Θαλής 2010-2011

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β ΜΚΔ16,32,248 8 = = . Από την υπόθεση έχουμε: α 86 υ 48 υ,όπου υ = ⋅ + = + ακέραιος με δυνατές τιμές από 0 μέχρι και 7 . Δοκιμάζοντας τις δυνατές τιμές του υ στην παραπάνω σχέση διαπιστώνουμε ότι μόνο για υ 1 = , ο αριθμός α 49 = που προκύπτει, είναι πολλαπλάσιο του 7. Άρα έχουμε α 49 = και β 8 = . 2 3 2

Transcript of Θαλής 2010-2011

  • ( ) 34

    106 79 . 3616532 - 3617784 - Fax: 3641025

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

    GREEK MATHEMATICAL SOCIETY 34, Panepistimiou (leftheriou Venizelou) Street

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

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

    71

    , 30 2010

    1. 2 3 5x 3 4 2 : 4 2= + 2 3 2y 4 5 4 7 3= + . () x y . () , x y . ()

    2 3 5x 3 4 2 : 4 2 9 4 8 : 4 32 9 32 : 4 32 9 8 32 33= + = + = + = + = . 2 3 2y 4 5 4 7 3 4 25 64 7 9 100 64 63 99= + = + = + = .

    () x, y . ( ) 33,99 33= , 33= . 2. , . 6. , 7, 16, 32 248. , 16, 32 248.

    16 32 24816 0 80 0 8

    ,

    ( ) 16, 32, 248 8= = . : 8 6 48 , = + = + 0 7 . 1= , 49= , 7. 49= 8= .

  • 2

    3. . . . 0 70= 0 130= , : ) . ) o . . / / : o1 70= = ,( , ). / / , : o o o1 180 130 50= = = . ( 1 , , ). ,, , :

    o o o o o o 180 180 180 70 50 60+ + = = = = . . , :

    oo

    1 70

    352 2

    = = = . , / / , : o1 1 I 35= = , 1 1 I , , I .

    1

    , : o

    o1

    50 25

    2 2= = = .

    E / / , : o2 1 I 25= = , 2 1 I , E, . 4. . 80 , 120 . 2600 . 10% , : . .

  • 3

    . 150% . . 120+80=200 , 2600:200=13 . 120 120 13 1560 = . 101560 156

    100 = .

    . x ,

    150 3xx100 2 = .

    3x 260080 x 120 2600 80x 180x 2600 260x 2600 x 102 260

    + = + = = = = .

    3 10 152 =

    , 120 15 1800 = 101800 180

    100 = .

  • 4

    1. ( )2x y 3 2+ = 6 44 63 3y w

    5 5

    = ,

    : 7x 10y 3w 87= + . ( )2x y 3 2 3 4 12+ = = =

    6 44 6 24 24 24 24

    2424

    3 3 3 3 3 5y w5 5 5 5 5 3

    3 5 1 1.5 3

    = = = = = =

    .

    :

    ( ) ( ) 7x 10y 3w 87 7x 7y 3y 3w 87

    7 x y 3 y w 87 7 12 3 1 87 84 3 87 0.= + = + +

    = + + = + = + = 2. , : () 4, () , () 5 , () 1. xyzw 1000 x 100 y 10 z w= + + + . , () w 0 4 8= , () z 0 2 4,= . , () y 1 5= . :

    x100, x124, x148, x500, x524, x548 . () 4500, 4524, 4548 , 0.

    3. 0 120= . x y , , , . 0 120= , 0 60= 2 3 , : . . . . . .

  • 5

    . 1

    o2

    120= = , o1 60= . o1 60= . .

    2

    . (x) E (y) , : 0 o1 3 90 120 90 30= = = =D D . : = ( ),

    o2 2

    120= = o1 3 30= = . , = . 60 = =D , 0 30= = , . . . H :

    2 2 2 = + ( )2 22 2 34= +23 12 4

    4 = = .

    H :

    2 2 2 = + , ( )2 22 3 2 32 = + 2 48 4 3 = = . 12 8 3+ . 12 , 3 2 3

    3+ .

    4. 2 . 1 ( )C , , , . 2 ( )C , . . ( ) ,, ( )C , ( )C , . . ( ) ( )1 2 1 2 ,

  • 6

    ( )( )12

    135

    .

    . x ( )C , x ( ) ,, .

    3

    () 2 2 2 2 2 2 2= + = = ,

    ( )( ) ( )2 2 2 2 2 ,, 2 2 = = = . () 1 2 , .

    ( ) ( ) ( ) ( )2 2 2 2 21 2 2 2 4 2 4 = = = . 2 , ,

    ( ) ( ) ( )2 2 2 2 22 2 4 4 = = = . ( )( )12

    2 4 4

    = : ( )( ) ( ) ( )12

    2 4 13 725 2 4 13 4 23 72 3,1304 4 5 23

    = > > > > , , 3,14 . () x 2< < ,

    ( ) ( )2 2 2 2 2 2 2 22

    2 2 2

    2 x x 2 x x

    3 32x 3 x x .2 2

    = = = = =

    .

  • 7

    1. -

    ( )22 x x 1x 1 x 1x 5x 14, .2 4 4

    = + < 2x 5x 14 = ( )x x 5 14 = . , x 14. { }x 1, 2, 7, 14 . 7 -2. , 2x 5x 14 0 = , 1, 5, 14= = = , 2 4= =81 x 7= x 2= .

    ( )2 2 2x x 1x 1 x 1 2x 2 x 1 x x 3x 3 x 12 4 4

    + < + < < < . x 2= . 2. ,, , , :

    4 3 2 2 2 2 2 3 2 4 2 2 2 2 4 2 2 = + + + .

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

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

    4 3 2 2 2 2 2 3 2 4 2 2 2 2 4

    4 3 2 2 2 2 2 3 2 4 2 2 2 2 4

    2 2 2 2 2 2 2 2 2 2 2

    2 2 2 2 2 2 2 2 2

    2 22 2 2 2 2 2 2 2

    2 2

    2 2

    2 2

    2

    2

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

    = + + = + = + ( )( ) .+ + +

    3. :

    x 4 x 1 2 5 x1 ,2 y 2 3y 3 3

    = = + .

    1 wy= , :

  • 8

    x 8w 2 x 2 8w x 2 8w x 2 8wx 4w 13 x 13 4w 2 8w 13 4w 8w 4w 13 2

    11 x 2 2 11 x 24x 2 8x 2 8w 4 .11 114w 11 11 w ww 4 44

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

    ( ) 4x, y 24,11

    = 4. ( )= . () . () , = < . . ( )= ( )= . = , . ( = ). , 1 2 = = . , 3 4 = 3 4 , 1 2 , .

    4

    : 1. = ( ). 2. = ( ).

  • 9

    3. 3 4 = ( 1 2 , ). = = . : 1. = ( ). 2. = ( ). 3. = ( ). = . : 1. = ( ) 2. = ( ) 3. = + = + = 1 1 = . , 1 1 = , , . , .

  • 10

    1. x, y, z

    2

    2

    2

    x y z x 2

    y z x y 2

    z x y z 2,

    = = =

    x y z 6+ + = x, y, z . : x 2, y 2 z 2 , 2 2 2x y z, y z x z x y. + + +

    2 2

    2 2

    2 2

    x y z x 4x 4 4x y z 4y z x y 4y 4 x 4y z 4

    x y 4z 4z x y z 4z 4

    = + = = + + = + = = + , (1)

    : x y z 6+ + = . x, y, z (1),

    ( )( )( )

    5x x y z 4 5x 6 4 x 25y x y z 4 5y 6 4 y 25z x y z 4 5z 6 4 z 2

    + + = = = + + = = = + + = = = .

    , x 2, y 2 z 2 , x 2> , x y z 6+ + > , . x y z 2= = = . 2. 1c (,) ( 1R = ) 2c (,) (

    2R = ). 1c (,) . 2c (,) . . . . = o 30= , .

  • 11

    . ( ) 1c (,) , o 90= . ( ) 2c (,) ,

    o 90= . , , / / = . o 90= = , / / . o 90= = .

    5

    . . o 30= , .

    2= = = ,

    , = , = , , . 3. x, y x y 4+ = , :

    ( ) ( )2 22x 1 2y 1 25x y+ ++ .

    ; :

  • 12

    2 24x 4x 1 4y 4y 1 25x y+ + + ++

    ( ) 1 1 : 4 x y 8 25x y

    + + + +

    : 1 1 1x y+ .

    .

    ( ) 1 1x y 4x y

    + + , x y 4+ = , ( ) 1 1 x y x yx y 2 2 2 4

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

    x y 2= = . ,

    ( )

    ( )2

    2

    1 1 x y1 1 xy 4 x 4 x 4 x 4x 4 0x y xy

    x 2 0, .

    ++ +

    x y 2= = , x y 2= = . 4. o( 90 )= . . , . E A , :

  • 13

    6 = .

    1 1 2

    = = .

    , 1 2 2

    = =

    1 1 2

    = = , 1 2 2= = . = , . :

    1

    2

    =

    , 1

    2

    =

    . 2

    = , = = ,

    . , . , , .

  • 14

    1.

    ( ) ( ) ( )3 3 32 2 22x 3x 1 x 3x 2 7 x 1+ + + + = .

    (1 ) 2 2a 2x 3x 1, b x 3x 2= + + = + + , 2a b x 1 = :

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

    3 33 3 2 2

    2 2 2 2

    2 2

    2 2

    2 2

    a b 7 a b a b a ab b 7 a b

    a b a ab b 7a 14ab 7b 0