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A gymnasiou 14 15 a tomos

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Page 1: A gymnasiou 14 15 a tomos
Page 2: A gymnasiou 14 15 a tomos

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Page 3: A gymnasiou 14 15 a tomos
Page 4: A gymnasiou 14 15 a tomos
Page 5: A gymnasiou 14 15 a tomos

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Page 6: A gymnasiou 14 15 a tomos
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Page 9: A gymnasiou 14 15 a tomos

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Page 10: A gymnasiou 14 15 a tomos

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Page 13: A gymnasiou 14 15 a tomos

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Page 14: A gymnasiou 14 15 a tomos

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Page 15: A gymnasiou 14 15 a tomos

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Page 16: A gymnasiou 14 15 a tomos

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Page 21: A gymnasiou 14 15 a tomos

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Page 53: A gymnasiou 14 15 a tomos

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Page 55: A gymnasiou 14 15 a tomos

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Page 56: A gymnasiou 14 15 a tomos

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Page 57: A gymnasiou 14 15 a tomos

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Page 71: A gymnasiou 14 15 a tomos

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Page 76: A gymnasiou 14 15 a tomos

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, .

;

:

:

, .

, , , .

+

+

:

.

+3 1:

2 .

.

+3 1 , :

2 .

Page 137: A gymnasiou 14 15 a tomos

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4: –

«A_En

.

(2)

,

{ 4, 3, … , +3, +4}.

.

+ +4 +3 +2 +1 0 1 2 3 4 +4 +8 +7 +6 +5 +4 +3 +2 +1 0 +3 +7 +6 +5 +4 +3 +2 +1 0 1+2 +6 +5 +4 +3 +2 +1 0 1 2+1 +5 +4 +3 +2 +1 0 1 2 30 +4 +3 +2 +1 0 1 2 3 41 +3 +2 +1 0 1 2 3 4 52 +2 +1 0 1 2 3 4 5 63 +1 0 1 2 3 4 5 6 74 0 1 2 3 4 5 6 7 8

;

.

5.

; ;

0 .

0.

Page 138: A gymnasiou 14 15 a tomos

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(3)

.

8/3/2012 100 378,68 9/3/2012 210,14 10/4/2012 1000,55 10/4/2012 125,20 12/4/2012 50,00 12/4/2012 500,00 12/4/2012 467,32 13/4/2012 1445,39 ;

13/4/2012. .

: (+2) + (+5) = +(2 + 5) = +7 ( 4) + ( 9) = (4 + 9) = 13

: ( 3) + (+7) = +(7 3) = +4 ( 8) + (+1) = (8 1) = 7

( 3) + (+7) 3 + 7

--- -

Page 139: A gymnasiou 14 15 a tomos

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4: –

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

: ( 12) + (+3) = (+3) + ( 12) [( 2) + (+7)] + ( 1) = ( 2) + [(+7) + ( 1)]

+ = = + : ( 13) + 0 = ( 13)

+ ( ) = : ( 12) + (+12) = 0

, 12 +12.

: +4 4 ,

4 .

1. : ( ) (+12) + (+15) ( ) 20 + 4

( ) ( 9,5) + (+6 ) ( ) + 1( ) ( 7) + 0 ( ) + +

( ) (+12) + (+15) = +(12 + 15) = +27

. ,

.

Page 140: A gymnasiou 14 15 a tomos

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4: –

( ) 20 + 4

= (20 4) = 16

. ,

-

.

( ) ( 9,5) + +6 = 9,50 + 6,25 = 3,25 -

.

( ) + 1 = 1 = 1 = 1 .

( ) ( 7) + 0

= 7 .

( ) + + = 0 . , .

2. :

( ) ( 1,3) + (+5) + (+3,2) + + (+1,3) + + ( ) ( 3) + (+1) + (+3,2) + ( 1,1) + (+1,3) + ( 5)

( ) , , 0.

, . ( 1,3) + (+5) + (+3,2) + + (+1,3) + + = ( 1,3) + (+1,3) + + + + (+3,2) + (+5) = 0 + 0 + 3,2 + 5 = +8,2

( )

. ,

. ( 3) + (+1) + (+3,2) + ( 1,1) + (+1,3) + ( 5) = 3 + 1 + 3,2 1,1 + 1,3 5 = 3 11 5 + 3,2 + 1,3 + 1 = 19 + 5,5 = 13,5

- - -

Page 141: A gymnasiou 14 15 a tomos

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4: –

1. : ( ) + ( ) +

( ) + ( ) + ( ) 0,3 + 0,15 ( ) 2,35 + 4,2

2. :

( ) (+8) + (+14) ( ) 3 + 25( ) ( 9) + (+15) ( ) +3 25( ) ( 13) + ( 15) ( ) 28 2( ) (+6) + ( 19) ( ) 24 + ( 30)

( ) (+16) + ( 16) ( ) 17 + 17

3. :

( ) ( 2,95) + ( 1,2) ( ) 1,20 + 2( ) + + ( ) 2 3 ( ) + ( 2,15) ( ) 1 + 4 ( ) 1 + ( ) 3 + 1

4.

:

5. ,

(+33) + = 16. , .

Page 142: A gymnasiou 14 15 a tomos

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4: –

6. :

( ) (+8) + ( 3,5) + (+3,5)

( ) ( 7,2) + (+3) + (+6,1) + ( 9) + ( 3)

( ) +61 + 29 41 29

( ) (+20) + (+14) + ( 1013) + (+6) + (+1013)

( ) ( 5) + + + (+12) +

( ) + + + + ( 3)

( ) 2 2 +

7. = 5 + 12 9 = +7 3 + 1,

+ .

8. , = 10, y = 7 z = 8.

( ) + | 1| ( ) 6 + |z| ( ) z + ( 5) ( ) y + z ( ) + + ( ) | + y|

9. : +3 +1 1 +1

0 2 1 3

10.

€500,75. ,

€315,45 €224,22.

11. : A = ( 2,4) + (+1,5) + ( 2014) + (+1) + (+2013) B = ( 3) + + ( 2,3) + += +

Page 143: A gymnasiou 14 15 a tomos

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4: –

12. , :

( )

.

/

( ) , .

/

( ) + = 0, , .

/

( ) .

/

13.

( – ). .

14. 6, 7, 5, 3, 1, 2, 4, 5, 9 , :

+ + = + + = + +

15. :

( ) 5 14 ( ) +2 3 5 + 12 15

( ) 4 + 3 15 5 ( ) 3 5 6 12 3

( ) 5 + 3 + 4 ( ) + 2 3 + 2 + 5

16. , : ( ) + = 0 ( ) + = ( ) + = 0 + =

Page 144: A gymnasiou 14 15 a tomos

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4: –

24

.

, :

, , .

5 .

Page 145: A gymnasiou 14 15 a tomos

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4: –

. . .,

. +2, 5.

o ,

(+2) ( 5) .

, (+2) ( 5) = (+2) + (+5) = +7

.

= + ( )

: (+3) ( 4) = (+3) + (+4) = +7 +12,5 (+2,3) = +12,5 + ( 2,3) = +10,2

-

. -

-

2 .

.

.

. .

Page 146: A gymnasiou 14 15 a tomos

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4: –

«+ : +( 4 + 2 1) = 4 + 2 1 = 3 « -

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

1. : ( ) (+9) (+12) ( ) 7,3 ( 15,14) ( ) (+3 4) ( 2 + 3 1) ( ) (+9) (+12) = (+9) + ( 12) = 3

+12,

, 12

( ) 7,3 ( 15,14) = +7,3 + (+15,14) = 22,44 15,4,

, +15,4.

( ) (+3 4) ( 2 + 3 1)

(+3 4) ( 2 + 3 1)

. = +3 4 + 2 3 + 1 = +3 4 = 1

. = ( 1) ( 3 + 3) = ( 1) 0 = 1

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2. 38,9 660,2 .

;

: 660,2 ( 38,9) = 660,2 + (+38,9) = 660,2 + 38,9

= 699,1

1. :

( ) ( 17) (+13) ( ) (+27) ( 8)

( ) (+5) (+7) ( ) 4 ( 19)

( ) ( 20) (+5) ( ) 19 ( 19) ( ) 0 ( 25) ( ) (+8) ( 10)

2. :

( ) (+1,2) (+2,6) ( ) 4,9 ( 1,9) ( ) ( 11) (+4,2) ( ) 1,5

( ) 15 14 ( ) 2

3. 3 :

( ) 10, 8, 6, 4, , , ( ) 10, 7, 4, 1, , ,

( ) 19, 14, 9, 4, , , ( ) 1 , 3, 4 , 6 , , ,

4. >, = , < ,

:

( ) 2000 . . . . . ( 2000) ( ) (+5) . . . . . (+5) ( 10)

( ) 2 + 3 . . . . . 2 4 ( ) 4 . . . . . 4 ( ) | 1| | 2| . . . . . ( 1) ( 2) ( ) 3 ( 2) . . . . . | 1 + 2|

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4: –

5. – N . 1962

235 , 2002 182 .

2002 1962.

6. , :

( ) ( 5) = +1 ( ) (+11) = +4

( ) 5 + = 13 ( ) +12 = 12

( ) | 11| + = +20 ( ) | 3 | = +10 ( ) + = ( ) 1 + = 2

( ) 2 + > 0 ( ) 7 > 0

7. :

( ) (+10) + ( 15) + ( 3) ( 1) (+7)

( ) ( 7) + ( 5) (+8) + ( 7) ( 15)

( ) 12 (11 3) + (5 7) (8 + 6)

( ) (13,7 2,6) + 14,8 ( 8,7 + 5)

( ) +

8. = 3 = + , :

= 3 + = | |

9. ,

. , .

( ) =

( ) =

( ) =

( ) + =

Page 149: A gymnasiou 14 15 a tomos

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4: –

10. . , ,

.

= ( 2,3 + 7,2) ( 5,9) = ( 104,99 4,8) ( 100 + 2,99) ( 100,3)

11. + ,

:

12. . . .. ,

, .

, 08: 00 ,

08: 00. ( )

. ( ) ,

17: 00;

Page 150: A gymnasiou 14 15 a tomos

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4: –

(1)

829,84 160 57

504 18

, , 5 .

, ,

. :

. : 4 ; 3 ;

, 20 ; , 45 ;

,

. : 4 ; 3 ;

, 10 ; , 40 ;

«A_En_4_Ginomeno_1

.

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4: –

(2)

«A_En_4_Ginomeno_2 .

.

. 0 .

.

. .

:

6. .

Page 152: A gymnasiou 14 15 a tomos

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4: –

+ : (+3) (+7) = +(7 3) = +21 ( 8)( 1) = +(8 1) = +8

: (+3)( 7) = (7 3) = 21 , , = ( ) = ( ) ( + ) = + ( ) = .

: ( 2)(+3) = (+3)( 2) [( 2)(+7)]( 1) = ( 2)[(+7)( 1)] ( 2)[( 4) + (+6)] = ( 2)( 4) + ( 2)( 6) = =

: ( 3)(+1) = ( 3) : ( 3)( 1)( 2) = 6 ( 1)( 5)( 4)( 2) = +40

(+3) (+7) (+3)(+7) ---

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4: –

0

=

:

4 .

1. : ( ) 3 ( 4) ( ) 2 (+2) ( ) 2 ( 3) ( ) 0 ( 2014) ( )

( ) 3 ( 4)

= (3 4) = 12

. , .

( ) 2 (+2)

= (2 2) = 4

. , .

( ) 2 ( 3)

= +6 . , .

( ) = =

. , .

( ) 0 ( 2014) = 0 . , .

( ) = + = +1

. , +1.

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4: –

2. 1000 .

30 000 , : ( ) 3 ; ( ) 3 ;

( ) 1000 , 3 : ( 3)( 1000) = +3000

, 30000 + 3000 = 33000 .

( ) 1000 , 3 : (+3)( 1000) = 3000. , 30000 3000 = 27000 .

3. , = 0,5. = +

= + = 0,5 0,5 + 0,5 = 0,5 (0,5 + 0,5) (0,5 0,5) = 0,5 1 0 = 0

-.

- -

.

4. : = ( 2)(+3)( 4)(+5) + ( 2)( 3)( 4)(+5)

. 2

. 3 . = ( 2)(+3)( 4)(+5) + ( 2)( 3)( 4)(+5)

= +(2 3 4 5) (2 3 4 5) = 0

, . , .

5. :

( ) 2( + 5) ( ) (4 6)

Page 155: A gymnasiou 14 15 a tomos

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4: –

.( ) 2( + 5) = 2 2 (+5) = 1 2 10 = 10

- , -

- .

( ) (4 6) = 4 ( 6) = 2 + 3 -

.

1. : ( ) 2 ( )

( ) 2 ( ) 3,2 5,1

2. : ( ) ( 7) (+2) ( ) ( 2) ( 5)

( ) ( 3) 0 ( ) 2 (+3) 3 ( 12) ( ) 5 ( 11)

( ) ( 6,2) ( 3) ( ) ( 1) 0

( ) 8 ( ) ( 2,1) ( 3,2)( ) ( ) 1

3. :

( ) (+3) ( 1) (+2) ( ) (+3) ( 1) ( 2)

( ) (+3) ( 1) ( 2) ( 2) ( ) (+3) (+1) (+2) ( 1)

( ) (+3) 0 (+2,5) (+1,2) ( ) ( 1) ( 1) … ( 1)

( ) + ( ) ( 1) (+1) ( 1) (+1) … (+1)

( ) 1 0 ( ) (+1) ( 1) (+1) ( 1) … (+1)

4. , :

+3 2,5 3 +7 1 +

Page 156: A gymnasiou 14 15 a tomos

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4: –

5. .

.

6. ,

:

( ) .

/

( ) - -

.

/

( ) .

/

( ) ( ) - .

/

( ) .

/

7. :

( ) 3 ( 6 + 8) ( ) 3 + 6 ( 2)

( ) 6 2 (5 + 4) ( ) 6 ( 2) ( 8)

( ) 2 ( 4) ( 1) ( 1) ( ) 3 ( 6) + 3 ( 8 + 2)

( ) 5 ( 4 + 3) 2 21 ( ) 2 ( 6 + 10) ( 15)

( ) ( 3 + 3) ( 3) ( 3) ( ) ( 1) ( 1) … ( 1) ( 1)

8. , :

( ) ( 3) = ( ) + + = ( ) = ( ) 2 = +1 ( ) ( 5) = 0 ( ) +2 = 2 (+1)( ) ( 3) = 1 ( ) ( 2012) = ( 1) ( 2)

( ) 1, 2, 4, 8, , , ( ) 2, 2, 2, 2, , ,

( ) 1, 10, 100, 1000, , , ( ) 1, 3, 9, 27, , ,

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4: –

9. 8 6 .

10. 2,5 , +9 , 0 , 1 , +10 , 2100. :

( ) , ( ) .

11. , = 1. = 2 + ( + 1) = (3 + 2)(1 )( + 0,5) = ( 2010)( 2011)( 2012) + ( 2012)( 2011)(+2010) = ( 2)(+3)(+5) + ( 2)( 3)( 5)

12. :

( ) 3( + 2) ( ) 3( 1) + 5( ) (6 + 32) ( ) 6 ( 5) + 7

( ) 5(3 2) + 7 ( ) (4 + ) ( + 2)

( ) 4( + 7) 6(4 ) ( ) 5( 11) + 2

13. 350 €6 €9.

( ) €6, :

i. €9 ,

ii. €6, iii. €9, iv. .

( ) 120 €6, .

14. := 2( 3 ) + 4(2 ) 2(3 2 ).

, + = 4.

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4: –

2 . ;

1 2 3 4 (+3) 3 ( 3) (+3) 9 ( 3) (+3) 27 ( 3) (+3) 81 ( 3) (+3) 243 ( 3) (+3) 729 ( 3) (+3) 2187 ( 3) (+3) 6561 ( 3) (+3) 19683 ( 3) (+3) 59049 ( 3) (+3) 14348907 ( 3) (+3) 43046721 ( 3) (+3) 3486784401 ( 3)

; , .

;

3 ;

Page 159: A gymnasiou 14 15 a tomos

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4: –

A > , > .

: (+7) = +343

, . < >

: ( 3) = +9 , . < < .

: ( 2) = 8

1. : ( ) 5

( ) ( 5) ( ) (+4) ( ) ( 1256)

( ) 5 , (5) .

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

.

( ) (+4) , (+4) .

( ) ( 1256) , ( 1256) (27) .

2. : ( ) ( 4) ( ) 4 ( ) ( 3)

( ) (+4,2) ( ) (+1) ( ) ( 1)

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4: –

( ) ( 4) = +(4 ) = 16

( ) 4 = (4) = 16

( ) ( 3) = 27( ) (+4,2) = 1( ) (+1) = +1

( ) ( 1) = +1

1. : ( 2)

6,3 2

2. :

3. :

( ) ( 1) = 1 ( ) ( 2) = 16 ( ) (+3) = 27 ( ) ( 1) = 1 ( ) 2 = 16 ( ) 6 = 36 ( ) = ( ) ( 2,7) = 2,7 ( ) (0,5) = 0,125

4. :

( ) ( 5) ( ) 2 ( ) ( )

( ) ( 1) ( ) ( 18)

( ) ( 3) ( ) ( 10)

5.

: ( ) 25 ( ) 8 ( ) 1

( ) ( 3) ( ) ( ) 2012 ( ) (+2,15)

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4: –

6. < , =, > . .

( ) ( 2) = 8 ( ) ( 5) = 5

( ) 3 = +9 ( ) ( 1) = 1

( ) ( 5) = +25 ( ) ( 1) = 1

( ) 3 = 1 ( ) =

( ) (3 5) > 0 ( ) ( 5 ) < 0

7. < , , > ,

:

( ) (+2) … … ( 2) ( ) ( 7) … … (+7)

( ) (+12) … … ( 12) ( ) (+113) … … ( 113)

( ) ( 2) … … (+2) ( ) (+4) … … ( 4)

8. ( 6) , 6 , ( 6) , 6

, .

9.

.

.

.

. .

10. = 1, : = 2 4 + 1 5

11. :

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

Page 162: A gymnasiou 14 15 a tomos

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4: –

(1)

, .

, 2 .

12 , ;

08: 00 2 ,

8 ;

03: 00, 08: 00 2 ;

(2)

12 15.

.

.

Page 163: A gymnasiou 14 15 a tomos

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4: –

.

+ : (+12) (+2) = +(12 2) = +6 ( 30) ( 5) = +(30 5) = +6

: ( 45) (+5) = (45 5) = 9 (+2,1) ( 7) = (2,1 7) = 0,3 = , 0

: ( 5) (+2) = ( 5) + = = = =

: = = = = = 5 – = = 0 , 0, 0

: = = =

--

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4: –

. , , 0

( + ) = + ( ) =

: [( 4) + (+6)] ( 2) = ( 4) ( 2) + (+6) ( 2)

1. : ( ) 12 3 ( ) 24 ( 3) ( )

( ) ( )

( ) 12 3 = (12 3) = 4 . , .

( ) 24 ( 3) = +(24 3) = 8 . ,

.

( ) = = 5 . ,

.

( ) = = +6 -

. . , .

( ) = · = = 2

, -

.

2. N 85 .

, = ( ) C ,

.

= , 0 = = , 0

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4: –

= ( ) = ( ) 85. = ( ) .

= .

= 65 .

3. : ( ) ( 3 + 1 + 2) ( 2)

( ) [ 52 + ( 6) ] ( 1) (8 9)

( ) .

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

. = (+1) 0 = +1

.

( ) ( )

. [ 52 + ( 6) ] ( 1) (8 9) = [ 52 + 36] ( 1) ( 1)= ( 16) ( 1) ( 1) = (+16) ( 1) = +16 + 1 = +17

. .

1. := = 20 5 = ( 4) ( 3) = ( 2) (+5) = 8 ( 10) = ( 12) ( 16)

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4: –

2. : ( ) ( 10) ( 2) ( ) ( 15) 5 ( ) 18 ( 3)

( ) ( 70) ( 7) ( ) ( 2,1) ( 3) ( ) ( 4,5) 9

( ) 50 ( 0,5) ( ) ( 12,24) ( 4) ( ) 80 ( 2)

( ) ( ) 1 : 5 ( ) ( 7) 3. 20 15.

4. +2 +4.

5. <, =, >,

: ( ) (+6) ( 2) . . . . . +5 2 ( ) ( 12) ( 3) . . . . . 12 ( 3)

( ) (+6) . . . . . 0 ( ) ( 1) . . . . . ( 1) ( 1)

( ) (+2) . . . . . 2 2 ( ) (+6) . . . . . + ( 10) 6. ,

: ( ) (+32) = (+8) ( ) ( 24) ( 6) = ( ) ( 25) = +5 ( ) (+24) = +2

( ) ( 3) = 4 ( ) ( 5 ) = 5

( ) ( 4) = ( ) (+3) = 9

( ) = ( ) = 2

( ) = +2 ( ) = 2

7.

.

= ( )

.

49 58 50 41 40 20

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4: –

8. : ( ) +1 ( ) 1 ( ) 0

9. :

( ) ( 12 4) ( 4) ( ) ( 9 + 3) ( 2 4) ( ) ( 2 + 3 1) ( 3 + 2) ( ) ( 2 5 1) ( 3 + 1)

( ) + + ( ) : ( 4 2)

( ) ( )

10. :

( ) ( 20) 2 ( 2) ( ) ( 21) + 6 3 ( ) 10 + 3 2 7 ( ) / ,

5.

11. , :

( ) ( 10). . . . . ( 2). . . . . (+1) = 21

( ) ( 5). . . . . ( 2). . . . . (+4) = 1

( ) 6. . . . . ( 7). . . . . (+2) = 44

( ) ( 8). . . . . ( 2). . . . . ( 2 ) = 9

12. :

( ) 45 35 ( 5)

( ) 15 ( 2) ( 1 + 3)

( ) ( 2) ( 3) ( 12) ( 1)

( ) 127 25 ( 1 + 2)

( ) 2 [ 3 2 ( 1 1)] ( ) 2 + 2 + ( 1)

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

( ) ( ) ( )

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4: –

13. , = 12, = 4 = 6:

( ) ( )

( ) ( + ) ( + 2) ( ) ( )( + )

( ) ( ) ( )

14. , 11: ( 5) + (+4) ( 2) ( 1) ( 3)

15. , .

.

16. = 2 = 3, :

( ) 3 + 2 + ( ) ( ) ( 1) +

17. : ( ) 3( 3) ( ) 6 ( 2 + 1) ( ) ( 4) + 3 1 (+2) ( ) (2 3) ( 5) + (3 5) ( ) ( 7) ( 7) + ( 3) + 5 1

( ) 7,25 ( 1 + 2) + ( 3) ( 4,2)

( ) ( 2) ( 2) ( 5) +2 ( 3)

( ) ( ) ( )( )

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4: –

1, , , … , .

1 1 0,111111111111 …. 0,5 0,1

0,333333333333 … 0,090909090909 …

0,25 0,833333333333 …

0,2 0,076923076923 … 0,1666666666666 … 0,0714285714285714 …

0,142857142857 … 0,066666666666666 … 0,125 0,625

.

,

2 / 5, .

.

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4: –

: 0,333 … = 0, 3 , 3 1. 34,23181818 … = 34,2318 , 18 2.

: 0, 3 34,2318

: = 0, 3125 , = 0, 416 ( 10 = ( , , 0).

= 2, 171

10 3, 172

-

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4: –

1. : ( ) 2 ( ) ( )

( ) 2 2 = 2 = 2,6.

( ) 5 8

, = 0,625

( ) ,

5 3 .

5,000 3

, = = 1,666 = 1, 6 3 1,666 20 18 20 18 20

2. 0, 3 .

= 0,333 … . 10 = 3,333 … 10. 10 = 3,333 … 0,333 … 9 = 3 =

-

. 0, 3 =

5,000 8 48 0,625 20 16 40 40 0

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4: –

3. 0,586 . = 0,58666 … . 100 = 58,666 … 100 -

. 1000 = 586,666 … 1000.

1000 100 = 586,666 … 58,666 … 900 = 528 = =

-.

0,586 =

1. :

( ) 4, 7 ( ) 2, 3

( ) 3,417 ( ) 0,28

2. 1 , . 3. :

( ) 1, 1 1,11111111

( ) 3,68 3,68

( ) 4,912 4,9102

4. :

0,69 0, 69 0,069 0, 6 0,69

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4: –

;

13 10

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4: –

.

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4: –

: 2 6 = 3 2 . 6 3 .

:

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

, .

.

+ = + = =

+ =

2 8 = +15 3

= 2

: 2 + 3 = 8 + 15 =

-

, ( ).

.

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4: –

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

.

4 + 3 = 5 5 3 4 5 + 3 = 5 3 2 = 8 . = . = = 4 .

2. : + 1 =

+ 1 =+ =

[3,2,1] = 6 - . = ( )

-. 4 + 6 = 3 2( + 3) .4 + 6 = 3 2 6

.4 3 + 2 = 6 6

. 3 = 12 33 = 123 . = 4 .

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4: –

3. 30 5 10 .

( ) .

( ) €2,40, .

( ) 5 ,

10 30 .

5 : 10 : 30

, 5 : 5

10 : 10(30 ) = 5 + 10(30 ) = 5 + 300 10 = 300 5

( ) = 240 300 5 = 240

5 = 240 300 5 = 60 = = 12

, 5 12, 10 18. 4. ,

= . 3

, 7

. , 38 .

= . , 2 3,

3(2 3) 7. = = = = 2 3 = = 3(2 3) 7 = 6 9 7 = 6 16

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4: –

= + + + + = 38 + 6 16 + + 2 3 + 2 3 = 38 + 6 + + 2 + 2 = 38 + 16 + 3 + 3 12 = 60 1212 = 6012 = 5

, = = 5 = = 6 16 = 14

1. . ,

( ).

( )

( )

Page 179: A gymnasiou 14 15 a tomos

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4: –

2. : ( ) 7 + = 5 ( ) 4 = 16

( ) 6 = 3 ( ) 3 4 = 19

( ) 11 = 4 1 ( ) 5 + 2 = 9 2

( ) 8 4 + 3 + 26 = 3 4 + 2 ( ) 2 + 4 = 2 + 4

3. : ( ) 5( + 1) = 3(3 1)

( ) + (5 7) = 5

( ) 5( 2) 2( 1) = 11 ( ) 4( 3) 5 = 0 ( ) 7( 3) 12 = 13

( ) 3( + 2) ( 2) = 4( + 1)

4. :

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( ) = 1 ( ) =

( ) = ( ) =

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5.

:

( ) 3 9 = 18 ( ) 2 = 6 ( ) 5 = 9 3 = 9 = 6 + 2 = 9 5 = 3 = 8 = 4

6. , 10.

7. , , = 5 + 9, = 7 + 5. .

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Page 181: A gymnasiou 14 15 a tomos

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( ) = ( ) ( 25) = 25 ( ) ( 4) = 64

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:

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5. :

, , ,

Page 182: A gymnasiou 14 15 a tomos

182

4: –

6. <, =, >, :

( ) 0 . . . . . 1 ( ) 5 . . . . . 6

( ) 5 . . . . . | 5| ( ) 5 . . . . . 12

( ) +5 . . . . . 5 ( ) 3 . . . . . 3

( ) 2010 . . . . . 2011 ( ) 3 2 1 . . . . . (+3)

( ) 9 + 2 . . . . . ( 1) ( ) ( 3 + 1) . . . . . ( 16) ( 4)

( ) 3 ( 2). . . . . 3 ( 2) ( ) 0 ( 1) . . . . . | 1| 1

7.

:

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: ( ) (+14) + = +5 ( ) ( 2,4) = 9,2

( ) ( 20) = +100 ( ) ( 8) = 1

( ) ( 2 1) = 1 ( ) (+2) = 4

( ) 12 + = 7 ( ) +2 = +7

( ) (+3 1) = 6 ( ) (+2 4) = | 2| ( ) = 2 + 3 ( ) = 2014 + 2014

9. , .

.( )

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

1 5

5 1 0 3

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10. , : = 3 + = 3 = 3 = 3

11. 48

28 .

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( ) ( 6) ( 3) + ( 4) ( ) ( 5) 4 ( 3)

( ) 8 ( 7) 8 7 ( ) 11 4 + ( 8) ( 3)

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

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

Page 184: A gymnasiou 14 15 a tomos

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4: –

15. : ( ) ( 1) 1

( ) : 1( ) ( 2) (+1) 7

( ) ( 1) 2 (+5 1)( ) ( 2) ( 2) ( 3) ( 3 + 4)

( ) ( 4 + 2) 4 [ (+5) ( 3) ] + ( 2)

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/

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/

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/

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/

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/

( ) .

/

17. :

. . i. = 0

. . ii. < 0

. iii. > 0

. . iv. = 1

v. + = 0

Page 185: A gymnasiou 14 15 a tomos

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4: –

18. < , = , >, . < 0, > 0 > 0 , :

( ) | | … … 0 ( ) … … 0

( ) … … 0

( ) … … 0 ( ) … … 0 ( ) … … 0

19. , ( , 0) :

( ) < 0,

:

( ) > 2, :

( ) ( 2) = 0, :

( ) ( )( )( ) > 0, :

20. o , : ( 1) + ( 1) + ( 1) + ( 1) = 0

21. = 3, : = 2 ( 5) 4 ( 2) 4 + 6 = + 5(3 ) 4 4(5 2 )

22. + = 5 = 3,

: ( ) 3 + ( 3 ) ( ) 5 ( ) + 4

23. : ( ) + 6 5 = 14 2 ( ) 16 = 14

( ) 2(y 1) = 3y + 4 ( ) ( 1) = 2

( ) 3 (2 + 1) = 1 ( ) 2 3(2 ) = 4 8

( ) = 2 ( ) =

( ) + = 1 + ( ) + = 2

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4: –

24. , .

25. 27 . 3

. .

26. 115 ( , ).

5 , , .

27. €160.

, €20 .

;

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€20 . .

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, 100 . 5 50,

100 . ;

30. . €20 , .

;

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4: –

1. = 11

+ ( 2) 2. , , , – > 0, = 222 < 0,

.

3. | + | = | | + | |.

.

4. . 100 7 , 0 3

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5. : ( 200 + 199) ( 199 + 198) + ( 198 + 197) ( 197 + 196) + + ( 2 + 1) ( 1) 6. , 34

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7. , : |2010 | = 2010 > 0

Page 188: A gymnasiou 14 15 a tomos

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4: –

8. ,

.

.

9. 4 ( + 3) = ( 2) 7 + 5 = 2 + 3, .

10. , :

( 1) + (+1) ( 1) + 1 + ( ) + ( )

11. = 1 = 2, : = 4 + 3( + ) + 3(3 + )

12.

, ier in i:

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190

:

Page 191: A gymnasiou 14 15 a tomos

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:

11

1. ( ) 258, 285, 528, 582, 825, 852

( ) 258, 285, 528, 582, 825, 852

( ) : 285 , : 8522. ( ) : 982 , : 203

( ) : 9832 , : 2039

( ) : 98320 , : 20389

3. ( ) ( )

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4. ( ) ( )

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0, 12 ( )

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8. ( ) 26,65 ( ) 10

( ) 7,82 ( ) 16,328 9. ( ) 140 ( ) 35,28

( ) 778 ( ) 184 ( ) 5148 ( ) 18,6

( ) 7,875 ( ) 18

10. ( ) 5700 ( ) 8200

( ) 46000 ( ) 0,09

( ) 110 ( ) 200011. ( ) 81 243 – 729 ( ) 83 65 47

( ) 6,5 7,1 7,7 ( ) 800 400 200

12. ( ) 100 ( ) 0,0574

( ) 0,01 ( ) 0,00001

( ) 57400 ( ) 0,53

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:

13. 0,045

14. ( ) , , ,

( ) , , ,

( ) , , ,

15. = 806600 = 80,66

= 8066 = 8066

16. ( )

i. 8 ii. 8 iii. 3 iv.

17. ( )

i. ii. iii. iv.

( ) 5654 ( ) 6120

( ) 37962

18. ( ) , ( )

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1. ( ) > ( ) <

( ) = ( ) <

2. : : : : 524

3. ( ) 2 ( )

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( ) ( )

( ) 10 ( )

( ) ( ) 5

4. 171000000 5. + + + +

6. = 0,9 = 1,6 = 2,5 = 0,56257. 20

8.

9. ( )

Page 193: A gymnasiou 14 15 a tomos

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:

10. : : :

1

1. ( ) = 20 , = 25

( ) = 26 , = 40

( ) = 20 , = 20

( ) = 18 , = 14

2. 1800

3. ( ) ( )

( )

4. 8

5. 140

6. 1

7. ( ) 135,2 ( ) €500,24

29

1.

= { , , , , }, ( ) = 5 = {86, 88, 90, 92, 94, 96, 98} ( ) = 7 = { } , ( ) = 0 = {18, 27, 36, 45, 54, 63, 72, 81, 90, 99} ( ) = 10 = { , , , , , , } ( ) = 7

2.

: : 1,2,3 : 11

3. ( ) ( )

( ) ( )

( ) ( )

( ) ( )

5. , , , ,

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:

37

1. ( ) = {3, 4, 5, 6, 7, 8} ( ) = {4, 6}

2.

3. ( ) = { , , , , , , , , , , } ( ) = { , , } ( ) = { , , } ( ) = { , , , , , , , , , }

4. ( )

( )

( )

5. = { , , , , } 7. ( ) . . { , , , } { , , , , } = { , , }

( ) {22 ,33 , 11} {11 , 22, 55, 33 , 66} = {11, 22, 33, 55, 66} 8. 8

9. ( )

3

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

( )

3. ( ) 5, 3 1,2 ( ) , , ,

4. . . = {1,2} = { , #} 5. ( ) . . = {2,4,6,8, … } ( ) . . = {1,2,3,4,5} 6. ( ) ( )

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:

( )

1. = { , , , } ( ) = 4 = { } ( ) = 0

= {7,0,1} ( ) = 3 = { , } ( ) = 2 = { , , , , , , } ( ) = 7

2.

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7. ( ) , , , , , ,

8. ( ) iv ( ) iii

( ) i ( ) ii

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9. , , , ,

10. ( )

( )

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1. 3 2. ( ) . . , = , = , …

Page 196: A gymnasiou 14 15 a tomos

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:

2

3. ( ) ( )

4. ( ) ( ) = ( ) + ( )

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

5. ( ) 45

6. ( ) ( )

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1. ( ) 3 ( )

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2. ( ) 49 : 7 : 2 ( ) 144 : 12 : 2

( ) 8 :2 : 3 ( ) 10000 : 10 : 4

( ) 243 : 3 : 5 ( ) 1 : 1 : 3

( )1234 : 1234 : 1

( ) 1 : 5 : 0

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3. ( ) 2 ( ) 7

( ) ( ) 10

4. ( ) . . 1 , 1 , 2014 , …

( ) 8 4 2 ( ) 9 3 81

( ) 16 4 2

5. ( ) 390625 ( ) 20736

( ) 117649 ( ) 4096

6. 1 < 5 < 2 < 5

7. 5

8.

: 21 2 = 1048576 €10484,76 20 €5242,38, . . . ,

€15000 .

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62

1. ( ) 2 ( ) 4

( ) 5 ( ) 8

( ) 8 + 1 ( ) 13 5

2. ( ) ( )

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3. ( ) 8 ( ) 2

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( ) 10 ( ) 81

10. ( ) 14 ( ) 5

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( ) 112 ( ) 26 11. ( ) 2 (5 + 2) = 11 23

( ) 22 (7 5) 2 = 2 3 4 12. 6, 2, 5, 3

57

1. ( ) 13333312. 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010

3. ( ) 6 ( ) 15

( ) 9 ( ) 127

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4. ( ) 1010 ( ) 1000

( ) 1110 ( ) 10001

( ) 1110 ( ) 101111

( ) 110100 ( ) 1000011 5. ( ) 100, 101, 110

( ) 1110, 1111, 10000 ( ) 1100, 1101, 1110

6. ( ) : 100 : 111 ( ) : 4 : 7

7. : 8, 9, 10, 11, 12, 13, 14, 15

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:

( ) 5 ( ) , 10

( ) 2 ( ) 3

4. ( ) = 4 ( ) = 6

( ) = 3

5. , ,

6. ( ) + 2 ( ) €14

( ) 2 ( )

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( ) €20

7. = 4 + 3

8. ( ) : 3 :

( ) : 3 + 7 : + 7

( ) : 3 6 : 6

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67

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( ) i. = 44 ii. = 77

2. ( ) 36 ( ) 9

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( ) 26 ( ) 29

3. ( ) : 8, 10, 13, 15, 20

: 44, 54, 69, 79, 104 ( ) = 5 + 4

4. ( ) 22 ( ) 42

5. ( ) = 20 ( ) = 4

( ) 32

6. = {5,10,15, … } = {1,3,5,7, … } = {5,9,13,17,21}

7.

= {2 } = { + 1 } = { } = { | < 5}

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:

8. ( ) : : = 100 + + 2 ( ) = 357 €3,57

– 72

1. ( ) ( ) 2. ( ) ( )

( ) ( ) ( )

3. ( ) = 9 + ( ) = ( ) = 4 ( ) = 13 + ( ) =

4. ( )

( ) 6. 2012, 2013, 2014

H 78

1. 10 , 11

2. ( ) = 7 ( ) = 30

( ) = 15 ( ) = 64

( ) = 3125 ( ) = 510

( ) = 57 ( ) = 5

( ) = 5 ( ) = 15

3. ( ) = 11 ( ) = 7

( ) = 3 ( ) = 6

( ) = 3 ( ) = 25

( ) = 15 ( ) = 2

4.

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

5. . . + 5 = 15, 5 = 50, + 2012 = 2022,

100 = 10, = 9 , …

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:

6. ,

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9. €13680

10. 10

11. €12

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14. : 2 : 8

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18. = 10

81

1. 2

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( ) = 5 ( ) = 9

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( ) 12 144 ( ) 2 4 16

5. ( ) 122 ( ) 0

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6. = 17, = 27, = 7

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:

7. ( ) 3 ( ) 6

( ) 15 ( ) 18

8. ( ) 111 ( ) 10010

( ) 1110101 ( ) 10110101

9. ( ) 6 ( ) 2 + 4

10. ( ) ( )

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( ) = + 1080

12. ( ) , 6 = 11( ) , 4 = 20

( ) , + 5 = 8

( ) , 7 = 12

( ) , 6 = 54

( ) , 4 + 12 = 26

13. 3, 9, 27, 81, … 14. : 36 : 12

15. 194, 195

17. 242, 243, 244

18. = 16 , : 4 , = 16

19. = 26, = 40

20. €4

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2. . .

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4. ( )

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:

724 = 1011010100( ) = 1324( ) ( ) . i. ii.

( ) . i. 736( ) = 1050( ) . ii. 9

5. ( ) 11110110( ) ( ) 10011111( ) ( ) 1100100( )

8. ( ) 580

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11. 2 ,2 , 9

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:

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1. ( )

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8. = 12, 24, 36, ….

1

1. ( ) 1520, 4404, 3850 ( ) 765, 4404

( ) 1520, 4404 ( ) 765, 1520, 3850

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3. ( ) 2 3 4 6 7 8 9 ( ) 0 2 3 6 8

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( ) 0 ( ) 4 5

4. ( ) 100 ( ) 100

( ) 102 ( ) 108

( ) 100 ( ) 105

5.

6. 0 3 9

7. 90

8. 102

9. 6

10. : 64 : 90

14. ( ) 37 , 111, , 3 , 37

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:

11

1. 56 = 2 7 , 108 = 2 3 , 73 = 73 , 84 = 2 3 7 , 65 = 5 13

: 73 2. 42

3. 30

4. . . 2 + 3 = 5

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13.

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3. ( ) ( )

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:

( )

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:

13

1. ( ) < ( ) <

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:

( ) 3 2 = 5

6. ( ) +8 ( ) 10,1 ( ) +20 ( ) +40

( ) +7 ( ) 3

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( ) 15,2 ( ) 1

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3. ( ) 2, 0, 2, ( ) 2, 5, 8 ( ) 1, 6, 11 ( ) 7 , 9, 10 ,

4. ( ) < ( ) <

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2. ( ) ( )

( ) ( )

3. ( ) + ( ) +

( ) + ( )

( ) ( )

( ) + ( )

( ) +

4. ( ) 25 ( ) 16

( ) ( ) ( ) 1 ( ) 1

( ) 27 ( ) 100

5. ( ) 25 = ( 5) = (+5) = (+25)

( ) 8 = ( 2) = ( 8)

( ) . . 1 = ( 1) = (+1) = ( 2012)

6. ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

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:

7. ( ) = ( ) =

( ) > ( ) =

( ) < ( ) >

8. ( 6) < 6 < ( 6) < 6

9. : 0 , 1 : 0 , 10

: 0 1 10. = 16

11. = 0

165

1. = 5, = 4 , = +1 , = , = , = + < < < < <

2. ( ) +5 ( ) 3

( ) 6 ( ) +10 ( ) +0,7 ( ) 0,5

( ) 100 ( ) +3,06

( ) 40 ( )

( ) ( ) +98

3. ±1 , ±5

4. . . 4, +2, +6

5. ( ) < ( ) >

( ) < ( ) =

( ) > ( ) >

6. ( ) +4 ( ) +4

( ) 5 ( ) +12

( ) +12 ( ) +25

( ) + ( ) 3

( ) 8 ( ) +2

( ) 24 ( ) . . …

7. : 45 : 50

8. ( )

( )

( )

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9. ( ) +4 ( ) +1

( ) 0 ( ) +4

( ) ( ) +

( ) ( ) +

10. ( ) 12 ( ) 19

( ) +9

( ) ( 20) [2 ( 2)], [( 21) + 6] 3, 10 + 3 (2 7)

11. ( ) ( 10) ( 2) + (+1) = 21 ( ) ( 5) ( 2) + (+4) = 1

( ) 6 ( 7) (+2) = 44 ( ) ( 8) ( 2) ( 2) = 9 12. ( ) 38 ( ) +19

( ) 6 ( ) +102

( ) 0 ( ) +4

( ) 1 ( ) 2 ( ) +7 ( ) 2

( ) ( )

13. ( ) 3 ( ) 2

( ) 3 ( ) 160

( ) 30 ( ) 7

14. ( 5) + [(+4) ( 2) ( 1)] ( 3)

15. ( ) ( )

( ) ( )

( )

16. ( ) +9 ( ) 19

17. ( ) 27 ( ) 7

( ) 16 ( ) 3

( ) 2 ( ) 4

( ) 31 ( )

172

1. ( ) ( )

( ) ( )

2. 1, 27 , 0,375 , 0,83

3. ( ) 1, 1 > 1,11111111 ( ) 3,68 > 3,68

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:

( ) 4,912 > 4, 102

4. 0,069 0, 6 0,69 0, 69 0,69

178

1. ( ) 2 ( ) 6

2. ( ) = 12 ( ) = 4

( ) = 3 ( ) = 5

( ) = 3 ( ) = 1

3. ( ) = 2 ( ) =

( ) = 19 ( ) = 2

( ) = 10 ( ) = 2

4. ( ) = 3 ( ) = 6

( ) = 2 ( ) = 3

( ) = 9 ( ) =

( ) = 8 ( ) = 2

6. : 5

7. : 57

8. : 7 cm , : 34 cm , : 10 cm , : 35 cm

9. : 140 , : 80, : 60

10. = 16 cm = 15 cm

11. : €19 : €7

13. : 10

14. 43

181

1. ( ) +6 ( ) 5

( ) +1 ( ) 5

( ) +6 ( ) 4

( ) +2 ( ) +60 ( ) +2 ( ) +5

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:

2. ( ) + ( ) +

( ) + ( ) +

( ) ( )

3. ( ) ( )

( ) ( )

( ) ( )

( )

4. ( ) 9 ( ) 1

( ) 16 ( ) 1

( ) ( ) 16

( ) 1 ( ) 25

( ) 1 ( ) 0

5. ( ) 1,5 ( ) 4, 3

( ) 1, 81 ( ) 0,875

6. ( ) > ( ) >

( ) < ( ) >

( ) > ( ) >

( ) > ( ) <

( ) < ( ) <

( ) > ( ) =

7.

8. ( ) 9 ( ) +6,8

( ) 5 ( ) 8

( ) ( ) +8

( ) +5 ( ) 5

( ) 3 ( ) 4

( ) +2 ( ) 0

9. ( )

( ) : 2 > 7 , | 2| < | 7| ( ) : |3 ( 1)| |3| | 1| ( ) : |3 ( 1)| 3 ( 1)

( )

10.

11.

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:

12. ( ) +12 ( ) +4

( ) +14 ( ) 17

( ) 0 ( ) 20

( ) +8 ( ) 6

( ) +10 ( ) 14

( ) +1 ( ) +

13. iii.

14. ( ) = 5 ( ) = ±12 ( ) = 0 , = ±1 , = ±2

15. ( ) 0 ( )

( ) +2 ( ) 33

( ) 19 ( ) 4

16. ( ) ( )

( ) ( )

( ) ( )

( )

17. A , i , ii , iii

18. ( ) > ( ) <

( ) < ( ) <

( ) < ( ) >

19. ( ) ( )

( ) ( )

21. = 26 = +7

22. ( ) +9 ( ) 25

23. ( ) = 2 ( ) = +3

( ) = 6 ( ) = 1

( ) = +1 ( ) = 2

( ) = 10 ( ) = +7

( ) = 6 ( ) = +2

24. 121

25. : 33 : 6

26. : 80 : 20 : 15

27. : €60

: €20 : €20

: €60

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:

28. : 10 : 40

29. : 30

30. : 36 : 12

16

1. 17

2. < 0

3.

4. : 30 10

5. 0

6. ( ) : 16 : 18

( ) 6

7. ,

8.

9. = 3

10. 4

11. 32

12. 19683

Page 218: A gymnasiou 14 15 a tomos

{ } =

{1,2,3,4, . . . } {0,1,2,3,4, . . . } {0, ±1, ±2, ±3, ±4, . . . } {1,2,3,4, . . . } { 1, 2, 3, 4, . . . } { / : ,

/

Page 219: A gymnasiou 14 15 a tomos

o

=

·

Ans

^

SHIFT 2nd F

AC

DEL

Abs

(-)

a b/c 23

S D a b/c

CLR

M+

M-

M

Page 220: A gymnasiou 14 15 a tomos

3 + 4 3+4 7 2,34 – 1,1 2.34 – 1.1 1.243 2 3 2 6 2 2 2 32 5 10 5 3 5 103 5000

3 4 2

2

2 3 4 4 4 2 5 5 25 2 (7 3) 2 (7 3) 8 2 (7 3) 2 8 2 ( 3) 2 ( 3) 5 | 3| | 3| 3 0,25

9 6 2

9 – 6 2 + 6

3 (9 6: 2) 6: (9 6: 2) 3 – 6 17

+ = 3 4

· – 2 3 4 1 · 1 =

3 x 2 =

2 5 =

5 P 3 =

3 a b/c 4 =

3 a b/c 4 =

SHIFT 23 2 3 4 =

2 a b/c 3 a b/c 4 =

4 =

5 X² =

= 3 – 7 ( x 2 )

x 2 = 3 – = 7 Ans

2 ( (-) 3 ) =–

= 3 (-) Abs

S D

a b/c SHIFT

= 2 ÷ 6 – 9

3 – 6 ÷ =