58248255 mathimatika b gymnasioy theoria askiseis

7/24/2019 58248255 Mathimatika B Gymnasioy Theoria Askiseis

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5/298

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(+3) + (+2) , (7) + (2).

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(+5) + (7) ,10 3

4 4

+ +

,

. ,

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

10 3 10 3 7

4 4 4 4 4

+ + = + = +



6/298

16.

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;

1. : + = +

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

.. ( ) ( ) ( )[ ] ( ) ( )[ ] ( )2 3 4 2 3 4 3 3+ + + + = + + + + = + =

, + () = 0.

.. (+5) + (5) = 0 + 0 = .

.. (+3) + 0 = +3, (6) + 0 = 6

. -

:

(7) + (5) + (+2) + (+7) + (1)

-

:

1. ( -

) .

2. .

3. -

.

.. ( ) ( ) ( ) ( ) ( )7 5 2 7 1 + + + + + + =

1 ( )7 ( ) ( ) ( )5 2 7+ + + + + ( )1+ =

2 ( ) ( ) ( )2 5 1+ + + =

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

, -

,

-



7/298

17.

- - -

.

.. ( ) ( ) ( ) ( ) ( )7 5 2 7 1

+ + + + + + =( ) ( ) ( ) ( )12 2 7 1 + + + + + =

( ) ( ) ( )10 7 1 + + + = ( ) ( )3 1 4 + =

,

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

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

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x

.

+ x = x = .

,

, :

= + ( ) = + ( ).

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.

-

( + ) ;

( ) ;

-

:

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



8/298

18.

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( + ), -

.

( ),

-

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,

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

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1. , -

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2. , -

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3. ( + ),

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4. ( ),

-

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9/298

19.

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32 3 5 6 4 54, , 3, 5, , 3, 5, , , 1, 2, , 5, 6, 3, 5, 6,

8 2 2 3 8 20

:32

48

= 3 1,52

=

52,5

2=

62

3=

40,5

8 =

50,25

20=

:

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. 8 . +4,5 .1

3 . 8,3 .

6

5+ .

7

8

.

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3 3 = +

. 8, 3 8, 3 = + .6 6

5 5+ = + .

7 7

8 8 = +

-

, -

, -

. -

.



10/298

20.

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.

:. 8 +8.

. +4,5 4,5

. 1

3

1

3+ .

. 8,3 +8,3.

. 6

5+

6

5 .

.

7

8

7

8+ .

(< )

( >).

. +5 .....+7 . 7,8 ..... +0,5 . 3,1 ..... 2,9 .1 1

2 ..... 24 2

. 1,8 .... 1,08 . 3 .... 8 . +13 .... +8 .1 2

.....8 3

. +5 8 .+13 > +8 .1 2

8 3 >

:

. 4 < ... < +2 . 5 < .... < 0 . 2 < .... < 1

. 1 < ....< 1 . 4 < .... < 5 . 0 < .... < 1/2

.4 < 1 < +2 . 5


11/298

21.

- - -

7, 6, 5, 4, 3, 2, 1, 0, +1, +2, +3, +4, +5.

:

. 2.

. 2.

. 4.

. +2.

. +4.

. +2.

. -5 +1.

. +1 +5.

. 4 .

. 2 : 3, 4, 5, 6, 7.. 2 : +3, +4, +5.

. 4 : 3, 2, 1, 0, +1, +2, +3, +4, +5.

. +2 : 3, 4, 5, 6, 7.

. +4 : +3, +2, +1, 0, 1, 2, 3.

. +2 :

7, 6, 5, 4, 3, +3, +4, +5.

. +1 5 : 4, 3, 2, 1, 0

. +1 +5 :+2, +3, +4, 2, 3, 4.

. 4 4. 4 4 = 4 4+ = : 4, + 4.

:

. 8 3 2 3 5 6+ + + + + + + + .1

4, 3 5,1 1, 8 72

+ + + + + +

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1

3 4 6

+ + + +

. 4 5 8 11 12 22 4 16+ + + +

+1/3 2/5 7 4 2 || +1/3 2/5 7 4 2

1/3 2/5 7 4 2

() () 1/3 2/5 7 4 2



12/298

22.

- - -

. ( ) ( ) ( ) ( ) ( ) ( )8 3 2 3 5 6 + + + + + + + + = : ( ) ( )8 3 + + ( ) ( )2 3+ + + ( ) ( )5 6+ + + =

: ( ) ( ) ( ) ( )2 6 8 5+ + + + + =

: ( ) ( )8 13+ + = , -

: ( )13 8 5 = . :

( ) ( ) ( ) ( )1

4, 3 5,1 1,8 72

+ + + + + + =

( ) ( ) ( ) ( )1

5,1 4, 3 1,8 72

+ + + + + + =

( ) ( ) ( ) ( ) ( )5,1 0, 5 4, 3 1,8 7+ + + + + =

( ) ( )5, 6 13,1+ + =

( )13,1 5, 6 7, 5 = . :

( )2 1 5 13 4 6

+ + + + = ( ... 12)

8 3 10 12

12 12 12 12 + + + + =

10 23 23 10 13

12 12 12 12 12

+ + = =

. 4 5 8 11 12 22 4+ + + + 16 =

5 8 11 12 22 16+ + + =5 11 12 8 22 16+ + + =

( )28 46 46 28 18+ = =

:

. .... + (12) = +1 . .... + (20) = 4

. (+25) + .... = 0 . (10) + .... = 5

. (+13) + (12) = + (13 12) = +1.



13/298

23.

- - -

+13.

. (+16) + (20) = (20 16) = 4. +16.

. (+25) + (25) = 0. 0.

.(10) + (+5) = (10 5) = 5. +5.

= x + y +z B = x + y + x = 2, y = +5,

z = + 1/2, = -2/3.

= x + y + z (1)

(1): ( ) ( ) 1

A 2 52

= + + + +

... (1,2) = 2

:4 10 1

A2 2 2

= + + + +

4 11 11 4 7A

2 2 2 2 2

= + + = + = +

B = x + y + (2)

: ( ) ( ) 2

B 2 53

= + + +

... (1,3) = 3

: 6 15 2B3 3 3

= + + +

6 2 15B

3 3 3= + + +

8 15 15 8 7B

3 3 3 3 3

= + + = + = +

.



14/298

24.

- - -

, .

:

i. ( ) ( )8 5+ + ( ) ( )6 5+ + + ( ) ( ) ( )8 6 8 6 2= + + = + = +

ii. ( )3 ( ) ( )2 3+ + + + ( )0 2 0 2+ = + + = +

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

iv. ( ) ( ) ( )4 6 7 + + + + ( )7+ ( ) ( ) ( )4 6 6 4 2= + + = + = +

:v. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )8 3 1 4 8 1 3 4 9 7 9 7 2+ + + + + = + + + + + = + + = + = +

vi. ( ) ( )5 2 + + ( )2+ ( ) ( ) ( ) ( )6 5 6 6 5 1+ + = + + = + = +

(+1)

(+2). .

:

A 18 9 2 18 7 11 5+ + + + +

:

( ) ( ) ( ) ( ) ( ) ( ) ( )A 18 9 2 18 7 11 5= + + + + +

( )A 18= + ( ) ( ) ( )9 2 18+ + + + ( ) ( ) ( )7 11 5+ + + +

( ) ( ) ( ) ( ) ( )A 9 11 2 7 5= + + + + + +

( ) ( ) ( )A 20 14 20 14 6= + + = + = +

:A 10 7 18 2 7 6 + + + + +

2 1 5 7B 1

3 4 6 12

= + + +

:

( ) ( ) ( ) ( ) ( ) ( )A 10 7 18 2 7 6= + + + + + + +

-

-

. ..

( ) ( ) ( ) ( )9 8 9 8+ = + + + ( ) ( ) ( ) ( )5 3 5 3 = +



15/298

25.

- - -

:

( ) ( )A 10 7= + ( ) ( ) ( )18 2 7+ + + + + ( )6+ :

( ) ( ) ( ) ( )A 18 10 2 6= + + + +

:

( ) ( )A 18 18 0= + + =

:

2 1 5 7B 1

3 4 6 12

= + + + + +

... (3, 4, 6, 12) = 12

:

12 8 3 10 7B

12 12 12 12 12

= + + + + +

:

12 7 8 3 10B12 12 12 12 12

= + + + + +

:

19 21 21 19 2 1B

12 12 12 12 12 6

= + = = =

: . x + (10) = 8 . x (2) = + 6

. x, 10 8. x 10 8.

: x = 8 (10) .

: x = 8 (10) = 8 + (+10) = + (10 8) = +2

.

. x + (+2) = +6. x (+2)

+6. x +2 +6.

: x = + 6 (+2) = +6 + (2) = + (6 2) = + 4.

: 8

x. !!!



16/298

26.

- - -

:

: + = 8

, , . -

x .

: 4 + x (5) = 8 : 4 + x + (+5) = 8

: x + (4 + 5) = 8

x + 9 = 8

x +9 8. x +9 8.

: x = 8 (+9) = 8 + (9) = 17.

+ = +16

: x + (2) (+6) = +16

x + (2) + (6) = +16

x + (8) = +16

x 8 +16. x 8 +16.

: x = +16 (8) = +16 + (+8) = +24

: + = 15

: 4 + (3) x = + 7

(4 + 3) x = + 7

7 x = + 7

x 7 +7. x +7 7.

x = 7 (+7) = 7 + (7) = 14

: + = x

2 +(+8) (1) = x

2 +(+8) + (+1) = x

2 + (+9) = x

+7 = x

:

17, +24, 14 +7

-

:

4 4 2 2 3 +8

5 +6 1

+ 8 +16 +7



17/298

27.

- - -

A 4 5 16 8 8 7 3 2 + + + + + B 12 9 20 14 15 5 + +

:

( ) ( )A 4 5 16 8 8 7 3 2= + + + + +

A 4 5 16 8 8 7 3 2= + + +

A 16 7 2 4 5 8 8 3= + + +

A 25 28=

A 3= :

( ) ( )B 12 9 20 14 15 5= + +

B 12 9 20 14 15 5= + + +

B 20 14 5 12 9 15= + +

B 39 36 3= = +

: A 10 11 + + +

( ) ( )A 10 11 = + + +

10 = + 11 + + 10 11 1= + = +

2 4

(): 3 x y 15 7 + + +

x +15.

: ( ) 3 x 15 y 7= + +

:

. () () . () (+) . () ()

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

. () (+) = + = . .

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



18/298

28.

- - -

1. :

8, +6, +4/8, 8/4, +20, 15, +10,5, +1 1 2

2. :

. 10,8 . 12,5 . + 7,3

. 5,6 .5

8 .

6

5+

3.

.

.5, 7, 0, 2, + 4, 8, + 1

. 30,5, 24, + 7,3 , + 7,1 , 24,4

.3 3 1 3 4 7

, 4 , 5 , 2 , 2 , 24 10 4 5 5 10

+ +

4. () () :

. 7 2 < . 7 2 < . 8 8+ < . 8 8+ <

. 5 9< . 4 0 < . 4 0 < . 0 6<

5. :

. 8. ... 3 + . 3. ... 8 . 0 .... 5 . 5. ... 7+ +

. 4,2....0 . 0 .... 0,1 . 7,8 .... 7,08 . 5,1.... 5

6. :

.1

102

.8

3+ .

15

6 . 11

. 13,5+ . 6, 7 .18

25

.6

7

+



19/298

29.

- - -

7. :

8. -

. ( )

. ( ) ( )4 12 16+ + + = + . ( ) ( )3, 2 2,3 0, 9 + + = .5 3

18 8

+ =

( ) ( )9 4 5 + + = ( ) ( )1, 6 2,1 0,5+ + = + 1 1 1

3 2 6

+ + = +

( ) ( )9 2 11 + = + ( ) ( )4, 4 5, 4 0,1 + + = + 3 1 2

5 2 3

+ + = +

( ) ( )8 15 23+ + = ( ) ( )6, 6 6, 6 12,12 + =

9. > , < = :

. ( ) ( ) ( )17 11 .... 23+ + + + . ( ) ( )15 15 .... 15 + + +

. ( ) ( )23 19 .... 2+ + + . ( ) ( ) ( ) ( )15 88 .... 25 98 + + + +

. ( ) ( )22 22 .... 0+ + .( ) ( ) ( ) ( )54 37 .... 24 67 + +

. ( ) ( )18 5 .... 22 + . ( ) ( ) ( ) ( )17 71 .... 25 25 + + +

10. :

x 4,5 +1/8 2/3 +11 12,3

x

x

x

+ +33 28 +54 70

+19

45

54

+62

+ 1/2 1/3 +3/4 +5/12

+1/2

1/3

1/4

7/12

+ 1,5 +0,1 7,2 +3,9

+2,2

3,9

+6,5

+0,1



20/298

30.

- - -

11. :

( ) ( ) ( ) ( )A 10 12 8 16= + + + +

( )1 1 11

B 22 3 6

= + + + + +

18 25 71 64 101= + +

( ) ( ) ( ) ( ) 3, 05 1,15 7, 3 4, 6= + + + +

( ) ( )2 7

14 185 10

= + + + + +

30 45 68 25 72= +

12.

.

13. :

( ) ( ) ( ) ( ) 1,8 2, 05 1,8 2, 05= + + + + +

( )1 1 6

8 4 22 2 3

= + + + + ( ) ( ) ( ) ( ) 7, 4 1, 5 7, 4 13, 2= + + + +

1 1 1 1 1

2 4 8 10= + +

14. :

. 2, 2 7, 8 18, 6 20,1 5, 5 + + .3 1 5 1

28 4 16 2

+ +

. 1 1 1 21 7 1 22 3 6 3 + +

. 8, 3 10, 2 12, 4 3, 75 6, 25 + +

. 45,18 68,44 14,02 68, 45 10,13 +

15. 1

x2

= , y 1= ,3

z2

= ,1

23

= 2= +

:

A x y z= + + B x y = + + x y = + +

x y z

= + + + E x y z

= + + + x y z

= + + + +



21/298

31.

- - -

16. :

. ( ) ( )2 5 + . ( ) ( )28 12 + . ( ) ( )16, 4 13, 6 .1 32 4

+

( ) ( )6 4+ + ( ) ( )33 40+ + ( ) ( )29,1 21, 9 + 5 2

6 3

+

( ) ( )14 10+ ( ) ( )52 32+ ( ) ( )26,1 24, 9 + 5 2

3 26 3

+

( ) ( )15 35 ( ) ( )60 60 ( ) ( )55,5 33, 3 2 4

1 35 5

+ +

17. = :

. ( ) ( )18 13 .... 18+ + + . ( ) ( )12 4 .... 12+ +

. ( ) ( )20 40 .... 20+ + . 19 19 .... 19+

. 37 37 .... 0 . ( ) ( )14 14 .... 0

18. :

. ( )x 12 1 + = + . ( )x 20 4 + = . ( )x 15 18+ =

. ( )x 8 10+ = + . ( )x 25 0+ + = . ( )x 7 7+ =

. ( )10 x 2+ = + . ( )6 x 20+ = . ( )x 2,5 9 = +

. ( )30 x 55+ = + . ( )8 x 12 = . 44 x 2 + = +

19. :

. ( ) ( ) ( ) ( )42 23 32 23 + + + . ( ) ( ) ( )8,35 9, 02 4,18 +

. ( )2 5 1

1

3 6 2

+ +

. ( ) ( ) ( ) ( )88 45 53 29+ + +

.3 3 4 7

7 5 2 64 20 5 10

+ + + +

. ( ) ( ) ( ) ( )5,33 8, 44 1, 29 4, 02+ + +

20. x y

.

( ) ( ) ( ) ( )A 18 x 15 60 100= + + +

( ) ( ) ( )B 14 y 16 9= + + +



22/298

32.

- - -

21. :

.

.

22. :

. ( ) ( ) ( )75 28 60 + + . ( ) ( ) ( )57 40 16+ +

. ( ) ( ) ( ) ( )15 26 26 15+ + . ( ) ( ) ( ) ( )220 140 510 400 + + +

23. :

i. , ii. .. ( ) ( )30 5 6 8 7 18 22 + + + . ( ) ( )44 28 12 10 11 + +

. ( ) ( )8, 22 4, 91 3, 47 5, 3 12, 88 + .1 1 7 4

3 2 8 52 5 10 5

+

24. -

.

( ) ( )A 25 38 17 55 20 64= + + +

( ) ( )

B 10 22,8 88, 2 40 60= + + +5 5 3 5

8 6 12 4

= + ( ) ( ) 200 160 800 230 500= + + +4 2 2 5

E 1 56 3 3 6

= +

( ) ( ) 2,1 4, 5 8, 7 5, 9 3, 5 6, 6= + + + +

3 5 1 1Z 1 7

4 12 12 3

= +

4 +22 +12 +15

+8 7 6

+8 10 13 8

10 14 +6 +11

( + ) ( ) ( ) + ( + )

2 8 5

9 6 10

4 0 4



23/298

33.

- - -

25. :( ) ( )A 20 x y 4 x y 16= + + +

( ) ( )B y x 12 x y 11= + +

( ) ( ) x y 6 x y 6= + + +

( ) ( ) x y 18 x y 18= + + +

26.

:

( ) ( )A 12,8 7 1, 4 7,5 8, 2 7, 4 = + + +

( )[ ] ( ) ( )[ ]B 40 30 15 18 25 30 16= +

( ) ( ) ( )[ ] 8 1 7 15 8 10= + +

( )[ ] 3 500 2000 500= +

27. A x y 10= B x y 20= + + + .

28. :. ( )20 x y 35 + x 10= y 8= + .

. ( ) ( )x y 10 x y + x 4= y 5= .

. ( ) ( )x y y x + + x 1= y 1= .

29. :

( ) ( ) ( )A 18 x y 2 x 20 y = + + x y 4 =

30. x + y (x + y);

. ( ) ( )x y + . ( ) ( )x y . ( )x y + . ( )x y

. ( ) ( )x y+ + . ( )x y . ( )x y + . ( ) ( )y x



24/298

34.

- - -

1

;

() ;

2

;

1

:

.( 8) + (+3) + (+2) + (3) + (5) + (+6)

. 2 1 5 713 4 6 12

+ + +

2

:

. ( )x 3 4 = + . ( )4 x 12 = +

3

: ( ) ( )A 2 x y 1 x 2y = + + ,

x 1= y 1= .



25/298

;

:(+2) (+4) (3) (2)

-

,

(+).

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

;

:

3 4 2

43

+

-

,

() .

( ) ( )3 4 12+ = ( ) 2 843 3

+ =

:

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

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



26/298

36.

- -

;

:

= [ ]

0 0 = 1 =

( ) ( ) = [ ]

( ) + = + ( ) + = +

( ) = ( ) =

[ ]

;

: +5, 3, 2/5.

+ 1

. -

. ,

+5 15

+ .

3 1

3 .

2

5

5

2 .

;

-

, -

-

(+)

()

.

1

x x 0= .



27/298

37.

- -

.

:

.

;

-

: 4

2

+

+

,60

5

,9

3

+

,6

2

+

.

, -

:

(+) .

() .

,

4

2 22

+= + =

+,

6012 12

5

= + =

,

9

33

+=

,

63

2

=

+.

, ;

:

, 0

.

, -

,

1

=

:

( )

( )( )

+= +

+ ( )

( )( )

+=

( )

( )( )

= +

( )

( )( )

=

+

, .

.



28/298

38.

- -

1. , -

( + ).

2. ,

-

( ).

3. +1 .

4. ,

( + ) ( ) -.

5. ,

( + ),

( ).



29/298

39.

- -

:

. 3 4 + . 3 4 + . 3 4 . 3 4

. 2 7 . 2 7 . 2 7 + . 2 7 +

. (+)

. ( ) ( )3 4 12+ + = +. ()

. ( ) ( )3 4 12 + =

. : ( ) ( )3 4 12 = +

. ( ) ( )3 4 12+ =

. ( ) ( )2 7 14 = +. ( ) ( )2 7 14+ =

. ( ) ( )2 7 14+ + = +

. ( ) ( )2 7 14 + =

:

. 3 8 4 2 + . 9 11 4 15 + +

. 0, 25 8 1, 5 1, 5 + +

.1 1 1 1

2 3 3 2

+ + +

. :

( ) ( ) ( ) ( )3 8 4 2 + = ( )24 8 24 8 32+ = + + = +. :

( ) ( ) ( ) ( ) ( )9 11 4 15 99 60 159+ + + = + =

. ( ) ( ) ( ) ( ) ( )0, 25 8 1, 5 1,5 2 2, 25 4, 25 + + = + =



30/298

40.

- -

.1 1 1 1 1 1 2 1

2 3 3 2 6 6 6 3

+ + + = + = =

:

.1

4 2 32

+ . 5 6 8 12 + . ] 8 5 3 +

, -

:

. ( ) ( )1 1 1 5 54 2 3 52 2 2 1 2 + = = =

. ( ) ( )5 6 8 12 5 10 50 + = + =

. ( )[ ] ( ) ( ) ( ) ( ) ( )8 5 3 8 5 3 8 15 15 8 7 + = + + = + = =

, , , , . .

( ) ( ) 7 2 14= + = , ( ) ( ) 4 2 8= + + = + ,( ) ( ) 9 2 18= + = ,

( ) ( ) 7 6 42= = + , ( ) ( ) 4 6 4= + = 2 , ( ) ( ) 9 6 54= = + .

:

10y 2y , ]B 10 4 y 2y , 4y y y 1 .

[ ( + ) = + ]

y 1:

( ) ( ) ( ) ( ) ( )A 10y 2y 10 2 y 10 2 1 12 1 12= = = = = +

( )[ ] ( )[ ]{ } ( )[ ] ( )B 10 4 y 2y 10 4 2 y 6 2 1= = = + =

( ) ( ) ( )8 1 8 8= = + =

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

:



31/298

41.

- -

.1

1, 8 3 1, 8

3

+

.1 1

2 2

2 2

+ +

.1 1 1 1

2 1 5 22 2 2 2

+ +

. 2003 2004 0 +

. 3, 4 2,1 6, 7 2,1 + . 6 8 2 16 +

. ( ) ( ) ( ) ( )1 1 1 3

1,8 3 1,8 1,8 3 1,8 3 13 3 3 3

+ = + + = + = + = +

. ( ) ( )1 1 2 22 2 02 2 2 2 + + = + + =

.1 1 1 1 5 3 11 5

2 1 5 22 2 2 2 2 2 2 2

+ + = + + + =

15 55 70 35

4 4 4 2

= + + + = + =

. , , ,

, 0 0 = 0 0 = . ( ) ( )2003 2004 0 0 + =. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3, 4 2,1 6, 7 2,1 7,14 14, 07 7,14 14, 07 + = + = + =

( )14, 07 7,14 21, 21= + =

. ( ) ( )6 8 2 16 6 22 132 + = = +

:

x , , :

( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )3 12 10 6 3 12 10 6 3 22 6 66 6 72 + = + + = + + = + =

( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )3 12 8 6 3 12 8 6 3 4 6 12 6 18 + + = + = + + = + =

( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )3 12 5 6 3 12 5 6 3 17 6 51 6 57 + = + + = + + = + =

x 3 (12 x) + (6)

10

+ 8

5



32/298

42.

- -

:

. 3 x 2 y

+

. 1 x 1 y

:

. ( ) ( ) ( ) ( )3 x 2 y 3 2 y x 2 y 3 2 3 y x 2 x y 6 3 y 2 x x y + = + + = + = +

. ( ) ( ) ( ) ( )1 x 1 y 1 1 y x 1 y 1 1 1 y x 1 x y 1 y x x y = = + = +

x 1

y 1+ :

A x x y 2 x y y y x + +

x y ,

,

.

( ) ( ) ( )A x x y 2 x y y y x= + + =

( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ]1 1 1 2 1 1 1 1 1= + + + + + + =

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

( ) ( ) ( )1 0 2 2 1 2= + + + =( )0 4 2 4 2 6= + + = + + = + . A 6= +

:

. 2 8 1 7 + .4 1

17 3 13 2

+

. 8, 4 3, 5 1, 2 + . 4 5 2 3 12 + +

. 3, ,

().

: ( ) ( ) ( ) ( )2 8 1 7 112 + =

. 4, .

(+).

: ( ) ( ) ( )4 1 204

17 3 1 343 2 6

+ = + = +



33/298

43.

- -

. : ( ) ( ) ( )8, 4 3, 5 1, 2 35, 28 + = +

. : ( ) ( ) ( ) ( ) ( )4 5 2 3 12 1440 + + =

:

. x x y x y y x+

, x 1

, y 1+ .

. x x y y x y + , x 1+ , y 1 .

x y -

, . :

. ( ) ( )( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]x x y x y y x 1 1 1 1 1 1 1 + = + + + + =

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 1 1 0 2 2 0 + + + = + =

. ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( )x x y y x y 1 1 1 1 1 1 + = + + + + =

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

:

2= + , 1= 2= :

( ) ( ) 2 1 2 = + = , ( )( ) 1 2 2 = + = [ ]

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

( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) 2 1 2 2 1 2 2 3 6 + = + + = + = + =

3= + , 1= + , 4= :

( ) ( ) 3 1 3 = + + = + , ( ) ( ) 1 3 3 = + + = + ,

( ) ( ) ( ) 3 1 4 12 = + + = , ( ) ( ) ( ) ( )[ ] ( ) ( ) 3 1 4 3 3 9 + = + + + = + =

:

( + )

+2 1 2

+3 +1 4



34/298

44.

- -

. 8 23 ... 16 3 1 .... = +

. 10 2 4 ... 5 6 .... + + =

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

.

.

.

( ) ( ) ( ) ( ) ( )8 23 16 3 1 8832+ = +

:

. ( ) ( ) ( ) ( ) ( )10 2 4 5 6 2400 + + =

. ( ) ( ) ( ) ( ) ( )4 5 2 3 5 600 + + + = +

:

. 1 1 1 1 1 1 . 1 1 1 1 1 1 + + + +

. .

: ( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 1 = +. ( ).: ( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 1+ + + + + =

:

. x y + + . x y

. ( ). .

. ( ). .

:

A x y x y x y x y + + + + + , x 2 y 2+ .

x y :

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )A x y x y x y x y= + + + + + =

( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]2 2 2 2 2 2 2 2= + + + + + + + + + =



35/298

45.

- -

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

( ) ( )16 16 32= + + + =

:

. 25 : 5 . 4 9 : 7 .1 1

:2 3

. 100 : 10 . 2, 8 : 7 . 35 :10

. 2, 25 : 1, 5 .

36 : 6 : 2

. 60 : 15 : 2+

:

. ( )25 : 5 5 = . 49 : 7 7 =

.1 1 1 3 3

:2 3 2 1 2

= = +

. ( ) ( )100 : 10 10 = + . ( )2,8 : 7 0, 4 =

. ( )35 :10 3, 5 = . ( )2, 25 : 1, 5 1, 5 =

. ( ) ( ) ( ) ( )36 : 6 : 2 6 : 2 3 = = + . ( ) ( ) ( ) ( )60 : 15 : 2 4 : 2 2 + = + =

:

1 1 1 1A 2 : 3

3 2 2 5

= +

: ( ) ( )2 3 1 1

A 2 : 3

6 6 2 5

= +

: ( ) ( )5 1 5

A 2 36 2 1

= +

10 15A

6 2= +

: 10 45 10 45 55

A6 6 6 6 6

= + = =



36/298

46.

- -

:

. 3 . 83

. 54

1

. :

. 3 1

3 .

. 8

3

1 3

8 83

=

. 5

4 1 4

5 5

4

=

.

:

. 3 8 55

4 5 10 +

.2 3 1

5 4 2

+

+

. :

( ) ( ) ( ) ( )3 8 55

0, 75 1, 6 5, 5 0, 75 1, 6 5, 54 5 10

+ = + = + + + =

( ) ( ) ( )2, 35 5,5 5, 5 2, 35 3,15= + + = + = +

.( ) ( ) ( )

( ) ( ) ( ) ( )

2 3 1 6 11, 2 0,125

5 4 2 5 8

+ + + = + = + = +

( )1, 2 0,125 1, 325= + =

:

. 2 3 4 48 + = . 5 1 100 = +

20 :

. ( ) ( ) ( )2 3 4 48 + = . ( ) ( )5 1 100+ = +

24 48+ = 5 100 = +



37/298

47.

- -

( ) 48 : 24= + ( ) ( ) 100 : 5= +

2= 20=

:

, , :

3= , 1= , 1= + .

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) : : 3 : 1 3 : 1 3 1 3 3 3 + = + + = + + + =

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

5= , 2= + , 5= + :

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) : : 5 : 2 5 : 5 5 2 2, 5 1 10 + = + + + + = + =

( ) ( ) ( ) ( )2, 5 1 10 12, 5 1 12, 5 1 11, 5 + + + = + + = =

: = 2[3 (5)] + (3) : (5 + 3)

( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 5 3 : 5 3 2 3 5 3 : 2 2 8 3 : 2= + + = + + = + + =

( ) ( )16 1, 5 16 1,5 14, 5= + + = =

3 1 +1

5 +2 +5

: : +



38/298

48.

- -

1. :

. ( ) ( )7 4+ . ( ) ( )8 10+ . ( ) ( )3 14+ . ( ) ( )21 3+

. ( ) ( )3, 5 2+ . ( ) ( )4,5 3+ + . ( ) ( )0, 4 9+ . ( ) ( )0,3 0,8+

.1 8

2 3

+

.1 5

12 6

.7 1

23 3

.1 2

3 5

2.

( ) ( ) ( ) ( ) ( ) ( )A 5 3 4 1, 5 8 0,5= + + + +

( ) ( ) ( ) ( ) ( ) ( )B 1,5 5 3 4 2 6= + + + +

( ) ( ) ( )

2 1 1 1 3 5

3 2 4

= + +

( ) ( ) ( )1 1 1

1 4 2 3 3 22 2 2

= + + + ( ) ( ) ( ) ( ) ( ) ( ) 5 3 4 3 7 4= + + +

3.

x 2= y 1= .

A 10x 2x= ( )B x 5x 3y y= +

( ) x y 5x 5y= + 6x x y 2y= + +E 6, 5x 7x 2x x= + 8y 4y 0, 5y 3, 5y=

4. :

.

+3 4 +5

+3 4 5

3 +4 5

+



39/298

49.

- -

.

5. :

( ) ( )A 3x 5y 2 4y 4x= + , x 1= y 2= + .

( )B 2x x 2y 4 = + , x 2= y 1= .

( ) 18x 24y 3x 5y= + , x 2= y 2= .

12x 8y 4y 8x= + + , x 1= y 1= .

6. :

1

2 , 3,

6

5+ , 4,

2

3,

1

7 , 8

7. 15 11

;

8. :

. ( )( )x 1 1 + . ( )( )x 1 1+ . ( )( )3 x 1 1 + +

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

9. :

. .

10. :

. ( )( )( )( )( )( )2 3 4 5 1 6 + + . ( )( )( )( )( )( )2 3 4 5 1 6+ + +

2 +1 3

+2 1 +3

2 +1 3

x y x y x (y) y x(y)x 2 (x y + x )

x 4 (8 x) 5

1

+2

3

+4

5

y y ( 2) + 10

1/2

+1/3

2/5

+5/6

1 1/2



40/298

50.

- -

.

( )1 4 1 2

82 3 3 4

+ + . ( )( )( )

( )0, 5 1, 5 2, 5 100 +

. ( )( ) 1 1

100 100100 100

+ +

11. :

. ( )( )( )( ) ( )( )( )( )4 0, 5 8 0, 5 2 3 1 5 + + + +

. ( )1 1 1 1 1 1

12 2 3 3 4 9

+ + +

. ( )( )( ) ( )( )( )2 3 4 2 3 4 + + +. ( )( )( ) ( )( )( )( )1 0,8 10 1 2 0, 9 10 + + +

12. .

. ( )( )( )( )1 4 ... 5 3 60 + = . ( )( )( )( )2 5 ... 7 4 280 + = +

. ( )( )( )( )1 1 1 ...1 1 + = + . ( )( )( )( )( )3 3 ... 3 3 3 243 + + =

13. x 1= , y 2= , 1= + .

( )( )( ) x x y x y x = + + ( ) ( ) ( )B 4 x x y x y = +

( )( )( )( ) x y x y x = ( )( ) ( ) 10 x y y x = +

14. () () .

:

. .

. 18 10 .

. +1

.

. (1)

.

. .

15. :

A x y z= ( )B x y z z y z= ( ) ( ) ( )( ) x y z x y z x y z= +



41/298

51.

- -

x 1

2

, y +3 z

+2.

16. :

17. x, y, z .

.( ) ( )( )

2 3 4 x 24 + = .( )( )( )( )

1 1 2 3 y 12 + = . ( ) ( )4 z 7 56 + = + . ( )( )( )2 3 4 x 96 + =

18. :

A x 2y= ( )B x y 4= x A y B= , x 1= y 2= .

19. :

. .

4 2

1 +4 2

+5 +1 3

+3 4 1

x y x y () x y x y (10) +2 1 2

+1 2 1

2 1 1

2 +2 +1

: 1 1 5 7 82 3 2 4 5

+ +

1

2

1

3

1

4

1

5

+

+

: +16 -32 +64 -12 -128

2

+ 2

4

+ 4



42/298

52.

- -

20. :

.3 :0,52

.

( ) ( )0, 25 : 0, 5 . ( )32 : 2080

.5 6

:6 5

. ( )1

1 : 152

. ( )16

: 1008

21. :

( )( )( )

( )

2 8 5A

10

+ =

( )( )( )

( )

( )( )

( )

4 21 2 7 9B :

8 3

+ =

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

8 3 8 1 3 :

5 2 5 +=

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

6 8 22

3 4 2 += + +

2 4 3E :

5 5 5

= +

( )( )

( )( )

10 3 4 : 100

2

+ +=

22. :

( ) ( ) 10 x 4 : 5 y= + + ( ) ( ) ( ) ( )B x : y x 4 : 3 y= +

( ) ( ) 3 x y : y 11 = + ( ) ( ) x y : x y= +

( ) ( )E x y : x y= + x 2= y 1= .

23. :

. 5x 60 = . 10x 200 = . 10x 40= . 2x 12 = + . 8x 16+ = + . 5, 5x 55=

.1

x : 42

= + .

x4

5= . ( )x : 7 2 =

. ( )x : 10 3 = + . 7x 49 = .x

28 =

24. 2 +18

9.

25. :

( )[ ] ( ) ( )A 6 4 3 18 : 9= + +

( ) ( ) ( )B 32 : 18 16 75 : 25= + +



43/298

53.

- -

( )[ ] ( ) ( ) 200 : 50 150 45 15 : 6= +

26. :

27. x, y, , .

i. ( )( )( )2 3 4 x 48 + = ii. ( )( )( )5 8 2 y 240 + + = +

iii. ( ) ( )( )5 10 3 300 + = + iv. ( )( )( ) 16 3 4 192 =

28. :

( ) ( ) xy x y : y xy : 2 = + x 2 y 3.

29. :

( ) ( ) ( )A 3 x y x 4 y : x 4= + + + +

( ) ( ) ( ) ( )B x x y y x y x y : x y= + + + +

x 2= y 1=

x y z x y : z (x + y) : (x + y) : z

8 +4 2 +1

6 +12 4 1

+14 2 +2 1



44/298

54.

- -

1

;

; .

2

;

, ;

1

:

. ( ) ( )1 1

4 22 4

+ + .

( ) ( )3 2 +

2

:

( ) ( ) ( ) 3 x y x 4 y : x 4= + ( ) ( ) ( )B x x y y x y x y= + + + + ,

x 2= y 1= .

3 :

( ) ( ) ( )

( ) ( )

2 3 8A

6 4

+=

+

2 4 3 3B :

5 5 5 2= + +



45/298

; ;

,

... ,

(

).

.

... =

;

; : 23, (3)2, (3)3.

-

-

.

.

32 8 0= > , ( )2 23 3 9 0 = = > , ( )3 33 3 27 0 = = 0, > 0

< 0

< 0, < 0



46/298

56.

- -

-

;

:

,

, :

+ =

.. ( ) ( ) ( ) ( )3 4 3 4 72 2 2 2 128+ = = =

,

-

:

:

= =

..6

6 5 1

5

55 5 5

5

= = =

, -

, :

( ) =

.. ( )2

2 23 4 3 4 = , ( )3

3 3 32 3 4 2 3 4 = ,

-

, : ( )

=

.. ( )3

2 2 3 63 3 3= =

.

,

.

.. 2 3 12 2 2 4 8 2 14+ + = + + = :

-

-

-

- , -

.

()=

(.)=

: =

= +



47/298

57.

- -

-

;

;

,

,

,

:

1

=

=

,

0 1

= 0

.

;

-

. :

( )

( )

0

: 1

1

+

= =

= =

= =

= =

;

:

56.000.000.000, 0,00000000012

.

, -

.

( ) :

. 10k



48/298


49/298

59.

- -

-

-

; - :

1

200,

21

14,

1

6

,

0 , - 2 5,

. - .

2 2 3 2

1 1 1 1 1

200 2 100 2 4 25 2 2 5 2 5= = = =

.

21 3 7

14

=

2 73

2=

1

6

6

2 5.

, -

.

,

.

1. -

.

2. :

( )

( )

0

: 1

1

+

= =

= =

= =

= =



50/298

60.

- -

-

:

.

3

4 .

21

2

.

2

5 .

31

3

. 24 .3

2 .5

1 . 02003

. 34 4 4 4 64= =. .

2 2

2

1 1 1

2 42

= + =

=

. ( )2 2

5 5 25 = + = +

.3 3

3

1 1 1

3 273

= =

. () , -

24 16 = .

. . ( )3

2 8 = .

. ( ) ( )51 1 1 = = +

. 1,0

1= .

02003 1= .

:

. 8 10 35 5 5 .2 8 5

2 2 2

+ = :

. 8 10 3 8 10 3 215 5 5 5 5+ + = =



51/298

61.

- -

-

. ( ) ( ) ( ) ( ) ( )2 8 5 2 8 5 152 2 2 2 2+ + = =

:

.2 3 2 3

3 3 3 3+ + .5 42 2

1 1 1 1+ +

:

. ( ) ( )2 3 2 3

3 3 3 3 9 + + = + 27 9 27+ 0=

. ( ) ( ) ( ) ( ) ( )5 42 2

1 1 1 1 1 1 1 1 3 1 2 + + = + + + = + + =

:

. 58 .3

1 .4

5 .7

12

. , . .

. , . .

. , . .

. , . .

< , > = .

.8

4 ..... 0 .7

3 ..... 0 .16

1 ..... 1

. . -

. . ( )8

4 0 >

. .

. . ( )7

3 0 < .

.( )161 1 = .

:

. 5 1

88

5

.

44 41

2 1 22

+

( ) = :



52/298

62.

- -

-

.

5 5

5 51 1 88 8 1 1

8 8 8

5 = = = =

. ( ) ( ) ( ) ( ) ( ) ( )

444 4 41 1

2 1 2 2 1 2 2 162 2

+ = + = = +

: .2

2

10

5.

3 320 : 5

=

:

.

222

2

10 102 4

55

= = =

. ( ) ( )33

33 3

3

20 2020 : 5 4 64

55

= = = =

:

.4 3 3

4 3 3

30 100 18

15 650 +

.3 33 2 2 3

14 : 7 36 : 12 28 : 14+

-

:

.( )

( )4 3 34 3 3

34 3

4 3 3

30 100 18 30 100 182 2 3

15 50 615 650

+ = + = + =

( )16 8 27 16 8 27 51 + = + + =

. ( ) ( )( ) ( )

3 2 3333 2 2 3

3 2 3

14 36 2814 : 7 36 :12 28 : 14

127 14 + = + =

( ) ( )3 2 3

3 3214 36 28 2 3 2 87 12 14

+ = + =

9 8+ + 9= +

: .3

21

.3

32



53/298

63.

- -

-

( )

= :

. ( ) ( )3

2 61 1 1 = = + . ( ) ( )

33 9

2 2 512 = =

:

3 2 3 7 2 6 4 : 12 +

,

, :( ) ( ) ( ) ( )[ ] ( ) ( )3 2 3 7 2 6 4 : 12 3 7 8 6 16 : 12 = + = + =

( ) ( ) ( ) ( )3 7 8 96 : 12 3 15 96 : 12 45 8 37= + + = + = + = +

:

.18

15

2

2.

28

26

4

4.

10

9

6

6

= :

.18

18 15 3

15

22 2 8

2

= = = .28

28 26 2

26

44 4 16

4

= = =

.10

10 9 1

9

66 6 6

6

= = =

:. 20 10 8 355 5 5 : 5 .

16 8 5 24 : 4 4 4

:

. ( )20 10 8 35 20 10 8 35 38 35 38 35 35 5 5 : 5 5 : 5 5 : 5 5 5 125+ + = = = = =

. ( )16 8 5 2 16 8 5 2 16 15 16 15 14 : 4 4 4 4 : 4 4 : 4 14 4 4+ + = = = = =

:



54/298

64.

- -

-

. 3 54 x 4= . 516 x 2= .

2

31x 3

3

=

. 3 54 x 4 = . 5 3 5 3 2x 4 : 4 4 4 16= = = = .

. 516 x 2 = . 5 5 4 5 4 1x 2 :16 2 : 2 2 2 2= = = = = . 16 2.

.

2

31 x 33

=

.

2 2 23 3 3 3 2 3 2 5

2 2

1 1 3x 3 : 3 : 3 3 3 3 3 243

3 3 1

+ = = = = = = =

.

:

x 2= + y 1= :

( ) ( )[ ] ( ) ( ) ( ) ( ) ( )22 2 3 2 2 32 3x y x y 2 1 2 1 2 1 2 1+ = + + = + =

( ) ( ) ( ) ( ) ( )2 2 3

1 2 1 1 4 1 1 4 1 4 5= + + = + = = + = + x 2= y 1= + :

( ) ( )[ ] ( ) ( ) ( ) ( ) ( )22 2 3 2 2 32 3

x y x y 2 1 2 1 2 1 2 1+ = + + + = + =

( ) ( ) ( ) ( ) ( )2 2 3

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

:

x 2 x 2A 2 : 16 32 : 2 =

, x 4 .

x 4 . :4 2 4 2

A 2 :16 32 : 2+ = . 16 32 2

. :

4 2 4 5 4 2 6 4 5 2 6 4 5 2 2 3A 2 : 2 2 : 2 2 : 2 2 : 2 2 2 2 2 4 8 4+ = = = = = =

:

x y (x + y)2 x2 y3

+ 2 1

2 +1



55/298

65.

- -

-

.10 10

5 5

. 2 34 4 . 0 110 :10

1

= :

.10

10

10 10

1 55 1

5 5 = =

.3

2 3 3 3 2 1

2 2

1 44 4 4 4 4 4

4 4

= = = = =

. 0 1 0 0 011 1

10 :10 10 : 10 : 10 10 1 10 101010 = = = = =

: , :

+ = : = .

:

.

2 31 3

3 2

.

3 43 1

:8 8

.2

2 43 3

+

+

.1

3 24 : 4

:

.

2 3 2 3 2 32 3 3 1 3

3

1 3 3 2 3 2 1 83 2 3 2 8

3 2 1 3 1 3 33

= = = = = =

:

=

.

3 4 3 4 3 4 33 4 1

3 3 4 3 3

3 1 8 8 8 8 8 1 1 1 1 1 1: : : 8 8

8 8 3 1 1 27 8 2163 3 8 3 3

= = = = = = =

. ( ) ( )[ ] ( ) ( ) ( )( )22 2 2

2 4 2 4 2 4 2 2 2 4

4

1 13 3 3 3 3 3 3

813

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

. ( ) ( ) ( ) ( ) ( )

11 1 1 1

3 2 3 3 2 3 2 5 5 1 5

2 5

1 1 14 : 4 4 : 4 4 4 4 4 4

10244 4

+ = = = = = = = =



56/298

66.

- -

-

:

x 2= + :

2 2

2

1 1x 242

= = =

2 3 1 2 3 11 1 1x x x 2 2 2

x 2 2

+ + + + = + + = 1

4 92

+ + 13=

x 3= :

( )( )

23

2

1 1x 3

93

= = =

( ) ( ) ( )2 3 12 3 11 1 1 1

x x x 3 3 3 9 27x 3 3 3

+ + + = + + = + =

1

3=

19 27

3+ 18=

x 2=

( )( )

22

2

1 1x 2

42

= = =

( ) ( ) ( )2 3 12 3 11 1 1 1x x x 2 2 2 4 8

x 2 2 2

1

2

+ + + = + + = + =

=

14 8

2+ + 4=

: x 4 x 2 x 2 x 3A 3 2 3 3 + = + + , x 2 .

x 2 :

2 4 2 2 2 2 2 3 2 0 0 1

2 1

1 1 1 1 1 3 4A 3 2 3 3 3 2 3 3

9 3 9 9 93 3

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

x x2 2 3 11 x x xx

+

+ +

2

3

2



57/298

67.

- -

-

:

. 68.000.000.000 . 0,00000000075

.68.000.000.000 = 68 1.000.000.000 = 68 109 6,8 1010.

.0,00000000075 = 7,5 10 10

:

. 518.000.000 0,000031 .102.000.000 : 170.000

.518.000.000 = 5,18 108

, 0,000031 = 3,1 105

.: 518.000.000 0,000031 = 5,18 108 3,1 105= 16,058 108 + (5)= 16,058 103.

.102.000.000 = 1,02 108, 170.000 = 1,7 105.

: 102.000.000 : 170.000 = 1,02 108 1,7 105 = 0,6 1013.

:

.2

84 10 .

36

3 10

.2

82 10

. ( ) ( )2 28 2 8 164 10 4 10 16 10 = =

. ( ) ( )3 3

6 3 6 183 10 3 10 27 10 = =

. ( ) ( )2 2

8 2 8 162 10 2 10 4 10 = =

:

8 8 8A 2,1 10 4, 2 10 1, 5 10 +

( )8 8 8 8 8A 2,1 10 4, 2 10 1, 5 10 2,1 4, 2 1, 5 10 4,8 10= + = + =

:

. 6,14 108 ..... 6,04 108 . 5,3 104 ..... 14,8 104

.6,14 > 6,04 6,14 108 > 6,04 108

.5,3 < 14,8 5,3 104 < 14,8 104



58/298

68.

- -

-

6,25 105

625 108.

: 6,25 105 < 625 108 .

6,25 105 = 625 103. 625 108: 625 103 = 1 1083= 105.

6,25 105 105 .

:

.1

3.

3

20 .

3

40.

8

5

.1

0,333333 0, 33

= = .3

0,1520

= .3

0,07540

= .8

1, 65

=

1,45 .

x 1, 45= . x 1, 454545...=

100: 100 x 100 1, 454545... = 100x 145, 4545...=

: 100x 145 0, 4545...= +

x : 100x x 145 0, 4545 x = +

: ( )100 1 x 145 0,4545 1,4545... = + : 99x 145 1=

99x 144=

144x

99

=



59/298

69.

- -

-

1. :

. ( )2

8 .3

5 .

31

3

. ( )7

2 .2

5

. ( )3

3 . 26 . ( )29 . ( )25 . ( )3

1

. 37 . ( )2

1,5 . ( )224 4 + . ( )1000

1 . 22,3

2. :

( )32 3 2

A 5 5 5 5 5= + + + +

( ) ( )5 24 5 3

B 2 2 2 2 2 2= + + +

( ) ( )3 22

8 8 8= + +

( ) ( )2 32 4

6 6 6 6=

( ) ( ) ( ) ( )8 9 105 9

1 1 1 1 1 1= + + +

3. :

. 18 9 5 63 3 3 3 . 5 17 3 64 4 4 4

. 10 9 8 38 8 8 8 . ( ) ( ) ( ) ( )7 9 4 52 2 2 2

.

10 9 20 15

1 1 1 15 5 5 5

. 7 8 5 14 206 6 6 6 6

4. > , < , = .

. ( )10

3 .... 0 . ( )25 .... 0 . ( )72 .... 0

. ( )19

1 .... 1 . ( )54 .... 0 . ( )81 ....1

. ( )5

10 .... 0 . ( )91 .... 1 . ( )47 .... 0



60/298

70.

- -

-

5. :

.8

8 177

.5

5 144

. ( ) ( )5 5

55 1 13 2

3 2

. ( ) ( )5 5

55 1 18 2

8 2

. ( )6

66 110 110

6. :

.

4

4

300

150.

2

2

735

73,5. ( )

3 380 : 40

. ( )4436 : 12 .

( )

3

3

250

50. ( )

66500 : 250

. ( ) ( )3 3

640 : 160 . ( )22

100 : 20 . ( )3 31024 :128

7. :

( ) ( )34 4 3 2 2A 20 : 10 36 : 12 25 : 5= +

( ) ( )2 22 2 2B 845 : 84,5 90 : 9 60 : 152= +

( ) ( )2 32 3 3 3 2 2

50 : 10 16 : 8 200 : 100 1000 : 200= + +

( )

4 5 4 3

4 5 4 3

15 68 128 512

3 34 64 128= + +

( )32 2 3

2 2 3 3

805 1000 60 40

80,5 100 30 20

= + +

8. :

. ( )32

2 . ( )65

1 . ( )2

52 . ( )22

8

. ( )2

23 . ( )2

24 . ( )

75

1 . ( )2

25

9. :

( ) ( ) ( )3

3 2 6A 2 8 2 8 : 4 = +

( ) ( ) ( )1000 22 22 2B 10 3 4 5 : 5 = +



61/298

71.

- -

-

( ) ( ) ( )[ ]23 2 4 12 8 :10 16 : 8 5 = +

( )2

22 2 2 31 5 1 45 :15 5

5

= + +

10. :

.

20

19

3

3.

18

16

5

5 .

( )

( )

17

14

5

5

.

2004

2002

7

7

.

8 51 1

:

2 2

. ( ) ( )10 8

1, 5 : 1,5 .25 23

3 3:

4 4

. 50 474 : 4

11. :

. ( ) ( ) ( ) ( )5 10 8 19

3 3 3 : 3 . 12 4 7 208 8 8 : 8

. 14 20 4 16 4 106 6 6 : 6 6 6 . ( ) ( ) ( ) ( )10 25 12 207 7 : 7 7

12. :

. 4 68 x 8 = . 10 143 x 3 = . 632 x 2 =

. 27 x 243 = . ( ) ( )8 107 x 7 = .3

2

x3

3=

.

9

31 x 22

+ =

. ( ) ( )10 12

10 x 10 = . ( ) ( )8 11

5 x 5 =

13. :

2 3

3 3

1 1 1

3 2 6A :1 2

15 5

+ =

( ) ( ) ( )

3 2 2

4 3 22

2 1 41

3 3 3B :2 41

=

14. :

18 x 14 y 10 x yA 2 : 2 128 : 2+ + + += +

20 16 x 22 y 10B 3 : 3 3 : 9 = +

( )x y10 x y x 12y 4 4 : 4 ++ + = x 2= y 1=



62/298

72.

- -

-

15. :

.

.

16. :

. ( ) 2

4 . 4 22 4 .

2 31 5

5 3

.

2 14 1

:7 7

. ( ) 2

2 33 3 + . ( )

23

2

.2 3

1 2:

3 3

. ( )5

31

.2 2

5 :10

. ( ) 2

4 5

8 8

. ( ) 2

4 3

6 6

.

2

2 18 4

17. :

.

2 2 21500 4 1500

400 400 8

.

10 5 4 19300 300 300 300

80 80 80 80

.

4 5 8 1

1000 1000 1000 1000:60 60 60 60

18. :

. .

2 2 3 3+ 42 3

2 1

+2 3

3 2

x x3 3 x2 ( x2)3 x2 x3

2

+3

3

2 2+ 3 4

1

2

+2

3 1+ 2 3

1

2

+2



63/298

73.

- -

-

19. :

( )2 4 42 1 1 1

33 8 2

= +

2 2 3

43 1 1 22 2 2

= + +

8 164 2

1 1

2 2

=

20. :

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

( ) ( )x 1 x 2x 1

1 2 3+ = x 1=

( ) ( )x x

x x 2 x 2= + + + x 3=

21. :

. 750000000 . 81.000.000.000 . 4.320.000.000

. 33.100.000.000 .0,0000000712 .0,000000000422. 0,00000358 .0,0000000848

22. :

. 800.000.000 75.000.000 . 350.000.000 1.600.000.000

. 73.000.000 0,00000045 . 0,00000004 0,00000005

.0,000000006 : 40.000.000 .4.000.000.000 : 8.000.000

23. :

. ( )248 10 . ( )364 10 . ( )572 10

. ( ) 4

63 10

. ( ) 6

42 10

. ( )

24

5 10

24. :

9 9 9 3,5 10 4,8 10 6,9 10= + 3 3 3 5,7 10 14,6 10 2,8 10 = +

8 8 8 1,7 10 4,3 10 7,25 10=



64/298

74.

- -

-

25.

.. 88,45 10 14845 10 . 63,2 10 81,6 10

. 72,4 10 51,2 10 . 91,8 10 79 10

. 54 10 42 10 . 86 10 101,5 10

26. Kg 3 109

4,5 1012Kg.

27. :

.1

4.

5

4 .

5

8 .

2

9

.4

3.

10

4 .

16

32.

6

20

28. :

. 0, 5 . 0, 6 . 0, 8

. 1, 5 . 1, 25 . 2,541

29. :

. 0, 5 0, 6 0, 8+ + . 1,5 1,25



65/298

75.

- -

-

1

;

2

;

: 3 2 51,3 10 , 4 10 , 15,8 10

1

:

2 2 2 2 3 1A 20 : 5 3 5 3 3 : 25 3 = + +

( ) ( ) ( )2000 2 3

2 2 2 2 26 5 8 8 : 8

= +

2

103 m .

378 ;

3

:

. 0,18 . 0,14 . 0,544



66/298



67/298

4 4 4 4 4

5 5 5 5 5

2

- -



68/298



69/298

-

;

;

(). -

x.

.. 3x + 5 = 2x + 9

= ,

( 3x + 5).

= ,

( 2x + 9). x (

3x 2x).

.

, x 4,

: 3 4 + 5 = 2 4 + 9.

17 = 17. .

4 ,

4.

-

; -

.

-

( ).

.



70/298

80.

- - -

-

-

.

.

1.

x + 8 (x + 3) + 7 = 3 (x + 2) 11

:

x + 8x + 24 + 7 = 3x + 6 11

:

x + 8x 3x = 24 7 + 6 11

:

6x = 36

:

x36

66

=

=

6.

!!!

-

.

. ,

.

2

:x 2 3x 1 8

53 2 3

+ + = +

... (2,3) = 6

:

62 x 2

3

+ 6 5 6+ =

3 3x 1

2

6+

2 8

3

( ) ( )2 x 2 30 3 3x 1 16+ + = +

-

-

x

( 6).

.

-

.

-

, -

.

-

-

-

-

.



71/298

81.

- - -

-

:

2x 4 30 9x 3 16+ + = + :

2x 9x 4 30 3 16 = + :

7x 21 = :

21x 3

7

= =

; -

; -

.

.

0 x , 0 = .

( )2x 3 2 x 3 1+ = + , : 2x 3 2x 6 1+ = +

2x 2x 3 6 1 = + 0x 2=

0 2= -

, .

x. 0x = 0.

x 1 4 2x 95

2 4 2

= +

: x 1 4 2x 9

4 5 4 4 42 4 2

= +

( )20 2 x 1 4 2x 18 = +20 2x 2 4 2x 18 + = +

2x 2x 20 2 4 18 + = + +

0x 0=

-

-,

!



72/298

82.

- - -

-

: 0 0= !

,

.

.

-

;

-

. :

,

.

()

-

x.

.

, -

.

, :

s

t= ,

, s

t .

s :

-

s.

. : s t= . t :

t. . :s

t

= .

: t, .

,

.



73/298

83.

- - -

-

, -

115 . ,

5

.

;

.

x . 5

. x 5 .

.

2x .

115 .

:

( )x x 5 2x 115+ + = : x x 2x 115 5+ + = +

4x 120=

120x

4=

x 30= 30 , 25

60 .



74/298

84.

- - -

-

:

. . . .

. 7.

. 2.

x.

. :1 x

x2 2

=

. : 6 x

. : 8 x

7 : 8x 7.

. :1

x 25

+

: . .

. .

. 2. 2 k. 2 :

2k + 2, (2k + 2) + 2

: 2k, 2k + 2 , 2k + 4

. + 1.

2k + 1. 2k + 1, :

(2k +1) + 2, (2k + 1 + 2) + 2

: 2 + 1, 2 + 3, 2 + 5



75/298

85.

- - -

-

: 5

.

x, :1

x 52

1

3 x 52

.

, , :

. 41.

. 6 .

. x, : 41 x.

. x, : 2x + 6

:

1

4

1

9 11

x :1 1

x x 114 9

= 1 1

x x 119 4

=

:

, 1

5 .

.

90 .

180.

:1

90 x x 1805

+ + =

1x x 180 90

5+ =

1x x 90

5+ =

5x x 450+ =

6x 450=



76/298

86.

- - -

-

450x

6=

ox 75= 75 90 75= 15.

:

. 4x 3 3 2x= . 3z 8 6z 10= . 7 2y 6y 4=

. 4x 3 3 2x

4x 2x 3 3

6x 6

6x

6

x 1

= + = +

=

=

=

. 3z 8 6z 10

3z 6z 10 8

3z 18

18z

3

z 6

+ = =

=

=

=

. 7 2y 6y 4

2y 6y 7 4

4y 3

3y

4

= + + = +

=

=

: . 3 x 4 2 x 1 4=

. 7 x 2 11 2 x 9x 1 =

. ( ) ( )3 x 4 2 x 1 4

3x 12 2x 2 4

3x 2x 12 2 4

x 10

+ = ++ = + = +

=

. ( ) ( )7 x 2 11 2 x 9x 1

7x 14 22 11x 9x 1

7x 11x 9x 14 22 1

9x 37

37x

9

= + + = ++ = + + +=

=

:

. 8x 2 3 x 1 3 2x 4 0 = . 2x 3 7 x 4 31=

. ( ) ( )8x 2 3 x 1 3 2x 4 0

8x 2 3x 3 6x 12 0

8x 3x 6x 2 3 12

x 7

7x

1

x 7

+ + =+ + = = +

=

=

=

. ( )2x 3 7 x 4 31

2x 3 7x 28 31

2x 7x 28 3 31

5x 0

0x

5

x 0

+ = ++ = + = +

=

=

=



77/298

87.

- - -

-

!!!

() 1. ()

.

5.

. !

: x 1 x 2 5 x

142 3 4

=

... 12.

12 - :

x 1 x 2 5 x12 12 12 12 14

2 3 4

+ + + =

( ) ( ) ( )6 x 1 4 x 2 3 5 x 168+ + + = 6x 6 4x 8 15 3x 168+ + + + = 6x 4x 3x 168 6 8 15+ + = + 13x 169=

13 : x 13= 13.

: . 2x 4 x 1 3 2x 1 5 =

. 11x 8 x 2 5x 3 14=

. ( ) ( )2x 4 x 1 3 2x 1 5

2x 4x 4 6x 3 5

2x 4x 6x 4 3 5

0 x 2

0 2

+ = + + = +

+ = + + ==

. ( )11x 8 x 2 5x 3 14

11x 8 x 10x 6 14

11x x 10x 8 6 14

0 x 0

0 0

+ = + ++ = + +

= + ==

. x

:x 3 x 1 2

5 3 15

=



78/298

88.

- - -

-

... 15.

15 -

:

153 x 3

5

15

5 x 1

3

+ 15=

2

15

( ) ( )3 x 3 5 x 1 2 + = 3x 9 5x 5 2 = 3x 5x 9 5 2 = + + 2x 16 =

2 :16

x2

=

x 8= 8.

: .y 3 y 5 1 y

2 6 2 3

=

.

10 5 8

35 10 5

=

. ... 6.

6 -

:

63 y 3

62

y 56

6

=

3 16

2 +

2 y

3

( ) ( )3 y 3 y 5 3 1 2y = +

3y 9 y 5 3 2y + = +

3y y 2y 9 5 3 = +

0y 7=

0 7= .

. ... 10.

10

:



79/298

89.

- - -

-

102 10

5

+ 10+

5

10

10=

2 8

5

10 3+

( ) ( ) ( )2 10 5 2 8 30+ + = + 2 20 5 16 2 30+ + = + 2 2 20 5 16 30+ + = + + + 5 31=

5 :31

5

=

31

5

.

:7x 3 1 3 6x 5 3x 2

14 7 2 7

=

... 14.

14 -

:

14 7x 3 1414

+2

1 147

= ( )7

3 6x 5 142

+2

3x 27+

( ) ( ) ( )7x 3 2 1 7 3 6x 5 2 3x 2 + = + +

( )7x 3 2 21 6x 5 6x 4 + = + + 7x 3 2 126x 105 6x 4 + = + + 7x 126x 6x 3 2 105 4 = + 125x 100 =

125 : 100x

125

=

4x

5=

4

5.

:

11 x2xx 7

32 2

5 10 5

=



80/298

90.

- - -

-

... 10.

10 -

:

102

1x

2 105

+

12x

3 1010

+ =

2

x7

2

5

+

1 1 x

2 x 2x 2 72 3 2

+ + = +

1 1 x

2x 2 2x 2 2 72 3 2 + + = +

12x 1 2x x 14

3 + + = +

1

2x 2x x 1 143

+ = +

44

3x3

=

3 :

443x3

= 44

x9

=

44

9.

3 9 x 17= , x

. .

( )3 9 x 17 = . x, 3 9.

0 x 17 = . : 3 9 0 =

3 9=

9

3=

3=



81/298

91.

- - -

-

, :

. 3x 2 4 6x= . 8x 4 11 x=

. 3x 2 4 6x = + 3x 6x 4 2 = + ( )3 6 x 4 2 = + 0 x 0 = . :

3 6 0 = 4 2 0+ =

3 6= 4 2=

6

3

= 2

4

=

2= 1

2

=

. ( )8x 4 11 x 8x x 11 4 8 x 11 4+ = + = =

. , -

:

8 0

8

==

11 4 0

11 4

411

==

=

5x 4 3x 2x 8 = . ,

:

2 1 5 2A 2 3 8 =

( )5x 4 3x 2x 8 = + . :

( )5x 4 3x 2x 8

= + 5x 4 3x 2x 8 + = + 5x 3x 2x 4 8+ = + 6x 12=

12

x6

=

x 2=

2. = 2. , , -

2 :



82/298

92.

- - -

-

( ) ( ) ( )2 2 1 5 22A 2 2 2 3 2 2 8 2

= + + =3 5 2

4 2 1 4 6 64 2 1 4 36 64 2 4 36 30= + = + = + =

, 1

E 2

= ,

. .

:

.

1E 2=

2 2= 1

2

2 =

2 =

2

=

PV = nRT . :

. . V . R .

.nRT

V

= .nRT

VP

= .PV

RnT

= .PV

TnR

=

:

2E 2 . .

2E 2 = +

22 = 22 =

2

2

=



83/298

93.

- - -

-

180

= .

.

.

= = . :

180

180

180

=

==

180

180180

=

==

;

x. x.

, :

x xx x 0

2x 0

0x

2

x 0

= + ==

=

=

,

.

7

3.

x .

: : 3 x

7 : 3x 7

: 2 x

3 : 2x + 7



84/298

94.

- - -

-

: 3x 7 2x 3 = + .: 3x 2x 7 3 = + x 10= 10.

15

2

13

3.

, .

x .

:

15 x 13 x

2 3

=( ) ( )3 15 x 2 13 x =

45 3x 26 2x = 3x 2x 26 45 + = x 19 =

x 19= 19.

, -

4. 42cm.

.

x 4x 4. 42cm. :

+ + + = 42

4x 4 x 4x 4 x 42 + + + =10x 42 8= +

10x 50=

x 5= x = 5 cm 4x 4 4 5 4 16 cm = = .

17%. 86 .

;

x . 17

x100

.



85/298

95.

- - -

-

, :17

x x

100

.

:17

x x 86100

=

100x 100 17

100x 86 100=

100x 17x 8600 =83x 8600=

8600x

83

=

x 103,6

103,6 .

. .

.

. 3220 . ;

x . x2

-

x

3. , :

x xx 3220

2 3 =

6 x 6 3 x

62

2 x

6 32203

=

6x 3x 2x 19320 =x 19320=

19320 .

9

17

6

5.



86/298

96.

- - -

-

x . :

9 x 6

17 x 5

+=

+

( ) ( )5 9 x 6 17 x+ = +45 5x 102 6x+ = +5x 6x 102 45 =

x 57 =x 57=

57.

.

7 -

4. .

x.

2x + 7 3x 4 .

360. :

( ) ( )x x 2x 7 3x 4 360+ + + + =x x 2x 3x 7 4 360+ + + = + +

7x 357=

357x

7=

ox 51=

: 51.

: 2 51 + 7 = 109

: 3 51 4 = 149

144 ,

.

x, 144 x.

: ( )8x 10 144 x= 8x 1440 10x= 8x 10x 1440+ =



87/298

97.

- - -

-

18x 1440=

1440x18=

x 80= 0 80 64.

80 . 3

. 1 .

192 . ;

x . , 80 x . 3x.

1(80 x). : ( )3x 1 80 x .

: ( )3x 1 80 x 192 =3x 80 x 192 + =4x 192 80= +

272x

4=

x 68=, 68 12 .

40Km/h. 4 -

70Km/h. ;

x .

x + 4 . ( )x 4 40 Km+ .

x . x 70Km .

. : ( )40 x 4 70x+ =40 x 160 70x+ =70x 40x 160 =30x 160=

160x

30=

x 5, 33=



88/298

98.

- - -

-

5,33 .

40cm.

5.

.

x . x

52

+ .

40cm, :x

x x 5 402

+ + + =

2x 2x 2+ + x

2 2 5 2 40+ =

2x 2x x 10 80+ + + =5x 80 10= 5x 70=x 14 cm=

14cm :

14

5 7 5 12 cm2 + = + =



89/298

99.

- - -

-

1. :

. 9.

. 13.. .

2. :

. 4.

. 7.

3. :

137

81 .

4. :

1

7

11.

5. :

30

.

.

6. :

. 2x 5 x 1+ = . 3y 11 13y 21+ = + . 6 2 7 4 = +

7. :

. ( )4x 5 2 x 4 6x 3 = + + . ( ) ( )2 x 1 5x 7 3 x 11 + = +



90/298


91/298

101.

- - -

-

18. , :

. 3x 5 7x 10+ = + . 11 12x 1 x5 3

=

19. ( )3 30 x 6 18 = + .

. -

: ( ) ( )3 10 2 5 71 6

+ = + + +

20. :2x 1 2x 7 3x 2 1

3 6 4 6

+ + = +

, :

( ) ( ) ( )4

4 A 2 1 7 4 3 1 50

= + +

21.

:21s t

2= . :

. . t2

22. h :

E mgh= , g m . :

. m . g . h

23. , q Q. r, :

2

qQF

r= . q.

24. x, y, z V xyz= (1) E 2xy 2xz 2yz= + + (2). (1) y z.

(2) x.

25. 5

17.



92/298

102.

- - -

-

26. 9

5

8

13.

.

27. 2m.

10m. 40m2,

.

28. , ,

5 .

1. 107 .

.

29. 330 3

5.

30. ,

2,7. 43 cm.

.

31. 528 8% .

, ;

32. :

. 92

. 64

33. 17 . 10

50 . 6 50 .

10 50 .



93/298

103.

- - -

-

1

;

;

2

;

;

,

;

1

: x 5 2x 5 9 x 1

10 5 5 10

+ + = +

2

: ( ) ( ) ( ) ( )7 1 3x 2 5x 13 20 x 5 3 4x 7 = + +

3

. B

18 . -

1. .



94/298



95/298

6 6 6 6 6

7 7 7 7 7

3



96/298



97/298

. (..).

. .. ;

. x

.

. :

-

. :

2= 2+ 2

.

,

.

. .. : 2 2 2

2

2

x 6 10

x 36 100

x 64x 8

+ =

+ =

=

=

. .. .

. .. ;

. 6 cm, 7 cm, 9 cm

.

.. :



98/298

126.

-

-

, -

.

2= 2+ 2 oA 90= .

. ..

,

.

.

.. :

2 2 2

9 6 7= + 81 36 49= + 81 85= .

. -

;

. ;

.

. :

1. 36 ......= 0,36 ......= 3600 ......=

2. 0 ......= 1 ......= ( )2

......=

. -

x ,

.

: x= 2

x = , 0> , x 0>: 0 0= .

.

.

. 1. 36 6= 0, 36 0, 6= 3600 60=

2. 0 0= 1 1= ( )2

= , > 0



99/298

127.

-

() () :

. oB 90 2= 22.

. AB

2= 2 2, o 90 .

. x, y, :

2 2 2x y 2 2 2y x

.(), oB 90= .. 2= 2 + 2 2= 22.

.(), 2= 2 2 2 = 2+ 2 o 90= .

. (), 2 2 2x y = + , ..,

x 2 2 2

y x = + , y .

.

.

: . x .

. y .

. .

. x

.. :

2 2 2x 3 4= + 2

x 9 16= + 2x 25= x 25= x 5=

. y

A x 5= = = .

.. : 2 2 2 = + 2 2 2

y x x= + 2 2 2y 5 5= +

2y 25 25= + 2y 50= y 7, 07



100/298

128.

-

.

2E x 25= = ( . )

36 cm

10 cm. :

. A .

. .

.

36 cm. : AB 36+ + =

10 10 36+ + = 20 36+ = 36 20= 16cm=

: 16 8 cm

2= =

.. :

2 2 2 + =

2 2 2 8 10+ =

2 100 64= 2 36= 6 cm=

.

: 2 16 6 96 48 cm2 2 2 2 = = = = =

= 10cm = 6cm.

: . ,

. ,

. .

. .. :2 2 2

+ = 2 2 2

6 10+ = 2 36 100+ = 2 100 36=

2 64= 64= 8cm= .

. :

2 8 6 48E 24 cm2 2 2 2

= = = = =

. :



101/298

129.

-

E

2

=

B 24

2

=

10 24

2

= 10 24 2= 10 48=

48

10

=

4, 8 cm=

= 8 cm,

= = 10 cm = 20 cm. :

. .

. .

. . -

: = = 8 cm.

58248255 mathimatika b gymnasioy theoria askiseis

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