P. Pooksombat [2011/10/14]
Version 1.0
ppooksombat.blogspot.com
1
1 :
2 : 1
3
, Contradiction Contrapositive
-
- Contradiction
()
- Contrapositive
A B ~B ~A
Without Loss of Generality (WLOG)
(
) WLOG
2
3 : 1
1. 1, 2, 3 3 + 12 + 2 + 3 = 0 1, 2, 3
= max{ 1 , 2 , 3 } < 1 + = 1,2,3
() z 3 = 12 + 2 + 3
||3 = |12 + 2 + 3|
, 12 + 2 + 3 1 2 + 2 + |3|
||3 1 2 + 2 + |3|
= max{ 1 , 2 , 3 } 1 , 2 , 3
||3 2 + +
||3 1 < 3, ||3 1 < { 2 + + 1}
2 + + 1 > 0 1 <
< 1 + #
2. , , , , > 0 + = + = =
(Contradiction) WLOG >
> >
> >
> >
+ > +
() >
>
= #
3
3. 1
A B, B C C A ( 3-cycle)
(Contrapositive) 3-cycle
3-cycle
By Contradiction, 3-cycle
M N k
WLOG M N
M k-1 N k
k N M N M
M k-1 N k
N M M
X X M, M N N X 3-cycle
#
( :
3-cycle
3-cycle
)
4
4. , , 0 + + = 3 + + = 0 4 < 0
, , 3 + + 2 + + + = 0
= = 3 32 , By Contradiction, 0 < 4
0
3 32 0
2( 3) 0
3 , , 3 + + 9 + + = 3
< 4
3 32 < 4
+ 1 2 2 < 0
< 1 , , < 1 + + < 3 + + = 3
4 < 0 #
5. , + + 1 | 2 2 + 2 1 + + 1
( )
By Contradiction, + + 1
2 2 + 2 1 = + 2 + 2 1
= + + 1 ( + 1) + 2
+ + 1 | 2
+ + 1 1
+ + 1 | + + 1 > | |
+ + 1 #
5
6. () [] ( () ) =
=
() []
, < < 0 > 0 () (, )
= 0 (, ) (, ) x
= 0
= 0 < 0 > 0
, WLOG > 0 x
By Contradiction, r = ( )
> 0 >
() () > 0 > ()
= #
( : WLOG > 0 < 0 )
7. , , 3 + 3 + 3 0
3 + 3 + 3 = + + 3 3 + + + + + 3
+ + = 0 3 + 3 + 3 = 3
By Contradiction, 3 + 3 + 3 = 0
+ + = 3 ( )3
0 = 3 ( )3
0 = ( )
= = =
3 + 3 + 3 0 #
6
4 : 2
2
Induction Strong Induction
- (Induction) ()
(1) ()
( + 1) ( (1)
P(2) (3) ... Induction
)
- (Strong Induction) Induction
(1), (2), ()
( + 1) ( (2) (3)
)
() 1
.
7
5 : 2
1. 1 + 2 + 3 + + = +1 2
n
(Induction) () 1 + 2 + 3 + + = +1 2
= 1,2,3,
(1) 1 = 1 2 2
() ( + 1)
1 + 2 + 3 + + = +1 2
+ 1 1 + 2 + 3 + + + + 1 = +1 2
+ + 1
1 + 2 + 3 + + + + 1 = +1 (+2)2
( + 1) () n
1 + 2 + 3 + + = +1 2
n #
(1) ?
(1) (2)
(1) (2) ?
() ( + 1)
(1) (2)
(3) ?
(2)
(3) (3)
(4)
8
2. cos cos n
(Strong Induction) () cos
= 2 = 1
cos 2 = 2 cos2 1 cos
= 3 , cos 3 = 4 cos3 3 cos cos
1 , (2), , () ( + 1)
cos + 1 = cos + = cos k cos sin sin
= cos k cos 12
(cos 1 cos + 1 )
cos( + 1) = 2 cos k cos cos 1
cos , cos 1 , cos cos( + 1)
cos n #
3. ! < 2 !2 2
() ! < 2 !2
(2) 2! < 4!2
2 2 < 6
() ! < 2 !
2
+ 1 ! < ( + 1) 2 !
2
+ 1 = 2 + 2 + 1 < 22 + 3 + 1 = 2+2 (2+1)2
+ 1 ! < 2+2 (2+1) 2 !
22=
2+2 !
2+1
( + 1) ! < 2 !2 2 #
9
6 :
1. 2 0, 1, , > 0 = 1,2, , 1
1 + ( + +1) = 1 +1 2
10
9. 1
12+
1
22+
1
32+ +
1
2 2
1
n
10. 3! | 3 ! 0
11.
12. () [] () = 0,1,2,
( : )
13. + 1
= 2 cos + 1
= 2 cos n
14. 1, 2, , = 1 2 2 2 2 + 1
( :
)
15. = + + 1 + 0 0 0 > 1 + 2 + +
16. P Q P Q
AB A, B P, Q
AB
17. = sin + sin , sine
curve 2 + 2
11
18. (Gamma Function)
= 1
0
1 ! = 1 ! > 0
( : (Integration by Parts), = )
19. cos21 sin21 /2
0=
2 +
(Beta Function)
B , = 1 1 1 1
0
B , = +
, > 0
( : , 0 )
( : = cos2 )
20. = ()
= 2
= = 2
= 2 =
() () = ()
() ( )
( )
= (()) = (()) ()
() = ()
21. 1 = 2/2 2 =
2/2 ,
= 12
(x2
2
2 )
12
22. sin + sin 2 + + sin =sin
2sin +1
2
sin2
23. cos + cos 2 + + cos =sin
2cos +1
2
sin2
24. (Bonus) 23 sin2 + sin2 2 + + sin2 cos2 +cos2 2 + + cos2
25. (Bonus) 1 1 1 ( 51
2 )
-
2011-10-15T02:04:09+0700Perathorn Pooksombat
