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! .
, 2012
!
( ( :
1. 1 1, ( 0, 1, 2, 3, 4, ... ( ( ( 7 (8 1) 1
(8 1) 1.
: >.
( > >.
(, (, ( 8 = 21 , > >.
2. , ( ( (, 1, 8, -1 (8 7) > 8 > 7
, > , (
7 -- C ) 7 >< (8 (< ).
7 7, 8
+
=
+
, 1 > ( 7 1, 7 > 1 8 7 (.
7, 7 + (5) = 75 = 2 , 5 + (7) = 57 =2 .
1. 1 = 1 = , 7 > > . 1 ( 7.
2. 0 = 0 = 0 , 7 > > .3. . = .4. . ( ) = ( ) . > 8 1 >
1 1 C C C, 1 ( > 8:
(+) (+) = (+) , (+) () = () , () (+) = () , () () = (+) . ( C > 1 (.
> C >, 1 ( 7 ( (, > > 1 8:
8 C > ( 7, 7 (+). 8 C > ( >, 7 (). ( 1 ( >.
7 1 (8 >) 7, !
( 1 1 >, > 7 -
8 > 7, ( 7 7 8 7 7.
7 > 1 (8
)
=
1
= 1
=
.
R8, > ' ( > 1 ( 1 7 ( > 8 8
' ( > 1 ( 1 7 ( > 8 ... ,
.
> 7 ( 7 1 1 7, 8 7 1
1 2 3 ... = 1+2+3+...+ .
7 > ' 7, 8 + = ( >, >
8 ' 7, 8 =
, ( >, >
, > ( 8 ,
> 1 7 , >
8 8 > 7
7 7 7 > 1 1 1 - 1 C.
7 > 8 ' 7, 8 = () , ( >,
> 1 >, > ( 7 1, 7 ( 7 2 C 7 (C ( 3 C 7 (C ( 1 C 7 (C ( ( 8:
22 = 2 2 = 4 , 8 7 7 > 2, > () !!
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17. ,
.
32 = 3 3 = 9 , 8 7 7 > 3, > () !! 14 = 1 1 1 1 = 1 , 8 4 1, > () !!>, 8, 1 7 > ( 8 7 C C ( | | .
1 8:
) | | = , > 0 , 8 > 8 > 1 1 ( ( >.
) | | = , < 0 , 8 > 8 > 1 1 ( (> .
) | 0 | = 0 .
8 ( > 1 1 >, (, > ( 7, ( > .
8 ( ( , > 0 .
7 >.
1.
8 02 = 0 , ( 0 = 0 .
2.R ( ( 1 1, > 7 >, 7> ( ->.
3.> > 8 ( 1 >:
) 8 ( 0 , ( 0 , 8 8 ( ( > > (1 8 ( >).
) 0 , > ( )2 =
2
= .
1 ( 1
> ( 7 (5)2 = 25 = 5 (8 (5)
2 = 52 = 5 ).
6. > > .R8, > 7 > 8 ( 7 7, > 1 ( 8 >!!
!!, ( ( )2 =
2
= ,
C ( 7, > 1 (.
, !!!
> 2
= , 0 .
,
!!!>
2 = | | .
2. = , , 0 .R8, > 1 C C, > ( 7 ( 8 (- > 1 C.
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20. .
7 > 8 ' 7, 8 = , 1 >, > 8 ( 1 C, > ( 7 ( 1 > 1 1, 8 1
1
2
3...
=
1
2
3...
,
1,
2,
3,...,
0 ,
7 > 7,
1
2
3...
=
1
2
3...
.
3.
=
, 0 , > 0 .
R8, > ( 7 ( 8 ( 7 ( 1 C.
7 > 8 ' 7, 8
=
, 1 >, > 7-
> ( 8 ( ( 7 8 ( 1 8 7 ( C, > (
( 7 C 8 , 12 , 18 ...
> 7 > > 1 C, ( 7, 8:
8 = 4 2 = 22 2 = 2 2 .
12 = 4 3 = 22 3 = 2 3 .
18 = 9 2 = 32 2 = 3 2 .
75 = 25 3 = 52 3 = 5 3 .
300 = 100 3 = 102 3 = 10 3 .
+ = + !!! (1 ' 7 !!).
= !!! (1 ' 7 !!).
H ,
7 1 C:
1 : 8 7 > ( -8 ( (, ).
> ( 8 8 7- ( , 7 7 >, (!
> 7, 8.
R 7 (:
2
2=
2 2
2 2=
2 2
22
=2 22
= 2 .
3
5=
3 5
5 5=
3 5
52
=3 55
.
1
2 2=
1 2
2 2 2=
2
2 22
=2
2 2=
24
.
3
5 3=
3 3
5 3 3=
3 3
5 32
=3 35 3
=35
.
2 : 8 7 8 (, ( 8 ( ' 8 (.
> ( 8 8 > 8 7 -8.
> 7, 8.
8 7 ( 7 C (+ )() = 22 .
! . + ( 8 7, >
4
2 2 +1=
4 (2 21)
(2 2 +1) (2 21)=
4(2 21)
(2 2)2 12=
4(2 21)
22 22
1=
4(2 21)7
.
2
2 3=
2 ( 2 + 3)
( 2 3) ( 2 + 3)=
2( 2 + 3)
22
32
=2( 2 + 3)
23= 2( 2 + 3) .
2
2 3 + 3 2=
2 (2 33 2)
(2 3 + 3 2) (2 33 2)=
2(2 33 2)
(2 3)2 (3 2)2=
2(2 33 2)
22 32
32 22
=
=
2(2 33 2)1218
=2(2 33 2)
6=
2(3 22 3)6
.
( ( 8, 8 () > 6 7 > ( (8 ( () 7 7 > 7:
1. = , + = + .R8, 1 ( .
2. = , = .R8, ' 1 ( ' .
3. = , = , 0 .R8, 1 ( 7 (, >, >.
4. = ,
=
, 0 .
R8, 1 ( , >.
( (4) ( 7 >:
> 1 ( (( 1 7. ( 8, ( 7 C (7 7 ,
( ( > C >-.
R 7 7 (.
2x + 4y + 6z = 2 x + 2 2y + 2 3z = 2(x + 2y + 3z) (> 7 8 > 2).
x2 + xy + 2xy 2 = x x + x y + x 2y 2 = x(x +y + 2y 2) (> 7 8 x).
x2y 2xy 2 + x 3y = xy x xy 2y + xy x 2 = xy(x 2y + x 2) (> 7 8 xy).
( > >, > 7 (, 1 > 1 7 7- > 7 8 7. 7 7 7 8 7 ( 7, > 7 -( > 1 ( 7 1 > 7 (7) > > 7. ( 8, 7 > 7 1 (.
C > > 7 ( 7 7 7 -( 7.
1.
x +y + 2x + 2y = (x +y)+ (2x + 2y) = (x +y)+ 2(x +y) = (x +y)(+ 2) . .
x +y + 2x + 2y = (x + 2x)+ (y + 2y) = x(+ 2)+y(+ 2) = (+ 2)(x +y) .
2.
3x312x 2 + 5x 20 = (3x 312x 2)+ (5x 20) = 3x 2(x 4)+ 5(x 4) = (x 4)(3x 2 + 5) .
3.
33 + 9 = (3)+ (3 + 9) = (3)3(3) = (3)(3) . .
33 + 9 = (3)+ (3+ 9) = (3)3(3) = (3)(3) .
3 . .C ( (
9x2 12x + 4 = (3x)2 2 3x 2 + 22 = (3x 2)2 .
x21 = x 212 = (x +1)(x 1) ( !)
1x2 = 12x 2 = (1+ x)(1x) ( !)
x2 4 = x 222 = (x + 2)(x 2) ( !)
4x2 = 22x 2 = (2 + x)(2x) ( !)
x29 = x 232 = (x + 3)(x 3) ( !)
9x2 = 32x 2 = (3 + x)(3x) ( !)
x216 = x 2 42 = (x + 4)(x 4) ( !)
16x2 = 42x 2 = (4 + x)(4x) ( !)
4x225 = (2x)252 = (2x + 5)(2x 5) .
169x2 = 42(3x)2 = (4 + 3x)(43x) .
x3 +1 = x 3 +13 = (x +1)(x 2 x 1+12) = (x +1)(x 2 x +1) ( !)
x3 1 = x 3 13 = (x 1)(x 2 + x 1+12) = (x 1)(x 2 + x +1) ( !)
1x3 = 13 x 3 = (1x)(12 +1 x + x 2) = (1x)(1+ x + x 2) ( !)
x3 + 8 = x 3 + 23 = (x + 2)(x 2 x 1+ 22) = (x + 2)(x 2 2x + 4) ( !)
x3 8 = x 3 23 = (x 2)(x 2 + x 2 + 22) = (x 2)(x 2 + 2x + 4) ( !)
8x3 = 23 x 3 = (2x)(22 + 2 x + x 2) = (2x)(4 + 2x + x 2) ( !)
4 . .
8 8 7 8 x2 + x + , 0 ( -
7 C), 1 7 > = 2 4 .
7 8 C:
) > ( R > 0, > C
x2 + x + = (x x
0)2 .
R8, > ( R = 0, > C ( (+ )
2 8
()2 , C 8 > ( R < 0, > C
8 8 7 (8 C 8 7) ( C.
7, .
( 8 7, .
) . ( C 1.
7 > 7 (...) 1 (8 (- C C > , 1 > ( 7 1 C.
( >, > 7 8 ( 7, !
C C 1
.
) ( ( 8 8 0 = >, 7 > ( -7 >, > ( ), 8 8 > ( 1 .
) H ( (() > ( ( (, > 7 7 1.
) = 0 , = 0 = 0 . > 8 1 7 1 ( C, ( ( ).
( > ( (- 7 > 7, ( > ' (, 7).
(: L = 2
L2
=22 =
L2
.
2. 1 1 P = 2x + 2y y (( y 7 > 7).
(: P = 2x + 2y P 2x = 2y
2y2
=P 2x
2 y =
P 2x2
.
3. 1 1 =
( +)2
(( 7
> 7).
(
=
( +)2
2 = 2 ( +)
2 2 = ( +) 2 = + 2 =
=2
=
2
.
1. .
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27. .
H , > ( (8 ( () 7- 7 > 7:
1. < , + < + .R8, 0 , < .R8, , > 7 ( 7.
( 8 ( 7 >:
> 1 ( (( ( > 7. ( 8, ( 7 C (7 7 , , > 7 ( 7.
( 8 ( 7 >:
> 1 ( (( ( > 7. ( 8, ( 7 C (7 7 ,
) 1 7 ( C, > 1 ( ( 7' C 1< 1 ().
: > ( > 7. 7 8 C :
8 8 ( 8 8 8. 8 8 ( 8 8 - 8 1 ( >.
1. ( 7
34
,47
.
= > (
34
>47
. 7 8 C 28 47 7 3 > 4 4 21 >16 , 1,
> > (
34
>47
.
2. ( 7
22
,35
.
= > (
22
> 0 .(, > 0 , > > .
2. < , > < 0 .(, < 0 , > < .
3.
>
>
, + > + .
R8, 1 >, ( 7, (
4.
, , , , >
>
, > .
R8, 1 >, ( 7 1 C C, ( 7- 7 > .
6. 2 0 , . !!!!!
1 7 1 8 (, > > -X Y R !!!!
( 1
7 8, ( ( ( >.
1 x2 , 8
x 2 + = 0 x 2 = x 2 =
.
7
( ( 7 ) ( 7 x2 = , >:
1. .
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32. ( )
.
33. .
34.
.
!!!
) ( > 0, > 1 ( ( ( x = 8 x = .
> > 7 x = .
) ( = 0, > ( ( 1 (1 ( = 2 4 , >:
) ( R > 0, > ( ( 1 x =
2
.
) ( R < 0, > ( C : C 7 , 7 1 7 ( 7.
) ( x 2 8.' >, 7 > ( , C ( 7.
) 7 7 ( ( 7 ( (, > 7 ( .
1. .
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7 C 8 ( , 8 ( 7 >!
> > (, 7 99%, ( > ( (, C 8 ,
0 , 0 0 .
7, C C (8 1) >, , 1 1, > 7 ( x =3 .
1. .
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35.
.
!!!
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