Θέματα και λύσεις...
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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 ...
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67
, 9 2006
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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
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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. .
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