Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/free-books/pdf/alfutova.pdf Äàòà èçìåíåíèÿ: Thu Mar 24 19:14:35 2005 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 12:38:14 2012 Êîäèðîâêà: IBM-866 Ïîèñêîâûå ñëîâà: redshift survey

. . , . . September 3, 2003


51 21.1 45

45

. . . . . .-- .: , 2002.-- 264 .
ISBN 5-94057-038-0
, , . , . . , , 1995-- 2000 - . . . .

21.1

ISBN 5-94057-038-0

c . ., . ., 2002. c , 2002.



, , . , . . , 1995 í 2000 . . , . . . . . . , , , . , - . . . , , . , , . . . , . , . , , ë¨. , . . , , , . .


-. . ë ¨ . , . , 18 ( . . . ), . . . , . . , . . , , , . . . TEX- . , , . : 117630, , . . , . 8 , ë ¨. . . alpha@pisem.net . . ustinov@mech.math.msu.su



, .
. 70 x ( , x) {x} x: {x} = x - [x] n! : n! = 1 § 2 § . . . § n 7 {xn } x1 , x2 , . . . , xn , . . . b|a b a 8 . . a.b a b 8 a b mod m a b m 53 ï a m 53 (ak . . . a0 )q q- 6 (a1 , . . . , an ) a1 , . . . , an 29 [ a1 , . . . , a n ] a1 , . . . , an 32 [a0 ; a1 , . . . , an ] 42 (n) 60 (n) n 34 (n) n 34 Fn 36 i i = -1 101 C 101 z z 101 arg z z 101 |z| z 101 e 73 = ( 5 + 1)/2 -- 39 151 ïn Ak k- n 16 Ak k- n 16 n Pn n 17 Ck k- n 17 n Ck k- n 17 n Cn 25 En n: En = 11 . . . 1 74
n

N Z Q [x]


1


1.
. , 1, , n, n+1, . 1.1. . , a b -- b = 0, q r , a = bq + r, 0 r < | b |.

1.2. . , q 2 n n = ak qk + ak-1 q 0
k-1

+ . . . + a1 q + a0 ,

a0 , . . . , ak < q. (. 3.125, 11.68.)

. ak , ak-1 , . . . , a1 , a0 -- n q- , n = (ak ak-1 . . . a1 a0 )q . (ak ak-1 . . . a1 a0 )10 = ak ak-1 . . . a1 a0. 1.3. {an } = a0 , a1 , . . . , an , . . . -- , T a
n+T

= an

(n

0).

, t, T t . 1.4. . , . (. 12.1.) 1) .


2. ,

7

2) . 3) 1 , . 4) , a, , k, , a k < n n, k a. 5) ( .) , 1 2, , n > 1, 2n n - 1, . 1.5. x , x + n xn + 7.46.) 1.6. x1 , . . . , xn . , (1 + x2 ) § . . . § (1 + x2 ) 1 n . (. 7.14.) 1.7. A1 , A2 , . . . , An , . . . A1 = 1, A2 = -1, An = -An
+2 -1

1 -- . , x

1 . (. xn

- 2An

-2

(n

3).

, n

2 2n

- 7A2 . n

2. ,
. n! ( ën ¨) 1 n: n! = 1 § 2 § . . . § n. , 0! = 1. 1.8 í 1.14 . 1.8. 1 + 3 + 5 + . . . + (2n - 1) = n2 . 1.9. 12 + 22 + . . . + n2 =
n(n + 1)(2n + 1) . 6


8

1. 1.10. 12 + 32 + . . . + (2n - 1)2 =
n(2n - 1)(2n + 1) . 3

1.11. 13 + 23 + . . . + n3 = (1 + 2 + . . . + n)2 . 1.12. 1 § 2 § 3 + 2 § 3 § 4 + . . . + n(n + 1)(n + 2) =
n(n + 1)(n + 2)(n + 3) . 4 n(n + 1) 12 22 n2 1.13. + + ... + = . 1§3 3§5 (2n - 1)(2n + 1) 2(2n + 1)

1.14. 1 § 1! + 2 § 2! + . . . + n § n! = (n + 1)! - 1. 1.15. . , n n = a1 § 1! + a2 § 2! + a3 § 3! + . . . , 0 a
1

1, 0

a2

2, 0

a3 a

3, . . .

1.16. a0 , a1 , . . . , an , . . . : a0 = 2, a1 = 3,
n+1

= 3an - 2an

-1

(n

2).

. . a -- b -- . b a, q , a = bq. a b, q -- a b. ëb a¨ b | a . . a . b (ëa b¨). , b = 0. b a, b a. , 1.17 í 1.24, n. . . 1.17. 10n + 18n - 1 . 27. . 133. n+2 2n+1 . 1.18. 11 + 12 . + 5n § 3n . 1.20. n3 + 5n . 6. . . 2n+1 . 1.21. 6 + 1 . 7. 1.22. 32n
+2

1.19. 25

n+3

+2

. . 17. .

. . + 8n - 9 . 16. . n . 9. 1.23. 4 + 15n - 1 . n . . 1.24. 23 + 1 . 3n+1 . 1.25. , n , 3n , 3n .


2. ,

9

1.26* . 1 2n n + 1 . , , . (. 2.34.) 1.27. 1-
x(x - 1) x(x - 1) § . . . § (x - n + 1) x + - . . . + (-1)n = 0. 1! 2! n!

1.28 í 1.36 n.
1 + 12 1 1.29. + 1 (2n)! 1.30. (n!)2

1.28.

1 1 1 + 2 + . . . + 2 < 2. (. 7.81.) 2 2 3 n 1 1 + ... + n. n 2 4n > . n+1

1.31.

1 1 1 13 + + ... + > n+1 n+2 2n 24
n

(n > 1). 1 + nx x > -1.

1.32. . (1 + x)n 1.33. 2 > n. 1.34.
1 § 3 § 5 § . . . § (2n - 1) 2 § 4 § 6 § . . . § 2n
n+1

1 . 2n + 1

1.35. n

> (n + 1)n

(n > 2). |x1 | + . . . + |xn |, x1 , . . . , xn --

1.36. |x1 + . . . + xn | .

1.37* . .
x1 + . . . + xn n

n

x1 § . . . § xn ,

x1 , . . . , xn -- . 1.38. 2m .
+n-2

mn, m n --

1.39. n : ) n! > 2n ; ) 2n > n2 . 1.40.
23 - 1 33 - 1 n3 - 1 §3 § ... § 3 3 2 +1 3 +1 n +1

(n

2).


10

1.

3.
1.41. 16 ½ 16 . , ë¨ . 1.42. I. ë ¨ 8 , . , , . , . ( )? (. 5.71.) 1.43. I I. 1, 2, 3. , 1- 3-. , 1- 3- ? ( 2- . , .) 1.44. I I I. , 1.42 : ? 1.45. , n n, . 1.46. , n n, . 1.47. . ) , , . , : , . 11 , 11 , . , ? ) , , n ? 1.48. - . , , . (. 3.72.)


3.

11

1.49* . . ) , , . : ë ?¨ , - ë¨. ? ( .) ) , , 1000? 1.50. n ë ¨, , ? 1.51. n , , . ? 1.52. n , , ? 1.53* . n ë ¨? ë ¨? 1.54. . , , . ( , .) 1.55. n-. , n- ( ) . , n- (n - 2). 1.56. 100 ½ 100 4 , 2 ½ 2 . , . 1.57. k . k . ( k ) . . , m? 1.58. . , - + = 2, -- , -- , -- .


12

1.

1.59* . . , , , . , . 1.60. . , , . 1.61. () 100 ?


2


1. ?
2.1. ) A, B C. A B 6 , B C -- 4 . c A C? ) D -- A D D C. A C? . a m , b ( a) -- n , ëa b¨ m + n . . a m , b ( a) -- n , ëa b¨ m § n . 2.2. C (c, )? 2.3. (30 ) . ? 2.4. 32 , -- 29 . : ? 2.5. 6 8 , . ? 2.6. - . , . -? (. 12.9.) 2.7. , 5? 2.8. , ?


14

2.

2.9. , ? 2.10. : , , ? 2.11. , , , . , 23 37. , . , ? ( 23 37 237.) 2.12. , (, 54345, 17071)? 2.13. , ? 2.14. 7 ? 2.15. ë¨, . ë¨ ? 2.16* . . . ( ). , ?

2.
( ). nk + 1 n - k + 1 . 2.17. , , , . 2.18. 70 , : 20 , 20 , 20 , -- . , , 10- ?


2.

15

2.19. . , , . 2.20. 2k + 1 , 1 2k + 1. , ? 2.21. , ? 2.22. , -- . , - . 2.23. 200 41, 42 43 , 600 300 300 . , 100 . 2.24. . , , . 2.25* . 51 ( 0). , 6 , 2 . 2.26. 1 101 . , 90 , 11 . 2.27. 2000 . , ? 2.28. 1002 , 2000. , , . , 1002 1001? 2.29* . , , . , , - .


16

2.

2.30. . , -- . , . ? 2.31. ( -- ). , , , . 2.32. , 11 , . 2.33. 6 , . . , , . (. 5.36.) 2.34. 1.26 .

3. ,
. M = {a1 , . . . , an } -- n . (ai1 , . . . , aik ) k- . k- , . ai1 , . . . , aik , . ai1 , . . . , aik , , . ï Ak Ak . n n 2.35. : ï ) Ak = n(n - 1) . . . (n - k + 1); ) Ak = nk . n n 2.36. 17 . 17 , ? . n- M = {a1 , . . . , an }.


3. ,

17

n Pn . 2.37. Pn = n!. 2.38. 8 , ? 2.39. . ? 2.40. , 17 ? 2.41. 7- , 1, . . . , 7. 2.42. ) 28 ? ) , ? 2.43. , ? 2.44. , 28 , 4 . ? , ? . M = {a1 , . . . , an } -- n . k- (ai1 , . . . , aik ), . k- , , . . Ck Ck . n n 2.45. . , ? 2.46. n . ? 2.47. n ë ¨. . ?


18

2.

2.48. a b A1 , A2 , . . . , Am B1 , B2 , . . . , Bn . , Ai Bj (1 i m, 1 j n), , ? 2.49* . . 9 . . , , , 6 ? , n , m (m n). 2.50. 7 , 9 . ? 2.51. ) Ck = n
n! ; (n - k)! k!

) Ck = Ck n n

+k-1

=

(n + k - 1)! . (n - 1)! k!

2.52. , Ck n k- n . 2.53. . (x + y)n = C0 xn + C1 x n n
n-1

y + C2 x n

n-2

y2 + . . . + Cn yn . n

Ck , n x + y. 2.54. ) ( 2 + 4 3)100 ; ) ( 2 + 3 3)300 ? 2.55* . , a n n, n + 1, nn + 1, nn + 1, . . . a. 2.56. : ) 10-; ) k- (k > 3)? 2.57. n- . . . ?


3. ,

19

2.58. . , . : ) ; ) ; ) ; ) ; ) ; ) ? , . . 2.59. . m ½ n -- , ë¨, n - 1 m - 1 ë¨. , ( (0; 0)) ( (m; n))? (. 2.77.) 2.60. 10 ½ 10 ½ 10, . O . , , , , O . , ? 2.61. , ; m . ? 2.62. 6 : ) 12; ) 24 ? 2.63. C, n . C A B , ) A B ; ) A B? 2.64. . , (x1 + . . . + xm )n =
k1 +...+km =n

C(k1 , . . . , km )x

k1 1

. . . xkm m

C(k1 , . . . , km ) C(k1 , . . . , km ) =
n! . k1 ! § . . . § km !

C(k1 , . . . , km ) .


20

2.

2.65. 10 , . ? ( , 10 .) (. 2.95.) 2.66. 6- , ? 2.67. m n , m > n. , ? (. 3.129, 11.84.) 2.68. 1 6. 20 , ? n m (n > m) , ? 2.69. 1 6. 20 ( )? 2.70. x1 + x2 + x3 = 1000 ) ; ) ? (. 11.67.) 2.71. 17 , , , , , ? . ,
C0 0 C0 1 C0 2 C1 2 C1 1 C2 2 1 1 1 1 2 1 1 3 3 1 ... ... ... ... ... ... ...

C0 C1 C2 C3 3 3 3 3 ... ... ... ... ... ... ...

(. 2.76, 2.77).


3. ,

21

2.72. 112 = 121 113 = 1331 ? 114 ? 2.73.


� 2.74. - . 2.75. n (a + b)n ? 2.76. ) C0 5 ) C0 n ) C0 n : + 2C1 + 22 C2 + . . . + 25 C5 ; 5 5 5 - C1 + . . . + (-1)n Cn ; n n + C1 + . . . + Cn . n n

2.77. : ) Cm Ck = Ck Cm-k ; r r-k m r +1 m ) Cm+1 = Cm + Cn +1 ; n n n 02 ) C2n = (Cn ) + (C1 )2 + . . . + (Cn )2 ; n n ) Ck +m = C0 Ck + C1 Ck-1 + . . . + Ck C0 ; n nm nm nm ) Ck = Ck-1 + Ck-1 + . . . + Ck-1 . n-1 n-2 k-1 n : , Ck -- k- n n ; , Ck -- n xk (1 + x)n ; ë ¨ 2.59. 2.78. . Ck-1 § Ck+1 § Ck n-1 n n
+1

= Ck n

-1

§ Ck+1 § Ck-1 . n+1 n

2.79. 120 . ? 2.80. (x + y)n 240, -- 720, -- 1080. x, y n.


22

2.

2.81. . , n n = C1 + C2 + C3 , x y z x, y, z -- , 0 x < y < z. 2.82. 10 14 . , . 2.83. m n ,
+1 -1 Cm+1 : Cm+1 : Cm+1 = 5 : 5 : 3. n n n 2.84. (1 + 3)100 ?

2.85. , 1, 2, 3, 4 5, : ) ; ) ; ) ? 2.86. 5 5 10- , ? 2.87* . n- . , . ? ? 2.88. .
1 2 1 12 1 60 1 1 1 6 1 30 1 2 1 12 1 60

1 6

1 5

1 4 1 30

1 3 1 20

1 3 1 20

1 4 1 30

1 5

1 6

, . ë ¨ . .


4.

23

, . , . 2.89. :
1 1 1 1 1 1 =++ + + + ...; 1 2 6 12 20 30 1 1 1 1 1 1 ) = + + + + + ...; 2 3 12 30 60 105 1 1 1 1 1 1 ) = + + + + + ... 3 4 20 60 140 280

)

2.90.
1 1 1 1 + + + + ... 12 30 60 105

. 2.91.
1 + 1§2 1 ) 1§2§3 0! ) + r! (r

)

1 1 + + 2§3 3§4 1 + + 2§3§4 3 1! 2! + - 1)! (r - 2

1 + ...; 4§5 1 1 + + ...; §4§5 4§5§6 3! + + . . . (r )! (r - 3)!

2).

. - , . (. [8].) 2.92. 10 15 . 4 . , ? 2.93. . , 5? 2.94. , 0 9. . , ) ; ) ? 2.95. 4 , 4 . , ) 2 : 2; ) 3 : 1; ) 4 : 0? (. 2.65.)

4.
2.96. . , , . , ,


24

2.

. , , , . , , , ? 2.97. . a1 , , a2 , , . ., ak , k . ? (, k .) 2.98. n A1 , . . . , An E j (x) -- , j (x) = 1, 0, x Aj , x E \ Aj (j = 1, . . . , n).

, (x) -- A = A1 . . . An , 1 (x), . . . , n (x) 1 - (x) = (1 - 1 (x)) . . . (1 - n (x)). 2.99. . |A1 A2 . . . An | = |A1 | + . . . + |An | - |A1 A2 | - - |A1 A3 | - . . . - |An-1 An | + . . . + (-1)n-1 |A1 A2 . . . An |, |A| A. (. 4.138.) 2.100. 100 28 , -- 30, -- 42, -- 8, -- 10, -- 5, 3 . ? 2.101. ABC 8 . ( A, B, C ), ABC? 2.102. 1 16 500, ) 5; ) 5 3; ) 5 3, 11?


5.

25

2.103. 1 33 000, 3 5, 11? 2.104. 1 1 000 000, , , ? 2.105. . 30 . , ? 2.106. 15 , ? 2.107. 6 2 3 2 . , - , 1 2 . 2.108. 5 9 1 . , , 1/9. 2.109* . 1 5 1/2 . ) , , 3/20. ) , , 1/5. ) , , 1/20. 2.110. , 2.109 ) ) 1/5 1/20 .

5.
{Cn } = {C0 , C1 , C2 , . . . } = {1, 1, 2, 5, 14, 42, . . . }. . n + 1 x0 , x1 , . . . , xn , n . Cn x0 § x1 § . . . § xn , . , n = 2 : x0 § (x1 § x2 ), (x0 § x1 ) § x2 , n = 3 5: x0 § (x1 § (x2 § x3 )), x0 § ((x1 § x2 ) § x3 ), (x0 § x1 ) § (x2 § x3 ), (x0 § (x1 § x2 )) § x3 , ((x0 § x1 ) § x2 ) § x3 .


26

2.

2.111. {a1 , a2 , . . . , a2n }, + 1 - 1, , a1 + a2 + . . . + a2n = 0, a1 , ? 2.112. (n + 2) ? 2.113. . 2, 3, 4, . . . . , . ( .) ? 2.114. . 50 , 2n . , -- 50 . , . , , ? 2.115. . {a1 , a2 , . . . , an } -- , + 1. , {a1 , a2 , . . . , an }, {a2 , . . . , an , a1 }, ..., {an , , a1 . . . , an
-1

a1 + a2 ,

...,

a 1 + a2 + . . . + a

2n

},

. : Cn = Cn 2n
+1

(4n - 2)!!!! 1 1 = Cn = , 2n 2n + 1 n+1 (n + 1)!

(4n - 2)!!!! = 2 § 6 § 10 . . . (4n - 2) -- , . (. 3.105.) 2.116. . , Cn = C0 Cn (. 11.92.)
-1

+ C1 Cn

-2

+ . . . + Cn

-1

C0 .


3


1.
. p , p > 1 p , 1 p. , 1 . , , . 3.1. . , . 3.2. , 17. 3.3. , 30 -- . 3.4. n > 2. , n n! . 3.5. p q, p2 - 2q2 = 1. 3.6. , n! + 1 n + 1, n + 1 -- . 3.7. , p = 4k + 3 . (. 4.127.) 3.8. , p = 6k + 5 . (. 4.128.) d 3.9. , n n.

3.10. n ? 3.11. 111, 1111, 11111, 111111, 1111111. (. 4.25.)


28

3.

3.12. , 1000 . 3.13. , n n , . 3.14. ) 5; ) 6 , ? 3.15. , ? 3.16. , 15 , d. , d > 30000. . , 2 -. 3.17. , 3, 5 7 -. 3.18. , . 3.19. , n > 2 2n - 1 2n + 1 . 3.20. n n4 + 4 -- ? 3.21. , P(n) = n2 + n + 41 n ? 3.22. {pn } -- (p1 = 2, p2 = 3, p3 = 5, . . . ). , pn > 2n n 5. n pn > 3n? 3.23. pn
+1

< p1 p2 . . . pn .

3.24. , p1 p2 . . . pn + 1 ? 3.25. . : e1 = 2, en = e1 e2 . . . en
-1

+ 1 (n

2).

en ? (. 4.79.) 3.26. . , an + 1 , . . 2 n = 2k . ( fk = 22k + 1 a. .


2. 3.27. , fn 2fn - 2.

29

3.28. , fn n > 1 . 3.29. . , an - 1 , a = 2 n -- . q = 2p - 1 . 3.30. Pn (x) = an xn + . . . + a1 x + a0 -- (n 1, an = 0). , x = 0, 1, 2, . . . Pn (x) -- ?

2.
. () a1 , . . . , an a1 , . . . , an , . a1 , . . . , an (a1 , . . . , an ). a1 , . . . , an 1, . 3.31. , a1 , . . . , an 0, . 3.32. m n, , . ? ? 3.33. p q . [0; 1] p + q . , , p + q - 2
1 , p 2 , p

...,

p-1 , p

1 , q

2 , q

...,

q-1 . q

3.34. 1 . , -- , -- -- . ? 3.35. 1.1 a b. , a = bq + r (a, b) = (b, r). 3.36. . m0 m1 -- , m1 > 0 m1 m0 . , k > 1


30

3.

a0 , a1 , . . . , ak-1 m2 , . . . , mk , m1 > m2 > m3 > . . . > mk > 0, ak > 1, m0 = m1 § a0 + m2 , m1 = m2 § a1 + m3 , m =m §a +m , 2 3 2 4 .............. m k-2 = mk-1 § ak-1 + mk , mk-1 = mk § ak , (m0 , m1 ) = mk . 3.37. , s k - 1 0 us , vs , us ms + ms+1 vs = d, d = (m0 , m1 ). , u v : m0 u + m1 v = d. (. 6.67.) 3.38. (a, b) = 1 a | bc. , a | c. 3.39. (1 . . . 1, 1 . . . 1).
m n

3.40. a b, , a § b = 600? 3.41. a1 , a2 , . . . , a49 a1 + a2 + . . . + a49 = 540. ? 3.42. 19 19 ? 3.43. 1 1000 . , 15- : 1, 15, 31, . . . , , . . ? 3.44. , (5a + 3b, 13a + 8b) = (a, b). 3.45. ? 3.46. , a, b c b+c a+c a+b , , = (a, b, c).
2 2 2


2.

31

3.47. 40 18. , , . ? , ? 3.48. x y 3x + 2y 23. , 17x + 19y 23. 3.49. , n: )
2n + 13 ; n+7

)

2n2 - 1 ; n+1

)

n2 - n + 1 . n2 + 1

3.50. n )
n2 + 2n + 4 ; n2 + n + 3 n4 + 1 ; n +n+1
2

)

n3 - n2 - 3n ? n2 - n + 3

3.51. n ) )
n3 + n + 1 n2 - n + 1

? 3.52. n > 1 n3 - 3 n - 1. 3.53. 3.54. , m = n : ) (am - 1, an - 1) = a(m,n) - 1 (a > 1); ) (fn , fm ) = 1, k fk = 22 + 1 -- . (. 3.39, 3.122, 6.69.) 3.55. , 22 - 1 n . pn
+1
n

3m - n , 5n + 2m

, m n .

3.56. , n 22 + 1.

3.57. , (a, mn) = 1 (a, m) = 1 (a, n) = 1. 3.58. , (a, b) = 1, (2a + b, a(a + b)) = 1. 3.59. , (a, b) = 1, a + b a2 + b2 1 2. 3.60. a b -- . , a, 2a, 3a, . . . , ba (a, b) b.


32

3.

3.61. (a, b) = 1 (x0 , y0 ) -- ax + by = 1. , x = x0 + kb, y = y0 - ka, k -- . 3.62. ax+by = c a, b, c? 3.63. ( ): ) 45x - 37y = 25; ) 109x + 89y = 1; ) 19x + 95y = 1995; ) 43x + 13y = 21; ) 10x + 2y + 18z = 7; ) 34x - 21y = 1. 3.64. , . 3.65. , , 120 ? 3.66. a b,
a+b 3 =. 13 a2 - ab + b2

3.67. , (a1 , a2 , . . . , an ) = 1, a1 x1 + a2 x2 + . . . + an xn = 1 . . a1 , . . . , an -- 0 . () , . a1 , . . . . . . , an [a1 , . . . , an ]. 3.68. ) [1, 2, . . . , 2n] = [n, n + 1, . . . , 2n]; ) (a1 , a2 , . . . , an ) = (a1 , (a2 , . . . , an )); ) [a1 , a2 , . . . , an ] = [a1 , [a2 , . . . , an ]]. 3.69. n . , , . ) , . ) . : (4, 6, 9) (2, 12, 9) (2, 3, 36) (1, 6, 36), (4, 6, 9) (4, 3, 18) (1, 12, 18) (1, 6, 36).


3.

33

3.70. c, ) 7x + 9y = c 6 ; ) 14x + 11y = c 5 . 3.71. c, 19x + 14y = c 6 ? 3.72. a b -- . (x, y), 0 x b - 1. N(x, y) = = ax + by. ) , c (x, y) (0 x b - 1), c = N(x, y). ) . , c, ax + by = c , c = ab - a - b. 3.73* . a b . , , ax + by = c n , c (n - 1)ab + a + b (. 1.48.) 3.74* . 81x + + 100y, x, y -- , -- . , . , . c (n + 1)ab - a - b.

3.
. , 1, ( ) . 3.75. 3.38. 3.76. , . 3.77. 100! ?


34

3.

3.78. n, 1999! 34n . 3.79. , n+1 2n 2n , 2n+1 . 3.80. , ) (a, ) [a, ) (a, a = p1 1 1 , . . . , s , b) = pmin(1 ,1 ) 1 b] = pmax(1 ,1 ) 1 b)[a, b] = ab. §. 1, §. §. . . . . . . . . § ps , b = p1 § . . . § ps , p1 , . . . , ps -- 1 s s . , s 0. : § pmin(s ,s ) ; s § pmax(s ,s ) ; s

3.81. : ) [a, (a, b)] = a; ) abc = [a, b, c](ab, ac, bc); ) (a, [a, b]) = a; ) abc = (a, b, c)[ab, bc, ac]. . 3.82. , (bc, ac, ab) . (a, b, c)2 . . 3.83. , (a, b, c)[a, b, c] = abc . (a, b, c) ½ ½ [a, b, c] abc? 3.84. ) 2 § 3 § 5 § 7 § 11; ) 22 § 33 § 55 § 77 § 1111 ? 3.85. k 1 6 , k . 3.86. (n) -- n = p1 § . . . § ps , (n) -- . : 1 s ) (n) = (1 + 1) § . . . § (s + 1); ) (n) =
p1 1 - 1 ps +1 - 1 §...§ s . p1 - 1 ps - 1
+1

3.87. n, , (n) = 6, (n) = 28. 3.88. n . ) 15; ) 81 . ? 3.89. n = 2x § 3y § 5z , , 30 , -- 35 -- 42 , . . f(n), , : 1) f(1) = 1; 2) f(m § n) = f(m) § f(n) (m, n) = 1.


3.

35

f(1) = 1 f(m § n) = f(m) § f(n) m n, f(n) . 3.90. (n) (n). 3.91. (n) 2 n. 3.92. . , . 3.93. (m, n) > 1. (m § n) (m) § (n)? (n). (. 4.144.) . n , (n) = 2n. , 6 28 -- : 1 + 2 + 3 + 6 = 2 § 6, 1 + 2 + 4 + 7 + 14 + 28 = 2 § 28.

3.94. . , 2k - 1 = p -- , n = 2k-1 (2k - 1) -- . 3.95* . . , n -- , n = 2k-1 (2k - 1), p = 2k - 1 -- . . . m n , m n , , n m. , m n , (m) - m = n, (m) = m + n = (n). 3.96. . , p = 3 § 2k-1 - 1, q = 3 § 2k - 1 r = 9 § 22k-1 - 1 -- , m = 2k § p § q n = 2k § r -- . . 3.97. , ) (n) > 3n; )* (n) > 100n? 3.98. . n = 2 p1 p2 , p1 p2 -- , , (n) = 3n. (n) - n = m,


36

3.

3.99. -- , d -- . , , d, .
d

3.100. , d
[] = . d d

3.101. . n! n! = p1 . . . ps . 1 s k =
n n n + 2 + 3 + ... pk pk pk

3.102. , p n! , n .
p-1

3.103. n : n = 2e1 + 2e2 + . . . + 2e
r

(e1 > e2 > . . . > er

0).
-r+1

, n! 2n-r , 2n

.

3.104. . p -- n p- : n = ak pk + ak-1 pk-1 + . . . + a1 p1 + a0 . , p , p n!, n, p ak . 3.105. 3.101 , 1 Cn (n 0) . (. 2.115.) 2n
n+1

3.106. ,

(2m)! (2n)! (m, n 0) . m! n! (m + n)! n! 3.107. r, n-r 2

n

1?

4. ,
. {F0 , F1 , F2 , . . . } = {0, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . . } F0 = 0, F1 = 1, F
n+2

=F

n+1

+ Fn (n

0).


4. ,

37

ë ¨ (1202 .) (). 3.108. . . , , , ? 3.109. , . , , . . , . n- ? (. 3.114.) 3.110. 6 , : § - §§ -- §- -§

, , , 12 . ? F
-1

3.111. , F-2 , . . . , F-n , . . . ? 3.112. . F
n+1 n-1

F

- F2 = (-1)n n

(n > 0).

n? (. 12.13.) 3.113. : ) F1 + F2 + . . . + Fn = Fn+2 - 1; ) F2 + F4 + . . . + F2n = F2n+1 - 1; ) F1 + F3 + . . . + F2n-1 = F2n ; ) F2 + F2 + . . . + F2 = Fn Fn+1 . 1 2 n 3.114. , n F
n+m

1m
n-1 m

0
m+1

=F

F + Fn F

.

: 3.109. , .


38

3. 3.115. ) F2n+1 = F2 + F2 +1 ; n n ) Fn+1 Fn+2 - Fn Fn+3 = (-1)n ) F3n = F3 + F3 +1 - F3 -1 . n n n 3.116. F
4 n+2

+1

; F F .

- Fn F

n+1 n+3 n+4

3.117.
1 2 + + ... + 1§2 1§3 F
n-1

Fn §F

.
n+1

3.118. . : ) 2 | Fn 3 | n; ) 4 | Fn 6 | n; ) 3 | Fn 4 | n; ) Fm | Fn m | n. 3.119. , m Fn (n 1), m. 3.120. , m Fk . , m | Fn , k | n. (n 3.121. , F 1) .
n-1

F

n

3.122* . . (Fn , Fm ) = F (. 3.141.)

(m,n)

.

3.123. 8 , . , . 3.124. n, 0 1, 1 . , Fn+2 . 3.109. 3.125. . , n, Fm ,
m

n=
k=2

b k Fk ,

b2 , . . . , bm 0 1, , bk bk+1 = 0 (2 k m - 1).


4. ,

39

: n = (bk . . . b2 )F . (. 12.14, 4.193 .) 3.126. . : Fn =
1+ 5 1- 5 = -- ë ¨ , = 2 2 n - n , 5

(ë ¨) -- . (. 11.43, 11.75.) 3.127. :
[(n-1)/2]

2n (. 4.129.)

-1

Fn =
k=0

Cn 2k

+1

5k .

3.128. , Fn , Fn = +
n 5 1 . 2 n 5

3.129. . : C0 + C1 -1 + C2 -2 + . . . = Fn+1 . n n n , , , , (. , II 2.67, 3.124, 11.44 11.45.) 3.130. : Sn = C0 - C1 n n (. 11.44, 11.45.) 3.131. 1 2, n? , n = 4, : 11111, 112, 121, 211, 22.
-1

+ C2 n

-2

- ...

3.132. x§
n+1

+ y § n = 1.


40

3. .

{L0 , L1 , L2 , . . . } = {2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, . . . } L0 = 2, L1 = 1, Ln+2 = Ln+1 + Ln (n 0). 3.133. , : ) Ln = Fn-1 + Fn+1 ; ) 5 Fn = Ln-1 + Ln+1 ; ) F2n = Ln § Fn ; ) L2 +1 + L2 = 5F2n+1 ; n n ) Fn+2 + Fn-2 = 3Fn . (. 9.79, 11.41.) 3.134. 1 2. , n (n 3)? , . 3.135. Ln . (. 11.77.) 3.136. ) )
4

5

7+3 5 - 2 11 + 5 5 + 2

4

9

7-3 5 = 1; 2 76 - 34 5 = 1. 2

, . 3.137* . : ) x2 - xy - y2 = 1; ) x2 - xy - y2 = -1. 3.138. ) , 2 4 5 m- . ) , F5t+2 (t 0) t + 1 . 3.139. 3.36 k . , m0 m1 m1 Fk+1 , m0 Fk+2 . 3.140. . m1 t . , m0 m


5.

41

k m0 m1 k 5t. 3.141. .
1 1 1 1 1 1 1 1 13 8 104 5 40 260 3 15 60 260 2 6 15 40 104 1 2 3 5 8 13 1 1 1 1 1 1 1

k Fn ,
k Fn =

Fn § Fn-1 § . . . § Fn-k+1 Fk § Fk-1 § . . . § F1

(0

k

n).

) , k k Fn = Fn-k . k-1 k ) , Fn Fn-1 k Fn-1 ( ) 2.77). ) , . 3.142* . . A1 , A2 , . . . -- , , (Am , An ) = A(
m,n)

(m, n

1).

, Ak = n
An § An-1 § . . . § An-k+1 Ak § Ak-1 § . . . § A1

. (. 8.89.)

5.
. a0 -- , a1 , a2 , . . . , an -- an > 1. n- ( ) a0 +
a1 + 1 1 a2 + . . . 1 + an

(3.1)


42

3.

( [a0 ; a1 , a2 , . . . , an ]). a1 , a2 , . . . , an (3.1). 3.143. 3.144.
129 147 . 13 111
n

Pn = [1; 1, . . . , 1 ]. P Qn
n

Qn ?

3.145. ? 3.146. . (. 2) . m0 ½ m1 (m1 m0 ) a0 m1 ½ m1 , m1 ½ m2 (m2 m1 ) a1 m2 ½ m2 , . . , . (. 3.157.) m1 /m2 . 3.147. n , n . 3.148. . a/b , a/b. a ½ b 3.146? 3.149. a0 -- , a1 , . . . , an -- . P-1 = 1, Q-1 = 0, P 0 = a0 , Q0 = 1, Pk = ak Pk-1 + Pk-2 Qk = ak Qk-1 + Qk-2 (1 (1 k k n); n).

, k = 0, 1, . . . , n : )
Pk = [a0 ; a1 , a2 , . . . , ak ]; Qk

) Pk Qk-1 - Pk-1 Qk = (-1)k+1 ; ) (Pk , Qk ) = 1. .
Pk = [a0 ; a1 , a2 , . . . , ak ] Qk

(k = 0, 1, . . . , n)


5. (3.1). 3.150. : ) Pk Qk-2 - P
k-2

43

Qk = (-1)k a

k

(k 1);

2);

(-1)k+1 P P ) k - k-1 = Qk Qk-1 Qk Qk-1

(k

) Q1 < Q2 < . . . < Qn ; )
P0 P P Pn < 2 < 4 < ... Q0 Q2 Q4 Qn P P ) 2k < 2l+1 (k, l 0). Q2 k Q2l+1

... <

P5 P P < 3 < 1; Q5 Q3 Q1

3.151. a b

a = [a0 ; a1 , a2 , . . . b

. . . , an ]. , ax - by = 1 c x y (Qn-1 , Pn-1 ) (-Qn-1 , -Pn-1 ). , ? 3.152. a/b , ax - by = 1, a) a = 101, b = 13; ) a = 79, b = 19. 3.153. . 365 , -- 366. n- , 100, , n 4. n- , n 100, , n 400. , , 1996- 2000- -- , 1997- 1900- -- . XIII. ë ¨, . , . , ë ¨ , . (. 12.7.) . [a0 ; a1 , . . . , an , . . . ] = a0 +
a1 + 1 1 a2 + . . . + 1 an + . . .

a0 -- , a1 , a2 , . . . , an , . . . -- .


44

3.

, , = lim
Pn = [a0 ; a1 , a2 , . . . , an ]. Qn
n

Pn , Qn

3.154. , = [a0 ; a1 , . . . , an-1 , n ], a0 -- , a1 , a2 , . . . , an-1 -- , n > 1 -- . , . 3.155. , [a0 ; a1 , . . . , an , . . . ] -- . , , , . , - . . 3.156. , = [a0 ; a1 , . . . , an , . . . ]. , = a0 +
1 1 1 1 - + - . . . + (-1)n q0 q1 q 1 q2 q2 q3 qn q n

+ ...
+1

3.157. {ak } {bk } : a1 = 1, b1 = 2, an+1 = min(an , bn ), bn+1 = |bn - an | (n 1). ) , an = 0 an 0 n . ) , cn = a2 + a2 + . . . + a2 1 2 n .


5.

45

3.158. . , 365,2420 . . . = [365; 4, 7, 1, 3, . . . ] ë ¨. -- , 366 . ? 1000 ? , , ? 3.159. . 1079 , . , [365; 4, 7, 1] . ? . [a0 ; a1 , . . . , ak , ak+1 , . . . , ak+T , ak+1 , . . . , ak+T , . . . ] = = [a0 ; a1 , . . . , ak , ak+1 , . . . , ak+T ] ak+1 , . . . , ak+T . a0 , a1 , . . . . . . , ak . [ a0 ; a1 , . . . , aT -1 ] . 3.160. : ) [ 5; 1, 2, 1, 10 ]; ) [ 5; 1, 4, 1, 10 ]; ) [ 2; 1, 1, 3 ]. 3.161. : 1 ) 2; ) 3; ) + 7.
2

3.162. A4. n, m, m 297 2< < .
n 210

3.163. . n, m, m 220 3< < .
n 127

(. [185].) . , .


46

3.

= b + c d -- , = = b - c d . 3.164. , -- . . . , . (. [5], [28].) 3.165. , 0,0001, ) = 2; ) = 2 + 5; ) = 3 + 7. 3.166. :
(1 + 2)n+1 - (1 - 2)n [ 2; 2, . . . , 2 ] = (1 + 2)n - (1 - 2)n
n +1

.

3.167* . . , -
p 1 < 2, q 2q

p -- . q

3.168. . , pn /qn (n 1) -- , -
pn 1 <2 qn 2qn



-

pn-1 1 <2 qn-1 2qn

.
-1

: p/q , -
p 1 < 2. q 2q

3.169* . , p q (q = 0), p 1 2- > 2.
q 3q

3.170. , k
a a
Fk+2 Fk+1

1 :
Fk-1

-1 = [aFk ; a -1

, . . . , aF0 ],


5. {Fk } -- .

47

3.171. , bx2 - abx - a = 0, a b -- , , 2. ? 3.172. , x = [ a; b ], -
1 . [ a; b ]

3.173. , = B , = =
A- D (-1; 0). B A+ D

3.174. , x = [a; b, c ], a - [ c, b ].


4


1.
4.1. m n -- . , mn(m + n) -- . 4.2. . , - , . , , -- . 4.3. -- . , -- , . 4.4. 10 15 . , , . 24 ? 4.5. (a8 h1). ? 4.6. 5 ½ 5 (. ). , ? ( .) 4.7. ( ), 64 ? . 4.8. . 6 . . , ? 4.9. 27 . , 27 .


1.

49

. , . - . 26 ? , , . 4.10. A, B C. , . 25 . ? 4.11. . 5 . ? 4.12. 1 , a b ab - 1 ? 4.13. 2n- A1 , . . . , A2n . P, . , P A1 , . . . , A2n . 4.14. 1, 2, . . . , 10. ë+¨ ë-¨ , ? 4.15* . 17- , , . , . 4.16. , 15 . , . 4.17. 101 , 50 , 1 . , , . ? 4.18. 7 . 7 -- . 4 . , ? 4.19. 4 ½ 4 + - ,
+ + + + - + + + + + + + + + + +


50

4.

, , , - ( , ). , , , . 4.20. . A, B C. . . , A 20 , B -- 21 , C -- 22 ? (. 5.78.) 4.21. ë¨. 32 ( . 1). -- , , . , 1 . , 5 , 2. (. 5.79.)

. 1.

: . A B , C -- . A, B C, B . 4.22. , . , , n2 . n ½ n. 2. n. ( ). , 0111, 2 ½ 2 3 ½ 3:
0 1 1 1 0 1 1 1 1 0 1 0 1

âá
. 2.


2.

51

, (n + 1) ½ (n + 1) , . , ?

2.
. . 4.23. p > 3 -- . , p2 - 1 . 24. 4.24. , , 3n , 37. 4.25. , 11 . . . 1 (1986 ) ) 8; ) 32 . 4.26. , 2001 2001 ) 23 + 1; ) 23 - 1 -- . 4.27. : . . . . . ) 241 - 1 . 83; ) 270 + 370 . 13; ) 215 - 1 . 20801. . 4.28. , p > 2
1 1 1 m = + + ... + n 1 2 p-1

p. 4.29. m n , m > n, m n n , m + n m - n. m : n. . 4.30. a, b, c , a + b + c . 6. , . . a3 + b3 + c3 . 6. . . . 4.31. , 1110 - 1 . 100. 4.32. , , 15? 4.33. : ) 3x2 + 5y2 = 345; ) 1 + x + x2 + x3 = 2y . 4.34. , 119
99

+ 219

99

+ . . . + 161

999

17.

4.35. , . , 13.


52

4.

4.36. , 1 2001 , . 7 77 77 . 4.37. , 77 - 77 . 10. . 4.38. x , x2 001 ( ). x ( ). 4.39. . 1, 2, 3 4. , . , ) 2004? ) 2005? 4.40. . 2 3, , 6. , 5, , 6. 4.41. , , 37, , 37. 4.42. , p -- 1 . Ck . p. p. k p - 1,

4.43. , 4.42: . Ck . n 1 k n - 1, n -- . n. 4.44. , p -- 1 . Ck-k+1 - Ck-2 -1 . p. ? . p-k p k p - 2,

4.45. , p -- , a b . (a + b)p - ap - bp . p. . 4.46* . : -- 51 , -- 49 , -- 5 . , . 105 ? 2. 4.47. , n , n, , n.


3.

53

4.48. , , . 4.49. 99 1, 2, . . . , 99. . 1, 2, . . . , 99. , , 99 . , .

3.
. m 1. a b m, m. a b (mod m). 4.50. : ) a b (mod 0); ) a b (mod 1)? 4.51. . , a b (mod m) c d (mod m), ) a + c b + d (mod m); ) ac bd (mod m). . m ï a m. a. ï 4.52. , a mt + a, t -- . ï ï 4.53. , a b , a b (mod m). . , . 4.54. , m x1 , . . . , xm m, m. 4.55. x1 , x2 , . . . , xm m. a b yj = axj + b (j = 1, . . . , m) m? 4.56. , , : , , , . ,


54

4.

. , m = 6 :
+ 0 1 2 3 4 5 012345 0 1 2 3 4 5 1 2 3 4 5 0 2 3 4 5 0 1 3 4 5 0 1 2 4 5 0 1 2 3 5 0 1 2 3 4 ½ 0 1 2 3 4 5 012345 0 0 0 0 0 0 0 1 2 3 4 5 0 2 4 0 2 4 0 3 0 3 0 3 0 4 2 0 4 2 0 5 4 3 2 1

m = 7, 8, . . . , 13. 4.57. a b (mod m) ac bc (mod m) ? 4.58. a b (mod m) ac bc (mod mc)? 4.59. 100 . 1 5 . , . . (. 5.81.) 4.60. ë¨ 8 . , 8, . . 2002 ? 4.61. - , 21 , -- 4 , 1999 . , 100 ? (: , , 3 , , .) 4.62. 6 , 15, 16, 18, 19, 20 31 . 5 , , . ? 4.63. , n2 3, 4, 5, . . . , 9. 4.64. , ax2 + bx + c = 0 -- , .


3.

55

4.65. , , 0. , , 0? 4.66. , , . 4.67. 22001 3, 5, 7, . . . , 17. 4.68. 7. , . , 7. 4.69. p , p, p + 10, p + 14 -- . 4.70. , p 8p2 + 1 -- . p. 4.71. , p p2 + 2 -- . , 3 p + 2 . 4.72. 6 , . 4.73. 77 . 4.74.
n2 + 1 n? 3
77

4.75. a b -- . , . . . . ) a2 + b2 . 3, a2 + b2 . 9; ) a2 + b2 . 21, a2 + b2 . 441. . . . . . 4.76. a, b, c d , a4 + b4 + c4 + d4 . 5. . . . , abcd . 625. . 4.77. a, b c , a3 + b3 + c3 . 7. , . . . abc . 343. 4.78. 17 21999 + 1. 4.79. en ? (. 3.25.) 4.80. . , 3, 5. 4.81. m -- n (n > 1). , ) m + 1; ) m - 1 . 4.82. n an = 5n2 + 10n + 8 3? 4?


56

4. 4.83. n n2 - 6n - 2 ) 8; ) 9; ) 11; ) 121? 4.84. n n2 - n - 4 ) 17; ) 289?

4.85. x, x 3 (mod 7), x2 44 (mod 72 ), x3 111 (mod 73 ). . 4.86. , 22225555 + 55552222 . 7. . 4.87. : ) 1 + 2 + 3 + . . . + 12 1 + 2 + 22 + . . . + 211 (mod 13); ) 12 + 22 + 32 + . . . + 122 1 + 4 + 42 + . . . + 411 (mod 13). ‡ ? 4.88. , 1k + 2k + . . . + 12k 13 k = 1, 2, . . . , 11. 4.89. , 6n + 11m 31, n + 7m 31. 4.90. , ax4 + bx3 + cx2 + dx + e, a, b, c, d, e -- , x 7. , a, b, c, d, e 7. 4.91. , P(x) = an xn + . . . + a1 x + a0 x = 0 x = 1 , P(x) = 0 . 4.92. , pp+2 + (p + 2)p 0 (mod 2p + 2), p > 2 -- . 4.93. : ) 8x 3 (mod 13); ) 7x 2 (mod 11); ) 17x 1 (mod 37); ) 80x 17 (mod 169). ax b (mod m), ax + my = b. 4.94. 1xy2 x12y, , 7. 4.95. ax b (mod m)? . 4.96. a ax 1 (mod p) a?


3. 4.97. . , p (p - 1)! -1 (mod p).

57

4.98. . , n>1 (n - 1)! -1 (mod n), n -- . 4.98 . . p > 2 -- . p- , p- ? (, , .) . 4.99. . , p -- , (p - 2)! 1 (mod p). 4.100. . , p p+2 - , 4((p - 1)! + 1) + p 0 (mod p2 + p). 4.101. , a1 , . . . , an ‘1 a1 a2 + a2 a3 + . . . + an-1 an + an a1 = 0. . . , n . 4. F(x1 , . . . , xn ) -- x1 , . . . , xn . , F(x1 , . . . , xn ) = 0 F(x1 , . . . , xn ) 0 (mod m) (m 1). (4.2) (4.1)

, m (4.2) , (4.1) . 4.102. , : ) x2 + y2 = 2003; ) 12x + 5 = y2 ; ) - x2 + 7y3 + 6 = 0; ) x2 + y2 + z2 = 1999; ) 15x2 - 7y2 = 9; ) x2 - 5y + 3 = 0; ) x4 + . . . + x4 = 1999; 1 14 ) 8x3 - 13y3 = 17.


58

4.

4.103. , . 4.104. . , Hn = 1 + n > 1 . 4.105. 1! + 2! + . . . + n! = m2 . 4.106. 2x - 1 = 5y . 4.107. (m, n) = 1, a b (mod mn) a b (mod m) a b (mod n).
1 1 1 + + ... + 2 3 n

4.
4.108. n, 10n - 1 ) 7; ) 13; ) 91; ) 819. 4.109. , . . ) 111 . . . 1 . 13; ) 111 . . . 1 . 17. . .
12 16

. p -- p a. ap-1 1 (mod p). 4.110. , (1 + 1 + . . . + 1)p . 4.111. p -- , p = 2, 5. , 111 . . . 11, p. : , , -- . 4.112. n n
2001

- n4 11?

4.113. , , 0 1. 4.114. p a, p. k -- , ak 1 (mod p). , p - 1 k.


4.

59

4.115. , : p -- , a ap a (mod p). 4.116. , a
12

+b

12

. + c12 + d12 + e12 + f12 . 13. .

. , abcdef . 136 . . 4.117. . p > 2 -- . p- a ? (, , .) . 4.118. 103 ) 5102 ; ) 3104 . 4.119. , 30239 + 23930 -- . 4.120. 25710
92

+ 1092?

4.121. , p -- , p = 2, 5, 1/p p - 1. , p - 1. 4.122. p -- . , 2p - 1 2kp + 1. 4.123. n -- , 17. , n8 + 1, n8 - 1 17. 4.124. , p . . 1 . . . 1 2 . . . 2 3 . . . 3 . . . 9 . . . 9 -123 . . . 9 . p.
p p p p

4.125. p > 2 a, p, x2 a (mod p). , a(
p-1)/2

1 (mod p).

4.126. , x2 + 1 p, p = 4k + 1. 4.127. 4.126 , p = 4k + 1. (. 3.7.)


60

4.

4.128. , p p = 4k + 1 x = ‘(2k)! x2 + 1 0 (mod p). 4.129. 3.127 , p Fp Fp+1 p. 4.130. p -- p > 3. , x2 + x + 1 0 (mod p), p 1 (mod 6). 6n + 1. (. 3.7.) 4.131. p -- p > 5. , x4 + x3 + x2 + x + 1 0 (mod p), p 1 (mod 5). 5n + 1. . (n) 1 n, n. 4.132. a) (17); ) (p); ) (p2 ); ) (p ). 4.133. (1) + (p) + (p2 ) + . . . + (p ), -- ? (. 4.149.) 4.134. (n) . a b
1, b + 1, ... (a - 1)b + 1, 2, b + 2, ... (a - 1)b + 2, 3, b + 3, ... (a - 1)b + 3, . . . . . . . . . . . . , , , , b 2b ... ab.

b? a? , . . m , ,


4.

61

ï . (, a m, a m.) 4.135. m? 4.136. x1 , x2 , . . . , xr m. a b yj = axj + b (j = 1, . . . , r) m? 4.137. (m, n) = 1, x y m n . , A = xn + ym mn. . 4.138. n = p1 . . . ps . 1 s (n) = n(1 - 1/p1 ) . . . (1 - 1/ps ) ) ; ) (. 2.99). 4.139. ) (x) = 2; ) (x) = 8; ) (x) = 12; ) (x) = 14. 4.140. 1 5 ? 4.141. ) (x) = x/2; ) (x) = x/3; ) (x) = x/4. 4.142. n : a) (n) = n - 1; ) (2n) = 2(n); ) (nk ) = nk-1 (n)? 4.143. ) (5x ) = 100; ) (7x ) = 294; ) (3x § 5y ) = 600. 4.144. , (m, n) > 1. (m § n) (m) ½ ½ (n)? (. 3.93.) 4.145. a = 2(a). 4.146. , n > 2, n -- . 4.147. n. 4.148. n . , n = 12 :
0 1 111 5 1 7 2 3 5 11 , ,,,, ,, ,,,, . 1 12 6 4 3 12 2 12 3 4 6 12


62

4.

d, d -- n? 4.149. . (d) = n,
d|n


d|n

, n

(. 4.133.) 4.150. . n n . n ? 4.151. : ) (m) (n) = ((m, n)) ([m, n]); ) (mn) ((m, n)) = (m) (n) (m, n). . . m 1 (a, m) = 1. a(m) 1 (mod m). (. 4.197.) 4.152. , 0001? 4.153. ) , m = pn ; ) . 4.154. , 751 - 1 103. 4.155. p > 2 -- . , . . 7p - 5p - 2 . 6p. 4.156. x, ax + b 0 (mod m), (a, m) = 1. 4.157. , . a) a5 - a . 30; . ) a11 - . 17 . 510; ) a73 - ) a - a . a a a: . . 66; . . . 2 § 3 § 5 § 7 § 13 § 19 § 37 § 73. .

4.158. , m . . n, 2n - 1 . m. 4.159. , n 2n! - 1 n.


5.

63

4.160. . , 561 : (a, 561) = 1, a560 1 (mod 561). , , . 4.161. a, a10 + 1 10.
4.162. . m = p1 1 . . . ps -- s m . (m) (p1 ), . . . , (ps ): 1 s

(m) = [(p1 ), . . . , (ps )]. 1 s , a , (a, m) = 1, a
(m )

1 (mod m).

5.
4.163. 3, 9 11. N N = an a
n-1

. . . a1 a0.

. . ) N . 3 an + an-1 + . . . + a1 . 9 a + a . ) N . n n-1 + . . . + a1 . . ) N . 11 ‘an an-1 ‘ . . . -

: . + a0 . 3; . . + a0 . 9; . . a1 + a0 . 11. .

4.164. , 100 , 100 100 , ? 4.165. 2, 4, 8, 5 25. 2, 4, 8, 5 25. 4.166. xy9z, 132. 4.167. 13xy45z, 792. 4.168. . N, . , , . . , , N. , N 9.


64

4.

4.169. 9 1234 . . . 500? ( 1 500.) 4.170. , 192021 . . . 7980 1980. 4.171. , abcd 99 , ab + cd 99. 4.172. {xn } : x1 = = 32001 , . x5 . 4.173. , , 225. 4.174. ? 4.175. a b . a - b? 4.176. , n > 6 -- , 1. 4.177. 8n . , , , , . , n = 2001? 4.178. : ) 4237 § 27925 = 118275855; ) 19652 = 3761225; ) 42971064 : 8264 = 5201; ) 5 371293 = 23. 4.179. , : , -- . ab § cd = effe. ? 4.180. , 230 , . 4.181. , 2 , 2? 4.182. 19. N 19: 1) N; 2) 2; 3) 1) 2) , , 19.


5. 4) 19, 19 | N, 19 N. .

65

4.183. 10n ‘ 1 . , 21, 7. 21? 4.184. x y xxyy ? 4.185. , 12 . 4.186. , N 5N , N 9. 4.187. , ? ) 30-; ) 20- , 1, 2, 3, 4, 5. , -- -- . . , 9, -

4.188. 1, 2, 3, 4, 5, 6, 7, . , . 4.189. . N N = an an-1 . . . a1 a0, ri -- 10i m (i = 0, . . . , n). , N m , M = an rn + an-1 + . . . . . . + a1 r1 + a0 m. 4.190. 3, 9, 6, 8, 12, 15, 11, 7, 27, 37. ri , ë¨, , 10i ri (mod m). 4.191. , 2 , 2. , 2 m > 1. 4.192. , : 1) 5 , 5; 2) 7 , , , 7.


66

4.

4.193. , , , , 3 9. 4.193 . ) 2, ) 3, ) 5, , .

6.
. m1 , . . . , mn , (mi , mj ) = 1 i = j, m = m1 . . . mn , a1 , . . . , an , A -- . x , x a1 (mod m1 ), .............. x an (mod mn )

(4.3)

A x < A + m. (. 6.51.) . ( ) . . 4.194. n an = n2 + 3n + 1 55? 4.195. : ) 1910 66; ) 1914 70; ) 179 48; ) 1414
14

100.

4.196. m1 , . . . , mn . , ab a a . a .. b (mod b (mod ....... b (mod m1 ), m2 ), ... mn ). (mod m1 § m2 § . . . § mn )


6.

67

4.197. m1 , . . . , mn . , x = (m2 m3 . . . mn )(m1 ) x 1 (mod m1 ), x 0 (mod m ), 2 ............. x 0 (mod mn ). 4.198. , x, (4.3). 4.199. . 4.200. x, : x 3 (mod 5), x 2 (mod 13), ) ) x 7 (mod 17); x 4 (mod 19). 4.201. , 2, 3, 5, 7 1, 2, 4, 6 . 4.202. , . 4, 5 6 , , 7 , . ? 4.203. 1000! 10250 . 4.204. a , a + 1 3, a + 2 -- 5, a + 3 -- 7, a + 4 -- 11, a + 5 -- 13. 4.205. m1 , m2 , . . . , mn . , x1 , x2 , . . . , xn m1 , m2 , . . . , mn , x = x1 m2 . . . mn + m1 x2 m3 . . . mn + . . . + m1 m2 . . . mn
-1 n

x

m1 m2 . . . mn . . 4.206. . , x , a1 , . . . , an , (4.3) m1 , . . . , mn . .


68

4.

4.207. , m1 , . . . , mn c . ,
m1 . . . m
n

ni /mi (y = 1, . . . , n). 4.208. , 454 2, 7 9? 4.209. , -- , -- , -- . 4.210. -. ) 625 , : 6252 = 390 625. x2 x (mod 10000)?

) , k 4 k -- 00 . . . 00, 00 . . . 01 , , -- : , . 4.211. . , ( 1 37) . , , , , . , 9 3 ½ 3, , 2 ½ 2. 4.212. . -- , , , , -- ( ), , , . 12- , 12 . , : , 0 1 -- ( ); , 2 3 -- ( ); , 4 5 -- ( );


6.

69

, 6 7 -- ( ); , 8 9 -- ( ). 60- 5 . , 5 .


5

, ,
1.
. , = m/n, m -- , n -- . . Q. , . . = 0,a1 a2 . . . ak b1 b2 . . . bn b1 b2 . . . bn b1 b2 . . . bn . . . , b1 b2 . . . bn -- , , . b1 b2 . . . bn , a1 a2 . . . ak -- , n -- = 0,a1 a2 . . . ak (b1 b2 . . . bn ). 5.1. : )
1 ; 7

)

2 ; 7

)

1 ; 14

)

1 . 17

5.2. a b , 0,aaaaa . . . = 0,bbbbb . . . 5.3.
1 = 0,0204081632 . . . 49

, 2? 5.4. . :
1 = 0,004115226337448 . . . 243


1.

71

5.5. : ) 0,(12) + 0,(122); ) 0,(3) § 0,(4); ) 0,(9) - 0,(85). 5.6. , , . 5.7. n n ? 5.8. ) = 0,101001000100001000001 . . . ; ) = 0,123456789101112131415 . . . ? 5.9. , , . 5.10. , 2. , , , . 5.11. n, n n+1 . 5.12. , [2k 2] (k = 0, 1, . . . ) . 5.13. 3 ) 17; ) 2 + 3; ) 2 + 3 + 5; : ) 3 3 - 2; ) sin 1 ; ) cos 10 ; ) log2 3. ) tg 10 ;
1 1 1

5.14. . , ) 8x4 + 4y4 + 2z4 = t4 ; ) x2 + y2 + z2 + u2 = 2xyzu; ) x2 + y2 + z2 = 2xyz; ) 3n = x2 + y2 . 5.15. , x3 + x2 y + y3 = 0 (0; 0). 5.16. ) ? ) ?


72

5. , ,

) ? 5.17. x2 + ax + b = 0 1 + 3. a b, , . 5.18. a, b, c -- . , a, b, c . 5.19. : 2 3+ 5.20.
3

5-

13 +

48.

6+

847 + 27

3

6-

847 = 3. 27

5.21. 17 : )
1 1 1 + + ... + ; 1+ 2 2+ 3 99 + 100 2 + 3/2 2 - 3/2 + ) ; 2+ 2+ 3 2- 2- 3

)

|40 2 - 57| -

40 2 + 57.

5.22. : 3 3 ) 20 + 392 + 20 - 392; 3 3 ) 5 2 + 7 - 5 2 - 7; ) x + 6 x - 9 + x - 6 x - 9 (9 10 + 24 + 40 +

x

18).

5.23. . 60.
a2 - b . 2

5.24. . : a‘ (. 7.15.) 5.25* . , . b=
a+ a2 - b ‘ 2 a-



2+



3+



5+



7+



11 +



13 +



17

5.26. a b loga b ?


1.

73

5.27. , sin x cos x , tg(x/2) . 5.28. . , -- . 5.29. ? 5.30. . , n = 4 n- . 5.31. , ( 2; 3) . 5.32. : 1 1 1 ; ) ) ) ; 3 3
1+ 2+ 3 1 ; ) a + 4 ab + b 1 ) ; 4 2+ 44+ 48+2 2+ a+ . 3 c

5.33. n ( 2 + 1)n - ( 2 - 1)n ? 5.34. : ) 2+ 2 + ... +
10

a 1 ) ; a+ b+ c 1 ; ) 3 1 - 3 a + a2

b+

2+

6=

1024

2+



3+

1024

2-



3;

) 2+ 2 + ... +
n

2+

2 = 2 cos

2
n+1

.

5.35. e. e e = lim (1 + 1/n)n . ,
n

) e = lim (2 + 1/2! + 1/3! + . . . + 1/n!);
n

) e = 2 + 1/2! + 1/3! + . . . + 1/n! + rn , 0 < r ) e -- . (. 11.73, 7.51).

n

1/(n!n);

5.36* . e . N , . , k .


74

5. , ,

, N > [k! e], , . (. 2.33.) 5.37* . {xn } {dn } = [ 2xn (xn + 1) ], dn = x2n+1 - 2x2n-1 (n 1). , 2 2 = (d1 , d2 d3 . . . )2 . ( 2 .) x1 = 1, x
n+1

2.
, . . n , n En = 11 . . . 1 .
n

. . 5.38. ,
10n - 1 = a1 a2 . . . an m

, 1/m 1/m = 0, (a1 a2 . . . an ). 5.39. , (m, 10) = 1, En , m. ? 5.40. {p/q} {10k p/q}? 5.41. , (m, 10) = 1, 1/m . . , . 5.42. 1/m, . 5.43. (n, 10) = 1, m < n, (m, n) = 1, t -- . . , 10t - 1 . n. , t m/n. ?


2.

75

5.44. , (m, 10) = 1, 9En /m, n- ( ) 1/m. , (m, 3) = 1 En -- , m, 9En /m . 5.45. , (m, 30) = 1, , 1/m 9. 5.46* . . 1/7 N = = 142857. : -- (142 + 857 = 999). , q > 5 p < q p/q 2n- N = N1 N2 , N1 + N2 = 99 . . . 9.
n

5.47* . . N = 142857 . : 2 § 142 857 = 285 714, 3 § 142 857 = 428 571 . . . , 1, 2, 3, . . . , 6 ; 14 + 28 + 57 = 99; N2 = 20408122449, 20408 + 122449 = 142857 = N. . 1/17, 1/19? . 5.48. L(m) 1/m. , (m, 10) = 1, L(m) (m). 5.49. (m, n) = 1. , m/n (m). 5.50. , (m1 , 10) = 1 (m2 , 10) = 1, L(m1 m2 ) = [L(m1 ), L(m2 )]. 1/m1 + 1/m2 ? 5.51. , . 5.52. , . 5.53. , , 5, 6 8 . 5.54. m m = 2a 5b m1 , (10, m1 ) = 1. k = max(a, b). , 1/m (k + 1)- , , 1/m1 . 5.55* . 1/107, 1/131, 1/151. ( , .)


76

5. , ,

3.
5.56. 1 , 3 , 9 , 27 81 . 61 , ? 5.57. , 1 . 10 1 ? 5.58. . -. . n . ? 5.59. 4 ) ) . ; ?

5.60. 4 . , 1 40 ? 5.61. ) . , . . , , 30 . , , 15 ? ) ( ) , ? 5.62. ) , . , , , , . , , , . ? ) , 2 ? 5.63. : ë ¨ ë 1¨. 0 ) 100; ) n? (. 6.77.) 5.64. . , x n. , , n = 16,


3. 15 x : x1 = x § x = x2 , n = 2e1 + 2e2 + . . . + 2e
r

77
16

= x § x § . . . § x, x4 = x3 § x3 = x16 . 0).

x2 = x1 § x1 = x4 ,

x3 = x2 § x2 = x8 ,

(e1 > e2 > . . . > er

, xn b(n) = e1 + (n) - 1 , (n) = r -- n. (. 11.88.) 5.65. l(n) -- , xn . n = 15 n = 63 , , n l(n) < b(n). 5.66. 1 31 5
1 9 17 25 3 11 19 27 5 13 21 29 8 12 24 28 7 15 23 31 9 13 25 29 10 14 26 30 2 10 18 26 11 15 27 31 3 11 19 27 6 14 22 30 16 20 24 28 7 15 23 31 17 21 25 29 18 22 26 30 4 12 20 28 19 23 27 31 5 13 21 29 6 14 22 30 7 15 23 31

, . , , ? , 1 63? 5.67. . ) 27 ( ). . , ( , , , . .). , . , . . , , ?


78

5. , ,

) , 3n (n < 9) ? 5.68. : 1, 2 3. , ë¨, ë¨ ë ¨. , ? 5.69. 1 200. , ) ë¨ ë¨; ) ë¨, ë¨ ë ¨? 5.70* . 1 200, ë¨ ë¨. ( ) . , ? 5.71. , A A = a0 + 2a1 + 22 a2 + . . . + 2n an , ak = 0, 1 -1 ak a .
k+1

= 0 0

k

n-1,

5.72. . 0 1 . -- (1/3; 2/3) , , . , , . ) . ) , 1/4 . )
2 2 2 2 ++ + + ... 3 9 27 81

. , -- . ) , x [0, 2] x = + , -- . 5.73. . 01101001100101101001 . . .


3.

79

. . . , , -- . ) 2001 ? ) , , ? ) , 01, -- 10. ) , . ) , n , n- ? (. 11.88.) 5.74. . , 28 - 1 . -- ë ¨ (. 1.42) . -- , 0 28 - 1. . 5.75. . n 1 n. , . , n = 10, : 2, 4, 6, 8, 10, 3, 7, 1, 9, 5. n J(n) . , ) J(2n) = 2J(n) - 1; ) J(2n + 1) = 2J(n) + 1; ) n = (1bm-1 bm-2 . . . b1 b0 )2 , J(n) = (bm-1 bm-2 . . . b1 b0 1)2 . 5.76. -. , n - m k (m k = n), m k . 1) m k m = (ms . . . m1 m0 )2 , k = (ks . . . k1 k0 )2

( ). 2) 2: (ms , . . . , m1 , m0 ) + (ks , . . . , k1 , k0 ) (ns , . . . , n1 , n0 ) (mod 2).


80

5. , , 3) (ns , . . . , n1 , n0 ) n: (ns . . . n1 n0 )2 = n. , 4 9 = 3,

4 = (100)2 , 9 = (111)2 , (1, 0, 0) + (1, 1, 1) (0, 1, 1)

(mod 2), (011)2 = 3.

, - : ) m m = 0; ) m k = k m; ) (m t) k = m (t k); ) n = 0 m1 m2 . . . ml = n, j (1 j l), mj n < mj . (5.1)

5.77. ë¨. . . () , . , . m1 , m2 , . . . , ml (5.1). ) , -, - n = 0. ) , - - n = 0. ) ë¨. ) , : 3, 4 5 ? 5.78. I I. - . , 4.20. {A, B, C} f, f(A) f(B) = f(C), f(A) f(C) = f(B), f(B) f(C) = f(A).

, ? 5.79. ë¨ I I. - ë¨ 4.21. 5.80. ë¨. , 6 ½ 8 = 48 . :


3.

81

- , , , . , , . ) . , , ? ) ? ) ? 5.81. . . 1 5 . , , ) ; ) . (. 4.59.) 5.82* . . 3 ½ n n n , :

, : . , : ) ; ) . n? 5.83. 4 . ( , , ). . 5.84* . 12 . , ( , , ). , .




82

5. , ,

5.85* . 13 . , 13 , -- . , , ?


6


1.
. x1 , x2 -- x2 + px + q = 0. x1 + x2 = -p, x1 x2 = q.

6.1. x1 , x2 -- x2 + px + q = 0. p q )
1 1 +; x1 x2

)

1 1 + 2; x2 x2 1

) x3 + x3 ; ) 2 1

1 1 + . (x1 + p)2 (x2 + p)2

6.2. f(x) = x2 + ax + b g(y) = y2 + py + q x1 , x2 y1 , y2 , a, b, p, q , R(f, g) = (x1 - y1 )(x1 - y2 )(x2 - y1 )(x2 - y2 ). f(x) g(y) . 6.3. x2 + px + q = 0 x1 x2 . , y1 , y2 : ) x3 , x3 ; ) 1 2
11 , ; x2 x2 1 2

) x1 +

1 1 ,x + ; x2 2 x1

)

x2 x1 , . x1 x2

6.4. x1 , x2 -- ax2 + bx + c = 0 Sn = xn + xn (n 0). 1 2 aSm + bSm
-1

+ cS

m-2

= 0 (m

2).

6.5. a x2 + 2ax + 2a2 + 4a + 3 = 0 ? ? 6.6. p q, Ax4 + Bx2 + C = A(x2 + px + q)(x2 - px + q)?


84

6. 6.7. a

x2 -

15 x + a3 = 0 ? 4

6.8. f(x) = x2 +px+q. p q f(p) = f(q) = 0? 6.9. p q x2 + px + q = 0 2p p + q? 6.10. a ) ax2 + (a + 1)x - 2 = 0; ) (1 - a)x2 + (a + 1)x - 2 = 0 ? 6.11. , y = 2x2 . 6.12. y = x2 + px + q, . , , , . 6.13. , x2 + 5bx + c = 0 x1 x2 , x1 = x2 , y2 +2x1 y+2x2 = 0 z2 + 2x2 z + 2x1 = 0. b. 6.14. , ax2 + bx + c bx2 + cx + a (a = 0) . . 6.15. a x2 + ax + 1 = 0 x2 + x + a = 0 ? 6.16. -- x2 + px + q = 0, -- x - px - q = 0. , x2 - 2px - 2q = 0.
2

6.17. (x; y), y = p2 + (4 - 2p)x - x2 . 6.18. (x; y), y = p2 + (2p - 1)x + 2x2 . 6.19. (x; y), (x - a)2 + (y - a)2 2 + a2 .


1.

85

6.20. , ) (x - a)(x - b) + (x - b)(x - c) + (x - a)(x - c) = 0; ) c(x - a)(x - b) + a(x - b)(x - c) + b(x - a)(x - c) = 0 -- . . x2 + px + q Opq (p; q). . a2 + ap + q = 0 , p2 - 4q = 0 -- . 6.21. , ? 6.22. a Opq a2 + ap + q = 0. , p2 - 4q = 0. (. 9.20.) 6.23. x2 + px + q = 0 x1 , x2 . Opq M(p; q), : ) x1 = 0, x2 = 1; ) x1 = x2 ; ) x1 0, x2 2; ) -1 x1 0, 1 x2 2. 6.24. a, 4x2 - 2x + a = 0 , x1 < 1, x2 > 1. 6.25. q, p x2 + px + q = 0 . 6.26. Opq p2 - 4q = 0 p + q + 1 = 0, -2p + q + 4 = 0 . , x2 + px + q = 0 (-2; 1). 6.27. (p; q) p2 - 4q = 0. . 6.28. a (a2 + a + 1)x2 + (2a - 3)x + (a - 5) = 0 1, -- 1? 6.29. , x2 + ax + b = 0 2 x + cx + d = 0 1. , x2 +
a+c b+d x+ =0 2 2


86 1.

6.

6.30. x2 + px + q = 0 p q -1 1. , . 6.31. a 1 (2 - a)x2 - 3ax + 2a = 0 ?
2

6.32. a (1 + a)x2 - 3ax + 4a = 0 1? 6.33. a (a - 1)x2 - - 2(a + 1)x + 2(a + 1) = 0 ? 6.34. m x2 - (m + 1)x + m - 1 = 0 ? 6.35. r, (r - 4)x2 - 2(r - 3)x + r = 0 , -1. 6.36. x, (2 - a)x3 + (1 - 2a)x2 - 6x + 5 + 4a - a2 < 0 a [-1; 2].

2.
6.37. . P(x) Q(x) -- , Q(x) . , T (x) R(x) , P(x) = Q(x)T (x) + R(x), deg R(x) < deg Q(x); , T (x) R(x) . . P(x) Q(x) P(x) = Q(x)T (x) + R(x), T (x) , R(x) -- . R(x) , T (x) -- , Q(x) P(x) (Q(x) | P(x)).


2.

87

6.38. . , P(x) x - c P(c). 6.39. , n n . 6.40. - n- , n ? 6.41. x1 , x2 , . . . , xn -- an xn + . . . + a1 x + a0 = 0. ) a0 xn + . . . + an-1 x + an = 0; ) an x2n + . . . + a1 x2 + a0 = 0? 6.42. P(x) = xn + a
n-1 n-1

x

+ . . . + a1 x + a0

x1 , x2 , . . . , xn , P(x) = (x - x1 )(x - x2 ) . . . (x - xn ). Q(x) = P(x) P(-x). , ) Q(x) 2n x; ) Q( x) x2 , x2 , . . . , x2 . n 2 1 (. 9.83.) 6.43. : ) x4 - 4x3 + 6x2 - 3x + 1 x2 - x + 1; ) 2x3 + 2x2 + x + 6 x2 + 2x + 1; ) x4 + 1 x5 + 1. 6.44. P(x) = x5 - 17x + 1 x + 2. 6.45. a P(x) = x x + 1? 6.46. P(x) = x a) x - 1; ) x2 - 1. 6.47. , P(x) = (x + 1)6 - x6 - 2x - 1 x(x + 1)(2x + 1). 6.48. P(x) 2 x - 1, 1 x - 2. P(x) (x - 1)(x - 2)?
81 27 1000

+ ax2 + 9

+x

+ x9 + x 3 + x


88

6.

6.49. , x3 + y3 + z3 + k xyz x + y + z. 6.50. n 1 + x2 + x4 + . . . + x 1 + x + x2 + . . . + xn-1 ?
2n-2



. m(x) -- . a(x) b(x) m(x), m(x). , a(x) b(x) (mod m(x)). 6.51. m1 (x), . . . , mn (x) (mi (x), mj (x)) = 1 i = j, a1 (x), . . . , an (x) . , p(x) , p(x) a1 (x) (mod m1 (x)), .................... p(x) an (x) (mod mn (x)) . , --

deg p(x) < deg m1 (x) + . . . + deg mn (x). (. 6.131 6.140.) 6.52. P(x) = (2x2 - 2x + 1)17 (3x2 - 3x + 1)17 . a) ; ) x. 6.53. a b P(x) = (a + b)x5 + abx2 + 1 x2 - 3x + 2? 6.54. . , . 6.55. R(x) xn + x + 2 x - 1.
2

6.56. x3 - 6x2 + ax - 6 = 0 3. . 6.57. a P(x) = xn + ax (n 2) x - 2?
2 n-2

6.58. p q x4 + 1 x + px + q?


2. 6.59. a P(x) = a3 x5 + (1 - a)x4 + (1 + a3 )x2 + (1 - 3a)x - a3 x - 1?

89

6.60. , x P(x - 1) = (x - 26) P(x). 6.61. xn - an-1 xn-1 - . . . - a1 x - a0 = 0, an-1 , . . . . . . , a1 , a0 0. , . 6.62. . , f(x) = an xn + . . . + a1 x + a
0

an , . . . . . . , a1 , a0 . 6.63. f(x) = an xn + . . . + a1 x + a0 ? 6.64. , a3 (b2 - c2 ) + b3 (c2 - a2 ) + c3 (a2 - b2 ) (b - c)(c - a)(a - b). . , . , P1 (x), . . . . . . , Pk (x) (P1 (x), . . . , Pk (x)). 6.65. , P(x) = Q(x) T (x) + R(x) (P(x), Q(x)) = (Q(x), R(x)). 6.66. . P(x) Q(x) -- , Q(x) Q(x) P(x). , s 1 A0 (x),


90 A1 (x), . . . , As (x) R1 (x), . . . , Rs (x) ,

6.

deg Q(x) > deg R1 (x) > deg R2 (x) > . . . > deg Rs (x) P(x) = Q(x) § A0 (x) + R1 (x), Q(x) = R1 (x) § A1 (x) + R2 (x), R (x) = R (x) § A (x) + R (x), 1 2 2 3 ....................... R (x) = R (x) § A (x) + R (x), s-2 s-1 s-1 s Rs-1 (x) = Rs (x) § As (x),

0,

(P(x), Q(x)) = Rs (x). ( 3.36.) 6.67. (P(x), Q(x)) = D(x). , U(x) V (x) , deg U(x) < deg Q(x), deg V (x) < deg P(x), P(x) U(x) + Q(x) V (x) = D(x). ( 3.37.) 6.68. P(x), Q(x) P(x) U(x) + Q(x) V (x): ) P(x) = x4 + x3 - 3x2 - 4x - 1, Q(x) = x3 + x2 - x - 1; ) P(x) = 3x4 - 5x3 + 4x2 - 2x + 1, Q(x) = 3x3 - 2x2 + x - 1. 6.69. (xn - 1, xm - 1). 6.70. a0 , a1 , a2 , . . . a0 = 0, a
n+1

= P(an ) (n

0), 0.

P(x) -- , P(x) > 0 x , m k (am , ak ) = a(m,k) . 6.71. x6 - x5 + x4 - x3 + 5x2 = 5, x6 - 2x5 + 3x4 - 4x3 + 2x = 0.

6.72. p 3x2 -4px+9 = = 0 x2 - 2px + 5 = 0 ? 6.73. P(x) Q(x) , (x + 1) P(x) + (x4 + 1) Q(x) = 1. 6.74. ( 3, . 92) P(x) Q(x), P(x)(x2 - 3x + 2) + Q(x)(x2 + x + 1) = 21.


2.

91

6.75. P(x) Q(x), P(x)(2x3 - 7x2 + 7x - 2) + Q(x)(2x3 + x2 + x - 1) = 2x - 1. 6.76. 6.77. . Pn (x) = an xn + an
-1

2n + 1 n(n + 1)

n n + 1?

x

n-1

+ . . . + a1 x + a0

(an = 0)

x = c , n . Pn (x) Pn (x) = (. . . (an x + an
-1

)x + . . . + a1 )x + a0 .

(. 5.63.) bn , bn-1 , . . . , b0 -- , Pn (c), bn = an , bk = c § b
k+1

+ ak

(k = n - 1, . . . , 0).

, Pn (x) (x - c) , bn-1 , . . . , b1 , b0 . , : Pn (x) = (x - c)(bn x
n-1

+ . . . + b2 x + b1 ) + b0 .

6.78. . :
2n+1

a

an+1 - bn+1 = (a - b)(an + an-1 b + . . . + bn ); + b2n+1 = (a + b)(a2n - a2n-1 b + a2n-2 b2 - . . . + b2n ). 2

6.78 . , n n
n-1

. . - 1 . (n - 1)2

6.79. . , Pn (x) (x - c):
n

Pn (x) =
k=0

ck § (x - c)k ,


92

6.

ck ck = (. 11.21.) 6.80. , x4 + 2x3 - 3x2 - 4x + 1 x + 1. 6.81. P(x + 3) x, P(x) = x4 - x3 + 1.
P
(k)

(x) k!

(0
x=c

k

n).

3.
. , . () . , . , , , , . , . 6.82. : ) x4 + 4; ) 2x3 + x2 + x - 1; ) x10 + x5 + 1; ) a3 + b3 + c3 - 3abc; ) x3 + 3xy + y3 - 1; ) x2 y2 - x2 + 4xy - y2 + 1; (. 9.8.) ) (a + b + c)3 - a3 - b3 - c3 ; ) (x - y)5 + (y - z)5 + (z - x)5 ; ) a8 + a6 b2 + a4 b4 + a2 b6 + b8 ; ) (x2 + x + 1)2 + 3x(x2 + x + 1) + 2x2 ; ) a4 + b4 + c4 - 2a2 b2 - 2a2 c2 - 2b2 c2 ; ) (x + 1)(x + 3)(x + 5)(x + 7) + 15.

6.83. x4 + x3 + x2 + x + 12? 6.84. , x4 + px2 + q . 6.85. :
(a + b + c)5 - a5 - b5 - c5 . (a + b + c)3 - a3 - b3 - c3


4. 6.86. , m (x + y + z)m - xm - ym - zm (x + y + z)3 - x3 - y3 - z3 .

93

6.87. a, b, c -- . , a2 (c - b) + b2 (a - c) + c2 (b - a) . 6.88. , a, b, c 1 1 1 1 ++= ,
a b c a+b+c

- . 6.89. , a + b + c = 0, 2(a5 + b5 + c5 ) = 5abc(a2 + b2 + c2 ). 6.90. . , (p, q) = 1 p/q -- P(x) = an xn + . . . + a1 x + a0 , . . ) a0 . p; ) an . q. . . , . (. 7.41.) 6.91. , 17 -- . 6.92. , cos 20 -- . 6.93. : ) x5 - 2x4 - 4x3 + 4x2 - 5x + 6; ) x5 + x4 - 6x3 - 14x2 - 11x - 3. 6.94. : ) x4 + x3 - 3a2 x2 - 2a2 x + 2a4 = 0; ) x3 - 3x = a3 + a-3 .

4.
. P(x) = (x - a)k Q(x), k 1 Q(a) = 0. a P(x) k. a --


94

6.

1, , 1, a . 6.95. , a 1 , P(a) = 0 P (a) = 0. 6.96. P(x) , R(x), , P(x), 1. Q(x) = (P(x), P (x)) R(x) = P(x) Q-1 (x). , ) P(x) R(x); ) R(x) . 6.97. R(x) , : ) P(x) = x6 - 6x4 - 4x3 + 9x2 + 12x + 4; ) P(x) = x5 + x4 - 2x3 - 2x2 + x + 1. 6.98. , P(x) = 1 + x + . 6.99. A B Ax ? 6.100. , x x = 1.
2n n+1

x2 xn + ... + 2! n!

+ Bxn + 1 x = 1
n+1

- nx

+ nx

n-1

- 1 n > 1

6.101. , P(x) , P(x) = an (x - x0 )n . 6.102. , n > 0 nx (x - 1)2 . 6.103. , n > 0 n2 x
n+2 n+1

- (n + 1)xn + 1

- (2n2 + 2n - 1)x

n+1

+ (n + 1)2 xn - x - 1

(x - 1)3 . 6.104. , n > 0 x
2n+1

- (2n + 1)x

n+1

+ (2n + 1)xn - 1

(x - 1)3 . 6.105. , P(x) = a0 + a1 x + . . . + an x
n


5. -1 : a0 - a1 + a2 - a3 + . - a + 2a - 3a + . 1 2 3 ............ - a1 + 2m a2 - 3m a3 (. 11.12.) 6.106. , P(x) = (x Q(x) = (x1 - 1)(x2 - 1) . . . (xm - 1). (. 11.95.)
n+1

95 m , .. .. .. + + (-1)n an = 0, + (-1)n nan = 0, ............. . . . + (-1)n nm an = 0.

- 1)(x

n+2

- 1) . . . (x

n+m

- 1)

5.
. x1 , x2 ,. . . , xn -- an xn + a
n-1

x

n-1

+a

n-2

x

n-2

+ . . . + a1 x + a

0

(an = 0). x1 + x2 + . . . + xn = -an-1 /an , x x + x x + ... + x 12 23 n-1 xn = an-2 /an , ...................... x1 x2 . . . xn = (-1)n a0 /an . . , , . 1 (x1 , 2 (x1 , .. n (x1 , x x . x , .. 2, . . ... 2, . .
2

. . . .

, , . ,

xn xn .. xn

) ) . )

= = . =

x1 x1 .. x1

+ x2 .. x2

x2 + . . . + x n , + x2 x3 + . . . + xn-1 xn , ............. . . . xn ,

. . F(x1 , . . . , xn ) : F(x1 , . . . , xn ) = G(1 , . . . , n ) (. [23].)


96

6.

G F , , , F , G . (. . 92). F(x1 , . . . , xn ) m a1 . . . an , 1 n (a1 + 2a2 + . . . + nan ) m. 6.107. : ) (x + y)(y + z)(x + z); ) (x2 + y2 )(y2 + z2 )(x2 + z2 ); ) x3 + y3 + z3 - 3xyz; ) x2 + x 2 + . . . + x2 ; n 1 2 3 3 ) x + y ; ) x4 + y4 + z4 . 6.108. , a+b+c = 0, a2 +b2 +c2 = 1. a4 +b4 +c4 . 6.109. x, y, z x + y + z = a,
1 1 1 1 ++=. x y z a

, a. 6.110. : x + y + z = a, x2 + y2 + z2 = a2 , 3 x + y3 + z3 = a3 . 6.111. a, x1 , x2 , x3 x3 - 6x2 + ax + a (x1 - 3)3 + (x2 - 3)3 + (x3 - 3)3 = 0. 6.112. , x3 + x2 - 2x - 1 = 0. 6.113. , x1 , x2 , x3 -- x3 - 2x2 + x + 1 = 0. , y1 = x2 x3 , y2 = x1 x3 , y3 = x1 x2 .


5. 6.114. c x3 + ax2 + bx + c = 0

97

a b, , . 6.115. , x3 + px2 + qx + r = 0 . p, q r , , , ? 6.116. ) , x + y = u + v, x2 + y2 = u2 + v2 . , n xn + yn = un + vn . ) , x + y + z = u + v + t, x2 + y2 + z2 = u2 + v2 + t2 , x3 + y3 + z3 = u3 + v3 + t3 . , n xn + yn + zn = un + vn + yn . 6.117. : x + y + z = 6, 1 1 1 11 ) ++=, x y z 6 xy + yz + xz = 11; x(y + z) = 2, ) y(z + x) = 2, z(x + y) = 3; ) x2 + y2 + x + y = 32, 12(x + y) = 7xy; 2 2 x + y = 7, )
y x 2

1 1 + = 1; y x 2 x + y + z = 1, ) xy + xz + yz = -4, 3 x + y3 + z3 = 1; x2 + y2 = 12, x + y + xy = 9.

)

6.118. a, b, c x4 - ax3 - bx + c.


98 .

6.

6.119. , a, b, c a + b + c = 0. , 2a4 + 2b4 + 2c4 -- . 6.120. ax3 + bx2 + cx + d = 0, , . 6.121. a b x3 + ax + b = 0 , ? 6.122. a, b, c -- , p -- , r R -- . , p, r, R, a, b, c.
1 1 1 1 + + = . ab bc ac 2rR

6.123. x + y = uv, u + v = xy. 6.124. ) 4x3 - 18x2 + 24x = 8, 4x3 - 18x2 + 24x = 9; ) 4x3 - 18x2 + 24x = 11, 4x3 - 18x2 + 24x = 12?

6.
6.125. c
(x - a)(x - b) (x - a)(x - c) (x - b)(x - c) +b +a = x. (c - a)(c - b) (b - a)(b - c) (a - b)(a - c)

6.126. c2
(x - a)(x - b) (x - a)(x - c) (x - b)(x - c) + b2 + a2 = x2 . (c - a)(c - b) (b - a)(b - c) (a - b)(a - c)

6.127. x1 < x2 < . . . < xn -- . f1 (x), f2 (x), . . . , fn (x) n - 1, fi (xi ) = 1 fi (xj ) = 0 i = j (i, j = 1, 2, . . . , n). 6.128. f(x) = f1 (x) + f2 (x) + . . . + fn (x),


6. fi (x) -- .

99

6.129. x1 < x2 < . . . < xn -- . , y1 , y2 , . . . , yn f(x) n - 1 , f(x1 ) = y1 , . . . , f(xn ) = yn . 6.130. A, B C -- P(x) x - a, x - b x - c. (x - a)(x - b)(x - c). . n - 1, x1 , . . . , xn ( ) y1 , . . . , yn , . 6.131. f(x) (x - xi )? (. 6.51). 6.132. f(x) 2, : ) f(0) = 1, f(1) = 3, f(2) = 3; ) f(-1) = -1, f(0) = 2, f(1) = 5; ) f(-1) = 1, f(0) = 0, f(2) = 4. 6.133. . . 12, 14 15 7, 5 11 . 13 ? 16 ? 6.134. . , 12, 14 15 5, 7 2 . 13 ? 6.135. 100 . , . , 100 . 6.136. z + ay + a2 x + a3 = 0, z + by + b2 x + b3 = 0, z + cy + c2 x + c3 = 0.


100

6.

6.137. a, b c -- . , x + ay + a2 z = 0, x + by + b2 z = 0, x + cy + c2 z = 0, x = y = z = 0. 6.138. f(x) = x
10

+ a9 x9 + . . . + a0 ,

f(1) = f(-1), . . . , f(5) = f(-5). , f(x) = f(-x) x. 6.139. P(x) = an xn + . . . + a1 x + a0 -- . , |3n 1. 6.140. , f(x) , n,
f(x) (x - x1 )(x - x2 ) . . . (x - xn )
+1

- P(n + 1)|, . . . , |31 - P(1)|, |1 - P(0)|

(x1 , x2 , . . . , xn -- ) n :
A2 An A1 + + ... + , x - x1 x - x2 x - xn

A1 , A2 , . . . , An . (. 6.51.) 6.141. x1 x2 a1 - b1 + a1 - b2 + . x1 x2 + +. a2 - b1 a2 - b2 ............. x1 x2 + +.
an - b1 an - b2 xn = 1, a1 - bn xn .. + = 1, a2 - bn

.. +

......... xn .. + = 1.
an - bn


7


1.
. z = = x + iy, x y -- , i -- , , -1; x z, y -- ( x = Re z, y = Im z). z x = 0, y = 0 . z = x - iy z = x + iy. C. 7.1. z = x + iy, z = x + iy . ) z + z ; ) z § z ; ) z/z . 7.2. : ) z + z = z + z ; ) z/z = z/z ; ) z § z = z § z ; ) (z) = z. . z = x + iy (x; y) Oxy . r = x2 + y2 z (r = |z|). , Oxy Ox (x; y), z (r = arg z). , arg z - . |z| = r, arg z = , z z = r(cos + i sin ). z. z = x + iy z. 7.3. : ) z + z = 2 Re z; ) z - z = 2i Im z; ) z § z = |z|2 . 7.4. : ) |z1 + z2 | |z1 | + |z2 |; ) |z1 - z2 | |z1 | - |z2 | ; ) |z - 1| | arg z|, |z| = 1.


102

7.

7.5. : ) 1 + i; ) 2 + 3 + i; ) 1 + cos + i sin ; 7.6. : ) |z| 1; 1; ) arg
z-i =; z+i 4

) sin

+ i sin 6 cos + i sin ) cos - i sin

; 6 .

) |z - i| + |z + i| = 2; ) Im )
< arg(z - i) < ? 6 3 1 1 <- ; z 2

) |z - i| ) |z| = z; )

) Re(z2 )

1;

) |iz + 1| = 3;

z-1 < 1; z+1

7.7. min |3 + 2i - z| |z|

1.

7.8. : ) , ; ) , ; ) , , ; ) 1 ( ) O, . 7.9. z, |z - 1 - i| = 2|z + 1 - i|. 7.10. . , |z - a| = k|z - b| k = 1 (a b -- ). 7.11. , z1 z2 |z1 + z2 |2 + |z1 - z2 |2 = 2(|z1 |2 + |z2 |2 ). ? 7.12. , aj , bj (1 (a1 + a2 + . . . + an )2 + (b1 + b2 + . . . + bn )2 a2 + b 2 + 1 1 a2 + b2 + . . . + 2 2 a2 + b2 . n n j n)


1.

103

7.13. , x + iy = (s + it)n , x2 + y2 = (s2 + t2 )n . 7.14. . : (a2 + b2 )(u2 + v2 ) = (au + bv)2 + (av - bu)2 . (. 1.6.) 7.15. , z = = a + ib w=‘
a2 + b2 + a ‘i 2 a2 + b2 - a 2

.

, , ? (. 5.24.) 7.16. ) 24 + 70i; ) 3 - 4i; ) 2 + i 2; ) 1 + i 3; 7.17. : ) z2 ) z2 ) z2 ) -7 - 24i; ) 12 - 5i.

+ z + 1 = 0; + 4z + 29 = 0; - (2 + i)z + 2i = 0; ) z2 - (3 + 2i)z + 6i = 0; ) z2 - (3 - 2i)z + 5 - 5i = 0; ) z2 - (5 + 2i)z + 5 + 5i = 0.

7.18. : ) z4 - 4z3 + 6z2 - 4z - 15 = 0; ) z4 + (z - 4)4 = 32; ) z3 + 3z2 + 3z + 3 = 0; )
1 - ix 1 + ix

= i.

7.19. x4 + px2 + q = 0, p2 - 4q < 0? 7.20. , |z| = 1 (z = -1), t z = (1 + it)(1 - it)-1 . 7.21. y(x) = |x + x2 - 1| (x -- ). 7.22. z . ) 2z2 ; ) 3z + z2 ; ) (z - i)-1 ; ) Rz + zn ( < R). 2 ) z + 3z ; ) z-3 ; ) (z - 2)-1 ; 7.23. z -1 - i, 2 - i, 2 + 2i, -1 + 2i.


104 a) z2 ; ) z3 ; ) z-1 ?

7.

7.24. . . , z = r(cos + i sin ): zn = rn (cos n + i sin n) (n 1).

n n- : wk = r
1/n

cos

+ 2k + 2k + i sin n n

(k = 0, . . . , n - 1).

(. 12.11.) 7.25. : a) i; ) 4 -1; ) -8i; ) 3 1 - i; ) 6 -1; )
8

i 3 - 1.

7.26. , wk (k = 0, . . . , n - 1), wn = z z n-. (. 8.2.) 7.27. , zn = 1 1, , 2 , . . . , n-1 . 7.28. ) z4 ) z2 ) z2 : ) z2 + |z|2 = 0; = z4 ; + |z| = 0; ) (z + i)4 = (z - i)4 ; + z = 0; ) z3 - z = 0.

7.29. s zn = 1, s -- . 7.30. : )
cos n = 1 - C2 tg2 + C4 tg2 - . . . ; n n cosn sin n ) = C1 tg - C3 tg3 + C5 tg5 - . . . . n n n cosn

7.31. a) (1 + i)n ; ) (1 + i 3)n ; ) )
1 + i 3 20 ; 1-i 3-i 1- 2

) (1 + cos + i sin )n ; ) ( 3 + i)n ; )
cos + i sin cos + i sin
n

.

20

;

7.32. x4 + x3 + x2 + x + 1 = 0.


1. 7.33. , x x4 + x3 + x2 + x + 1 = 0. 7.34. : ) cos
4 6 2 + cos + cos ; 7 7 7
44

105 +x
33

+x

22

+x

11

+ 1 = 0

) cos

2 4 6 § cos § cos . 7 7 7

7.35. ) , P(x) = (cos + x sin )n - cos n - x sin n x2 + 1. ) , Q(x) = xn sin - n x2 - 2x cos + 2 . 7.36.
n-1 -1

x sin n + n sin(n - 1)

) x ) x ) x ) x

2n

- 1 = (x2 - 1)
k=1 n

x2 - 2x cos

k +1 ; n 2k +1 ; 2n + 1 2k +1 ; 2n + 1

2n+1

- 1 = (x - 1)
k=1 n

x2 - 2x cos x2 + 2x cos
k=1

2n+1

+ 1 = (x + 1)
n-1

2n

+1=
k=0

x2 - 2x cos

(2k + 1) +1 . 2n

7.37. , , cos nx = Tn (cos x), sin nx = sin x U
n-1

(cos x),

Tn (z) Un (z) -- n. n = 0, 1, 2, 3, 4, 5. . Tn (z) Un (z) . 7.38. , Tn (x) Un (x) T0 (x) = 1, T1 (x) = x; U0 (x) = 1, U1 (x) = 2x,

Tn
+1

(x) = 2xTn (x) - Tn

-1

(x),

Un

+1

(x) = 2xUn (x) - U

n-1

(x).

(. 11.80.)


106

7.

7.39. , 2Tn (x/2) , -- . 7.40* . , cos = 1/3. ? 7.41. (. 6.90), , p/q Q cos(p/q) = 0, ‘ 1/2, ‘ 1, cos(p/q) -- . 7.42. ,
n n-1

cosn x =
k=0

ak cos kx,

sinn x = sin x
k=0

bk sin kx,

a0 , . . . , an , b0 ,. . . , bn-1 -- . n = 2, 3, 4, 5. sinn x n
n n

sinn x =
k=0

ck cos kx, -- sinn x =
k=0

dk sin kx.

n , n -- , 5. 52 5

7.43. , sin = 3/5. , sin 25

7.44. P0 (x) = 1, P1 (x) = x, P2 (x) = = x2 - 1, . . . P
n+1

(x) = x Pn (x) - P

n-1

(x).

, P100 (x) = 0 100 [-2; 2]. ? 7.45. :
1 + i tg 1 - i tg
n

=

1 + i tg n . 1 - i tg n

7.46. , z + z-1 = 2 cos , zn + z-n = 2 cos n. zn + z-n y = z + z-1 ? (. 1.5.) 7.47. x = cos Tn (cos ) = cos n, U
n-1

(cos ) =

sin n . sin

, x = sin ? 7.48. a, -- (a, b) = 1. , b ( a + i b)n (a; b) = ( ‘ 1; ‘ 1), ( ‘ 1; ‘ 3), ( ‘ 3; ‘ 1).


1.

107

. n (n 1), n ( ) . (. [20], [217].) 7.49. f(x) a + ib. , a - ib f(x). (. 7.82.) 7.50. , , . 7.51. . a b -- . a + ib n ea+ib = lim 1 + .
n

n

: ea
+ib

= ea (cos b + i sin b).

, sin x cos x : cos x =
e
ix

+e 2

-ix

,

sin x =

e

ix

-e 2i

-ix

.

(. 5.35, 11.73 12.12.) 7.52. , z1 , z2 ez1 ez2 = ez1 +z2 . (. 11.73.) 7.53. , . 7.54. ln z z? 7.55. az ? (. 12.12.) 1 7.56. i -1 = (-1)1/i 23 . 7.57. z = e2i/n a ) 1 + za + z2a + . . . + z(n-1)a ; ) 1 + 2za + 3z2a + . . . + nz 7.58. ) : cos + . . . + cos n =
sin(n/2) cos((n + 1)/2) ; sin(/2)

7 2 2 = cos + i sin . n n
(n-1)a

.


108 ) : sin + . . . + sin n. (. 8.11.) 7.59. :

7.

sin + sin 3 + . . . + sin(2n - 1) = tg n. cos + cos 3 + . . . + cos(2n - 1)

7.60. : ) cos2 x + cos2 2x + . . . + cos2 2nx; ) sin2 x + sin2 2x + . . . + sin2 2nx. 7.61. (1 + i)n , : ) C000 - C2 0 + C4 0 - . . . + C100 ; ) C1 - C39 + C5 - . . . - C99 . 99 99 9 99 100 10 10 1 7.62. ) : C0 - C2 + C4 - . . . = 2n/2 cos n n n ) : C1 - C3 + C5 - . . . n n n 7.63. ) : 1 + C3 + C6 + . . . = n n ) : C1 + C4 + C7 + . . . ; n n n 7.64. : C1 - C3 + C5 - . . . = n n n
1 3 1 9 2 3
n (n-1)/2

n . 4

1n n 2 + 2 cos . 3 3

C2 + C5 + C8 + . . . n n n

sin

n . 6

7.65. : ) 1 + a cos + . . . + ak cos k + . . . (|a| < 1); ) a sin + . . . + ak sin k + . . . (|a| < 1); ) cos + C1 cos 2 + . . . + Cn cos(n + 1); n n ) sin + C1 sin 2 + . . . + Cn sin(n + 1). n n 7.66.
k

lim 1 +

1 1 cos x + . . . + k cos kx . 2 2

7.67. z1 , . . . , zn -- , < arg z < + . ,


1. ) z1 + . . . + zn = 0; ) z-1 + . . . + z-1 = 0. 1 n

109

7.68. z1 , z2 , . . . , zn -- . z = 1 z1 + 2 z2 + . . . + n zn , 1 , 2 , . . . , n -- , 1 + 2 + . . . + n = 1. 7.69. ,
1 1 1 + + = 0, z-a z-b z-c

a, b, c -- , a, b, c, ( ). 7.70. f(x) = (x - a)(x - b)(x - c) -- a, b, c. , a, b, c. 7.71. 1 , . . . , 1 , í . f(x) -- n n . M . . . , n . , M.

7.72. n ) x2n + xn + 1 x2 + x + 1? ) x2n - xn + 1 x2 - x + 1? 7.73. , a n (a + 1)2n+1 + an + 2 a2 + a + 1. 7.74. n (x + 1)n + xn + 1 : ) x2 + x + 1; ) (x2 + x + 1)2 ; ) (x2 + x + 1)3 ? 7.75. n (x + 1)n - xn - 1 : ) x2 + x + 1; ) (x2 + x + 1)2 ; ) (x2 + x + 1)3 ? 7.76. (x - 1) | P(xn ). , (xn - 1) | P(xn ). 7.77. P(x) = x Q(x) = x6 + x5 + x4 + x3 + x2 + x + 1, , n 7.
6n

+x

5n

+x

4n

+x

3n

+x

2n

+ xn + 1


110

7.

7.78. (z - 1)n = (z + 1)n . ? 7.79. , a(z - b)n = c(z - d)n , a, b, c, d -- , . (. 7.10.) 7.80. , n > 1
n-1

m=1

1 n2 - 1 = . 3 sin (m/n)
2

7.81* . . ) , n > 1
(n-1)/2

m=1

1 2 2 - = 2 6 2n m

(0 < < 1).

) :

m=1

1 2 . = 2 6 m

7.82* . , = a2 (x) +

. P(x) x . a(x) b(x), P(x) = b2 (x).

2.
: Ta -- a; Sl -- l ( l); R -- A ; A Hk -- A k. A 7.83. : 0, 1 - i, 1 + i w= +
1 2 i z? 2

7.84. w = z3 ?


2.

111

7.85. : ) w = z + a; ) w = 2z; ) w = z(cos + i sin ); ) w = z? 7.86. w = f(z) l Ox? 7.87. w = f(z) : ) H2 T3+4i ; ) R/4 ; ) H2 H1/2 ; -1 i 1 O ) R/4 R/4 R/4 R/4 . ) T3+4i H2 ; ) Hk ; 1 -i -1 i A O O = (0; 0) -- . : (f g)(z) = f(g(z)). 7.88. H2 i O. 7.89. . , : H
k2 A2

H

k1 A1

=

Ta , k1 k2 = 1, Hk , k1 k2 = 1, A

a A1 A2 , A A1 A2 k = k1 § k2 . 7.90. A(0; 0), B(0; 2), C(2; 2), D(2; 0) : ) w = iz; ) w = 2iz - 1; ) w = z2 ; ) w = z-1 . 7.91. 2 < Re z < 3 : ) w = z-1 ; ) w = (z - 2)-1 ; ) w = (z - 5/2)-1 ? 7.92. ) |z - a - bi| = a2 + b2 w = 1/z; ) |z - a| = R w =
2

2aR . z - a2 + R2

7.93* . n- . , ) n2 ; ) n ctg ;
2n

) n

n/2

.


112

7.

. - , w=
az cz az w= cz + + + + b , d b , d

(7.1) (7.2)

= ad - bc = 0. 7.94. (7.1) (7.2) , = = ad - bc = 0? . C C, 1 = , C = C {}.
0

7.95. , - . 7.96. , - (7.1) w = R/z.


8

+
1.
8.1. , , n- , . 8.2. :
2 1 - cos =; 5 5 2 1 1 1 ) = + ; sin(/7) sin(2/7) sin(3/7)

a) cos

) sin 9 + sin 49 + sin 89 + . . . + sin 329 = 0. (. 7.26.) 8.3. ) cos
4 7 cos cos ; 9 9 9

) cos

3 5 + cos + cos . 7 7 7

8.4. cos 36 cos 72 . 8.5. ) , , 36 (. . ). ) 2. 8.6. a) ) ) 0 < x < 90 : 13 - 12 cos x + 7 - 4 3 sin x = 2 3; 2 - 2 cos x + 10 - 6 cos x = 10 - 6 cos 2x; 5 - 4 cos x + 13 - 12 sin x = 10.
1 1 + arctg = . 2 3 2

8.7. : arctg 1 + arctg 8.8. : ctg 30 + ctg 75 = 2.


114

8. +

8.9. x, y, z -- xyz(x + y + z) = 1. (x + y)(x + z). 8.10. x, y, z 5 x, y, z 8. S = 2x2 y2 + 2x2 z2 + 2y2 z2 - x4 - y4 - z4 ? 8.11. xk cos x + cos 2x + cos 3x +
1 = 0. 2

2 cos xk ? (. 7.58, 8.88.) 8.12. ay + bx = c, cx + az = b, bz + cy = a. ? (. 8.83.) 8.13. a, b, c, x, y, z , x2 + xy + y2 = a2 , y2 + yz + z2 = b2 , x2 + xz + z2 = c2 . xy + yz + xz a, b c. (. 9.16.)

2.
. . , z1 z2 , z1 + z2 . 8.14. z1 z2 -- . z, : z - z1 z -z = 0; ) arg 1 = 0. ) arg
z - z2 z - z2

. V (z2 , z1 , z0 ) =
z2 - z0 z1 - z0


2.

115

( ) z2 , z1 , z0 . 8.15. , , z0 z1 z2 , V (z2 , z1 , z0 ) z2 , z1 , z0 . 8.16. , z2 , z1 , z0 , V (z2 , z1 , z0 ) -- ,
ï ï z - z2 z0 - z2 =0 . ï ï z1 - z2 z1 - z2

8.17. , , z1 z2 -- z,
ï z - z2 z - z2 = . ï ï z1 - z2 z1 - z2

8.18. , Bz - B z + C = 0, C -- . 8.19. , , z0 , z1 , z2 , z3 ( )
V (z0 , z1 , z2 ) z - z2 z0 - z3 =0 : . V (z0 , z1 , z3 ) z1 - z2 z1 - z3

. W (z0 , z1 , z2 , z3 ) =
V (z0 , z1 , z2 ) V (z0 , z1 , z3 )

( ) z0 , z1 , z2 , z3 . 8.20. . z1 , z2 , z3 , z4 -- , - (7.1) z1 , z2 , z3 , z4 . , W (z1 , z2 , z3 , z4 ) = W (z1 , z2 , z3 , z4 ). 8.21. W (z1 , z2 , z3 , z4 ) (7.2)?


116

8. +

8.22. - . , - . 8.23. , ( ) Azz + Bz - B z + C = 0, A C -- . 8.24. , (8.1) w = z + u w = R/z . - . . S O R , A, O, A , OA OA = R2 /OA. O , , , O. S O R2 , S -- . 8.25. , w = 1/z . 8.26. - w =
az + b w = z cz + d

(8.1)

) i R = 1; ) Rei R; ) z0 R. 8.27. . , . 8.28. (8.1). (7.1) A zz + B z - B z + C = 0, A C . A , B C A, B C.


2.

117

. A R O |OA|2 - R2 . 8.29. , w Azz + Bz - B z + C = 0 ww +
B B C w - w+ . A A A

8.30. . , w, S1 S2 , . S1 S2 . 8.31. . S1 , S2 S3 . , Q, . Q S1 , S2 S3 . 8.32. . a1 , a2 a3 z z = 1. , h = a1 + a2 + a3 a1 , a2 a3 . 8.33. . a1 , a2 a3 z z = 1. , e = h/2 1/2 a1 a2 a3 , , a1 , a2 , a3 h. 8.34. . , m = (a1 + + a2 + a3 )/3 a1 a2 a3 . 8.35. . , , . 8.36. . u -- z z = 1 u1 , u2 , u3 -- , u a2 a3 , a1 a3 , a1 a2 a1 a2 a3 . ) , u1 , u2 , u3 u1 = (a2 + a3 + u - a2 a3 /u)/2, u2 = (a1 + a3 + u - a1 a3 /u)/2, u3 = (a1 + a2 + u - a1 a2 /u)/2.


118

8. +

) , u1 , u2 , u3 . 8.37. 4 . , . , 4 .

3.
8.38. : ) sin 20 sin 40 sin 60 sin 80 ; ) cos 20 cos 40 cos 60 cos 80 . 8.39. : cos
2 3 4 5 6 7 cos cos cos cos cos cos = 15 15 15 15 15 15 15 1 2
7

.

8.40. : cos a § cos 2a § cos 4a § . . . § cos 2n 8.41. : ) sin
2n + 1 sin ) sin 2n ) cos 2n + 1 ) cos cos 2n 2 3 n sin § . . . § sin ; 2n + 1 2n + 1 2n + 1 (n - 1) 2 3 sin § . . . § sin ; 2n 2n 2n 2 3 n cos cos § . . . § cos ; 2n + 1 2n + 1 2n + 1 (n - 1) 2 3 cos § . . . § cos . 2n 2n 2n
-1

a.

sin

8.42. : tg 20 § tg 40 § tg 80 = 8.43. : cos
x x x x x 1 cos 2 cos 4 cos 8 cos 16 = . 31 31 31 31 31 32

3.

8.44. , sin = 1/5 sin(2 + ). : tg( + ) = 3/2 tg . 8.45. -- , 3 sin2 + 2 sin2 = 1, 3 sin 2 - 2 sin 2 = 0.


3. , + 2 = /2. 8.46. : ) sin 15 =
6- 2 , 4 -1 + 5 ) sin 18 = , 4 30 - 6 5 - 8 6+ 2 ; 4 10 + 2 5 cos 18 = . 4

119

cos 15 =

8.47. : sin 6 =
6+2 5

,

cos 6 =

18 + 6 5 + 8

10 - 2 5

.

8.48. : + + + ) sin + sin + sin - sin( + + ) = 4 sin sin sin ;
2 2 2 + + + ) cos + cos + cos + cos( + + ) = 4 cos cos cos . 2 2 2

8.49. : tg + tg + tg -
sin( + + ) = tg tg tg . cos cos cos

8.50. , , , tg + tg + tg = tg § tg § tg . 8.51. , + + = , sin + sin + sin = 4 cos
cos cos . 2 2 2

8.52. ) f1 (x) = a cos x + b sin x; ) f2 (x) = a cos2 x + b cos x sin x + c sin2 x. 8.53. cos x + cos y = a, sin x + sin y = b. cos(x + y) sin(x + y). 8.54. , cos x . 8.55. n y = cos nx § sin 3? 8.56. f(x) = A cos x + B sin x, A B -- . , f(x) x1 x2 , x1 - x2 = k (k -- ), f(x) .
5 x n


120 8.57. ,

8. +

a1 cos(1 + x) + a2 cos(2 + x) + . . . + an cos(n + x) x = 0 x = x1 = k (k -- ) , x. 8.58. f(x) = = sin6 x + cos6 x. 8.59. sin4 x + cos4 x = a. 8.60. sin x + sin 2x + sin 3x = 0. 8.61. tg x + tg 2x + tg 3x + tg 4x = 0. 8.62. -- a cos x + b sin x = c. , c2 - . =2 cos2 2
2 a +b

8.63. : x sin + y sin 2 + z sin 3 = sin 4, x sin + y sin 2 + z sin 3 = sin 4, x sin + y sin 2 + z sin 3 = sin 4. 8.64. : ) arccos sin -
7

; ) arcsin cos

33 . 5

8.65. , : ) cos arcsin x = 1 - x2 ; ) sin arccos x = 1 - x2 ; ) tg arcctg x =
1 ; x

) ctg arctg x =

1 ; x

) cos arctg x =

1 ; 1 + x2 x ) cos arcctg x = ; 1 + x2

x ; 1 + x2 1 ) sin arcctg x = . 1 + x2

) sin arctg x =

8.66. : ) arctg x + arcctg x = ; ) arcsin x + arccos x = .
2 2

8.67. : ) arcsin(-x) = - arcsin x, ) arccos(-x) = - arccos x. 8.68. arctg x + arctg ? 8.69. : arctg x + arctg y = arctg
x+y + , 1 - xy 1 x


3.

121

= 0, xy < 1, = -1 , xy > 1 x < 0, = +1, xy > 1 x > 0. 8.70. : 4 arctg
1 1 - arctg =. 5 239 4

8.71. : arctg
1 1 1 1 + arctg + arctg + arctg = . 3 5 7 8 4

8.72. : arctg
x x x + arctg + . . . + arctg 2 2 1 + 1 § 2x 1 + 2 § 3x 1 + n § (n + 1)x2

(x > 0).

8.73. : arctg
r r r + arctg + . . . + arctg 1 + a1 § a2 1 + a2 § a3 1 + an § a

,
n+1

a1 , a2 , . . . , an+1 r (a1 > 0, r > 0). 8.74. , {Fn } arcctg F2n - arcctg F2n+2 = arcctg F2n+1 . (8.2) arcctg 2 + arcctg 5 + arcctg 13 + . . . + arcctg F
2n+1

+ ... =

. 4

8.75. , x > 1 : 2 arctg x + arcsin 8.76. arcsin
x2 - 8 x = 2 arcsin - . 8 4 2 2x = . 1 + x2

8.77. : arcsin 1 - x2 , arccos x = - arcsin 1 - x2 , 8.78. : arcsin x + arcsin y = arcsin(x

0 - 1

x x

1; 0.

1 - y2 + y 1 - x2 ) + ,


122

8. +

= 1, = 0, xy < 0 x2 + y2 1; = -1, = -1, x2 + y2 > 1, x < 0, y < 0; = -1, = 1, x2 + y2 > 1, x > 0, y > 0. 8.79. , 0 < x < 1 = 2 arctg + = . 8.80. arcsin cos arcsin x 8.81. , 0 arccos sin arccos x.
2 1+x , 1-x 1 - x2 , 1 + x2

= arctg

cos sin > sin cos . 8.82. sin 2 arctg
1 5 - arctg . 5 12

8.83. . ,
b c a = = , sin sin sin

++=

(8.3)

: a = b cos + c cos , b = c cos + a cos , c = a cos + b cos . (. 8.12.) 8.84. , (8.4) 0 < < , 0 < < , 0 < < , a > 0, b > 0, c > 0 (8.3). 8.85. . , (8.4) a2 = b2 + c2 - 2bc cos , b2 = a2 + c2 - 2ac cos , (8.5) 2 2 2 c = a + b - 2ab cos , (8.4) (8.5) . 8.86. . , , A, B, C. (8.4)


3.

123

(8.7) (8.6), (8.8) ( ). , , . , cos = cos cos + sin sin cos A, cos = cos cos + sin sin cos B, cos = cos cos + sin sin cos C, (8.6)

, , , , A, B, C 0 . , sin A sin B sin C = = . (8.7)
sin sin sin

8.87. . , (8.6) cos A = - cos B cos C + sin B sin C cos , cos B = - cos A cos C + sin A sin C cos , cos C = - cos A cos B + sin A sin B cos , tg
A+B+C- = 4

(8.8)

tg

p p- p- p- tg tg tg , 2 2 2 2

2p = + + . 8.88. . : ) )
3

cos cos

2 + 7 2 + 9

3

cos cos

4 + 7 4 + 9

3

cos cos

8 = 7 8 = 9

3

3

3

3

3

5-337 ; 2 339-6 . 2

(. 8.11.) 8.89. uk =
sin 2nx § sin(2n - 1)x § . . . § sin(2n - k + 1)x . sin kx § sin(k - 1)x § . . . § sin x

, uk cos x. (. 3.142.) 8.90. uk . : ) 1 - u1 + u2 - . . . + u2n = 2n (1 - cos x)(1 - cos 3x) § . . . § (1 - cos(2n - 1)x); ) 1 - u2 + u2 - . . . + u 1 2
2 2n

= (-1)n

sin(2n + 2)x § sin(2n + 4)x § . . . § sin 4nx . sin 2nx § sin 2(n - 1)x § . . . § sin 2x


9


1.
9.1. , ) p 0 x3 + px + q = 0 ; ) p < 0 ; ) p < 0 . 9.2. , z3 + Az2 + Bz + C = 0 z = x + x3 + px + q = 0. (9.1) 9.3. , ) x3 + px; ) x3 + px + q; ) ax3 + bx2 + cx + d . 3 3 9.4. 2 + 5 + 2 - 5 = 1. 9.5. x3 + x 2 + x = - . 9.6. , x3 + ax2 - b = 0, a b b > 0, . 9.7. a b, x3 + px + q = x3 - a3 - b3 - 3abx?
1 3


1. 9.8. a3 + b3 + c3 - 3abc . (. 11.74.) 9.9. a b x3 - a3 - b3 - 3abx = 0.

125

. 9.10. , (a2 + b2 + c2 - ab - bc - ac)(x2 + y2 + z2 - xy - yz - xz) = = X2 + Y 2 + Z2 - XY - YZ - XZ, X = ax + cy + bz, Y = cx + by + az, Z = bx + ay + cz. 9.11. . x3 + px + q = 0: x=
3

-

q + 2

p3 q2 + + 4 27

3

-

q - 2

p3 q2 +. 4 27

9.12. x3 + x - 2 = 0 . 9.13. , 3 13 = 5 2+7- 5 2-7 .
2

. 9.14. a x3 - x - a = 0. 9.15. x3 - x - = 0.
33 2

? 9.16. , x1 , x2 , x3 -- x3 + px + q = 0, x2 + x2 x3 + x2 = x2 + x1 x3 + x2 = x2 + x1 x2 + x2 = -p. 2 3 1 3 1 2 (. 8.13.)


126

9.

. f(x) -- n 2, f(x) = an (x - 1 ) . . . (x - n ) -- f(x) . D(f) f(x) : D(f) = a2n-2 (j - l )2 . n
1 j
D(f) , f(x) , D(f) = 0. 9.17. . x3 + px + q = 0 x1 , x2 x3 . p q D = (x1 - x2 )2 (x2 - x3 )2 (x3 - x1 )2 . 9.18. , 4p3 + 27q2 = 0 x3 + px + q = 0. 9.19. a b, x3 + ax2 + 18 = 0, x3 + bx + 12 = 0 , . . 4p3 + 27q2 = 0 Opq x3 + px + q = 0. ap + q + a3 = 0, , a, . 9.20. Opq ? , , , ? (. 6.22.) 9.21. Opq (p; q), x3 + px + q = 0 ) ; ) ; ) ; ) . 9.22. Opq (p; q), x3 + px + q = 0 1. 9.23. Opq (p; q), x3 + px + q = 0 ,


1.

127

(a; b). , , , a = -2, b = 4. 9.24. . 4p3 + 27q2 < 0, x3 + px + q = 0 ( ), , , . . ) , p < 0 9.1 x = kt 4t3 - 3t - r = 0 (9.2) t. ) , 4p3 + 27q t1 = cos = arccos r. 9.25. ) x3 - 3x - 1 = 0; ) x3 - 3x - 3 = 0. . 9.26. , f(x) = x3 + ax2 + bx + c , f (x) = 3x2 + 2ax + b , . 9.27. , x3 + px + q = 0, x3 + p x + q = 0 , (pq - qp )(p - p )2 = (q - q )3 . 9.28. ) , 4p3 + 27q x = y +
2 2

0 (9.2)
+ 4 , 3

, 3

t2 = cos

+ 2 , 3

t3 = cos

0 9.1 (9.3)

ay3 - 3by2 - 3ay + b = 0 y. ) , (9.3) y1 = tg
, 3

y2 = tg

+ 2 , 3

y1 = tg

+ 4 , 3


128 : sin =
a2 b , + b2

9.

cos =

a . a + b2
2

9.29. . 4- . ) , 4 x4 = Ax2 + Bx + C. (9.4)

) (9.4) x4 + 2x2 + 2 = (A + 2)x2 + Bx + (C + 2 ). (9.5) , -A/2 (9.5) ( x). (9.5), (9.4).

2.
9.30. x2 + y2 = 1, 4xy(2y2 - 1) = 1. 9.31. y = 2x2 - 1, z = 2y2 - 1, x = 2z2 - 1.

9.32. , x y , 0<
x-y 1 < . 1 + xy 3

9.33. 2 2 x + y = 4, z2 + t2 = 9, xt + yz = 6, , x + z .


2. 9.34. ) 1 - x2 = 4x3 - 3x; ) 1 - x = 2x2 - 1 + 2x 1 - x2 ; ) x +
x -1 x2

129

=

35 ; 12

)

1 - |x| = 2x2 - 1. 2

9.35. {hn } : h1 = ,
k=1

1 2

hn


+1

=

1-

1-h 2

2 n

(n

1).

hk < 1,03.

9.36. [0; 1] 8x(1 - 2x2 )(8x4 - 8x2 + 1) = 1? 9.37. |x1 | 1 | x2 | 1.
2 2

1 - x2 + 1

1-x

2 1-

x1 + x2 2

2

.

9.38. |2x - 1 - 4x2 | = 2(8x2 - 1). 9.39* . x, y z xy + yz + xz = 1. , , , , + + = x = tg(/2), 9.40. x + 3y = a) y + 3z = z + 3x = 2 2x + x y ) 2y + y2 z 2z + z2 x y = tg(/2), z = tg(/2).

: 4y3 , 1 3 x+ =4 y+ x 4z3 , ) xy + yz + xz = 1; 4x3 ; = y, 2y 1 1 - x2 = § = z, ) 1 + x2 1 + y2 1 xy + yz + xz = 1. = x;

1 y

=5 z+

1 , z

- z2 , + z2

9.41. xy + yz + xz = 1. :
x y z 4xyz + + = . 1 - x2 1 - y2 1 - z2 (1 - x2 )(1 - y2 )(1 - z2 )

9.42. :


130 tg x § tg z = 3, tg y § tg z = 6, x + y + z = . 9.43. : y = x(4 - x), z = y(4 - y), x = z(4 - z). 9.44. :

9.

1 + 2x 1 - x2 + 2x2 = 1. 2

3.
. - . , y = f(x) = = f1 (x) x, f2 (x) = f(f1 (x)), f3 (x) = f(f2 (x)), . . . , fn (x) = f(fn-1 (x)) , , . . . , n- f(x). xn = fn (x0 ) . . , , . , , , . (. [7].) 9.45. . 1 . . , , . . , , , 100 2/3 . 1/3 . 1 . 9.46. 2. {xn } : x0 = 1, , lim xn =
n

x

n+1

=

1 2 x+ 2n xn

(n

0).



2. (. 9.65.)


3.

131

9.47. , x0 = -1? 9.48. . , {xn }, x0 = 1, x
n+1

=

k 1 x+ , 2n xn

(n

0),

. . 9.49. a k > 0 . {an } a0 = a, an
+1

=

1 k a+ 2n an a- k a+ k

(n

0).

, n
an - k = an + k
2n

.

9.50. a0 a1 . {an } a + an-1 an+1 = n (n 1).
2

an a0 , a1 n. 9.51. I. ) , 3 x (x > 0) , x. . {yn }, y0 -- x, , , y0 = yn ,
n +1

=



xy

n

(n 3 x.

0).

lim yn =

) . 9.52. I I. ln x x = 1 1.
x 0

lim

ln(1 + x) ln(1 + x) - ln 1 = lim = 1. x (1 + x) - 1 x 0


132

9.

N. 9.51, . 9.53. . , , f(x) = x, . x0 , {xn } xn+1 = f(xn ) (n 0). , x = lim xn , f(x) , : f(x ) = x . . . Oxy f(x) -- y = x. A0 (x0 , f(x0 )), A1 (x1 , f(x1 )), . . . , An (xn , f(xn )), . . . , -- B0 (x0 , x0 ), B1 (x1 , x1 ), . . . . . . , Bn (xn , xn ), . . . B0 A0 B1 A1 . . . Bn An . . . . 9.54. : ) f(x) = 1 + , x0 = 0, x0 = 8; ) f(x) =
1 , x x 2
n

x0 = 2; x0 = ;
5 2

) f(x) = 2x - 1, x0 = 0, x0 = 1,125; ) f(x) = -
3x + 6, 2

) f(x) = 2 + 3x - 3, x0 = 1, x0 = 0,99, x0 = 1,01; x ) f(x) = 1 + x, x0 = 0, x0 = 8; ) f(x) =
x3 5x2 25x - + + 3, 3 2 6

x0 = 3.

9.55. {an } a1 = 1, a
n+1

= an +

1 a2 n

(n

1).

, ? 9.56. {an }
n

lim

an

+1

-

an 2

= 0.

, lim an = 0.
n


3. 9.57. a1 , a2 , . . . , ak ,
n

133

lim (xn + a1 x

n-1

+ . . . + ak x

n-k

)=0

{xn }, lim xn = 0. ,
n

P() = k + a1 k-1 + a2 k-2 + . . . + ak 1. 9.58. : ) x ) x ) x
n+1 n+1 n+1

=

= xn , x0 = a (0; ); sin = a + x, a > 0, x0 = 0.

1 , 1 + xn

x0 = 1;

9.59. ? -- , a b , a : b = b : (a - b). , , . , ; . , , . . , . . 9.60. 3 a. {an } : a0 = a > 0, , lim an =
n

an 3

+1

=

a 1 2an + 2 3 an

(n

0).

a.

9.61. x3 - x - 1 = 0. 9.62. {an } a1 = 1, a
n+1

=

3an 1 + 4 an

(n

1).


134 , ) {an } ; ) |a1000 - 2| < (3/4)1000 .

9.

9.63. , a1 = 2, a
n+1

=

an a2 +n 2 8

(n

1).

9.64. . , f(x) [a; b] , |f (x)| q < 1. , f(x) = x [a; b] x . , : |x
n+1

- xn |

| x1 - x 0 | § q n ,

| x - x n |

| x 1 - x0 | §

qn 1-q

(n

0).

9.65. , {xn } 9.46 : xn = [1; 2, . . . , 2 ] (n
2n -1

0).

|xn -



2|. (. 9.81)

9.66. 9.48 ? 9.67. 2 2x1 = x2 , 1 + x2 1 2x2 2 = x3 , 1 + x2 2 2x2 3 = x1 . 2
1 + x3



k -

9.68. : 2 3 y = 4x + x - 4, 2 z = 4y3 + y - 4, 2 x = 4z3 + z - 4. 9.69. {xn } : x1 -a, xn+1 = a + xn .


3.

135

, {xn } . . 9.70. . , f(x) x = f(f(x)) x = f(x) . 9.71. a + a + a + x = x. 9.72. - . a b -- , a > b. {an } {bn } : a0 = a, b0 = b, a
n+1

=

an + bn , 2

b

n+1

=

an b

n

(n

0).

, . - a, b ²(a, b). 9.73. - . a b -- , a < b. {an } {bn } : a0 = a, b0 = b, a
n+1

=

2an bn , an + bn

b

n+1

=

an + bn 2

(n

0).

) , . - a b. ) , a b. ) a = 1, b = k. {bn } {xn } 9.48? 9.74. - . - a b {an } {bn }, a0 = a, b0 = b , a
n+1

=

2an bn , an + bn

b

n+1

=

an b

n

(n

0).

(a, b). , (a, b) ²(a, b) (. 9.72)
1 11 =² , . (a, b) ba


136 9.75. 1 - 1 - ... 1 - 1-

9. x2 = 1 x2 = 2 ... x2 -1 n x2 = n x2 , x3 , .... = xn , x1 .
100

9.76. 0,01 x {xn }, ) x1 [0; 1], xn+1 = xn (1 - xn ), (n > 1); ) x1 [0,1; 0,9], xn+1 = 2xn (1 - xn ), (n > 1).

-

9.77. , f(x), (x0 ; f(x0 )) Ox x0 -
f(x0 ) . f (x0 )

9.78. . f(x) = 0 x
n+1

= xn -

f(xn ) , f (xn )

( x0 , f(x) = x2 - k x0 > 0 k, lim xn = k. n xn+1 xn ? 9.48.

). -

9.79. . f(x) = x2 - x - 1. , ) x0 = 1; ) x0 = 0? ? xn . 9.80. p q -- p2 - 4q > 0. , :


3. ) y0 = 0, y
n+1

137
q p - yn q =p- zn

=

(n

0);

) z0 = 0, zn+1 (n 0). y , z x2 - px + q = 0. 9.81. . , =p-
p- q q q p- ...



=
p-

q q p- q ...

. 9.80, x2 - px + q = 0. , : x
n+1

= xn -

x2 - pxn + q x2 - q n =n . 2xn - p 2xn - p

, x0 , x1 , x2 , . . . . (. 9.65.) 9.82. f(x) = 0. f(x) = x(x - 1)(x + 1) x0 , f(x0 ) = x0 x2 = x0 . 9.83. . P(x) = xn + an
-1

x

n-1

+ . . . + a1 x + a0

x1 , x2 , . . . , xn , |x1 | > |x2 | > . . . > |xn |. 6.42 Q(x) n, x2 , x2 , . . . , x2 . 1 2 n P(x). . P0 (x), P1 (x), P2 (x), . . . , P0 (x) = P(x) k k Pk (x) x2 , . . . , x2 . 1 n Pk (x) = xn + a(k)1 x n- , ) lim (-a
k (k) 1/2k n-1 n-1 ( + . . . + a1k) x + a (k) 0

.

)

= x1 ; ) lim

k

-

a a

(k) n-l (k) n-l+1

1/2k

=x

l

(1

l

n).


138

9.

9.84. . , , x2 -x-1. x1 x2 , | x1 | > | x2 | ? 9.85. . 1/2. n-. Pn ( ) pn ( ). ) P4 , p4 , P6 p6 . ) , : 2Pn pn P2n = , p2n = pn P2n (n 3).
Pn + pn

) P

96

p96 . 3
1 10 <<3 . 71 3

9.86. .
2 = 1 § 2 1 1 + 2 2 1 § 2 1 1 + 2 2 1 1 + 2 2 1 ... 2

9.87. x0 , x1 , x2 , . . . x0 = 1, x
n+1

=a

xn

(n

0).

a, . a? 9.88. a1 , a2 , a3 , . . . a1 = 1, a
n+1

= an +

1 a2 n

(n

0).

, ) ; ) a9000 > 30; a ) lim n . 3
n

n

9.89* . (xn , yn , zn ) (n
6 x1 = 2, y1 = 4, z1 = , 7

1) :

x

n+1

=

2xn , x2 - 1 n

yn

+1

=

2yn , y2 - 1 n

zn

+1

=

2zn z2 - 1 n

(n

1).


4.

139

) , . ) (xn , yn , zn ), xn + yn + zn = 0?

4.
9.90. . , , , . 4 / , 3 / 6 / -- . ? . , n x1 , . . . , xn . . , x1 . x2 . , . . 9.91. x - 3y + 2z - t = 3, 2x + 4y - 3z + t = 5, ) 4x - 2y + z + t = 3, 3x + y + z - 2t = 10; x + 2y + 3z - t = 0, x - y + z + 2t = 4, ) x + 5y + 5z - 4t = -4, x + 8y + 7z - 7t = -8; x + 2y + 3z = 2, x - y + z = 0, ) x + 3y - z = -2, 3x + 4y + 3z = 0; x + 2y + 3z - t = 0, x - y + z + 2t = 4, ) x + 5y + 5z - 4t = -4, x + 8y + 7z - 7t = 6.

9.92. . , 1, -- 2.

)

)


140

9.

9.93. 4 . . 1/4 . . , , 1/4 . 2. , ) ? ) , ? 9.94. . , , . ax + y = a2 , ax + y = a3 , ) ) x + ay = 1; x + ay = 1; ) ) ) ax + ay = a2 , x + ay = 2; (a + 1)x + 8y = 4a, ax + (a + 3)y = 3a - 1; a2 x + (2 - a)y = 4 + a2 , ax + (2a - 1)y = a5 - 2; ) ) ) ax - ay = ab, 2ax - y = a; ax + by = a, bx + ay = b; |a|x - y = 1, x + |a|y = a.

9.95. ? 9.96. , , (1, 1, 1, 1) (1, 2, 2, 1) . 9.97. x + y + z = 0, x + y + z = 0, x + y + z = 0. . , , . 9.98. :


4. 2 3 2x + 3y = 5, x + ay + a z = a , ) x - y = 2, ) x + by + b2 z = b3 , x + 4y = a; x + cy + c2 z = c3 ; x + ay = 1, x + y + z = 1, ) 2x + 4y = 2, ) ax + by + cz = d, 2 bx + 4y = 2; a x + b2 y + c2 z = d2 ; ax + by = a, ax + by + cz = a + b + c, ) (a - 2)x + y = 3, ) bx + cy + az = a + b + c, x + y = 1; cx + ay + bz = a + b + c.

141

9.99. : x1 + x2 + x3 = 0, x1 + x2 + x3 + x4 = 2a1 , x2 + x3 + x4 = 0, x + x - x - x = 2a , 1 2 3 4 2 ............. ) ) x1 - x2 + x3 - x4 = 2a3 , x + x + x = 0, 99 100 1 x1 - x2 - x3 + x4 = 2a4 ; x100 + x1 + x2 = 0; x1 + 2x2 + 3x3 + . . . + nxn = a1 , x + y + z = a, x + y + t = b, nx1 + x2 + 2x3 + . . . + (n - 1)xn = a2 , ) ) ........................ x + z + t = c, 2x1 + 3x2 + 4x3 + . . . + xn = an . y + z + t = d; 9.100. 13 . , 12 , , . , , , : ) ; ) ; ) () . 9.101. , a1 - 4a2 + 3a3 a2 - 4a3 + 3a4 ............. a99 - 4a100 + 3a1 a100 - 4a1 + 3a2

0, 0, . 0, 0.
100

a1 = 1; a2 , . . . , a

?


10


, ( ), .

1.
10.1 í 10.37 . 10.1. x + 1/x 2. 10.2. .
a2 + b2 2 a+b . 2

10.3. (a + b + c + d)2 10.4. 10.5. 4 (a + c)(b + d) ab
3

4(a2 + b2 + c2 + d2 ). ab + cd.

a + 3b . 4 a + 2b + 3c + 4d 10
10

10.6. ab2 c3 d4 10.7. x2 + y2 + z2 10.8. x + y + 1
2 1 2 2 2 3 2 2

.

xy + yz + xz. xy + x + y.
2 5

10.9. x + x + x + x2 + x 4 10.10. x + y + 8 10.11.
4 4

x1 (x2 + x3 + x4 + x5 ).

8xy.
a+b+c . 3

3 1/a + 1/b + 1/c
1 2 x 4 x

10.12. (ab + bc + ac)2 10.13. 2 +2 2§2 10.14. ab + bc + ac 10.15.

3abc(a + b + c).
6x

.

0 a + b + c = 0.

x+y < 1, |x|, |y| < 1. 1 + xy


1. 10.16. x y
2 2

143

x + y, , + = 1 (, > 0). abc(a + b + c). 16abc.
2

10.17. a b + b2 c2 + a2 c2 10.19. (a + b + c + d + 1)
2

10.18. (a + 1)(b + 1)(a + c)(b + c)
2

4(a + b + c2 + d2 ), a, b, c, d [0; 1].

10.20. x4 + y4 + z2 + 1 2x(xy2 - x + z + 1). 10.21. ( x + y)8 64xy(x + y)2 (x, y 0). 10.22. (a + b)(b + c)(a + c)
2 2

8abc.
2

10.23. (a + b + c)(a + b + c ) 10.24. a (1 + b ) + b (1 + a ) 10.25. a4 + b4 + c4
3 3 3 2 4 2 4

9abc. (1 + a4 )(1 + b4 ).

abc(a + b + c). abc(a + b + c). ab(a + b) + ac(a + c) + bc(b + c).
. . + an . . + bn z 1+ x
1kn 3

10.26. a b + b c + c a 10.27. 2(a + b + c ) 10.28. 10.29. 10.30. 10.31. 10.32. a1
a a1 + . min k b1 + . 1 k n bk x y 1+ 1+ y z b c a ++ 3. b c a a b + + b+c a+c a 1 1 + + b+c a+c a
3 3

max

ak . bk

8.

c +b 1 +b

3 . 2 9 . 2(a + b + c)

10.33. 3(a1 b1 + a2 b2 + a3 b3 ) a2 a3 , b 1 b 2 b 3 . 10.34. , a1 a2 ... an ,

(a1 + a2 + a3 )(b1 + b2 + b3 )

b

1

b

2

...

bn ,

a1 b
k1

+ a2 b

k2

+ . . . + an b

kn

(k1 , k2 , . . . , kn -- 1, 2, . . . , n), a1 b1 + a2 b2 + . . . + an bn , -- a1 bn + a2 b
n-1

+ . . . + an b 1 .


144 10.35. . ,
a1 b1 + a2 b2 + . . . + an bn n a1 + a2 + . . . + a n
n

10.

§

b1 + b2 + . . . + bn . n

10.36. : a+b+c
a2 + c 2 b2 + c a2 + b2 + + 2c 2b 2a
2

a3 b3 c3 + + . bc ac ab

10.37. . , a1 a2 + a2 + . . . + a2 1 2 n
a a an 1 + 2 + . . . + n+ n-1 1+ 0 2+ 1
2

a2

...

a

n

0

.

10.38. (1+x1 ) . . . (1+xn )

2n , x1 . . . xn = 1.

10.39. , n 1 1 1 + + ... + > 1.
n+1 n+2 3n + 1

10.40. , n
1 1 1 + + ... + n+1 n+2 2n

1/2 3/4. 10.41* . p, q,
ap bq + . p q 11 + = 1. pq

, a b ab

10.42. x2 + (1 - x2 )2 + 1 x2 + (1 - x3 )2 + . . . + 2 x
2 2n

+ (1 - x1 )2 .

10.43. n : n+1 n ) n n! ; ) )
n 3 n e
n

n

2 nn < n! < ; 2 nn < n! < n . e


2. 10.44. , x 0; 0< (. 7.81.)
2

145

1 1 - 2 < 1. sin2 x x

10.45. , m n m n, n m 3 3. 10.46. 22 ? 10.47. :
1 + n+1 n ) 1+ 2 1 1 ) <§ 15 2 13 ) § § . . 24
.2 ..

,

)

1 1 1 + ... + n+2 2n 2 1 1 + ... + n n 2 2 -1 3 99 1 § ... § <; 4 100 10 99 1 .§ <. 100 12

(n (n

1); 1);

10.48. , ++ = 1 y z x .

x

y

z

2.
10.49. . , f1 (x), . . . , fn (x), [a; b]. :
x[a;b]

min f1 (x) + . . . + min fn (x)
x[a;b]

x[a;b]

min (f1 (x) + . . . + fn (x)).

10.50. :
b2 b2 1 + ... + n a1 an (b1 + . . . + bn )2 . a1 + . . . + an

10.51. ) í : (c1 d1 + . . . + cn dn )2 (c2 + . . . + c2 )(d2 + . . . + d2 ); 1 n 1 n

) :
a1 + . . . + a n
n

a2 + . . . + a2 n 1 ; n


146

10.

) : b1 + . . . + bn n .
1/b1 + . . . + 1/bn n

10.52. :
b1 + . . . + bn a1 + . . . + an
b1 +...+bn

b1 a1

b1

...

bn an

bn

.

10.53. , : a1 + . . . + an ) n a1 . . . an ;
n

) ) c

b1 + . . . + bn n
b1 1

b1 +...+bn

b

b1 1

...b

bn n

;

. . . cbn n

c1 b1 + . . . + cn bn , b1 + . . . + bn = 1.

10.54. . , A B, : . k(1) , k(1) , k(2) , k(2) . B A B A , N A, , k(1) § N. A k(1) = 2, k(1) = , k B A
3 2
(2) A

= ,k

4 3

(2 ) B

= 3.

, N, , , ? ( k(1) , kB1) , A (2 ) (2) kA , kB .

3.
. -- f(x), [a; b]: = {(x, y) : x [a; b], y = f(x)}. f(x) , T , T . . () () .


3.

147

10.55. , f(x) [a; b], x1 , x2 [a; b] 1 , 2 , 1 + 2 = 1 : f (1 x1 + 2 x2 ) > 1 f(x1 ) + 2 f(x2 ). 10.56. . , f(x) [a; b], x1 , x2 , . . . , xn (n 2) [a; b] 1 , 2 , . . . , n , 1 + 2 + . . . + n = 1, : f(1 x1 + . . . + n xn ) > 1 f(x1 ) + . . . + n f(xn ). 10.57. , x1 , . . . , x : sin
x1 + . . . + xn n
n

[0; ]

sin x1 + . . . + sin xn . n

10.58. : ) n(x1 + . . . + xn ) ( x1 + . . . + xn )2 ; )
n3 (x1 + . . . + xn )2 1 1 + ... + 2 ; x2 xn 1

) nx1 . . . xn

xn + . . . + x n ; 1 n

) . (x1 + . . . + xn )
1 1 + ... + x1 xn

n2 . 12.

10.59. , x + y + z = 6, x2 + y2 + z2

10.60. . p q -- , 1/p + 1/q = 1. , a1 b1 + a2 b2 + . . . + an b
n

(ap + ap + . . . + ap )1/p (aq + aq + . . . + aq )1/q . 1 2 1 2 n n

. = 0 x1 , . . . , xn S (x) =
x + . . . + x n 1 n
1/

.

: ( = -1), ( = 1), ( = 2). 0 S0 (x) = n x1 . . . xn .


148

10.

10.61. , : S
-1

(x)

S0 (x)

S1 (x)

S2 (x). S (x).

10.62. , < = 0, S (x) 10.63 . , < 0 < , S (x)
-0 *

S0 (x)

S (x),

lim S (x) = lim S (x) = S0 (x).
+0

10.64. , < , S x1 = x2 = . . . = xn .

S ,

4.
10.65. ) x4 + ) x3 + ) x4 + ) x5 + : y4 + z4 x2 yz + xy2 z + xyz2 ; y3 + z3 3xyz; y4 + z4 + t4 4xyzt; y5 x3 y2 + x2 y3 .

. -- , x, y z. , = (k, j, i), k j i, . T (x, y, z) = T(k,j,i) (x, y, z) T (x, y, z) =
{a,b,c}={k,j,i}

xa yb zc

( {a, b, c} 3! {k, j, i}). 10.65 : ) T(4,0,0) (x, y, z) T(2,1,1) (x, y, z); ) T(3,0,0) (x, y, z) T(1,1,1) (x, y, z); ) T(4,0,0,0) (x, y, z, t) T(1,1,1,1) (x, y, z, t); ) T(5,0) (x, y) T(3,2) (x, y). 10.66. T ) x4 y + y4 x x3 y2 + x2 y3 ; ) x3 yz + y3 xz + z3 xy x2 y2 z + y2 z2 x + z2 x2 y.


4.

149

. , = (1 , . . . , n ) ë¨ n , k- k , -- .

s = 1 + 2 + . . . + n . 10.67. T : ) (3, 2); ) (3, 2, 1); ) (3, 3, 0, 0); ) (4, 1, 1, 0). 10.68. ) s = 4; ) s = 5; ) s = 6; ) s = 7. . = (1 , . . . , n ) = ( s = 1 + . . . + n , ( ), : 1 1 , 1 + 2 1 + 2 , ..................... 1 + . . . + n-1 1 + . . . + n-1 , 1 + . . . + n = 1 + . . . + n . s,
1

, , , , . , (4, 2, 1) (3, 2, 2), 4 3, 4 + 2 3 + 2, 4 + 2 + 1 = = 3 + 2 + 2. 10.69. , = (1 , 2 , 3 ) = (1 , 2 , 3 ) , ( ) (k - 1, j + 1, i) (k, j, i) - (k - 1, j, i + 1) (k, j - 1, i + 1) .

èç æ å â ä â ã â á

æå ä ã â ã â á

, . . . , n ) -- = 1 + . . . + n . -


150

10.

10.70. s = 4 , (4, 0, 0, 0), (1, 1, 1, 1). 10.71. ) , (4, 1, 1) (3, 3, 0) , -- . 6? ) s = 7. 10.72. T (x, y, z) y, z. , . T (x, y, z) x,

10.73* . . = (1 , . . . , n ) = (1 , . . . , n ) ,

-- . , , x1 , . . . , xn T (x1 , . . . , xn ) T (x1 , . . . , xn ).

10.74. . ë¨ , (n, 0, . . . , 0) n (1, 1, . . . , 1)? 10.75. : ) x4 y2 z + y4 x2 z + y4 z2 x + z4 y2 x + x4 z2 y + z4 x2 y 2(x3 y2 z2 + x2 y3 z2 + + x2 y2 z3 ); ) x5 + y5 + z5 x2 y2 z + x2 yz2 + xy2 z2 ; ) x3 + y3 + z3 + t3 xyz + xyt + xzt + yxt. 10.76. 10.36 . ?


11


1.
. {bn } = b1 , b2 , . . . , bn , . . . {bn } , {bn }: bn = b
n+1

-b

n

(n = 1, 2, . . . ).
)

(, b0 = 0.) . 11.1. ) n2 ; ) n(n - 1); ) nk ; ) Ck . n



11.2. {an } {bn }, bn = an , (n = 1, 2, . . . ). Sn {an } Sn = a1 + a2 + . . . + a {bn }? 11.3. {an } , an = n2 . , 12 + 22 + 32 + . . . + n2 . 11.4. 13 + 23 + 33 + . . . + n3 . 11.5.
n n

1 F
2
k

=3-

F

2n -1

k=0

F

2n

(n

1).

) -- , , .


152

11.

. f(x) f(x + 1) - f(x) f(x) . : n f(x) = ( (, 0 f(x) = f(x)). 11.6. , Q(x) -- m + 1, P(x) = Q(x) -- m. 11.7. , P(n) m Q(n) m + 1 , : Q(n) = P(n) Q(0) = 0. 11.8.
n n-1

f(x)) (n > 1)

n f(x) =
k=0

Ck (-1)n n

-k

f(x + k).

11.9.
n

f(x + n) =
k=0

Ck k f(x). n

11.10. f(x) -- m. , m < n, n f(x) = 0. m f(x) = 0? 11.11.
n

Ck (-1)k 1 - n
k=0

k n

n

. m < n -

11.12. , m 1 :
n

(-1)k km Ck = 0. n
k=1

(. 6.105.) 11.13* . y0 , y1 , . . . , yn , f(x) m < n :
n

f(k)yk = 0.
k=0

(11.1)

, yk = (-1)k Ck , -- . n


1.

153

11.14. , : )
bn 1 =- ; bn bn bn+1

)

an bn

=

bn an - an bn . bn bn+1

11.15. (an § bn ) an bn . . 11.15 . . n-
n (n)

f(x)g(x)

=
k=0

Ck f(k) (x)g( n

n-k)

(x).


n n



f(x)g(x) =
k=0

Ck k f(x) n

(n-k)

g(x).

()

11.16. y = ex , y (x) = y(x) y(0) = 1. {an } , ? 11.17. . . , :
n-1 n-1 n-1 x

f(x)g(x) = f(n)
x=0 n-1 x=0

g(x) -
x=0

f(x)
z=0 n-1

g(z),

f(x)g(x) = f(n)g(n) - f(0)g(0) -
x=0 x=0

g(x + 1)f(x).

11.18. {an } , a ( 11.19. : xex dx.)

n

= n2n .


154
n

11.
1 ; k(k + 1) 1 ; 2 k -1 1 ; k(k + 1)(k + 2) (k - 1) 2k ; k(k + 1)
n

)
k=1 n

)
k=1 n

)
k=2 n

)
k=1 n

4k + 1 ; k(k + 1)(4k2 - 1) k-1 ; k!

)
k=1 n

)
k=1

k! k.

)
k=1

11.20. :
n n n

)
k=1

k2 q

k-1

; )
k=1

k sin kx; )
k=1

k2 cos kx.

. Cn x x(x - 1) . . . (x - n + 1) Cn = , x
n!

x. x n . 11.21. . ) , f(x) n f(x) = d0 C0 + d1 C1 + . . . + dn Cn . x x x Ck x x. , . (. 6.79.) ) , d0 , d1 , . . . , dn dk = k f(0) (0 k n). 11.22. . f(x) n x = 0, 1, . . . , n. , f(x) = d0 C0 + d1 C1 + . . . + dn Cn , x x x d0 , d1 , . . . , dn -- . 11.23. f(x) = x3 - x 2 f(x). , . , f(x) . 6 x. . 11.24. , f(x) n x = 0, 1, . . . , n, x.


2. 11.25. ) q -- f(x) = cqx + an xn + . . . + a1 x + a0

155

x = 0, 1, 2, . . . , n + 1. , x 0 f(x) . ) ) f(x) m 1 x = 0, 1, 2, . . . , n + 1. , f(x) m x 0. 11.26. , n f(n) = 22n - 9n2 + 21n - 14 27. 11.27* . n ‘12 ‘ 22 ‘ 32 ‘ . . . ‘ n
2 -1

-

+ -, 0? . f(x, y) . f(x, y) , , f(x, y) = (f(x + 1, y) + f(x - 1, y) + f(x, y + 1) + f(x, y - 1)). 11.28. f(x, y) g(x, y) -- . , a b af(x, y) + bg(x, y) . 11.29. f(x, y) -- . , x f(x, y) = f(x + 1, y) - f(x, y) y f(x, y) = f(x, y + 1) - f(x, y) . 11.30* . . f(x, y) -- , M , (x, y) Z2 |f(x, y)| M.
1 4

, f(x, y) .

2.
. a0 , a1 , . . . , an , . . . , p q an
+2

= pan

+1

+ qan

(n = 0, 1, 2, . . . )

(11.2)


156

11.

( ) . x2 - px - q = 0 (11.3) {an }. 11.31. , a0 , a1 , {an } . 11.32. , {an } = bxn 0 (11.2) , x0 -- (11.3) {an }. 11.33. (11.3) {an } x1 x2 . , a0 , a1 c1 , c2 , an = c1 xn + c2 xn (n 0). 1 2 11.34. (11.3) {an } x0 2. , a0 , a1 c1 , c2 , an = (c1 + c2 n)x
n 0

(n

0).

11.35. n- , (n 0): a) a0 = 0, a1 = 1, an+2 = 5an+1 - 6an ; ) a0 = 1, a1 = 1, an+2 = 3an+1 - 2an ; ) a0 = 1, a1 = 1, an+2 = an+1 + an ; ) a0 = 1, a1 = 2, an+2 = 2an+1 - an ; ) a0 = 0, a1 = 1, an+2 = 2an+1 + an . 11.36. 1 + 2 , : (1 + 2)1 = 1 + 2 = 2 + 1, (1 + 2)2 = 3 + 2 2 = 9 + 8, (1 + 2)3 = 7 + 5 2 = 50 + 49, (1 + 2)4 = 17 + 12 2 = 289 + 288. an bn (1 + 2)n = an + bn 2, (n 0). (. 11.59.)


2.

157

) an bn (1 - 2)n . ) a2 - 2b2 = 1. n n ) {an } {bn }? ) ), n- {an } {bn }. ) an , bn 2. 11.37. : 2+ 3= (2 + 3)2 = (2 + 3)3 = (2 + 3)4 = 4+

3, 49 + 48, 676 + 675, 9409 + 9408.

. 11.38. , (x + y 5)4 + (z + t 5)4 = 2 + 5 x, y, z, t. 11.39. a2 - 3b2 = 1. n n 11.40. , Qn , Q0 = , Q1 = , Qn
+2

= Qn

+1

+ Qn
n

(n

0),

F

Ln .

. Fn (x) (n 0) F0 (x) = 0, F1 (x) = 1 Fn+1 (x) = x Fn (x) + Fn-1 (x) (n 1). , Ln (x) L0 (x) = 2, L1 (x) = x, Ln
+1

(x) = x Ln (x) + L

n-1

(x) (n

1).

11.41. . . x = 1? , (. 3.133):


158 ) ) ) ) )

11. Ln (x) = Fn-1 (x) + Fn+1 (x) (n 1); Fn (x) (x2 + 4) = Ln-1 (x) + Ln+1 (x) (n 1); F2n (x) = Ln (x) § Fn (x) (n 0); Ln (x)2 + Ln+1 (x)2 = F2n+1 (x)(x2 + 4) (n 0); Fn+2 (x) + Fn-2 (x) = (x2 + 2)Fn (x) (n 2).
+1

11.42. Fn , (. 3.144.)

(x)/Fn (x) L

n+1

(x)/Ln (x) (n

1)

11.43. . (. 3.126 11.75.) 11.44. , U
n

x 2

=

F

n+1 (ix) n

i

,

2T

n

x 2

=

Ln (ix) . in

11.45. Fn (x) Ln (x). 3.129 3.130 . 11.46. -. ABC, . A A n ? 11.47. ABCDEF, . ) A C n ? ) , , D? ) -. A, D . . , n ? ? 11.48. , p > 2 , 2+ 2 + ... + 2+ 2+p=
2n

-

-2n

.

n

11.49. , , , 6 . . n- ? 11.50.


2. ) (6 + 35)1999 ; ) (6 + 37)1999 ; ) (6 + 37)2 1000 .
000

159

11.51. , n [(1 + 2)n ] n (mod 2). 11.52. , an = 1 + 17n2 (n . 0)

11.53. {xn } {yn } : xn = x
n-1

+ 2yn

-1

sin2 , yn = yn

-1

+ 2xn

-1

cos2 , x0 = 0, y0 = cos .

xn yn n . 11.54. . . , , , 1/5 , . ; . , . ? . a0 , a1 , . . . , an , . . . , b0 , . . . , bk-1 an
+k

=b

k-1

an

+k-1

+ . . . + b0 a

n

(n = 0, 1, 2, . . . ),

(11.4)

( ) k- . xk - bk-1 xk-1 - . . . - b0 = 0 {an }. 11.55* . n- k- , a) x1 , . . . , xk ; ) x1 , . . . , xm 1 , . . . , m ? 11.56. (11.3) (11.2) x1,2 = a ‘ ib = re‘i . , c1 , c2 an = rn (c1 cos n + c2 sin n).


160

11.

11.57. n- , (n 0): a) a0 = 0, a1 = 1, an+2 = 4an+1 - 5an ; ) a0 = 1, a1 = 2, an+2 = 2an+1 - 2an ; ) a0 = 1, a1 = 2, an+2 + an+1 + an = 0; ) a0 = 1, a1 = 8, an+2 = 6an+1 + 25an . 11.58. (11.4) a) an = n2 ; ) an = n3 ? 11.59. (1 + 2 + 3)n = pn + qn 2 + rn 3 + sn 6 (n 0). : p p p ) lim n ; ) lim n ; ) lim n . (. 11.36.)
n

q

n

n

rn

n

s

n

3.
. F(x) = a0 + a1 x + a2 x2 + . . . + an xn + . . . (11.5)

. , , , , , . . (11.5) F (x) = a1 + 2a2 x . . . + nan x
n-1

+ ...

11.60. : ) (1 + x + x2 + x3 + . . . )(1 - x + x2 - x3 + . . . ); ) (1 + x + x2 + x3 + . . . )2 ; ) 1 + x +
x2 xn + ... + + ... 2! n!

1-x+

(-x)n x2 - ... + + ... . 2! n!

11.61. . , a0 = 0, F(x) F-1 (x) = b0 + b1 x + . . . + bn xn + . . . , F(x)F-1 (x) = 1. 11.62. : ) (1 + x)-1 ; ) (1 - x)-1 ; ) (1 - x)-2 . . {an } = a0 , a1 , . . . -- . F(x) = a0 + a1 x + . . . + an xn + . . .


3. .

161

11.63. F(x) -- {an }. an =
F
(n )

(x) n!

.
x=0

11.63 . , p Fn (U ,
F3n F3 F5n Fn (11) = F5 F7n Fn (29) = F7
p-1

(

Fnp 5 , )) = 2 Fp

Fn (4) =

(p = 3); (p = 5); (p = 7).

11.64. : ) an = n; ) an = n2 ; ) an = Cn . m 11.65. : ) C1 + 2C2 + 3C3 + . . . + nCn ; ) C1 + 22 C2 + 32 C3 + . . . + n2 Cn . n n n n n n n n 11.66. an -- x1 + . . . + xk = n (11.6)

F(x) -- an . : ) F(x) = (1 + x + x2 + . . . )k ; ) F(x) = (1 - x)-k . 11.67. , , an -- (11.6) . an , 11.63. (. 2.70.) 11.68. : (1 + x + x2 + . . . + x9 )(1 + x ½(1 + x (. 1.2.) 11.69. . , n ) Ck n = (-1)k Ck - n
+k-1 100 10 20

+x
900

+ . . . + x90 )½
1 . 1-x

+x

200

+ ... + x

)... =

;

) (1 + x)-n =



Ck n xk . -

k=0


162

11.

11.70. : ) an = Cm+n ; ) an = Cm . n m 11.71. . , 1 000 000 000 000 999 999. , . N -- . : ) (1 + x + . . . + x9 )3 (1 + x-1 + . . . + x-9 )3 = x27 + . . . + a1 x + N + + a-1 x-1 + . . . + x-27 ; ) (1 + x + . . . + x9 )6 = 1 + . . . + Nx27 + . . . + x54 . 11.72. . . Exp(z) = 1 + z +
z2 zn + ... + + ... = 2! n!

k=0

zk . k!

11.73. : ) Exp (z) = Exp(z); ) Exp(( + )z) = Exp(z) § Exp(z). (. 7.52.) 11.74. a, b c a = 1 + C3 x3 + C6 x6 + . . . , n n b = C1 x + C4 x4 + C7 x7 + . . . , n n n c = C2 x2 + C5 x5 + C8 x8 + . . . . n n n , a3 + b3 + c3 - 3abc = (1 + x3 )n . (. 9.8.) 11.75. , F(z) = F0 + F1 z + F2 z2 + . . . + Fn zn + . . .
z 1 1 1 = - , 1 - z 1 - z - z2 5 1 - z 1+ 5 1- 5 = ,= . 11.63, 2 2

F(z) =

. (. 3.126 11.43.)


3. 11.76. , 0,1 + 0,01 + 0,002 + 0,0003 + 0,00005 + 0,000008 + 0,0000013 + . . . .

163

11.77. L(z) = L0 + L1 z + L2 z2 + . . . + Ln zn + . . . . , Ln . (. 3.135.) 11.78.


)
n=0

Fn ; 2n



)
n=0

Ln . 2n

11.79. . F(x, z) = F0 (x) + F1 (x)z + F2 (x)z2 + . . . + Fn (x)zn + . . . L(x, z) = L0 (x) + L1 (x)z + L2 (x)z2 + . . . + Ln (x)zn + . . . 11.80. . (. 7.38):


FT (x, z) =
n=0

Tn (x)z ,

n

FU (x, z) =
n=0

Un (x)zn .

11.81. , , :
n-1 n-1

)
k=0 n-1

2k xk ; k 2k ;
k=0

)
k=0 n-1

k2 2k ; k sin kx.
k=0

)

)

11.81 . (11.2) a0 a1 . . .


164

11.

. , 3 = 1 + 2 3 = 2 + 1 . 11.82. p(n) -- n. : p(0) + p(1)x + p(2)x2 . . . = (1 + x + x2 + . . . ) . . . (1 + xk + x = (1 - x)-1 (1 - x2 )-1 (1 - x3 )-1 . . . ( , p(0) = 1.) 11.83. n . ak -- , k. ak . , , . , 5, 3, 3, 2, (5, 3, 3, 2) (4, 4, 3, 1, 1) (5, 3, 3, 2). 11.84. , n 2n-1 - 1 , , , . , 24-1 - 1 = 7 4: 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 2 + 1, 2 + 1 + 1, 2 + 2, 1 + 3, 3 + 1.
2k

+ ...)... =

11.85. n , , . , 6 : 6, : 1 + 5, 3 + 3, 1 + 1 + 1 + 3, 1 + 1 + 1 + 1 + 1 + 1. 1 + 5, 2 + 4, 1 + 2 + 3,

d(n) n , l(n) -- . : ) d(0) + d(1)x + d(2)x2 + . . . = (1 + x)(1 + x2 )(1 + x3 ) . . . ; ) l(0) + l(1)x + l(2)x2 + . . . = (1 - x)-1 (1 - x3 )-1 (1 - x5 )-1 . . . ; ) d(n) = l(n) (n = 0, 1, 2, . . . ). ( , d(0) = l(0) = 1.)


3.

165

11.86* . - . , 4, . 148. 11.87. an (1 + qx)(1 + qx2 )(1 + qx4 )(1 + qx8 )(1 + qx16 ) . . . = a0 + a1 x + a2 x2 + a3 x3 + . . . (. 5.64.) 11.88. n- (1 - a)(1 - b)(1 - c)(1 - d) . . . = = 1 - a - b + ab - c + ac + bc - abc - d + . . . (n = 0, 1, 2, . . . )? (. 5.73.) 11.89. (1 - 4x)- 2 = 1 + 2x + 6x2 + 20x3 + . . . + an xn + . . . 11.90. x y x = y + y2 + y3 + . . . + yn + . . . y x. 11.91. x y x=y+
y2 y3 yn + + ... + + ... 2! 3! n!

1

y x. 11.92. C(z) =
n=0

Cn zn -- -

{Cn }. , C(z) = zC2 (z) + 1, C(z). (. 2.116.) 11.93. 2.115,


(1 - z)1/2 =
n=0

Cn (-z)n , 1/2

Cn = 1/2
(1/2)(1/2 - 1) . . . (1/2 - n + 1) n!

-- ( . 154).


166

11.

4.
. k l gk,l (x) gk,l (x) =
(1 - xl+1 )(1 - xl+2 ) . . . (1 - xl (1 - x)(1 - x2 ) . . . (1 - xk )
+k

)

. 4 ,

11.94. gk,l (x) 0 .
hk+l (x) , hk (x) § hl (x)

k+l

11.95. gk,l (x): ) gk,l (x) =

hm (x) = (1 - x)(1 - x2 ) . . . (1 - xm ) (h0 (x) = 1); ) gk,l (x) = gl,k (x); ) gk,l (x) = gk-1,l (x) + xk gk,l-1 (x) = gk,l-1 (x) + xl gk-1,l (x); ) gk,l+1 (x) = g0,l (x) + xg1,l (x) + . . . + xk gk,l (x); ) gk,l (x) -- x kl. gk,l (x) . . , , . 11.96. gk,l (x) x = 1. , gk,l (x) , x = 1. gk,l (1) = Ck+l . k , ) í ) 11.95 x = 1? 11.97. Sl (x) = g0,l (x) - g1
,l-1

(x) + g2,

l-2

(x) - . . . + (-1)l gl,0 (x).

11.98. Pk,l (n) n k , l. : ) Pk,l (n) - Pk,l-1 (n) = Pk-1,l (n - l); ) Pk,l (n) - Pk-1,l (n) = Pk,l-1 (n - k); ) Pk,l (n) = Pl,k (n); ) Pk,l (n) = Pl,k (kl - n). P 11.99. fk,l (x) -- k,l (n): fk,l (x) = Pk,l (0) + xPk,l (1) + . . . + xkl Pk,l (kl).


4. ) : fk,l (x) = fk-1,l (x) + xk fk,l-1 (x) = fk,l-1 (x) + xl fk-1,l (x).

167

) , fk,l (x) gk,l (x). 11.100. , k l gk,l (x) , xkl gk,l (1/x) = gk,l (x). : Pk,l (n) 11.98. 11.101. , Pkl (0) + Pkl (1) + Pkl (2) + . . . + Pkl (kl) = Ck+l , k .


12


12.1. , . , , . , n ( 1 n). 1 n - 1 . 2 n . 2 n - 1 -- , . n . , , ? (. 1.4.) 12.2. . :
9 - 25 9 25 =-; 6 + 10 6 10 8 - 50 8 50 =-. 2+5 2 5

? 12.3. 100, .
16 1 =. 64 4

12.4. log 16 +
16 15

= log 16 + log

16 ; 15

log

64 64 - 8 = log - log 8. 7 7

? 12.5. a b sin a + sin b = sin(a + b)?


169 12.6. 12 21 (144 441). ? : 441 ? 12.7. . , , . , (. 3.153). 12.8. , , . , . , ? 12.9. - 2.6. 12.10. x (x - a)(x - b)(x - c) . . . (x - z). 12.11. ë1 = -1¨. , , . :
1 = -1 -1 1 -1 = 1 1 = -1 -1 1 = -1. 1 -1 1

, : -1 = i2 = -1 § -1 = (-1)(-1) = 1 = 1. - ? (. 7.24.) 12.12. , . 7.51, , sin x , cos x -- : sin x = cos x =
e
ix

-e 2i

-ix

=
-ix

(e

2i x/(2)

)

e

ix

+e 2

=

(e

2i x/(2)

)

- (e-2i )x/(2) 1-1 = = 0; 2i 2i -2i x/(2) + (e ) 1+1 = = 1. 2 2

? (. 7.55.) 12.13. ë65 = 64 = 63¨. . ,


170

12.

, , , :

, ë64 = 63¨? (. 3.112.) 12.14. -- . 3.125 . , . , , 30 . 30 : 30 = 21 + 8 + 1 = F8 + F6 + F2 = (1010001)F . , F7 + F5 + F1 = 13 + 5 + 1 = 19 = (101001)F . -- 19 . ( -- 18,46 .) . , . , n n 100, 2/3 . 12.15. S : S = 1 - 1 + 1 - 1 + 1 - ... (12.1)

(12.1), , S: S = 1 - (1 - 1 + 1 - 1 + . . . ) = 1 - S S = . S (12.1) : S = (1 - 1) + (1 - 1) + . . . = 0 + 0 + . . . = 0; S = 1 - (1 - 1) - (1 - 1) - . . . = 1 - 0 - 0 - . . . = 1.
1 2


171 , , S: S = -1 + 1 - 1 + 1 - 1 + . . . = -1 + (1 - 1) + (1 - 1) + . . . = -1. , , S: 1 S = = 0 = 1 = -1.
2

S ?


, ,
1
1.15. 1.14. 1.16. an = 2n + 1 (n 0). 1.26. n = 1 . , n 1 n + 1. , . 1 2n + 2 n + 2 . 2n + 1 2n + 2, . n a1 , . . . , an . n + 1, n + 1 | 2n + 2. , a1 , . . . , an n + 1, . n + 1 1 2n, . : n + 1 , 2n, , -- 2. 1.27. x = 1, 2, . . . , n. 1.37. ë ¨ (. 1.4), n n - 1. n n = 2k . 1.40.
2 1 1+ . 3 n(n + 1)

1.42. 28 - 1 . 1.43. 38 - 1. 1.44. 2 § 37 - 1. 1.45. n , n + 3 ( ). n = 3k, n = 3k + 1, n = 3k + 2, n, .


, ,

173

n 6, -- 4, -- 8. ( 3 ½ 3, 2 ½ 2 5 ½ 5. 1.47. ) 2 (11 = 1 + 1 + 3 + 6). ) (n + 1)2n - 1 . 1.48. 8, 9 10 . , . 1.50. 1 + n(n + 1)/2. 1.51. n2 - n + 2. 1.53. (n3 + 5n + 6)/6. 1.57. k + mk - m. 1.58. . 1.59. . : . , , . 1.61. an n . , an = a a +
n-1 , 4

n -- ; n -- .

n-3

0,
100

= 24 + 23 + 21 + 20 + . . . + 3 + 2 = 208.

2
2.1. ) 24. ) 28. 2.2. 9 § 106 . 2.3. 103 § 303 . 2.4. 27/25 . 2.5. 65 § 84 + 64 § 85 . 2.6. 40. 2.7. 9 § 104 § 2. 2.8. 9 § 105 - 56 . 2.9. 9 § 109 - 9 § 9!. 2.11. 360. 2.12. 9 § 102 2.13. 9 § 107 § 5. 2.14. 37 . 2.15. 54 . 2.16. . : .


174

, ,

2.18. 38 2.19. n , 0 n . , n , -- 0. 2.20. k + 1 , . k + 2 , , k + 1 1 2k. 2 k - 1 . 2.21. 16 2 ½ 2. 16 , . : 16. 2.22. 50 . 2.23. N(k, l) N(k, r) -- k- . N(k, l) + N(k, r) = 200 (k = 41, 42, 43); N(41, l) + N(42, l) + N(43, l) = 300; N(41, r) + N(42, r) + N(43, r) = 300. , () (). , N(41, l) N(41, r), N(42, l) N(42, r), N(43, l) N(43, r).

N(41, l) + N(42, l) + N(43, r) = 300 - N(43, l) + N(43, r) 100.

2.24. . 2.19. 2.25. . m 6 m 10. m A6 , 6- . ( ) A5 , . . . , A1 . A1 . -- A2 , . . 2.27. 999. 2.28. , 1 2n - 2 (n 3) n + 1 , , . n = 3 . , n = k


, ,

175

n = k + 1. k + 1 1 2k - 2, . , 2k - 1 2k. k 1 2k - 2. (1, 2k - 2), (2, 2k - 3), . . . , (k - 1, k), , . 2k - 1. 1002 1001, . 1000, 1001, . . . , 2000. 2.30. n .
n(n - 1) . 2

(0, 1, . . . , n - 1) . 1 . 2.31. . . . . , . , . , . 2.33. 6- . 3 , , . . , . , . 2.34. n {1, 2, 4, 8, . . . , 2k {3, 6, 12, 24, . . . , 3 § .......... {2n - 1, (2n - 1)2, (2n - 1)4, . . , . . . }, 2k , . . . }, .. . , (2n - 1) § 2k , . . . }.

[n + 1; 2n]. , (n + 1)- . 2.36. 17!. 2.38. 8!. 2.39. 16!. 2.40. 16!/2. 2.41. 28 § 6! § 1111111. 2.42. ) 28!; ) 28! - 27 § 2 § 26!. 2.43. C4 . 9


176

, ,

2.44. C4 , C3 . 28 27 2.45. 2 § C7 + 1 § C6 . 10 10 2.46. C2 . n 2.47. C2 . n 2.48. C2 § C2 . m n 2.49. , 5 . . . C5 9 . 4 , . , 6 , . Cm-1 n - m + 1 n . 2.50. C5 § C5 . 9 7 2.54. ) 26; ) 51. 2.55. n = 2a - 1. 2.56. ) C2 - k. k 2.57. n. 2.58. ) 5!; ) 6!/2; ) 8!/3!; ) 11!/(2! § 3!); ) 11!/(4! § 2! § 2!); ) 13!/(2!)4 . 2.59. , m ëx¨ n ëy¨ : Ox, ëx¨, Oy, ëy¨.
(m + n)! = Cm m m! n!
+n

= Cn m

+n

.

2.60. , 9 x, y z. : 27!/(9!)3 .
(m + 1)(m + 2) 2 . 2 1 24! 1 2.62. ) C6 . ) § . 2 12 4! (6!)4

2.61.

2.63. A B C a, b c. 3n . 2.65.
32! . 10! § 10! § 10! § 2! § 2

2.66. 6- 9876543210. : C6 . 10


, ,

177

2.67. (m + 1)- (m - 1 ) n , . : Cn +1 . m 2.68. C5 ; Cn-1 . m-1 19 2.69. C5 . 25 2.70. ) C2 9 ; ) C2002 . 1 99 2.71. C5 . 22 2.72. 113 = C0 103 + C1 102 + C2 101 + C3 100 , 114 = 14641. 3 3 3 3 2.73. 27 . 2.74. . , , ( 2.88). Ck n ) n 11.69. 2.75. n = 2k - 1. 2.76. ) 35 ; ) 0; ) 2n . 2.77. ) : m r, m k. : k r, m - k. 2.79. 36 . 2.80. x = 2, y = 3, n = 5. 2.81. ë¨ . n C3 , . z C2 . y C1 . x 2.82. 3 C30 = 120. 8 , 1 8 § 14 = 112. 8 . 2.83. m = 3, = 2. n 2.84. C6 0 4( 3)64 . 10 2.86. 2 § 5! § 5!. 2.87. , . , . Tn Tn = C4 . Kn -- , n n- , Km
+1

- Km = Tm

+1

- Tm + m - 1.


178

, ,

m 2 n, , Kn Kn
+1 +1

= Tn

+1

+

n(n - 1) . 2

= C4 n

+1

+ C2 = n

n(n - 1)(n2 - n + 10) . 24

2.88. , , , . , Ck , n
1 . (n + 1)Ck n

1 1 1 + = k (n + 1)Ck-1 (n + 1)Ck n § Cn-1 n n -1

. 2.90. , (. 2.88)
1 1 1 - =, 6 12 12 1 . (r - 1)! § (r - 1) 1 1 1 - =, 12 20 30 1 1 1 - = , ... 20 30 60

2.77 ). 2.91. ) 2.92. C4 /C45 . 10 2 2.93. 5/90 = 1/18. 2.94. ) 1/103 ; ) 1/102 . 2.96. , . 2.97. a1 + a2 + . . . + ak . 2.100. 20. 2.101. Na -- , BC . Nb , Nc , Na,b , Nb,c , Na,c Na,b,c . N . N = 63 , Na = Nb = Nc = 62 , Na,b = = Nb,c = Na,c = 6, Na,b,c = 1. : 63 - 3 § 62 + 3 § 6 - 1 = 53 . 2.102. ) 13200; ) 8800; ) 8000.


, ,

179

2.103. 1600. 2.104. 998 910. 2.107. S -- , Si -- i- (i = 1, 2, 3), Si,j -- , i- j- (1 i < j 3), S1,2,3 -- , . S - (S1 + S2 + S3 ) + (S S
1,2 1,2

+S

1,3

+S

2,3

)-S

1,2,3

0.

+S

1,3

+S

2,3

S1 + S2 + S3 - S = 3.

S1,2 , S1,3 S2,3 1 2 . 2.108. 2.107. 2.109. ) : S- Si + S
i,j

-

S

i,j,k

+

S

i,j,k,l

-S

1,2,3,4,5

0.

(13.1)

, . , S1 - S
1,i

+

S

1,i,j

-

S

1,i,j,k

+S

1,2,3,4,5

0.

, Si - 2 S
i,j

+3

S

i,j,k

-4

S

i,j,k,l

+ 5S1,

2,3,4,5

0.

(13.2)

(13.1) (13.2), Si,j,k . , (13.2) (13.1): 3S - 2 S
i,j

Si +

S 2

i,j

-

S

i,j,k,l

+ 5S1

,2,3,4,5

0.

Si - 3S = 5 - 3 = 2.

i j Si,j 1/5. ) (13.1) (13.2), Si,j . 2.110.
12 24 123 145 13 25 124 234 14 34 125 235 15 35 134 245 23 45 135 345


180

, ,

. , 1 2 , . , 1/5 1/20 -- . 2.111. x0 § x1 § . . . § xn "(" +1; " § " -1 " § " +1; ")" -1.

2.112. x0 § x1 § . . . § xn , x0 , x1 , . . . , xn ( x0 § x1 § . . . § xn ). 2.113. 2.111. 2.115. , {a1 , . . . , an }, , : , . , . , 2.103 a0 = 1. 2.116. x0 § x1 § . . . § xn , .

3
3.1. p1 , p2 , . . . , pn -- , N = p1 ½ ½ p2 § . . . § pn + 1 ,x . 3.2. 2 19. 3.5. 2q2 = (p - 1)(p + 1). , p -- . q -- . q = 2. p = 3. 3.7. p1 = 3, p2 = 7, . . . , pn -- . N = 4p2 § . . . § pn + 3. p1 , p2 , . . . , pn , 4k + 3. 3.8. 3.7. N N = 6p1 § . . . § pn + 5.


, ,

181

3.9. a n b, a § b = n. a b n. 3.10. n -- . 3.9. 3.11. 111 = 3 § 37; 1111 = 11 § 101; 11111 = 41 § 271; 111111 = 3 § 7 § 11 ½ ½ 13 § 37; 1111111 = 239 § 4649. 3.12. 1001! + 2, 1001! + 3, . . . , 1001! + 1001. 3.14. ) 5, 11, 17, 23, 29; ) 7, 37, 67, 97, 127, 157. 3.15. a , a > 1. ak = a + kd k = ma a, . 3.17. 3. 3.18. 5. 3.20. n4 + 4 = n4 + 4n2 + 4 - 4n2 = (n2 - 2n - 2)(n2 + 2n - 2). 3.21. n = 40 n = 41. n = 0, 1, . . . , 39 P(n) . 3.23. p1 §2 § . . . § pn - 1. 3.24. 2 § 3 § 5 § 7 § 11 § 13 + 1 = 30031 = 59 § 509. 3.25. e5 = 1807 = 13 § 139. 3.26, 3.29. 6.78. n n+1 2n 3.27. 2n > n + 1, 2n+1 | 22 . 22 - 1 | 22 - 1. n+1 2n fn | 22 - 1 | 22 +1 - 2 = 2fn - 2. 3.28. 3.19. 3.30. x = 0 a0 . a0 = p -- . P(x) x1 = p, . x2 = p2 , x3 = p3 , . . . , P(xj ) . p. P(xj ) = p . (j = 1, 2, . . . ) P(x) , . : . 3.32. (m, n) + 1, m + n - (m, n). 3.33. OABC, , O(0; 0), A(q, 0), B(q, p), C(0, p). OB x + y = k (k = 0, 1, . . . , p + q), p + q . 3.34. 180 . 3.35. (a, b) (b, r) . . 3.38. (a, b) = 1, u v au + bv = 1. c, acu +


182

, ,

+ bcv = c, a . a , a | c. 3.39. m > n m = nq + r, r < n, ( 1 . . . 1 , 1 . . . 1 ) = ( 1 . . . 1 , 1 . . . 1 ).
m n n r

, , , m n , , , (m, n) . 3.40. 10. 3.41. 10. 3.42. 19 § 19 - 360 = 1 . 3.43. 500. 3.45. . 3.47. 20 , 9 . 3.49. , . 3.50. ) (n2 + 2n + 4, n2 + n + 3) = (n + 1, 3), , (n + 1, 3) > 1. : n = 3k - 1. ) (n3 - n2 - 3n, n2 - n + 3) = (n2 - n + 3, 6n), 2, 3, n. . , n = 3k n = 3k + 1. (n2 - n + 3, n) = (n, 3), n n = 3k. : n = 3k, n = 3k + 1. 3.51. ) , . ,
n4 + 1 n+3 = n2 - n + 2 . n +n+1 n +n+1
2

, n = -3, 0 < |n2 + n + 1| < |n + 3|. n 0 < n2 + n + 1, : n + 3 0 n + 3 < 0. , n2 2. n = -1, n = 0, n = 1 . : n = -3, -1, 0. ) n = 0, 1. 3.52. n = 2, 3. 3.53. 17. 3.54. ) m n (am - 1, an - 1) = m-n n = (a - 1, a - 1).


, ,

183

(m, n) 0. , 3.39. , 3.39 . , , a. ) fn+1 = f0 f1 . . . fn + 2. 3.55. . 3.57. 3.38. 3.58. 3.57. 3.64. , , Fn Fn+1. 3.66. 7 2. 3.68. ) . 3.73. 3.72, (x; y) N(x, y) = = ax + by. , N(x, y) = c (13.3)

(b; -a). , c, (13.3) n + 1 , nab + a + b. (xk ; yk ) = (1 + kb; 1 + (n - k)a) (0 k n).

, c, (13.3) n , (n - 1)ab + a + b. c, (13.3) n , (n - 1)ab + a + b c nab + a + b.

3.74. 4140.5. 3.76. 27. 3.77. 24. 3.78. 123. 3.81. 3.80, , . ) max(, min(, )) = ; ) min(, max(, )) = ; ) + + = max(, , ) + min( + , + , + ); ) + + = min(, , ) + max( + , + , + ). 3.84. ) 32; ) 3 § 4 § 6 § 8 § 12.


184

, ,

3.86. , n = p1 § . . . 1 . . . § ps s d = p1 . . . p 1 s
s

(0



1

1 , . . . , 0



s

s ).

3.87. n = 12. 3.88. ) 28; ) 160 169. 3.89. x = 6, y = 5, z = 4. 3.90. (n) (n) 3.86. 3.91. n a, b , a § b = , n n. 3.93. (m § n) < (m) § (n); (m § n) < (m) § (n). 3.94. (n) 3.86. 3.95. n n = 2k-1 b, b -- , k 2. (x) -- , (n) = (2k-1 )(b) = (2k - 1)(b). , (n) = 2n = 2k b, (2k - 1)(b) = 2k b. b = (2k - 1)c, (b) = 2k c = b + c.

c = 1, b b, c 1. (b) 1 + b + c, (b) = b + c. c = 1, (b) = b + 1, b = 2k - 1 -- . 3.29, k. n n = 2p-1 (2p - 1), p b = 2p - 1 -- . p = 2, 3, 5, 7 n = 6, 28, 496, 8128. 3.96. 220 284; 17296 18416. 3.98. 120. 3.100. , d. 3.102. .


, , 3.103, 3.104. p = (ak pk-1 + . . . + a1 ) + (ak pk-2 + . . . + a2 ) + . . . + ak = = ak (pk-1 + . . . + p + 1) + . . . + a1 = =
(n - ak - . . . - a1 - a0 ) (ak (pk - 1) + . . . + a0 (p0 - 1)) = . (p - 1) (p - 1)

185

3.107. . n n = 2k - 1 . 3.108. 377 . 3.109. an -- , n- . a1 = a2 = 1. , (n + 1)- n- , n- . an+1 = an-1 + an . an = Fn-1 . 3.110. 233 . 3.111. F0 = 0, F1 = 1 Fn+2 = Fn+1 + Fn : F-n = (-1)n+1 Fn . 3.112. . 3.113. . 3.114. : n- . 3.116. 1. 3.117.
F3 Fn+2 - . F 1 § F2 Fn § Fn+1

3.118. ), ), ) Fn 2, 3 4. ) 3.114. 3.119. F1 , F2 , . . . m. . 0 ( F0 m) . 3.122. ) 3.35. ) (Fm+n , Fm ) = (Fm , Fn ), 3.114. 3.123. ) 3.113 Fn + F
n+1

+ ... + F

n+7

=F

n+9

-F

n+2

.
n+8

, F - Fn+2 < Fn+9 .


n+9

-


186

, ,

3.124. . 3.109. . 3.125. n ë¨ : n Fm , n. 3.126. , Fn , , F0 = 0, F1 = 1 Fn+1 = Fn + Fn-1 . 3.127. , . 3.129. . 3.124 2.67. 3.130. Sn = 0, n 2, 5 (mod 6); Sn = 1, n 0, 1 (mod 6); Sn = -1, n 3, 4 (mod 6). : Sn = 3.131 3.134 3.135 3.136
sin (n + 1)/3 (n + 1) 2 = sin . sin /3 3 3

. Fn+1 . . Fn-2 + Fn = Ln . Ln = n + n . .
n

-1

.
5 5

Ln + F 2

n

- (-1)

kk

Lk - F 2

k

= 1.

3.137. ) (1, 0), (2, 1), (5, 3), . , (xn , yn ) = ‘ (F2n+1 , F2n ), n -- . (. 3.111.) , (F2n+1 , F2n ) , : F2n+1 - 2 - F2n F2n+2 = 1. (. 3.112.) , . , , (x, y), x 1 y 0. , x y y < x 2y. , (x, y) . . . (x - y, 2y - x) (x, y) (2x + y, x + y) . . .


, , : 0 < x = x - y < x, 0 y = 2y - x < y.

187

, y = 0, x = 1, (F1 , F0 ). . . . (F1 , F0 ) (F3 , F2 ) . . . (F
2n+1

, F2 n ) . . .

, (x, y) = (F2n+1 , F2n ) = (xn , yn ). ) (xn , yn ) = ‘(F2n , F2n-1 ), n -- . 3.138. ) Fn+5
21 F ,F 2n
n+3 k n

8Fn .

3.141. 3.142. 3.143. 3.144.

k-1 k-1 k k ) F = Fn-k+1 Fn-1 + Fk-1 Fn-1 = Fn-k-1 Fn-1 + Fk+1 Fn-1 . k , An Ak-1 Ak -1 . n-1 n ) [11; 3, 4]; ) [1; 6, 6].

F

n+1

F

.

n

3.147. 3.113, ). 3.149. ) k = 0, 1 . . , k. k + 1. k + 1 k- ak ak + 1/ak+1 : [a0 ; a1 , . . . , ak , ak+1 ] = [a0 ; a1 , . . . , ak + 1/ak+1 ]. [a0 ; a1 , . . . , ak ] = [a0 ; a1 , . . . , ak , ak+1 ] =
(ak + 1/ak+1 )P (ak + 1/ak+1 )Q
k-1 k-1

Pk aP =k Qk ak Q

k-1 k-1

+ Pk-2 , + Qk-2 + Pk-2 P = k+2 . + Qk-2 Qk+2

(13.4)

) . ) ) , Pk Qk . 3.151. ) 3.149. 3.152. ) xk = 4 + 13k, yk = 31 + 101k (k Z); ) xk = -6 + 19k, yk = -19 + 79k (k Z). 3.153. 400 ë¨ 400 ,
97 § 366 + 303 § 365 97 = 365 + 400 407


188

, ,

, 26 , . : 3323 . 3.157. 1, 2 3.146. 3.158. 128 . 8 3.159. [365; 4, 7, 1] = 365 . 33 , 33 , , . , 8 33 . 5000 . , - . : 1 = (365,24219879 - 0,0000000614 § (n - 1900)) , n -- 3.160. 3.161. 3.163. 3.164. . ) 33; ) 34; ) (1 + 17)/2. ) 2 = [1; 2]; ) 3 = [1; 1, 2]; ) 1/2 + 7 = [3; 6, 1, 6, 5]. 97/56. , = [a0 ; a1 , . . . , ak-1 , ] . 3.149 ) (13.4), . ak , =
pk-1 + pk-2 , qk-1 + qk-2

: 2 q
k-1

+ (q

k-2

- pk-1 ) - pk-2 = 0.

, -- . , = [b0 ; b1 , . . . , bm , ], .


, , 3.166. , pn =
(1 + n 2)
+1

189

- (1 - 22

n 2)

+1

p1 = 2, p2 = 5 pn+2 = 2pn+1 + pn , qn =
(1 + n 2) - (1 - 2)n 22

-- q1 = 1, q2 = 2 qn+2 = 2qn+1 + qn . , pn /qn -- n- [2] = 1 + 2. 3.167. , -- pn /qn -- . , n qn q qn+1 . n p/q n- . , . , , p/q (-; pn /qn ), (pn /qn ; pn+1 /qn+1 ), (pn+1 /qn+1 ; ) ( , pn /qn < pn+1 /qn+1 ).
1 p p p > - > -n q q qn 2q2 1 qq
n

q :
1 qn qn

n

> 2q, n.
pn+1 p - qn+1 q

=
+1

pn+1 p - n= qn+1 qn

+

p p -n q qn

1 q qn

+
+1

1 , q qn

q qn + qn+1 , n.
1 p p 1 p > - > - n+1 , q q qn+1 q qn+1 2q2 1 p 1 pn 1 pn p - - + - +2 qqn qn q qn q qn qn+1 2q q q q q qn+1 > 2q 1 + n . 1 < + qn+1 2q 2q 2q p , , = q a = , b p a = q b a p 1 b

= 1.
pn . qn



-

q

bq

.


190

, ,

3.169. , p/q p 1 2- < 2.
q 3q

3.167, p/q p/q = Pn /Qn . 2-
|P P p P = lim n+k - n = lim q Qn k Qn+k k
n+k



2.

Q n - Pn Q Qn Qn+k

n+k

|

.

an = |Pn+k Qn - Pn Qn+k |. a0 = 0, a1 = 1 ak = 2ak-1 + ak-2 (k 2). an 2: ak = Qk-1 (k 1), d=
Qk-1 1 QQ = 2 § n k-1 . Qn Qn+k Qn+k Qn

,
Qn Qk-1 Qn+k 1 . 3

Qn . , , 3.114: Qn+k = Qk-1 Qn+1 - Qk-2 Qn .
Qn Qk-1 = Qn+k Q
k-1

Qn Qk-1 Qn+1 - Q

k-2

Q

=
n

Q

n+1

1 /Qn + Q

k-2

/Q

.
k-1

,
Qn+1 Qn
Fk+1

5 2



Q Q

k-2 k-1

1 . 2
Fk+2

3.170. a a - 1. 3.171.
ab + a2 b2 + 4ab = [a; b]. 2b

-1

3.173. = [a0 ; . . . , an ] f(x) = = qn x2 + (qn-1 - pn )x - pn-1 . . f(0) = -pn-1 < 0 f(-1) = pn - pn-1 > 0, (-1; 0).


, ,

191

4
4.3. a b -- , a2 + b2 = c2 , c2 2 4. 4.5. . . : . 4.12. . 4.18. , . 4.19. . 4.20. , , A B . A, C B, C. : B. 4.21. 1.

. 1.

. 2.

, , 4.20, , , A. - ( . 2). , A 5 , . 4.22. . , ( ). , , . , -- 4. , 0111. . 4.23. p2 - 1 = (p - 1)(p + 1). p - 1 p + 1 2, 4. 4.24. 111 = 3 § 37.

â á âá á â âá â

á â á â á

â á â âá â á á

á á â âá âá â

â á â á â

á â âá á â âá á


192

, ,

4.25. ) 8 11 . . . 1 (n ) n = 1, 2, 3, 6, 331, 662, 993. ) 111 111 = 1001 § 111 = 3 § 7 § 11 § 13 § 37. . . . . + 1 . 2k + 1, 23k - 1 . 23 - 1.

4.26. 23 4.28.

k

1 1 p + = . k p-k k(p - k)

4.29. m/n = 5/2. 4.30. c = 6k - a - b 2 3. 4.31. , 11n . 4.32. b c -- . : (b, c) = (8, 17), (35, 40), (36, 39) (112, 113). : 4. 4.33. ) (x, y) = (‘5, ‘9), (‘10, ‘3). ) (1 + x)(1 + x2 ) = 2y . : (x, y) = (0, 0), (1, 2). . 4.34. k1999 + (17 - k)1999 . 17. . 4.35. . , (, 765765). . abcabc = abc § 1001 = abc § 7 § 11 § 13, 13, , , 13 . abcdef (abc = def). defabc. , . 13, abcdef + defabc = (abc + def) § 1001 = (abc + def) § 7 § 11 § 13. 13 . 4.36. , 9. 4.39. ) , 2004 § 4 10 = 1 + 2 + 3 + 4. ) . 401 (1, 2, 3, 4) (4, 3, 2, 1). 4.42. Ck = p
p! . 0 < k < p, (k!(p - k)!, p) = 1, k!(p - k)!

p .


, ,

193

4.43. n p -- . n n = p m, (m, p) = 1. (. 3.101) , p Cp n - 1, n Cp . n 4.45. 4.42. 4.46. . , , 5. 4.47. , a1 , a1 + a2 , . . . , a1 + a2 + . . . . . . + an n, n. 4.48. , 10 , 2, 3, 5, 7. . , 5 2. 2 3, -- 5 7. 9 . 4.49. 1, 2, . . . , 99 50 49 . , . 4.50. ) . ) a b. 4.51. (a + c) - (b + d) = (a - c) + (b - d), ac - bd = c(a - b) + b(c - d). ï 4.52. , a b, ï b a (mod m) b - a = mt. , a mt + a. ï ï 4.53. a = b. c, . , t1 t2 c = a + mt1 , b = a + mt2 . a - b = m(t2 - t1 ), b a (mod m). 4.54. m. 4.55. (a, m) = 1. 4.57. (m, c) = 1. 4.59. , , , 1 6. 4.62. x , -- 2x, 3. 4.64. 5 8, .


194

, ,

4.68. b § 105 + a 10a + b. 7 5b + a 3a + b. 5b + a 5(3a + b) (mod 7). 4.69. , 3. p = 3. . . 4.70. p = 3, 8p2 + 1 . 3. 4.71. , 4.69, 4.70, 3. 4.72. 5. 5, , , 4. : 5, 11, 17, 23, 29. 4.73. 3. 4.74. n2 3 0, 1, . 4.75. a2 + b2 0 (mod 3) a b 3. m = 7 4.78. 24 -1 (mod 17). 4.79. . en 5. 4.80. a2 + b2 = c2 . 4.81. ) m + 1 3 (mod 4). ) m - 1 2 (mod 3). . . 4.82. an . 3 n = 2 + 3k; an . 4 n = 2k (k Z). . . 4.83. ), ), ) n; ) n = 3 + 11k (k Z). 4.84. ) n = -8 + 17k (k Z); ) . 4.85. x = 17 + 73 k (k Z). 4.91. , x P(x) 1 (mod 2). , P(x) . 4.93. ) x 2 (mod 13); ) x 24 (mod 37); ) x 5 (mod 11); ) x 15 (mod 169). 4.94. 1652 6125. 4.96. a ‘1 (mod p). 4.97. 2 p - 2 , a b ( a 4.96) , ab 1 (mod p). (p - 1)! p - 1 (mod p). 4.98. n -- . p -- n (p < n), (n - 1)! 0 (mod p), (n - 1)! -1 (mod p). 4.99. (p - 1)! + 1 = (p - 2)!(p - 1) + 1 = (p - 2)!p - ((p - 2)! - 1).


, ,

195

4.100. p -- , 4((p - 1)! + 1) + + p 0 (mod p). , p + 2 -- , 4((p - 1)! + 1) + p 0 (mod p + 2). p -2 (mod p + 2) p + 1. p(p + 1) -2(p + 1) = -2((p + 2) - 1) 2 (mod p + 2). 2(p - 1)! : 2(p + 1)! 4 (p - 1)! (mod p + 2). p + 4: 2((p + 1)! + 1) + (p + 2) 4((p - 1)! + 1) + p (mod p + 2). p + 2 -- , (p + 1)! + 1 0 (mod p + 2) , , 2((p + 1)! + 1) + (p + 2) 0 (mod p + 2), 4((p - 1)! + 1) + p 0 (mod p + 2). 4.101. , a1 a2 + a2 a3 + . . . + an-1 an + + an a1 0 (mod 4). aj (j = 1, . . . , n). +1, n 0 (mod 4). 4.102. m, ) m = 4; ) m = 3; ) m = 7; ) m = 8; ) m = 5; ) m = 5; ) m = 16; ) m = 13. 4.103. 5: (n - 2)2 + (n - 1)2 + n2 + (n + 1)2 + (n + 2)2 = 5(n2 + 2). n2 + 2 5, 5(n2 + 2) . 4.104. 1, 2, . . . , n , . . . 4.105. , , , 10. 4.106. x = 1, y = 0. 4.107. 3.38. 4.108. 6. 4.111. 10p-1 - 1 = 99 . . . 9 0 (mod p). 4.115. a = 0 . , a 0. 4.42


196

, ,

(a + 1)p ap + 1 (mod p). , , .
ap - a , p p a -a . : + a. p

4.117. a

4.118. ) 1; ) 9. 4.119. 31. 4.120. 1093. 4.122. q -- 2p - 1. 2p - 1 0 (mod q) 2q-1 - 1 0 (mod q) ( -- ). 3.54 ). 4.123. n16 - 1 = (n8 - 1)(n8 + 1). 4.129. 3.127, , Fp 5(p-1)/2 (mod p), 2Fp+1 1 + 5(p-1)/2 (mod p). , 5(p-1)/2 1 (mod p) = 10k ‘ 1 5(p-1)/2 -1 (mod p) = 10k ‘ 3. 4.130. x = 1 xp-1 - 1 0 (mod p) 3 x - 1 0 (mod p). 3.54. 4.131. xp-1 -1 0 (mod p) x5 -1 0 (mod p). 4.132. ) 16; ) p - 1; ) p(p - 1); ) (p ) , p. 1 p p . (p ) = p-1 (p - 1). 4.133. p . 4.134. , b, (b) . (a) , a. (. 4.55.) 4.135. (m). 4.136. (a, m) = 1 b m. 4.139. ) x = 3, 4, 6; ) x = 15, 30, 20, 24, 16; ) 36; ) . 4.140. , (m) = 2. 1 5 (mod 4), 3 5. 4.141. ) x = 2 ; ) x = 21 32 (1 , 2 1); ) . 4.142. ) n; ) n; ) n. 4.143. ) x = 3; ) x = 3; ) x = 2, y = 3. 4.144. p, m n, (m § n) 1 - 1/p, (m) § (n) -- (1 - 1/p)2 . (1 - 1/p) < 1, (m, n) = 1 (m § n) > (m) § (n). 4.145. 8, 12.


, , 4.146. . n -- , k = n/2 4.147. S(n) -- n>1
1 d = n 2

197 k/n, (n - k)/n. k/n = (n - k)/n , k/n n/2. . , S(1) = 1. =
1 2

(d,n)=1 1dn

(d,n)=1 1dn

n-d d + n n

1=
(d,n)=1 1dn

(n) . 2

: S(1) = 1, S(n) = (n)/2 n > 1. 4.148. (d). 4.149. 4.148. , n . , d, (d) . 4.150. (n)/2. 4.151. ) , , m n . m = p , n = p ( 0). [m, n] = m = p , (m, n) = n = p . ) , (p
+

) (p ) = (p ) (p ) p .

, , (p ) = p (p - 1). (. 4.132.) 4 4.152. 3(10 ) - 1 0 (mod 104 ). 4.155. 2 3 . p . 4.156. -a(m)-1 b. 4.158. n , , (m). . . 4.159. n = p1 . . . ps . , 2n! - 1 . pj j 1 s j = 1, . . . , s. mj = pj j . 4.160. 561 = 3 § 11 § 17, a560 1 (mod p), p p = 3, 11, 17. . 4.161. , a, (a, 10) = 1. (10) = 4, a10 + 1 0 (mod 10) a2 +1 0 (mod 10).


198

, ,

a = ‘1, a = ‘3, , a ‘3 (mod 10). : a ‘3 (mod 10). 4.162. , (a, m) = 1 a
(m )

1 (mod pj j ) (j = 1, . . . , s).



4.166. 3993, 3597, 6797. 4.167. 8, 9 11. : 1 380 456. 4.169. . , 1 + 2 + . . . + 500. 4.170. 9 11 8079 . . . 2019. 4.173. 11 111 111 100. 4.175. 9. 4.179. 11. 4.180. 3. 4.181. . 9. 4.182. 10a + b 0 (mod 19) a + 2b 0 (mod 19) . 4.184. xxyy 11. , x0y 11. : 882 = 7744. 4.185. 12, , 3. 3, 3. 9. 4. , 36. 27, 27§12 = 324. , , 18 ( 9 27). 18 § 12 = 216. 108, 144, 180, 216. : 108. 4.187. ) : , 6. ) : 1, , 6. 55, . 4.188. 9. 4.189. aj 10j aj rj (mod m), M N (mod m). M N m . 4.191. an qn + . . . + a1 q + a0 an + . . . + a1 + a0 (mod m)


, , an (qn - 1) + . . . + a1 (q - 1) 0 (mod m),

199

ai , q - 1 0 (mod m). , m = 2 . : q = 1 + mk (k 1). 4.192. 21. 4.194. 5 11. : n = = 6 + 55k, k Z. 4.195. ) 1910 66 r. 19 1 (mod 2), 19 1 (mod 3), 19 -2 (mod 11), , r 1 (mod 2), r 1 (mod 3), r (-2)10 (mod 11). , (-2)10 1 (mod 11). r = 1. ) 11; ) 17; ) 36. 4.197. . 4.198. x = a1 x1 + . . . + an xn , xj = 4.200 4.201 4.203 4.204 4.207 . . . . .
m1 . . . m mj
n (mj )

(j = 1, . . . , n).

) x = 58 + 85k (k Z); ) x = 78 + 13 § 19k (k Z). 2 § 3 § 5 § 7 - 1. 2 § 10249 . 15803. n = 2.
c a b = + m1 § m2 m1 m2

c = a § m1 + b § m2 . , , c a § m1 + b § m2 (mod m1 § m2 ) 4.196. n 4.208. 45486. 4.209. 215 § 310 § 56 . 4.210. ) x(x - 1) 0 (mod 24 ) x(x - 1) 0 (mod 54 ). x 0, 1 (mod 24 ) x 0, 1 (mod 54 ). 104 4 , -- x = 0 x = 1. -- x = 0625 x = 9376, . : x = 9376. ) . , 0 1 ë¨ x1 = . . . 8212890625, x2 = . . . 1787109376. 4.211. m1 = p2 , . . . 1 . . . , m37 = p2 , p1 , . . . , p37 -- . 37


200 ??.
(p - 1)! + 1 (p - 1)! - (p - 1) = -1 p p

, ,

5
5.1. ) 1/7 = 0,(142857); ) 2/7 = 0,(285714); ) 1/14 = 0,(714285); ) 1/17 = 0,(0588235294117647). 5.2. a = 0, b = 0 a = 1, b = 3. 5.3. 1/49 = 0,(020408163265306122448979591836734693877551). 2/102 + 4/104 + . . . , 1/49. 5.4. 1/243 = 0,(004115226337448559670781893). 5.5. ) 15926/111111 = 0,(143334); ) 4/27 = 0,(148); ) 14/99 = = 0,(14). 5.7. n = 2 § 5 . 5.9. , , . 5.11. n = 21 § 51 , n + 1 = 22 § 52 . n = 1 n = 4. 5.12. . 5.16. ) . ) 2 + (- 2) = 0. p ) ( 3) = 2. = ,
q

2 = 2 . , . , 2 = 2 2. 5.17. x = 1 + 3, (4 + a + b) + (a + 2) 3 = 0. 3 -- , 4 + a + b = 0 + 2 = 0 : a = -2, b = 2. a. 5.18. a, b, c , q p q( b - a) = p( c - b) b(p + q) = p c + q a. , ac -- . , a c -- . 5.19. 2 + . 6 5.21. ) 9 = 100 - 1; ) 1; ) -10. 5.22. 4; 2; ) 6. ) ) 5.23. 2 + 3 + 5. 5.24. .

3p = 4q .


, ,

201

5.25. : b1 , . . . , bn -- , a1 , . . . , an -- , , (13.5) b1 a1 + . . . + bn an = 0. a1 = 1, a2 + . . . + an . 13.5 p1 , . . . , pm , a1 , . . . , an . 5.26. a b , a = dm b = dn d, m n. 5.28. . , , . 5.29. , , , , tg 60 = 3 Q. / 5.31. (x; y) (u; v). (x - 2)2 + (y - 3)2 = (u - 2)2 + (v - 3)2 , 2(u - x) 2 + 2(v - y) 3 = u2 + v2 - x2 - y2 . . 5.32. ) 6.82 ). 5.33. n. 5.36. 2.33, . k = 1 . . , k - 1, k. N . N-1 N1 = ,
k

, x -- n , x n. N1 N1
[k ! e ] k

>

[k ! e ] k! e = = [(k - 1)! e]. k k

(. 3.100.) - N1 , . , , k - 1 . N1 > [(k - 1)! e], ,


202

, ,

, . 5.38. 0,(a1 . . . an ) = , a1 a2 . . . an =
a1 a2 . . . a n . 10n - 1

10n - 1 1 , = 0,(a1 a2 . . . an ). m m

5.39. 10k(m) - 1 0 (mod m) k 1. 9 § Ek(m) 0 (mod m). m 3, Ek(m) 0 (mod m). m 3 9, m E3k(m) E9k(m) . 5.40.
p q

= 0,a1 a2 . . . ak a

k+1

. . . ,

10k p q

= 0,a

k+1

...

5.41. r0 = 1, r1 = 10 - m[10/m]. . . -- , 1 m . rk 10k (mod m), 10. (m, 10) = 1, 10(m) 1 (mod m), r(m) = r0 = 1. , , (m). 5.42. 11, 33 99 -- 99, 9. 5.43. 10t 1 (mod n), r0 rt (. 5.38) , r0 = m rt 10t m (mod n). m/n t. , T -- , rT = r0 ( 5.41) r0 10T r0 (mod n). r0 = m (m, n) = 1, r0 . , 10T 1 (mod n). 5.45. 5.38. 5.46. t = 2n -- . 5.43, 10t 1 (mod q). 10n -1 (mod q)
p + q 10n p q

= 1.
p = 0,N1 N2, q 10n p = 0,N2 N1, q

N1 + N2 = 99 . . . 9.
n

5.47. 1/7 : r0 = 1, r1 = 3, r2 = 2, r3 = 6,
r0 r = 1, 7 7

r4 = 4,

r5 = 5,

r6 = 1,

...

2§
r0 r = 2, 7 7

3§

4§

r0 r = 4, 7 7

...


, ,

203

, 1/7 + + 2/7 + 4/7 = 1 . , N N = = (106 - 1)/7. N2 = (106 - 1)2 /49. , N2 ,

k=1

N2 1 106 - 1 142857 = N2 6 = = . 6k 49 7 10 10 - 1 142857 7 1 = 0,(142857), 7



=

N2 N. 5.48. . 5.41. 5.51. abcdef = 3 § fabcde. , abcdef: = 0,(fabcde) = 10 § = f,(abcdef) = f + 3 § . = . f. 7 5.52. , . 5.56. 81 + 9 + 1 = 61 + 27 + 3 5.57. . . 5.58. 2n - 1. 5.59. ) 24 - 1 = 15; ) (34 - 1)/2 = 40. 5.60. 1, 3, 9 27 . 5.61. ) , -- . 30 . , . ) 24 - 1. 5.62. ) -- , . ) 9. 5.63. ) n = 2k1 + 2k2 + . . . + 2km (k1 > k2 > . . . > km 0), k1 + m. 5.65. b(15) = 6, l(15) = 5: x1 = x2 , x2 = x1 § x = x3 , x3 = x1 § x2 = x5 , x4 = x2 = x10 , x5 = x3 § x4 = x15 . 3 l(63) = 8 < 10 = b(63).
f 1 § 0,(abcdef). 3


204

, ,

5.66. . , 23, 1, 2, 4 16. 5.67. . 5.71. A , A m = A/2 ë¨ . A , a0 = ‘1 a1 ; A - a0 4 A m = (A - a0 )/4 ë¨ a0 . A m. 5.72. 0 1 0, 1 2. , ) -- , 1. 5.73. ) 1; ) ; ) n- 2 (n) ( n). 5.74. ë ¨ 0 7. 1 n, , . k- , , n- ë ¨ k. 5.75. ) 1, 2, . . . , 2n, . 1, 3, 5, . . . , 2n - 1 . k- 2k - 1. , , J(n), 2J(n) - 1. 5.76. ) n = (ns . . . n1 n0 )2 , ns = 1. m1 , m2 , . . . , ml s- . mj -- , mj n < mj . 5.77. ) n 0 m1 , m2 , . . . , ml , n. , . ) 5.76 ), j (1 j l) mj n < mj . j- mj -(mj n) , -. ) , n = 0. -- . , ë¨, . ) 1, 4, 5.


, ,

205

5.78. , f(A) = 3, f(B) = 5, f(A) = 6. , - , 5 = f(B). 5.80. ) ë¨ ë¨ 4 . , , , : 2, 5, 1, 4 (2 , 5 -- , 1 -- 4 -- . 5.81. ) m1 , m2 , . . . , ml , r1 , r2 , . . . , rl -- m1 , m2 , . . . , ml 6. n = r1 r2 . . . r
l

-- - 6. n = 0, ; -- . m1 = m2 = . . . = ml
-1

= 1,

ml -- .

(13.6)

( .) : 13.6, l , , - n 1; l rl = 1, , . , n = 0. , . 5.83. 1- 2- , 1- 4-. 5.84. -, : 001, 010, 011, 012, 112, 120, 121, 122, 200, 201, 202, 220. , 0 ( 001, 010, 011, 012), - , 2 (200, 201, 202, 220). ë0¨, 0. ë2¨-- 2. -- 1. 001, 200, 201, 202 ( , 0), -- 120, 121, 122, 220 ( , 2). , . 010, 020, 200, 220 012, 112, 122, 202 (, ) . -- , . :


206

, ,

- , . , , . - . . 5.85. 13- 111 . 5.84.

6
6.1. ) - ; ), )
p q p q
2

-

2 ; q

) -p(p2 - 3q).
p2 - 2q 1 y + 2 = 0; q2 q

6.2. b2 - abp + bp2 + a2 q - 2bq - apq + q2 . 6.3. ) y2 + p(p2 - 3p)y + q ) y2 +
p(p + 1) (q + 1)2 y+ = 0; q q
3

= 0; ) y2 -

) y2 +

2q - p2 y + 1 = 0. q

6.5. -- 18. a = -3. 6.7. . : -5/2, 3/2. 6.8. (p, q) = (0, 0), (1, -2), (-1/2, -1/2). 6.9. . : p = 2/3, q = -8/3. 6.10. ) a ( - , - 5 - 2 6) ( - 5 + 2 6, 0) (0, + ) ) a = 1, a = 3. 6.11. y = -1/8. 6.12. D(0; 1). 6.13. b = 1/10. 6.14. x = 1. 6.15. . : a = 2. 6.17. , y = 4x - 2x2 . : y < 4x - 2x2 . 6.18. y x2 - x 6.20. ) 10.7. ) 10.12. 6.21. , , . 6.28. , = 1 . a (-2 - 11; x -2 + 11). 6.30. -
1+ 5 1+ 5 , . 2 2


, , 6.31 6.33 6.34 6.35 6.37 . . . . . a (16/17; 2). a [-1; 1] {3}. m = 0. r (-; 5/2) (4; 9/2). x, f(a) = -a2 + a(4 - 2x2 - x3 ) + (2x3 + x2 - 6x + 5)

207

a [-1; 2] . , . . x, f(a) 0 a [-1; 2] ()

f(a) -- , , () f(-1) 0, f(2) 0. x(x - 1)(x + 2) 0, (x - 1)(x + 3) 0. , x (-, : : x (-, -2) (0, 1) (1, 6.38. P(x) x [-2, 0] {1}. , -2) (0, 1) (1, +). +). - c r:

P(x) = (x - c)T (x) + r. x = c, r = P(c). 6.40. . 6.42. Q(x) = (x - x1 )(-x - x1 ) . . . (x - xn )(-x - xn ) = (x2 - x) . . . (x2 - x), 1 n Q(x) x Q( x) -- n. Q( x2 ) = P(xk )P(-xk ) = 0, k x2 , x2 , . . . , x2 Q( x). 1 2 n 6.43. ) x4 - 4x3 + 6x2 - 3x + 1 = (x2 - x + 1)(x2 - 3x + 2) + 2x - 1; ) 2x3 + 2x2 + x + 6 = (x2 + 2x + 1)(2x - 2) + 3x + 8; ) x4 + 1 = (x5 + 1) § 0 + x4 + 1. 6.44. , P(-2) = 3. 6.45. , P(x) x + 1 P(-1) = a + 10. 0 a = -10.


208

, ,

6.46. ) P(1) = 5. ) . P(x) = (x2 - 1)T (x) + ax + b x = 1, x = -1, a = 5, b = 0. : 5x. 6.47. , P(0) = 0, P(-1) = 0, P(-1/2) = 0. 6.48. R(x) P(x) (x - 1)(x - 2) R(x) = ax + b. P(1) = 2, P(2) = 1. a = -1, b = 3. : R(x) = 3 - x. 6.49. k = -3. 6.50. , n
x2n - 1 xn - 1 xn + 1 : = 2 x-1 x+1 x -1

x. , , -1 xn + 1. n -- . 6.51. n. 6.52. ) P(1) = 1. )
P(1) - P(-1) 1 - 517 7 = 2 2
17

.

,
P(1) + P(-1) 1 + 517 7 = 2 2
17

.

6.53. P(x) x2 - 3x + 2 = (x - 1)(x - 2) P(1) = 0 P(2) = 0. (a + 1)(b + 1) = 0. a = -1, b = -1. a = -1, b = 31/28. b = -1 a = 31/28. : (-1, 31/28), (31/28, -1). 6.55. 6.46, R(x) = = [x(1 - (-1)n ) + 7 + (-1)n ]/2. 6.56. -- 1, 2, 3. 6.57. a = - 4. 6.58. 4 + 1 = (x2 - i)(x2 + i) = (x2 - 2x + 1)(x2 + 2x + 1) = x = (x2 - 2ix - 1)(x2 + 2ix - 1). -- .


, ,

209

6.59. P ) = 0 a3 - 4a + 3 = 0. (1 -1 ‘ 13 . : a = 1,
2

6.63. f(-x). 6.69. 3.54. : x 6.70. Pn (x) = P(P(P . . . (P(x)))).
n

(m,n)

- 1.

, am = Pm (a0 ) am = = Pm-k (ak ) m k. Pn (x) = an + x Qn (x), Qn (x) -- , m k (am , ak ) = (P
m-k

(ak ), ak ) = (a

m-k

+ ak Qn (ak ), ak ) = (a

m-k

, ak ).

3.39 6.71. x = 1. 6.72. p = 3. 6.73. P(x) = -
(x - 1)(x2 + 1) 1 , Q(x) = . 2 2

6.74. P(x) = ax + b, Q(x) = cx + d. , , (a + c)x3 + (-3a + b + c + d)x2 + (2a - 3b + c + d)x + 2b + d = 21. , a + c = 0, - 3a + b + c + d = 0, 2a - 3b + c + d = 0, 2b + d = 21. a = 4, b = 5, c = -4 d = 11, P(x) = 4x + 5, Q(x) = = -4x + 11. 4 5 4 11 6.75. P(x) = x + , Q(x) = - x + .
21 21 2n + 1 1 1 6.76. =+ . n(n + 1) n n+1 21 21

6.78. . 6.80. 1 + 4(x + 1) - 3(x + 1)2 - 2(x + 1)3 + (x + 1)4 . 6.81. P(x + 3) = 55 + 81x + 45x2 + 11x3 + x4 . 6.82. ) (2-2x+x2 )(2+2x+x2 ); ) (-1+2x)(1+x+x2 ); ) (1+x+x2 )½ ½ (1 - x + x3 - x4 + x5 - x7 + x8 ); ) (a + b + c)(a2 - ab + b2 - ac - bc + c2 ); ) (x + y - 1)(1 + x + x2 + y - xy + y2 ); ) (1 + x - y + xy)(1 - x + y + xy);


210

, ,

) 3(a+b)(a+c)(b+c); ) -5(x-y)(x-z)(y-z)(x2 -xy+y2 -xz-yz+z2 ); ) (a4-a3 b+a2 b2-ab3+b4 )(a4+a3 b+a2 b2+ab3+b4 ); ) (1+x)2 (1+3x+x2 ); ) (a - b - c)(a + b - c)(a - b + c)(a + b + c); ) (2 + x)(6 + x)(10 + 8x + x2 ). 6.83. x4 + x3 + x2 + x + 12 = (3 - 2x + x2 )(4 + 3x + x2 ). 6.84. , p2 - 4q < 0, x2 + q/x2 + p t = x + q/x . 6.85. 5/3(a2 + b2 + c2 + ab + ac + bc). 6.86. , (x + y + z)3 - x3 - y3 - z3 = 3(x + y)(y + z)(x + z). , (x + y + z)m - xm - ym - zm x + y. , , y = -x. , (x - x + z)m - xm - (-x)m - zm = 0. x + z y + z . 6.87. . 6.88. a + b = 0, . , a + b. , a + c b + c. , . 6.89. c = -a - b. 6.91. 6.90, x2 - - 17 = 0 ‘1 ‘17. . x2 - 17 = 0 . 6.92. , = cos 20 4x3 - 3x = 1/2, . 6.93. ) x = 1, 3, -2; ) x = -1, 3. 6.94. ) x4 + x3 - 3a2 x2 - 2a2 x + 2a4 = (2a2 - x2 )(a2 - x - x2 ); ) x3 - 3x - a3 - a-3 = (a + 1/a - x)(x2 + x(a + 1/a) + a2 + 1/a2 - 1). 6.97. ) x2 - x - 2; ) x2 - 1. 6.98. , (P(x), P (x)) = 1. 6.99. A = n, B = -n - 1. 6.106. , , 3.142. 6.107. ) 1 2 - 3 ; ) 1 (2 - 32 ); ) 1 (2 - 32 ); ) 41 2 3 + 1 1 + 2 2 - 23 - 2 - 23 3 ; ) 2 - 22 ; ) 4 + 41 3 + 22 - 42 2 . 12 2 3 1 1 1 2 1 6.108. 2. 6.109. (a - x)(a - y)(a - z) = a3 - a2 1 + + a2 - 3 .


, ,

211

6.110. (0, 0, a), (0, a, 0), (a, 0, 0). 6.111. a = -9. 6.112. x3 - 5x2 + 6x - 1 = 0. 6.113. y3 - y2 - 2y - 1 = 0. 2 1 6.114. c = - a3 + ab. 27 3 6.115. , x1 , x2 , x3 , , (x1 + x2 - x3 )(x1 - x2 + x3 )(-x1 + x2 + x3 ) > 0. p, q r, p3 - 4pq + + 8r > 0. 6.116. ) , x, y u, v . , x, y u, v . 6.118. x4 - ax3 ; x4 - ax3 - x + a; x4 - x3 + x - 1; x4 + x. 6.119. 6.107 ). 6.120. (c + d)(b + c + d) = ad. 6.121. b = 0, a < 0. 6.123. (x - 1)(y - 1) + (u - 1)(v - 1) = 2. 6.124. , : . 6.125. x = a, x = b, x = c. 6.127. fi (x) =
(x - x1 ) . . . (x - xi-1 )(x - xi+1 ) . . . (x - xn ) . (xi - x1 ) . . . (xi - xi-1 )(xi - xi+1 ) . . . (xi - xn )

6.128. f(x) = 1. 6.129. f(x) = y1 f1 (x) + . . . + yn fn (x). , n n + 1 , . 6.130. R(x) 2, R(a) = A, R(b) = B, R(c) = C.

6.129: R(x) = A
(x - c)(x - a) (x - a)(x - b) (x - b)(x - c) +B +C . (a - b)(a - c) (b - c)(b - a) (c - a)(c - b)

6.131. f(xi ) = yi . 6.132. ) 1 + (3 - x)x; ) 1 + (1 + x)2 ; ) x2 .


212

, ,

6.133. 1 17 . 6.134. 2 14. 6.135. . 6.136, 6.137. a, b c. 6.138. f(x) -- , f(-x) . (. 6.129) f(x) = f(-x). 6.139. P(x) , P(0) = 1, . . . , P(n) = 3n . , P(n + 1) < 3n+1 . 6.140. f(x) = f(x1 )
(x - x2 ) . . . (x - xn ) (x - x1 ) . . . (x - xn-1 ) + . . . + f(xn ) . (x1 - x2 ) . . . (x1 - xn ) (xn - x1 ) . . . (xn - xn-1 )

6.141. f() =
( - a1 ) . . . ( - an ) x1 xn + ... + =1- . - b1 - bn ( - b1 ) . . . ( - bn )

7
7.4. ) . ) . ) , . 7.5. ) 2 cos + i sin ; 4 4 ) 2 + 3 cos + i sin = 2 cos cos + i sin ;
) 2 cos 2 12 cos + i sin 2 ) (1/ 2) cos + i sin 4 4 12 ; 2 12 12 12

;

) cos 2 + i sin 2. 7.7. 13 - 1. 7.8. ) Re x < 0; ) 0 < arg z < ; ) | Re z| < 2; ) |z| < 1 Im z
2
2 2 2

0.

7.9. (x + 5/3) + (y - 1) = (4/3) . 7.11. . 7.13. . 7.14. ) 7.2.


, ,

213

7.16. ) ‘(2 - i); ) ‘ 1 + 1/ 2 + i 1 - 1/ 2 ; ) ‘(7 + 5i); ) ‘( 3/2 + i/ 2); ) ‘(3 - 4i); ) ‘((5 - i)/ 2).
-1 ‘ -3 7.17. ) z = ; 2

) z = -2 ‘ 5i; ) z = 2, i; ) z = 3, 2i;

) z = i + 2, 3 - i; ) z = 3 + i, 2 + i.

3 2 (1 ‘ i 3); 7.18. ) z = -1, 3, 1 ‘ 2i; ) z = -1 - 3 2, -1 + 2 k + , (k = 0, 1, 2, 3). ) z = 2, 2 ‘ 2 6i; ) x = tg 16 4

7.19. 6.6. 7.20. t z: t=i
1-z . 1+z

t, z, . z = cos + i sin . 1 - z 1 + z : 1 - z = 2 cos
+ cos 2 cos 1 + z = 2 cos 2 + + + i sin , 2 2 + i sin . 2 2

t, , t -- : t = tg(/2). , t , z = -1. 7.21. , [-1; 1], x x = sin t (t [-/2; /2]). 7.29. 0, s = kn, n, s = kn. 7.30. . 7.31. . 7.34. ) -1/2; 1/8. 7.35. ) , P(i) = P(-i) = 0 . ) ), , Q((cos ‘ sin )) = 0. 7.37. , V. 7.39. , 2Tn (x/2). 7.40. = m/n. cos(360n) = 1. cos(360n) = = T360n (cos ), , T360n (x) - 1 x = cos = 1/3. f(x) = 2T360n (x/2) - 2


214

, ,

x = 2/3. 7.39, f(x) 1, -- . , f(x) . 7.41. T360q (x) - 1. 7.44. Pn (x) = 2Tn (x/2), Tn (x) -- . 7.45. . 7.46. z + z-1 = 2 cos z. 7.47. x = cos , sin = cos( - /2). : n = 2k, Tn (sin ) = (-1)k cos n, n = 2k + 1, Tn (sin ) = (-1)k sin n, U
n-1

U

n-1

(sin ) = (-1)k+1

sin n . cos

(sin ) = (-1)k

cos n . cos

7.49. f(z) = 0, f(z) = f(z) = 0. 7.50. . : (x - a - ib)(x - a + ib) = x2 - 2ax + a2 + b2 . 7.51. lim 1 + n ea :
n

a + ib n
n n

n

,
n

lim 1 +

a + ib n

n

= lim 1 +
n

a n

lim 1 +

ib a+n

. ,
n

lim 1 +

ib a+n

n

= lim 1 + i tg
n

b a+n

n

.

:
n

lim 1 + i tg

b a+n

n

= lim cos
n

nb nb + i sin a+n a+n

= cos b + i sin b.

7.52. 7.51.


, ,

215

7.54. ln z . z = |z|ei , ln z wk = ln |z| + i( + 2k) (k Z). ewk = eln
|z|

§ ei(

+2k)

= |z| (cos + i sin ) = z.

7.55. az = ez ln a , 7.51, -- 7.54. 7.56. i- z = ei = e(+2k)i e+2k , k -- . z = -1 i -1 = e(1+2k) . , , k = 0.
sin(n/2) sin((n + 1))/2 . sin(/2) sin 2nx cos(2n + 1)x sin 2nx cos(2n + 1)x ; ) n - . 7.60. ) n + 2 sin x 2 sin x

7.58. )

7.61. ) -250 ; ) 249 . 7.62. ) 2n/2 sin 7.63. )
n . 4 (n - 2) (n - 4) 1n 1n 2 + 2 cos ; 2 + 2 cos . 3 3 3 3

7.67. ) z1 , . . . , zn arg z = + /2. ) z-1 , . . . , z-1 - < arg z < 1 n < 2 - . 7.69. z abc, z - a, z - b, z - c . 7.67, . ). 7.70. f (x) = f(x) + + z-a z-b z-c 7.69. 7.72. ) n 1, 2 (mod 3); ) n ‘1 (mod 6). 7.74. ) n = 6k ‘ 2; ) n = 6k - 2; ) . 7.75. ) n = 6k ‘ 1; ) n = 6k + 1; ) . 7.76. x = 1 -- P(xn ), xk = cos(2k/n)+i sin(2k/n) (k = 0, . . . , n-1). P(xn ) (x - x0 ) . . . (x - xn-1 ) = xn - 1. 7.77. 7. 7.78. zk = i tg(-/2 + k/n) = i ctg(k/n) (1 k n - 1).
1 1 1


216

, ,

: Cn n 1 (z1 , . . . , zn
-1 -1 n-1

z

+ Cn n

-3 n-3

z

+ . . . = 0.

) = 0,

2 (z1 , . . . , zn

-1

)=

Cn n Cn n

-3 -1

=

(n - 1)(n - 2) . 6

, 6.107, ), z2 + z2 + . . . + z2 1 2 n 7.80.
m m 2 m = 1 + ctg n = 1 - i ctg n sin n
2 -1

=-

(n - 1)(n - 2) . 3

1

2

,


n-1

m=1

m= sin2 n

1

n-1

1 - i ctg
m=1

m n

2

n-1

=n-1-
m=1

i ctg

m n

2

.

7.78, - ,
n-1

(n - 1)(n - 2) . 3

m=2

m= sin2 n

1

n-1

1 + ctg2
m=1

m n

=n-1+

(n - 1)(n - 2) n2 - 1 = . 3 3

7.81. , 0 <

1 1 - 2 < 1 x (0; /2) sin2 x x

(. 10.44) 7.80. 7.82. P(x) , z1 , z1 , . . . , zn , zn (. 7.49).
n

(x - zk ) = a(x) + ib(x).
k=1


n

(x - zk ) = a(x) - ib(x).
k=1

P(x) = a2 (x) + b2 (x).


, , 7.85. 7.86. 7.87.

217

) a; ) 2; ) . w = z § e2i . i+1 ) w = 2(z + 3 + 4i); ) w = 2z + 3 + 4i; ) w = (z - i) + i; 2 ) w = k(z - A) + A; ) w = z - 2; ) w = -z + (1 + i)(2 - 2). 7.89. H
k1 A1

: w = k1 (z - A1 ) + A1 ;

H

k2 A2

: w = k2 (z - A2 ) + A2

w = k1 § k2 § z + k2 (1 - k1 )A1 + (1 - k2 )A2 . k1 § k2 = 1, . k1 § k2 = 1, k1 § k2 , A k1 § k2 § z + k2 (1 - k1 )A1 + (1 - k2 )A2 = k1 § k2 (z - A) + A. 7.93. ) 7.58, ); ) 7.58, ); ) 8.41, ) í ). 7.96. (7.1) w= = ad - bc = 0.
a - , c c(cz + d)

8
8.1. , n-2 , , n . 8.2. ) 8.1 , . ) A1 A2 . . . A7 . M -- A1 A4 A2 A5 . A1 MA5 A2 A3 A4 . ) 8.1 , .
5+1 5-1 = ; cos 72 = =- . 4 2 4 2 21 - 21 2 14 - 21 8.6. ) x = arccos ; ) x = arccos ; ) x = arccos . 24 3 20

8.4. cos 36 =


218

, ,

8.7. OAB, OKLM, O(0, 0), A(1, 2), B(3, 1), K(0, 2), L(3, 2), M(3, 0). 8.8. 30 . 8.9. x = p - a, y = p - b, z = p - c. : 2. 8.10. . : 3 § 54 S 3 § 84 . 2k (k = 1, 2, 3); x3 + x2 - 2x - 1 = 0. 8.11. xk = 7 8.12. , x = cos , y = cos , z = cos . 8.13. x2 + xy + y2 = a2 x, y, a 120 . : xy + yz + xz =
a+b+c . 2 p(p - a)(p - b)(p - c) , 3

p =

8.14. ) , z1 z2 , [z1 ; z2 ]. ) , z1 z2 . 8.21. W (z1 , z2 , z3 , z4 ) = W (z1 , z2 , z3 , z4 ). 8.26. ) w = i + 8.28. C = Ab 8.38.
1 ; z+ i

) w = Rei +

ï ï ï ï ï ï ï ï A = Aaa + Bac - Bac + Ccc, B = Aab + Bad - Bcb + Ccd, ï + Bbd - Bbd + Cdd. ï ïï ï b ) 3/16; ) 1/16.
2 4 7 cos cos cos 15 15 15 15

R2 z - Re

-i

; ) w = z0 +

R2 . z - z0

8.39. cos cos
3 6 cos . 15 15

8.40. sin a. 8.41. )
2n + 1 ; 2n

)

n ; n-1 2

)

1 ; 2n

)

n . n-1 2

8.43. 32 sin x2 =
31 (2n + 1), n = 33l + 16 (n, l Z). 33

x . : x1 = 2n, n = 31l; 31

8.45. : sin 2 =
3 sin 2, 2

3 sin2 = 1 - 2 sin2 = cos 2.


, , cos( + 2) = cos § 3 sin2 - sin §
3 sin 2 = 0. 2

219

8.46. ) sin 15 = sin(45 - 30 ) cos 15 = cos(45 - 30 ); ) 8.4. 8.47. sin 6 = sin(60 - 54 ) sin 54 = = cos 36 . 8.48. ) sin + sin sin - sin( + + ). ) . 8.49. tg + tg + tg tg + tg + tg = 8.50 8.51 8.55 8.58 8.59 . . . . .
sin( + + ) + sin sin sin . cos cos cos

+ + = k. ) 8.48. n = ‘1, ‘3, ‘5, ‘15. -- 1, -- 1/4. 1 = (sin2 x + cos2 x)2 = sin4 x + cos4 x + 2 sin2 x cos2 x.

8.63. x = 2(cos + cos + cos ) + 8 cos cos cos ; y = -2 - 4(cos cos + cos cos + cos cos ); z = 2(cos + cos + cos ). 8.64. ) 8.65. = 1- 8.66.
9 ; 14

) -

. 10

) y = arcsin x (-/2 x /2). sin y = x, cos y = sin2 y = 1 - x2 , , cos y 0. . : -
< arctg x < , 2 2

0

arcctg x

.

-
3 < arctg x + arcctg x < . 2 2

sin(arctg x + arcctg x) = 0. , -
3 < arcsin x + arccos x < 2 2


220

, ,

sin(arcsin x + arccos x). 8.68. /2 x > 0 -/2 x < 0. 8.69. , arctg x + arctg y arctg
x+y , -- . 1 - xy

, tg(arctg x + arctg y) = - < arctg x + arctg y < , 0 ‘1. . 8.70, 8.71 8.69. 8.74. ctg(arcctg F
2n

x+y x+y = tg . 1 - xy 1 - xy

- arcctg F

2n+2

)=

F2n F2n+2 + 1 =F F2n+2 - F2n

2n+1

.

(8.2) . n 1 , arcctg 2 + arcctg 5 + arcctg 13 + . . . + arcctg F 8.75. = 2 arctg x + arcsin
2n+1

+ . . . = arcctg 1 = /4.

2x . , 1 + x2

0 < 3/2 sin = 0. 8.76. 0 x 4. 8.80. arcsin cos arcsin x + arccos sin arccos x = /2. 8.82. 8.69 2 arctg 1/5 = arctg 5/12. : 0. 8.83. a = b cos + c cos sin = sin cos + cos sin . . 8.84. (8.4) : b=
c(cos + cos cos ) , sin2

a=

c(cos + cos cos ) . sin2

, 1 - cos2 - cos2 - cos2 - 2 cos cos cos = 0.


, ,

221

cos + cos cos = sin sin , cos + cos cos = sin sin , + + = , a sin = c sin , b sin = c sin . 8.86. cos A = sin2 A =
1 - cos2 - cos2 - cos2 + 2 cos cos cos , sin2 sin2 cos - cos cos . sin sin

sin2 A 1 - cos2 - cos2 - cos2 + 2 cos cos cos = . 2 sin sin2 sin2 sin2

, , , A, B, C ,
sin2 A sin2 B sin2 C = = . 2 2 sin sin sin2

, , , A, B, C 0 , sin A sin B sin C = = .
sin sin sin

9
9.2. z = x + , x2 A + 3. = -A/3. 9.3. ) f(x) = x3 + px -- , . ) f(x) = = x3 + px + q f(x) = x3 + px , . ) 9.2 , f(x) = ax3 + bx2 + cx + d f(x) = x3 + px + q . . 9.5. 2x3 + (x + 1)3 = 0. 9.6. x1 x2 + x1 x3 + x2 x3 = 0, x1 x2 x3 = b > 0. 9.7. a b a3 + b3 = -q, a3 b3 = -p3 /27.


222

, ,

a3 b3 y2 + + qy - p3 /27 = 0. a3 , b 3 = -
q ‘ 2 q2 p3 +; 4 27

a, b =

3

-

q ‘ 2

q2 p3 +. 4 27

9.8. : a3 + b3 + c3 - 3abc = (a + b + c)(a + b + c2 )(a + b2 + c). 2 -- 1:
-1 + i 3 = , 2 -1 - i 3 = . 2
2

9.9. x1 = a + b, x2 = a + b2 , x3 = a2 + b, 2 -- 1 ( 9.8.) 9.10. a2 + b2 + c2 - ab - bc - ac = (a + b + c2 )(a + b2 + c), x2 + y2 + z2 - xy - yz - xz = (x + y + z2 )(x + y2 + z), (a2 + b2 + c2 - ab - bc - ac)(x2 + y2 + z2 - xy - yz - xz) = = (X + Y + Z2 )(X + Y2 + Z) = X2 + Y 2 + Z2 - XY - YZ - XZ. 9.11. , 9.7 x1 9.9. , , , a b . 9 x x = a + b, , a b a § b = -p/3. a b 9.7, x1 = a + b, x2 = a + 2 b x3 = 2 a + b, -- 1. 9.12. x0 = 1. x3 + x - 2 = (x - 1)(x2 + x + 2), . x0 = 1: 1=
3

1+

1+

1 + 27

3

1-

1+

1 . 27


, , 9.13. x3 + 3/4x - 7/4, = 1. 9.14. a
2 33

223

-;

2 2 3333

3 , a =
2 2 -- 1 . 3333

= ‘ -- a - ; /

9.15. x1 = 2/ 3. , x3 - x - = x -
2 33 2 3

x2 + +

2 3

1 . 3

. , , . : x1 = 2/ 3, x2 = x3 = -1/ 3. 9.16. x1 + x2 + x3 = 0. 9.17. D(x3 + px + q) = -4p3 - 27q2 . 9.18. 9.17. 9.19. a = 1, b = 2, x1,2 = 1 ‘ -5. 9.25. ) xk = 2 cos
1 + 6k (k = 0, 1, 2); 9 1 + 12k (k = 0, 1, 2). ) xk = 2 cos 18

9.27. , (p - p )x + (q - q ) = 0. (13.7)

q , q , x3 (q - q) + x(pq - qp ) = 0, x2 (q - q) + pq - qp = 0. (13.8)

(13.7) (13.8) x, . 9.29. 9.5 : ; . : D() = (C + 2 )(A + 2) - B2 = 0, -A/2.

. . , D(-A/2) = -B2 0, lim D() = ,



224

, ,

-A/2. 9.30. x = cos t, y = sin t, t [0; 2].
4k 8k 2k , cos , cos (k = 0, . . . , 4), 9 9 9 2k 4k 8k (x, y, z) = cos , cos , cos (k = 1, 2, 3). 7 7 7

9.31. (x, y, z) =

cos

9.32. . 9.33. x = 2 cos , y = 2 sin , z = 3 cos , t = 3 sin . 1 9.34. ) x = cos t. : x - ,
2+ 2 2

,-

2- 2

2

2

; ) : x {5/3, 5/4}; )

x = sin t, t [-/2; /2]. t t = /5. : x = sin(/5). ) x = cos t. 9.35. hn = sin 2, hn+1 = sin . h1 = sin(/6),


S =
n=1

sin

. 3 § 2n

sin x < x (x > 0)
1 S< + 2

n=2

3§2

n

=

1 + < 1,03. 2 6

9.36. x = cos t, t [0; /2]. 8 cos t cos 2t cos 4t + 1 = 0. sin t, , sin 8t + sin t = 0 sin(7t/2) cos(9t/2) = 0. , , [0; /2] 3 . : 3. 9.40. ) y = 2x 1 - x2 2y z= 1 - y2 x = 2z 2
1-z

x = tg , (-/2, /2) , y = tg 2, z = tg 4, x = tg 8. tg = tg 8, , = -4
k 2k 4k k , 7

x 3. : (x, y, z) = (tg , tg , tg ) (-3 k 3). 7 7 7 9.43. a = 2 - x, b = 2 - y, c = 2 - z, a = 2 cos .


, ,

225

9.44. x = sin t, t [-/2; /2],
1 + sin 2t = cos 2t. 2

, sin 2t = -1 sin 2t = 1/2, t [-/4; /4]. t = -/4 t = /12. : x - , . 2 2 9.46. dn dn = xn - 2. {dn } dn
+1

1

2-

3

=

d2 n 2( 2 + dn )

(n

1).

n , 0 < dn < 1, dn , : 0 < dn+1 < dn /2. d2 = 3/2 - 2 , lim dn = 0, lim xn = 2.
n n

9.47. - 2. 9.48. 9.46, dn dn = xn - k. dn
+1

=

d2 n 2( k + dn )

(n

0).

{dn } . , 0. 9.49. . 9.50. an =
2a1 + a0 a - a0 + (-1)n 1 n-1 . 3 3§2

9.51. , : y0 = 9.52.
k



x

yn

+1

=
1/2k

x3 yn

(n

0).

lim

ln N N

1/2k

= 1,
-1
k


k

ln N = lim 2k (N1/2 - 1).


226

, ,

ln N k , . 9.59. OAKB -- , , O(0; 0), A(a; 0), K(a; b), B(0; b). x=
5-1 aq , 1 - q4

y=

aq4 , 1 - q4

q = = -. 2 9.60. 9.46, dn dn = an - 3 a. dn
+1

=

2d3 + 3d2 3 a n n 3( 3 a + dn )2

(n

1).

n0 dn0 ( n = 2 a), dn+1 dn
+1

<

3d3 + 3d2 3 a d2 n n = n < dn 2 3 3 a + dn 3( a + dn )

(n

n0 ).

{dn } . , 0. 9.61. x2 = 1 + 1/x. , x1 = 1, xn+1 = 1 + 1/xn , . ( ) , . (. 9.78.) 9.62. 0 < 2 - an < (3/4)n . 9.63. 0 < q < 1 , ( n) an+1 < qan . 9.64. . 9.67. (0, 0, 0), (1, 1, 1). 9.71. 9.69 , a + x = x. , a - x1,2 = . 4 2 9.72. bn < bn+1 < an+1 < an . ²(a, b) a b
1 1‘ 1 + 4a


, , :
/2

227

²(a, b) = 2
0

dx a2 sin2 x + b2 cos2 x

-1

.

9.73. ) , : an bn = ab (n 0). . 9.74.
1 a
n+1 2n +1

=

1/an + 1/bn , 2
2n

1 = bn+1

1 1 § . an bn

9.79. ) xn = F ) xn = F

/F

= [1; 1, 1, . . . , 1] ;
2n 2n -1

-2n +1

/F

-2n

= -F

/F

2n

= [-1; 2, 1, . . . , 1] .
2n -1

9.80. {yn } {zn } x2 - px + q = 0. , q. 9.81. , pn /qn
p2 - q § q2 p2n n n = . q2n qn (2pn - p § qn )

9.82. f(x) , x0 = 0, x1 = -x0 . , x2 = -x1 = x0 . x1 = x0 - x0 (x2 - 1) = -x0 . x0 = ‘ 2. 0 9.83. : P1 (x) = x2 - 3x + 1 = x2 - L2 x + 1, P2 (x) = x2 - 7x + 1 = x2 - L4 x + 1, P3 (x) = x2 - 47x + 1 = x2 - L8 x + 1, Lk -- . : Pk (x) = x2 - L2k x + 1. x1 = lim (L2k )1/2 = , x2 = - lim (L2k )-1/2 = . k k 9.85. ) 4, 2 2, 2 3, 3; ) an = tg(/n), bn = sin(/n), pn = nbn , Pn = nan .
k k


228 9.86. sin x = 2n sin
n

, ,

x x x x x cos cos 2 cos 3 . . . cos n 2n 2 2 2 2

lim 2n sin

x =x 2n



x x x x = cos cos 2 cos 3 . . . sin x 2 2 2

, x = /2, . 9.87. a y = ax y = x. : a = e1/e , lim xn = e. 9.88 9.89 9.91 9.92 9.93 9.94 . . . . . . a3 > 3n a3 - 3n. n n x1 + y1 + z1 = x1 y1 z1 . ) (2, -1, -3, -4). ) 13 ½ 11; ) 61 ½ 69. ) (4/9, 5/9, 1/2, 1/2); ) (8/13, 6/13, 6/13, 6/13). ) a = 1, (x, y) = (t, 1 - t) (t R); a = -1,
a2 + a + 1 a ,- . a+1 a+1
n

; a = ‘1, (x, y) =

) a = 0, (x, y) = (2, t) (t R); a = 1, ; a = 0, 1, (x, y) =
a2 - 2 2 - a , . a-1 a-1 4(a - 2) 1 - a , . a-3 a-3 a5 - 2a4 + 2a2 - a + 6 a6 - a2 - 2a - 4 , . 2(a2 - 1) 2(a2 - 1)

) a = 1, (x, y) = (2 - 4t, t) (t R); a = 3, ; a = 1, 3, (x, y) =

) a = 0, (x, y) = (t, 2) (t R); a = ‘1, ; a = 0, ‘ 1, (x, y) = ) a = 1, = (t, t - 1) (t R); ) a = 0, (x, y) = (1/2 + t, t)

(x, y) = (t, 1 - t) (t R); a = -1, (x, y) = a = ‘1, (x, y) = (a2 + 1, -a). (x, y) = (t, 0) (t R); a = 1/2, b = 1/2, (t R); a = 1/2, b = 1/2, ;
a - b a(1 - 2b) , . 2a - 1 2a - 1

a = 0, 1/2, (x, y) =

) a = b = 0, (x, y) = (t1 , t2 ) (t1 , t2 R); a = b = 0, (x, y) = (t, 1 - t) (t R); a = -b = 0, (x, y) = (1 + t, t) (t R); a = ‘b, (x, y) = (1, 0). ) a 0, (x, y) =
a2 - 1 2a ,2 ; a < 0, (x, y) = (0, 1). a +1 a +1
2


, ,

229

9.95. . x y -- , x + (1 - )y . 9.97. , (1, 1, 1).

10
10.1. (x - 1)2 0 x. 10.2. . 10.3. 10.2. 10.4. . 10.5. . 10.6. . 10.7. (x - y)2 + (y - z)2 + (x - z)2 0. 10.8. . x1 x2 , . . . , (x1 /2)2 + x2 10.9. (x1 /2)2 + x2 5 2 x1 x5 . 10.11. . 10.12. 10.7. 10.13. : , . 10.14. a2 + b2 + c2 + 2ab + 2ac + 2bc = (a + b + c)2 = 0. 10.15. 1 + xy . , (1 ‘ x)(1 ‘ y) > 0. 10.16. . 10.17. a2 (b-c)2 +b2 (a-c)2 +c2 (a-b)2 0. 10.18. . 10.21. ( x - y)4 0. 10.22. a(b-c)2 +b(a-c)2 +c(a-b)2 0. 10.23. . 10.24. (1 + a4 )(1 - b2 )2 + (1 + b4 )(1 - a2 )2 0.


230

, ,

10.25. 10.7 10.17. 10.27. a3 + b3 a2 b + ab2 . 10.29. 10.22. 10.30. . 10.31. 10.27. 10.32. ( 10.11). 10.37. . 10.41. x y , p= = a
1/x

x+y , x
1/y

q=

x+y . y

,=b
x


x+y

x+y

+ y x+y

x y .

:
xx
+y

+ y x+y

x+y
x+y



x+y

...

x+y



x+y

...

x+y

= x y .

x yn n 10.43. ) ( n n!)2 = n (1 § n) . . . (n § 1) n = n. . 10.44. x (0; /2) x < tg x, 1 1 1 1 - 2< - 2 = 1. 2 2

sin x

x

sin x

tg x

10.46. m, n, k

2, m(
k

nk )

(mn )k .
1 1 > (1 n+k 2n

10.47. )

k

n - 1).

) 2 (0 k
24 100 § ... 35 101

n). )



1246 98 § § § ... . 2357 99

)
79 99 § ... . 8 10 100


, , 10.48. 10.50. 10.52. 10.53. = an = 1. 10.55. 1 x1 + 2
xyz § § = 1. yzx

231

fk (x) = ak x2 + bk x. fk (x) = ak ex - bk (x + 1). ) b1 = . . . = bn = 1. ) a1 = . . . = ) a1 = b1 c1 , . . . , an = bn cn . l(x) -- f(x) x2 . , ,

1 f(x1 ) + 2 f(x2 ) < 1 l(x1 ) + 2 l(x2 ) = l(1 x1 + 2 x2 ) = f(1 x1 + 2 x2 ). 10.56. . n = 2 10.55. , n 2 n + 1. = 1 + . . . + n > 0.
1 + . . . + n = 1. n = 2,

f(1 x1 + . . . +

n+1 n+1

x

)=f

>

n+1

f(x

n+1

1 x + . . . + n xn + 1 1 ) + f x + . . . + n xn . 1

n+1 n+1

x

>

, , f
1 x + ... + nx 1
n

>

1 f(x1 ) + . . . + n f(xn ).

f(1 x1 + . . . +
n+1 n+1

x

) > 1 f(x1 ) + . . . +

n+1

f(x

n+1

).

10.58. : ) f(x) = x; ) f(x) = 1/x2 ; ) f(x) = enx ; ) f(x) = 1/x. 10.60. y = xp , (1 x1 + 2 x2 + . . . + n xn )p 1 =
n 1 , . . . , n = 1 + . . . + n 1 + . . . +
n

1 x p + 2 xp + . . . + n xp . 1 2 n

(1 x1 + 2 x2 + . . . + n xn )p 1 x1 + 2 x 2 + . . . + n x
n

(1 + . . . + n )p-1 (1 xp + . . . + n xp ) 1 n

(1 + . . . + n )1/q (1 xp + . . . + n xp )1/p . 1 n


232

, ,

k = bq , xk = ak b1-q (k = 1, 2, . . . , n). k k 10.63. f(x) = ex ln x1 , . . . , ln xn . ex , x [-1; 1]: ex = 1 + x + x S (x) =
e
ln x1 2

(|| < 1).

+ ... + e n

ln xn

1/

= 1+
1/

ln(x1 . . . xn ) + 2 A n
1/

1/

,

S0 (x) = e

n ln(x1 ...xn )

= 1+

ln(x1 . . . xn ) + 2 B n

,

A = 1 (ln2 x1 + . . . + ln2 xn ); B =
0

2 2 ln (x1 . . . xn ); |1 |, |2 | < 1. n2
1/

lim

S (x) 1 + (/n) ln(x1 . . . xn ) + 2 A = lim S0 (x) 0 1 + (/n) ln(x1 . . . xn ) + 2 B

= 1.

10.71. ) ; ) (5, 1, 1) (4, 3, 0); (4, 1, 1, 1) (3, 3, 1, 0). 10.72. : x = y = z = t; x = y = t, z = 1; x = t, y = z = 1 t. 10.73. , , ë¨ . , = (1 , 2 , 3 , . . . , n ) = (1 - 1, 2 + 1, 3 , . . . , n ), 1 - 2 2. x1 x2 x3 . . . xn in i1 i2 i3 x1 -1 x2 +1 x3 . . . xn . i1 i2 i3 in T (x1 , . . . , xn ) T (x1 , . . . , xn ) , : x
3 i3 1 i1

x

2 i2

A+x

1 i2

x

2 i1

A;

x

1 -1 2 +1 i1 i2

x

A+x

1 -1 2 +1 i1 i2

x

A,

0) , (A = x . . . x . , x
1 i1

n in

x

2 i2

+x

1 i2

x

2 i1

-x

1 -1 2 +1 i1 i2

x

-x

1 -1 2 +1 i2 i1

x

=

= (x

1 -1 2 i1 i2

x

-x

1 -1 2 i2 i1

x

)(x

i1

- xi2 ) -x

0, x

1 - 1 > 2 x , xi1 - xi2 .

1 -1 2 i1 i2

x

1 -1 2 i2 i1


, ,

233

10.75. 10.75. ) x4 y2 z + x2 y4 z - 2x3 y3 x = x2 y2 z(x - y)2 0.

) (5, 0, 0) (2, 2, 1) : (5, 0, 0) (4, 1, 0) (3, 2, 0) (2, 2, 1), : x5 + y5 - x4 y - xy4 = (x4 - y4 )(x - y) 0, x4 y + xy4 - x3 y2 - x2 y3 = xy(x2 - y2 )(x - y) 0, 32 x y + y2 z3 - x2 y2 z - xy2 z2 = y2 (x2 - z2 )(x - z) 0.

11
k-1

11.1. ) n2 = 2n + 1; ) n(n - 1) = 2n; ) n

k

=
j=0

Cj nj ; k

) Ck = Ck +1 - Ck = Ck-1 . n n n n 11.2. Sn = bn+1 - b1 . 11.3. an . an an = An3 + Bn2 + Cn A, B, C an = n2 . 11.4. an an = An4 + Bn3 + Cn2 + + Dn, an = n3 . 11.5. , m 1/F2m = Fm-1 /Fm - F2m-1 /F2m . 11.6. Q(x) = xm+1 . Q(x) = xm+1 - xm = (m + 1)xm + . . . 11.8. n. 11.10. 11.6. m f(x) = m!am , am -- f(x). 11.11. n!/nn . 11.13. 11.12, yk = (-1)k Ck n . , {yk } . , ( : y(1) , . . . , y(1) y02) , . . . , y(2) . 1 2 , 0 n n


234

, ,

11.1 {y(1) } k {y(2) } f(k) = kn : k
n n

kn y(1) = 1 , k
k=0 k=0

kn y(2) = 2 . k

y(3) = 2 y(1) - 1 y(2) (k = 0, . . . , n) k k k ,
n

f(k)y(3) = 0 k
k=0

f(x), deg f(x) n. f(x) , f(k) = y(3) (k = 0, . . . , n). k
n

(y(3) )2 = 0, k
k=0



y(3) = y(3) = . . . = y(3) = 0, 2 1 n

{y(1) } {y(2) }. k k 11.12. 11.8 11.10. 11.15. (an bn ) = an+1 bn + bn an = an bn + bn+1 an . 11.16. an = 2n . 11.15 . n = 1 (f(x)g(x)) = f(x + 1)g(x) + f(x)g(x) = f(x)g(x) + f(x)g(x + 1) . . () n.
n+1

(f(x)g(x)) = n (n f(x)g(x)) = n (f(x)g(x) + f(x)g(x + 1))


n n n+1



(f(x)g(x)) =
k=0

Ck k f(x)k - k + 1g(x+k)+ n
k=0

Ck n

k+1f(x)n-k

g(x+1).


, ,

235

,
n



n+1

(f(x)g(x)) =
k=0

Ck k f(x) n
n+1

n-k+1

g(x + k) +

+
k=1 n+1

Ck-1 k f(x) n

n-k+1

g(x + k) =

=
k=0 n+1

(Ck + Ck-1 )n f(x) n n Ck n
k=0

n-k+1

g(x + k) =

=
1 ; n+1

+1

k f(x)

n+1-k

g(x + k)

11.19. ) 1 - )
2n+1 - 1; n+1

(3n + 2)(n - 1) 11 1 ; ) - ; 4n(n + 1) 22 (n + 1)(n + 2) 2 1 1 ) 1 - + ; ) 1 - ; ) (n + 1)! - 1. 2n + 1 n+1 n!

)

11.21. ) n f(x) g(x) = d0 C0 + d1 C1 + x x + . . . + dn Cn k f(0) = k g(0) (0 k n). x x = 0, 1, . . . , n, . 11.22. f(x) x = 0, 1, . . . , n, dk , dk = = k f(0) (. 11.21), . 11.27. n = 4k + 1 n = 4k + 2, . . n = 4k + 3 n = 4k. , n = 7, -- n = 8. , n = 3 n = 4 . n2 - (n + 1)2 - (n + 2)2 + (n + 3)2 = 4 , , + -, 0. , n, n + 8k (k 0). n = 7, 11 12. , , 0 m, , m = 8. : 1 + 4 - 9 + 16 - 25 - 36 + 49 = 0; 1 - 4 + 9 + 16 + 25 - 36 - 49 - 64 + 81 - 100 + 121 = 0; 1 - 4 + 9 + 16 + 25 - 36 + 49 - 64 + 81 - 100 - 121 + 144 = 0.


236

, ,

11.28, 11.29 . 11.30. x f(x, y) = f(x + 1, y) - f(x, y) y f(x, y) = f(x, y + 1) - f(x, y),

. x f(x, y) . , M = = sup(x,y)Z2 f(x, y). Z2 K (n ½ n), x f(x, y) > M/2 V . , f(x, y) K Ox M § n/2. f(x, y). 11.31. . 11.33. 11.32, {an } = ci xn (i = 1,2) i c1 , c2 (11.2), . , c1 , c2 , a0 = c1 + c2 , a1 = c1 x1 + c2 x2 . , {an } {c1 xn + c2 xn } 1 2 . 11.31, . 11.35. ) an = 3n - 2n ; ) an = 1; ) an =
1 1 1+ 2 5 1+ 5 2
n

+

1 1 1- 2 5

1- 5 2

n

=F

n+1

;

1 ) an = ((1 + 2)n - (1 - 2)n ). 22 11.36. ) (1 - 2)n = an - bn 2. ) a2 - 2b2 = (an - bn )(an + n 2) = (1 + 2)n - 2)n = 1. 2 b (1 n n ) (an + bn 2)(1 + 2) = (an+1 + bn+1 2) , an bn an+1 = an + +2bn , bn+1 = an +bn . an+2 -2an+1 -an = 0, bn+2 -2bn+1 -bn = 0 (n 0). 1 1 ) an = ((1 + 2)n + (1 - 2)n ), bn = ((1 + 2)n - (1 - 2)n ).
2 22

) an = n + 1;

11.38. . 11.41. , V. , ), ) ) . . (. 11.31.) ) ), -


, ,

237

, , . , c F
1 x + x2 + 4 11.43. Fn (x) = 2 2 x +4 2+4 n x+ x Ln (x) = + 2
2n+4

(x) = (x2 + 2)F

2n+2 n

(x) - F2n (x). -
x-

x - x2 + 4 2

x2 + 4 2
n

n

;

.
n-2k

11.45. F

n+1

(x) =
k0

Ck n

-k

x

n-2k

, Ln (x) =
k0

(Ck n

-k

+ Ck-1 -1 )x n-k

.

11.46. an , bn , cn , dn , en , fn A n A, B, C, D, E, F . , bn = fn , cn = en . , bn+1 = an + cn , dn+1 = 2cn , an+1 = 2bn , cn+1 = bn + dn . , xn+4 - 5xn+2 + 4xn = 0 (n 0). a0 = 1, a2 = 2, a2n = (2 + 4n )/3. 11.47. ) cn+4 - 5cn+2 + 4cn = 0 (n 0) (. 11.46) c0 = 0, c2 = 1, c2n = (4n - 1)/3. ) , D, cn+2 = 3cn (n 1). c2n = 3n-1 n-1 (n 1). a2n = 2 § 3 . ) , n , D, . n : P n: P
2k+1 2k

=

a

2k

+ c2k + e 22 k 2a

2k

=

3 4

k-1 k-1

(k

1).

=

b2

k+1 + f2k+1 2k+1

2

=

2k

+ c2k + e 22k+1

2k

=

3 4

k k

(k

1).

D . n = 2k + 1
c
2k + e2k 2k+1

2

=

2 § 3k-1 . 22k+1


238

, ,




N=
k=1

(2k + 1)

2 § 3k-1 . 22k+1


1 f(t) = 3

k=1

3k 2k t 22k

+1

=

t3 . 4 - 3t2

f(t) t = 1: N = f (t)|t=1 = 4. : 4 . 11.49. (3n - (-2)n )/5. 11.50. . 11.52. x2 - 17n2 = 1 (x1 + n1 17)k = xk + nk 17 (k 1). 11.53. xn = 1/2 sin [(1 + sin 2)n - (1 - sin 2)n ], . yn = 1/2 cos [(1 + sin 2)n + (1 - sin 2)n ] 11.54. a0 -- , ak -- , , ë¨ k- . a
k+1

= 4/5(ak - 1) (k = 0, . . . , 4).

, {ak } 5ak+1 - 9ak + 4ak-1 = 0 (k = 1, . . . , 4). 11.33 , ak = c1 + c2 (4/5)k (k = 0, . . . , 5).

, c1 = -4. ak k 0 5, c2 c2 = 55 n. n = 1 a5 = 45 - 4 5, 55 - 4. : 55 - 4. 1+i 1-i 11.57. ) an = i/2(-(2+i)n +(2-i)n ); ) an = (1+i)n + (1-i)n ;
2 2

) a3n = 1, a3n+1 = 2, a3n+2 = -3; ) an = i((3 - 4i)n - (3 + 4i)n ). 11.58. 3 n2 = 0 4 n3 = 0.


, , 11.59. n = (1 + 2 + 3)n = 1 n = (1 - 2 + 3)n = 2 n = (1 + 2 - 3)n = 3 n = (1 - 2 - 3)n = 4

239 11.36. 2 3, pn + qn 2 + rn 3 + sn 6, pn - qn 2 + rn 3 - sn 6, pn + qn 2 - rn 3 - sn 6, pn - qn 2 - rn 3 + sn 6.

(1, 1, 1, 1), (1, -1, 1, -1), (1, 1, -1, -1), (1, -1, -1, 1), pn = q r
n

n

sn lim
n

pn = 2, qn

1n ( + 41 1 = (n 1 42 1 = (n 1 43 1 = (n 1 46 p lim n = n rn

n + n + n ), 2 3 4 - n + n - n ), 2 3 4 + n - n - n ), 2 3 4 - n - n + n ). 2 3 4
n

3, lim

n

pn = 6. sn
-1

11.65. f(x) = (1 + x)n =
k=0 n

Ck xk . f (x) = n(1 + x)n n

=

=
k=0

kCk x n

k-1

. x = 1, . ):
n

kCk = f (x)| n
k=1

x=1

= n § 2n

-1

.

. ):
n

k2 Ck = (xf (x)) | n
k=1

x=1

= n(n + 1)2n

-2

.

: ) n § 2n-1 ; ) n(n + 1) § 2n 11.67.
F
(k)

-2

.
1 (1 - x)n

(x) = Ck n k!

+k-1

+k

an = Ck +k-1 . n 11.68. , .


240

, ,

11.69. , 11.63. 11.70. -- , xm :


Cm m
n=0

+n

xn =

1 (1 - x)m


+1

,
n=0

Cm xn = n

xm (1 - x)m
6

+1

.

11.72. 11.71 , N x
27


-6 6

1 - x10 1-x

.

N (1 - x ) (1 - x) (1 - x10 )6 =
k=0

10 6

x:

(-x)10k Ck ; 6


(1 - x)-6 =
k=0

xk Ck+k-1 . 6

N = C0 § C27 - C1 § C17 + C2 § C72 = 55252. 6 32 6 22 6 1 11.73. . zn Exp(( + )z) Exp(z) § Exp(z). ,
n

( + )n , -- n!
n

k=0

n-k 1 k § = k! (n - k)! n!

Ck k n
k=0

n-k

.

( 2.53). 11.74. , a3 + b3 + c3 - 3abc = (a + b + c)(a + b + 2 c)(a + 2 b + c) ( 9.8). a + b + c = (1 + x)n , a + b + 2 c = (1 + x)n , a + 2 b + c = (1 + 2 x)n . a3 + b3 + c3 - 3abc = [(1 + x)(1 + x)(1 + 2 x)]n = (1 + x3 )n .


, ,

241

11.76. z = 1/10. 10/89. 11.77. L(z) =
2-z . 1 - z - z2

11.78. ) 2; ) 6.
z 2 - xz , L(x, z) = . 2 1 - xz - z 1 - xz - z2 1 - xt 1 11.80. FT (x, z) = , FU (x, z) = . 1 - 2xt + t2 1 - 2xt + t2

11.79. F(x, z) =

11.81. ) f(x). , , f(x) =
(2x)n - 1 . 2x - 1

) 11.65 f (1) = 2n (n - 2) + 2. ) , 11.65 ), (xf (x)) |
x=1

= f (1) + f (1) = 2n (n2 - 5n + 8) - 8.

) 7.58 ),
n-1

g(x) =
k=1

=

sin((n - 1)/2)x § cos(nx/2) . sin(x/2)

-g (1): -g (x)|
x=1

=

n sin(n - 1)x - (n - 1) sin nx . 2(1 - cos x)

11.83. . 11.84. 2.67. ,
1 1 = = 1 + x + 2x2 + . . . + 2n 1 - x/(1 - x) 1 - x - x2 - x3 - . . .
-1 n

x + ...

11.87. an = q(n) , (n) -- n. 11.88. (1 - ax)(1 - bx2 )(1 - cx4 )(1 - dx8 ) . . . , x = 1. a = b = = c = d = . . . = 1. 11.87 (-1)(n) .


242

, ,

11.89. an = Cnn . 2 11.90. x = y/(1 - y), y = x/(1 + x). , y = x - x2 + x3 - x4 + . . . + (-1)n
+1 n

x + ...

11.91. y = x - x2 /2 + x3 /3 - x4 /4 + . . . + (-1)n+1 xn /n + . . . 11.92. 2.116 , zn C2 (z) . 11.93. C(z) = zC2 (z) + 1, C(z) =
1- 1 - 4z 2z

( ë¨ C(0) = 1). C(z) = -
1 2


Cn/2 (-4)n zn 1
n=1

-1

,

Cn = (-4)n

1 2

+1

+ Cn/21 = 1

(2n)! . n!(n + 1)!

11.94. , , :
g0,0 (x) g1,0 (x) g2,0 (x) g3,0 (x) g3,1 (x) g2,1 (x) g3,2 (x) g1,1 (x) g2,2 (x) g3,3 (x)

, V. 11.95. ) ) ). ) k ). ) l ). 11.96. gk,l (x) (1 - x)k . gk,l (x) x = 1 2.77. 11.97. Sl (x) = 0 l Sl (x) = (1 - x)(1 - x3 ) . . . (1 - x
l-1

)=

h

l/2

hl (x) (x2 )

l. . , S0 (x) = 1 S1 (x) = 0. , Sl (x) = (1 - x
l-1

)S

l-2

(x).

) 11.95, Sl (x) = (1-x
l-1

)g0,

l-1

(x)-(1-x

l-2

)g1,

l-2

(x)+. . .+(-1)l

-1

(1-x0 )gl

-1,0

(x).


, ,

243

(1 - xl )gk,l (x) = (1 - xk+l )gk,l-1 (x) , : Sl (x) = (1 - x
l-1

)(g0,

l-2

(x) - g1,

l-3

(x) + . . . ) = (1 - x

l-1

)S

l-2

(x).

11.98. ) . ) n = a1 + a2 + . . . + aj , j k, ai l n kl-n = (l-a1 )+(l-a2 )+. . .+(l-aj )+l+. . .+l kl-n, l - ai , , , l, k - j. , ? 11.101. 2.59

12
12.3. 16/64, 19/95, 26/65, 49/98. 12.5. sin
a b a+b sin sin = 0. 2 2 2

: a = 2k, b = 2l, a + b = 2m. 12.7. , 400- 7. 12.9. . 12.10. (x - x). : 0. x 12.12. 2 . - , ln z -- . 12.13.

12.14. 1,609 . . . , = 1,618 . . .



. . -- .: , 1990. ., . . -- .: , 1965. ., . . -- .: , 1965. . ., . ., . . . -- .: , 1959. . . . -- .: , 1960. . . . -- .: , 1969. . ., - . ., . . : - 10 . -- .: , 1998. . ., - . ., . . : - 11 . -- .: , 1998. . . -- .: , 1971. . . -- .: , 1972. . -- .: , 1974. . . . -- .: , 1967. . ., . ., . . . -- .: , 1998. . ., . ., . . . -- .: , 1977. . . -- .: , 1992. ., . ?. -- .: , 2001. . . . -- .: , 1979. . . -- .: , 1966. . . -- .: , 1970. . . . . -- .: , 1987. . . . -- .: , 2000. . . : . -- .: , 1962. . . . -- .: ., 1937. . . . -- : - . . - . . . , 1954. . . . -- .: , 1996.

[1] [2] [3] [4] [5] [6] [7]

[8]

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]




245

[26] . . . -- : ë ¨, 2000. [27] . ., . ., . . -- .: , 1948. [28] . . . -- .: , 1978. [29] . . -- .: , 1970. [30] . . . -- .: , 2000. [31] . . . -- .: , 1963. [32] . ., . ., . ., . . . : . -- .: , 1988. [33] . ., . . . -- .: - . -, 1995. [34] . ., . ., . ., . . . -- .: , 1986. [35] . ., . ., . . . -- .: , 1999. [36] . ., . . . -- .: , 1986. [37] . ., . . . . . -- .: , 1996. [38] . ., . ., . . . -- : , 1994. [39] : . -- .: , 1977. [40] . . . -- .: , 1940. [41] . . . -- . -- .: , 1937. [42] . . . -- .: , 1970. [43] . . . -- .: , 1957. [44] . ., . ., . ., . . . -- .: , 1962. [45] ., . . . 1. -- .: , 1978. [46] . . . -- .: , 2001. [47] . . . . 1 í 4. -- . [48] . ., . ., . ., . . . -- .: - - , 1997. [49] . . . . 1 í 3. -- .: . . . , 1998. [50] . ., . ., . . : 3 . . 1. . -- .: , 2001.


246
ë¨



[51] . ., . ., . . . . -- . 22. -- .: , 1982. [52] . . -- . -- . 64. -- .: , 1988. [53] . . . -- . 56. -- .: , 1986. [54] . . -- . 83. -- .: , 1992. ë ¨ [55] . . . -- . 35. -- .: , 1968. [56] . . . -- . 39. -- .: , 1963. [57] . . . -- . 6. -- .: , 1984. [58] . ., . . . -- . 21. -- .: , 1961. [59] . . . -- . 47. -- .: , 1969. [60] . . . -- . 5. -- .: , 1983. [61] . . . -- . 34. -- .: , 1960. [62] . . . -- . 1. -- .: , 1950. [63] . . . -- . 13. -- .: , 1979. [64] . . . -- . 59. -- .: , 1986. [65] . . . -- . 3. -- .: , 1974. [66] . . . -- . 43. -- .: , 1979. [67] . . . -- . 60. -- .: , 1989. ë¨ ë¨ 30 (). -- .: ë¨, 2000. (. ë¨. 1). . l // 10. 1978. . // 1973. . 11. 1980. ., . // 1995. -- // . . . 4. -- .: ë¨, 1999. (. ë¨. 5). ., . 3. 1991.

[68] [69] [70] [71] [72]

--

5. // 2. -- //

[73]



[74] [75] [76] [77] [78] [79]

247

[80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90]

[91]

[92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104]

. // 3. 1991. . // 9. 1971. . // 2. 1980. ., . // 3. 1973. ., . ? // 6. 1995. . ? // 3. 1971. -- // . . 4. -- .: ë¨, 1999. -- (. ë¨. 5). . // 5. 1971. -- // 1. 1990. . -- // 7. 1974. ., . // 2. 1974. . // 8. 1973. . // 10. 1982. ., . // 9. 1970. . // 8. 1979. . // 4. 1975. . // 8. 1970. . // 1. 1970. ., ., . // 9. 1983. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). . // 6. 1992. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). . // 3. 1983. . // 2. 1977. . // 4. 1975. . // 4. 1989. . // 9. 1972. . // 9. 1978. ., . // 5. 1978. . // 6. 1972. . // 2. 1980. . Ck , , // 2. 1973. n . // 11/12. 1988. . // 9. 1973. ., . // 1, 1983. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2).


248



[105] ., . -- -- // 1. 1986. [106] ., . // 1. 1985. [107] ., . // 1. 1982. [108] ., . // 6. 1981. [109] ., . /7 7 // 2. 1996. [110] . // 6. 1980. [111] . // 11. 1970. [112] . // 10. 1989. [113] . // 1. 1971. [114] . // 10. 1978. [115] . 4. 1975. [116] . , . . . // 8. 1988. [117] . // 2. 1986. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [118] ., . // 5. 1984. [119] . ? // 5. 1977. [120] . // 4. 1987. -- . // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [121] . // 7. 1978. [122] . // 6. 1990. [123] . // 3. 1988. [124] . ë ¨ // 9. 1976. [125] . // 10. 1972. [126] . ë ¨ // 6. 1983. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). [127] . . . . // 9. 1970. -- // 1. 1995. [128] ., . // 9. 1990. [129] . // 12. 1983. [130] . // 5. 1989. -- // . ( ). -- .: ë¨, 1995. -- (. ë¨. 3). [131] . // 9. 1979. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [132] . // 1. 1984.




249

[133] ., . // 7. 1985. -- // . . . 3. -- .: ë¨, 1999. -- (. ë¨. 3). [134] . // 8. 1971. -- // 5. 1991. [135] . ? // 7. 1986. [136] . // 4. 1998. [137] . // 6. 1999. [138] . // 6. 1991. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [139] . // 6. 1992. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [140] . // 2. 1974. [141] . // 4. 1997. [142] . // 5. 1970. [143] . // 10. 1977. -- // . . . 3. -- .: ë¨, 1999. -- (. ë¨. 3). [144] ., . // 3/4. 1993. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [145] . // 6. 1972. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). [146] ., . x2 = x // 11. 1989. [147] . ? // 3. 1978. [148] . ? // 11, 12. 1980. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [149] ., . // 4. 1990. [150] ., . // 6. 1984. -- // . . . 4. -- .: ë¨, 1999. -- (. ë¨. 5). [151] . // 2. 1972. [152] . , // 7. 1977. -- // 6. 1994. [153] ., . , . . . // 9. 1976. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). [154] . // 1. 1974. -- // 3. 1984. [155] . í // 8. 1987. [156] . // 4. 1987. [157] . ? // 7. 1972.


250
[158] [159] [160] [161] [162] [163] [164]


. 1600 // 4. 1973. . // 4. 1998. . ? // 4. 1994. ., . , // 9. 1981. . // 9. 1972. ., . e // 8. 1979. . // 1. 1981. -- // . . . 4. -- .: ë¨, 1999. -- (. ë¨. 5). ., . // 5. 1980. ., . // 10. 1978. ., . // 1. 1978. -- // . . . 3. -- .: ë¨, 1999. -- (. ë¨. 3). ., . // 12. 1984. . // 11. 1989. . // 5, 6. 1999. . // 5. 1975. ., . ë -- ¨, ë¨ // 7. 1984. . // 9. 1978. . // 11. 1974. ., . // 5. 1983. . // 12. 1979. . // 3. 1971. . // 6. 1973. ., . // 12. 1985. . // 3. 1982. -- // 2. 1983. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). . // 3. 1984. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). . // 4. 1982. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). . // 4. 1972. . // 9. 1984.

[165] [166] [167]

[168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180]

[181]

[182]

[183] [184]



[185] [186] [187] [188] [189] [190] [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208]

251

[209]

[210] [211] [212] [213]

[214] [215] [216] [217] [218]

. , 220 /127 3? // 11. 1978. . // 5. 1972. ., . 30 // 3. 1992. ., . // 2. 1994. . // 9. 1976. . // 2. 1998. . , // 12. 1991. . - // 9. 1988. . // 4. 1987. . // 3. 1991. . // 11. 1991. . // 3. 1988. . 2 e // 6. 1996. . -- // 6. 1997. . // 6. 1996. ., . // 7. 1979. . // 5. 1979. . // 2. 1997. ., . // 1, 3. 2000. . // 4. 1974. . // 6. 1973. . // 2. 1971. . í // 1. 1972. . // 7. 1991. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). . 4. 1992. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). . . . III. -- .: ë¨, 2001. -- (. ë¨. 4). . 10n ‘ 1 // 4. 1987. . // 10. 1988. . , // 6. 1990. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). . , , // 7. 1982. . // 6. 1994. . ë¨// 9. 1978. . // 2. 1990. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). // 8. 1989.


252



[219] . // 7, 10. 1983. -- // ë¨. ( ). -- .: ë¨, 1994. -- (. ë¨. 2). [220] . // 12. 1976. [221] . // 3. 1997. [222] . // 7. 1983. [223] . , , , // 8. 1981. [224] . sin nx cos nx // 6. 1986. [225] ., . // 6. 1970. [226] ., . // 6. 1971. [227] ., . // 12. 1973. [228] . // 1. 1976. -- // . -- .: ë¨, 2000. -- (. ë¨. 6). [229] . // 9. 1984. [230] . // 6. 1988. -- // . . . 3. -- : ë¨, 1999. -- (. ë¨. 3). [231] . // 10. 1979. [232] . // 9. 1979. -- // . . . 4. -- .: ë¨, 1999. -- (. ë¨. 5). [233] . // 3. 1980. -- // . . . 4. -- .: ë¨, 1999. -- (. ë¨. 5). [234] . // 9. 1990. [235] . // 2. 1971. -- // 1. 1992. [236] . // 2. 1974. [237] . // 7. 1984. [238] . // 9. 1981. [239] . ë¨ // 11. 1971. [240] . // 2. 1997. [241] . // 6. 1995. [242] . // 12. 1984. [243] [244] [245] [246] [247] [248] . . . . . . ë¨ // 21. 1998. ë¨ ë¨ // 3. 1998. 12 // 51. 1997. // 25. 1999. - // 48. 1997. - // 20. 1998.





, . . . . , , . 1. . . . . . 2. . . . . . . (). . . . . . [ -. .] 3. . . . . . . . . - . [ , . . .] 4. . . . . . . . . , , . . . [ .] 5. . . . . . . . [ . .] 6. . . . . . . . . . . . -


254



. . . . . . . . . í . . [ . .] 7. . . . ; . . . . . . [ . .] 8. . . . . [ . .] 9. . . . . . ( ). . . . . . [ .] 10. . . . . . n- . . . . . 11. , , . . . . . . . . . . . .





. , . , . , . 1 : [16], [38], [51]. 1 : [32], [35], [52]. 2 , : [7], [32], [35], [36], [58], [60], [65]. 3 : [1], [6], [9], [36], [58], [195]. 2 : [6], [14], [29]. 1 ?: [8], [30], [38], [51], [113], [148], [170]. 2 : [26], [35], [36], [38], [46], [51], [93], [177]. 3 , : [8], [11], [13], [19], [30], [38], [39], [40], [51], [66], [71], [84], [101], [113], [165], [170], [173], [224], [225]. 4 : [30], [51], [104], [170], [236]. 5 : [13], [74], [121], [232], [246]. 3 : [5], [13], [33], [34], [36], [37], [38], [42], [49]. 1 : [10], [16], [24], [30], [54], [80], [115], [116], [120], [192], [152], [154], [157], [159], [171], [176], [187], [215], [238]. 2 : [9], [16], [24], [30], [35],[43], [59], [99], [104], [158], [175], [203], [209]. 3 : [39], [54], [103], [104], [134], [192], [160], [203], [137]. 4 , : [9], [11], [41], [47], [54], [57], [83], [193], [172], [184], [196], [198], [235], [237]. 5 : [9], [24], [26], [28], [57], [88], [89], [175], [226], [243]. 4 : [5], [13], [33], [37], [38], [42], [49], [50]. 1 : [9], [26], [216]. 2 : [26], [142], [177], [229], [203]. 3 : [24], [26], [79], [82], [105], [114], [125], [138], [142], [144], [150], [162], [167], [203]. 4 : [10], [15], [24], [114], [125], [144], [152], [188], [203], [238].


256



[ [

[

[ [

[

[

[

[ [

[

5 : [9], [10], [56], [69], [211]. 6 : [138], [146], [221]. 5 , , : 1 : [5], [10], [16], [18], [26], [30], [32], 33], [37], [41], [48], [51], [136], [100], [119], [140], [151], [163], [167], [203], [219], 227], [228], [197], [199]. 2 : [15], [37], [127], [188], [218]. 3 : [1], [13], [34], [123], [191], [200], 210], [231], [244], [245], [247], [248]. 6 : [7], [21], [25], [32], [41]. 1 : [35], [43], [48], [50], [75], [90], [91], [94], [97], [98], 135], [139], [141], [147], [183], [212]. 2 : [20], [23], [30], [50], 81], [111]. 3 : [22], [30], [43], [50], [222]. 4 : [20], [23], [30], [43], [212]. 5 : [22], [23], [44], [48], [50], [156], [168], [194]. 6 : [23], [30], [101], [194]. 7 : [8], [20], [31], [63], [77], [126], [217]. 1 : [23], [32], [39], [41], [43], [47], [50], [51], [76], [107], 112], [122], [129], [178], [180], [186], [204], [208], [213], [224]. 2 : [17], [182]. 8 + : 1 : [26], [48], [109], [169], [242]. 2 : [17], [31], [77], [214]. 3 : [41], [44], [50], [230]. 9 : [41]. 1 : [25], [43], [44], [50], [97], [124], [126], [145], 153], [174], [181], [189], [190], [212], [230], [239]. 2 : [72], [130]. 3 : [19], [23], [34], [51], [55], [67], [92], [95], [110], [143], [161], [205]. 4 : [32], [36], [43], [50], [61], [64], [132], 135]. 10 : [2], [27], [32], [34], [38], [48], [49], [51], [179]. 1 : [3], [35], [41], [43], [47], [50], [53], [73], [78], 86], [87], [117], [128], [131], [139], [155], [164], [201], [202], [206], [207], [234], 240]. 2 : [73] 3 : [3], [60], [149], [233]. 4 : [133], [179]. 11 : 1 : [10], [12], [13], [30], [37], [47], [70], [85], [101], [108], 166]. 2 : [12], [13], [47], [57], [62], [100], [106].




257

3 : [6], [19], [30], [45], [47], [57], [101], [102], [107], [112], [118], [165], [220], [223]. 4 : [21], [30], [41]. 12 : [4], [9], [13], [39], [96].





I. A E I N P B Z K O X o H T M²

I I. 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 9 10 2 3 1 3 6 10 15 21 28 36 45

5 8 1 4 10 20 35 56 84 120

13 21 1 5 15 35 70 126 210

34 55 1 6 21 56 126 252

89 144 1 7 28 84 210

233 377 1 8 36 120

610 987 1 9 45

1597 2584 1 10

4181 6765 1

I I I. , , n 2n 3n Cn n! 0 1 1 1 1 1 2 3 1 1 2 4 9 2 2 3 8 27 5 6 4 16 81 14 24 5 32 243 42 120 6 64 729 132 720 7 128 2187 429 5040 8 256 6561 1430 40320 9 512 19683 4862 362880 10 1024 59049 16796 3628800

IV. (40 ): 2= 3= 5= , 1,41421 35623 73095 04880 16887 24209 69807 85697 . . . 1,73205 08075 68877 29352 74463 41505 87236 69428 . . . 2,23606 79774 99789 69640 91736 68731 27623 54406 . . .



= 3,14159 26535 89793 23846 26433 83279 50288 41972 . . . e = 2,71828 18284 59045 23536 02874 71352 66249 77572 . . . = 1,61803 39887 49894 84820 45868 34365 63811 77203 . . . (40 ): 2 3 5 e = 1,01101 01000 00100 11110 01100 11001 11111 10011 . . . = 1,10111 01101 10011 11010 11101 00001 01100 00100 . . . = = = = 10,001 11,001 10,101 1,1001 11 10 00 10 10 11 1 110 00 00 11 00 11 01 11 11 01 11 00 01 11 11 00 11 01 10 10 01 111 00 110 10 101 00 11 100 11 10 01 11 01 10 01 01 11 00 10 11 00 10 00 00 10 00 10 10 111 1 100 0 100 0 11 11 111 010 101 111 0 1 0 . . . . . .. .. .. .

259

V. 1. : T T T T T T T T
0 1 2 3 4 5 6 7

(x) (x) (x) (x) (x) (x) (x) (x)

= = = = = = = =

1, x, 2x2 - 1, 4x3 - 3x, 8x4 - 8x2 16x5 - 20 32x6 - 48 64x7 - 11

+ 1, x3 + 5x, x4 + 18x2 - 1, 2x5 + 56x3 - 7x,

U0 U1 U2 U3 U4 U5 U6 U7

(x) (x) (x) (x) (x) (x) (x) (x)

= = = = = = = =

1, 2x, 4x2 - 1, 8x3 - 4x, 16x4 - 12x2 + 32x5 - 32x3 + 64x6 - 80x4 + 128x7 - 192x5

1, 6x, 24x2 - 1, + 80x3 - 8x.

2. : F F F F F F F F F F
0 1 2 3 4 5 6 7 8 9

(x) (x) (x) (x) (x) (x) (x) (x) (x) (x)

= = = = = = = = = =

0, 1, x, x2 x3 x4 x5 x6 x7 x8

+ + + + + + +

1, 2x, 3x2 4x3 5x4 6x5 7x6

+ + + + +

1, 3x, 6x2 + 1, 10x3 + 4x, 15x4 + 10x2 + 1,

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9

(x) (x) (x) (x) (x) (x) (x) (x) (x) (x)

= = = = = = = = = =

2, x, x2 x3 x4 x5 x6 x7 x8 x9

+ + + + + + + +

2, 3x, 4x2 5x3 6x4 7x5 8x6 9x7

+ + + + + +

2, 5x, 9x2 + 2, 14x3 + 7x, 20x4 + 16x2 + 2, 27x5 + 30x3 + 9x.

3. : 1 1 1 1+x 1 1 1 1 + x + x2 1 + x + x2 1 1 + x + x2 + x3 1 + x + 2x2 + x3 + x4 1 + x + x2 + x3

1

1


260
VI. 4. : cos( ‘ x) = - cos x; cos sin( ‘ x) =



sin x;

‘ x = sin x; sin ‘ x = cos x; 2 2 3 3 cos ‘ x = ‘ sin x; sin ‘ x = - cos x; 2 2 tg ‘ x = ctg x; ctg ‘ x = tg x. 2 2 5. : cos(x ‘ y) = cos x cos y sin x sin y; sin(x ‘ y) = sin x cos y ‘ sin y cos x; tg(x ‘ y) = tg x ‘ tg y ; 1 tg x tg y ctg(x ‘ y) = ctg x ctg y 1 . ctg y ‘ ctg x

6. : sin 2x = 2 sin x cos x; cos 3x = 4 cos3 x - 3 cos x; cos 2x = 2 cos2 x - 1 = 1 - 2 sin2 x; sin 3x = 3 sin x - 4 sin3 x = sin x (4 cos2 x - 1); sin 4x = 4 sin x cos x (2 cos2 x - 1); ctg 2x = ctg 3x = ctg2 x - 1 ; 2 ctg x ctg3 x - 3 ctg x . 3 ctg2 x - 1

cos 4x = 8 cos4 x - 8 cos2 x + 1; tg 2x = tg 3x = 2 tg x ; 1 - tg2 x

3 tg x - tg3 x ; 1 - 3 tg2 x

7. : sin tg x =‘ 2 x =‘ 2 1 - cos x ; 2 cos x =‘ 2 1 + cos x ; 2

1 - cos x sin x 1 - cos x ; = = sin x 1 + cos x 1 + cos x

x 1 + cos x 1 + cos x sin x = = = . sin x 2 1 - cos x 1 - cos x 8. : ctg sin 2x = tg 2x = 2 tg x ; 1 + tg2 x 2 tg x ; 1 - tg2 x cos 2x = ctg 2x = 1 - tg2 x ; 1 + tg2 x 1 - tg2 x . 2 tg x



9. : - + cos ; 2 2 + - cos - cos = -2 sin sin ; 2 2 ‘ cos ; sin ‘ sin = 2 sin 2 2 sin( ‘ ) sin( ‘ ) tg ‘ tg = ; ctg ‘ ctg = . sin sin sin sin 10. : cos + cos = 2 cos 1 [cos( 2 1 cos cos = [cos( 2 1 sin cos = [sin( 2 tg + tg tg = ctg + sin sin = - ) - cos( + )]; - ) + cos( + )]; - ) + sin( + )]; tg . ctg

261

11. : sin2 x = sin3 x = 1 (1 - cos 2x); 2 cos2 x = cos3 x = 1 (1 + cos 2x); 2

1 (3 cos x + cos 3x); 4 1 1 sin4 x = (cos 4x - 4 cos 2x + 3); cos4 x = (cos 4x + 4 cos 2x + 3). 8 8 12. : a sin x ‘ b cos x = = = a2 + b2 § a 2 2 a + b § (sin x a2 + b2 § sin(x
2

1 (3 sin x - sin 3x); 4

a b sin x ‘ cos x + b2 a2 + b2 cos ‘ cos x sin ) = ‘ ),

=

b . , = arcsin a2 + b2 sin x ‘ cos x = 2 § sin x ‘ . 4 13. : sin x = a, |a| cos x = a, |a| 1 x = (-1)n arcsin a + n 1 x = ‘ arccos a + 2n (n Z); (n Z). (n Z); (n Z);

tg x = a x = arctg a + n ctg x = a x = arcctg a + n


262
VI I. 10 20 30 40 50 60 70 80 90 0 100 400 900 1600 2500 3600 4900 6400 8100 1 121 441 961 1681 2601 3721 5041 6561 8281 2 144 484 1024 1764 2704 3844 5184 6724 8464 3 169 529 1089 1849 2809 3969 5329 6889 8649 4 196 576 1156 1936 2916 4096 5476 7056 8836 5 225 625 1225 2025 3025 4225 5625 7225 9025 6 256 676 1296 2116 3136 4356 5776 7396 9216



7 289 729 1369 2209 3249 4489 5929 7569 9409

8 324 784 1444 2304 3364 4624 6084 7744 9604

9 361 841 1521 2401 3481 4761 6241 7921 9801

VI I I. 275 . 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 10 10 10 10 11 12 13 13 13 14 15 15 16 16 17 17 18 19 19 19 19 21 22 22 22 1 3 7 9 3 7 1 7 9 9 1 7 3 7 3 9 1 1 3 7 9 1 3 7 9 23 23 24 25 25 26 26 27 27 28 28 29 30 31 31 31 33 33 34 34 35 35 36 37 37 3 9 1 1 7 3 9 1 7 1 3 3 7 1 3 7 1 7 7 9 3 9 7 3 9 38 38 39 40 40 41 42 43 43 43 44 44 45 46 46 46 47 48 49 49 50 50 52 52 54 3 9 7 1 9 9 1 1 3 9 3 9 7 1 3 7 9 7 1 9 3 9 1 3 1 54 55 56 56 57 57 58 59 59 60 60 61 61 61 63 64 64 64 65 65 66 67 67 68 69 7 7 3 9 1 7 7 3 9 1 7 3 7 9 1 1 3 7 3 9 1 3 7 3 1 70 70 71 72 73 73 74 75 75 76 76 77 78 79 80 81 82 82 82 82 83 85 85 85 86 1 9 9 7 3 9 3 1 7 1 9 3 7 7 9 1 1 3 7 9 9 3 7 9 3 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 49 51 61 63 69 87 91 93 97 03 09 17 23 29 51 53 63 71 81 87 93 01 13 17 23 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 14 14 14 29 31 37 49 59 77 79 83 89 91 97 01 03 07 19 21 27 61 67 73 81 99 09 23 27 14 14 14 14 14 14 14 14 14 14 14 14 14 14 15 15 15 15 15 15 15 15 15 15 15 29 33 39 47 51 53 59 71 81 83 87 89 93 99 11 23 31 43 49 53 59 67 71 79 83 15 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 17 97 01 07 09 13 19 21 27 37 57 63 67 69 93 97 99 09 21 23 33 41 47 53 59 77



6 2 128 -- 29 í 33, 40, 42, 179 -- -- 84 í 89 -- 174, 182 253 -- - 13, 166 19 99 18, 21, 106, 182, 212, 235 106, 149 í 152 23, 156, 232 108 84 -- -- 6 147, 148, 162, 228, 236, 237 -- -- 147, 148 123 -- 124 124 -- 83, 203 43 -- -- 45 í 46 -- -- 45 -- 73 í 74 -- -- 69 -- -- 73 -- 42 -- () 41 í 47 71 -- 78 -- 37 -- 12 -- 35 39, 130 ë¨ 50, 79 -- ë¨ 79, 201 -- ë¨ 79, 201 -- 132 114 129 68 -- 43, 166 -- 45 -- 45 45 53 99 -- -- 110 149 í 153 -- -- 150 -- -- n 150 n- 102 -- 101 -- 91 -- -- 91 -- -- 91 -- 63 17 í 23, 163 -- -- 41 -- 41, 184 - 113 -- -- 114 155 -- 156 50, 79, 188 135 -- 124 -- , 75


264
-- 136 129 -- 135 -- 6 í 12, 165 -- 88, 89, 228 -- 133, 222 -- 70 81 í 98 -- 163 í 164, 254 -- 96 í 98 -- 155, 160, 231, 254 -- 108 -- 93 í 96, 146 í 148 -- 155, 160, 231, 254 -- 152 -- 103, 155, 160, 210, 254 -- 93 77 99 146 -- -- 147 -- -- 148 87 -- -- -- 29 32 42 140 í 148 -- 9 -- 145 -- 145, 226, 227 -- 142 -- í 143 -- 9, 144, 148 -- -- -- -- 143 -- 145 -- 148 -- 146 í 148 -- 142 - 78 í 80, 201


100 115 115 108 104 -- -- 33 113 -- -- 113 -- 112 - 109 í 110 -- 108 í 110 108 16, 146 69 -- 45 - 13, 166 108 153 í 157 -- 40 -- 77 -- 36 87 -- 13 -- 13 69 -- 45 151 -- 108 í 110 19, 23 63 í 65 -- -- 2, 4 8 63 -- -- 19 64 -- -- 3 9 65 -- -- 10k n ‘ 1 64 -- -- 65 14 í 16, 52, 58, 190, 198 157 í 163 -- -- 160 -- -- -- 160 -- -- -- 160 -- -- 162 -- -- -- 160 -- -- -- 160



83, 124 115 -- 115 115 115 42 -- 161 í 162 16 81 5, 73 í 74 108 -- 157 í 163 42 í 43 -- 53 -- 37 21 53, 67 -- -- 60, 67 -- 65 í 66 -- 21 -- -- 65 -- -- 36, 73, 75 í 80, 201 í 202, 236, 254 -- -- 73 í 74 -- -- 6 -- -- 75 í 80 -- -- 8 -- -- 38, 167 -- , 136 í 139 17 53 í 68 -- 56 - 132 -- - 132 -- 9, 145, 148 -- 145 -- - 133 -- 9, 145, 148 -- 145 -- 145 í 146 114 88, 205 159, 234

265
84, 203, 204, 206 -- 46 -- 128, 222 -- 93 í 96, 212 -- -- 81, 203 -- 57 -- , 57 -- í 107 -- 27 -- 86 -- -- -- 65 í 68, 196 -- 57 -- 120, 214 -- -- 120 -- 46 -- -- 222 -- 40 -- 46 -- 57 -- 153 -- 38 -- 91, 104, 210 -- 93 -- 109 -- 104 -- -- 33 -- 19, 58 -- 120 -- -- 120 -- 58 í 62, 192 í 195 -- 11, 35, 58 í 62, 195, 198 48 61 -- 37, 183 -- 101 , 22, 174 -- 20, 22, 39 í 41, 175, 253 99 126 í 127 -- 255 í 256 101


266
122 í 126 -- -- 124 -- 153 í 157 83 -- -- 124 7, 253 n- 154 -- 26, 163, 236 -- 39, 155, 160, 182 -- 23 í 25, 60, 175 í 176 -- 121, 214 -- -- 128 -- 123, 124, 218 í 219 -- 36, 181, 189 -- 102, 105, 209, 210 -- 185 -- , 152 -- 121 -- 71 -- 89 -- 89 -- 105 (n) 76, 200, 236 -- (n) 34 í 35, 181 -- (n) 34, 61 -- 35 -- () 144 -- 153 -- 211 -- 33 í 36 -- 105 -- 157 í 164 -- -- 160 -- -- -- 160 -- -- -- 160 -- -- -- 162


160 -- -- -- 160, 235 -- (n) 60 í 62, 66, 67, 193 í 194, 198 10, 78, 200 63 67 -- 57 -- 35 -- , en 28, 55 -- 45 -- 69 í 73 -- 62 -- Cn 25 í 26, 163, 253 -- 99 -- 99 í 110 -- Ln 40, 135, 155, 160, 183, 223 -- 29, 35, 181 -- 69 -- 27 í 29, 55, 257 -- -- 28, 57 -- 69 í 73 -- 35, 63, 181 -- 27 í 28 -- fn 28, 31 -- , Fn 36 í 41, 47, 119, 134 155, 160, 182 í 184, 222, 253 5, 108, 135, 253 -- 2 46, 73 -- 1/7 74 -- 2 128, 131, 154, 253 -- e 5, 72, 73, 142, 198, 223, 253 -- i 5, 99 -- 69 -- , 5, 39, 160, 213, 253 19 151, 159 -- 105



. . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 1. . . . . . . . . . . . . . . . . . . . . 2. , . . . . . . . . . 3. . . . . . . . 2. 1. ? . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . 3. , . . . . . . . 4. . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . 3. 1. . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . 4. , . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . 4. 1. . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . 4. . . . . . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . 6. . . . . . . . . . . . . 5. , , 1. . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . 3. . . . . . . 6. 1. . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . 4. . . . . . . . . . . . 5. . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . 7. 1. . . . . . . . . . . . . . . . . . 2. . . . . . . . 8. + 1. . . . . . . . . . . . . . . 2. . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . ...... ...... ...... ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 6 6 7 10 13 13 14 16 23 25 27 27 29 33 36 41 48 48 51 53 58 63 66 70 70 74 76 83 83 86 92 93 95 98 101 101 110 113 113 114 118

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01335 24.03.2000 . 27.6.2002 . 60 ½ 90 1/16. 1. . . . 16,5. 2000 . 121002, , ., 11 ë¨. 248640, . , . , . 5.
ë ¨, ., . 11. . 241í72í85. E-mail: biblio@mccme.ru