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( ) 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

70 , 21 2009

B 1 a = 4 2

1 3 5 b = 5 + , : 5 2 2 1 A = a : b 2009 b . 5a

5 2 4. (i) ; (ii) , , 39 50, 4 1. 5 3 AB , 1400 1 6, . () . () . 5 4 1 1 , , 4 5 1 , 80 8 . , , 12 , : () ; () ; 5

( ) 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

70 , 21 2009

1 , :

( 1) 2 v +1 ( 1)3v A = 4 ( 1) + 2 7 . 5 5v

5 2 O 5 2. . 5 3 1 , 2 , . 1 4, 2

( ) : y = 2 x (0,6).

() . () , xy , 1 , 2 2 xx . 5 4 r1 , r2 , r3 0 < r1 < r2 < r3 . 1 r1 , r2 , 2 r2 , r3 .

r2 r1 = r3 r2 r3 = 3r1 ,

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

1 2 , .

( ) 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

70 , 21 2009

1 75. . 5 2 ,

4 2 + 4 + 2 2 + +1 , = 2 + 2 34. 5 3 . () , , = = , . () ( ), , = = , . 5 4 . ( ) . ( ) . ( ) , , . : , . . , , o ; ; 5

( ) 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

70 , 21 2009

1 , , :

4 1 1 + + . + 2 2 , C (O, R) . 1 , 1 , 1 , , 2 , 2 , 2 , , , 2 1 2 12 1 1 2 , 12 1 2 . 5 3 x, y x 2009 y 2009 :

5

x 2009 + y + 2009 =

x+ y +1, 2

=

x y+2 . 2 5

4 :

( x + y )3 = z 2 x y 3 ( y + z ) = x 2 y z ( z + x )3 = y 2 z x .

( ) ,

5

( ) 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

70 , 21 2009

1 . x, y

2 x 2 + 3x ( x 2 ) + 11x 10 y = 2015 2 f : 5

:

f ( x f ( y )) f ( y f ( x )) = 2 f ( f ( x ) f ( y )) , x, y

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

. 5

3

000

000 ,

2

00 00 , 2 . : 33 33 . 5 4 . , C (O, R) 1 , 1 , 1 R R , , . C1 ( 1 , ) , C2 (1 , ) 2 2 R C3 (1 , ) . C1 , C2 , C3 ( ) 2 2 , 2 , 2 , , . 1 2 , 12 , 1 2 . 5

( ) 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

70 , 21 2009

B 1 a = 4 2

1 3 5 , : b = 5 + 5 2 2 1 A = a : b 2009 b . 5a

1 4 11 20 11 9 3 5 3 5 8 = 5 = 5 = 5 4 = 1, a = 42 = = = b = 5 + 5 1 5 5 5 5 2 2 2 2 2 : 1 9 2009 1 9 1 9 1 76 31 A = a : b 2009 b = :1 1 = :1 1 = 1 = 1 = . 9 5 5a 5 9 5 9 45 45 5 5 2 4. (i) ; (ii) , 39 50, 4 1. (i) (ii) : = 4, , = 4 + 1 = 4 + 2 = 4 + 3 . = 4 + 1, : 39 < 4 + 1 < 50 38 < 4 < 49 9,5 < < 12, 25 , , = 10 = 11 = 12 = 41 = 45 =49.

.

1

3 AB 1400 1 6, . ) . ) . ) : = 1800 + = 1800 1400 = 400.

(

)

:

= = = , = 6 + 6 = 1400 = 200. 1 6 : = 200 = 1200 .

= + = 1400 , : 1 6

1) . , 1800 . : (1) = + = 90 + . 2 = 400 , = 1800 1200 = 600 , (1) = 500 .

(

)

4

, ,

1 1 , 4 5

1 80 8 . , , 12 , : ) ;

2

) ; (1 ) 1 1 1 23 23 . ) + + = 4 5 8 40 40 12 . 40 23 17 . 80 -12=68 = 40 40 40 : 17 40 68 : = 68 = 4 40 = 160 . 40 17 1 ) 160 12 = 32 12 = 20 . 5 2 ) x , , , : x x x + + + 80 12 = x , 4 5 8 10 x + 8 x + 5 x + 3200 480 = 40 x 17 x = 2720 x = 160. x 160 ) 12 = 12 = 20 . 5 5

1 , : (1) 2 v +1 (1)3v A = 4 (1)v + 2 7 . 5 5

(1)3 (1) 2 v +1 (1)3v (1) v A = 4 (1) + 2 7 = 4 (1) + 2 7 5 5 5 5v

2 7 ( 1) 7 2 13 ( 1) 2 = 4 (1) = 4 ( 1) = , 5 5 5 5 5 : 13 2 11 , = = . 5 5 , = 3 . v

2 O 5 2. .

3

5 2, = 5 + 2 , . , , , . , = 2 + 1, . , = 5 ( 2 + 1) + 2 = 10 + 7 , 10 7, , 7. 7 , 7. 3 1 , 2 . 1 4 , 2 ( ) : y = 2 x (0,6). ) . ) , xy , 2 xx . ) 1 y = 4 x , 2 y = 2 x + , (). (0,6), 6 = 2 0 + = 6 . 2 y = 2 x + 6. ( 3,12 ) .

2) 2 x ( 3, 0 ) , 3, 12. :

4

() =

1 3 12 = 18 .. 2

4 r1 , r2 , r3 0 < r1 < r2 < r3 . 1 r1 , r2 2 r2 , r3 .

r2 r1 = r3 r2 r3 = 3r1 ,

( 1 ) , ( 1 ) ( 2 )

( 2 ) 1 2 , .

3 2 2 ( 1 ) ( r2 r1 ) ( r2 r1 = = ( 2 ) ( r32 r22 ) ( r3 r2

)(r + r ) = r + r , )(r + r ) r + r2 1 2 1 3 2 3 2

(1)r1 + r3 , , 2

r2 r1 = r3 r2 . r2 = r3 = 3r1 r2 =

r1 + 3r1 = 2r1 . (1) 2 ( 1 ) r2 + r1 3r1 3r 3 = = = 1= . ( 2 ) r3 + r2 3r1 + 2r1 5r1 5

, r2 = ( 2 ) ( 1 ) =

r1 + 3r1 = 2r1 2

( r32 r22 )

( r22 r12 )

=

2 = 3r1 = 3 . 2 2 2 ( 3r1 ) ( 2r1 ) 5r1 5

( 2r1 ) r12 2

5

1 75. . x , x 2 10 x = 75 x 2 10 x 75 = 0 x = 15 x = 5 . , 15. 2 4 2 + 4 + 2 2 + +1 , = 2 + 2 34. .

(2 )

2 2

+ ( 22 )

+2

2 2 + 0 ( 2 2 ) + ( 2 + 2 ) 2 2 + 0 ( 2 2 2 + 2 ) 02 2 2

2 2 2 + 2 = 0 2 4 = 1 4 = 0.

= 2 + 2 = 2 + 4 + 2 = 2 ( 24 + 1) = 17 2 = 34 2 1 ,

34, . 3 . () , , = = , . () , , = = , . () ( = , = ) , = . , = , . , = = 90

6

. , = ,

4

5() () , = . , : = = , . (). , :

7

= , , = , . = . , , , . 4 ( ) . ( ) . ( ) ( ) . : , . . , o ( ) ; ; x , . ( ) : x x2 x3 12 (1) -- + = 1 5 x = 12 + 6 6 x = 3 4 6 5 x x2 x3 14 (2) -- + = 1 5 x = 12 + 8 6 x = 4 3 6 5 (3) --

x x2 x3 17 + = 1 5 x = 12 + 9 4 x = 3 6 4 5 x x2 x3 + = 1 5 x = 12 + 12 4 x = 4 4 6 3 x x 1 x 16 + = 1 5 x = 12 + 4 x = 4 3 6 5 x x 1 x + = 1 5 x = 12 + 3 x = 3 3 4 6

(4) --

(5) --

(6) --

-- x = 4 ( (4)). -- x = 3 ( (6)). (4) ( ), .

8

x x , . 4 x2 . x 2 6 x 3 x3 . (4). 3 (6) ( ), ( ). x x , . 3 x 1 x 1 . 4 x . x , 6 (6). .

1 ( , , :

4 1 1 + . + + 2

+2

,

(1)

0 4 1 1 + , +

(

).2

(2)

4 1 1 4 + 2 2 + 4 ( + ) 0 ( ) . + + (1) (2) 4 1 1 + + . + 2

9

2 . , C (O, R) . 1 , 1 , 1 , , 2 , 2 , 2 , , , 2 1 2 12 1 1 2 , 12 1 2 . , : = = = R .

6 2 1 , : R (1) . 2 1 = = 2 2 12 , : R (2) . 12 = = 2 2 R . 2 (1) (2) 11 2 2 , .

10

12 2 1 , . 3 . x, y x 2009 y 2009