfAB(0) 6 = N AB9eGZJ,bDC,/>DFEG,Te;.CAGAILCk f (r) = r(r + 2001) LAB.

#

&

r∈

#

&

* 4 8

;A:6BH\>RE3DFEIH\>z6By;93?;,P.VAILDFEG, Q;RAIZ[yBDCHqAB6G>V?/uYAIy;.O.C,/4K;,P.a> D1A3eG.CAG?BZJ,/9>VAILpDFEG, O~a~a~~qDC4PZ\H\46Y$4bDFEG,/94PDCH N 4PZ M Z uG9eGH\4KQp,/>o,/6G4bDCH N AQp~qDC4PZuQ $4bu ª/ª QG]jHXB,P6sHJ6 ªI ª S k

k`H\6;Kg4/Z[Z

3

KIHq]jHDe;AB>5HDCHXB,H\6ID1,/]P,/.a>DFEB4bD4/.C,

DCH\9:,P>DFEG,V>Fy;9³AILBDFEG,PH\.KIHq]jHDC>Pk

)+* , 8 * 08 / 1 ,:1 4 1 * ,:,8 +8 *4 /N¶5*8 4 7 °, P 0138 8 * 4 /;8 1 4 1 *4 ° 6 / /38u¢ 4 7 8 8 * 8 SU 4, , u« ¶ 84 0 © ) 9,;* P+* -1 01 8 8 87 8 4 ¦ ¨ © N« ¬ 4 ¬ 1:,L¯ 1 ° - 1;8 41 8T 1./ s¬ 0/38 ,3*0*o²

`¬ 8x1 8 1:, 124 1D³ x¨ 1;/;8 -O /,bD ?;,z4 KBHJ]jH[DeAB>FH[DCH XB,dHJ6IDC,]P,P.5H\93eBZ\H[,P>cD1A ,¨By;4bDC+HqAB10b 6gHJ9+ eGZ[Hqc,P>=DFEB34(a 4PD H\>| KBH XBH\>5HJ?BZJ,z?/u k 34

8b + 11c b P7 HJ6 N , H\>V4KI22a Hq]jHDRQ;= FDdEB4jXB, Q 4/6;K:DFEG,P6 p kzpEIH\>z]jHXB,P>dy;>d11 DFEG, b b = 0 c = 2a 6By;9:?G,/.a> Q Q Q4/6;K k{~qD N 4n 6 ?G,gXB,/.aH\,K{DFEB4bDz,P4 N EYAILDFEG,P>R,sH\>T4 408 >RAIZ[yBDCHqAB6k 102 204 306

4

§

k=/,jD 4/6;K ?;,TDdAILpDFEG,geBZJ4/6;,BQ4/6;K3Zq,jD ?;,TDFEG,g9HJK;FeAIH\6IDOAIL k/,bA D ?G,@B 4TZ\H\6;,BQB46;KsZq,jD 4/6;K ?G,zDFEG,@eG.CA v, N DCHJABM 6G>AIL 4/6;K AB6IDCA k AB R >o>Fy;9H\6;]drDFEB4PD Q QI46;K 4/.C,T6;AD SN AIZ[Z\H\6;,/4/.RQIeG.CA/XB,VDFEB4PDDFEGA , N HJ. N y;B 9 N HJ. N ZJ,=rAIL A M D1.CH\46;]jZJ, EB4> DFEG,=>5493R,T.a4KIH\y;>-4>DFEG, N H\. N y;9 N H\. N Zq,=AIL k

¡

)+* , 8 ! 1;7µ 89, + / 1 ,:,;8 + * -+8 4 ¢ 4 7 8 4 9,;* * - 1L 01 8P 8 87 8 ¬ t8 8O 84 / 0 / 1:, ,38³ * , -+/21;*4 AM R

BSM

A

M

B

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

R

M

S

¡

/,bD ?G,-DFEG,eG.CA v, N DCHJAB6@AIL AB6ID1A kEG,/6 HJ>DFEG,9sHqKG5e;AIHJ6IDAIL M0 M r M0 RS N, H\>`DFEG,T9sHqKG5e;AIHJ6IDÀIL 4/6;K k q ~ DLAIZ[ZqADFEB4bD 0 M AB M M ⊥ RS 4RM S HJ>|H\>RAB> N ,bZqP, >`FHJ6

&

# &

RM = M S

46;K ?G,ODFEG, N H\. N y;93.a4KBH[HGAIL 46;K Q/.C,/>5e, N DCHXB,bZuwk hVA<sQ/ZJ,bD ρb 4AM R 4BM S ,@EB4jXB, ρa RM SM ª Q k 2ρa =

};yBD

2ρb =

sin(∠RAM )

∠RAM + ∠SBM = 180

◦

#q>FHJ6

N,

& MBEG,P6

sin(∠SBM )

AR k BS

sin(∠RAM ) = sin(∠SBM )

ª

I.CAB9

#vS & Q0# ª & QB46;K# & QAIL LAB.oAIZ\yBDCHJAB6G>AILDFEG,, Iy;4PDCHJAB6 3 n x −3x+n = 0 4.C,zHJ6IDC,]P,P.a>/k

)+* , 8 0138 8 * 4 /389124 1L *4 °6 / /;8 ¢ 4 7 8 4 ,3* P +* - 1 01 8P 8 87 8 x¬ 8x1 8 * 4 /;8 1 4y³ * ,.-0/213*4

¨

~LpDFEG,g]jH XB,/6:, Iy;4PDCHJAB6{EB4/>4/63H\6ID1,/]P,/.|.CAGAD QIDFEG,/6 H\>4/6 3 −α HJ6IDC,]P,P.D1AGAwkpEG,P.C,PLAB.C,IQIDFEG,VXI4PZ\y,P>OAIL DFEB4PDR,/,0@9sy;n>FD=4PZ\Z ?G,TH\6I+D1,/3α ]P,/.a>Pk /,bD kpEG,T]jHXB,P6n , Iy;4PDCHJAB6xH\>`DFEG,/6 k7bD1.a4/HJ]PEDF 3 f (x) = x − 3x f (x) = −n LAB.O>REGA-HJ6 N .C,P4>FHJ6;]=AB6 4/6;K Q −1] 46;KsK;, N .C,/4/>5H\6;]zAB6 kÂB.C,APfXB,/.oQ 4/6;(−∞, K kp[1, EIy;+∞) >jQDFEG, [−1, 1] , Iy;4PDCHJAB6 EB4>ODFEB.C,/,x.C,/4PZw.CAGfA(−1) DC>H[L4/= 6;Kt2AB6BZ u:fH[L (1) = −2kdEG,/.C,bLAB.C,BQ f (x) = −n Q Q Q Q ky;.aDFEG,P.193AB.C,BQAB6;,gAILDFEG,THJ6IDC,]P,P.||n| >RAIZ[≤ yBDCHq2AB6G>EB4/>-DCA3?;, n ∈Q {−2 −1 0 1 2} QpAB. GDFEIy;>bQ Q Q Q Q kg+HJ.C, N D N EG, N 0H\6;] −1 1 DFEG,@KG,/>Fn >REGADFEB4bD HJ.C,/∈Kg{f XI4/Z[(−1) y,/>-4/.C,f (0) f (1)} 46;= K {−2 k 0 2}

¨

¨

M

n = −2

n=2

«

k-;.CA/XB,VDFEB4PDRQIH[L

H\>-e;.aHJ93,BQDFEG,P6

KIHXIHqKG,/>

k

)+* , 8 01;8 8 * 4 /389124 1L *4 °6 / /;8 ¢ 4 7 8 - / * U 1L¶ 08 41 8T 1 d¦0 ¦ -84 * 1 8 +8 4/2124 9,;* * - 1L 01 8P ¬ x8 8 871 8 8 U 4 7B1 ¶ ¦ + 8¨? ³ s 4* ,.©-0 N/2« 13* t4 ¬ 4F¬ 1:,L¯ 1 0° - 1;8 4+1 8 1;/ F¬ 0/38 ,3*0*x²

m = 5 n + 3n + 1

12

n

4

}BuYDFEG,$+H XBH\>5HJAB6 qZ ]PAB.aH[DFEB9tQoAB9:,3HJ6IDC,]P,P.a> 12q 4 6;K >Fy N E{DFEB4PD kt~v6 DFEG,sLAIZ\ZJnAo,/>d/k

%¤

?u $ k r H\>OAGK;KNMDFEB4PD`H\>bQ r = 2k + 1 kEG,/6

?u $ k r = 2 Q r = 6 QGAB.

m = 512q+2k+1 + 3n + 1 ≡

?u $ k r = 4 AB.

r = 10

m = 5n + 312q+4k+2 + 1 ≡ r=8

52

MDFEB4PD`H\>

6q

52

34

3q

34

r = 4k

k

jQ

7PHJ6 N ,

HJ>@e;.aHJ93,{4/6;K

m QB46;K KIHXIHqKG,/> mk ≥ 9 r=0 12 n

0≤k≤2

kEG,/6

32 + 1 ≡ 32 + 1 ≡ 0 (mod 5)

56

k

k

5 + 1 ≡ 5 + 1 ≡ 0 (mod 3)

MDFEB4PD`H\>bQ r = 4k + 2 o,/>@4?;APXB,{4.C,:eAB>5>5HJ?BZJ,;k

* 4 8

pEB4bD N AB93eBZq,jD1,P>DFEIHJ>`6By;9:?G,/.AILBDFEG, kpEIH\>`H\>`4 N 4/Z[ZBLAB.>oAIZ\yBDCHJAB6G> .C,/4K;,P.a>xREY>RAIZ[yBDCHqAB6G>VFDa>HJ6Irj =¡¢£c¤¥j¦s¢£§v¨r©/rP¦ k ,O6;,,/KzuBAIy;. N AB6ID1.CHJ?ByBDCHJAB6G>AILB6BH N ,>oAIZ\yBDCHJAB6G> 46;KW]P,P6;,/.a4PZ\H F4PDCHJAB6G>bQ-e;.C,bL,/.a4?BZ u^3AILODFEG,n4/e;e;,/4/.a46 N ,$AILODFEG, e;.CAG?IZq,P9nk

U

x

p

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.................................................................................................... ............................................................................. ............. ............... ............... ........... . .......... ............ ............ .......... .................. ......... ....... ......... ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ .................. ..... . ........ ....... ............................ .... ....... ...... . . . . . . . . . . . . . . ...... ....... ....... .. .... ..... . . . . . ..... . . . . . . . . . . ..... . .... ... .. ... . . ..... . . . . . . . . . . .. .. .... .... . . ... .... . . . . . . . . . . . .... .... . . 4 ... .... ... .. . . .... . . ....... . .. .... .. .. . .... . .. . . . . . .... . . .... .. ... .... ... .. . .... . . . . . ... ... ... ... . .. . ... . . . ... . . . ... ... . .. . 0 ........ . . ... . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . .. ................... ... ........ . ... . . . ... .. .. .. .. . . . . . . . . . . . . ... ... ... .......... . .. .. . . ... . ...... ............ . .. . . . . . . . . . . .. .. .. .... ...... . .. ... ... . . .. .. .. .. .............. . . . . . . . . ... .. . .... ..... .. .. .. . . ........ .. . . . . . .. . . . . . . . . .. .... .... . .. . . . .. ... . .. .. .. .. .. 0 . . . . . . . . . ... . . . .. .. .. .. .... ... .. . .. .. . .. .... .. ..... . . . . . .. . . . . . ... . .... .. ... .. ... . .. .. .. .. ... ... .. . . . . . . . . . ... . . . .... .. ... .. ... .. .. .. ... .. . . .. ... .... . . . . . . .. . . . .... . . . .. .. .......... .. . . . .... ... . . . . . . . . . . .. . . . . ... . ..... . ................ .. .. .. .. . . . . ... .... ....... . ... . .. .. .. .. .. .... ..... ...... . . ........ . .. . . . . . . . . . . .... . . . .. .. .... ... .. .. .. .......... ... .... . . . . . . .. . . . . . . . . . . . . . . .... . . .......................................................................................................................................................................... . ... ... . . . . . . . . . . . . . . . . . . . . ... . . 3 . . . ... ... . .. ... . . .. . . . . . . .. ... . .... . .... ... ... . . . . . . . .. .. ..... .. . . . . . . . . . . . . .. ... .. . . . .. .. ... .. .. .. ... .. ... ... ... ... .. ... .. .. ... .. .. .. ... . .. .. .. ... ... . . ...... ... ... .... .. .. . . . . .. . . . . . . . . .. .. .. .. ..... .. .. ... .. .. .. .... .. . .. .. .. ....... ... .. .. .. ... .... . .. .. . ... .. .. ..... .. ......... ... .. .. ...... ... .. ......... ... .. . . .. .. ....... . ... . . ............ . . . . . . . . . . . . . . . . .... .. .. . ............... ... ................................................... .. .. .. .. .. .. .. ... ... ... ... ... ... ... .... ... .... ... ... .... .... .... .... .... .... . . . . . . . . .... .... . .. .. .... .... .... .... .... .... .... .... .... .... ..... .... .... .... .... .... .... .... ..... . . . . .. ...... ...... . . . . . ...... ...... ... ......... ...... ....... ....... .. ....... ...... ........ .............. ........ ......... ... .... ........ ......... ......... .................. ......... . . . . . . . . . . ........... . . . . . . . . ........... .............. ........ ................ ............ 2 ............................ .......................................... 1 ......................................................................... .............

C

c

F

B

S

S

c

A

G

E

! "$##% '&$(&) &$(+*,&-!-. /0&-!1 23&$4'5$21 .1! c

c

D

B

c

U

H H $ Hk G KBH XBHJK;,P> AB 4 N1N AB.CKIHJ6;]dD1AgDFEG,=]PAIZqKG,/6>o, *0*¯ /PD1,P6;K S B DCAxH\6ID1,P.a>R, N D c 4bD B k-}, N 4Py;>R, 0

N DCHJAB6k

0

1

AS = F S = AE =

1 AB 2

=

1 FB 2

Q

o,,{DFEB4PD 4/6;K 4.C,nAB6$DFEG, e,P.1e;,/6;KIH N yGZ\4.x?BH\>R, N D1AB.gAIL Q-4/>s4.C, FA S0 46;K k0 pEB4bDBH\>bQ Q S0 Q Q 0 Z[Hq,=AB64TZ\H\6;,BQB4/6;K B

S S

B B

k

AB = AB 0 = F B = F B 0 = 2AF

BAIZ[ZqA5yGZDHJ6 S Q/ 0 4 NFN AB.CKBH\6;]D1A@DFEG, S BEB4bXB, P] AIZqKG,/6T>o, N DCHJAB6kHJ6G4PZ\Z upQb>FHJ6 N , QrS = AG : GB G AB 4 N1N F AB.CSKIHJ6;= ]dD1F AgG DFEG,=]PAIZqKG,/6S >o, NSDCHJAB: 6SB k

#

&M

$ k S;k pEIH\>tF>FDC,/e N AB6G>FD1.ay N DCHqAB6WHJ6XBAIZ XB,/>DtAILVDFEG, 8 137 1L 7.L1 QrPE=Iv#E#>G=r>HzAB=rZ#ZMOKTKTGUW[=E?AO=@X\ KzQRM=o>`MZ'=AN^ >IO=H ^ =OMOIEP]?WMOAMWEP@{HoH| AB=OMP=%IfHjv#X>EKN="MO>OEuQ0Zqp KT=MW^ GKT@HNG[HsAm=\ Q}yuMO=VEPfH Hx=ABKN=%=OM9HtIZzE>>AB[Q0=A @=^=9^ KTKNGMO]?HEuMOqp %EMWrH~@HrABrZ=KH,= ^ q=O]'=Îs=O]'>=Y9KNMtAAw MOMOAEPwE?MOWE?A>PWMW=A@MWHH@HZGm K =y^ qq MO@B=^ =DEP]'A HoAB==KNAHMdMHAB]#=d= > q IXKN@B{ MOGOMOEuE}Eqp MPMW]']?@=HM H HGWAN>PIX@BGOnEMVHKAB=^ =]?ZM KTq GIX@B[GOAE=\ Q}HG=f> H| =MWKNM`@BINZz=,>PKT[=A QR@W=`^ =]?K qMO@BE=HzKjAWMo=AMOMOE#E?">=`MOK ]'q}=fAMr=KZKNKS@B@X=E Kq @ =OZ#ItMWAB"=fMOHGWKIXAN@X>EIX@BGOzEsGWALZ]?KH=ÎBH=%=@XE?E# IO=O^ ]'=?= m Aw E?@BU=KNHT@BI=s^ ]'= 9GOE#ILK=%^ MWA u]w MUGW@XKrILKNMP]#>@BI#AB=HZKTG[A=\ Q}=%Hm

%

(?)

§ ¬`§ k *P * 8~¡ * 8 ¡ 0° -1 x ¡ 8* 4+1 8 1;/8 ¡ * , +/;8 7µ 41´ -8 8

0/ ,3*4 8 7 89,3*4 8 `¦ %P4 8

k

p4PZ N yGZJ,/.

k !n12 n 2 Y k + n2 k lim ln 2 n→∞ n k=1

§ ¬ ª k *P * 8t¡ "1Lµo¡ , 84 7 98 P * * -¶41;8 k y;6{eAIH\6IDB ¨ y,bZ N AB6\¨By,sKB46G>Zq,geBZJ4/6tKBy:D1.aHJ4/6;]jZq, ABC k7AIH[D D Q E 7AIH[D ,bD ZJ,/>e;AIPHJ6IDa>9H[Z\HJ,Pyx.C,/>5e, N DCH\Lq>|K;,P> N A D¦,/¥ > Q ,bD kI7PH ,/>1DZJ, N ,P6ID1.C, K;,@F]b.a4jXBHD¦,@ ¥ KIysD1.CH\46;]jZJ, ABC QB93AB6ID1.C,/./B¨ y, BC CA AB G 0 ≤ 3P G+P A+P B +P C −2(P D +P E +P F ) ≤

1 (AB +BC +CA) 2

§ ¬ « k *P * 8t¡ "1Lµo¡ , 84 7 98 P * * -¶41;8 k ÂB6ID1.C,/.VÏy,BQweAIy;.DCAIyBDz,/6IDCHJ,/.deAB>FH[DCH[L QpH[Z,5/HJ>1D1,sy;6ne;AIZuB6 AB 9:, n K;,@KG,]b.,= ¥ 4Py:93AIHJ6G> 8n QD1,bZÏy, (2n+1)2

X

|P (k)| < |P (0)|

k

Q

P (x)

k

§ ¬P¬ k *P * 8t¡ "1Lµo¡ , 84 7 98 8/ 1 4 124 7 P * * -¶41;8 k 7AIH[D y;6eAIH\6ID`H\6ID¦,/¥ .aHq,by;.|KIysD1.CH\46;]jZJ, kp+X,/¥ >FHq]b6;AB6G>-.C,P>oe;, N DCH XB,/93,/6ID P N A D,P¥ > e;4/. Q ,bD ZJ,/>`KIHJ>1DC4/6 N ,P>`,/6ID1.C, ,bDZJ,/> ABC Q ,jD Q,jDeG4. Q d d d ,jD Zq,P> KBH\>FDa46 N ,/>,/6ID1.C, ,jDPZq,P> N A D¦,/¥ > Q BC ,jD CA kGH\6G4/ZJAB ,/93,/6IDRQ>oAIH[D D Q D ,jD D Zq,P>-9:,P>5y;.C,P>O.C,P>oe;, N DCH PXB,/>OK;,P>O4/6;]jZq,P> A B Q C P ,bD k P P BP C CP A AP B ÂB6ID1.C,/.I ¨ y, k=1

a

b

B

B

c

A

C

A

C

da db sin

1 (PA 2

+ PB ) + db dc sin ≤

1 (DB DC 4

1 (PB 2

+ PC ) + dc da sin

1 (PC 2

+ PA )

sin PA + DC DA sin PB + DA DB sin PC )

k

U

§ ¬ ® k *P * 8t¡ ,;/ ) 4 * 8 )

k

7AIH[D y;6gD1.aHJ4/6;]jZq,z4/y=4/6;]jZq,P>|6;AB6AG?IDCy;>-,jD Zq,z.a4juBAB6K;,z>RAB6 N ,P. N Zq, N HJ. N AB6G> N .aH[DjABC kV7PH Q ,bD >RAB6IDZq,P>{ZqAB6;]jy,by;.a>YKG,/> N ADR,/> ABeGeAB> ,/>t4Py(46;]jZJ,/> .C,/>5e, N DCH\Lq> Q a,bD b QB93cAB6ID1.C,/. Iy, A B

¦¥

¨

C

abc k 4R2

a cos3 A + b cos3 B + c cos3 C ≤

§¬

*P * 8t¡ ,;/ ) 4 * 8 ) k

7AIH[D

¨

Q

Q

Q

x1 x2 . . . x n

ÂB6ID1.C,/. Iy,

¥

¥

k

n K;,P>6;AB9:?B.C,/>. ,,bZJ>OeAB>FH[DCH[Lq>>54PDCH\>5Lq4PHJ>546ID Q

k

xi = 1

i=1

n X

i=1

xn i (1 + xi ) ≥

n 2n−1

n Y

(1 + xi )

k

i=1

§ ¬ k *P * 8 ¡ 8O 8 , 124 1L¶1./ * +/21;*4 , '« 1µ 7µ *0* , *¯ +/Nµ 8O¶0/2137 4 )+7.1384 7 8 )* -,L 138 k

¥ ¯¨ @¤ ¥ 6£ b c ∈ A b ≤ c a=b+c #J4 & ÂB6ID1.C,/.¨ByE¤ H\ZI,5/HJ>1D1,KG,/>KBH\>FDCH\6 N DC> a Q a Q . . . Q a ∈ A QjD1,bZJ>6Ïy, k a + a + ··· + a = 0 #? & ? /,.,/¥ >FyGZ[Da4PD N HJoKG,/>5>5y;> .C,/>1D1,/1DF1H\Z{6,, ¥ N ,/>5>o4PHJ.C,P9:,P6ID^XB.a4/Ht>FH{ZK¤ AB6AB9:,jD Zr¤ y;6BH N HD¦,@ ¥ KG,TZ\4=.C,/eG.,P¥ >R,P6IDC4bDCHqAB63K;, a N AB9393, a = b + c ¢ § ® k *P * 8t¡ "1Lµo¡ , 84 7 98 P * * -¶41;8 k 7AIH[D ,bD KG,Py86;AB93?G.C,P>.,¥ ,bZJ>s>o4bDCHJ>FLq4/H\>o4/6ID k 0 ≤ a ≤ ≤ b ≤ 1 ÂB6ID1.C,/.I ¨ ay, b #J4 & 2(b − a) ≤ cos πa − cos πb M

y;6$,P6G>R,P9:?IZq,;6BHKG,:6;AB93?G.C,P>@. ,,bZJ>zDC,PZ Iy,D1AIyBD eByGHJ>5>R, 7AIH[D y;6BHXBA Iy,/93A,/6ID|> , N .CH\.C,=>oAIy;>|ZJ4TLAB.193, QGA y Q ,bD a ∈ Ak

A¨

1

1

2

2

k

k

1 2

#? &

(1 − 2a) cos πb ≤ (1 − 2b) cos πa

k

§ ® k *P * 8x¡ -, 1 -j¢ ,3* 1 '/2, 4/2137 41 8T 1./ *07 +/;*4 j¢ ° ) k ¥ DC4/6IDYK;AB6G6 ,W ¥ y;6ieAIH\6ID KB46G>YZq,We;.C,P9HJ,/. Ïy;4KB.a46IDRQ=AB6i>54/HD Ïy,WZq, >R,/]b9:,P6ID #KIH[DxKG,ÊIH\ZJAB6 K;,tZqP & 6¤ ,/>1DVAB6;e;]j4/y> ,bNy;AB.g6G9s>FD1HJ.a6By H\N 93DCHq4P?IZqZq,I,3Q`4jeGXB4, >5N >o4/Z\436IDg.,/£ eG]j4Zq. ,:P,jDd,jZq,D`¡vN AIABHJ93]b6GeG446I>PD k Zq,P>=4r,/>=KG, N AGAB.CK;AB6G6 ,/¥ ,/>= w,Pe,P6;KG4/6IDRQZJ,:>R,/]b9:,P6ID=K ,b¥ 6BH\>o>546ID #q4jXB, N ZJ,/>@K;,by 4r,/> y;6YD1.CH\46;]jZJ,{KG4/6G>=Zq, e;.C,P9HJ,/.jI ¨ y;4KB.a46IDRQKG,@e,,/¥ .aHJ9S,b£ D1.C,=9H\6BHJ94/ZvQBZ\yGHvQ;,P>FD N AB6G>FD1.a& y N DCHq?IZq,Gk+-AB6G6;,P.y;6;,

D1,bZ\ZJ, N AB6G>FD1.ay N DCHqAB6k

U

§ ® O k *P * 8R¡ 01 4 7 8 4/:1.- -,;8 7 - 41 8T 1./P8¡ 8 13* 13*

* -¶41;8 k 7AIH[D y;6;, LAB6 N DCHJAB6 N AB6IDCH\6By,Ye;AB>5HDCHXB, K ,P¥ ;6BHq, >Fy;. ,jDTD1,bZ\ZJ, Ïy, f kÂB6ID1.C,P.¯I ¨ yE¤ H[Z;6¤ ,5/HJ>1D1,Ve;4/>|K;,dLAB6 N DCHqAB6xe;(0, AB>5H∞) DCHXB, QIK;,byTLAIH\> lim inf f (x) > 0 # N AB6IDCHJ6By,P9:,P6ID & KIH\,/¥ .C,P6IDCHJ4?BZJ,BQjK ,b¥ 6BHJ,->Fy;. [0, ∞) ,bD>o4bDCHJ>FLq4/H\>o4/6ID g g+f ◦g = 0 k § ® § k *P * 8o¡ -4 * 7 * *'¶58* ¡ ´ -8 41 8T 1./P8 ¡ 8 , , * ,:1 ,:, * , 1 `¦ %P4 8 k 7AIH[D y;6TD1.aHJ4/6;]jZq,O.C, N Da46;]jZJ,4jXB, N ZK¤ 4/6;]jZq,KG.CAIHD,P6 k7AIH[D Z\4V?IHJ>o >R, N D1.CH N ,|Zr¤ 4OAB 6;]jZJ, Qb4jXB, N >Fy;. k/7AIHD ,jD ZJ,/>eBHq,/KG>K;,PO>e;,/.Ce,POO 6;KBH N yGZJ4PHJ.C,P> E ,bD .C,/>5e, N DCHXB,P>V4?B4/O H\>o>,¥ ,P>VKGO , >5y;AB .-ZJ,/> N A D,P¥ D > kp7AIH[D Q O OA OB F = OO ∩ DE QG,jD k G = AE ∩ O D ÂB6ID1.C,/.B ¨ y, ZJH,tD1=.aHJ4/BD 6;]jZq, ∩ O E ,/>1D@y;6^D1.CH\46;]jZJ,Y.C, N Da46;]jZJ, HJ>oA N ,P£ ZJ, 4bXB, N F GH Zr¤ 46;]jZJ,@KG.CAIHD|,P6 k F § ® ª k *P * 8o¡ -4 * 7 * *'¶58* ¡ ´ -8 41 8T 1./P8 ¡ 8 , , * ,:1 ,:, * , 1 `¦ %P4 8 k +|46G>|y;6xD1.CH\46;]jZJ, QB>oAIH[D Zq,Te;AIHJ6ID|KN¤ H\6ID1,P.a>R, N DCHqAB63KG,TZ\4@?BH\>o>o, N D1.aH N , HJ6ID,P¥ .CHJ,Py;.C,tKG,:ZK¤ 4/6;]jZq, ABC 4jXB, N ZJ, N A A D, ¥ k7AIH[DT.C,P>oe;, N DCH XB,/93,/6ID ,bD Zq,P> eGHJ,KB>gK;,P>@e;,/.Ce,P6;KBH N yGA ZJ4PHJ.C,P>@H\>o>Fy,/>@KG, BC >5y;.VZJ,/> N A D¦,/¥ > ,jD B k|^C AB6ID1.C,P. ¨ y, BB ,bD CC >R, N AIy;e;,/6ID>5y;.ZJ4@EB4PyBAD1,by;.H\>o>Fy,=K;, A k AC AB B § ®P« k *P * 8t¡ )¶588 4 u£,;/ 0*'¶"¶589, P k 7AIH[D y;6xD1.CH\46;]jZJ,T4jXB, N Zr¤ 46;]jZJ, ,jD`ZK¤ 4/6;]jZq, kG7AIH[D y;6@eAIH\6ID>Fy;ABC .wZq, N A D, ¥ KG,>RAB.D1, 90 k7AIAHD < 60 Zq,e;AIHJ6ID>5y;P. ¨ yGH>R,gD1.CAIyIXB,4/AB I y;>5>5H>Fy;.ZJ43?BH\>o>o, N D1.aH N ,xHJ6ICP D,P¥ .CHJB ,Py;.C= ,360 K;,sZK¤ 4/6;]jZq, D k7PHZK¤ 4/6;]jZq, CP QI9:AB6ID1.C,P.B ¨ y, CP ,/>1D`y;6;,@KB.CAIH[DC,zD1.aHJ>o, N D1.aH N ,=K;,TZK¤ 4/6;A]jZq, ACB k CBD = 30 x→∞

00

0

0

0

0

0

0

0

0

0

0

0

0

◦

◦

◦

◦

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

§ ¬`§ k *P * 8 * 8 ¡ 0° -1 x ¡ 8* 41 8T 1./ 0/ * , 1;/ 8 7 41;7 8

0/ ,.-4 7 89,3*4 )?1 4

k

p4PZ N yGZ\4PDC,

k !n12 n 2 Y k + n2 k lim ln 2 n→∞ n k=1

§ ¬ ª k *P * 8 !1Bµ`¡ , 84 7 98 *

*'¶41 k

/,bD ?G,s4/6uteAIH\6IDH\6{DFEG,seBZJ4/6;,AIL k3/,jD Q Qw4/6;K KG,/6;ADC, DFEG,:9sHqKG5e;PAIHJ6IDa>gAIL Q Q4/6;K Q.C,P>o4ABC e;, N DCH XB,PZ ukn~L D JH E >zDFEG, N ,PF6I1D .CAIHqKnAIL CA AB G QBeG.CA/XB,zDFEB4bBC D

4ABC

0 ≤ 3P G+P A+P B +P C −2(P D +P E +P F ) ≤

1 (AB +BC +CA) 2

k

UI

§ ¬ « k *P * 8 !1Bµ`¡ , 84 7 98 *

*'¶41 k

BAB.46u3e;AB>5HDCHXB,THJ6IDC,]P,P. Q;e;.CAPXB,=DFEB4bD`DFEG,/.C,g,5/HJ>1DC>O4ge;AIZuB6;AB9H\4/Z Q P (x) AILwK;,/]b.C,,=4bD`ZJ,/4/>FD QB>5y N EgDFEB4bn D 8n

(2n+1)2

X

|P (k)| < |P (0)|

k

§ ¬P¬ k *P * 8 !1Bµ`¡ , 84 7 98 4 1 4 124 7 *

*'¶41 k k=1

/,bD ?G,g4/63HJ6IDC,/.aHqAB.|eAIH\6IDOAILpDFEG,=D1.aHJ4/6;]jZq, kT+-,P6;AD1,g?/u Q Q a KGd DFEG,{KIHJ>1DC4/6 PN ,P>TLq.CAB9 D1A{DFEG,:>FHqKG,/> Q Q ABC Q.C,P>oe;, N DCH XB,PZ upQ4/6;dK8 ,/b6;AdDC, c PKBH\>FDa46 N ,/>|Lq.CAB9 BCD1A@CA ?/u Q Q DFEG,= DFEG,dXBAB ,/.DCH N ,/> Q Q QI.C,/>5e, N DCHXB,bZuwk DA DB DC P y;.DFEG,/.zZJ,bD Q Q4/6;K KG,/6;ADC,DFEG,:9:,P4>Fy;.C,/>gAIL A B C Q Q4/6;K PC ∠BP C ∠CP A QB.C,P>oe;P, AN DCH XBP,PBZ uk ∠AP B ;.CA/XB,VDFEB4PD 1 (PA 2

da db sin

+ PB ) + db dc sin 1 (DB DC 4

≤

1 (PB 2

+ PC ) + dc da sin

1 (PC 2

+ PA )

sin PA + DC DA sin PB + DA DB sin PC )

§ ¬ ® k *P * 8 ,./ )4 * 8 )

k

/,bD ?G,@4g6;AB6;oAG?DCy;>R,TD1.CH\46;]jZJ,z kd~L Q Q 4.C, DFEG,TZq,P6;]rDFEBABC >AIL;DFEG,=>5HJK;,P>OABe;e;AB>5HD1,T4/6;]jZq,P> Q Q B Q .C,P>oe;, N DCH XB,PR Z upQBeG.CA/aXB,zbDFEBc4bD A B C

a cos3 A + b cos3 B + c cos3 C ≤

§¬ k

*P * 8 ,./ )4 * 8 )

/,bD ;.CA/XB,VDFEB4PD

Q

Q

Q

x1 x2 . . . x n

abc k 4R2

k

?;,Ye;AB>5HDCHXB, .C,P4/Z-6By;9:?G,/.a>>o4bDCHJ>FL[uBH\6;]

n Q

xi = 1

i=1 n X

xn i (1 + xi ) ≥

n 2n−1

n Y

(1 + xi )

k

*P * 8 8 89,:1 4 1B¶1;/ * +/21;*4 , '« 1Lµ 7µ *0* , *¯ +/Nµ 8O¶0/2137 4 )+7.1384 7 8 )* -,L 1 k

§¬

k

i=1

i=1

¨

/,bD ?G,@4@;6BH[DC,@>R,jD-AIL.C,P4/Z6By;9:?G,/.a>d>Fy N EsDFEB4bDO,P4 N E HJ>-y;6BH Iy,PZ u a∈A ,5e;.C,P>o>FHq?IZqA ,T4> QFDOKBH\>FDCH\6 N D,bZq,P9:,P6IDC> Q Q Q >Fy N E3DFEB4bD a1 a2 . . . a k ∈ A k

a1 + a 2 + · · · + a k = 0

#? & ? +-AG,P>^DFEIHJ>(6;, .C,/eG.C,/>o,/6IDa4PDCHJAB6

N ,/>5>o4/.CH[Zu³EGAIZqKH[L:H[D$H\>W6;AZqAB6;]P,P.^4/>o>Fy;9:,/KDFEB4PD ,/4 N E H\>|y;6BH Iy,

a=b+c

¨ ¢

k

k

U

§®

*P * 8 !1Bµ`¡ , 84 7 98 *

*'¶41

k /,bD

#J4 & #? &

a

46;K

b

?G,=.C,P4/Z6By;93?;,P.a>O>54PDCH\>5L[uIHJ6;]

2(b − a) ≤ cos πa − cos πb

M

(1 − 2a) cos πb ≤ (1 − 2b) cos πa

0≤a≤

1 2

k ≤b≤1

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