• • • • • •
28th
a
−
a>b ab 1 (ab + 1)2 + (a
b ab + 1
a
−b
a+b 2
− b)
n n 2 (n2 3
W 1 W 1 D
W 2
D
W 2 AB
P
− n)
AB W 2
AD
P A
C
B
M
BC ∠DP M = ∠BDC
P (x) = ax3 + bx2 + cx + d
{
}
{| | |
|}
min d, b + d > max c , a + c
−
P (x) = 0
[ 1, 1] = 60◦ E BE = CF = BC E K
ABC AB EF K
AC
∠A
n
(n
− 1)
2
F ACE ∠BAC
P (x, y) x Q(x, y) P (x, y) = 3x4 y
−
P (x, y) 2x y + 5xy + x 5 Q(x, y) = 3x4 y x1 x2 y1 y2 2 3
2
−
y 2 3
− 2x y
Q(x1 , y1 ) > 0 , Q(x2 , y2 ) < 0
{(x, y)|P (x, y) = 0} S
S M
M (2a + 1)
(2b + 1)
× (2c + 1)
a b
2a + 1
2b + 1
A
c
2c + 1
n
A A n
A A
A
×
S x, y
∈ S
S S S
x
y
S S
n
•
2n 1 x j +2n −1 = x j
−
j
•
0
2n
≤t
1
x1 , x2 ,...
−1
< t2 < ... < tk < n
× x × ... × x s<2 −1
x j +n = x j+t
j +t2
1
n
2 n −1
xixi+s =
i=1
• • • •
(0, 1)
−
(1, 0) ( 1, 0)
xy yz
−
(0, 1)
zx
−1
j +tk
j
S
P (S )
n
S f : P (S )
• •
A
⊆ S
A, B
−→ N
f (A) = f (S
⊆ S
− A)
max(f (A), f (B))
≥ f (A ∪ B)
f
n n
2
n +1
2
k +1
2
n +1
ABC BC
D
E
L K AT BC
LK
M C M E
MD T A = T M
||
B AB T
AC
n>2 an
a1 a2
∈N a1 a2 , a2 a3 ,...,an a1 n>1
n
F
n
F
m
F m+
a a 2
≤n N
a,b,c
• •
b, c
A
gcd(a,b,c) = 1
∈A
∈A
a
∈A
a = b, c
f : R x, y
∈R
F
|
a bc
−→ R
f (x + f (x) + 2f (y)) = f (2x) + f (2y) OB = OC
OB C ω
O A AC H ω1 J I
OB
B
ω1
ABC
ω2 K OB C P
h1 +
h1 , h2 , h3
ω ABC
AB
P
h2 =
h3
C ω ∠KJH
p
≤
k p f (x) x f (x) A0 (x), A1 (x),...,Ak (x)
pk
k
f (x) =
(x p
i=0
i k−i
− x) p
Ai (x)
k >p n l n B A A
A
∩B = ∅ l
B
l n+1 k
(a2 + 1)(b2 + 1)(c2 + 1) +
∈R
+ (c2 + 1)(d2 + 1)(a2 + 1) +
(b2 + 1)(c2 + 1)(d2 + 1) (d2 + 1)(a2 + 1)(b2 + 1)
≥ 2(ab + bc + cd + da + ac + bd) − k A B
AB
a,b,c,d
C
ABC P A = P A P B = P B
BC AC
P P P C = P C
P P f : N af (a) + bf (b) + 2ab f (a) = a
−→ N
a, b a
∈N
∈N
(a + b)2 + (ab
2
− 1)
= a2 b2 + a2 + b2 + 1 =
2
⇒c
= (a2 + 1)(b2 + 1)
(a2 + 1, b2 + 1) = 1 p p a2 + 1, p b2 + 1 = p a2
|
|
⇒ | −b p|ab − b p|b + 1 (ab + 1, a − b) = 1 p|a + b = p a
| −b
p a
2
⇒ | −b p|a + b p|ab + 1
2
a2 + 1
2
b2 + 1
a2 + 1 A
B
C
d ABC
C AB
2 d
A
2
B
T
≤ × n
T
2
3
4
⇒ T ≤ 23 × n(n − 1)
=
PD AB ∠DP M = ∠BP C AB NA2 = ND
∠BDC = ∠BP C
N
∠DP B
× NP = NB
=
∠M P C
2
AP = ∠P AB ∠P BC = ∠P DC = 2
PB = ∠P BA ∠P CB = 2 P CB P BA P M ∠DP B = ∠N P B = ∠MP C
d
AB
d
AB 2
P N
N
|α| ≤ 1 P (α) = 0 =⇒ aα + bα + cα + d = 0 =⇒ aα + cα = −(bα + d) =⇒ |α||aα + c| = |bα + d| =⇒ |aα + c| ≥ |bα + d| (∗) [0, 1] f (x) = |ax + c| f (x) ≤ max{|c|, |a + c|} |aα + c| ≤ max{|c|, |a + c|} α
P (x)
3
2
2
3
2
2
2
2
2
b+d
d 0 bx + d
|
≤x≤1 |
bx + d > 0 g(x) =
2
|bα + d| ≥ min{|d|, |b + d|} (∗) max{|c|, |a + c| } ≥ |aα + c| ≥ |bα + d| ≥ min{|d|, |b + d|} ≥ min{d, b + d} 2
2
{| | |
|} ≥ min{d, b + d}
max c , a + c
α T BC = BE
BF ∠BC E =
CE ∠ABC
∠CBF
2
∠CT F = ∠BC E + ∠CBF
∠BC E + ∠BE C = ∠ABC
=
=
∠ABC + ∠ACB
2
∠ACB
2
= 60◦
ABTC ∠EB F = ∠ACE = ∠AKE ∠ABF = 180 ∠EB F = 180 ∠AKE = ∠AKF ABKF ∠EB K = ∠CF K ∠BE K = ∠F CK BE = F C KB E KF C KC = KE = KC = KE AK ∠BAC
−
−
⇒
≤ i ≤ n) | |≤
Ai (2 Ai
n(n−1) i(i−1)
Ai
i
Ai
|A | ≤ (( )) = n 2
i
i
2
m
Ai n(n−1) i(i−1)
| |
| |
m = A2 + ... + An
≤ n(n − 1)( 2(2 1− 1) + ... + n(n1− 1) ) = n(n − 1)(1 − n1 ) = (n − 1)
2
(x, y) (0, 0) (x1 , y1 ) (x, y) Q
•
Q(x, y) = 0
Q(x1 , y1 ) (x, y)
(x2 , y2 )
•
(x2 , y2 )
k
t >0
t Q(x, y) t>0
•
Q(x2 , y2 )
k = deg(Q) Q(tx,ty) Q(x, y)
Q k
t>0
(s1 , s2 )
Q(tx,ty) =
(t1 , t2 )
∈R (x, y) ∈ R M
2
A > S
Q(x, y) Q(x1 , y1 )
P (Ax,Ay)
+
Q(x, y) = (0, 0) Q(x, y) degQ = k degQi = i (0 i
S
∈R
+
≤ ≤ k − 1)
Q0 ,...,Qk−2 , Qk−1
P = Q + Qk−1 + Qk−2 + ... + Q0 L(A) = P (Ax, Ay) F (A) = Q(Ax, Ay) Qi
F i (A) = Qi (Ax, Ay)
Q
L(A) = P (Ax,Ay) = Ak Q(x, y) + Ak−1 Qk−1 (x, y) + ... + Q0 (x, y)
≤ ≤ k − 1)
Qi (x, y) (1 i L(A) Q(x, y) = 0
Q(x, y)
A k
A
Q(x, y)
→ +∞
(x1 , y1 ) (l1 , l2 ) (x2 , y2 ) (m1 , m2 )
P (l1 , l2 ) > 0, P (m1 , m2 ) < 0 M
x = ay + b P
P y = ax + b
∈R
+
(2a +1, 2a+1) a (2c + 1, 2c + 1)
c
b (2b + 1, 2b + 1) (x, y) (x + 2a + 1, y + 2a + 1) (x + 2b + 1, y + 2b + 1) (x + 2c + 1, y + 2c + 1) (x + 1, y + 1) 2a + 1 r(2a + 1) + s(2c + 1) = 1
∈Z
r, s
2c+ 1
(x + 1, y + 1) = (x + r(2a + 1) + s(2c + 1), y + r(2a + 1) + s(2c + 1)) (0, 0)
(x, y) a
x +y a+b
b
a+b+1
(2a + 1, 2b + 1, 2c + 1) = 1 A b
2
1
R S R A + an Bi = A + ai xij = ai + a j Bi j1 = j i1 = i ai ai a j a j
⊆
S + b = x + b x S A + a1 A + a2 n A (1 i n) a j + ai Bi ai + a j B j B j x Bi B j x = xij ai +a j = x = a j +ai x xij = a j a j = ai ai Bi B j
{a , a ,...,a }
2
∈
∈ ∩
1
1
2
≤ ≤
2
3
∈ ∩
1
1
∈ ∩B ∩B
x Bi i2 = i3
1
ai = ai
{
n
i2
i3
a1 ,...,an
∈
| ∈ } ∈
−
1
−
1
− ∩
ai +ai = ai +ai 1
2
1
3
a1 ,...,an
{ | ∈ S }
S.b = xb x i n)
≤
b
∈C
S
⊆C
ai .a j = a j .ai i1 i2 i3 = ai ai
ai ai 1
2
3
i4
ai1 ai3
4
=
Bi = A.ai , (1
∈B ∩B i
j
ai4 ai2
ai ai 1
ai ai O 2
3
i1 i2 i3
i4 Bi B j B1 ,...,Bn
4
∩
S
S ai x, y z
S x,y,z
S
S
x , y , z z
S
d
∈ S
k
∈N
S t
u k
×t
(t + 2d)
× (t + 2d)
× ≥
t t (t+1) u k (t + 2d) (t + 2d)
×
2
≤
u
≤
(t+2d+1)2
(t + 1)2 t2
k
us
≤ ≤
s us
2
≥t
≤ (t + 2d)
(t + 2d + 1)2 (t + 2d)2
≤ ≤
uk us
S 2
⇒
(t + 1)2 (t + 2d)2
2
k s
≤ ≤
(t + 2d + 1)2 t2
2
1) ⇒ lim (t(t++2d) ≤ ks ≤ lim (t + 2dt + 1) ⇒ k = s 2
t→∞
···
2
t→∞
[x]i = xi x = (x1 , , xn ) n k 0 < i1 < i2 < < ik n [x]i [x]i [x]ik = 1 (1, 1, X = X , 1)
··· − {−1, 1}
1
2
x∈X
n
[x]i = [y]i 1
2n
n s = i1
x
−1 1 ≤ i ≤ 2 −1
(1, 1,
···
(xi xi+1
, 1)
···
xi = xi−n+t , xi−n xi+s
···
×···×
xi−1 , xi−n+tk xi
··· , x
xi−1 ,
··· , x
i−n
xi+s−1
i+s−n
xi+s n 2
0 < u1 < u2 <
···
n
xi−1 , xi+s−2 ,
2
··· 1
··· , x
1
n
xi = x2n −1+i xi xi+s
xi+s−1 , , xi+s−n−1
n y [x]i [x]i [x]ik = 0
x∈X
n
xi+n−1 )
X
n
[x]s = [y]s
1
··· ≤ − { ··· }
xl = 1 < u p n
···
x p = xl−u
l
≤ ×x
1
l−u2
∈Z
l
i−n
∈Z
×···×x
l−up
2n −1
xi xi+2
i=1
i
n
xi
n+1 n
i
n + 1 + 2n
−2
xi xi+s
n
n
{−1, 1}
··· (1, 1,
, 1)
n
36 12 12
12
xy
xz 24 xz
xy
xy xy 12
xz
xz 24
−
−
− −
(x,y,z ) = (0, 1, 1), (0, 1, 1), (0, 1, 1), (0, 1, 1) 12
yz
yz
yz
12 24 + 12 = 36 B
A1 A2
Am
≤ max{f (A ), f (A ),...,f (A )} f (A ∩ B) ≤ max{f (A), f (B)} f (A ∩ B) = f ((A ∪ B) ) = f (A ∩ B ) ≤ max{f (A ), f (B )} = max{f (A), f (B)} f (B)
1
2
c
c
Am Am i m 1) i l = i
f : P ((S
c
c
A1 A2
Am−1
A j
∈ A ∪ A ∪ ... ∪ A {i} = A ∪ A ∪ ... ∪ A ∪ ... ∪ A a = {i, l} i
A1 A2 Am−1 l A1 A2 i i l
∈
S
c
f (A1 ) < f (A2 ) < ... < f (Am ) A1 A2 Am−1
∈ {} ≤ ≤ −
i Am (1 j
m
∪
1
2
2
2
m−1
m
m
l
− {i, l}) ∪ {a}) −→ N ∈ Domain(f ) A , A ,...,A m−1 ≤n−1 m≤n 1
1
f
m−1
f (A∆B)
≤ max{f (A), f (B)}
P (S )
∅
∆
{|
Bi = x f (x)
A
≤ i} (1 ≤ i ≤ m)
{1, 2,..,m} f | ||| ( )| |B | B B ... B x∈B ⇒x ∈B |B | ≥ 2 , |B | ≥ 2 , ..., |B | ≥ 2 =⇒ 2 ≥ 2 =⇒ m ≤ n P (S )
Bi
B1
P S
1
i
c
1
1
1
2
2
m
f
2
m
1
m
n
m
n f (A) := max min(A),min(Ac)
{
{ }
min X
}
X n n2 + 1 n2 + 1
k2 + 1
n2 + 1 k k2 + 1
|
k +1 n an = 22 + 1
(am , an ) = 1
k2 + 1 k2 + 1 n2 + 1
|
l2 + 1 k 2 + 1 2
= P (S )
an m = n
H
ABC T A AH T A ω A EH ( ∠M BE = 90 ∠C = ∠HAC = 2 MD LK T M ω
⊥
ω BM = EM
−
ω) M
⇒
M E
AH ∠M EB = ω
ω
T A2 = T M 2 =
⇒ T A = T M n = 2k a1 < a3 < ... < a2k−1 < a1
n a1 a2 < a2 a3 < ... < a2k−1 a2k < a2k a1 n
a1 ,...,a2k+1
b1 = x b1 b2 = 1, b2 b3 = 2,...,b2k+1 b1 = 2k + 1 1 3 b1 b2 = 1 b2 = b3 = 2x b4 = , 2x x
⇒
⇒
⇒
1×3×···×(2i−1) 1 x 2×4×···×(2i−2) 2×4×···×(2i) x 1×3×···×(2i−1)
×
b2i = b2i+1 =
2 1
∈R
+
···
≤i≤k ≤i≤k
b2k+1 b1 = 2k + 1
× 4 × · · · × (2k) x 1 × 3 × · · · × (2k − 1) 2
2
ai = bi
= 2k + 1
⇒x=
× × · · · ×
(2k + 1)!
1
3
(2k + 1) (2k)
×4×···× a a , a a , ··· , a a 2
1 2
2 3
n 1
× 4 × · · · × (2i) × x× (2k + 1)! = 2 ×···× (2i) (2i +1) ×···×(2k +1) a2 +1 = 1 × 3 × · · · × (2i − 1) 1 × 3 × · · · × (2i − 1) 1 × × (2k + 1)! = 1 × 3 ×···× (2i − 1) (2i) ×···× (2k) a2 = 2 × 4 × · · · × (2i − 2) x 2
i
i
ai
n = 2k + 1 i
F
m : yi = y j [x, y]
1, 2,...,n A = [x1 , y1 ], [x2 , y2 ], ..., [xm , ym] 1 i, j m : xi = x j xi = x j
{ ∀ ≤
≤
}
[i, j]
i, j
∈ {1, 2,...,n} ∀1 ≤ i, j ≤
X, Y
F
F
∈ − {x}
x / X
∈ − {y } F
y / Y [b, y]
x
∈ X − {x}
y
[x, y]
∈ Y − {y}
[x, a]
B = [x, y], C = [x, a], D = [b, y] B
⊆ C, B ⊆ D
B B, C
⊆B D ⊆ B C, B ⊆ D D ⊆ B, C ⊆ B C, D E ∈ A D
B
D
B D, B C B D
A1
F
⊆ C, B D
⊆ B ⊆ E
D
A2
F |A | + |A | ≥ a A A m + |A | ≤ n m+ ≤n A −{1} 1
2
1
|A| + |A | ≤ n m + |A | ≤ n |A | + |A | ≥ a |A | ≥ |A | ≥ 1∈ 1∈A /A a = 1, a b ∈ A, b b|a |A| ≥ 4 a, b ∈ A c ∈ A − {a, b} 1
1
1
2
a
1
a
2
2
2
2
a
2
1 < b < a
a b
|
ab
(a, c) = (a,b,c) = 1 A
−{a, b}
∈
d A (d, a) = 1
b ac b ac e A e ad
|
|
d ac
|
dc
| |
∈
|
∈
d e=b
b ad ac e = c (e,d,c) = e > 1
|
b ac
e
c = d
∈ A − {a, b} e|d|c e
c
|A| = 4
e A
d
− {a, b}
b|ac b|ad ⇒ b|a(b,c,d) = a ⇒ b|a ⇒ a = b b|ab |A| ≥ 4 {b, c} {d, e} a|bc,a|de n≥4
a
n
b,c,d,e
p
2
A
a
∈A
n
|
pd
>n pe
2
|
|
ed
|
|
p d
|
p (a,c,d)
pc b=d
| | |
a bc a be a ba (a,b,c) = 1 a bc b ac c = pr
|
|
⇒ |
a b(c,e,a) = b
|A| ≤ 3
|
c ab
|
ac
r q (a, b) = (a,b,c) = 1
(a, b = 1 c = ab r=r ( p, q, r) = 1
{
}
A = a,b,ab
a = pq
c ab
{
}
A = a,b,c q b pc
|
|
2
| ⇒ pr | pq.qr ⇒ r |q r
q = 1
a,b,c
|
|
b ac bc c c ab c = ab (r , q ) = 1 q>1 r r a = pq b = qr c = pr s ( p, q, r) s (a,b,c)
|
f (x)
OR t
x=y=t
a, b r r
|
|
(a, b) = 1
b = qr
|
|
{
A = pq, qr, rp
∈R
}
( p, q, r) = 1
f (t) = 0
⇒ f (t) = 2f (2t) ⇒ f (2t) = 0
(1)
(0) + 2f (y)) ⇒ f (f (0) + 2f (y)) = f (0) + f (2y) ⇒ f (f −f (t + 2f (y)) x = t ⇒ f (t + 2f (y)) = f (2y) = f (0) 2f (y) c∈R f (f (0) + c) − f (t + c) = f (0) x=0
(1)
c=t c=0 c = f (0) x = 0, y = 0
(1)
⇒ ⇒ ⇒ ⇒ (2)
f (f (0) + t) = = f (f (0)) f (2f (0)) f (3f (0))
f (0) f (0)
(2) (3)
= 2f (0) (4) = 2f (0) (5)
x = 0, y = f (0)
(3), (4)
(5)
⇒
f (3f (0)) = 3f (0) f ( 0) = 0 x= 0 f (2y) f (x + f (x) + 2f (y)) = f (2x) + f (2f (y))
f (2f (y)) = 2f (y)
R f (x + f (x) + y) = f (2x) + f (y) (6)
y=0 f (x + f (x)) = f (2x) (7) y = f (x + f (x)) = f (2x)
(6)
f (x + f (x) + y) = f (x + f (x) + f (x + f (x))) = f (2x + 2f (x)) = f (x + f (x) + (x + f (x))) = f (2x) + f (x + f (x)) = 2f (2x)
f (x + f (x) + y) = f (x + f (x) + f (2x)) = f (2x) + f (f (2x)) f (f (2x)) = f (2x)
f (2x)
∀a ∈ R : f (a) = a B C K P
CF K B
P
BB BC HN HC
CC
ω BE
N
E P
BC
F N
K B
BT Q
BK QB T N =1 . . K Q B T NB T N NB
=
CH HS
BK CH B T = . K Q HS QB C H
P BK CH C U = . K Q HS SC B T QB
=
H
B N
X Y
C U SC
BC AB
BE
K HC K P
N CF BC
P
∠CHN
= ∠CXN = XHCNI
B C
∠CIN
B P I Y K
J
∠J HA
= ∠J IC = ∠J Y H
J Y H AC
AC
H C
Y
Y
H
Y = Y
C H
J Y H ω
J Y H K XJ AB
Y
ω
ω
∠K J H = ∠ACI + ∠ABI =
180◦
− ∠BAC 2
= ∠BI C = ∠AHJ + ∠AK J AH AK J Y H K J X J Y H ω1 ◦ ◦ K = K ∠IJH = 180 ∠ICH = 180
−
∠XDP + ∠XEQ
=
DX
2
+
∠XF R ◦
− ∠XDP =
XE
2
F D+DX
60 ∠XDP = α ◦ ∠XF R = 60 + α
−
=
2
−
2
∠DOE
J Y H
K J X
K J X ω2 ∠IBK = ∠IJK
J = J IJ
= 60◦
DX
FD
= 2 = ∠F OD = ◦ ∠XEQ = 60 α 2
−
= (2r sin α)sin α = 2r sin2 α P X = XD sin α RX = XF sin ∠XF R = XF sin(60 + α) = 2r sin2 (60 + α) QX = XE sin ∠XEQ = XE sin(60 α) = 2r sin2 (60 α)
⇒
√
RX =
√
2r sin(60 + α) =
√
AX AX AX
X , B , C A B C
−
2r(sin α + sin(60
− α)) =
AEOD A X, B , C X
√
−
P X +
X
QX
A
√ XR = √ XQ+√ XP
√ XR XR = XR
=
√ XQ + √ XP
X
X
ABC
deg(Ai ) f (x) = (x p
k
≤ p −
k=1 1 i
≤ k−1
− x) A (x) + g(x), deg(g) ≤ kp − 1 k−1 g(x) f (x) = (x − x) A (x) + (x − x) A (x) + ··· + p A (x) ⇒ p|( x p− x ) A (x) + ( x p− x ) A (x) + ··· + A (x) p
k
h(x) =
k−1
p
k
p
k
k−1
p
k−2
k−1
xp −x p
Ai (x + ap) x a
0
k−2
(I)
0
h p
g(x + ap)
k−1
k−1
p
≡ (x + ap) p− (x + ap) ≡ x − px − ap ≡ h(x) − a ≡ A (x) (mod p)
|
p (h(x)
i
p
p yk−1 Ak−1 (x) +
|
x k 1 p (mod p) pBi(x) (III) (I) k >p
k−1
− a)
··· + A (x)
(mod p)
Ak−1 (x) +
···
(II)
0
y
− ≤ −1
p Bi (x) f (x) = (x p
i
p−1
Ai k−1
p−1
− x) − p
∀x, i : A (x) ≡ 0
(x p
− x)
Ai(x) = (III)
k−1
n l
G
G
n
l G
0 = yG =
{
l
yA + yA + ... + yAn n
A = A1 , A2 ,...,An
1
}
2
⇒y
A1
G
+ yA + ... + yAn = 0 2
n+1
G, G1 ,...,Gn
G
n
O
i i
A A1 ,...,Ai−1 , A , Ai+1 ,...,An OG O OG OG Ai OGi X = Ai , Ai , Ai ,...,Aik n+1 O O G Gi
{
1
= A +A +...+A n G n +A +...+A n −A i +A A i Gi = 1
2
2
n
G
⇒ −
O
Gi
OGi
{A
, Ai ,...,Aik Y = A X i1
}
2
1
Ai
2
−
}
O
i = Ai G
− A = 2 (A − O) i
n
i
n
O Ai G A1 G, A2 G, , An G
O
···
OG
O Gi G j GGi G j OAi A j OAi A j TG = n2 OG S TO
Gi
−GG −→ = −A−→O 2
GGi G j TO = n2 T S
2
i
n
i
n
T
Ai A j
1 2 GO GO/GT n/2 + 1 = = 2 = = GS GS/GT n /4 1 n/2 1 n 2
−
G
−
O
−
OG GS
S
O G
−2 n−2
⇒
Ai A j
≥ ≥
(a2 + 1)(b2 + 1)(c2 + 1) = (a2 + 1)(b2 + 1)(c2 + 1)
cyc
((a + b)2 + (ab (a + b)c + ab 1 2 ab 4 sym
a=b=c=d=
=
√ 3
−
−
2
− 1) )(c
2
+ 1)
k
2 n−2
−1
A B C
2
P
G B = P = B P P 1 P XP = 12 XP S T P 1 A A P
P 1
A B C
X
NT = 23x P 1 P
N S =
P 1 S + P 1 T 2x = 2 3
N
⇒ N P = x3 , N P = 4x3
1
N P P
P 1 P P P
P P 1
N −1
2
− A) − B) − C )
2 2 2
= (P = (P = (P
−A , B = −B , C = −C
2
2
+ P 2P P P
2
2
2
2
2
2
2
2
XP XP
P P
2
2
2
2
2
+ P 2P = =2 X
2 A
4
2
2 B
2
2 C
4
4
P (1) + A. P (2) + B. P (3) + C.
X
+P 2P 3
X
a=b=p
|
p p + f ( p) s
2
p
af (a) = p
2
− A ) ⇒ P + A − 2A. P = P + − B ) ⇒ P + B − 2B. P = P + P = P + − C ) ⇒ P + C − 2C. − B ) = (2P + P ).−→ (1), (2) ⇒ (A BA −→ − C ) = (2P + P ).− (2), (3) ⇒ (B CB −→ −CB −→ BA 3 4 3 4
2k−1
−GK −→
K
G = A
−1
2
P P
(P (P (P
1 2
x
G ABC P 1
1 4
|
ABC = 12
P 1 C C P
|
⇒
1
x = P P 1 P P
P 1 T = x3 P 1 S = x
1
s
∈ N, p s
|
p f ( p) p b = p2k
2 p( p + f ( p)) a N af (a) 2k−1 p af (a) k N 2k−1 2k 2k p s + p (f ( p ) + 2a)
∈
∈
p
≤ f ( p)
p
≤
p2k−1 (s + p(f ( p2k ) + 2a)) p af (a) p > f (1) p f ( p) 2 pf ( p) + f (1) + 2 p < ( pf ( p) + 2) f (1) + 2 p < 4 pf ( p) + 4 4 pf ( p) 4 p > 2 p + f (1) f ( p) p pf ( p)
≤
⇔ ≥ ≥
|
( pf ( p))2 < pf ( p) + f (1) + 2 p < ( pf ( p) + 2)2 pf ( p) + f (1) + 2 p = ( pf ( p) + 1)2
f (1) + 2 p = 2 pf ( p) + 1
|
p f ( p) k
≥2
f ( p) = k 2 p
pf ( p)
3 p > f (1) + 2 p = 2 pf ( p) + 1
f ( p) = p
k
∈N
4 p + 1
f (1)
p
f (1) = 1 p f ( p) = p N a f (a) = a b = p p 2 2 2 (a + p) = af (a) + p + 2ap [af (a) + p + 2ap] (a + p)2 = af (a) a2 2(a + p) 1 2 (a + p) 2(a + p) 1 af (a) a2 a f (a) = a
∈
−
|
− |≥
−
−
−
