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: , 1. .

2. , , . , , , - , 3, , 4 , 5 . - 6, , , , . 7.

, , . 8 9 . 10. ; . , , , , , .

:

. ... .... ( ), ,....

, , 286 , 10-15 , , .

. ( )

, 986

1 , , . ( . , , ,26)2 - . , .3 . , , , 81, 15-254 >1.5 . , ( )6 . , , 58-59 , 142 ..7 . , , 142 . , , .8 , , . . .9 . www.live-pedia.gr 10 , . , , , , . , , . . http://oasigr.5.forumer.com/a/_post503.html ( )

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( ) . ;

. . . 189

11 , , , . - , 12

. , ( )

, 1002, 30-35

. , 13

, 1072, 25-30

: , , 14, , , . . , , , .15. , 16. , , - - 17.

, ,

11 . 152 e . (153 )12 . www.padamakos.gr http://users.otenet.gr/~adampan/ / , , , , 5 .. , , . : . ; ; ;13 : , ( ) ( ).14 , .. , , . 18, 19 .15 .16 , .17 : ,

, 1084, 40 , . 21 , .

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18, 19. , , 20 . ,

. . , . 118

; , , 21. , . ; ; ; 22; ; 3,4,5 ; 23 ; ;

:

. . , , , , ,

, 1079, 30-40 1080, 20

, . ; , , ; , . , , 24, , , , , .

18 / , , .19 ( - )20 .21 . , , 9922 , , ( 7 , . . 7 , 7 )23 , , , - : , . , , , . : , , .24 e . !

3

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, . ( , )

, , - , . . ;

, , .

. , . , .

: , , (3,4,5) . , , . , 25. , . , . , ( ) , . , . . !

= hc

e 2(

1371 )

e, h, c, G e=1.602*10 -19 C, h=6.626*10 -34 J.s, c=2.998*10 8 m/s, G=6.673*10 -11 m3.Kg -1.s-2

: , , , 26, - - , , , . , , , 35. T , , , , .

1 (1,3) (3,4,5)1 (1,3) - 27 (3,4,5) ( ),

: , : 1,3,5 28 1,3,3 1,3,4 ( 29) , (3,4,5).

25 . . . 6 1,2,3 1+2+3=626 - : , 27 , , 57-58-5928 , , . , 51, 20-25 .[ : 1 3 , 5( , 1,3,5), 4 ( 1,3,4), . 1,3,3 , 3 . .29 , , 16, 5. (= )

4

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30 . 1. .

, .. , , 29, 25. . , , . + , + ( - - ) . , 3.

( ) 31

, . 396 b. , , o T . . , , ( ), . 405-406 , 51 55. , . 30,31 - .

, , , , ( )

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

.

30 , ,114. , , , .31 ( ) . , , . , .

. , 16-17 15

1

. , 1 . , 3, ( )

(1,3) ,

29, 30, 35, 37

1 1 1

. ( )

1, 3, 5 1,3, 31,3 ,4

5

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:

(,,) ,, , (,,)=1 2+2=2 ,

( , ) 1.

=3 : 3 2+2222

213

213

+=

(3,4,5)

=5 (5,12,13) . .

6

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2222

2 14

14

+=

+

( = .4)

/ ( , )

( , , ) ( + )/4 +(-1) ( * )

1 1,3 5,4,3 (5+3)/4=2= 2.1 2 2

2 3,5 17,8,15 (17+15)/4=8= 2.2 2 6

3 5,7 37,12,35 (37+35)/4=18= 2.3 2 6

4 7,9 65,16,63 (65+63)/4=32= 2.4 2 10

5 9,11 101,20,99 (101+99)/4=50= 2.5 2 10

6 11,13 145,24,143 (145+143)/4=72= 2.6 2 14

7 13,15 197,28,195 (197+195)/4=98= 2.7 2 14

! , , . . 7 , 32.

( ) 33, . , (1,3) . . . . (1) - (3). , (1,3) .

, , .

. , : , , (2)

32 . , 391 F33 , . Highs, . , - , , . . . , , , , .

8

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(2) =2

22 + , =2

22 = .

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34, -

. : ,

- , . 30, 20-25. ,

, .. , , . , , 35. , , , , , , . . 36 , 10, , 386. , 5. , () . 5, , 5 , : .

. . 37, , , . , . , , , , 38. . , , , . . , 39.. . , , , , . . 40;|=1 ||=2, |||=3 , ||||=4, =5 , |=6, ..., =10, |=11, ||=12,. |||=18,

5 . , , ,

34 . . . , . ( = ). , , , , ! ( . ( ), . , 11)35 ( ) , , 391 C36 37 , .38 . , , . , , . , , 385-38639 , , , , 391 D40 - 1 | .

10

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. , .

, , , , 1 3, , (1,3) . . | |||,

, , . . 1 3. . ;

41 . , , , , , , , ! , , .

. , 1, .( ) . , . . 189. , . ( . 44, F 309)

, , , " " , , 1, (1,3) , 6. : , , 391 c.

, , ,

. , , , 81, 15-25

41 ...

,

.

11

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12

(1,3)

72 o

216 =

216 3

0 72

3 , ( 63)

72=

; !

!

,

!

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( ) .

, , (),

, . . , . : 1 (1,3) (3,4,5) , (1,3) (3,4,5) (2) . (3,4,5) , (3 , 4 , 5 ) , : (3,4,5), (6,8,10),(9,12,15), (12,16,20), (15,20,25), (18,24,30), (3,4,5). : 1 (1,5) (5,12,13) : (5,12,13), (10,24,26), (15, 36, 39), (20, 48, 52), (25, 60, 65), .,

(5,12,13).

, , : .

( ) ( ) .

- , , , (5,12,13) :(10,24,26), (15, 36, 39), (20, 48, 52), (25, 60, 65), .. .

, , , , - .42 , , - . ;

, . , ! , , , - , . () , .

E ( .)

3 : 42 . . , 60 87. () .

13

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31. (,,) :

=+ ()

43

( )

33. () [] ( ). (,,) ( , ) :

+ ,

21,

21 22 (1, ).

():22

2..

2..

...

+=

+ . . 254 , . .

, .

53 - .

() []:( )

( 5,4,3 ) (5,4,3) 44 - ( ) - ( ) - (17,8,15) :

=+

:

53

81715 =+ 5

32515 = [] : [ ]

22

25.325.15

25.325.15

5.3.25.15

+=

+ 752+180 2=195 2 - (13,

12, 5), (75,180,195)=15

- (37,12,35) :

=+ :

75

123735 =+ 7

54935 = [] :

22

25.735.49

25.735.49

5.7.49.35

+=

+

245 2 + 840 2 = 875 2 - (25,24,7) 3 (875,840,245)=35, (65,16,63) (41,40,9) (101,20,99) (61,60,11) (145,24,143) (85,84,13) (197,28,195) (113,112,15)

, , - , , ! www.padamakos.gr http://users.otenet.gr/~adampan , - , , , , , , .

43 . !44 () ()

14

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

(2) .

=2

22 + , =2

22 = . 2=2+2 , . .( + )( - ) 24 . , 3, : . .( + ).( - )= .( .3).( .4).( .2)= .24 . .( + ).( - )= .( .3).( .2).( .4)= .24, 1, + - 2 .4 ( . 1)

. , 3, =3+1 =3+2 =3 +1 =3 +2, :. =3+2, =3 +2. =3+2, =3 +1. =3+1 , =3 +2. =3+1, =3 +1 , + =3+2+3 +1= .3, - =3+1-3 -1=3 -3 = .3 - =3+2-3 -2=3 -3 = .3 + , - .3, . ( + ).( - )= .24 2

{ . . + = .3, : . ( + )( - )= . [( .3).( .2)]( .4)= .24 . ( + )( - = . [( .3).( .4)]( .2)= .24,

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( + ),( - ) 2 4 ( 1), - = .3},

E=( + )( - ) /4 = .24/4= .6

. . , . 46 .-

.- . 2002 .

12 = = = 12 , 12 , 12 .

: , 6, , .

: 3, 24

: , , , , ., , , , , , , , ,, (3,4,5) , 6, - , , .

.

5. (2). :..= . .

2.

2

2222 +. 5 .

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

=5 +1 =5+1, 2- 2= ( + )( - )= .5 =5 +1 =5+2, 2+ 2= .5 =5 +1 =5+3, 2+ 2= .5 =5 +1 =5+4, 2- 2= .5

5 .

, 5.12=60 45

45 ., , , 6, 12

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: . ! ! - !

1 2 3 4

, , , . . . , , . , , .

A

A

5

3

46, . , . .

: =3,2 =1,8 =1,08 =1,44 =1,92 =2,56 =1,44 =2,4567 =0,77776 =1,3824 =1,3824

: 16049382,37776,04567,22

1==

=

1=1,777777777

777777,13824,14567,122

2==

=

46 ,

17

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2=1,333333333

13824,13824,12

3==

=

3=1 1, 2, 3 . : 100( 1. 2- 3)=137,0368741 !

, , .

. , , .

, ,

.

. - . . , . ;

. . ; . , , .

, . . . ,

Highs , , , , , .

, , , , . , , . .

. - - , , , . .

, , , , , . . 47 , , , , .

- - , , , .

; - . , , .

47 , , 39, 5

18

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. . , , , , , . , () ( ), Highs. 48

. , . . . 49, , , , , . , . , , , , . 156 . 50.

. . 51 . .

, . , , , . ,

. , ( * ) . , , . . , , , , , , . . . , , . , . , , . , . ,

, . , , . .

(**) , , , ,

37, 10-11 . , .

48 49 13, 222b 15-16. . . 41, . 50 51 . 84-85

19

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( * ) ( ), (1,3 ). ,

(**) .

, , 52. . ( ).

, . (2) , . . , , . . . () () . ,

. , , .

52 . , , . . +2 -2. (+2-2=0). ( ). , : , , , .

. , . , , . . , , , . . . , . . .