Magazine I

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    09-Dec-2015
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math magazine from greece which include math rpoblems with solution for high school

Transcript of Magazine I

  • :

    4o MA 2002

    : : : : : 1, 42 100, . (0431) 022988 , , : 73, 49 100, . (0661) 033243 : " " 27 () 54621, . 0310 263163, FAX 0310 240595

  • 5

    7 , .

    32 , , .

    39 , , Thbault.

    44 , .

    50 , .

    57 , -.

    63 , 3 .

    67 , .

    77 , y = 1/x.

    81 , .

    89 , , - .

    111 , N. Lobachevsky.

    122 , .

  • ,

    " " " -" , , - .

    - , , , -. , - , , .

    :

    " " - , , , , , - .

    , , , , .

  • * , , . - , , - . "", , - . - .

    * ... , 53-54, - 2000, 17.

  • 8 , 4

    - , - , .

    , - .

    , , , -:

    -, x - x 0 () xx. - xx, .

    : )

    - , , ,

    ) 30 - " ", - .

  • , 4 9

    - ( : x= , x< , x> , ) - , - , :

    x= x = x =

    x< < x <

    x> x < x > , : ,

    1990 ( , . , . , ..) -. . , -, , , - . , - . - "" , , - . - " " : x2 < 4 x < 2 x < 2 ( ). "" , - "-" .

  • 10 , 4

    , "" , , , , - , . - , , . , " "1, . : ) .

    ) .

    ) . , , , 1999-2000, , -

    1 - , () - , -. ( ) - , , . - , , - .

  • , 4 11

    . . , , . - , , . , 0 () .

    x x

    0

    , - . , - , .

    : " ;", : 0 .

    ; : , = 0. , .

    , , : . = .

    x x

    =

    =

    0

    :

  • 12 , 4

    , , - , . :

    x 2 = 3, x + 1 < 4, 2x 3 5,

    x 3 + x + 4 = 10, 5x 6 < 7

    "" .

    , , .

    ,

    x 2 = 3 x, 2 3 -.

    x + 1 < 4

    x + 1 = x (1), - x, 1 4 .

    2x 3 5

    x, 2x 3 5 .

    x 3 + x + 4 = 10

    x, x 3 4 10 .

    5x 6 < 7

    x, 5x 6 7 . x -.

  • , 4 13

    , :

    ) x + 1 < 4, x (1) < 4 ,

    ) : 5x 6 < 7. , , x . - 5x 6 - . .2

    , . .

    ) x 2 = 3

    :

    x 2, , - x 2. : "

    2 . . , - ... 29/4/1997, " ": - "" , . - . , , . , , - , - .

  • 14 , 4

    x 2 = 3" : " x, 2 3 . ".

    3 , - 2, 3 . . - . 1 5, , x 2 = 3.

    ) x + 1 < 4

    :

    x + 1 x (1). x, 1 4 . .

    3 , , - . , , , , .

    +

    3 .

    3 .

    x = 1 2 x = 5

    . : 2 3 = 1 . : 2 + 3 = 5

    +

    4 .

    4 .

    5 x 1 x 3

    . : 1 4 = 5 . : 1 + 4 = 3

  • , 4 15

    , 1, 4 . . , - 5 3. x , . x : 5 < x < 3.

    , . x + 1 (x + 1) 0. x + 1 < 4 (x + 1) 0 < 4. :

    x 4 < x + 1 < 4, 5 < x < 3.

    ) 2x 3 5

    :

    +

    5 .

    5 .

    2x 2 3 8 2x

    . : 3 5 = 2 . : 3 + 5 = 8

    +

    4 .

    4 .

    4 x + 1 0 x + 1 4

  • 16 , 4

    E x, 2x , 2x, . : 2x 2 2x 8 : x 1 x 4.

    ) i. - , M - , x -

    : x2+= .

    ii. : x + 4 = x 2 iii. : x + 4 < x 2

    :

    + M

    x

    i. , = :

    x = x 2x = + x2+= .

    ii. : x (4) = x 2.

    + M

    4 x 2

    x 4 2. - x , , (i), :

    x = 4 22

    + , x = 1.

  • , 4 17

    iii. x (4) < x 2.

    + M

    4 x 2 x 1

    - . x , - (: ). x < 1.

    ) x 3 + x + 4 = 10

    :

    . . - , , . -: , :

    x + x = , > 0.

    "" . , - . , - , x + x = + = , , - , - () x.

  • 18 , 4

    .

    + = :

    + ( + ) =

    + ( + ) =

    ( + ) + =

    + =

    2 =

    = 2

    x

    2 2+ =

    :

    x + x = (4) x2 2+ = , > 0

    4 , , - : x 3 + x + 4 = 10

    . :

    3 ( 4) 10 1x , x 52 2 2

    + = = .

    4 : < < + , , , . : < < .

    +

    A

    x 2

    +

  • , 4 19

    xx (2, 0) (4, 0). ( ;) xx, (x, 0), :

    (i) MA + MB = 2 (ii) MA + MB = 1

    "" (i) (ii), - . ;

    5 - , , - "-", . - 5 - . , - - , , , .

    +

    5 .

    5 .

    2

    11

    21

    29

    . : 2

    115

    2

    1 =

    . : 2

    95

    2

    1 =+ K

  • 20 , 4

    - . -, , test . , - , , . , , test . - .

    :

    1 - : ) x, - 4x 3 5 -. ) x, - 4 6 . ) x, - 2 6 . ) x, - x 4 2 12 .

    2

    ) 4x + 3 > 5.

  • , 4 21

    () ; - ;

    ) () () .

    ) xx , x, : " - 1000 5008". .

    ) x : x 1999< 2000;

    test, - 55 , :

    % 48 87,3 48 87,3 45 81,8 41 74,5

    % - "-"

    41 74,5 31 42 76,4 32 39 71,0 34 44 80,0

    , test: ) -

    , . ) , , -

    "-" , .

  • 22 , 4

    : ) "-"

    - .

    ) , - ( ), (< >) test.

    , - .

    - , ( = 55 ). . , - . - . . , , . , , - . , , .

  • , 4 23

    x x 0 , - : x x.

    : R [0, +), (x) = x. , , :

    x, x 0

    x x, x 00, x 0

    >= 0 x> x < x > , xR > 0

    :

    = = , 0 + +

    (

  • 24 , 4

    ) , - . :

    . 2 + 8 7. "": - . , - . , - , - . 5 x, x .

    , : x= x, x . , - x, - . "-", , - .

    , , x= x, x.

    - , - . , , 5, +5. : - . - 5, +5.

    ,

  • , 4 25

    - .

    :

    x + 5= 2, x 1 2 3 + = , x 8< 2, x + 9> 5, :

    x= x = x = x< < x < , x> x < x >

    , , . , - -, . , , , - " ". , -, -, " ". - . , , , - , . "" - . : , , "" -

  • 26 , 4

    . . , - . ""