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e I \ sR e 0>H:ELAO5 7C4 e6>HAELAD7H:8b47EGF6>H:8> elELK 5MAO>P 465 QR8C_ (?)
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K
ABCD DE = EF = F C BG = GH = HC L M N
E G AE
F H
DC BC DG
AF KN k CD
f (n, k)
DH DG
k
f (3, 7) = 0 f (3, 6) = 1
DH
n n=3
f (3, 4) = 6
f (2007, 1) + f (2007, 4) + f (2007, 7) + · · · + f (2007, 4012)
16
16
R 0 Oo # + \E mEAE0 2 U ; 1Q ; , O#$1Q+ 6 R 24$u2H;)0 01°'& 682C-7f$ 0G9 2C0@ACJ=-I9 0w6@9 -I9 L9 -240EbR0[25$'{ AF682C-7$ 9)6@(f9 & JQa)0@AE$'(I& 2C-7$ 9)E I \ &^ R 0 Oo # + CQ# 9 #$> R ¹ 619%9 * 01°'&)-I(I682C0@RAF68f(10G242CAF0@-76@AC9 J=O[-I(+9 0f0V(.-+2H08;)ET0-79z9 2H& ;)J>0 a ?)08(IA619 $'0v{"? $'-.2H-7;9'2F2E "%$V$'{D-.2FEs"y -2H0@;=A2C-I-+9':40@240@E^O8680@2sA ?L:4$'$)-I$ 9'AH2CEP * -I9)682C0@E"%;'-+:<;5(I-70z-7952H;)0z-79'2C0@AF-+$ Av$'{L2H;)0324AC-I619 O[(70 R I \ n R 0 so $# + Q *HJ# 9 =# 41 # >@ Z S7) # 0 s,C #0 +0% 6 +174E ! s1 E :(I4-7-70AC:4$ (709,¹ "2H;)-Eb2H0%°';5& EHAF61246DAFAC68*'0-+-IO8& $';DEf{2BE<$'-7EH{L* -+(+*)03080@9 (7E}0@O9 $'2HO; {2H2H; ;)610%9 -7Eb*=Ev0@:C-I:49)240@$ Eb9':HAv24AFAF61-+68a)9 (L01*,6@*,29 "%EFO[M (+$J>0Vy $'J>08{ AF0G2C24-7AF:H-+08:CK'6@E E<(.(.(&L-+M0%:<-I;%$ 9)2H9,E<; -7* 2H68;)0z2^026"%:4Eb-7$AC0@:4:Cy &)240@$ (I61AA2CAB-+$':46@{0@AC6 E: $'{L2H;)0VE01:C2C$ A[RfR fR 0[R 24R08A4R J,R -7R9 03R 2HR ;)R 0%R 0HR ®@6DR :CR 2^R y'68R (IR &L0VR R $'{R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R I \ s R 0 Oo :2# =Q c# ;0 1D8- ,+5 1 7 # HJ# R B619)EB& 9=AC0@:C2C6@0G9 2vO[(+(+0%0 : $D 2 0 E $ 089>9=*'24-ACy'$'-7-7EE 0z? (70V6@AF: 2C$D-70@ 2 E0 E 01E OG68(+080@E 9524AC$'-IEf?)61A2C-+08E 01E OG68(+08E R ACfF80@6w0VE-.(+2 $ EAF$ E AF2C0>°'&L0Q(70V:<9 ; $ 61J>° K &La)0QAC0%089 *){+0V61?L9'2z$ E<08E<9w-7a 0GAC2-. 0(I-:< 2 $'0@E -Efy *)0=0%6@* &x-IEH? 24(IAF& R)-+a'EV &L0G*)E 0@2C08A0@&DAC®LJ=Rva -I»'9 $ 610@9 A3Asa)(7$ 0H6B®19)y'0@E J=6@6 ? (708?)& (70 AT0@Ks*)9 EH0 P R I \ R 0 Oo :2# }! E # = JH;1 hC #$- s1 m1o R Eb? $ 68A(I-I249 0*)S °'ACAC&L$ $'0zJ>&'(7y 0 0z0@E? Az$'AC-+E$)0808*'9'-¼<&)2L0>-.2C2$'?)* & 6@08E^(.-7Ef*)9 :<08* ;'&DAC-I®$ ¶#JQ01ACE 0@OG08EB68EG&D9 KT® $ :<; 9,6f? 61Eb9 :40G&)& -.¼<(I9uEf0)R80G:H$ U#2^J=$ (76JQ? EACa'$ 08-+J>9)08961J>9'Mz2z061Eb*)[email protected]¼<-I0>(1:<*);':<-.0v;'¶s-I24AC¶#0G08(7ACEfE0@E[* 9 KT0$ * JQ:<0,; a 6D24AC°'0G0@(I&LE[(700o I \ R 0 Oo :2# \E mEAE0 2 U ; 1Q ; , O#$1Q+ # R * 0( 0@E ° & 0GE 6G2C2C0@-+AC$ J=9 -I9 0@A & 9z0@9)E0@J>a (70v-I9 L9 R -@*)0BE$'(I& 2C-7$ 9)Es0@939 $ J>a)AC08EAF6G2C-+$ 9)9 0G(7E I \ &^ R 0 Oo :2# D; 2 Q_o# + # Q# > R B0G2 619)Ev(+0z?)RD(I 619#0GE 2CKD0@& AC95J=24-IAF9 -70@6@A 9 (+O[0v(+0 9 $ 0@E ° JQ&)a -.(7AC6G0f 2 *)0@E AF0B68? (L$'6V-79'* 2F0GE &D®* 0zEb6f08? Ef:4$)E$ $ J>AC* J>$ 9)0G99 2FEB0@E 0@6@E&D®V0@9'?L2C$'@- 0@? -I9'AC082CEE E<-2C& 0@E E6? (\ -79'208E AC-708& AB*'&,24AC-I619 O[(70 R I \
(α, β)
α2 + β 2 = α 3 + β 3
(0, 0)
(0, n)
(x, y)
s
60◦
r
s/r
DE = EF = F C AE M
N
ABCD BC
DC
BG = GH = HC K L DG DH
DG DH AF KN k CD
f (n, k)
n=3
f (3, 7) = 0 f (3, 6) = 1
k
n
f (3, 4) = 6
f (2007, 1) + f (2007, 4) + f (2007, 7) + · · · + f (2007, 4012)
(α, β)
α2 + β 2 = α 3 + β 3
(0, 0)
(0, n)
(x, y)
I \ n R 0 Oo D2# Q *$HO#:9 =# 41}5,) #$ 6 O2# # >@ Z + # # J O#% #55) #$1 4E ! s1 E 95:46@A4A0zE *)0z: $D20E s 08EH2-79)E:HAC-2^EFM JG0[E 24AC-7° &L08JQ089'2* 619)EN& 95E01:C2C08& A^* 03:H08AC:4(+0
Eb *:H08$'0fAC-7AF0@:46[(+9'0)M 2VRs$ E<9 & 0GE A2C0G0@(70@2#ACJ=E5*#-I?)619 619 0@AO[A2C(7-+0(I0863E568y'&VAC6@01:H(7:C08082C9'& -.(I24Av-7ACOG0<0 9 ®@* 0@610E,:C240V*'& * 0 KGEb* 0@:C024240G0G& R (IA%(70f0GE2V$ AF(70@2C0fE5° * &L0G0 &D®x*)0868&D& ®}24AC* 0@0fE5EEH0@& EAVE$ (J>6@ACJ>:50G* 2F0E
60◦
r
s/r
!W" $'& ( j,0/?,'1 I \&\s R
0 Oo
J R
h! 85 Z6 A1 > , s1 8
# +
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1 2
b
w
Pb
Pw
Pb =
b b−1 · b+w b+w−1
Pb + P w =
Pw =
1 2
w w−1 · b+w b+w−1
b−1 w w−1 1 b · + · = b+w b+w−1 b+w b+w−1 2
2 b(b − 1) + w(w − 1) 2
2
2b − 2b + 2w − 2w b2 − 2bw + w 2 (b − w)2
= (b + w)(b + w − 1)
= b2 − b + w 2 − w + 2bw = b+w = b+w
b > w d = b−w d2 − d − 2w = 0 w = 21 d(d − 1) b = w + d = 21 d(d + 1) (b, w)
n(n + 1) 2
∞
n=1
= 1 3 6 10 15 . . .
I TZGoTZGJI {JR d ~ rDn S N'M R I5ZT S M b TZL6K'd R NL b S M V R M' l:RZW M AS NmF qGHGJI b S M V R M b S M S N ]
A I \&\ R 0 so $# + o#$ Q#CE BAB6 > '1f8 Z,# , 7) # 0 s,C1 (Z : 1 R @F 0G2 N a 0,6=? $ E<-2C-y 05-I9'240@O80@A}EH&L:<;Q2H; 682 N +( 086Gy 08Ez6=AC0@J=6@-I9 * 08A3$'{ 2 $ A 4
"%» AC;)$@0@y 9=05* 2H- ; y 68-72* 0@2H*=;)0@a@ACM 0>0<®@6@-79 E42}*-72H9';)2C0801ACO8008AF61E AC03-79'6@2C9 01* O808AFE "-2H;619 * E<&L:<;%2H; 6G2 "%;)0@AC0 -7Ez9 $D2 R * - y -IE<-7a (70Va@M R #( O6 9 8Q A#$;5#0 s # > 5 ,)*6; 7 O0 Q1 #( O6 9 8Q A#$;5#$1 7 4 7C > 1 m1o m1 >@+ 7# + CQ#/# + R R-IEQ0b0y 0@619AC056@O[9 -y * 089Q-7E>2H9 ; $D6G22 F@0G2 619 * a)0-I9'240@O80@AFE3EH&L:<;=2H; 682 $ A R # S ) ; 8 0 C A 8 0 l { $ C A ' 0 K ]l* Z8a)-7- 9 y $D-I:42HE<0;-7a 0b(7y 0w08-7Eva@9wM 9 $ $DA}2Ra)* $D- Z8y 2H-7;¤-I9 E<:4-7$)a 0 * (70*1`Ha@K M -I"%E=;'0bK@-+y "%:<08;Q0z9#-IEK J>010}? (I2H-7; 0@61E}689 2 *2H; 6G-72 Ev9 $D6@2J,9 * * & - y EH25-IE<; ; -7a 6[6[(7y y 0z00Xa@2H2HM ;);)0,0wE<]+E0@J>:405680& ? Eb? 6@036@ACAC--.2\{-2\MsM R "%0@AC0V* - y -IE<-7a (70Va@M KD2H;)089,2H;)0E6@JQ0"%$'&)(7*>a)0z24AF&L0{l$ A `
x
a
b
y N = x2 + 27y 2 2 N = a + 3b2 b
3
N = x2 + 27y 2 N ≡ 4 (mod 6) N N x2 27y 2 x y N 3 x 3 3 x2 + 27y 2 = N a = 21 (x + 9y) b = 12 (x − 3y) x y x+9y x−3y a b a2 + 3b2 = N b 3 x a b Jq RZWR RZWR N GmGHL q RZW TZGJIBKHL MBGONHT T4K ~sl M_LL R d ]
x y N ≡ 2 (mod 6) 3
3
I \&\ R
0 Oo
2 ;
x
R
\E m EAE0 U 1 Q ; , O#$1 Q+ 6
# +
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P (x) = x3 − 70x2 + 1629x − 12600
l
w
l
w
P (x) = x2 − 5x + 2
P (x) = (x − l)(x − w) = x2 − (l + w)x + lw
x2 − 5 + 2 = x2 − (l + w)x + lw
l+w = 5 lw = 2 2(l + w) = 2(5) = 10
lw = 2
2
Z8; -76G9 y :40 0 ]la1K ` F@K'0[612 9 K* 8K 1661AC9 03* 2H;)0za)0B¼<082HAC;)$)0}0@* Ef-I$'JQ{ 089)E<-7$ 9)EN$'{12H;)0 AC0@:C2C6@9 O[&)(76@AAC$)$ J-79JQ0[24KDAC"%0@E80 R l w
l w h
P (x)
S#;'& E[K
h
P (x) = x3 − 70x2 + 1629x − 12600
= (x − l)(x − w)(x − h)
= x3 − (l + w + h)x2 + (lh + lw + wh)x − lhw
R
R r12H;'& EGKD2H;)0E<& AF{+6D:40 J K
x3 − 70x2 + 1629x − 12600 = x3 − (l + w + h)x2 + (lh + lw + wh)x − lhw
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lh + lw + wh = 1629 lhw = 12600 2(lw + lh + wh) = 2(1629) = 3258 2 lwh = 12600 3 I TZGDTZGJI {JR d ~ r7I V RZW7HS rJ b lmS Nhx W M {ZS L R F qGHGJI b S rJT RZW M b 'K W6
*R r u S N d M q JS W d:j ] k R T TUb b FZ ] S N q Gmx S I_GOT ZR W d R TUb 5
>
I \&\ R
0 so $# + O7 1 O7 1 yE
0 C
S
+ : O+/! E Q # 1 7) # s, ?
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P ....................... ......... ..... .......... ..... .... ..... ....... ... ... .... ... ... .. ... . . . . .. ... ... .. .. .. ........................................... .... . ... . . ... . .. ... .. .. . . .. . . .. .. ... ... ..... . . ... .... ... ... . ..... ... . . ..................................................... ..................................
1
P Q
R
O
r
P QR
QR M
r−
+
q
TS
N
r2
−
1 4
S
Q
R
P
.............. ................... ........... ............................ ......... .......... .. .. .. ........ ....... ... ... ... ....... ..... ... ..... ..... ..... ..... . . . . . ... ... .... . ... . . . . . ... .... .... ... .. . . .... . . ... .. . .. . ... . . . . . ... .... ... .. ... . . . ... .. . ... .. . . . . .. ... ... . .. ... ... . . ... .. ... . . . ... . ... . . .. ......................................................................................... .... .. .. ... .. ... . .. . ... ... . . . . . ..... ... ... .. . . . ... . .. ... ..... ... ..... ... ... . . . . . ... . . . . .. . . . . . . . . . .. ..... ..... .. ... .... .. ... .. .. ... ... .... .. .. ... ... .... ... ... ... . ... . . . . . . . . . . .. .. .... . .... .... .... ..... ..... .... .... .... . . .... .... ... ..... ..... .... .. .. ..... ...... .... ....................................................................................................... ........ ............ ......... .....................................................
O MN √ 3 OM = r − P M = r − 2 q 2 ON = r 2 − 21 OM + ON = 1 √ 3 2
T
M
T
O
r
N
Q
= 1
S
R
r
q
r2 −
1 4
r 2 − 41 √ r 2+ 3
√
= 1 + 23 − r √ 2 = 1 + 23 − 2r 1 + √ = 1 + 3 + 34 + 14
{+AC$ J"%;'-+:<;%"%0V:4$ 9 :4(I&L*)0z2H; 682
R
K
√ 3 2
+ r2
r=1 I TZGfTZGJI {OR d ~ rc5I V RZW(HS r4b$ lmS N x W M Z{ S L R F q GHGJI b S Or T ZR W M b:sK W6
HR r ] M N G W6WRO L TZGJIBKHL MBGONHT T4K ~sl M_LL R d ]
K Jq RZWR RZWR L 7G
I \&p^ R
C 9
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# +
Q#
#$>
n
n
n
A B C D
1≤A
E
n
n
A B C D
E
n
n
A
B
C
D
E
15
1
2
3
4
5
16
1
2
3
4
6
17 17
1 1
2 2
3 3
4 5
7 6
18 18 18
1 1 1
2 2 2
3 3 4
4 5 5
8 7 6
19 19 19 19
1 1 1 1
2 2 2 2
3 3 3 4
4 5 6 5
9 8 7 7
20 20 20 20 20
1 1 1 1 1
2 2 2 2 2
3 3 3 4 4
4 5 6 5 6
10 9 8 8 7
n = 20
E =8
20
n J q RZWR RZWR N GmGHL q RZW ZT GJIBKHL MBGONHT T4K s~ l M_LL R d ]
A
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0 Oo # +
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1
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6
k 1 2 3 ... k
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k
2 1 2 1 7 1 3 5 15
4
1
3
7 21
5 6 1 2 3 6 1 2 5 10 1 3
3
Q#c so $# *E
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8
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3
15
5
6
14
1 2 7
18
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4
1 2 3 4 5 6 8 9 18
D
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ti 6= ti−1 0 1 1 0 0 1 0 1
ti
t2 t4 t6 t7 t8 1≤p≤q
0
0
i
1
p
1
q p=5
q=7
M
M
M
10 1000
M
10
1000
m ≥ 0
m
0
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m
1
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1
M
q
p+q
0 0 1 1 0 0 ... 0 1 1 ... 1
M
p p+q
p+q−1
M
p
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p + q − 1 = 2p − 1 0 1 p < q
0 1 0 1 ... 0 1 p = q M = p + q − 1 = 2p − 1 0
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1
p
2p
2p
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2p
p+q
2 tk = 0
p
M = 2p
1
p
p+q p
k p+q k tk−1 = 1 tk = 1 tk−1 = 0 tk = 0 tk−1 = 1 p+q−2 p+q−2
0
k
p−1
p−1
2
p+q−2 p−1
q−1
tk = 1
p+q−2 p−1
tk−1 = 0 k
k
p+q−1
2(p + q − 1)
2(p + q − 1) p+q p
p+q−2 p−1
=
=
p+q−2 p−1
2(p + q − 1)(p + q − 2)! (p − 1)!(q − 1)! (p + q)! p!q!
2(p + q − 1)!p!q!
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=
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a
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b
θ
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A1
.... . .... .... ... .... .... .... . . . . .... .... .... .... .... ... .... .... .... . . . . .... .... .... .... .... 1 2 ... .... .... ... .... .... . . .... . . . . . . .... .... .... .... .... .... .... .... .... .... .... .... .... .... ... ... . . .... . . .... . . . .. ... ..... ....... ..... ...... ......... .... ... .... .... ... ...... ... ..... ... ....... .... ...... .... ... .. ... .... .... . . . .. . . . . . . . ... . . . . . .... .......... .... .... ................................................................................................................................................................................................................................................... .... .... .......... ....
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c
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θ
a
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B
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OP = OQ =
O
A
ab a+b
a
b
θ
4A1 AB 4P OB A1 A = AO = a OB = b
AA1 OP = OB AB
ab a+b ab OQ = a+b
OP =
a OP = b a+b
B1 BA
B = (−b, 0) a(1 + cos θ), a sin θ B1 (b cos θ, b sin θ)
O = (0, 0) A = (a, 0) AA1 = OA = a A1 A (−a cos θ, a sin θ) 2 B2 −b cos θ, b sin θ P
BA1
OB2
P =
ab cos θ ab sin θ , a+b a+b
AB1
Q
Q =
QOA
−
ab cos θ ab sin θ , a+b a+b
OP = OQ =
ab a+b
OA2
. T
) -.6@2(.()-IE2H;)6@$ ? ¹ Eb?)Ev0AC"%$ "%? 0V;)AFa -7$,6G0@24?)O[0 AC-7$@9=2C$y'61-+(+*)9 $)01$D$)*,2H/z;)& 08a EA^6D":<M /[email protected]@$@;,Avy 08?)61As9 AC$)2H*=;)a 6V0(70@9 @J=D0[EG"³1K) :4y 9 $ $'& J>(I& J>J>J>a 0@0%089'AF$'2FEBE {N$'6@c {'9 d 2H*=;)e'0 Eg$'8Q(IhB& j+ J2Ck -7m>$ #$9) np E 61q of9 *q#r32H; t 6@ni9 / K ¹¯VAH$'/@&L61* *D6 M ¹ ¹ 9 (2 $'& 9 µª$ y'"E -+0 E *'-IFG&¤&)-71Ez& AC* EI&)68¼<- P[²L61ACAC0@AC$ ¹S $ :<;'E;'-I(.-+(+$,0861Z1Ef08-IZ8J=-79 - 0GM'{+6 6D/1$ ? $'&)(+$ E WuWu-+-+O[:<;)&L0G0G(^( ¹²L6GJQ2C6808-I9 (.(+O[0 & 6@(Us$8y'61E N!\$ 016@$'9)¶#9 AC-70bE M #6¹ 82FR E<#6-7/11-I9 E * 6@(.( ²L6D-7:
A;A !CW_ª-79>µ<61?)619R=WwMQ2H; 619 /1EO8$>24$ ¹ 9 *DM1FG-.&sKsU#6@9)6D*'-76@9¤S 0861J F@086D*)0@AK#{l$ A :H$'(.(+0@:C2C-79 O2H;)08J {l$ Av$'& A& E0 R _&_ !#" %$& !(' %$*<;>)Q=@?>I+ $FH !GJ,-IK; 2I+LMANFP$&OQ !C X.$*)/'0)21 I4356$*7 A D B E C E = 8:9 ' G: c i j $1_ R @F 0G2 n a 06w?L$ EH-.2C- y 0¤-I9'240@O80@AKf619 *Y(+0[2 x ≤ x ≤ · · · ≤ x a)0wAC086@(
9 & JQa)0@AFE8R ]76G`¤» AC$@y 032H; 6G2 R ]la1`>EbZ10@;)° $1&L" 089 2H; :H0)682R 2H;)00@° & 68(I-2\M>;)$'(+* E -.{s619 *>$ 9 (M,-I{ K K -IE}6@9>61AF-.2H; J>0G2C-7: $ a R F@0[2 R a)0w2H;)0uEb0[2=$'{36@(.(? $ E<-2C-y 0xAC086@(9 & J>a 08AFE@R¥1-79 *¢68(I( {7& 9 :C2C-+$ 9)E R R 2H; 682vE<682C-IE<{.M%2H;)0{l$'(.(+$1"-I9 O:4$ 9 * -2C-+$ 9)Eq ]76G` K K R r {l$ Aw6@(.( ]la1` {l$ Av68(I( R 1
n P
i,j=1
|xi − xj |
2
≤
2
n
n 2(n2 − 1) P (xi − xj )2 3 i,j=1
x1 . . . x n
+
f :
+
→
+
√ √ √ f (xyz) + f (x) + f (y) + f (z) = f ( xy)f ( yz)f ( zx) x y z∈ + f (x) < f (y)
1≤x
1
$ s 4R F@0G2 a 0B6 ? $ E<-2C-y 0-79'2C01O808AbKG619 *3(70G2 Eb?L0@$ ° EH&L-.2C80 - y 9 0:H08ACEz0@$'68({?L9 $ & EHJ>-.2Ca - y 080AFE=AC0@E<68&L(s:<;u9 & 2HJQ; 6Ga)2 0@AFE8RzZ8& ? ? $ Eb0 {l$ A6861(I( 9 * I- EK 65Eb0@° &L08a)9 R0v:H205F@"%0[$'$ 2 { K K R » AC$8y 0z2H; 6G2 n
2 zi+j
M = max{z2 . . . z2n }
M + z2 + · · · + z2n 2n
2
≥
(x1 , . . . , xn ) (y1 , . . . , yn ) (z2 , . . . , z2n ) ≥ x i yj 1 ≤ i j ≤ n
x1 + · · · + x n n
/[8 h i 8 j d
y1 + · · · + y n n
X _ ROF@0G2 x K . . . K x 6@9 * y K . . . K y a 0VAC086@(L9 & J>a 08AFE@RF@0[2 a 0z2H;)0J>6G24AC- ®z"-.2H;=089'24AC-70@E ( -.{ x + y ≥ 0 r 1 1
n
1
n
R
A = (aij )1≤i,j≤n
.- { R Z808& (70@? J>? 0@$ 9'Eb02FEV2H; -79u682 086D:<;w-IEBAC6@$19 "¥619 *u08J>6D:<6G;x24AC:H- ®3$'(."& J>-2H;59u0@$'9'{ 24AF-+08E-7E%01K °'& EH6@&L(^:<2C;%$¤2H2H; ;)6G0X2T2H:H;)$ 0zACACE<0@& E<J ?L$ $'9 {)*'2H-79 ;)0O E<& J{l$ AN2H;)0J>6G24AC- ® Rf» AC$8y 0z2H; 6G2 R X a Rv@y 0@AM,?L$'-I9'2^"-2H;-79'2C01O808AB:4$)$ AC* -I9)682C0@EB-I952H;)0? (76@9 0z-7E2H;)0V:H089'24AC0%$'{6%*'-7E: "-.2H;,AF61* -.& E R ]76G`¤* » -.AC¶$@08y AC0Y0@9'2H2B; 6G*'2>-7E2H:4;)E8R0@AC0 0<®@-7E42CE6@901°'&)-I(I682C0@AF68(324AC-I619 O[(70Y"%;)$ E0uy 0@A2C-+:40@E¤(.-+0_-I9 ]la1`E<» -7AC* $@0@y PH0(70@2H9 ; O6G2H2B;=0bOGy AC0808A68M=2C0@0@A^° &)2H; -.(7616G9 2408AF6@( R 24AC-I619 O[(70"-2H;Vy 0@A2C-+:40@Ev-I9=* -.¶08AC0@9'2v* -IEb:CEf; 61E X s R F@0[2 :H$ 9 *'-.2C-7$ 9)E q a)0p2H;)09 & JQa)0@AY$'{,-79'2C01O808AFE 2H; 6G2uE6G2C-7EH{.M 2H;)0p{l$'(I(7$1"-79 O ]I-` (+086D*'-79 O¼<08AC$)0@K)EfE68$(I(72H$1; "%68012 *[r ; 6@E}0<®@61:C2C(M *'-+O[-2CE}]I-79Q*)01:4-IJ>68(9 $D2C6G2C-+$ 9D`4KD"-2H; ]7-.-`2H* ;)- y 0z-IE<*'-7-+a O[(7-0V2CEBa@$'M { :4R 6@9,a)03?L08A4J,& 240@*-I95EH&L:<;6}"V6[MV2H; 6G2T2H;)0bMM'-+0G(+*%619%-79'2C01O808A » AC$@y 032H; 682 {l$ Av0by 0@AM,?L$ EH-.2C- y 0z-79'2C01O808A R aij =
i
0
j
xi + yj < 0
n×n
B
0 1
B
A
A=B
1/1000
96
f (k)
n
0 ≤ n < 10k
n
k
n 11
f (2m) = 10f (2m − 1)
9c 8 hHc jt
m
a R F@0[2 ABC a 03619%-7E$ Eb:408(70@E^24AF-76@9 O[(+0 "-.2H;
K8"%;)$ E0-79 :40@9'24AC0-7E R 2461AC9 -I61* 9 O[(7K 0 AC0@E619 :4-7* AC:4(70%K $'AC08{LE2H? ;)0103:C242CAC- y -I61089 ( MO[(7R0 » AC$8y 02HR ; 6822H;)0(I-I9 0@E 619 * -I9'2408AFEb0@:C2f$ 9=2H;)0 Gs R%'6D:<;Q? 68-7A}$'{^$ ? ? $ E<-240E<-7* 08E3$'{^6,:H$ 9Dy 0H®Q;)0H®@6DO8$ 9¤; 61E}2H;)05?)AC$ ?L08AF2MQ2H; 6G2 2H2H;);)008-IA*'-7(7E40@2C9 6@O9 2H:H; 0¤E8R a » 0[2AC"%$8y 010z089x2H; 2H6G;)2v086@-IA(.()J=2H;)-70V* PH6@?L9 $'O[-I(+9'082CE}E%$'-7{ E2H01;)°'0%& ;)6@0<(N®@2461$ O8$ 9=61AC0V012C-I°'JQ& 086@EV(R 2H;)0¤EH& J $'{ F@0G2 P a)0¤6? $'-79'2%$ 9u2H;)0:4-IAC:4& JQ:4-IAC:4(+0$'{v2H;)0QAC24AF-76@=9 O[BC (+0 AIB ( M -I9 OQ-I9)E<-7* 0Q2H;)I0
ABC
P
E
F
CA
P
G
CB AB DF EG
ABC
√ 3/2
AB D CA CB
:4-7AC:4(7 R0@EF@RT"0[S#2 -.;)2H0%;%:4* -I-IAC61:4J>(+a)030G032F2C6160@9 AF24EO8AC-I08619'9 2O[2C(7K $,0 "6@(.-.()2HK 2H;%; EbAC0801J=06@Eb-IAC?L080308J=*)AC-I-7AFJQ:46[-7"VAC0[:42495(7080@A$ E 9; 2H6@6@;)Ef9 0z*%AF6D$'-7*'9)& -IAF2F& 61E<E -7* * -.0& RfE $'» { AC2H$@R;)y S#0 03;)242H03AF; -7Eb6@68089 2 J=O[(+-70 P
ABC
s
r
BC CA AB
ABC
s s < t ≤ + 2 2
1−
f hi$cj ) c 8 jt
√ 3 r 2
t
R
a R ¹ 9¤-79'2C01O808A -7EzE<6@-7*X2C$a)0 -I{ -IEz9 $D2 2H;)0=E° & 6@AC0>$'{N6@9¤-79'2C01O808A[R {l$'(I(7$'$1{z"2H-7; 9 ACOB0@?)0 AC$ * ?L-IEH082CAF-I29 MW:Cq2XO8$):4$)6@*p9%a -I9'0 24AC0@0@O8?)0@ACAF0@EQE"%0@9';)2C$ 01E*0 K -7? f9pAC0G$)2C-7*'9 0@&LAC J=:C9 2v-.I- C2 9 I- 08EB0}( 2HM68;)(I0J=(1-IEb9'61°'9D24& 0@M 61O80@AC"V0VAFE6G$'M {#EG61K"9>6@-E¤$)2H;3*)6 *,2H;)EH-I0f9'& 24Jº 0@O80@AR s RtF@0G2 a)0=6>? AF-7J>0=9 & JQa)0@ARw» AC$@y 052H; 6G2}2H;)08AC0Q0H®1-IEH2FE6>?)AC-IJQ0=9 & J>a 08A E<&L:<;2H; 682bK'{l$ Av0by 0@AM5-79'2C01O808A KD2H;)0V9 & J>a 08A -7Ef9 $D2B*'-y'-7EH-+a'(+0Va@M R R S;)0YE01°'&L0@9 :40 K K K -7EX*)08 9 01*6@E>{l$'(I(7$1"VE q K36@9 * ;5+
n
|n|
m
m
p
q
np − p
n
q
2H;)0@9 *'-y'-+*)0@E l{ $ A R R«» AC$@y 0w2H; 6G25-I{36@9p$)*)*¢? AF-7J>0 * - y -7* 08E K RF@0[2 a)0,65? AF-7J>0,9 & J>a 08AbK#6@9 *Q(+0[2 a 056,E0G2}$'{T?L$ EH-.2C- y 0%-79'2C01O808AFE2H; 6G2 E6G2C-7EH08Ev2H;)0z{l$'(I(7$1"-79 Oz:H$ 9 *'-.2C-7$ 9)E q ]I-`2H;)0 Eb0[2$'{ ?)AC-IJQ0}* - y -IEb$ AFE^$'{12H;)00G(+08JQ089'2CET-I9 :H$ 9)EH-7E42CE^$'{ 08(70@J>0@9'2FEr ]7-.-`5{l?L$ 08AVAC{l610@9D:C2MY9 $ 9 ? P<$10@"%J=0@? A2R M_E<&La Eb0[2$'{ KN2H;)0?)AC$)* &L:C2$'{B-.2FE,0G(+08JQ089'2CE-IE59 $D2%6 ; 682-IEv2H;)0(I61ACO8 08EH2v? $ EEH-+a'(+09 & J>a 08AB$'{s08(70@J>0@9'2FEB-79 o
a0 a1 a2 . . . ak+1 = 2a2k − 1 k ≥ 0 2n+3 p2 − 1
a0 = 2 p an
p
A
A
p−1
A
p
A
?)WY&L68a'2H(I;)-IEb08²;)J>080@{l*6G$ 2CAC-+-70%:C9X6@2C2H(& ;)ªBA40Q9 (M -Iª 9 J>:CO%2C? $)24-7$Q61a * 08EbA $' (.11& 112CD-+$ D 9)q 8QE "% JD 0,BD#$O[ -R=y 05Sr ;)6=10=1ACD08? 6DAC*)$)q 0@a'@¨A(+08 E J·:4R $ -IEzJ>{+J>AC$ 0@J 9'22H$ ;)9¤0Q6,Z8-IEb9 $'OG(.61& ? 2C$ -+$ AC90 Rv1y'68(I& 6G240 R 8Q>?>@#
>
2002 X k=1
2002 X k! k · k! 2003! − − 2002 k k 2 2 2 k=1
6
&( )*,+ Q Z + # 1 }5,) #$ O
6Q#$1 D sQ 1
S#;)0V? &La (.-7E;)01*=E$'(I& 2C-7$ 9=0@E42C61a (.-7E;)0@Ev2H;)0z{l$ A4J,&)(76 n X (k − 1)k! (n + 1)! = k 2 2n − 1 i=1
1 }
E
a@M¤& E<-I9 O-79 *'&L:C2C-+$ 9w$ 9 RX² & 2}-2}-IE0@6@E<-70@A}6@9 *J>$ AC0=0@9 (.-+O8;D2C0@9 -I9 O24$¤$)a)E0@Ay 0 n
D 2H; 682bK EH-79 :40
(k − 1)k! = (k + 1)k! − 2(k!)
KD"%0V; 6Gy 0
n n n X X X (k − 1)k! (k + 1)! k! = − k k k−1 2 2 2 i=1 k=1 k=1
K
"%;'-+:<;240G(+08Eb:4$ ?L08Ev2C$2H;)0VE42C6G240@*=AF-+O8;D2HP<; 619 *=EH-+*)0 R 09 $1"iE;'-I{I2v2C$>AC086D*)0@AFE[)Eb$'(.& 2C-+$ 9)Ef2C$> ? 68AC2C$)-7$ a'9)(+086@J>(^ªBE ( -7M)9=J=2H?);)-I0 6D* $8$'y { 0@²J>$ a E<9 08A-76=@6@D9 ;*D 8Q J #$ 619 *X2H;)0=Z108(701:C2C-7$ 9¤S)0@E42}{l$ A 2H;)0 © ¯V08AF¼<0@O8$@y'-79)6 11' K O[- y 0@9=68f2 @D; D q A5@ R n OR F@0G2 K K 619 * a)0VAC0@68(L9 & JQa)0@AFE E<&L:<;2H; 6G2 6@9 * * ] -.¶-7E^08AC6}0@AC9 08:46@0 (Da ?)0[612AF"%6@JQ01080[924082H;)A`<0BRfJ>f60[®12408-IJ=A4J,& J-79 0B6@9 2H;)*z0vJ,y'-76@9 (.&L-IJ=0& $'J{12H?L;)0}$ E
z
x+y+z = 3
a
2 H
xy + yz + xz = a
I- 2 G JH
x Z O# + $ Z,7E
# J O# } J $#*1 $ % Z,#*1 '1 6 O# '1 7 s# # J#5E # ; ) # }J $#*
) # + 4 ;05+5
2H;)0VEFM)E4240z08J ; 6Gy 0
6@9 *
a
yz = a − x(3 − x)
y+z yz
y
8
R!+2^-7Ev0@6@EFMV24$zy 0@AF-I{.MV2H; 682
K K "2H;)-.0=2H;z& & 9 9 -7° /D&L9 0>$1"V:H$ 9)9 E *'-.2C61-7$ 9 9X* "%K10>; ; 61EN6[y 60,AC082C$6@('E<E68$'2C(I-I& E<2C{.MX-7$ -I9E -I{ 619 *%$ 9 (M-.{ R^S; 682-IEGK 6@9 * y+z = 3−x
5)$
= s = p
R¯0@9 KT:4$ 0 A K
s2 − 4p ≥ 0 (3 − x) ≥ 4(a − 3x + x2 )
z
2
R S#;)0Y*'-I¶s0@AC089 :H0ua 0[2"%01089 2H;)0uJ=6[®@-7J,& J 6@9 *¢J,-79 -IJ=& J£?L$ E<E<-7a (70¤y'6@(.&L0@EQ$'{ 01°'& 6@(IE -I{#6@9 *>$ 9 (M5-.{ K1"%;'-+:<;5-7J=?)(.-+08Ev2H; 6G2 R \ RS AF-76@9 O[(+0 61619 9 O[O[(7(70@0@E8E RS#;)089w61*)9 AF* 6["-IEz2H;)O[0,- y (.0@JQ-79 9X0@0-I0G9¤2H2B; 6,2H6G;)23?)0(I:H61E<$ 9 -79)* 0)9 08R=0@E :CB2CEAF6G2H" ;)610=2H9 ;)? * 0=$'-7a'9'-72FEE}01KL"%:CAC2C0@;)$ E
3(x − 1)2 ≤ 4(3 − a)
1−2
q
a≤3
1 − 13 a ≤ x ≤ 1 + 2
q
4 1−
8
ABC
ABC
1 a 3
q
1 − 31 a
a = −9
=8
ACB
x
AB BAC BC AB AC M N 2M N = BM + CN ) # + ;#H Y>@# ; @8Q5)$ s18 A; 5# Z 1 B, Z 1 $ Q+ # s# E Q Q+5 B1m- >@+ # 1Y8 H1 o E #f;) # # : Z,,# %H$ Y>@ # ; 8Q)$ '1 >@+ # + # # + ,, E
2 > C
AC
0
9
p
H
C #
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∠ACB AC E
AB
P
Z
P
PX = PY + PZ
D
"
?0AC s$1K 0@ :CAC 2C0@-+E<$ F@?L9)0G0@2EV:C2C$'-{y 0G6@(9 Ms*$ R 9'24F@$Q0G2a 2H0u;)0=2H;)61E<-709 * * 08$ E A2H;)$)a)O80$ 6@2H9)9 ;)6@*0 ( E2H<$ ; -7AF* 2H6808;)2 E $)O8$ 9)68(Y6@9 ? * AC$1610@9 :C2C* -+Kz$ AC9)08E E? $'01{ :C2C- y 08(R $ M9'R 24$« 2H$D;)2C00 6"%19 03* ; 6GF@y 0G0 2 RRBZ8S-I9 ;):H0@0 9 Kr E<-IJ=-.(76@AC( M#K#EH-79 :40 RS#;)089 K "%0,6@(IEb$¤; 6[y 0 E<-I9 :H0 619 * ¹2H;)Ex0-7? 9³$' -72H9'$1;)2F"³0¢EB"%"%(+08;)03J>0@2CJ=AC& 0AC6)9,2HKV;)2C"%05$0¢a'2H-7;)E(+0%0[012:CO[2C- $ y AF0@E9=61$'? 9 { AC* $)a'(+08Jxa)0 R 61AC0@9 E<* ?L0@:C2C-y 0G(MsR JQF@0@0[0G22v2H;)0%a)E<0>-7* 6@089DE Mw?L$'-I9'6@2z9 $ * 9¤2H;)0 K Eba 0@0VOG$ JQ95082H9';)2 0Va'-7ERT01:C] 2CF8$ 6GAv2408$'Ab{ Kb"%0"-I(.(@K'AC6101Ef°'&)AC0@-7AC° 0 &)-IAC012C*$ -7952H;)0VF@?)0GAC2 $)a (70@JwR ` * 089 $D240 2H;)0¢61AC086p$'{6 24AC-I619 O[(70 RZ8-I9 :H0 K"%0 Eb0@0z2H; 6G2 6@9 * R
S
D
T
.. ......... .. ........... .... ... ... ............ ....... ..... ......... ....................... ........... .... ................................... ........... .... .. .. .... .... ...................................... . . . ..................... . ........ ................... ..... .............................................................. . ... ...... . . . .................................................... ...... ....... ...................... .... ..... ...... ....... . . ............. . ...... . . ... . ........ ...... ........................... ... .. ... ...... .. ... ...... ................ ....................... ..... .... ..... ...... . . . . . . . . . . . . ... .... . . . ............. ...... ......... . . . . . ...... . . . . . ... ... . . . ............ . ... . ...... ............ ... .... ............ ..... .... ... ............ ... .... ... ............ ............ ........... ............ ...... ... ... ............. .... ............ ...... . . . . . . . . . . . . . . . . . . . . . . . ... .. . . . . . ..............................................................................................................................................................................................................................
BC AB DR = DS ET = EU r = DP/DE 0
Y
.. ..............
U
E
S
.. ..............
AC
A .
BC U
E
Z
P
D
B
R
X
T
R
P Y + P Z = (1 − r)DR + rET = P X
DR k P X k ET
DP/P E = r/(1 − r) D
A
E ∠ACB AC
P
DE
P
∠BAC
.... ... ...... .. ........... .. ...... ...... .. ... ...... ..... ... .... ....... ... .... ........... . ... . ...... . . ...... . . . ...... .. .. . .. .... ............. ...................... ...... ........................................................................................................................................ ...... ......................... ..... ...... ... ... ...... ... ...... ... ...... ..... ... ...... ... ...... ... ...... .... ... ...... ...... ..... ... ...... ... ... ...... .... ...... ... ...... .. . . .. ...... ..................................................................................................................................................................................................................
Y
.. ..............
AB
∠ABC
K
C
Z M D
E N
P
[U V W ] B X UV W M N k BC [M BP ] = [M XP ] [P CN ] = [P XN ]
Rv¯0@9 :40 K
C
R Z8-79 :402H;)0x61AC086Y$'{ J=6GM a 0u0H®D?)AC0@E<Eb0@*¢a@M KB2H;)0x6@AC0@6u$'{ @ a M ' K 1 6 9 * H 2 ; G 6 v 2 ' $ { a @ M @ K % " 0J=6GMV"VAC-240 2H;)0V6Da)$@y 0V01°'& 682C-7$ 9=61E R ]` 6"%19 0%*$)KLaDa@ 2CM,68$1-72H"95;)02H61;)(+E<0V08E<J>& * J>08J=E<06)-IACK 2H01; *,68AC2 0@EH&)(.(.2[-+08R E$ 9XRv2HZ8;)&L0=a)E4a 2C-I-.Eb2C0@& :C2C24-I$ 9 AO$'-I9'{ 24$>] `N6@9 R%*QS#*';)-y'089-+*'-79 OVa@M K p RN!l{ -IET6}9)6G2C& AF6@(D9 & J>a 08AbK@?)AC$@y 0B2H; 6G22H;)0 9 & J>a 08A -7Ef9 $D2v6V?L08AC{l0@:C2vEb°'& 61AC0)R I # J O# } J- $,#H61 $- % 2G Z,#*1 Z JHO#E E<&L:<;2H¹¹ ; E68:HEH2 $ & J=JQJQ0'Ks$ {l9,$ A*'-2Hy';)-7E0=$ ? A& $'A4{L? 6@$ 9DEbM%0>2$'"%{$5:H$ $'9'{)242HAF-I;)Ef610z* 6V-79 :C?L& 2C08-+J>$ AC{l9#a 0@K#08:C2BAF2HE ; E6G° 2 & 6@AC0 -7EzK R 6>9)682CK& AF68(^K 9 & JQa)0@A -7$'E{ 682 6@J>9 *5$ EH682 2NJ>R $ ¹ EH(72TE2$"%KL$561J>:C61$ 9,9 a OV032H;)J=08&)Eb0(2C-7? (+089 Ef& $'JQ{ a)0@RAF!+E[2NK#{l6G$'2}(.(+$1J>"V$ E^EH2v2H; 2"%682N$X682^:46@(790@6@a)EH205J,$'&){)(.2C2H-I;)?)08(7Eb0@0E [M BP ] + [P CN ] = [M XP ] + [P XN ] = [M XN ] 4M BP 1 CN · P Y 2
4P CN
4M XN
1 BM 2
1 MN 2
· PZ · PX
BM · P Z + CN · P Y = M N · P X P P X = 2P Y
∠BAC
PZ = PY PY
(n+1)(n+2) · · · (n+10)
n
n
(n + 1)(n + 2) . . . (n + 10)
n + 1 n + 2 . . . n + 10
9
7
10
5
6
A
?9 619 & AF-7*JQ9 :4a)-I?)0@AF(70 EKD; 2H6[;)y 080,AC0V$ 0<9 ®@0,{l-7$ E4$'A2 {E2H$ K ;)JQ00{l$ ?LAC$ J>EHK -.E 2C- Ky 0zK -I9'K 240@O8K 0@AFE"-2HK#; 616@9 9 * * 6@9 EH&LR5* :<² ;M>2H; 2H;)68205K ?)-7O8K 01$ K 9 P<;)R $'(+0 $1" KD-.61(I-9 2C*V-+082HE};'{l& $ EGA K R 0@¹ E(I0,Eb$:4$ KG9 "%* 0z-2C; -+$ 6[y 9)0 E z R S ) ; 5 0 L ? $ < E < E 7 a E G 6 C 2 7 H E . { M I 9 O H 2 ) ; 61:46@AC0>Eb0 61Ez{l$'(I(7$1"VE q K6@9 *x2H;)K08AC08{l$ AC0>2H;)K 0XEb0@° &L089 K :H0 K^K 6@9 * K K Rw!9u0by 0@A-IM E :H08$ -2H9';)2F0@6@A -I9 01*x$ -IA 9x2H;)R0QZ8Eb-79 0[2 :40V9 $=K J>K $ AC0z2HK ; 6@9Q$ R¢9 ² 0& {+612z:C2H24;)$ 08Av9x$'{ 2H;)0QEb$ 0@A ° &L089 :46@:H0X9QJ,a)0V& ? EH2VAC08:HEb$ 089'9'2F2v6@-I-I99 2H6%;)? 0z0@?)AF{lAC01$):C* 2v&LEb:C°'2 & 61AC0'K1"%;'-+:<;=:4$ 9'24AF6D*'-+:C2FE}$'& ANK8-79 "%-2C0z-768:H(L$ 619 E<:4E<(.&L& * J=0}? 2H2C; -7$ 6892R2H;'-7Ev?)AC$)* &L:C2^-IEv9 $D2 _ R F@0G2 K K#6@9 * a)05AC0@68(9 & JQa)0@AFEzEH&L:<;>2H; 682 R,» AC$8y 02H;)0 -79 0@° & 68(I-2\M R
x2 2x2 3x2 6x2 i j ∈ {1 2 . . . 10} i<j n + i = kx2 2 n + j = ky x y k ∈ {1 2 3 6} k(y 2 − x2 ) = j − i 1 ≤ k(y 2 − x2 ) ≤ 9 kx2 = n + i ≥ 2 (x, y, k) (1, 2, 2) (1, 2, 3) (2, 3, 1) (3, 4, 1) (4, 5, 1) kx2 ≤ 16 n + 1 n + 2 . . . n + 10 {2 3 . . . 25} 11 17 11 17 (n + 1)(n + 2) · · · (n + 10) a b
a 2 + b2 + c2 = 1
c
a2 b2 c2 3 + + ≥ 1 + 2bc 1 + 2ca 1 + 2ab 5
9 H +H;- 0 C- 2G H j
H 2
) # + A; 5#* Y>@# ; 8Q5)$ s1 8 ;#$ 4 1 B, Z 1 6 # J O# } J $,#H61 $ % Z,#*1 4 J# Y)*,+ s+Q 1 45+ # 61 # ,#$ J A; 7) # s, 1 6 >? * 1 U1 o )*, 4 ;05+5 '1 6 O# '1 7 s# # J# Q+ 0# +Q 6 yE H 1& 5) # 41 m1 o E # ; ,) # s+Q $ Z,7E
!9_y'-+0"$'{ 2H;)0-79 0@° & 68(I-K)2\M 0G2C:GR+K1"%0Eb0@0z2H; 6G2 619 *_2H;)0x$)a)E0@Ay'6G2C-+$ 9_2H; 6G2 2
2xy ≤ x2 + y 2
2
1 + 2bc = a + (b + c) > 0
a2 b2 c2 + + 1 + 2bc 1 + 2ca 1 + 2ab
F@0G2
R a)0%* 0GL9 0@*=61E
≥ =
a2 b2 c2 + + 2 2 2 2 1+b +c 1+c +a 1 + a 2 + b2 2 2 2 a b c + + 2 − a2 2 − b2 2 − c2
R
R^S;)0@9
@6 9 * R R EH-79 O}µ089)Eb089# E!9 01°'& 6@(.-.2M#K1"%0%O80[2 S#;'& E[K -IE :H$ 9Dy 0H® K E<-I9 :H0 R K KL619 * O[- y 0@E r 2HS); 6168/12-I9 -IEGO K R $D2C0!q0@° & 68(I-2\M5-7E $ 9 ( M=? $ EEH-+a'(+0z-.{ R a R F@0[2 6@9 * a 0>*'-I¶s0@AC089'23?)AC-IJQ0,9 & JQa)0@AFE8R>Z1$'( y 052H;)0,{l$'(.(+$1"-I9 O5EFM EH2C0@J·$'{ 01°'& 682C-7$ 9)Ef-I952H;)0VE0G2B$'{L-79'2C01O808AFE q K R f : (0, 1) →
f 0 (x) =
f
f
x=a
2
2 (2 − x)2
f (x) = x/(2 − x)
f 00 (x) =
4 (2 − x)3
f 00 (x) > 0 x+y+z f (x) + f (y) + f (z) ≤
y =b
3
2
3
z =c
2
f (a2 ) + f (b2 ) + f (c2 ) ≥ 3f 1
a2 b2 c2 + + ≥ 3· 3 2 − a2 2 − b2 2 − c2 2−
a2 = b 2 = c 2 =
p
q
z+p z−p + x y z+p z−p − y x
= q
= q
1 3
1 3
=
3 5
1 3
"D
I
# J O#
$D2C0z2H; 6G2
-
} J $,#H61
- 2G
$ %
JH
Z,#*1
Z O#E
RNU(70@6@AC-I9 O2H;)0%*)0@9 $ J,-79)6G24$ AFE[K1"%0%O80[2 K R ² M, AFEH2vEH&La'24AF61:C2C-79 O2H;)0%0@° & 6G2C-+$ 9)E[KD2H;)089>61* *'-79 O2H;)0@J¤K1"%0%O80[2 K ]` R ]` 0-71WuE °'&)? &)(.AF-2Cy'-7-IJ>68?)(+(0[08M `<9'-IRv9 2^SOx24;'$ ] & EG`K a@M * - y 6@-79 * * 08E & EHRT-79 SOw;'Rf-7] Z8Ef-I`H0@9 KB° :H"%0& 6G0u2C-+O8$ 0[92 -7J=619 ?)*(.-+08E-I2HE}; ?)6G2AC-IJQ*'0'-K)y'--+2*)2H0@;)E 08KB9>"%]+{lEH;'$'-7-7(.9 :<(+:4$1; 0 "V-IEE 2H; 682 RT*'S#-y';)-+0*)EF0@M)E E42408R J F@-I0GEf2 9 $1" 619 * KL"%;)0@AC0 619 * 61AC05-79'2C01O808AFE6@9 * K ]` R ]A ` 'AC$ J¦] A `HK1"%0%*)01*'&L:H0z2H; 6G2 *'-y'-+*)0@E K E6[M RTS#;'& E[K K ]D ` R ] ` Z8-79 :40 -7Ef?)AC-IJQ0'K'-.2N{l$'(I(7$1"VEB{+AC$ J¦] D `2H; 682 K R /l c >R 'AC$ J«] `HKD"%0; K16G"%y R ;'0 -7:<;,-IEf6VEbK'$'"%(.& ;'2C-+-+:<$ ;,95-7{lJ=$ A ?)(.-+0861EB9D2HM>; 68$)2 *)*=?)AC-IJQ0)6@9 R * RBS;)0@9 /l ck R ¹ Es-I9zU#6@Eb0 Kb"%0f$)aD2C68-7R 9 619 * RsS#;'& E[K R /l c R R (I-.08EN2C$ ) ' ° & 8 6 C 2 7 $ , 9 ] ` < E I > J ? RS#;)089 J,& EH2a)0z6z? AF-7J>0*'-y'-7E$ A $'{ K'-79"%;'-7:<;=:46@Eb0 R /l c&m R ¹ EB-79>U61E0 K1"%0zL9 *5R 2H; 6G2 J=& E42B* - y -7* 0 RTS#;)089 R S#K';){l$ 08AvAC086@{l9D$ M=AC0'?)KzAC-I2HJQ;)0 0 Eb$'(.& 2C-+$ K 9)61E¤9 * 6@AC0 K K-.{ K K {l$ A^619DM?)AC-IJQ08E 6@9 * EH&L:<;V2H; 682 *'-y'-+*)0@E ]."%;'-7:<;%-IJ>? (I-70@E^2H; 682 `
xy 6= 0
(z + p)y + (z − p)x (z + p)x − (z − p)y zy xz + py
= px = qxy
xz 2 + xp2 = qpx2 p z q 6= p q z = pc a c
z z + p2 = qpx p2 qpx x x = pa 2
p a 6= 0
= qxy = qxy
cy pac + y
a
= pa = qay
y
cb
pc + b
p
y = ab
= p
= qab
(b, c) ∈ ±(p, 1) ±(1, p)
p
(b, c) = (p, 1)
qa = 2
q=2
(x, y, z) = (p, p, p)
a=1
p
(b, c) = (−p, −1)
q=2
(x, y, z) = (p, −p, −p)
a=1
(b, c) = (1, p)
p2 + 1 = qa
q
2
p +1
(x, y, z) =
(b, c) = (−1, −p)
(x, y, z) =
q=2
(x, y, z) ∈
p(p2 + 1) p2 + 1 , , p2 q q
p(p2 + 1) p2 + 1 ,− , −p2 q q
(x, y, z) ∈
p>2
p
p2 + 1
q
2
p(p + 1) p2 + 1 , , p2 q q
q
q
(p, p, p) (p, −p, −p)
p(p2 + 1) p2 + 1 ,− , −p2 q q
p2 + 1
p 6= q
1 s RF@0[2B2H;)0y 08AF2C-7:H08E3$'{s2H;)05:4$ 9Dy 0<®=°'& 6D* AC-.(76G2408AF6@( ABCD 6@9 *>2H;)0%-I9'2408AFEb0@:C2C-79 O ?L$'{B$'-I2H9';)2 0S° & $'61{*)-.AF2F-IE(I68*'2C-70@61AFO868$ ( 9)6@(IEa)0>-79'612C9 01*O808Az?L$'2H;)-I9'02CEV6@AC-I0@96w2H;)$'0Q{v24?)AF(I-7616@9 9 0)O[R(+0 @F 0[2 P R³a 0=» AC2H$8;)y 0Q0X6@2HAC;)0@60 ABS {l$'(I(7$1"-79 O-79 0@° & 68(I-2\M3q ABCD P 1
R I 5)$ 2 H > C # # + ,, *E 2H0 ;)c0@h 9 h LR^!l{ K K8619 R * 6@AC0 9 $ 9 P<:H$'(.(I-I9 0@6@A#?L$'-I9'2CE#"-2H;3-79'2C01O808A:4$)$ AC* -I9)682C0@E[K 0 s F@0G2 ; 6[y 0:4$)$ AC* -I9)682C0@E KG"-2H; K KG{l$ A K K RS;)0@9 R ¯V089 :H0'K r@2H; 682-IEGK R F@0G2 K KD619 * a)02H;)0z61AC0861Ef$'{)24AC-I619 O[(70@E KK K L K C A 8 0 E ? 1 0 C : C 2 y 8 0 ( M f R 0 ; [ 6 y 0 a 0@:468& 6@EbE 0 * $)08E ; 61Ex2H{+;)AC$ 0pJ´E6@2HJQ;)00;)a)0G6@-+EbO80 ;D2w{+AC$ R J Z82H-7;)J,0-I(I61a)AF6@(EbM0 K R S;'& KEGEHK-79 :40 R§(70@J=S#J>;)086 9 R @ a = M H 2 ) ; 0 ¯V089 :H0'K K & E<-I9 O32H;)0 ¹ W B BW¡!9 01°'& 6@(.-.2MR^S;)0@9 √ p √ 2 P ≥ P1 + 2 4 ;05+5 '1 6 O# '1 7 s# # J#$1 >@+ # +
X1 X2 [X1 X2 X3 ] ≥ Xi
1 2
X3
(ai , bi ) a1 2[X1 X2 X3 ] = a2 a3
2[X1 X2 X3 ] ≥ 1
[X1 X2 X3 ] ≥
P2 P3 P4 CDS ADS 4BAS AS 4BCS P4 /P3 = AS/CS P1 = P2 P4 /P3 ≤ 2P2 P4 p
2P1
ai bi ∈ b1 1 b2 1 ∈ b3 1
i=1 2 3
∗ +
1 2
BCS P1 /P2 = AS/CS
CS P1 /P2 = P4 /P3 P3 ≥ 21
A................................................................................................B . .. . ... ..... ....... ... ... ....... P1 ...................... ......... .... .. . ..
.... ... .. ... .... ....... ... .... ............ ... ... ...... ... ... ... 4 ....................... .... ... . 2 . . . . . . ... . .... ... .... . . . . . ... . ... . . .... ... ... .... ... ............. ... .... ... ......... .. .... 3 ........................ .... .... .......................... . . . ... ........................... ..................................... .....
P
D
P
S P
C
p ≤ 2 P 2 P4 ≤ P 2 + P 4
p ≥ P1 + P3 + 2P1 √ √ P1 + ≥ P1 + 21 + 2P1 =
P = P 1 + P2 + P3 + P4
S)61/1-I9 OE° & 6@AC0VAC$)$D2CE O[- y 0@Ev2H;)0%*)0@EH-7AC0@*=AC08E<&)(2[R
√
2 2
2
R
s$ 9 O5 WY0H6G®@2H2#;)"%0@J=0 (+68$)2C$)-7:4/368(#6G2TªBEb( M)$'J=(.& ?)2C-+-I$ 6D9)*=ETO[{+-ACy $ 08J9=68$'2& A 1AC0@161 * D 08q AFA5E^D©24$V ?)R AC$)a (70@J=Ev$'{D2H;)0 A ¯V$ 9 O \ RND-I9 *z68(I(@?L$ EH-.2C- y 0v-79'2C01O808AFE EH&L:<;2H; 6G2L2H;)0 01°'& 682C-7$ 9 ; 61Ef?L$ EH-.2C- y 0z-79'2C01O808AvEb$'(.& 2C-+$ 9)E8RB²0VEH& AC0z2C$=O[- y 0V6V? AC$)$'{CR
I
z
E<682C-IE
# J O#
n≥1
-
n
} J $,#H61
- 2G
$ %
x3 +y 3 +z 3 = nx2 y 2 z 2
Z,#*1
JH
Z O#E
a)036@9%-I9'240@O80@ATEH&L:<;2H; 682s2H;)0@AC030<®@-7E42T?L$ EH-.2C- y 0}-I9'240@O80@AFE K K16@9 * R ]` x y
x3 + y 3 + z 3 = nx2 y 2 z 2
D© X-.2H;)$'& 2f(7$ EE3$'{TK1O8"%0@9 ;'08-7AF:<6@;5(.-.(+208M#6DK)* "%Ev024$ J>6[MX6@EEH& JQ0V2H; 682 R 'AC$ J¦] `HK1"%06@(IEb$,; 6Gy 0 R K "%;'-+:<;=(+086D* E 24$ E<-I9 O ] $D`^2C0V619 2H*5; 682H2 ;)089Q] `HK1"%0VO80[2 2 2 2
nx y z ≤ 3x
R3S;)0@9>"%05; 6[y 0 ]` K'-7J=?)( M -I9 O2H; 682 ]`
x ≥y ≥z
3
ny 2 z 2 ≤ 3x
x2 (ny 2 z 2 − x) = y 3 + z 3 > 0 ny 2 z 2 ≥ x + 1
(y 3 − 1)(z 3 − 1) ≥ 0
1 + y3 z3 ≥ y3 + z3
S#;'& E[K & EH-79 O] `HK1"%0V* 0@* &L:40z2H; 682
1 + y 3 z 3 ≥ x2 (ny 2 z 2 − x) ≥ x2
R
]A `
R
R ]D ` /l c >R 'AC$ J¦] A `HK1"%0R Eb0@0z2H; 6G2 K 6@9 *52H;)0%O[- y 0@9>0@° & 6G2C-+$ 9M -708(7*)E R /l ck R S#;)RB08¯9 0@9 :40 K R $ A R RS#;'-IE32C$)O80G2H;)08Af"-2H;] D `f-IJ>? (I-70@E2H; 6G2 $ $ AA K@K'& -.2NE<-I-79 E O0861] D EF`HMK824"%$V03y ; 0@6[AF-Iy {.0M2H; 682 -IEf{l6V$ ACEb:H$'08(.E & 2C-+$ 9R R G K % " ' ; + < : ; S#619 ;)*¤089 2H;)0>O[-y 089uK "%0@° ;'& -+6G:<;Q2C-+$ -IJ>9w? AC(I0@-7* 0@&LE :42H0@; Ez6822C$ 6@9 * R>R5S;)!+2 0@9 {l$'(I(7$1"VJ=E & 2HE4; 2z68* 2 - y -7* 0 KK "%S#;);'08-+:
y=z=1
x=1
n=3
yz > 1 y 4 z 4 ≥ 2y 3 z 3 > 1 + y 3 z 3 2 n <9 n=1 n=2 n=1 (x, y, z) = (3, 2, 1) n=2 y 3 z 3 (4yz − 9) ≤ 9 yz = 2 y = 2 z = 1 x3 + 9 = 8x2 x2 x=3 n=2 n
n=1
n=3
k ≥ 4
F (x) 0 ≤ F (c) ≤ F (0) = F (1) = · · · = F (k + 1) $,#H61 $ % Z,#*1 Z O#E
I
yz ≤ 2 x ≥ 2 9
k
- - 2G JH -7!H9XR#WY»RL68²S#2C-=$ ;':E A4-IK 9)EE4?)¼CRsAC24$)0G» -7a 0[9#24(70@KAC$@J¦ y'- -7:E E K {+2AC$ J C2H K;) N0= &)!C-7W_a 08ª AF2bKD?Las© R E;)¹ $ K)AFR 2C'©Z@(.? -7E4RNAF2[-7R$ 9 AfO8¹ 08zA[EbRLR$'N(.& +2C&L-+$ /1-=9Q:E K :C 3619XR8µ
c = 0 1 ... k + 1 # J O# } J
_ RTS#;)08AC0V61AC0
61AC0682(70@6@EH2
I
9*2# ; >? #
8Q> o# Q+ Z >
? ?L68$'-7-IAF9'E 2C$'EB{L-79)2H;)EH-+08*)Eb0V0$ ?LAB$'-I$ 9'9=2CE 6%; :46[-IACy :4-I(+9 0OV"*'-.-72HE4;52C6@AF9 6D:H*'08-IE& E 6G2vJ>Rf» $ ACEH$@2 y 03R 2H; 682N2H;)08AC0
212 2001 # J O# } J
-
$,#H61
- 2G
$ %
Z,#*1
JH
Z O#E
1
1
61O[-9 y O[08(79X0?L$'$'{0z-I9'J=2C6GE MKv-7E ? a@LMu6@AF9 2CAF--6G242C0'M -+$ K E9%"%{+AC2H0$ ;)J|J=06G*'2HMX-7;)E0¤/z:<;)-7:H$)9'08$ 2C9'$ E240VAC0 2H:H;)$ 09 AFOG$'6[AF{M)&L2HE}0@;)9'Eb02$,*'E2H01-7; E:C6G/2C2 $ R_AF9 EGZ8$ K'-I9 9 0805:H6D0>:<$';2H{#;)"2H0¤;)-2H09 ;? & 6V$'J>-7:49'a 0@2F089'E}A%24AF(I$'68-70 {( 6
60◦
O
1¨ $ 9XEb0@65:C24AF$ 6[M#AFE[KKD0<2H®1;):40%0@?'*'2B-7E42H2C; 6@689 2v:H0V2H;)a 050[2:H"%089'012408AC9>0 6@9DM5J>26["%MX$,a ? 05$'$ -79'9 2F05EB$'-7Ef{s682H;)2v0J>? $ $'EH-72 9'2FE@RORF@!0G9X2 0@61:
6
1
6 6 P
n1
1 2
i
1
n1 . . . n6 O
ni = 212
i=1
ni (ni − 1) N
1
N ≥
6 X
i=1
1 1 ni (ni − 1) = 2 2
X 6
n2i
i=1
− 212
f (x) = x2 6 2 1 1 X 1 1 2 ni − 212 = N ≥ (212) − 212 > 3639 2
6
2
i=1
6
N ≥ 3640
ABC AB = 20 AC = 21 BC BD = ∠DAE ) # + >?>@# + Y OA 1 U Z 0 ; 1 8Q)$ '1 8 6 ;#$ 4 1 A 1 $ Z 5 )*, 4 ;05+5 '1 6 O# '1 7 O# # O# 0 + # s# o E # ; ,) # Q Q+5 B }E E
BC = 29 8 EC = 9
D
2 22 H 9H 9 " H C H
4 J# A; 5#* Y>@# ; Q s s1 h Q# '1 7 s# # J# E Q Q+5 B1 - >@+ #$ 178 H1
A ..... ........ ....... .....
... ............. ...... ........ .... .. .. .... ... .... .... ........ . . . .. ..... .. . ... .... .... .... .... ... .... . .... ... .. .. .... .. ... .... .... . . . . . .... . ... .. . . . .... . . .. . .... ... . . . . . .... .. . .. . . . . .... . . ... . .. .... . . . . . ... .... . ... . . .... . . . ... . .. .... . . . . . .... .. . .. . . .... . . . . .. . .. .... . . . . . . .... ... . .. . . .... . . . .. .... ... .. . . . . . .... .. . .. . . .... . . . . .. . .... .. . . . . . . .... ... . .. . .... . . . . . . . .........................................................................................................................................................................................................................................................................
α β γ
20
21
β +γ
F@0G2
S# ; 6G;)y 080AC08{l$ AC0'K ² & 2
B
α+β
@r K E<-IJ=-.(76@AC( M#K1EH-79 :4K)0 6@9 * 8
D
α = ∠BAD β = ∠DAE ∠AEB = α + β
K EH-79 :40
12
E
KG"%03BR ; Z86G-7y 9 0 :40 9
γ = ∠EAC CA = CD
BA = BE ∠ADC = β + γ
RNUs$ 9)E01°'&L0@9'2C( M#K
180◦ = (α + β) + (β + γ) + β = (α + β + γ) + 2β
α + β + γ = 90◦
202 + 212 = 292
KD"%0 R
C
R
β = 45◦
R
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A
A A
A
B C
C
B
A 3
B
4
A
B
B
C
C
A 1+2+2·2=7 A
B
C
B C
B C
D 1 + 3 + 3 · 2 = 10
B C D
D
10
10
J......q.....
Aq
Aq
....... ........... ..... ....... .. ...... ... .... ...... .. ... ... .. .... ......... .. ... ...... . .. ... .... .. ... .............. ... .. ... .... . . . .. .... .. ....... .. ... ............. . . . .. ........ . . .. . ... ... ..... ... .... .............. .... ...... ... .. .. .... ... .... ... ......... ... .... .. .... . . . . . . . . . . . . . . . .... . .. ... ... .. .... . .................................................................................................................................................................... ....... . .. .. ....... ... .... ............... .. ........ .............. .. ....... .. .. ... ................. .... .. ......... .. ........ ................ ... . .... . ........ ... .. .. .............. ............................... . . . .. . .. .. ...................... ... .................................... ....... .... .... .. ............. ... .... ............. ....... ... . . . . . . . . . . ....... .. .... ......... ... ... . . ....... .. .. ....... .. . ............. .......... .. ..
I q
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Hq
q
D
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q
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E
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q
F
E
B
q
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8
A B
C
A
B
C
D
3+1+1 = 5 D B C D 8
A
B C
4+1+1+2 =8
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p(p + 3) + q(q + 3) = n(n + 3)
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m m(m + 3) ≡ 1 (mod 3) 3 - m m(m + 3) ≡ 0 (mod 3) 3 | m p(p + 3) + q(q + 3) ≡ n(n + 3) (mod 3) p q 3 p=3 n ≥ q+1 n(n + 3) − q(q + 3) ≥ 2q + 4 > 3(3 + 3) q≤7 q≤7 q=2 q=7 (p, q, n) = (2, 3, 4) (p, q, n) = (3, 2, 4) (p, q, n) = (7, 3, 8)
(p, q, n) = (3, 7, 8)
a1 a2 a3 a4 . . .
a1 = 1
a2 = 1
9 - R 2G
a3 = 1
H 2 2 2
n≥3
an+1 an−2 − an an−1 = 2
JH 2 E H
an n≥1 ) # + >?>@# + Y OA 1 U 4 0 Z;1 Z O#,#$ J s# } J $#*1 $ % Z,#*1 4 J# + 5 )$ Z 5; 0+5 s1 6 O# '1 7 s# # J#5E #m; ) # Y sB6 : Z,,# %H$DE
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an = 2an−1 − an−2
an = 4an−1 − an−2 an
n
n
n≥1
0
pH C U;
√ 3 a b c 3 abc + + ≥ √ 3 1−a 1−b 1−c 1 − abc
H q2 0 2 2 > H &H H p +H; +H +H 6 m
- 2G H 2H 0 C 2
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(0, 1)
a+b+c 3 a+b+c 3
a b c + + ≥ 3 · 1−a 1−b 1−c 1−
a=b=c
≥
√ 3
abc √ 3 1 − abc 3
sR¹
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i i + 1 ... n −1 n 1 2 ... i − 1
i
1
n
n
n
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p+1
1 n−k a1
3n(n + 1)/2 3(n + 1)/2 n 1 2 ... n k+1
2 ... k n − k + 1 ... n − 1 a2 ... ak
k+1 k+2 n 1 ak+1 ak+2
... n ... n − k − 1 ... an
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k+1
k
k k+2 n k a1 a2 . . . ak+1 k+1 = p+1 3(n + 1)/2 i k
n − 1 + k = 2p + k
ak+2 . . . an
k+1 = p k
n = 2p + 1
1 2 k+1 k+2 2k + 1 2k − 1
/l ck R n = 2p + 1 6@9 *
ai
k+1=p+1
... k . . . 2k ... 3
k+1 2k + 1 1
0VJ=6GM5)(I()2H;)0%OGAF-+*,61EB{l$'(.(+$1"VER q
1 k+3 2k + 2
k+2 1 2k
k + 3 . . . 2k + 1 2 ... k 2k − 2 . . . 2
k+1=p
2 ... k k + 4 . . . 2k + 2 2k ... 4
k+1 2k + 3 2
k+2 1 2k + 3
k+3 2 2k + 1
... ... ...
2k + 1 k+2 1
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n
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a b c d
a+c=d
a(a + b + c + d) = c(d − b) 1 + bc + d = bd
- 2G 2 H E 2H 2 2
JH
n
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;
W_$ AC0%O8089 0@AF68(I( M#K1"%0"-I(.(*)0G2C0@ACJ=-I9 0V6@(.( -79'2C01O808AvEb$'(.& 2C-+$ 9)Ef$'{L2H;)0VEFM EH2C0@J K ]` K ]` R ]` !l{ 619 K* -.2-7E50@6@EFMx24$uEb0@R 0Q2H; 6G22H;)0$ 9 (MYE$'(I& 2C-7$ 9)E{l$ A 61AC0 "%;'-+:<;=F@:40G6@2N9>& a EB061E-7950 2H;)0z{l$ A4RvJ 'AC$ J ] `T619 *=] `HK@"%0; 6Gy R0 EH-79 O] `^61OG6@-I9#K1"%0%O80[2 K R ]A ` S#;)089#K -I9y -70[" $'{#] `HK R ]D ` K8"%;'-+:<; :46@E<9Q-I9 a)O30V] AAC0`"V6@9 AC-*,2F24] 08D 9=`#-76195E ] `HK8"%0z$)a'2F6@-I9 R^S;)0@AC0G{l$ AC0 K K 6@9 * R S#;'& E[K -7E6@9%-79'2C01O808A rG2H; 682T-IEGK K K RS#;)03:H$ J=?)(70G2C0}2F6Da'(+0 $'{#Eb$'(.& 2C-+$ 9)Ef:46@9>9 $1"±a 0V0@6@E<-.(M=:4$ 9)EH24AF&L:C240@*Wq a+c = d a(a + b + c + d) = c(d − b) 1 + bc + d = bd
d = 0 (1, 1, −1, 0) (−1, −1, 1, 0) d 6= 0 b(a + c) = (c − 2a)d
(a, b, c, d)
a(b + 2d) = c(d − b)
c = 2a + b
d = 3a + b
1 + b(2a + b) + 3a + b = b(3a + b) b + 1 = a(b − 3) b 6= 3 a =
b+1 4 = 1+ b−3 b−3
4 b−3
b − 3 ∈ {±1 ±2 ±4}
a 1 −1 5 −3 3 −1 2 0
b 1 −1 4 2 5 1 7 −1
c −1 1 14 −4 11 −1 11 −1
d 0 0 19 −7 14 −2 13 −1
%"!l{L;'"%-+:<0%;59 -7$1J="±?)(.AC-+0108°'Ev&)2H-7; AC6G02 2H; 6G2 K K K a 0V? AF-7J>0@E[K'2HR;)0@9 a b c d n = 2 · 7 · 11 · 13 = 2002
(a, b, c, d) = (2, 7, 11, 13)
K
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R f 0@9 $DpE2C0¢H&La@:<M;52H·2; 6GH2 ;)0 Eb0[2$'{%AF682C-7$ 9)6@(V9 & J>a 08AFE@Rºf0[2408A4J,-79 0 6@(.({7& 9 :C2C-+$ 9)E K{l$ A6@(.( K VR
f :
→
f x + f (y)
9 - R 2G
x y∈
= y + f (x)
H 2 2 2
JH 2 E H
) # + >?>@# + Y OA 1 U 4 0 Z;1 Z O#,#$ J s# } J $#*1 $ % Z,#*1 4 J# + 5 )$ Z 5; 0+5 s1 6 O# '1 7 s# # J#5E #m; ) # Y sB6 : Z,,# %H$DE
1Z 0G2F2C-I9 O -I9,2H;)0VO[-y 089Q:4$ 9 * -2C-+$ 9#K1"%0zL9 *2H; 682 {l;)$ 0@Af9 68:40 (I( K VR!l{ {l$ $ JiABE"%$ JQ;'-+0 :<;%"%K 0VO80[2 K 2H;)089 R^S;'& E[K -7EB-I9D01:C2C- y 0 Rr ' K + { C A {S#l$ ;)A086@9 (.( Z10G2F2C-I9 VRO NZ8-79 :40 {l$ -IAv9X-768Ev2H(I;)(-79D0=0@O[:C- y 2C-%R0@y 9w0'K@:4"%$ 9 0V* O8-2C0[-+2 $ 9#K "%0=$)a'2F6@-I9 K 619 *2H;'& E[K R $1",K'{l$ A68(I( K K R a ¯V0089 61:H9D0'MQK?La@$ MuEH-.6@2C9Y- y 00861AFEF6GMw2C-+-I$ 9 9)* 68&L(s:C9 2C& -7$ JQ9#a)K 0@Ar R;)Z10@0[9 2F:42C0 -79 K O {l$ Az"6@-(.2H( ; K R K $1"%R "¦05(+; 0[6[2 y 0 $1"³"%09 $D2C0z2H; 6G2bK'{l$ Av68(I( K K 619 *x2H%R;)08ACW_08{l$ $ ACAC0'01K$8y 0@AK#a@M¤E0G2F2C-I9 O {l$ AV68(I(R K "%0 R_9 S*>;'2H& ; EG68K 2 08'-AC2H$ ;)J|0@A !+6D2a)-7$@Efy 9 0'$1Ks" "%080Q61{l; EF$ M6[Ay 2C6@0$=(.( :<;)01:
f f (y) = f (0) + y x y ∈ f f (x) = f f (y) x=y f f x + f (0) = f (x) x = x + f (0) f (0) = 0
x=0 y ∈ f (x) = f (y) f (0) + x = f (0) + y y = 0 x∈ f f f (y) = y y∈ x y∈
f (x + y) = f x + f f (y)
= f (x) + f (y)
f (nx) = nf (x) n ∈ ∗ x = r/s r s∈ ∗ sf (x) = f (sx) = f (r) = rf (1) f (x) = (r/s)f (1) = xf (1) x∈
x
0 = f (0) = f (x − x) = f (x) + f (−x)
x ∈
f (−x) = −f (x) = −xf (1) f (x) = xf (1) x = f (1) f f (1) = f (1)f (1) f f (1) = 1 f (1) = ±1 f (x) = x x∈ f (x) = −x x∈ f (x) = x f (x) = −x x
α=2+
x √ 3
bxc
αn − bαn c = 1 − α−n
9 - 8 2G
H 2 2 2
n = 0 1 2 ...
JH 2 E H
) # + >?>@# + Y OA 1 U 4 0 Z;1 Z O#,#$ J s# } J $#*1 $ % Z,#*1 4 J# + 5 )$ Z 5; 0+5 s1 6 O# '1 7 s# # J#5E #m; ) # Y sB6 $ Z,7E
S#:46@;)9 0@:H$ 0GAC(# 0@$'J¤0>& K12[; 2HRv6G;)¯Vy 00 08$)9 * :H*0'K ?L$1"%08AFEf$'{ -IEfI-16 99,2H-7r#;)9'0z2H2C;)010
0 < α−n < 1√ √ n n 2+ 3 + 2− 3
αn + α−n = bαn c + 1
α=3+
√ 5
A n ^ R F@0[2
1AC6 0@9 E<* ?L0@:C2C-a)y 0¤0G(Ms2H;)Rv0w!l{ ?La $'0=-I9'652CE,24AC68-I2619 "%O[;'(70-7:<"%;Y;)2H$ ;)E0¤K 0=Eb-7EH9 ;)-+:4$1*)-I"³AC0,:4(+(72H0w0@; 9 68$'O2 { 2H; Ez61AC0=6824$'(I(&L-7:<9'R ;)2C0801E O808AFEGKs616@9 9 *¤* (70G2 K ABC
D ABC BC AC |AD 2 − BE 2 | ≤ 2 AC = BC ) # + >?>@# +Y OA 15U Z 0 ; 1 Z O# + 5)$ Z ;50+5 s1 6 O# '1 7 s# # J#5E # ; ,) # Y sB6 $ Z,1 >@+ # + Q#/# + ,,* E E
2 H 0%; 6[y 0 619 *
2 9 89 "
JH > C
CE = CD = (a + b − c)/2 AD 2
R ² M2H;)0VUs$ EH-79 0F86G",K
(a + b − c)2 − b(a + b − c) cos C 4 2 (a + b − c) a2 + − a(a + b − c) cos C 4
= b2 +
S#;'& E[K ² M52H;)0VUs$ EH-79 0F86[",K1"%0V; 6Gy 0 BE 2
=
BE 2 − AD 2 = a2 − b2 − (a + b − c)(a − b) cos C
BE 2 − AD 2
K 6@9 *>;)089 :H0'K
R
R
a2 + b2 − c 2 2ab a2 + b2 − c 2 2 2 = a − b − (a + b − c)(a − b) 2ab a−b 2 2 = a (−a + b + c) + b (a − b + c) + c2 (a + b − c) 2ab
cos C =
R
$'{ O80@9 08$ AF6@AN(.2H-.2;)M#0%KG? "%K1& 0}"%A4? J>;'$ -+6[:<EbM%;50%6@-I$'EfE{EH-I& :HJ>$ JQ? 9'0$ 24AFE61EH* -+a'-7:C(+2C03-+$ R{l$ 9#!lA{ KD-I"%9'240V0@61O8E<0@K8E
b
a= 6 b a < b+c
c=1
a
BE 2 − AD 2 =
(b + 1)2 (c − 1) + b2 (c + 1) + c2 (2b + 1 − c) 2(b + 1)b
2 ≤ c ≤ a + b − 1 = 2b f (c) = c2 (2b + 1 − c) 2 ≤ c ≤ 2b
BE 2 − AD 2
!l{
k≥2
BE 2 − AD 2
KD2H;)0@9 = ≥
≥ =
c≥2
b
k=1
f (2) = 4(2b − 1)
(b + 1)2 (1) + b2 (3) + 4(2b − 1) 4b2 + 10b − 3 = 2(b + 1)b 2b2 + 2b 3(2b − 1) 2+ > 2 2b2 + 2b
R
k (b + k)2 (c − k) + b2 (c + k) + c2 (2b + k − c) 2(b + k)b 2 2b2 + 2kb + k2 (b + k)2 + b2 = > 2 2(b + k)b b2 + kb
R
K 6:H$ 9'24AF61* -7:C2C-+$ 9R}¯0@9 :40 K r 2H; 682-IEGS#K ;'& E[K)-79Qa)$D2H;>R :46@Eb08EGK ]+6@9 *=EbS#$); $ 6G9D2 `:4{l$ $ J>A? ? AC(+$)0[24a'08(+Ef08J>2H;)Ev0 2H; 8Q682B J ; #$6[ y 0V{l$ 6@A? 2H? ;'0@-IE 6@AC-I01EEH*,&L-I0 9 Rf1Z110@D9 *>9 JQ& J>0zM a $'08& AFE}AB$'9 {L-7:H2H0%;)0 E$'8Q(I& 2C J-7 $ #$9) E R AC = BC
BE 2 − AD 2 > 2
a=b
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1¨
c ? : c b 8": 8
+?L] 9 $ 0HEH®@-.22C`B-7¹ $ ? 9)(7?LEf6[$ M 61EH0@9 -.A2C* -7Ks$ $ 9XABB -I2H? 9¤;)$ 0,E<E<&L-E2C:<01-+;X$ :49)$ 6>E[9 K OG*AC6@08]7JQ? E? AC050b01-Iy':CEG-+2CK)$'- y "& 08EC-.( `B2HM;¤? R (7a)6[0@M E40@2}AR?)(IS#6GM;)K08Eb0G05-.2H6@;)AC080=A}$'6%{I2C"0@-79X9X(I{l6D$ a)Af082H(.(+;)0@0* LAFE4B 2 b 8": 8 Rv!9,2H;)0VOG61J>0%*)0@E:HAC-7a 0@*=61a $8y 0 K 619 * r R}2H;)0 B ?L$ EH-.2C-7$ 9)Ef6@AC0 R}2H;)0 B ?L$ EH-.2CK-7$ 9)EN61AC0 K K K K K K K @ 6 9 * R RS#EH2F; 616GA2T2CE[(+K108616G9 y 08* E 619 * K8"%;'-7:<;K86@"%AC0;'-+":<;%-I9)6@ENAC0{l$ "A -I9)KGEN"%{l$ ;)A$)0by K80@"%A^;)E4$)2C6@0bAFy 2F08E@A R 1 6 9 * S#619 ; *6G2f-7E[K 6@AC0 B ?L6@$ 9 EH*-.2C-7$ 9)E@R 61AC0 B ?L$ EH-.2C-7$ 9)EGK "%;'-I(70 0 s 0& E0-I9 * &L:C2C-7$ 9R^S)$,EH2F61A2N2H;)0z-79 *'&L:C2C-+$ 9#K1"%0V:H$ 9)EH-+*)0@A R "-.(I(a)0>6Q(7$ EE[K P 619 * -IEV6,63""-79-79#{l$ KDA {l$ AN2Ha@;)Mu0*)9 080< ®89 2B-.2C? -7(7$ 6[9#M KT0@A61R 9 *u;)089 :H0'K P Eb;)032F6D/108-7E}Eva)6$D"2H;=-79a {l08$ 61AT9)2HE[;)K)0zO89 $'0<-79 ®82O? 24$(76[M 0@AKD-I{;)R 0O8$)0@EN24$ K'61E-7E -.{ Px-7!lE{2H;)K)08a AC& 0%2B6@$ AC039 ( +M & EH2 $'{La 2H;)080@61J9)E -IAC{#08EJ>;)0z68-7-79 E -I9 ODR K 2H;)0 AFEH2v? (76[M 0@AN2C61/10@Ev2H;)0@J¦6@(.(L-.{s;)0 PY² & 2 K"%;'-.(+6@0Q9 *2H;)0x$ 9 ( M_?L6@$ ACE<0xE<-7a a $D(702H;xJ>"$@-Iy 9)08E5E5{l{l$ $ A A R 6@AC!l{0 EH2F61A2CE[KK ;)0wO8$)0@EKf$ 2C$ A K1"%;'-+:<;,61AC06@(.(L9 0<®82HPO=9 6[24*$MQKG"%AC080 JQ9 $801y 0@0%*zKT& 2C"%$V?>;'O82C-I$,$(70 62H; a AC-0@2s05{7& a A082H61;)-I9)E0@E8A6 KR -I{ 2H;)03LAFE42?)(I6GM 08A^2F6D/108E $ A B ? K)$ $ E<-AN2C-+2C$ $ 9#K12H;)0zEb0@:H-.{L$ 2H9 ;)*50?) (I6GAFEHM 208?)A(I616Ga M (70803AN2C2C$,61/1O80@$%E 2C$ R
N
P
P
8k+1 (e, e)
8k+5 (d, d)
N 8k+7 (d, d) 8k+7 (e, e) 8k+6 (d, e) 8k+6 (e, d) 8k + 5 (e, e) 8k + 3 (e, e) 8k + 3 (d, d) 8k + 2 (d, e) 8k + 2 (e, d) 8k+1 (d, d) 8k+4 (d, e) 8k+4 (e, d) 8k (d, e) 8k (e, d) 8k +4 (d, e) 8k (e, d) N 8k (d, e) P
d
e
8k +4 (e, d)
k=0
0 (d, e) 1 (d, d)
e
1 (e, e)
2 (d, e)
1 (e, e)
2 (e, d)
0 (d, e)
3
d
2
4 (d, e) 1 (e, e) 1 (d, d)
e
4 (e, d)
d e
5 (e, e)
4 (d, e) 3 (d, d)
6 (d, e) 5 (d, d)
6 (e, d)
7 (e, e) d 5 (d, d)
N
d 3 (d, d) 2 (d, e)
4 (e, d) 2 (d, e)
d
5 (d, d)
4 (e, d)
e
4 (e, d)
P
7 (d, d)
8 (e, d) e 7 (e, e) 6 (e, d) 5 (e, e) 4 (e, d) 5 (d, d) 4 (e, d)
5 (d, d)
8 (d, e) d
k=1
9 (d, d)
P
3
4 (e, d)
8 (d, e) 5 (d, d)
2
9 (e, e)
1
N –positions P–positions 8k+8 (e, d) 8k+8 (d, e) @ 8k+7 (d, d) Q@ @ 8k+6 (e, d) PQ PQ P@ Q PR s q @ Q P 8k+5 (d, d) 8k+4 (d, e) 8k+4 (e, d) @ 8k+3 (e, e) Q@ @ 8k+2 (d, e) PQ PQ P@ Q PR s q @ Q P 8k+1 (e, e) 8k (e, d) 8k (d, e) @ 8k−1 (d, d) Q@ @ 8k−2 (e, d) PQ PQ P@ Q PR s @ Q q P 8k−3 (d, d) 8k−4 (d, e) 8k−4 (e, d)
N –positions
.....................................................................................................................................................................................................................................................................................................................................................................................................................
8k+7 (e, e) 8k+6 (d, e) 8k+5 (e, e) + ) 8k+3 (d, d) 8k+2 (e, d) 8k+1 (d, d) + ) 8k−1 (e, e) 8k−2 (d, e) 8k−3 (e, e) + )
D -7O[& AC0 qTU#$ 9 * 089)Eb0@*>OG61J>0z24AC0@0 'ACRx$ J 'AC$ J;)08AC0XK $ 9#KN":4$ -79)?1E5M"2H;)-.2H0>; E424AF682C01ObM{+6@AC9 $ *¢J {+AC$ a J 08619)E%a)61:"? -.(+2H0'; K {+AC$ J Us;)0@:$ EH$ -.2C9 -7($ MQ9"R -I9)¹ 9 (I( -79 $DO52H;)JQ08$8A%y (70@01E8OGR568(fSJQ;)0@$8ACy 0,0@E[-IE K {+$'AC{$ "%Ji;'08-+U#:<6D6@;,:<9>; 2H6@;)9D08M ACB 0$ ?L9 $ 610VEHAC-.02CEH-7& 2H$ ; ? 9#AC? KG01((70%M=0@61{+6%AC*3$ O824J«08$ 9 0@0@61AF68B:<( ;?L2H$ ;)EH01-.B 2C$ ? -7A$ $ M59)E<E-{l2C$ r-+Av2H$ ;)9>EH0b&LM61:<9 ; 6@*=AC0 {l2$'"%9 & $D$=AB29 E$';)$ & $19 2C:H"VPC$ "9J>-I9)-I0 9z9 32H-79 ;)OGOV6@0fJQLJQO[08$8& E y oAC0@0 E R c cjc d c? D("3M 9 ²0@AF(+0@/@61J=?sK#µ$); 9xU#$ 9D"V6GMKT619 * -+:<; 6@AC*x^&'MK 6; 2 9 @D0 A R CQ# >? &H 6 9 K 0@* -2C-+$ 9#K ¹ R vR» 0[2408AFEGK 682C-7:? +HZ o > 0 + s1 K » $'(.-.2C01:<; 9 -7/@6)K)ZG2[Rs» 0[2408AFEba'& ACODK A R f08? 6@AF24J>0@9'2B$'{^WY682H ;)089 J>- y 6G0@2CAF-+EH:C-.E2#M>Z-7:<$'; G{#612CU#6GAC682C*>-7(+E4OGT2C6@-+&'A:CMM E U#68(+OG6@AM#K ¹ ²KDS 5U#6@9)6DO* A6
10 (d, e) d 8 (d, e) 11 (d, d) 8 (d, e)
9 (e, e)
10 (e, d) e
8 11 (e, e)
9 (e, e)
8
8k + 3 8k − 5
N
N
N
8k + 4 8k − 4 P
8k + 5 8k − 3
8k + 6 8k − 2
8k + 7 8k − 1
P
[email protected]
A)
£ "
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8 wzE77H7HED:?:i5C7?;=;R;fFG7i : w_ ELK547H:? rK @C57Cr5C;H5BF6> ED>HelAELK ELK57HM:ADHFGQR8C7H5 Ao7H<: eD|5 7Ev7H@HAOAOas5C;B 5 8 Av>CAoF64D5 @?AEGF67H>H:i:;R_ 7HA|EGFG:Nk>C4L;f8 E65 FG:R> eELK5p:fF6c 5 AD8OF6EG
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Eb-7-I0@EE*Q2H2H;);)?)0B0vAC$)E<E<6161a J>J>(70@0B0vJ=616@EEsEs61?)?)ACACAC0%$)$)a a AC0@(7(70@0@? JJ 0@6G@2CE "D$' { 6"1 @E -.D2H;'-IA 9%¨'©q 2H;) ©0}1A 1(7K6@DEH 2s©82q "% A KL$V 61-79 KE<E<*>A &L ? 08D EGAC $)KGR="%a'(+Z8008-7J 9 61:4AC00> AC680@(I? ¨(s(72H61 ; 1:4AC-71019 D0>O *'2Hq;)& D08?)Ji(.-+ :CK-I689%D2C-72H$ ;'¨ 9)-I EEV-I-I; EfE6GEH2H&Ly ;)0 0>0R $ E<¹ 619 J>9D( MM0%61E? $'?)AC(I?L& $)082Ca'61-7(+$ AC080@9)J * E 2H{la $ ;)0A $ 24Eb2HAC0B;)080>? 682CAC$ 01$)AC*¤a'-7O[(+6@08-79)EzJ>68EET(^$'; ?)(I& 6GACy 2C$)-70fa $ (7689)0@(7E}J=AC0824E 6D$ *1Mz¨D6@¨'? © ? R6@0@9 ªB6@* AC& 01Af[* 2Hr8¨ ; 6@D 6@9D9 Mz"/DEb-IE(.$'(^2C(.a & $x2C0,-+Wu$ -7OG9)-+9 :<E;)$ 240GAC$}(B012H*²L;)K6G0 E<2C$ 68-I9 -IAF(.-+:H(+O[0,0-I9)Eb{l$ 6@$'(A}(.& a)2C-+AF$ -79 ¨ 9)O[E"-I9 2C-I(.$O ( 2H;'-7Ev24$,$'& Av6G2F24089'2C-+$ 9R p n S\ Rf#08?)(I6D:40@J>0@9'2[R 0 Oo # +> Y 5+ \,617 /! $#$1D8 m1o R E<&L:<;2HF@; 0G682 2 K K16@9 * a)R 0619DMV?L$ EH-.2C- y 0}AC0@68( 9 & J>a 08AFEGK1619 *V(70G2 a 0}63AC086@('9 & JQa)0@A ]76G`¤» AC$@y 032H; 6G2 R ]la1`» AC$@y 032H; 6G2bK'-I{ K'2H;)0@9 R ]l:F` ? » AC$@y 0V$ Av* -IE?)AC$@y 0 R p n Sa R #08?)(I6D:40@J>0@9'2[R 0 Oo # + y ,,#*1 45+ # 61 HJ# 2 $# #9HOQ+5 ; $H; A1 6Q+ 41 R F@0G2 *)0@9 $D2C0}2H;)0}& 9 -.2FEB*'-+O[-2N$'{ RD-I9 *68(I( ? $ E<-2C-y 0 -79'2C01O808AFE EH&L:<;2H; 682 R
a b c 0
(b +
c)p
+
p
b
(c +
a)p
+
c
b)p
≥
(c + a)p
≥
(a +
1
2p
a1−p + b1−p + c1−p
a1−p + b1−p + c1−p
p = 1/3
a
(a + b)p
+
b
(b + c)p
+
c
1
2p
√ √ a c b 1 √ +√ a+ b+ c +√ ≥ √ √ c+a a+b b+c 2
an
n
(4n)(3n)
n P
i=1
ai ≥ 4n
(2n)n
A
fR #08?)(I6D:40@J>0@9'2[R 0 Oo # +> +H; #HQ BA#$1 05# 1 4 HJ# R a)0f6 ?)AC-IJQ0'K86@9 *z(70G2 a)0f619DM3-I9'240@O80@A#EH&L:<;2H; 6G2 R » AC$@y 03F@2H0G; 2 682 q ]76G` K ]la1` R p&\^ n R 0 Oo # +I \E QEo-/# Q+ # 0 1 7 s+ #$ -hBAA1Y8 > #HA+H4 s61 h1 6 >@# >@ ; J 4 y E Q > R ^- y 0@9Q?L$ EH-.2C- y 0%-I9'240@O80@AFE 619 * K:H$ 9)EH-+*)0@Av2H;)0AC086@(sJ>$ 9 -+: ? $'(M 9 $ J=-I6@(IE 619 * "-.2H;9 $ 9 PH9 01OG6G2C-y 0>:H$) 0 X:4-70@9'2FE@R=0 61AC0z-79'2C0@AC08EH2C01*,-I9"%;)0G2H;)08A 619 * E<682C-IE<{.M%2H;)0%:4$ 9 * -2C-+$ 9 R ]76G`¤» AC$@y 032H; 6G2-I{ 619 * 6@AC0%a)$D2H;=$)* *K'2H;)08AC0V6@AC0V9 $,EH&L:<;=? $'(M 9 $ J=-I6@(IE@R ]la1`» AC$@y 032H; 6G2-I{ KD2H;)08AC0V61AC09 $,E<&L:<;,? $'(M 9 $ J=-I6@(IE@R ]l:F`=2HZ1;);)0@$1AC"0V* 2H; $=6G0H2®1{l-I$ EH2AzE<08&L6D:<:<;,; ? $'(M 2H9 ;)$ 0@ACJ=0,-I6@-7E(IE@6@R 9-79 9 -.2C0>E0G23$'{Ty'68(I&L08E%$'{ {l$ A "%;'-7:<; ]l*1`» AC$ $@A y 0R 2H; 6G2v2H;)0:4$)0 Q:4-+089'2CE -I9X0by 08AMQEH&L:<;>?)6@-IAf$'{? $'(M 9 $ J=-I6@(IE}6@AC008-2H;)0@A X-I(.(I-I61]lJ U#$ J>ª?)R µ8R 61` R±AC0VW_? AC$ $)Ea'0@(+A08K J 1 1) # -79X- *DQ"V+5 s6@# AC+*5µ8Rs²L 6@CAHQa)# 0@>?68&s K#&WuH & 9ACAF8-6G M> Z AR # s(;I61#(J> Ki/1-I9#"%K);)6@089 AC*0 p&\^S\ R 0 Oo # +> & E ! E0 >@# # 1 6>?>@#H1CQ# #CQ# Z Q+ R 2C2H6@; 9 682O80@9'F@2s0G2 (I-I9 0Ba 2C03$ 66@:46G9 -IAC2 * :4(+0 4R "F@-0G2H2;AF616@* 9 -.{l& $ * E AEbK1$ a)(7J>0G0 2 0?L$'?L-I$ 9'a)EH2C03-.EN2C61- $'y 9D{ 0MAC?L$ 0@$'9%68-I($ 9'9 ?)2& ?L$ J>$ 9 EHa -.2C08KD0 AF6@E EH9 -+*)*0@(7E^6@0G9 $'2 * { a)0ROE<&L2HF@;):<0G;0 2 65$'{ :4a)$ 09)EH6@612F9 9D61*M,9'2v? {l$'R $ -7Av9'2v6@(.$'(#{ EH&L:<* ;=-.¶?L08$'AC-I0@9'9'2CE2N{+AC$ K)J 6@9 *QRN* Z10[;)2408$1A4"³J,2H-79 ; 0682 2H;'-IE}:H$ 9)E42C6@9'2y'68(I&L0-79,2C0@ACJ>-7EE p&\^Sp R 0 Oo # +> & E ! E0 >@# # 1 6>?>@#H1CQ# #CQ# Z Q+ R 2CJ=6@-79 * O8PH0@?L9'F@$'2B0G-I29'2C$2^$'{ 6Ga 2 0%2HRs;)6@S#059 ;)*Q* 0}-IE(61M 01J>-IOG9 0GJ>O2C0@0@$ 9'A 9=2F$'E 2H{T;)650EbE086@6@J=JQ9 -7* 0%:4-7E
p = 2n+1
1≤s≤n
s
2s + 2k − 1 4 ≡ 1 (mod p) 2s − 1 k=0 n−s P 2s + 2k 4s 2s − 1 ≡ −1 (mod p) s
n−s P
k=0
P (x) =
m P
m
ai x
i
Q(x) =
i=0
n P
n
b j xj
j=0
P
Q
P (x)Q(x) =
m+n X
xk
k=0
m
n
m=n
m
0
n
1
m=n=5
Γ
Γ AB = mr
P
m
KL
r A B C AC = nr Γ A P
A
Γ
t
t
A m n cot ∠AP B + cot ∠AP C
n
AB Γ B BD K A B
Γ
D
AB
AC
L
Γ AD Γ M N M L = LK = KN
C
A p&\^O_ R 0 Oo # +> D 2 C }# JH *#$1 Q Z 9 $)$1o>? 56 R F@0G2 A K J ∈ M (R ) K@"%;)08AC0 J -IEN2H;)03J=6824AF- ®z6@(.( $'{'"%;)$ Eb030@9'24AF-+08Ev6@AC0 1 K
61J>9 6G*w24AC(7- 0G®V2 $)aD2C68-79 R50@*%Rw{+Z1AC0[$ 2 J a@MAC08?)K^(I6@6D9 :4-I*w9 O3{l$ 08A 6D:<;08(70@J>K 0@9'K 2T-I9ACK $1" KN* 08"9 -.$D2H;240Q2H;)a@0fM y'6@(.&L0 2H;)0 R » AC$@y 032H; 682 R p&\^Sa R 0 Oo # +> D 2 C }# JH *# Q+ Z6 &Y H 1Q Z 9 $)$1o>? 56 R $'{s2H;)0F@? 0G2$'(M O8$ 9Rz $ A a)06,K :4]I$ 2HK ;)9Dy 0u09 2H0@(+;)0[*052 a@:HM089'a)AC24008AC$'JQ61-+9D*X$8M>y $'-I-I9 {#9'O¤2H24;)08y 0AC0@-7$ ?LA24Af$'0H(?L® M $'O8-I$ 9'9 2 2H{+AC; $ 68J 2N2H;)0(.-79 08E `Rv6@!lAC{ 0%:4$ 9 -IEv:4& 2HA4;)AC0V089'AC0G2v¬{l$ 0@:CA 2C-+$ 9>$'{ K K 2H; ACK $'&LR O8;2H;)0V?L$'-I9'2 K'? AC$8y 0 p&\^&s R 0 Oo # +> D 2 C }# JH *#$1 Q Z 9 $)$1o>? 56 R F@0G2 a)0V6V?L$ EH-.2C- y 0z-79'2C01O808Av6@9 * 6VAC0@68(L9 & JQa)0@AR » AC$@y 032H; 682 n×n
b ∈
B = bJ A
k = 1 2 ... n
Ak
k
det(A + B) det(A − B) = (det A)2 −
n X
b
det Ak
k=1
!2
A1 A2 · · · A n P k = 1 2 ... n Gk A1 A2 · · · Ak−1 Ak+1 · · · An A1 A2 · · · A n Bk Gk Ai Bi i = 1 2 ... n
n
Ak
P
x
bxc2 + x +
1
2
n−1 2 + ··· + x + = bnxc2 n
-I{#6@9 *>$ 9 (M5-.{ K1"%;)08AC0 *)0@9 $D2C0@Ev2H;)0%OGAC08682C0@E42-79'2C01O808A{7& 9 :C2C-+$ 9R p&\^ R 0 Oo # +> 0# ,#$ }E 1 6 T7) # 0 s,C1 4 5+5 1D8 m1o R 2CF@6@0G9 2 O80@9'F@a 2s0G0 224$2H6u;)60:HJ=$ ?)9DAC-I9 $1y $ 0H01®_A^:C:42C°'-I-7AC$ & :4956D&)* (7$'6@AC{ -.A(76G6124AC08$ : AF9'6@2C( $ 682#R 2HF@;)0G02T; ? 66Gy $'Eb0-7089'J=2F-2CE -7E=:4-7EHACK-+:4*)(70f0@K@E "61-9 2H;* K K@AC0@61E<EKf?L* 0@61-I:C9 612C*J>-y 0G0G2C(Ms0@A R :4& 2 X-.682H;)2 $'& 26@& 9 EH* -79 O=K@24"AF-+O8-.2H$ ; 9 $ J>a 0G0[24A2M"%Kv01?)08AC9 $@y 0>612H9 ; * 682 R a'-7E01:C2FE 619 *x2H; 6G2 a -IEb0@:C2CE R p&\^ R 0 Oo # + 5#$1 45+ # 61 6JHO#$7 9 # 5#HJ +5 H;Q 1 6Q+ 41 R K1"%0VD; -I6G9 y *>0 2H;)05(I61ACO808EH2f-I9'240@O80@A E<&L:<;=2H; 682f{l$ A 6@(.(? $ E<-2C-y 0AC0@68(9 & JQa)0@AFE K K R p&\^ R 0 Oo # +> 2 sB,#y8 H, s#*1}5,) #$ 6 O , # 1 o>? 56 R F@0G2 a)0,6>:H$ 9Dy 0H®={7& 9 :C2C-+$ 9w$ 9¤619X-79'2C0@Ay'68( R= $ A K K K K(+0[2 Rff08 9 0 Rf» AC$@y 032H; 682 R n
bnxc = bxc
E AC
H
PH
b·c
AP QC ABC B AC H A
K
A B
K BE
AP P Q C PQ
QC
∠P EQ
∠AP B
k
a b
c
a3 + 3 b3 + 6 c3 + 12 > k(a + b + c)3
f
ai ∈ I
n 1 P a= ai n i=1
I
i = 1 2 ... n
n X X n(n − 2) n ai − a j f (a) + f (ai ) ≥ f a+ 2 2(n − 1) i6=j n i=1
A! p&\ n ^ R
> 2
0 Oo
6
sB,#y8 H, s#*1 }5,) #$ O
# +
» AC$@y 032H; 682bK'{l$ A6@(.(LAC0@68(9 & J>a 08AFE K K K a1 a2
n X
R K1"%0V; 6Gy 0
, # 1 o>? 56
√ i . . . an ∈ √12 , 2
n X 3 2 ≥ ai + 2ai+1 a + ai+1 i=1 i
K
h
"%;)0@AC0z2H;)0E<&La Eb:HAF-7?'2CEf61AC032C61/10@9=JQ$)*'&)(+$ R p&\ n&n R 0 Oo # +>/ ? >@02 Y Oo #$ R E"%<& ;'?)-+?L:<;Q$ F@E-I0G0XE32 ? 2H; 0@6GAC?L2 089 * -7:4a &)0V(76@65Af-.°'2F24& Eb$ 0G6D(I* { AC; -.(7616GK)E524"[email protected]@9_2H;( -I"%9 :4;'-7-7(AC:<M :4;=-I(79 0 -IO5REf-7$ F@9)9>0GE2:H2HAC;)-70,a 0@E<*=61a)J>-79>0X0562HEH;)-+:40*)-7AC0,* :4(7-I$'0 61{ J>0GR}2C10@& A%6@AFE$'2H;){ 08A R F@0G2 -79'2C0@AFE01:C2 6G2 619 * 682 R R » AC$@y 032H; 682 "%01a E<-24 0'K)+ a & S#2N;'2H-I;)Ez0V?)9)AC$)6@a JQ(70@0VJ$'{L:42H6@;)JQ0V0?)-7AC9'$ 2C?L$>$ E2H;)0@Av0=; 619 Ef- y EH&L0@a)AFEHE-.012M°'&L$'0@{B9'#2C( 01M>O[-Ia 9)0@6)0@ 9,E(+WY$ E4682[2HR ;¤ U#0@9'24AF68( p&\ n \ R 0 Oo # + 3! 2# =&h2 $% Q J O# O Q+ 4 H "HJ H C:2Q #$ s 17) # 0 s,6 0 B #9 H4+HZ + # 8 6 Z 1 Q HO#H, 1}$ R {l$ A R » AC$@y 03F@2H0G; 2 682 K K K a 0AC0@68(s9 & JQa)0@AFE3E<&L:<;=2H; 6G2 R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR p n S\ R##0@J=?)(I6D:40@J>0@9'2[R 0 so c2# Y 5+ \,617 /! $#$1D8 m1o R 9 $ JQa Z1AC0V$'-.A 2 0@E 08()K 240G(0G° 2 &L0 *)0@EX9 $ JQa RAC0@E¤A 01E 0G(7EX?L$ EH-.2C-.{+E¤° &L0G(+:4$ 9 ° &L08EGK}0G2>Eb$'-2 & 9 ]76G`XW_$ 9'24AC0@Av° &L0 R ]la1`¤W_$ 9'24AC0@Av° &L0'K)EHK)68(+$ AFE R ]l:F` ? Us$ 9 A4J>0@Av$'&=-79 A4J>0@AB°'&L0 R i=1
n
ABCD ABCD
Γ
EF
BD E EF M AC S AS : SC = EM : M F
BD
a1 a2 . . . a n n Y
( 2k )1/2 ak n(n+1)
k=1
Γ A
BD
ak ≥ 1 v u n uX 2 ≤ exp t ln ak
1≤k ≤n
k=1
a b
c
p
0
a
(b +
c)p
+
b
(c +
a)p
+
c
(a +
b)p
≥
a)p
≥
1
2p
a1−p + b1−p + c1−p
a1−p + b1−p + c1−p
p = 1/3
a
(a +
b)p
+
b
(b +
c)p
+
c
(c +
1
2p
√ √ a c b 1 √ +√ a+ b+ c +√ ≥ √ √ c+a a+b b+c 2
A;A p n Sa 0 so 3 2# 5#$1 2# 45+ 6 761 -2 HO # R # 8 0 > J ? 7 ( 1 6 H : 8 0 Q J 8 0 ' 9 [ 2 R #9HOQ+5 s# 6JHO#$7 9 # 1 6Q+ 41 R 0@E EH-+OG9 $ 9)E%?)61A a (70X:<;'-I¶#AC0X* 08EV& 9 -.208E E* 0 (4n) R S AC$'&'y 0@A24$'& E
(+08E 0@9'2C-70@AFEf?L$ EH-.2C-.{+E 240G(7E °'&L0 R p n R##0@J=?)(I6D:40@J>0@9'2[R 0 so c2# +H; #HQ 6 AB,#*105# 1 4 HJ# R ° &L0 Z1$'-.2 R#W_$ 9'24AC& 0@9=AB9 °'&L$ JQ0Ha q AC0? AC08J=-70@AK)0[2Eb$'-2 & 9>089'2C-+08Av°'&L08(7:H$ 9 °'&L0z2C08( ]76G` K ]la1` R p&\^ n R 0 Oo y2# \E QEY-/# Q+ # 0 1 7 O+ # -hAB1h8 > #HA+H4 s61 #$ > 2# >@ s#/+ # J 4 yE Q6 > 6 R D2F619'2Q* $ 9)99 0E * 08E¤9 $ JQa AC0@E089'2C-+08AFE¤?L$ EH-.2C-.{+E 0[2 K}$ 9³:4$ 9)E<-79* 08? AC0Y(+08E ?L$'( M)9 $ J>0@E,A 01E 0G(7E& 9 -.2F6@-IAC0@E 0G2 6Gy 0@:,*)0@E E:H$)6G2C 0 -7XEH{+:46@-7-I0@AC0z9'2F(7E 6%:49 $ $ 9 9* -9 2C-+01E $ OG9 6G2C-I{+E8R F86 °'&L0@E42C-+$ 90@E42_*)0³E6[y $'-7AxE<- 0G2 ? 08&'y 089'2 R ]76G`X?LW_$'$ ( M)9'9 24AC$ 0@J>A,0@° E8&LR 0uEH- 0G2 E$ 9'224$'& EQ*)08&D®Y-IJ>?)6@-IAFEGK-I(}9# M¢6u?)61EQ* 0w2C08(IE ]la1`¤W_$ 9'24AC0@Av° &L0VEHK'-I(L9# M=6%?)61E * 03240G(7Ef? $'(M 9 $ JQ08E@R ]l:F`X(+W_08Eb$ °'9'&L2408AC(.0@(+A=08E ° *)&L0z0Y2C? 08$'(IE & ? A5$'24($'M & 9 2 $ JQ08KfE -.0<(f®@M-7E4246u089'& 2[9 R 0u-79 9 -.2 0YE * 0y'6@(708& AFEX* 0 ? $'& A ]l*1`¤$'W_& $ 9'R 24AC0@AN° &L0* 619)E2C$'& 2403? 68-7AC0z* 0}240G(7Ev?L$'( M)9 $ J>0@E[K1(70@EB:4$)0 Q:4-+089'2CEvEb$ 9'2 s($ & I? 61J>/1]lU#-I9#$ KJ>6@?)9 61*QACX08R A¢` -I(.(I6@-I&«61J«?)ACª$)R a µ8@(RT0@? W_J>$ 0 Eb081 Ab1K ,* ) 61# 9)- E ) *1+5"V O# 61+ AC*µ8 RwCQ²L# 61>?A4a 08&6@H&s (K8-Wu & BA4,AF#$6[M«;5# Z RK p&\^S\ R 0 so c2# c&fE ! E0 >@# #$ 1 ,6>?>@#*A1 9 % Q 2 R 6 ? 0@9 Z1$'-.2 RTZ10[& $'2 9-2 :40@AC:40[(720f*)0B*)AFK6G08M &D$ ®Q9 0G? 2 $'K -79'2F*)& E0893E<&D& ®=?LA $'9 -I$ 9'KJ>2s*)°'a)0=&LAC? 0808(76@Ez:HAF$ A23019 E 0G0G°'23(7&LE*#0B?L E<6@$ & & EHA 24-.2CAC-.0>{+K[E@0G* R2 0 W_(7$ 69'K2C242C6@AC089 0@(IO8EVA}0@° °'9'&L&L2C000 K)*'-7E42C-79 :C2FE * 0 K 0[2B9* 0GE 2C0@AC65J=& -I9 9 0@0%ABy':4680G(+2F0G2C& 03Ay':H68$ (+0G9)& E42CAB6@0@9'9,2405{l$ ?L9 $':C& 2C-7AB$ 249>$'& * E0 (70@E36@? 9 $'* -79'2FE R EH& A (3n)(2n)
n
n P
n
i=1
n
ai ≥ 4n
p = 2n + 1 1≤s≤n n−s P 2s + 2k − 1 4s ≡ 1 (mod p) 2s − 1
s
k=0
4
s
n−s P k=0
2s + 2k 2s − 1
≡ −1 (mod p)
P (x) =
m P
m
ai x
i
n
Q(x) =
i=0
n P
b j xj
j=0
P
m+n X
P (x)Q(x) =
Q
xk
k=0
m
n
m=n
m
n
0
1
m=n=5
Γ r A Γ A B C AB = mr AC = nr m n cot ∠AP B + cot ∠AP C Γ A
Γ
t
t
A
P
m
n
A!D p&\^Sp R
6
0 so
9 2 R
c 2# c&fE ! E0 >@# #$ 1 , >?>@#*A1
% Q
E<J=& -.AL(I-7(I086v&5Z12F61$'*)9 -.0 2 O8089'240 RD(+6 ?Z10,$'* -2 0@-I619 JG0G0G? 242 0GAC20=EH-.(7*2C0@& E& 0BE 9w*)* 08*)&&D0@®@J J,-@08? 0@-+JQP<J>:H08080fEAC::4?L$D(+0$'2-I0BE 9'* 2CRsE0 Z1*#$' -.-729'2C° 0@&LAFE& 0 019:C2C?LK[-7$ 0G$'29-I9'EAC2$'08-.° E2 ? &L010G:C(+(+2C:40v-.$ {+?L9 Ev$'° *)-I&L9'0@0 E2 Eb0@OG0[JQ2 089'2CEHE & A 0GK 2 J>$ 9'246GAC0@y AB0@: °'&L0 RZ8- 0G2 E$ 9'2B(+08ER ?)AC$101:C2C-7$ 9)EAC0@E? 5,# R 0? @E OG6@6@A &D® Z16 ? $'(7-.632 KDJ>0G2^6GK 24EbAC$'-7-:H2 0z$)a'2C0@R59 R&L» 0HR $ 6z? EK $ ? $9)6@& E?AF2C-IAN0@* E40 2^(76z0@J=0[92b68KDAC24? 0@AFJ=-+$':4& 0?)A (I* 6 $ :H 9'619'2T2 24$'K & ?)EK 61(+A08SE RL08KE @( W_0@E KDJ>$ 9* 9'0@08E 249'E
Γ
A
B K L AD Γ
BD AC KL
B
1
Ak
Γ
D
AB
A J ∈ Mn×n ( ) b∈
J B = bJ A
k = 1 2 ... n k b
det(A + B) det(A − B) = (det A) −
C
M N M L = LK = KN
2
p&\^Sa R
Γ
JH
n X
det Ak
H 2
!2
R
R 61* A4&Xa ?L-24$'AFZ16@( M $'-IACO8-.0>2$ 9 * 0 &w?L$'( M O8$ 9 0 R& » 9 $'& ?LA $'( M O8$ 9 0uK :4$ K 9Dy ]I(+0<0%®1K 0'? KB$'K#(0GM Eb2O8$'$ Eb-2$'9 -02 $)a'(72C0>0@& 9 9 :4&¤0@?L9'0@24$'9XAC-I0>9'082*)9 0>-I(+9'0by'2 OG0@AFE6@6GAF9'-+y'2B0G-.& 2(+00EA Eb6 ? $ J=K JQJ>0[$ 2 9'24AC0@AB* °'0 &L0(+08E *)AC$'-2408E `RsZ8- Eb$ *99'0@2vE EH:H-+$ OG9 9 :405$'(I& 6=AF61A0G9'E ¬2C0@0HE®1-7? $ $'9w& *)A 0 ? K 6@AK AF6@? ? K $ AFR 2 p&\^&s R 0 so c2# \2 }# JH *#$1Q 4 2 5) 1 0 >? 5,# R Z1$'-.2 & 9=9 $ JQa AC0V? $ E<-2C-I{#0G2 & 9>9 $ J>a)AC0VA 01E 0G(FRW_$ 9'24AC0@AB°'&L0 0 so
c 2#
\2
A1 A2 · · · A n
Ak
P
k=1
}# *#/# Z @&Y 1 Q 4 5) 1 0 >? 5,# P Gk
k = 1 2 ... n A1 A2 · · · Ak−1 Ak+1 · · · An A1 A2 · · · A n Bk A i Bi
n
Gk i = 1 2 ... n
x
1 bxc + x + n 2
2
n−1 + ··· + x + n
2
= bnxc2
E<-0[2vEb0G&)(+08JQ089'2vE
b·c
.......
AP QC
B
C AC
P EQ
E H
K
B H
AP P Q ABC
.... ....
AC A K BE
QC A PQ
PH
AP B
k
a b
c
a3 + 3 b3 + 6 c3 + 12 > k(a + b + c)3
A) p&\^ R
0 so
9
0
(
Z1$'-.2 & 9 0%{l$ 9 :C2C-+$ 9X:4$ 9Dy 0<®10EH& AB& 9>-79'2C0@Ay'68(I(70 RT» $'& A Eb$'-2 R ªf9>*90GE L9 -2 RW_$ 9'24AC0@AB°'&L0 f
ai ∈ I
a=
n(n − 2)
p&\ n ^ R
2
f (a) +
n X
i=1
0 so
R
c 2# OA# 8 H, s#*1 7) # s, / 2# + # ,# A1 0 >? 5,#
9
1 n
n P
I
K K K K
i = 1 2 ... n
ai
i=1
n
f (ai ) ≥
2(n − 1)
0
X i6=j
f
a+
ai − a j
(
n
R
c 2# OA# 8 H, s#*1 7) # s, / 2# + # ,# A1 0 >? 5,#
R
W_$ 9'24AC0@A^° &L0'K1?L$'& As24$'& EN(70@E9 $ J>a)AC08EA0@E 08(IE K K K K $ 9=6 K $ &=? (+08EB-79 *'-+:40@EfEb$ 9'2v:468(+:4&)@( 0@E EfJQ$)*'&)(+$ R p&\ n&n R 0 so c2# h y so $#5 m >@# R ?)(.& ? EB0@AC°'?LZ1&L$'080 -.9 2 * -7:4&)(768-7AC0G? $ 6 ?& E9Y"E 01? ° *)& 0}K61(I*)&)AF-7-IP<EH(I68J-.2C2 & 0@08? 0=J>EAC0>*'0}-7&x& 9)E9,J :H:4AC0@-0@J>2AC:4*)0>(70}6@: 9)-7$D9)EV2E0=E& :H9YAC*)-2[0 :4R'0@Z1AC$':4(7-02 °'&LR^0 ªf(7039xR* E<-I& 61?)JG?L0G? $ 24:HEAC$'0X03& ? * * 000 *'-7W_E$ $ 9)9'E 24AC0@0@9 AB°'&L0 0[2 0@9 R R (\ 9 -y 08AF+E<- 2 0%E U#*)0>0=?)#AC01$)O[a -I@( 9)0@? 6)J>K 0QJ=0@6@E4-I23EB(+610?)9 ? $ 6@ACJ¥&u*'EH&Q& A?)(+AC0=$ ?LE<-$ 24E0,08& -I9'A24608A4*'9 -70[E<2? 6@:H08AC&>9'24?)AF6@61(A*)(I6V0E<WY&)-246G0)2H;xR * 0 p&\ n \ R 0 Oo 2# ! =2# (&h2 $% Q J O# Of# Z JH HO 6 $H, :C: 2 Q# O 1 }5,) #$ 6 O2# 0 #H; Q#/+ # 8 6 , ; #*1 Q HJ#* #*1 J ; # R ?L$'& A R W_$ 9'24Z1AC0@$'AB-.2 °'&L0 K K K *)0@EN9 $ J>a)AC08EA 01E 0G(7ET2C08(IE° &L0 R a1 a2 . . . a n ∈
n X
i=1
M
BD E AC S AS : SC = EM : M F
Γ EF
k=1
2k n(n+1)
BD
ak ≥ 1
a1 a2 . . . a n
( ak
n
n Y
1 √ √ , 2 2
n X 3 2 ≥ ai + 2ai+1 a + ai+1 i=1 i
ABCD ABCD
Γ EF
)
1/2
v u n uX 2 ≤ exp t ln ak k=1
A BD
1≤k≤n
A ©
H m +H
}
@H 2 - 9 v2 9 9
y O ,#$> f#) # mo# J>? #$7 , $# +QE # # + Y : :0,#$ $# + , O s+ # , D A Z , # $ Z, m # O; 7 0 O ,#$> E
24S#$>;)08$'AC& 0Av-7686DE2F/@2C6,&L0@a:49'(I2C6@ -7-I$ J«R 9>¹ J>6(I- M * 610b* -BXy#0VK:42H&)²L; (61682\/1M52 &X"ZG-2C2H6G;52402H;) 09 -7{lE-0@y E46G0@242CACAF& EH-7AC:C-.0122CM#(*>M=KE-7²L$'9 61(I:H& /1AC2C&s08-761K$ EH¹9,-79 ¼<2C08OV$ AH{la 1$ 6@'A - ©"l61D 9Q 1; 161DE a ACq $'A &L 6@O8 9 ;D * 2 R {l-7E0@6G& 2CE& 01AC*=01r@*Q2H-I;'9=E-IEf$'2H;)(I:4& 0%(7682C-7E-7$ Ji$'9#(I& KL-7Ev2C; -79 $ 6@$DE}9#2vK)-I9 :4-2B$ {l$ A4-IA4EACJ>0@-I:C9 012[* *>RT0@SJQ01;)*>00EH2H?)24; AFAC-+68$):C2va 2C( -7(7M,9X0@J-7089 6D:HJ>:
x≥0
a>b>0
ax − b x
AP
4ABC
AQ
AQ2 = AP 2 + |AC − AB|2 BP = CQ AP
AP
BP = CQ
∠BAC
b=c
AP
AP 4ABC
AQ
AQ
P AQ2 = AP 2 = AP 2 + (b − c)2
@6 9 * S#;)08AC08{l$ AC0'K
∠BAC
∠BAP = ∠CAQ 6= 21 ∠BAC ∠BAP + ∠CAQ 6= ∠BAC ∠BAC Q AP = AQ BP = CQ b 6= c x = CQ = P B
x 2 (a + b2 − c2 ) a x c2 + x2 − 2cx cos B = c2 + x2 − (a2 + c2 − b2 ) a
AQ2
= b2 + x2 − 2bx cos C = b2 + x2 −
AP 2
=
(b − c)2 = AQ2 − AP 2 =
1−
2x a
b2 − c 2
K
"%;'-+:<;>EH-7J=?)(.-IL0@EB2C$ R¯0@9 :40 K 619 *52H;)00V? EAC;)$)$1$'"±{-IE 9 0<:H®8$ 2J=2H?); (76G0G2B2C0 ?)R 61A2f]+aD`T-7Ef9 $D2N24AC&L0%]+a & 2B68(7J>$ EH2^24AC&L0`RF@0G2 Z8-79 :40 K1"%0VO80G2 a(b − c) = (a − 2x)(b + c)
BP =
b a−x CP = = c x PB
ac b+c
AP 2 = c2 + BP 2 − 2cBP cos B = c2 +
ac b+c
R
2
−
x = CQ
c(a2 + c2 − b2 ) b+c
R
R
K
A)¨
¹ (7E$K Z8&La)E42C-.2C& 2C-I9 O32H;)08Eb0z-79'2C$
AQ
2
2
2
= x +b −
a2 + b2 − c 2 a
K1"%0VO80G2
R
x
AQ2 = AP 2 + (b − c)2 2 c (b2 − c2 )(b + c) + a2 b a + b2 − c 2 2 x − x+ =0 2
"%;'-+:<;5{+6D:C2C$ AFEB-79'2C$
a
(b + c)
K
R 2¯V "%$ 08A$=9 6=:H?L0'$ :4K $ EH-.9 2C-7:H$ AC0G9)2CEf0,{l$ 0<®@A $ 6@A J>? 2H; (+0'6GK2f-.6@{ AC0%EFM J>J>K 0G24AF-+:R "K -612H$D9 ;=24*0}AC0@2HE<; ?L680@2T:C2H2v;)K 2C0@$52HE;)02H0@;)29 "%0%$6@(E2C-.$'2C(I&L& *)2C-70$ $ 9)9E^M -76@089 (7** R $ A R
ac x+ b+c
a2 + b2 − c 2 ac x− + a b+c
= 0
x = 2b cos C − BP Q BC a = 5 b = 4 c = 3 BP = 15/7 CQ = 15/7 CQ = 149/35 I TZG/TZGJI {JR d ~ r F g g h F [ v X j P b NHM {ORZW TUM_L_ryG DF SOWS URJ{ GJbYF SOWS URJ{ GJb ^ GOTNHM S/S Nd k RZW Ra G { M N SZu/ i ^ 5^ b N'M {JRZW TUMBL r G ^ R M W KHL b ^ R M W KHL b [HRO~sS NGON u wyj k g*[?^ 0Qj [H['g b i F4b W T4K4I M N R N a r l N S TUMAK l b j_NHN*T ~sW K
b' KJTZL W M S4u x g g* ] \i0i b G*K R N'b WS N UR,u Q [ 'k g$ 6 v i v b G l S P N R TZL M b G lmS N'M SZu*S N doL q R V W G4V GOT RZWZ] ^ MBGJI S N'M {JRZW TUMBL rJb [OS w M WS d S b 5b F u 'jU x = BP
D
"!# %$'&)(%*,+.-/+0'12435768 9:8#;:<=11'>@? ACB.D.EGF3+
√
3 2
PA
2
FA
EA
A
. ....... .. ...... .. .. ... .. ... .... ... .... ...... .. ..... .. ... .... ... .... ... .. .... .. .. .... ... ... . . . .. ..... . ... ......................... ....... ... . . . .... . . . . . . . . ... . ......... ....... .... . . . . . . . . ... . ..... .... .......... . .... ................... ......... . . . .... . . . . . ......... .................. ....... .... . .... . . . . . . . . ............. .... ... ...... ... . . . . . . . . . . . . . . . . . . . .............. .... . ... ...... . . . . . . . . . . . . .... . . . . . . . . . . ................. .. ...... . . . . .... . . . . . . . . . . . . . . . . . ............ ... . ..... .... ...... . . . . . . . . . . . . . . . . . . . . ............ .. ... . ..... ...... . . . . . . . . . . . . . . . . . . ............ ....... .. ....... ...... . . . .. . . . . . . . . . . . . . ............................................................................................................................................................................................................................................................ . .
S
B
. ...........
F
E
............
P
D
C
A F@0G2 a)0V2H;)0?L$'-I9'2 6G2v"%;'-7:<;=2H;)0%(.-79 0 JQ0@0G2FE 2H;)0%(I-I9 0 R]\!l{s2H;)08Eb0 (I-I9 0@E}6@AC0%? 6@AF6@(.(+0G(\K 2H;)089=2C61/10 2C$Qa)0%682v-79 9 -.2MR `F@0[2 619 * R ² M¤W_089 08(I6@& E[DS#;)0@$ AC0@J 61?)?)(.-+0@*,-I9 K1"%0V; 6Gy 0 S
EF
BC
BF x= FA
S
4ABC
2H; 682-IEGK
CS =y BS
CS = BS − BC x
CE EA
r
K1"%0V$)a'2F6@-I9 R $1"³"%06@? ? (MXW_0@9 0G(768& EGDS;)01$ AC08J -79 2C$=O80G2 x
R^¤AF-.2C-I9 O
BF AE CS · · = −1 F A EC SB
y=
]`
BC = x−y BS
4ABD
BF AP DS · · = −1 F A P D SB
2H; 682-IEGK
DP DS =x PA BS DP PA
RT¤AC-2C-79 O
DS = DB + BS
r
6@9 *,& EH-79 O] `HK1"%0VO80G2
DB + BS BC DB DB = x + x = (x − y) +x BS BS BC BC DB + BC DB DC BD x −y = x +y BC BC BC BC
R
= x =
8Z -79 :40 -7Ef2H;)0J=-7* PH?L$'-I9'2 $'{ K "%0; 6Gy 0 KL6@9 *X] `{l$'(.(+$1"VE -7J=JQ0@X* -I-.682H2C;%08( 2HM;)R 0z& EH& 6@( 9 $D2C6G2C-+$ 95{l$ ANE<-7* 08EB619 *,619 O[(70@Ev-7924AF-76@9 O[(+0 K@"%0z 9 *#K & E<-I9 OV] `HKD2H; 6G2 D
DC BD 1 = = BC BC 2
BC
ABC
DP PA
= =
1
a cos B
+
a cos C
=
1
·
2ac cos B + 2ab cos C
R 2C$=$)aD2C68-79
2 b cos A c cos A 2 2bc cos A 2 2 2 2 2 2 1 a +c −b + a +b −c a2 · = 2 b 2 + c 2 − a2 b 2 + c 2 − a2
0V9 $1" & E032H;)03"%08(.(+P
4AD 2 = 2b2 + 2c2 − a2
K {+AC$ J"%;'-+:<;%"%0E01032H; 682 -I{#6@9 *>$ 9 ( M,-.{ R . .4R I 9Q0 1 0 8Q>?> 5?8Q B,#*;#*1 7# c- 5) # 1 1B} R 2Ha ;)0,0,2HO[;)- 0>y F@0G0@J,29¤-+*):4$ PKL056@9 "* -.(I(^?)AC-7$@Ey 010,°'2H&)R3; -6Gy'Z8268-79 (+08:49'0 K 2B24$ K 61AC0Q:4$'(I(.-79 0861ARK#"%F@0G;'2 -7:<; K -7J=?)(.-+08F@Ev0G2H2 ; 682 -IEv2Ha ;)0}0V2H?L;)$'0z-I9':4$)2 $ AC* * -I9)0G68L2C9 00@24*,AF-7-76@959 2HO[;)(+0z0$'? { AC$)2H;)a'0z(+08;)Jx$ R J>$)O80@9 0@$'& EBa)6@AM :H089'24AC-7: :H$)$ AC*'-79)6G240BEFM)E42408JKb"-2H; K K86@9 * R DP 2a2 = PA 4AD 2 − a2
DP/P A = 1
AD =
√ 3/2 a
a = BC b = CA √ c = AB 4AD 2 = 2b2 + 2c2 − a2 AD = 3/2 a b2 + c2 = 2a2 Q AD Q E F Q P 4ABC A = (1 : 0 : 0) B = (0 : 1 : 0) C = (0 : 0 : 1)
D1
S#;)089
K
Rv¯V089 :H0'K K
D = (0 : 1 : 1) Q = (2 : 1 : 1) E = (b − c cos A : 0 : c cos A) F = (c − b cos A : b cos A : 0) 2 1 1 Q 0 c cos A det E = det b − c cos A c − b cos A b cos A 0 F
KL6@9 *
= (b2 + c2 − 4bc cos A) cos A = 2(b2 + c2 − 2bc cos A) − b2 − c2 cos A
"%;'-+:<;,JQ08619)Ev2H; 6G2 K K 6@AC0%:4$'(I(.-79 0861AR
= (2a2 − b2 − c2 ) cos A = 0
D "D D D
K
Q E F I TZG(TZGJI {JR d ~ rcwyj k g*[h^ 0Qj [H['g b G*K R NHb WS N 6R,uo kojx FHL S L R NHM {ORZW TUM_L_r4b JG4V$I M N'b w ib F uh j k0 n j ] k g FsF4b S N q G [ sk g* U v i FJb W T4K4I M N R N a r l N S TUMAK l b j_NHN*T ~sW K
b KJTZL W M S4u Qj MBL rJbi5T SJ
sS b S V S N u Oi g$[ F k [ i5F ^ g*X b ^ S rJTUMBd R b v *b FZ u S N d:L q R M N G W6WRO L TZGJIBKHL MBGON ]
sjBF4b0w M TTZG*K W MF'G*KHL q RZW N x S IBGOT RZW d R TUb b F u kjwh g '7 5bQ SJ
sS L T4K
M V W G4V GOT RZWZ] 4q RZWR S ToGON R
r : %! %$'&w(%*V+.-/+0N1:2 35 x gK"1 E "DFOD.x|EIE1'A`/+%}b1'>A *D> V1 P:QNR ! ABC! S Q e _ e ibRgQ[ e _@'mOQY RfX e _@'mZQHqX|RVU ! iXOfgi@WH! iVQ._^RfgQ O P:QNRRVUQ mIXI_@Q:c AO BO e ! _@Y CO WHQQ'RzRVUQ iXIfgimZQ.c aL]bfzRVUQ e ] _@RY QABO c
Q:iV]b_@YTRgXIWQ e R A B ! e _@Y C ! fgQ.ck\@QigRgX lbQ.m n BCO @ Y . Q @ _ V R U Q % \ . Q g f XIWQ'RQ.f P:QNR |XYCAO _e @Y [XY Z] RVUQ e fgQ e ]^a%RVUQ RfgX e _@'mOQ XY Z fg]:lbQ RVU Z| Re e BC + CA + AB = 1 |BCA | |CAB | |ABC | S [BCO] · [BCA ] + [CAO] · [CAB ] + [ABO] · [ABC ] = [ABC] ;b+:Ey}FOx+ >C35r68 9:8%;<=1:1N>? A"B.D.EGF3+
0
0
0
0
0
◦
◦
0
0
0
0
0
◦
0
◦
.......................................................... ......... ............... ........ ......... ........ ....... ........ .... .......... . . . . . . ...... . . . .. ..... ..... .. .... ...... .... ..... .. . . . . . . ... .... .... ... .. . . . . .... .. .... .. .. . . ... . . . ... .... ... .. ... . . . .... ... ... . .. . . . . ... .... .. .. .. . . . .. . .... ... . .. .. . . ... . .... .. ... .. . .. ... . .... .. .. .. . . ... . .... ... ... .. . ... . .. .... ... . ... .. .... .... ... ... . . . ... . . . . . . . . . . . ........................... .......................................... ..... . . ... . . .. . . . . . . . . ......... ... ... . ... .... ... .......... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .......... ....... . .... . . .. ........... . . . .. . . . . . . . . . . . . . ....... .. ..... . .... . .. ..... . . . . . . . . . . . . . . . . . . . ....... .. .. ... . ....... ............. .. ..... ... ... ....... .. ....... ...... .. ........... ............. .. .. ....... ...... ... ..... ....... ... ....... ...... .... ... ................ ............ .. .. . . . . . . . . . . ......................................................................................................................................................................... .. ... ..... .......... .. ....... .. .. ....... ... ... .. ... .. ................ ... .. .... ... .... ........... . . . ... . . . .......... .. .... . ... . ............ ... ....... .... .. ... ............. .. .............. . .. ........ ... .. ..... .. .... ............... .. ..... ............................................................. ... .. .. ..... .. ... .. . . . . . . . . .. ..... .. .. .. .. ...... .. . . ... . ... ..... .. .. ..... ... .. .. ... ..... .. .. .. ..... ... .. .. .. ..... .. ... .. ... . . . . . ..... .. ... .. ... ..... .... ..... .... .... ...... .... ..... ... .. .... .... ..... .. .. ... ..... . . . . . . . . . . . . ..... .............. ....... ........ ........ .......... ........ 0 ......... .............. ................................................
2B
◦
0
2
ρ sin
0
0
0
0
0
0
0
0
0
0
D BC
2ρ sin 2A
sin 2A
R
#0@EH&)(.2B]+6[`^9 $1" {l$'(.(+$1"VEv-IJ>J>01*'-76G240G(MsR $ A>619 ? * 6@AF2X]+aD`4Kz2H;)0 ]"F8-.6G2H"¡ $'-7AC{%:4& Z8J=-I9 AF0@6DE¤*'-I6@- ? ? (I-761019 ** 2C$¢Ks2HAC;)0@0E2C-y J>0G($ M19`B-7E<J=-7* ?)0 (.-+08E32H; 6G$'2 { ; 4 : K)$ A R ]` 0V9 $D2C0z2H; 6G2 R !9)Eb08AF2C-I9 O32H;)0y'6@(.&L0%$'{ {+AC$ J¥] `T-7952H;'-IEB(76@EH2v01°'& 682C-7$ 9>O[-y 08Ef& E R ªB{:4$'& AFEb0'K2H;)05619)68(+$)O8$'& Ez01°'& 682C-7$ 9)E;)$'(+*Q{l$ Af2HK1;)"%0=0z$DL2H9 ;)*08A}2H; ? 68682-7AFEz$'{^61AC0861E8RZ8-79 :40 |BCA0 |
=
=
2ρ(sin 2A + sin 2B + sin 2C)
sin 2A + sin 2B + sin 2C BC
4ABC 4A0 BC 2R sin A = 2ρ sin 2A
R
R
ρ =
1 BO 2 2
[BCO] · [BCA0 ] =
ρ
2 cos A
· BC cos A
1 BC 2
· BA0 sin 2B
= R sin A cos A · 2ρ2 sin 2A sin 2B sin 2C
ρ
[BCO] · [BCA0 ] =
1 4 R 2
sin 2A sin 2B sin 2C tan A
tan A + tan B + tan C = tan A tan B tan C X [BCO] · [BCA0 ] = 21 R4 sin 2A sin 2B sin 2C(tan A + tan B + tan C)
=
D
1 4 R sin 2A sin 2B 2 2
R
sin 2C tan A tan B tan C
= (2R sin A sin B sin C)2 = [ABC]2
D
I TZG/TZGJI {JR d ~ r F g g h F [ v X j P b NHM {ORZW TUM_L_ryG DF SOWS URJ{ GJbYF SOWS URJ{ GJb ^ GOTNHM S/S Nd k ZR W Ra G { M N SZu Ji k v X] k g g$ b XWS N d R x WS M W M R b? ^*u Q [ 'k g$ U v i FJb W T4KJI M N R N ` i b ^ M_GJI S a r l N S TUMAK l bj N*NHT ~sW K A
bH KJTZL W M S4u Ji g*[ F k [ i5F ^ g*X b ^ S rOTUM_d R b v *b FZ u x g g* ] mi N'M {JRZW TUMBL rJb [OS w M WS d S b b F u[ j ski bx GJI
c G lml KJNHM_L_r GJI I RaHR bQ7M N L RZW k S'{OR N'b [ b F u 'jU i v b G l S P N R TZL M b G lmS N'M SZu S NdmL q R V W G4V GOT RZWZ]
j
D
# b!# %$'&)(%*,+.-/+0'12435768 9:8#;:<=11'>@? ACB.D.EGF3+
√
3 2
A ....... ....... .. ... ... ... .... .... .... . . . . . ... .... .... ... .... ... .. ... .. ... .... ... . . ... . ... .. . . .. . . . .. ............ . ...... . . . . ......... ......... .... . . . . . . . . . . . . . . . ........ .. ......... ... .. . . . . . . . . . . . . ......... ... .... ... .................. .... ......... .... .............. ... ... ........ ......... . . . . . . . . . .. . . . . ........ .. ..... . . . . . . . . . . . . . . . . . . ........ ..... . ... ............... . . . . . . . . ........... ........... . . . . . . . .............................................................................................................................................
F
E
G
B
D
C
DD
F@0G0V2 E424ACa 0@9 03O2H2H;);)0%089,:40@2H9';)240AC$'? -7AC*$)RNa' (+08E<J -I9 O {+AC¹ $ J?L$'(. (+-.$ {9 -.2C& $ EG1 -IS{#;)6@019 $ *>AC08$ J9 (K1M,"%-.0V{DO8R 0G2 G 1 2
BE 2 =
c2 + a2 −
2 1 b 2
1 2 c 2
+ 12 a2 − 14 b2 =
1 4
S#;)08AC08{l$ AC0'KG"%0; 6[y 0 R $1"2H;)0z{l$'(I(7$1"-79 O3E42C6G2408JQ089'2CEf61AC0V01°'&)-y'68(+089'2 q BG · BE =
1 2 c 2
BF · BA =
° & 61*)AF-I(I682C0@AF68(
• ∠BEC = ∠AF C •
r
1 2 c 2
=
1 (2c2 6
2 BE 2 3
R
S#;'& E[K
1 6
2c2 + 2a2 − b2
R $D2C0}68(7E$32H; 6G2
R
r
DK 2H; 682-IEGK r ]7& EH-79 O ¹ ? $'(I(7$ 9 -I& E[1S#;)0@$ AC0@J` r
+ 2a2 − b2
• 2AD 2 = 2a2 − 21 a2 √ • AD = 3/2 a
=
2c2 + 2a2 − b2
-IE :M :4(.-+: r
AEGF
• BG · BE = BF · BA •
=
b2 + c2 = 2a2
-I{#6@9 *>$ 9 (M5-.{
R
√ ∠BEC = AF C AD = 3/2 BC I TZG/TZGJI {JR d ~ r F g g h F [ v X j P b NHM {ORZW TUM_L_ryG DF SOWS URJ{ GJbYF SOWS URJ{ GJb ^ GOTNHM S/S Nd k RZW Ra G { M N SZu wyj k g*[ ^ Q0j ['[Hg b G*K R NHb WS N 6R,uf kjAx 'j_FJb\w M TTZG*K W MoFHG*K'L_q RZW N?F'L S L R N'M {JRZW TUMBL rJb JG4V$I M N'b*w ib FZ u$ k0 5[Hg F ] n jAwyj v}v j g bJN aHR I_GQF'L S L R NHM {ORZW TUM_L_r4bHF S N0N a'R IBGJb zb F u j k n@j ] k g FsF4b S N q Ghx S IBGOT RZW d R TUb 5b FZ u Oik v X] k g g$ b XWS N d R x WS M W M R b ^*u Q [ 'k g$ U v i FJb W T4KJI M N R N a r l N S TUMBK l bQj N*NHT ~sW K A
b KJTZL W M S4u7Xg i 5g : ] s v n ['[ b k SJl d R NHb b F u g$g ` 0j [ bkoGON a GON a b qHM N SZu g n [ vD] w *bn G {ORZW bxJb F u Oi g$[ F k [ iF ^ g$X b ^ S rJTUMBd R b v *b FZ u ^ i ^ F g* g *b [HR GON'M S b v ,b F u 0j koj0wh g '7 5b SJ
sS L T4K
M M_L_r4b iT SO
OS b S V S N u [ j skD i b x GJI
D G lml KON'MBL r GJI I RaHR b }M NL RZW k S'{OR N'b [ b FZ u sj i v b G l S P N R TZL M b G lmS N'M SZu S NdmL q R V W G4V GOT RZWZ]
6D
D
D
D
D
D
,
@ ^!b .& (%*,+.-/+0'12 3 5bD.0,x|EG1#u *
FG+ D ,1 Adv>bx:1N*y0,xGFL5r+{`(EG+:x1'0NFOxOA +
a + (3 − x)b + xc
≥ 1
S ? fg].lbQRVUQ e S ]:lbQXO_@Q:b e mIXRGnwaL]bf 0 ≤ x ≤ 1 ;b+:Ey}FOx+ >CF+ e 3 5#FLK1T-*V+.-/+0N1'*gA/1g.-D>"2@123 5#FLK11:2xyF+ * P:QNR y = 3 − x UQ._ ! S n#cVRf e XZ.U RgaL]bfGq e fgYHi]bWH\bbR e RgXZ]b_c ! qQ%_@YwRVU e R ! a c A + =
a + by + cx
c + dy + ax
A+B
D"
"%;)0@AC0 6@9 * 6@AC0%*)08 9 01*=6@Ef{l$'(.(+$1"VEq A
B
A B
Z8-7J,-I(I61AF(MK
= = = =
K
a(c + dy + ax) + c(a + by + cx) (a2 + c2 )x + (ad + bc)y + 2ac (a + by + cx)(c + dy + ax) − A (ab + cd)xy − ac(1 − x2 ) + bdy 2
b d C + = b + cy + dx d + ay + bx C +D
K1"%;)08AC0
16 9 * S#;'& E[K'2H;)0VO[-y 089=-I9 01°'& 6@(.-.2M>:C619>a)03"VAF-.2F2C0@9=6@E C D
"%;'-+:<;5-7E 0@° &)- y'6@(70@9'2N2C$ Z8-79 :40 F@0G2
= (b2 + d2 )x + (ab + cd)y + 2bd = (ad + bc)xy − bd(1 − x2 ) + acy 2 A A+B
+
C C+D
619 *
AC ≥ BD
E = 2ac(x + 1) + (ad + bc)y
6@9 *
R
R
≥ 1
R
K ]`
F = 2bd(x + 1) + (ab + cd)y
R
K 619 * Rv¯0@9 :40 KD24$5? AC$8y 0%] `HK'-.2E<& Q:H08EB2C$,? AC$8y 0z2H; 682 "%0%; 6[y 0 R ]` KG619 * KK "%619 0}* ; 6[y WY0 6D/@-79 ON2H;)0fEH&La)E42C-.2C& 2C-7$ K 9)E R $1" K K K = (a2 + c2 − 2ac)x = (a − c)2 x ≥ 0 = (b2 + d2 − 2bd)y = (b − d)2 y ≥ 0 C≥F
A−E C −F
A≥E
EF ≥ BD
p = ac q = bd r = ad+bc s = ab+cd E = 2p(x+1)+ry F = 2q(x+1)+sy B = sxy −p(1−x2 )+qy 2 D = rxy − q(1 − x2 ) + py 2
@6 9 * ² M>*'-7AC0@:C2Ba & 2N240@* -7$'& EfJ>6@9 -7? &)(76G2C-+$ 9)EB& E<-I9 O EF BD
= 4pq(x + 1)2 + rsy 2 + 2(ps + qr)(x + 1)y = rsx2 y 2 + pq(1 − x2 )2 + pqy 4 − (pr + qs)(1 − x2 )xy + (ps + qr)xy 3 − (p2 + q 2 )(1 − x2 )y 2
2(x + 1)y − xy 3
619 *
y =3−x
K1"%0VO80G2
R
= y(2(x + 1) − x(3 − x)2 )
= y(2 − 7x + 6x2 − x3 ) = (1 − x)(2 − 5x + x2 )y
(1 − x2 )2 + y 4 − 4(x + 1)2 = y 4 + (x + 1)2 ((1 − x)2 − 4) = y 4 + (x + 1)2 (x2 − 2x − 3) = y 4 + (x + 1)2 (x + 1)(x − 3) = (3 − x)3 − (x + 1)3 y
R
= (26 − 30x + 6x2 − 2x3 )y = 2(13 − 15x + 3x2 − x3 )y = (1 − x)(13 − 2x + x2 )y
DA
¯V089 :H0'K ] `T-IE 01°'&)-y'68(+089'2N24$ (p2 + q 2 + rs)(1 − x2 )y 2 + (pr + qs)(1 − x2 )xy + (ps + qr)(1 − x)(2 − 5x + x2 )y ≥ 2pq(1 − x)(13 − 2x + x2 )y
R ]` O[- y 0@E !l{ K)01°'& 6@(.-.2M>;)$'(7*)EB-79>] `Rv!l{ KD2H;)089>* - y -7* -I9 OVa@M R ]A ` Z8-79 :40 @6 9 * Ks"%0X$)a'2F6@-I9 RTS;'& EGK Rw0>6@(IEb$x; 6[y 0 K K @6 9 * R Us$ 9)E01°'&L0@9'2C( M#K x 6= 1
x=1
(1 − x)y
(p2 + q 2 + rs)(1 + x)y + (pr + qs)(1 + x)x + (ps + qr)(2 − 5x + x2 ) ≥ 2pq(1 − x)(13 − 2x + x2 ) √ 0 ≤ x ≤ 21 5 − 17 2 − 5x + x2 ≥ 0 √ √ r = ad + bc ≥ 2 pq s = ab + cd ≥ pq p2 + q 2 + rs pr + qs ps + qr
≥ 2pq + 4pq = 6pq √ ≥ 2(p + q) pq ≥ 4pq √ ≥ 2(p + q) pq ≥ 4pq
(p2 + q 2 + rs)(1 + x)y + (pr + qs)(1 + x)x + (ps + qr)(2 − 5x + x2 ) ≥ 6pq(1 + x)(3 − x) + 4pq(1 + x)x + 4pq(2 − 5x + x2 ) = 2pq(9 + 6x − 3x2 + 2x + 2x2 + 4 − 10x + 2x2 )
K 0@E42C61a (.!+-72E;'{l$'-79 (I(7O$1"V] A Ev`R 2H; 6G2B] `HK 619 *>;)089 :H0%] `HK);)$'(7*#K):4$ J>? (+0[2C-79 O2H;)0? AC$)$'{CR = 2pq(13 − 2x + x2 )
GONHI r ] )
D
I TZGYTZGJI {JR d ~ r [ sk g* 6 v i F4b
W T4KJI M N R N a r l N S TUMBK l b0j_NHN*T ~sW K
b5QKOTZL W M S
(
V SJW L
v G(TZGJIAK'L M_GON LBGmV SOW L S T WRO6R M {JR d ] JqKOTUbM_L WRZlmS M N*TDG4V R N ] 4q R V W G4V GOT RZW Tq G R d √ b ~ KHL M ToL W K R GON'I rDM G W L_q S L bM N S' L b L_q R M N R K S I M_L_r ? EF ≥ BD x = 1 0 ≤ x ≤ 1 5 − 17 G lml:R N L R dDL_q S L LBG}Tq G AC ≥ BD AG W 0 ≤ x ≤ 1 M T0dOM K4IBL RJ{JR NM NYL_q R}BS 2T R d = 0 ] TUM N a7S G l VK'L RZW bHq R q S T {JRZW M R d}L_q R L W K'L_qG L q R5a M {JR N7M N R K S I MBL r 0q R N 1 5− √17 < x < 1 AG WlmS N*r l M I I M_GON*T G AS T R TM N 0qHM q b b b SJWR V GOTUM_L M {OR'] S N G*KOTV W G4V GOT 2R dT RJ{ORZWS I WR I S L R d7V W G ~ I RZl T ]
a b c d
D
z r : @ ^! .& (%*V+.-/+0N1:23 5 xZK`DN EL5 o1N> 1'A "*D:0'+b'A`/+
e S
n k+1 X X
k=1 i=1
n k+1 X X
k=1 i=1
2 k 2k i−1 k 22n+1 −2 = 2n + 1 2k 2k 2 2i − 2 (2i − 1) n
2 2n + 2 k 2k (2n + 3) i−1 k n+1 = −3 2k 22n+1 22k 2i − 2 i
D"D
I 9
]+6[`v $ A
Q0 1 0
K)* 0GL9 0
8Q>?> 5 ?8Q B,#*;#*1
619 *
|t| < 1 arcsin t f (t) = √ 1 − t2
g(t) =
!+2B-7E0861EFM,2C$5y 0@AF-I{.M,2H; 682 a)$D2H; 6@9 * ? AC$)a'(+08J q 619 * E<-I9 O2H;)0%a'-79 $ J,-768(E0@AF-+08E f (t)
dy (1 − t ) − ty = 1 dt 2
1 arcsin(2x) = 2
619 *>;)089 :H0'K
1 f (2x) 2
ªf9,2H;)0V$D2H;)0@Av; 619 *K
=
=
x 0
7# c- 5) # 1
k=0
g(t)
R 1 }
R E<682C-IE<{.M,2H;)0V{l$'(I(7$1"-79 Oz-I9 -.2C-I6@( y'68(I&L0 RB¯0@9 :40 K R K1"%0VO80G2 K ∞ X
22k t2k+1 2k (2k + 1) k
y(0) = 0
√
Z
1 = 1 − 4x2
f (t) = g(t)
∞ P
j=0
2j 2j j x
∞ X dz 2j 2j+1 1 x = √ 2j + 1 j 1 − 4z 2 j=0
∞ X 2j
∞ X
! 2j 1 x2j+1 x2j 2j + 1 j j j=0 j=0 ! ∞ k X X 1 2j 2k − 2j x2k+1 2j + 1 j k − j k=0 j=0 !
R
]`
R ]` Z8Us-7$ 9 J=:40 ? 6@AC-I9 OV:4$)0Q:4-+089'2CE[K1"%K#02H;) 0=9 *5?L$12H"%; 6G082 AEb08AC-70@E3-I9x] `f619 *w] f` J=& E423a 0,-7* 089'2C-+:C6@(R 1 2
1 f (2x) 2
k X
j=0
=
g(2x) =
∞ X
k=0
1 g(2x) 2
24k x2k+1 2k (2k + 1) k
1 2j 2k − 2j 22k (2j + 1) j k−j
=
=
² M52C08(70@E:H$ ? -79 ODK@"%0%$)aD2C68-79 n X
k X
k=1 j=0
1 22k (2j
+ 1)
2j j
2k − 2j k−j
=
=
"%;'-+:<;5-7E 0@° &)- y'6@(70@9'2N2C$2H;)0%O[- y 0@9,-7* 089'2C-.2MR
22k 2k (2k + 1) k n X
22k+1 22k−1 − 2k + 1 2k − 1 k k−1
2k+1
R
2k−1
2 2 2k + 1 − 2k − 1 k=1 k k−1
22k+1 −2 2n + 1 n
K
D1
]la1`f² M52H;)0Va -I9 $ J=-I6@(LE0@AF-+08E ¯V089 :H0'K ∞ X
k=0
1−
k X
∞ P 1 = √ 1 − 4x j=0
2j j j x
K1"%0VO80G2
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1
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xk j + 1 j k − j j=0 ∞ ∞ X X 1 1 2k + 1 k 1 2j j−1 = x = x −1 = √ 2x 2 j k 1 − 4x k=0 j=0
R
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=
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−
2k (2k + 1) k 22n−1
R
22k (j + 1) j k−j 2k + 2 2k (2k + 1) k n X (2k + 3) k + 1 = − 22k+1 22n−1 k=1
k=1 j=0
(2n + 3)
2n + 2 n+1
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D
D
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la
b
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b `a
`b
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h
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h cos α
h sin 2α
α
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d
d
c
r
r
r
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... ....... .. ... ... .. .. ... .. ..... ...... . .... .. .. .... .. ... ... ... ..... .... .. .. .... .. ... . .... .. .. . .... ... .. ... . . ... .... .. . ... .... .. . . .... .. .. . . ... ................................ . . . . .. . . . . . . . ......... .. ..... ......... ..... . .. .... ... .. . . ... . ....... ... ....................... ......... .............. .... ... ...... . 2 ... ... .. .... ... . ...................................... . . .. . . . . ... ... 1 ..... .. ...... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ........ .. .... . . . . .... . .... .... . .. ... . . . . . ... ... .. ..... .. ...... .... .... ..... .. ...... ...... .. . ........................................................................................................................................................................................................
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r
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d = AHB 0
c +h−c 2
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R K1"%0V; 6Gy 0
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C
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h = AH c0 = BH
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b0 = CH
h − r 1 − r2 =
b+c−a = d 2
d
∠A
AI
d r ABC
r
A
d
0
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• ∠A
h − r 1 − r2 < r
d
h − r 1 − r2 = r
d=r
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r + r1 + r2 ≤ h sin A
tan
h − r1 − r2 = d = r cot r
A r = 2 d
A 2
∠A
D
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a
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−
b2
+
c2
=
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c
A
4a4 + 9b2 c2 − 16m2b m2c
mb
√ mc = 12 2a2 + 2b2 − c2
=
1 2
√ 2a2 − b2 + 2c2
4a4 + 9b2 c2 − 16m2b m2c = 4a4 + 9b2 c2 − (2a2 − b2 + 2c2 )(2a2 + 2b2 − c2 ) = 4a4 + 9b2 c2 − 4a4 − 2a2 (2b2 − c2 ) + (2c2 − b2 ) + (b2 − 2c2 )(2b2 − c2 ) = −2a (b + c ) + 9b2 c2 + (2b4 + 2c4 − 5b2 c2 ) = −2a2 (b2 + c2 ) + 2(b2 + c2 )2 = 2(b2 + c2 )(b2 + c2 − a2 ) 2
2
2
R E
4a4 + 9b2 c2 − 16m2b m2c ∠BAC
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k=1
rk
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k
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k
k
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k
ak
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k=1
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k=1
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k=1
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Rk
2 sin Bk
k=1
!n1
n Y ck
+
k=1
n Y
+
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k=1
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n
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k=1
sin A + sin B + sin C ≤ 3 sin n Y ak R k k=1
!n1
n Y
k=1
+
ak
!n1
n Y bk R k k=1
+
n Y
k=1
!n1
bk
√ sin A + sin B + sin C ≤ 3 3/2 A B C
1 (A 3
n Y ck R k k=1
+
!n1
[0, π] √ + B + C) = 3 sin(π/3) = 3 3/2
+
n Y
k=1
!n1
ck
!n1
2 ≤ n
≥ 3
√ ! √ 3 3 n· = 3 3 2
n Y
k=1
ak b k c k
1 !3n
s
rs
p
s(s − a)(s − b)(s − c) s (s − a)(s − b)(s − c) r = s s s 3 1 (s − a) + (s − b) + (s − c) 1 s 3 s ≤ = = √ s 3 s 3 3 3 √ s ≥ 3 3r abc = 4Rrs R ≥ 2r
R 6@]\¯(IEb0@$5AC0=; 6G"%y 00Q; 6Gy 0>& Eb0@*w2H;)610 9 ¹* W B B W !]9 '0@&)° (7& 0@A68 (IE-2\!Mx9 0@6D° OG68& -7689(I-R 2\`¢M1`SRN;'Us& $ E J>a -I9 -79 O36@(.( 2HR ;)0@E00
' AC0@EH&)(.2FEGK1"%0V$)a'2F6@-I9 n Y
ak
k=1
!n1
√ √ abc ≥ 4(2r)r(3 3r) = (2 3r)3 n Y
+
bk
k=1
!n1
n Y
+
ck
k=1
!n1
K1"%;'-+:<;%"%0z& Eb0z-I9>] ` q n Y
≥ 3
√
(2 3rk )
k=1 n Y
√ = 6 3
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k=1
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rk
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b! :& (%*V+.-/+0N1:235 + K@D:
0
(M M 0 )2
= M A2
[BM 0 C]
+ M B2
[CM 0 A]
+ M C2
[AM 0 B]
[ABC] [ABC] [ABC] 0 0 0 0 [BM C][CM A] 2 [CM A][AM B] − AB 2 + BC [ABC]2 [ABC]2 [AM 0 B][BM 0 C] + CA2 [ABC]2
u"+
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0
0
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[ABC]
0
[ABC]
0
[ABC]
; S#;)089
−−−→0 −−→ −−→ −−→ M M = αM A + β M B + γ M C
(M M 0 )2
Z8-79 :40
K'619 *"%0V; 6Gy 0
−−−→ −−−→ = MM0 · MM0 = α 2 M A2 + β 2 M B 2 + γ 2 M C 2 −−→ −−→ −−→ −−→ −−→ −−→ + 2αβ M A · M B + 2βγ M B · M C + 2γαM C · M A = α 2 M A2 + β 2 M B 2 + γ 2 M C 2 + αβ(M A2 + M B 2 − AB 2 ) + βγ(M B 2 + M C 2 − BC 2 ) + γα(M C 2 + M A2 − CA2 ) = (α + β + γ)(αM A2 + βM B 2 + γM C 2 ) − αβAB 2 − βγBC 2 − γαCA2 α+β+γ =1
R
K1"%0%:4$ 9 :4(I&L*)0z2H; 682
(M M 0 )2 = αM A2 + βM B 2 + γM C 2 − (αβAB 2 + βγBC 2 + γαCA2 )
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T
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Q e c im e XIWQ:Y m −R < 0 O A = 90 zQ.fgQ XOc YbX e WHQNRgQ:fkRVU^@c ! m = R ! e _@Y m − R = 0 BC e
a
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B0C 0 OM
A
B0
A
C0 ∠M OA > ∠OM A ma − R > 0 A B0 ma − R = 0 B0B C 0C ma − R < 0
C0 A
A
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