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kV/,jD ?;,zDFEG,=HJ6 N ,/6ID1.C,gAIL D1.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 u k EG,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 1 H %$
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$/,bD ?;,346n4/.F?IH[D1.a4/.vuneAIH\6IDTAIyBDC>FHqKG, k{ EG, B1 DC4/6;]P,/6IDE Lq.CAB9 D1A DCAIy N EG,/> 4PD 4ze;AIHJ6ID k|/,bD ?G,z4ze;AIHJ6ID`AB6=E DFEG,VZ[HJ6;, E kTBAB.O,P4 N C >5y N EBDFEB1 4bD E E:1e;AB>5HDCHXBB,=2HJ6IDC,]P,P. QpK;,b6;, AIL B C B C = C B iHJ>d?GAIy;6;K;B 1 1 1 1 1 2 i+1 HJ6DC,/.C93>dAIL HJ6DFEIH\>934/6G6;,/.rk=;.CA/XB,TDFEB4PDDFEG,@>R, By,P6 N , ,/K: H\6 B {B } i i DFEG,=eGZ\46;,Gk
¨
r
§
k~v6x4VD1.aHJ4/6;]jZq, QBKG,P;6;, D1Ag?;,dDFEG, N H\. N Zq,Da46;]P,P6IDD1A Q/D1A Q4/6;K ABC AC D1A3DFEG,gHJ6 N H\. N Zq,sAILw DFEG,gD1.aHJ4/6;]jZq, Ca Qp4/6;K{ZJ,bD ?G,gDFEG,x.a4KIH\y;AB >zAIL k+-,b6;, ra 46;K HJ6{DFEG,3>5493,g<=4buwk ;ABC .CA/XB,xDFEB4PD Q <@EG,/C.Ca, HJ>VDFEG, r r r + r + r ≥ 4r r a b c HJ6Gb.a4KBH[y;>cAILDFEG,zD1.aHJ4/6;]jZq, k
ª
ABC
k /,jD 4/6;K ?;,:HJ6IDC,]P,P.a>g>5y N EYDFEB4bD k /,jD ?G,t4t>Fy?G>o,bD=AIL kFDa>d4/6 Q nQ u nQ P ({1 . . . n}) HJ6IDC,]P,P. >5y N E@DFEB4PD 46;K Q Q Q F G∈F k-;.CAPXB, t 1 ≤ t ≤ n {t t + 1 . . . t + k − 1} ⊆ F ∩ G DFEB4PD k n−k
«
|F | ≤ 2
k$IAB.T,RXB,P.vu8.C,P4/Z`6By;93?;,P. KG,P;6;, Q <@EG,P.C, {1H\.1.a− K ,P6;AD1,P>dDFEG,xLq.a4 N DCHJAB6G4/ZeG4.DVxAIL k:;.CAPhxi ; XB,gDF= EB4PDRmin({x}, Q LAB.d,RXB,/.vu{ 4bDCx}) HqAB6G4PZ6By;93?;,P{x} . 46;K:,RXB,P.vu3e;AB>5HDCHXB,=.C,P4/Z 6By;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.V<@EIH N E <@,teGy?IZ\H\>REG,/K^4t>RAIZ[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 `¤ 4 1 pEG,zLJy;6 N DCHJAB6g<@EIH N EgHJ>HqKG,/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
hVA<Zq,jD ?G,T4/6uADFEG,P.LJy;6 N DCHqAB6s>54PDCH\>5L[uIHJ6;]ODFEIHJ> N AB6;KBHDCHqAB6 k ,oEGA<DFEB4bD 9y;>1D?;f,T?GADFEx4KGKBHDCHXB,z4/6;Ks9yGZDCHJeBZ\H N 4bDCHXB,IQ/<@EIH N EgH\93eBZ\HJ,/>`DFEB4bD LAB. f f (x) = x 4/Z[Z k x∈ 4/6;K H\6 S Q<@,=]P,bD k-~L LAB.>oAB9:, 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 QI<@,@>o,,zDFEB4PD kpEIy;>jQ HJ9eGZ[Hq,P> y f (y) 6= 0 x=0 f (x) = 0 k x=0 GyBDaDCHJ6;] Qp<@,:]P,jD QLAB.V4PZ\Z k x = 1 f 1 + yf (1) = f (1) + f (y) y ∈ ~L Q;<@,g94bu N EGAGAB>o, kdpEIH\>z]jH XB,/> f (1) 6= 1 y = 1/ 1 − f (1) 1 + yf (1) = y EG,/6 N ,BQb<@,dAG?DC4PHJ6 LAB. N HJ6;] k pEIH\> 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 Q<@, 1=0 x=1 AG?IDa4/H\6 LAB.`4/Z[Z k f (1) = 1 f (1 + y) = 1 + f (y)
y∈
# &
M
G40,=46u
x 6= 0
kpEG,P6
f (x) 6= 0
k7,bDaDCH\6;]
f (x + 1) = f (x) + xf
, N AB6 N Z\yKG,VDFEB4PD
x 6= 0
f
Q<@,@]P,jD
1 f (x)
LAB.`4PZ\Z
1 x
=
46;K
x 6= 0
f
y∈
k-p,/eBZJ4 N HJ6;]
1 + yf (x) f (x)
x
# &
7PHJ6 N ,
= f (x) + xf
f (x + 1) = f (x) + 1
LAB.`4/Z[Z
f x + yf (x)
# &
,8DFEG,/6 y;>o, vS
DCA AG?DC4PHJ6
1/f (x)
= f (x) +
¨
# &
HJ6 S
?/u
y/f (x)
#ª &
Q
# ª &
HJ6
]jH XB,/>
# &
Q H\6
y
#ª &
?/u
= f (x) + 1 +
1 + yf (x)
x f (yx) f (x)
k
k
x f (yx) f (x)
k7PH\6 x f (yx) f (x) Iy;4PDCHJAB6tHJ>OXI4/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 <@EIH N E@HJ>4]b4/H\6@XI4PZ\HJKgLAB.4/Z[Z x 6= 0 y∈ 46;Ksy;>FHJ6;] Q<@,=AG?IDa4/H\6 f x + 1 + yf (x)
?/u
k-p,PeGZ\4 N H\6;]
# &
H\6 vS Q<@,@]P,jD
k
1 f (x)
x 6= 0
f (x + y) = f (x) + xf
XI4/Z[HqKxLAB.4PZ\Z
y = 1/f (x)
N,
x 6= 0
Qd<@, ]P,jD
Qw>FHJ6 N , k kO>5H\6;]DFEIHJ>xH\6 =ª 0jQ <@,]P,jDf4K;(0) KIH[DCH = XBHDc0u
# &
f (x + y) = f (x) + xf (y)f 1/f (x)
= f (x) + f (y)
Q
LAB.-4PZ\Z Q kO EIy;>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\ZJAf<=>DFEB4bD 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<=.CHDaD1,P6H\6R<=4/.CK;A 6pQwBAIZJ4/6;KpQ2Fy;6;, B ªIª 46;K]jHXB,P6H\6 ª/ BS ª S k
ªk 4
6¥
6nH\6ID1,/]P,/. 46;Kn4{eG.CH\9:, 4/.C,:>Fy N E DFEB4PD KBH XBHJK;,P> Q4/6;K 1 p KBH XBHJK;,P> 3 kn7G E >A< DFEB4PD H\>pDFEG,=> By;4/.C,@AILp4/6snHJ6IDC,]P,P.jk p − 1
7PHJ6 N , KIHXIHqKG,/> Q<@,=K;,/KBy N ,TDFEB4bD 4/6;K k p−1 N p ≡ 1 (mod n) n≤p−1 ~qDLAIZ[ZqA<=>`DFEBn4bD k7PHJ6 , H\>-e;.aHJ93,BQ<@ ,=EB4bXB, k n−1
p
gcd(n − 1, p) = 1
P
¤
7PHJ6 N , KBH XBHJK;,P> 3 QGHD|LAIZ[ZqA<=>-Lq.CAB9 _4/y;>5> 2 pEG,/AB.C,/9&DFEBp4PD KBH XBHJK;n,P> −21 = (n − k 1)(n /,bD +2 n + 1) Qw<@EG,/.C, H\>@4 n +n+1 n + n + 1 = kp eAB>FH[DCH XB,zHJ6IDC,]P,Pp .jkpEG,P6 k|^AB.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<@,T4.C,=K;AB6;,Gk
«
< H[DFE N HJ. N y;9 N ,P6ID1.C, T Qpe;AIHJ6IDa> 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
= 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,sP4bbQ<@,=EB4jXB,
k
2ab − (a2 + b2 ) cos C = 8R2 sin A sin B − 4R2 cos C(sin2 A + sin2 B) = 4R2 cos(A − B) − cos(A + B) − cos C(sin2 A + sin2 B) = 4R2 sin2 C cos(A − B) = c2 cos(A − B)
Q
<@EG,/.C,V<@,=EB4jXB,Ty;>R,/KxDFEG,zHqKG,/6IDCHDcu cos(A − B) cos(A + B)
~qDLAIZ[ZqA<=>`DFEB4bD 7PHJ6 N , OP 2
pEG,P.C,PLAB.C,IQ
k=
= cos2 A(1 − sin2 B) − sin2 B(1 − cos2 A) k = cos2 A − sin2 B
k 1 2 cos(A − B)
∠AOC = 2B
4/6;K
OA = OC
Qb<@,VEB4bXB,
∠OAP = 90◦ − B
k EIy;>bQ
= OA2 + AP 2 − 2OA · AP cos(90◦ − B) = R2 + k2 a2 − 2kRa sin B 1 1 2 2 2 = 4R k + sin A − sin A sin B 4k2 k 2 2 2 = 4R k cos (A − B) + sin2 A − 2 cos(A − B) sin A sin B = 4R2 k2 cos2 (A − B) + sin2 A − cos(A − B) cos(A − B) − cos(A + B) = 4R2 k2 (sin2 A + cos2 A − sin2 B) = 4R2 k2 cos2 B OP = 2kR cos B
k`7PH\9H[ZJ4/.CZ upQ
OQ = 2kR cos A
k
k|hVA<sQ
OP · CQ + OQ · CP = 4kR2 (cos B sin A − k sin B cos B + cos A sin B − k sin A cos A) = 4kR2 sin(A + B) − k sin(A + B) cos(A − B) Q = 4kR2 sin C − 21 sin C = 2kR2 sin C = R · kc = OC · P Q
~qDLAIZ[ZqA<=>Lq.CAB9 BD1AIZJ,/9gu >|pEG,/AB.C,/9³DFEB4bD Q Q Q 4.C, N AB6 N u N Z\H N k P e;AIQHJ6IDaC 7PH\6 N , QODFEG,= > 4/6;K EB4jXB,zDFEG,=>o4/9:, ApEIy;>jQ;BZJ,bDaDCH\6;] eA<@,P.`oe;, N AP DDCAs·DFAC EG, N HJ= . N y;BQ 9 N HJ·. N BC ZJ, AIL k| ?G,=DFEG, Q U N ,P6ID1.C,3AIL 46;K ?G,xDFEG,s.a4KIH\y;>TAIL Q<@Γ,EB4j4CP XB, 2 2 2 2 k3~qD ρ Γ DFEG,ze,P.1e;,/6;U N yGZ\4−.?IρHJ>o, = N DCABU.`B LAIZ\ZJA<=>DFEBΓ 4PD Q46;K H\>-AB6@ KIH A AIL − ρk EIHJ> .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 Qr<@EIH N Ez93,,jD AILDFEG,-eG.CAG?BZJ,/9 KG.a4j<(DFEG,-e;,/.Ce,P6;KBH N yGZJ4/. ?IHJ>o, N DCABP .a>AIQ L 4PD kO EG,/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 u k
* /;8
BC
P
Q
* 4 8
hV,F/DT<@,Y9:APXB,tD1A8DFEG,^$4bu ª/ 6By;93?;,P.sAIL-DFEG, 4/6;K(.C,P4KG,/.a> >RAIZ[yBDCHqAB6G>|DCAe;.CAG?IZq,P93>AIL DFEG,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 DCHqAB6g<@EIH N Es>54PDCH\>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
EB4/>,F/4 N DCZ ugDFEB.C,,T.C,/4PZ .CAGADa>-4/6;K EB4>DFEG,T>o4/9:,dXI4PZ\y,T4PD`eAIH\6IDC> 7PHJ6 N , a> uG99:,jD1.CH N 4f?;I A yBD QAB6;,3AILDFEG,.CAGADa>T9sy;>FDV?;f, k /,bD ks EG,3ADFEG,/. D<@A{.CAGADa>bQ 4/; 6 K 29Q 9y;>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. k EG,/.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<=6 4PD 9H\6ByBD1,P>@e;4/>FDT e;9 4/6;Kn.C,/4 N EG,K D1A<=6 4bD x >5493,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 <=4bD N EwkOlTA<946u9H\6ByBD1,P>dKIHqKsH[D`Da40,V 2oAGEB6sD1A]PA Lq.CAB9³D1A<=6 D1AgDCA<=6
HJ>4XI4/Z[y,VLAB. q4bDaDC4PHJ6;,/K=<@EG,P6 kpEG,P.C, N 4/6G6;AD?G,z4/6u pEIy;>jQ ZJ4/.C]P,/.2001 XI4PZ\y,zLAB. QB>FHJ6 N , dn KBH XBHJK;,P> k n = 1000
«
dn
dn
&
2001
k`~qDwH\>]jH XB,/6zDFEB4PD DFEG,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
/,bD KG,/6;ADC,gDFEG,s6By;93?;,P.VAIL 5,P.CAG,/>V4bD-DFEG,,/6;K AIL k= EG,/6 f (N ) , Iy;4/Z\>pDFEG,|f6B(N y;93)?;,P.wAIL/Lq4 N D1AB.a> AIL DFEB4bD 934juz?;,`LAB.193,KdH\6 k N 7PHJ!6 N , 10 N ! 10 = 5×2 46;K:DFEG,/.C,s4/.C, N ZJ,/4/.CZ u{L,r<@,/. >dDFEB4/6 >bQ HDOLAIZ\ZJA<=>DFEB4bD H\>T, By;4PZwD1A3 DFEG, 5 2 f (N ) 6By;9:?G,/.|AIL Lq4 N DCAB.a>OAIL H\6 k EG,/6 ?/u4V<@,PZ[Zq506;A<=6xLAB.19syGZJ4
¨
5
1 · 2 · 3···N
∞ X
f (N ) =
k=0
N 5k
}Bu>1D1.a4/HJ]PEDCLAB.<=4/.CK N AB93eByBDC4bDCHqAB6G>jQ<@,T;6;KxDFEB4bD f (109) =
109
f (110) =
110
<@EIH\ZJ, 7PHJ6 N ,
5
5
+
109
+
100
25
25
&
k
= 21 + 4 = 25
= 22 + 4 = 26
¨
Q k
H\>46gHJ6 N .C,P4>FHJ6;]LJy;6 N DCHJAB6pQ/<@,V>R,/,dDFEB4bDDFEG,z.C, IyGHJ.C,/K@XI4PZ\y,zAIL
® k 4 N H\. N Zq,xeG4>5>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 4PD e;AIHJ6IDa> 46;K C Q.C,P>oe;, N DCH XB,PZ u k7PABCD 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
N9 k pEG,=4/6G>1<@,P.HJ> 4 KIHJ>1DC24/6 N , ^AB.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/>@>oEGA<=6YHJ6 DFEG,{KIHJ4]b.a49Ykn7PH\6 N , H\>DC4/6;]P,/6IDD1A=DFEG, N HJ. N ZJ,z4bD QP<@,zEB4bXB,
....... .........
C
d
N
v
D
M
C
u I.CAB9DFEG,/>o,zD<@As, Iy;4PDCHJAB6G>bQ<@,=]P,jD /DFEB4PD`H\>bQ k BC · DC = d2 [ABCD] = d2
k3~v6;KG,,/KpQ<@,se;.CAPXB,xDFEB4bDd4/Z[ZwDFEG,xHJ6IDC,]P,P.a>VLq.CAB9 DCA pEG,s46G>C<@,/.OH\> 24 H\6 N Z\y;>FHXB,T4/.C,=4bDaDC122 4PHJ6G4?BZJ,;k 145 wAB6G>FHqKG,/.-DFEG,xLJy;6 N DCHJAB6 <@EG,/.C, k@ EG,/6 f (t) = 144t + 1441−tQ<@,@EB4jXBt, = sin2 x H\Lp4/6;K k7PHJ6 N , 0 0 0≤t≤1 f (t) = (ln 144)(144t − 1441−t ) =0 AB6BZutH\L H\L`4/6;KnAB6BZ u H[L 1 Qp<@f,(t) >R,/,s DFEB4bD H\> 1 k{7PHJ6 N , 0 t = 2 f (t) > 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{<@EG,/6 AIL 1 π Q LAB.z,5/4/93eBZq, Q46;KtDFEG,34?B>RAIZ[yBD1, f f = 24 x = 934rH\9y;9 2AIL HJ> q4bDaDC4P4HJ6;,/Kg<@EG,/6 QILAB.|,F/49eGZJ, k f f (0) = f (1) = 145 x=0 7PHJ6 N , HJ>4 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<@,/,/6 46;K HJ>`4PZJ>oA=4PDaDa4/H\6G4?IZq,IQ46;KxAIy;. 24 145 N ZJ4PHJ9³LAIZ\ZJA<=>/k
pEG,@,FeG.C,/>5>5HJAB6g<@,T4.C,VD1.vuBH\6;]dD1A,RXI4PZ\y;4bD1,T934ju3?G,=.C,r<=.CHDaD1,P64> n X (k − 1)k!
2k
k=1
−
(n + 1)! Q 2n
¨
<@EG,/.C, k ,s=H\>g, By;4PZD1A LAB. −1 ,RXB,/.vue;nAB>5= HDCHXB2002 ,zH\6ID1,/]P,/. k n hVADC,xDFEB4PDODFEG, N Z\4/H\9 H\>dD1.Cy,sLAB. k7Py;e;e;AB>R,6;A<DFEB4PD 4/6;K DFEB4PDDFEG,z, By;4PZ\HDcu@H\>XI4/Z[HqK@LAB. k n ,OREGA< DFEB4PDH[DEGAIZqKB>`LAB. n k`≥~v6;2K;,/,K Q
¨
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=A<946u<=4buB>V4.C,@DFEG,P.C,gDCA{4/.1.a4/6;]P, 5 HqKG,/6IDCH N 4PZ .C,K Q 5 HqKG,/6IDCH N 4PZ?BZ[y,BQ 46;K HJK;,P6IDCH N 4/ZO]b.C,,P6 934/.F?IZq,P>xHJ6 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 8 7 8
k
pEG,P.C,=4.C, <=4buB>/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<@ 4K 4 N ,P6ID94.1?BZJ,/>|46;K 4K 4 N ,P6ID94.1?BZJ,/>Pk 3 pEIy;>jQ<@,=EB4jXB,zDFEG,TLAIZ[ZqAo,/2>
¨
#&
¡
¡
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
<=4juG>ODCA{4/.1.a4/6;]P,
JH[H , N EGAGAB>o, AB6;, N AIZqAIy;.=4/6;K ?B.C,/40 HD=H\6ID1AYDFEG,:D<@A8eAB>5>5HJ?BZJ,teG4.DC>jQ <@EIH\ZJ,V<@,zZJ,/4jXB,zDFEG,=ADFEG,/.D<@A N AIZqAIy;.a> y;6BH[DC,K qk pEG,P.C,n4.C, DFEG,P6 ]b.CAIy;e;>:AILd94.1?BZJ,/>Pki EG, D<@A eG4.DC>3AILODFEG, ?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/<@,TEB4jXB, vS Q QGAB. S Q Q;AB. ª Q k ,=EB4bXB, 4 <=4juG>Pk 3
¡
# & # & # #JH[H\H & , N EGAGAB>o,VD<@A
&
M
\¤
3 · 3 · 2 · 2 = 36
¤
N AIZqAIy;.a>-46;K ?B.C,/40 DFEG,/9Yk pEG,P.C,x4/.C,=DFEG,/6 ]b.CAIy;eG>zAIL 94.1?BZJ,/>Pk ,geBZJ4 N ,=DFEG,=D<@A3eG4.DC>VAILpDFEG,sS ?B.CAG0,/6 N AIZJAIy;.OHJ6:D<@5At6;AB6;F4K 4 N ,/6IDd>oe;ADC>jQw4/6;K:<@,KGA:DFEG,s>o4/9:,xLAB.-DFEG, ª N EGAIH N ,P>/k ?B.CAG0,/6 N AIZJAIy;.jk ,=EB4bXB,
JH X , ?G.C,P40 B4PZ\Z N AIZqAIy;.a>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
<=4buB>/k
HJ6
O
hVA< <@,dDCy;.16xDCA{I4.D}:AIL;DFEG,=7PHJ6;]b4/eAB.C,x$4PDFEG,P934bDCH N 4/Z M ZuB93eBHJ4K]jH XB,/6 ª S/ k
ªI k|/,bD
2002
Q Q Q 4/6;K Q Q Q ?;,V.C,P4/ZB6By;9:?G,/.a>|?;,jD<@,,P6 4/6;K b1 b2 . . . b n 1001 n n P HJ6 N Z[y;>5H XB,;k7Py;eGeAB>o, P k ;.CAPXB,zDFEB4bD 2 2a1 a2 . . . a n
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
,V|H\6;, Iy;4/Z[H[Dvu k BAB.,/4 N E Q<@,=EB4jXB,
h ADC,zDFEB4bD|, By;4PZ\HDcuA NFN y;.a>|HJ6 U V 46;K3AB6BZ usH[L
a3i 4 Q + (a2i + b2i ) bi 5
≤
21 10
a2i −
2 5
b2i
k
M
#U &
H\L 46;KAB6BZuxH[L AB. /DFEB4PDHJ>jQIH[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
Q<@,@]P,jD
n n n n X a3i 21 X 2 2X 2 17 X 2 Q ≤ ai − ai = a b 10 i=1 5 i=1 10 i=1 i i=1 i
4>OK;,P>5H\.C,Kwk Iy;4/Z[H[Dvu3A N1N y;.a>-H\Lp4/6;K3AB6BZuxH\LQILAB.,/4 N E AB.
n P
b2i =
k pEG, N AB6;KBHDCHqAB6
i n P
i=1
QG,PHDFEG,/. a2i
(ai , bi ) = (1001, 2002) n P N 46g?G,d.C,r<=.CHDaD1,P6g4/> = b2i
Q<@EIH N ExHJ>
k EIy;>bQ
i=1
2 2 2 2 1001 p = 21LABn.EB4/Z[L AIL 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> <@EIH\ZJ, LAB.DFEG,@ADFEG,P.|EB4PZ\LCk i (ai , bi ) = (2002, 1001)
|NFN AB.CKBH\6;]OD1ATDFEG,V.C,P4.C.a46;]P,P9:,P6IDHJ6;, By;4PZ\HDcu Q/DFEG,V>5y;9H\>9sHJ6BH\9H 5,Kz<@EG,/6 Q Q Q 4/6;K Q Q Q HJ>OH\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
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2n + 1 0
D1A
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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
<@EG,/.C,zDFEG,zZJ4/>FD`>FDC,/esy;>R,P>DFEG,V<@,bZ\ZJo06;A<=6sHJK;,P6IDCH[Dvu m m k j = ]jm hVA< N AB6G>5HJK;,P.V4t]b.CAIy;e$AIL ?GA/uB>=46;K H\.CZ\/> − k:j ,x<=46IDdDCA >R,bZq, N D
2n + 1
n
2n + 1 4n + 2 <=4juG>TD1A8KGA n
e;,/.a>oAB6G>@Lq.CAB9DFEIH\>x]b.CAIy;e k8 EG,/.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;ewQP<@,d>o,,ODFEB4bD DFEG,OD1ADa4/ZB6By;93?;,P.AILI<=4juG>HJ>`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−→ >
~qDOH\>V.C,P4KIH\Z u N EG, N 0,/K:DFEB4bD-DFEG,gLJy;6 N DCHJAB6 H\>V4x>RAIZ[yBDCHqAB6 k ,zREGA-y;6BH Iy,;kz/,bD ?G,@4634r.1?B7→ HD1.ar(r 4.vu3>o+AIZ\2001) yBDCHJAB6 kV+-,/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 kn EG∈,/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
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#
&
r∈
#
&
* 4 8
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`¬ 8x1 8 1:, 124 1D³ x¨ 1;/;8 -O /,bD ?;,z4 KBHJ]jH[DeAB>FH[DCH XB,dHJ6IDC,]P,P. kpEG,P6 Q Q46;K abc4.C,xDFEG,: Qp<@EIH N E & k EIHJ>|100a a b + b + c) >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;= <@,x 9sy;>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[yBDCHqAB6 k 102 204 306
4
§
k=/,jD 4/6;K ?;,TD<@A3eAIH\6IDC>dAILpDFEG,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
/,bD ?G,-DFEG,eG.CA v, N DCHJAB6@AIL AB6ID1A k EG,/6 HJ> DFEG,9sHqKG5e;AIHJ6IDAIL M0 M r M0 RS N, H\>`DFEG,T9sHqKG5e;AIHJ6ID`AIL 4/6;K k q ~ DLAIZ[ZqA<=>DFEB4bD 0 M AB M M ⊥ RS 4RM S HJ>|H\>RAB> N ,bZqP, >`
#q>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# & Q<@,=AG?DC4PHJ6
ρa = ρ b
k
# &
k
N ,IQ
# &
¨
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¡
9 P$ k S (|k/lTAILq>FDC,bDaDC,/.oQ 4 7PH\93eBZq,|pAB6G>FD1.ay # ªIª & Q; ;k ª
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1 (AB +BC +CA) 2
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X
|P (k)| < |P (0)|
k
Q
P (x)
k
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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
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1 (PC 2
+ PA )
sin PA + DC DA sin PB + DA DB sin PC )
k
U
§ ¬ ® k *P * 8t¡ ,;/ ) 4 * 8 )
k
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4ABC
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k
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vn =
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# &
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#
# &&
2a 2b Q = −2b + 1 + 2a b−a+2
x =
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x =
¨
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a
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4/6;K
¨
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n > 2 T f_PQG A1G A J B1 B2 . . . B n K 2 . . . An HCIG5 O O=FGBOmNd F,LSBNLFCSrGlgG dNS G F U K Y Y YH Y S = {1 2 . . . n} e] HCTQT i j ∈ S O O=FGBO RFMF!o G cN FMF F F G$P N N SBN#RFG G PdG F#SBF U K H X T I T V I I X VV I YH Y Ai H IJ Bj YH SBF G F ODG^GrfmN PO6GBP GeO O@F;G O N \ G F G F#SBFcF6o PO6G O B B . . . B H H T H J IX U K H V I Y I Y H 1 2 n N F LGlvO O@F;G G G O F=RF FCSNd$F F F G OP N N I I V U K T V I IgX V,V I T ⊆ S Y H Y H HCI IvIUV)K f_PQG F Nd G F_O O=FGBO A1 A2 . . . A n Y HX!Y Y U K IgHCI
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25 54
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4
¨
¨
<@EG,/.C,zDFEG,zHJ6;, By;4PZ\HDcuxLAIZ\ZJA<=>O?/us464eGeGZ[H N 4PDCHJAB63AILDFEG, _| ~v6;, Iy;4/Z[H[Dvu k eAB>5>5HJ?BZJ,xLAB. Q Q Gy;.16BH\6;]3?B4 N 0sD1A:DFEG,eG.CAG?BZJ,/9tQ <@,6;ADC,xDFEB4bDdHDHJ>z6;ADd AB. DCA , By;4PZ k~L QpLAB.,5/4/93eBZq,IQ DFEG,P6 Q;<@EIH N E{9:,P46G>daDFEB4bbD c 1 QG4 a N = 1 b = c = 0 AB6ID1.a4KIH N DCHJAB6 k EIy;>bQ Q Q kO/,jD
¨
ab + bc + ca = 0
a b c ∈ [0, 1)
Q(a, b, c) = = =
hVADC,zDFEB4bD
?u $ k k
Q
1 1 1 , , 3 3 3
=
1 (k 8
(1 + ka)(1 + kb)(1 + kc) (1 − a)(1 − b)(1 − c)
k3 abc + k2 (ab + bc + ca) + k + 1 ab + bc + ca − abc k k2 abc + 1 2 k + (k + 1) ab + bc + ca − abc
+ 3)3
4/6;K
Q 0, 21 ,
1 2
= (k + 2)2
k
k 2 ,e;.C≤ APXB5,xDFEB4PD Q 43>FD1.a4PHq]PEDF 1 1 1 k7PHJ6 N , , y; , k4/+ 3 4PZ\3HDcuxHJ>O, BG LAB.<=4/.CK N 4PZ N yGZ\4PDCHJAB6>oQ(a, EGA<=>b,DFEBc) 4PD≥ DFEIH\Q >|HJ6;3, B y H XI ZJ,/6I1D> D1A 0
¨
¨
k2 (ab + bc + ca − 9abc) + 27(ab + bc + ca − abc) ≤ 8
k
# &
;S/ pEG,TDC,/.C9 >5HJK;,:AIL pEIy;>jQ
# &
# & !# &
HJ6XBAIZ XBH\6;] 2 H\>O6;AB6;56;,/]b4PDCH XB,BQGH\6xXBHJ,j<AIL ª k|7PHJ6 N , 2 QBDFEG,TZq,bL\D 4/6;K HJ>=4bDz93AB>FkD L AIkZ[ZqA≤ <=>z5Lq.CAB9 vS k 8 4(ab + bc + ca) − 9abc k 1 1 1 1 3 HJ>DFEG,T9H\6BHJ9sy;9XI4/Z[y,@AIL
?u $ Qk k ,≥ ,5 k = (k + 3) ,ge;.CAPXB,@DFEB4bD b,Dc) DFEIHJ>|HJ6;,B ¨ y;4PZ\HDcuxHJ>O,¨ByGQ(a, H XI4/ZJ,/6I D1A ≥ Q 0, 3 2
3
3
# &
Q
8
1 1 , 2 2
kV7PHJ6 N ,
k+1 > 0
QG<@,@;6;K3DFEB4bD
k 1 + 4 abc − (ab + bc + ca) + k2 abc ≥ 0
# &
pEIH\>{EGAIZJKG>:?/u S G ? , N 4/y;>o, 2 kfpEIy;>jQ HJ>sDFEG, k ≥ 5 Q 0, 21 , 12 = (k + 2)2 9H\6BHJ9sy;9XI4/Z[y,@AIL k
