算幾不等式的證明(ii)與（iii - ntnu.edu.tw
TRANSCRIPT
Microsoft Word - 1701HPM
1998 10 5 5 http://math.ntnu.edu.tw/horng
(II)III
aaa n ...21 n naaa ...21 )
122 n 224 n 3n
328 n 7,6,5n
2 aa
321
4321
121 ... 1
0,0 xfx xf
1 1
1
0,0 )1(
1
nnn aaaaaat ..... 21
21 xaaa n ...21 0xf
1
1
2
3 1 x kx 23 )23(1 xxxf m mk ,
2
aaa n ...21 AM n naaa ...21 GM)
(George Pólya,1887 – 1985)
naa ,1 naa 1 M
nn aaMaaM 11 )(
)()( 11 2
naaa ,..., 21 n
M M GMAM 1a
na )(, 1 MaaM n
()
xexf x 1 x 1)( 1 xexf
1)( xexf 0)1(,01 ff Rxxf ,0)( xf
() 1x x xex 1
1x =
naaa ,..., 21 GMAM
0M 1 xex
n
1. PDF e-mail [email protected]
2. 3. e-mail [email protected]
4. http://math.ntnu.edu.tw/horng/letter/hpmletter.htm 5.
HPM ( Boston Consulting Group) Mainz
()
()
()
()
(1775) ()
(1830)
HPM
3
R 'r
2 2 ' 2 6r R
' 2R r (1)
Stewart 4
2 2 2( ) ( ' 1) 2( 1) ( ' 3)[( ') 2( ' 1)]R r r r r r r r (2)
(1)(2) 'r
2 2 2( ) (3 ) 2( 1) (5 )[( 2 ) 2(3 )]R r R r R r R R (3)
1 1 1 1
1 1 2
2
1
(4)(3)
4 3 22 7 4 11 6 0R R R R
3 17 3 3 17
1, , , 2 2 2
3 http://www.museum.city.ichinoseki.iwate.jp/icm/ 4 Stewart Matthew Stewart 1746 ABC BC a CA b AB c
D BC BD m CD n AD d 2 2 2mb nc a d mn
HPM
2 22 = 2 2 2( ) 2
2 22 0
2
( )
2 2 2 2r b x R (5)
Stewart
2 2 2( )( ) ( )( ) (2 )[( ) ( )( )]b r a x b x a r b r x a b b r b x (6)
1 1 1 1
r x a a
a r x
2 2 22 2 2 23 3 ( ) ( ) (2 ) ( )( )
rx x rx r Rr Rx r x R x R r R r x R x R r
r x r x r x
x R r
2 2 2
3 3 3
R Rr R Rr r
R r R rrx x R R r R r Rr
R Rrr x R r R rr R r
8 7 6 2 5 3 4 4 3 5 2 6
22 2 2
2 3 4 5 6 9R R r R r R r R r R r R r
R r R r
2 2 2
33 3 2 R Rr
r rrx r R r Rr rR r R Rrr x R r
r R r
22 2
9 12 10 4R r R r R r Rr r
R r
(8)
9 8 2 7 3 6 4 5 5 4 6 3 7 2 8 9 10
23 2 2
2 13 12 22 12 4 12 2 2R r R r R r R r R r R r R r R r Rr r
R r R r
R r x R r R r R r
2 22 2 2
3 R Rr R Rr
Rr R r R r R r Rr rR r R rRr Rx r x
R Rrr x R r R rr R r
6 2 5 2 4 4 3 5 2 6 7 8
22 2 2
9 6 5 4 3 2R r R r R r R r R r Rr r
R r R r
R r R r
(8)
9 8 2 7 3 6 4 5 5 4 6 3 7 2 8 9 10
23 2 2
2 13 12 22 12 4 12 2 2R r R r R r R r R r R r R r R r Rr r
R r R r
,a b x ,a b
( , , ) 0f x a b ab a b a b
http://www.museum.city.ichinoseki.iwate.jp/icm/
HPM
HPM
HPM
20
GöttingenFelix Klein, 1849-1925
Hermann Weyl, 1885-1955
Max Born, 1882-1970
Zahlbegriff und Quadratur des Kreises
HPM
HPM
log3
10
HPM
10:00
13:3-14:10 (
1998 10 5 5 http://math.ntnu.edu.tw/horng
(II)III
aaa n ...21 n naaa ...21 )
122 n 224 n 3n
328 n 7,6,5n
2 aa
321
4321
121 ... 1
0,0 xfx xf
1 1
1
0,0 )1(
1
nnn aaaaaat ..... 21
21 xaaa n ...21 0xf
1
1
2
3 1 x kx 23 )23(1 xxxf m mk ,
2
aaa n ...21 AM n naaa ...21 GM)
(George Pólya,1887 – 1985)
naa ,1 naa 1 M
nn aaMaaM 11 )(
)()( 11 2
naaa ,..., 21 n
M M GMAM 1a
na )(, 1 MaaM n
()
xexf x 1 x 1)( 1 xexf
1)( xexf 0)1(,01 ff Rxxf ,0)( xf
() 1x x xex 1
1x =
naaa ,..., 21 GMAM
0M 1 xex
n
1. PDF e-mail [email protected]
2. 3. e-mail [email protected]
4. http://math.ntnu.edu.tw/horng/letter/hpmletter.htm 5.
HPM ( Boston Consulting Group) Mainz
()
()
()
()
(1775) ()
(1830)
HPM
3
R 'r
2 2 ' 2 6r R
' 2R r (1)
Stewart 4
2 2 2( ) ( ' 1) 2( 1) ( ' 3)[( ') 2( ' 1)]R r r r r r r r (2)
(1)(2) 'r
2 2 2( ) (3 ) 2( 1) (5 )[( 2 ) 2(3 )]R r R r R r R R (3)
1 1 1 1
1 1 2
2
1
(4)(3)
4 3 22 7 4 11 6 0R R R R
3 17 3 3 17
1, , , 2 2 2
3 http://www.museum.city.ichinoseki.iwate.jp/icm/ 4 Stewart Matthew Stewart 1746 ABC BC a CA b AB c
D BC BD m CD n AD d 2 2 2mb nc a d mn
HPM
2 22 = 2 2 2( ) 2
2 22 0
2
( )
2 2 2 2r b x R (5)
Stewart
2 2 2( )( ) ( )( ) (2 )[( ) ( )( )]b r a x b x a r b r x a b b r b x (6)
1 1 1 1
r x a a
a r x
2 2 22 2 2 23 3 ( ) ( ) (2 ) ( )( )
rx x rx r Rr Rx r x R x R r R r x R x R r
r x r x r x
x R r
2 2 2
3 3 3
R Rr R Rr r
R r R rrx x R R r R r Rr
R Rrr x R r R rr R r
8 7 6 2 5 3 4 4 3 5 2 6
22 2 2
2 3 4 5 6 9R R r R r R r R r R r R r
R r R r
2 2 2
33 3 2 R Rr
r rrx r R r Rr rR r R Rrr x R r
r R r
22 2
9 12 10 4R r R r R r Rr r
R r
(8)
9 8 2 7 3 6 4 5 5 4 6 3 7 2 8 9 10
23 2 2
2 13 12 22 12 4 12 2 2R r R r R r R r R r R r R r R r Rr r
R r R r
R r x R r R r R r
2 22 2 2
3 R Rr R Rr
Rr R r R r R r Rr rR r R rRr Rx r x
R Rrr x R r R rr R r
6 2 5 2 4 4 3 5 2 6 7 8
22 2 2
9 6 5 4 3 2R r R r R r R r R r Rr r
R r R r
R r R r
(8)
9 8 2 7 3 6 4 5 5 4 6 3 7 2 8 9 10
23 2 2
2 13 12 22 12 4 12 2 2R r R r R r R r R r R r R r R r Rr r
R r R r
,a b x ,a b
( , , ) 0f x a b ab a b a b
http://www.museum.city.ichinoseki.iwate.jp/icm/
HPM
HPM
HPM
20
GöttingenFelix Klein, 1849-1925
Hermann Weyl, 1885-1955
Max Born, 1882-1970
Zahlbegriff und Quadratur des Kreises
HPM
HPM
log3
10
HPM
10:00
13:3-14:10 (