ευκλειδησ 18 01-2014

Download ευκλειδησ 18 01-2014

Post on 13-Apr-2017

240 views

Category:

Documents

2 download

Embed Size (px)

TRANSCRIPT

  • ( ) 34

    106 79 . 210 3616532 - 2103617784 - Fax: 210 3641025

    e-mail : info@hms.gr www.hms.gr

    GREEK MATHEMATICAL SOCIETY 34, Panepistimiou (leftheriou Venizelou) Street

    GR. 106 79 - Athens - HELLAS Tel. 210 3616532 - 2103617784 - Fax: 210 3641025

    e-mail : info@hms.gr www.hms.gr

    74

    , 18 2014

    B

    1

    22 0

    4 3 2

    2 4 1 1 77 16 60 19 7: 3 2000 : 4 :3 9 3 3 3 81 3 3 9

    = + + + = +

    .

    22 0

    4

    2 4 1 1 77 4 4 1 77: 3 2000 : : 9 1 33 9 3 3 3 9 9 3 81

    4 4 1 77 4 77 4 77: 10 : 9 1.9 3 3 81 9 81 81 81

    = + + + = + + +

    = + + = + = + =

    3 2

    16 60 19 7 308 (60 3 19) 7 308 3 74 : : :81 3 3 9 81 27 9 81 27 9

    308 7 308 7 3159 35.81 9 9 9

    = + = + = + =

    += + = = =

    2 90 = = , 800 = , , 800 = . :

    2810 800 2+ . 796 . x , x 10.

    . ( 1 1000 ) , 800 x x = = + , = 800 . , , 800x = = , 800 = = = . , 800 2 = .

  • 2

    1 , : ( ) 800 800 800 2 2 1600 800 2x x x x = + + + + = + +

    ( ) ( )2 800 800 400 800 800 3200002

    xx x x+ = = + = + .

    : 2 1600 800 2 2810 800 2 2 1210 605x x x+ + < + < <

    800 320000 796000 800 476000 595x x x+ > > > 595 605x< < x 10, 600x = . 600 , =1400 = 800 600 320000 480000 320000 800000 + = + = , 800 .

    3 = 30 = . - 90 = . , . . : () . () = . ()

    180 180 30 75

    2 2

    = = =

    75 = = , 1 90 75 15 = = = .

    75 30 45 =+ = = = = .

  • 3

    2

    ()

    2 2 2 22 2 = + = = , :

    22 2

    = = .

    , = . , - = 30 = = . 60 = , . = ,x = =

    2 2x

    = = , : 22 2 2 2

    2 2 2 2 22 3 2 2 6 .4 2 4 4 3 3x xx x x

    = = = = = =

    4 55% . - . -

    , 711

    .

    60. .

    , 55% .

    7 55 3511 100 100 = , 35% -

    . ( )55 35 20% = - , . - ( )100 55 20 25% = ,

  • 4

    25% 60 .

    10060 24025

    = .

    , :

    :

    = 55100

    x = .

    = 45100

    x = .

    = . = .

    = . = .

    - , :

    = , :

    55( ) ( )100

    x + = + = ,

    7 55 3511 100 100

    x x = = ,

    20(55 35)100 100

    x x = = = ,

    ( ) 2545 20 60 240100 100

    x x x = = = =

    2 x y . - . . 55% , :

  • 5

    55 9( ) 100 55 55 9 11 (1)100 11

    x x y x x y x y y x= + = + = = .

    - , :

    (2)x a x a = = + .

    711

    .

    :

    (2)7 7( ) (3)

    11 11a a x a + = = .

    () 60 , : 60y = + .

    ( ) a , : (2) (1),(3) 9 7 560 6 60 60 60 132

    11 11 11y y a a y x x x x x x = + + = + + + = + + = + = = .

    132 , 9 9 132 10811 11

    y x= = = ,

    240 .

    1 :

    33 2 21 1 3 4

    3

    1 243 81: , , 3 , 33

    x x x y x y x yy y y

    + = + = = + = =

    .

    ( )( )

    ( ) ( )3 33 3 9 3 33 4 9 12 13 4 124

    43 3

    4 4

    3 1 3 3 1 1: : 3 3 : 33 3 3 3 33

    1 1 3 1 823 : 3 1 .3 3 3 81

    + +

    = + = + = + + = + = + = =

    ( ) ( )3

    2 23 4 5 6 4 8 1 4 4 43

    4 4 4

    4 4

    1 1 3 1243 3 11 81 3 3 3 3 3 3 3 3 3 3 3 1 28.1 13 3 3

    3 3

    ++ + + +

    = = = = = = + =

    3 43 3 27 81 108. = + = + =

    2 ( ) 6 4 216 16 1P x x x x= + ( ) 4 24 5 1Q x x x= + . () ( )P x ( )Q x .

  • 6

    () ( )( ) ( )

    25 12

    P xx

    Q x= + .

    ()

    ( ) ( ) ( ) ( )( )( )( )( ) ( )( )( )( )( )

    6 4 2 4 2 2 4 2

    2 2 2 2

    16 16 1 16 1 1 16 1 1

    4 1 4 1 1 4 1 2 1 2 1 1 1 .

    P x x x x x x x x x

    x x x x x x x x

    = + = = =

    = + = + + +

    ( ) ( ) ( )( )( ) ( )( )( )( )

    4 2 4 2 2 2 2 2

    2 2

    4 5 1 4 4 1 4 1 1

    4 1 1 2 1 2 1 1 1 .

    Q x x x x x x x x x

    x x x x x x

    = + = + =

    = = + +

    () ( )( ) ( ) ( )

    2 2 2 25 51 4 1 1 1 1,2 2

    P xx x x x x

    Q x= + + = + = =

    ( ) 10 1.2

    Q x x x

    . 3 ,x y x y> , 2014. 97. ,x y . 2014x y=

    ( )( ) 3

    2014 97, 97 11 1917, 97 1 2

    1 3 71, 97 1 2.

    y y yy y

    y y

    = + >

    + = > +

    + = > +

    y 31917 3 71= 97, - 3 71 213y = = 23 71 639y = = 33 71 1917y = =

    213y = , 1801 8x = = . 639y = , 1375 2x = = . 1917y = , 97 1917,x y= < = .

    4 = 30 = . - 90 = . - , . = , : () . () . ()

    2 2 2 22 2 = + = = ,

  • 7

    : 22 2

    = = .

    , 30 = , x = , 130 22

    x x= = = = ,

    :

    ( )2

    22 2 2 2 2 62 3 32 66

    x x x x x + = = = = = .

    4

    ()

    = 180 30 752

    = = =

    ,

    75 30 75 45 = + = = . , , . , = . = = .

    180 2 180 2 75 30 = = = = ,

    130 22

    = = =

    330 2 32

    = = = .

    :

    ( )1 3 = + = + . = :

    ( )2 3 3 1 . + = + =

  • 8

    1

    ( )( )( )84

    21 3 1 3 1 3

    x =+ + +

    4 2y = .

    1x + y . 8 3 1 - ,

    ( )( )( )( )( )

    ( )( )( )( )

    ( )( )( )

    8 8 88

    8 84 4 4

    2 3 1 2 3 1 2 3 13 1

    1 3 1 3 1 3 3 1 1 3 1 3 3 1 1 3 3 1x

    = = = =

    + + + + + + .

    8 481 3 4 2x y+ = < = = .

    2 x -:

    ( ) ( )11 0 2 5 03

    xx x

    .

    1

    1 0 1 3 3 1 3 2 4.3

    xx x x

    (1)

    ( )( )2 5 0x x : ( )( ) ( ) ( )

    ( )2 5 0 2 0 5 0 2 0 5 0

    2 5 2 5 , 2 5.

    x x x x x x

    x x x x

    2 2 2x x x 5 5 5x x , 5 2 2 5x x (2) (1) (2)

    2 2 4.x x=

    3 AB ( = > ). 1( , )c ( ) A . 2 ( , )c ( ) A 1( , )c . 3c . () , , . () .

    () :

  • 9

    (1) = = ( 1( , )c ) (2) = = ( 2 ( , )c )

    : 902

    = =

    : 902

    = = .

    . :

    () , () = , () 902

    = = .

    , (1)

    = = ,

    2 902

    = = . (3)

    5

    = = , ( ) ( ). :

    1 902

    = = = (4)

    (3) (4) :

    2 1 902

    = = = = , (5)

    , , , , - . , -, , . , , .

  • 10

    90 90 180 ,2 2

    + + = + + =

    , . () , : 1 90

    = . 1 1

    ( )3 , ,c = , :

    1 1 90 = = . 1 , , :

    1 1 = + ( )1 1

    90 902 2

    = = = ,

    . = , . 4 ,a b

    2 24 2 12 5a b a b+ = + . a b+ ,a b - . s a b= + , :

    ( ) ( )22 4 2 12 5a s a a s a+ = + . ( )2 25 8 10 4 12 5 0a s a s s + + = (1)

    (1) a , , :

    ( ) ( ) ( ) ( )( ) ( ) ( )

    2 22 2 28 10 20 4 12 5 0 4 4 5 5 4 12 5 0 4 20 0

    5 0 0 5 0 0 5 0 0 5.

    s s s s s s s s

    s s s s s s s

    = + + +