Θέματα και λύσεις...

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  • parmenides51 : facebook : http://parmenides51.blogspot.gr

    : http://parmenides52.blogspot.gr

    / ()

    2006-2007 67 / 09-12-2006 2006: - 2007-2008 68 / 24-11-2007 2007: - 2008-2009 69 / 01-11-2008 2008: - 2009-2010 70 / 21-11-2009 2009: - 2010-2011 71 / 30-10-2010 2010: - 2011-2012 72 / 19-11-2011 2011: - 2012-2013 73 / 20-10-2012 2012: - 2013-2014 74 / 19-10-2013 2013: - 2014-2015 75 / 01-11-2014 2014: -

    2005-2006 66 / 21-01-2006 2006: - 2006-2007 67 / 20-01-2007 2007: - 2007-2008 68 / 19-01-2008 2008: - 2008-2009 69 / 17-01-2009 2009: - 2009-2010 70 / 23-01-2010 2010: - 2010-2011 71 / 15-01-2011 2011: - 2011-2012 72 / 21-01-2012 2012: - 2012-2013 73 / 12-01-2013 2013: - 2013-2014 74 / 18-01-2014 2014: - 2014-2015 75 / 17-01-2015 2015: -

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    2006 2015 ...

    : 20 - 01-2015 parmenides51

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  • 67

    , 9 2006

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    , ' ..., 34, 106 79 , , , , .

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    10. .. ... , , , . , .

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    ( ) 34

    106 79 . 3616532 - 3617784 - Fax: 3641025

    e-mail : [email protected] www.hms.gr

    GREEK MATHEMATICAL SOCIETY 34, Panepistimiou (leftheriou Venizelou) Street

    GR. 106 79 - Athens - HELLAS Tel. 3616532 - 3617784 - Fax: 3641025

    e-mail : [email protected] www.hms.gr

  • , 9 2006 , ()

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    ( ) 34

    106 79 . 3616532 - 3617784 - Fax: 3641025

    e-mail : [email protected] www.hms.gr

    GREEK MATHEMATICAL SOCIETY 34, Panepistimiou (leftheriou Venizelou) Street

    GR. 106 79 - Athens - HELLAS Tel. 3616532 - 3617784 - Fax: 3641025

    e-mail : [email protected] www.hms.gr

  • ( ) 34

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

    e-mail : [email protected] 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 : [email protected] www.hms.gr

    67

    , 9 2006

    1. :

    54111 264 (15 ) 5 :12 :11 16

    = + + 2. 100 18 2 10; 3. 6% 0 4% . .

    9 36

    =

    4. = x .

    .

  • ( ) 34

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

    e-mail : [email protected] 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 : [email protected] www.hms.gr

    67

    , 9 2006

    1. x

    2. 2 02 + + =

    =10,

    : 2 2 2( ) ( 2 )

    2 = + +

    3. p , 27p + 1 . 4. , ,

    1 13 10 3.2 3

    a a + =

  • ( ) 34

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

    e-mail : [email protected] 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 : [email protected] www.hms.gr

    67

    , 9 2006

    1. 5 20 . 10 . 1090. . 2. ( ) 33 2 2x x + = + .

    3. , , :

    2

    3

    + + + +

    4.

    >

    ,

    . = . : = .

  • ( ) 34

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

    e-mail : [email protected] 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 : [email protected] www.hms.gr

    67

    , 9 2006

    1. 2 (2006 1) 2007 0x x + + = , . 2. = 4, = 2 = 1 = 3 . .

    3. 2 3 20082 2 2 ... 2 = + + + + .

    30 .

    4. ) : 3 3 32 3 19+ >

    ) : 3

    1 13 3

    382 .2 3

    x x + =+

  • ( ) 34

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

    e-mail : [email protected] 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 : [email protected] www.hms.gr

    67

    , 9 2006

    1. :f ( ( )) ( )f f x y x f y+ = ,x y .

    :h ( ) ( ) ( )h x f x f x= + . 2.

    13 3 4 1 0x xx x+ = .

    3. 1 2, 0, 2z z

    .

    : 2 2

    2 21 21 2 1 22 2 2Re( )

    z zz z z z

    + + + .

    4. , , .

    = = , ,

    .

  • ( ) 34

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

    e-mail : [email protected] 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 : [email protected] www.hms.gr

    67

    , 9 2006

    1. (111 144 :12) :11 1 (111 12) :11 1 99 :11 1 9 1 10 = + = + = + = + =

    2. 100 0 10

    0, 2 0. 2 5 10 15. 5 15. 10 .

    10 2 8 10 100. + =

    3. : 6 4 2

    100 100 3 = = .

    29 3 6 3 33 12 4 363

    = = = =

    4. x = . x//B =

  • 2 -

    = = . = = = , 0180 + + = , 060 = = =

    0 60 = = = .

    1. 0 0 180 3 4 180 7x x x = = =

    0 0 180 2 6 180 8x x x = = = , , :

    0 180+ + = 0 0 0 0 0180 7 180 8 5 180 10 180 18 .x x x x x + + = = =

    2. 2

    2 2 2 2 2 2 2 2( 2 ) ( ) 42 4 = = = =

    2 2( ) 10 100.= = =

    3. =27p+1. p=2 =27 2+1=55=5 11 , p2 o 27p .

  • 3

    4. 2

    1 13 10 92 3

    a a + =

    ,

    2 2 2 29 100 10 94 9

    a a + + =

    2 2 2 29 100 9 10 1, .4 9

    a a + = =

    1. ,,,, . :

    10( ) 1090 + + + + = 109 (1) + + + + =

    : < < < < . 20 ,,,, :

    20 20 21 > > 21 21 22 > >

    22 22 23 > > 23 23 24 > >

  • 4 -

    110 + + + + , (1). , . 2. :

    ( )

    2 3

    2 3

    2 3

    3 2 22 3 2

    2 3 2

    x xx x

    x

    + = +

    =

    =

    ( )2 2 0 2 0 0 2 = = = = 1. =0 2. =2 3. 0 2

    3

    2

    3 22

    x

    =

    ( )( )( )

    ( )2 22 1 12

    x x + +

    = =

    3. :

    2 2 2

    2 2 2 3 3 3 ,

    + + + + + + +

    2 2 2

    0

    + +

    ,

  • 5

    2 2 21 0,2

    + +

    . 4.

    = =-= .

    =

    =, = 1 2 =

    1= ,

    =.

    =

    =, = 1 2= =

    , =. . , =. = =||=| ()|=||=, >.

  • 6 -

    1. 1 2,x x ,

    1 2 2006 1 (1)x x + = + 1 2 2007 (2)x x = . (2) 1 2,x x . 1 2x x+ , (1). 2.

    ( )2 2 23 1 4 2+ = = .

    030 ,2

    = =

    0 060 30 .

    = =

  • 7

    , 12

    = 1 322 2

    = = .

    1 34 3.2 2

    = = .

    2MB B = 33 2

    2= =

    3.

    2 3 4 4 2 3 4 2004 2 3 4

    4 8 2004

    (2 2 2 2 ) 2 (2 2 2 2 ) ... 2 (2 2 2 2 )30(1 2 2 ... 2 ) .30= + + + + + + + + + + + + =

    = + + + + =

    4. ) : 2

    + > ,

    , : 3 3

    3 3 62 3 2 3 62+

    > =

    3 3 62 3 2 6+ > .

    6 26 32 6 19, 2 6 19 , 384 361 .

  • 8 -

    ) 3

    3 3

    382 3

    =+

    323 3

    3 3

    19 2 2 2 2.2 3

    = < <

    1. 0x = ( ( )) ( )f f y f y y = . 0y = ( ( )) (0)f f x x f x = .

    ( ( )) ( ) ( ( )) (0),f f x f x f f x x f x = =

    ( ) (0)f x x f = ( ) (0) (1) ( ) (0) (2),f x f x f x f x x = = +

    (1) (2) : ( ) ( ) 2 (0),f x f x f x + = .

    h .

    2. .

  • 9

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