.., ..

September 3, 2003

51 21.1

45

.. ..45 .

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

, . , . - .

, , 19952000 - . ...

21.1

ISBN 5-94057-038-0c .., .., 2002.c , 2002.

-, , - . , . .

, 1995 2000 .., .. ..-. . , , , .

, - . ..-, , . , -, . . .

, - . , .

-, , . , . . , , , -. .

-. .

- . , .

-, 18 ( . ..), . .., .., .., , , . .. TEX- -.

, -, . - : 117630, , . . , . 8, .

..alpha@pisem.net

..ustinov@mech.math.msu.su

, .

.N Z Q 70[x] x ( , -

x){x} x: {x} = x [x]n! : n! = 1 2 . . . n 7{xn} x1, x2, . . . , xn, . . .b | a b a 8a ... b a b 8a b mod m a b m 53a m 53(ak . . . a0)q q- 6(a1, . . . , an) a1, . . . , an 29[a1, . . . , an] a1, . . . , an 32[a0; a1, . . . , an] 42(n) 60(n) n 34(n) n 34Fn 36i i =

1 101

C 101z z 101arg z z 101|z| z 101pi e 73 = (

5+ 1)/2 39

151Akn k- n 16Akn k- n 16Pn n 17Ckn k- n 17Ckn k- n 17Cn 25En n: En = 11 . . . 1

n

74

1

1. . ,

1, , - n, n+1, .

1.1. . , a b b 6= 0, q r ,

a = bq+ r, 0 6 r < |b|.

1.2. . , q > 2 n

n = akqk + ak1q

k1 + . . .+ a1q+ a0,

0 6 a0, . . . , ak < q. (. 3.125, 11.68.). ak, ak1, . . . , a1, a0 n q--

, n = (akak1 . . . a1a0)q. -

(akak1 . . . a1a0)10 = akak1 . . . a1a0.

1.3. {an} = a0, a1, . . . , an, . . . -, T

an+T = an (n > 0).

, - t, T t .

1.4. . , - . (. 12.1.)

1) - .

2. , 7

2) - .

3) 1 - , .

4) , - a, , k, , a 6 k < n n, k > a.

5) ( .) , - 1 2, , n > 1, 2n n 1, .

1.5. x , x + 1x . ,

n xn + 1xn

. (. - 7.46.)

1.6. x1, . . . , xn. ,

(1+ x21) . . . (1+ x2n) .(. 7.14.)

1.7. A1, A2, . . . , An, . . .

A1 = 1, A2 = 1, An = An1 2An2 (n > 3).

, n > 2 2n+27A2n .

2. , . n! ( n )

1 n:

n! = 1 2 . . . n. , 0! = 1.

1.8 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

.

1.13. 12

1 3 +22

3 5 + . . .+n2

(2n 1)(2n+ 1)=n(n+ 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 6 a1 6 1, 0 6 a2 6 2, 0 6 a3 6 3, . . .

1.16. a0, a1, . . . , an, . . . :

a0 = 2, a1 = 3, an+1 = 3an 2an1 (n > 2).

.. a b . b -

a, q , a = bq. a b, q a b.

b a b | a a ... b (a b). , b 6= 0.

b a, b - a., 1.17 1.24, -

n.1.17. 10n + 18n 1 ... 27.1.18. 11n+2 + 122n+1 ... 133.1.19. 25n+3 + 5n 3n+2 ... 17.1.20. n3 + 5n ... 6.1.21. 62n+1 + 1 ... 7.1.22. 32n+2 + 8n 9 ... 16.1.23. 4n + 15n 1 ... 9.1.24. 23n + 1 ... 3n+1.1.25. , n ,

3n , 3n.

2. , 9

1.26*. 1 2n n+1 . , , .(. 2.34.)

1.27.

1x

1!+x(x 1)

2! . . .+ (1)n

x(x 1) . . . (x n+ 1)n!

= 0.

1.28 1.36 - n.

1.28. 112

+1

22+1

32+ . . .+

1

n2< 2. (. 7.81.)

1.29. 11+

12+ . . .+

1n>n.

1.30. (2n)!(n!)2

>4n

n+ 1.

1.31. 1n+ 1

+1

n+ 2+ . . .+

1

2n>13

24(n > 1).

1.32. . (1+x)n > 1+nx x > 1.1.33. 2n > n.

1.34. 1 3 5 . . . (2n 1)2 4 6 . . . 2n 6

12n+ 1

.

1.35. nn+1 > (n+ 1)n (n > 2).

1.36. |x1 + . . .+ xn| 6 |x1| + . . .+ |xn|, x1, . . . , xn.

1.37*. - .

x1 + . . .+ xnn

> nx1 . . . xn, x1, . . . , xn .

1.38. 2m+n2 > mn, m n.

1.39. n :) n! > 2n; ) 2n > n2.

1.40.

23 1

23 + 1 3

3 1

33 + 1 . . . n

3 1

n3 + 1(n > 2).

10 1.

3. 1.41. 16 16

. , .

1.42. I. - 8 , . , , .

, . - ( )? (. 5.71.)

1.43. II. - 1, 2, 3. , - 1- 3-. -, 1- 3- ?( 2- . , .)

1.44. III. -, 1.42 : ?

1.45. , n n, .

1.46. , n n, .

1.47. . ) -, , . , : , . 11 , - 11 , . , - ?

) , - -, n ?

1.48. - . , , . (. 3.72.)

3. 11

1.49*. . ) , , . : ? , - . ? ( .)

) , , 1000?

1.50. n -, , ?

1.51. n , , . ?

1.52. n , - , ?

1.53*. n ? ?

1.54. . -, , . ( - , .)

1.55. n-. , n- ( ) - . , n- (n 2)pi.

1.56. 100 100 4 , 2 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 , a b m+ n .

. a m -, b ( a)n , a 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. , . - ?

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 , , - .

- Akn Akn .

2.35. :) Akn = n(n 1) . . . (n k+ 1); ) Akn = nk.

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- , -, .

-.

- Ckn Ckn .

2.45. . , ?

2.46. n . - ?

2.47. n . - . - ?

18 2.

2.48. a b A1,A2, . . . , Am B1, B2, . . . , Bn . -, AiBj (1 6 i 6 m, 1 6 j 6 n), , -?

2.49*. . 9. . - , , - , 6 ?

, n , m (m 6 n).

2.50. 7 , 9 -. ?

2.51.

) Ckn =n!

(n k)!k!; ) Ckn = Ckn+k1 =

(n+ k 1)!

(n 1)!k!.

2.52. , Ckn - k- n .

2.53. .

(x+ y)n = C0nxn + C1nx

n1y+ C2nxn2y2 + . . .+ Cnny

n.

Ckn , x+ y.

2.54.

) (2+

43)100; ) (

2+

33)300?

2.55*. , a n, n + 1, nn + 1, nnn + 1, . . . a.

2.56. :) 10-; ) k- (k > 3)?

2.57. n- . - . - . ?

3. , 19

2.58. . , . : ) ; ) ; ) ; ) -; ) ; ) ?

, . - .

2.59. . m n , -, n 1 m 1 . , ( (0; 0)) ( (m; n))? (. 2.77.)

2.60. 10 10 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)xk11 . . . x

kmm

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 , , , , , ?

. ,

C00

C01 C11

C02 C12 C

22

C03 C13 C

23 C

33

. . . . . . . . . . . . . . . . . . . . .

1

1 1

1 2 1

1 3 3 1

. . . . . . . . . . . . . . . . . . . . .

(. 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. :) C05 + 2C15 + 22C25 + . . .+ 25C55;) C0n C1n + . . .+ (1)nCnn;) C0n + C1n + . . .+ Cnn.

2.77. :) Cmr Ckm = CkrC

mkrk ;

) Cm+1n+1 = Cmn + Cm+1n ;) Cn2n = (C0n)2 + (C1n)2 + . . .+ (Cnn)2;) Ckn+m = C0nCkm + C1nCk1m + . . .+ CknC0m;) Ckn = C

k1n1 + C

k1n2 + . . .+ C

k1k1.

: , Ckn k- n ; , Ckn xk (1 + x)n; 2.59.

2.78. .

Ck1n1 Ck+1n Ckn+1 = Ckn1 Ck+1n+1 Ck1n .2.79. 120

. ?

2.80. (x + y)n 240, 720, 1080. x,y n.

22 2.

2.81. . , n

n = C1x + C2y + C

3z,

x, y, z , 0 6 x < y < z.2.82. 10 14 . -

, .2.83. m n ,

Cm+1n+1 : Cmn+1 : C

m1n+1 = 5 : 5 : 3.

2.84. (1+3)100

?2.85. ,

1, 2, 3, 4 5, :) ;) ;) -

?2.86. 5

5 10- , ?

2.87*. n- . , . ? - ?

2.88. .11

12

12

13

16

13

14

112

112

14

15

120

130

120

15

16

130

160

160

130

16

, - . - . - .

4. 23

, . , .

2.89. :

) 11

=1

2+1

6+1

12+1

20+1

30+ . . . ;

) 12

=1

3+1

12+1

30+1

60+

1

105+ . . . ;

) 13

=1

4+1

20+1

60+

1

140+

1

280+ . . .

2.90. 1

12+1

30+1

60+

1

105+ . . .

.2.91.

) 11 2 +

1

2 3 +1

3 4 +1

4 5 + . . . ;

) 11 2 3 +

1

2 3 4 +1

3 4 5 +1

4 5 6 + . . . ;

) 0!r!

+1!

(r 1)!+

2!

(r 2)!+

3!

(r 3)!+ . . . (r > 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, x Aj,0, 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| . . . |An1 An| + . . .+ (1)n1|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, a1 + a2, . . . , a1 + a2 + . . .+ a2n

?

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 . . . , an1},

. :

Cn = Cn2n+1

1

2n+ 1= Cn2n

1

n+ 1=

(4n 2)!!!!

(n+ 1)!,

(4n2)!!!! = 2 6 10 . . . (4n2), . (. 3.105.)

2.116. . -,

Cn = C0Cn1 + C1Cn2 + . . .+ Cn1C0.

(. 11.92.)

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.)

3.9. , n d 6

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 < p1p2 . . . pn.

3.24. , p1p2 . . . pn + 1 -?

3.25. . :

e1 = 2, en = e1e2 . . . en1 + 1 (n > 2).

en ? (. 4.79.)

3.26. . , an + 1 , a ... 2 n = 2k. ( fk = 22

k+ 1

.

2. 29

3.27. , fn 2fn 2.

3.28. , fn n > 1 .

3.29. . , an 1 , a = 2 n.

q = 2p 1 .

3.30. Pn(x) = anxn+. . .+a1x+a0 - (n > 1, an 6= 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 > 0m1 - m0. , k > 1

30 3.

a0, a1, . . . , ak1 m2, . . . ,mk , m1 > m2 > m3 > . . . > mk > 0,ak > 1,

m0 = m1 a0 +m2,m1 = m2 a1 +m3,m2 = m3 a2 +m4,. . . . . . . . . . . . . .

mk2 = mk1 ak1 +mk,mk1 = mk ak,

(m0, m1) = mk.3.37. , s k 1 0 -

us, vs , usms +ms+1vs = d, d = (m0, m1). , u v :

m0u+m1v = d.

(. 6.67.)3.38. (a, b) = 1 a | bc. , a | c.3.39. (1 . . . 1

m

, 1 . . . 1 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

2,a+ c

2,a+ b

2

)= (a, b, c).

2. 31

3.47. 40 18. - , , . ? , ?

3.48. x y 3x + 2y 23., 17x+ 19y 23.

3.49. , - n:

) 2n+ 13n+ 7

; ) 2n2 1

n+ 1; ) n

2 n+ 1

n2 + 1.

3.50. n

) n2 + 2n+ 4

n2 + n+ 3; ) n

3 n2 3n

n2 n+ 3?

3.51. n

) n4 + 1

n2 + n+ 1; ) n

3 + n+ 1

n2 n+ 1

?

3.52. n > 1 n3 3 n 1.

3.53. 3m n5n+ 2m

,

, m n .

3.54. , m 6= n :) (am 1, an 1) = a(m,n) 1 (a > 1); ) (fn, fm) = 1,

fk = 22k+ 1 . (. 3.39, 3.122, 6.69.)

3.55. , 22n 1 n - .

3.56. , pn+1 6 22

n+ 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

a2 ab+ b2=3

13.

3.67. , (a1, a2, . . . , an) = 1,

a1x1 + a2x2 + . . .+ anxn = 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 6 x 6 b 1. N(x, y) == ax+ by.

) , c (x, y) (0 6 x 6 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 6 c 6 (n+ 1)ab a b.

(. 1.48.)

3.74*. 81x ++ 100y, x, y , . , - . , .

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 = p11 . . . pss , b = p11 . . . pss , p1, . . . , ps, 1, . . . , s, 1, . . . , s > 0. :

) (a, b) = pmin(1,1)1 . . . pmin(s,s)s ;) [a, b] = pmax(1,1)1 . . . pmax(s,s)s ;) (a, b)[a, b] = ab.

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) [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 = p11 . . . pss , (n) . :

) (n) = (1 + 1) . . . (s + 1); ) (n) = p1+11 1

p1 1 . . . p

s+1s 1

ps 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) 6 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 = 2k1(2k 1) .

3.95*. . , n - , n = 2k1(2k 1), p = 2k 1 .

- .

. m n , m n , , n m. , m n ,

(m) m = n, (n) n = m,

(m) = m+ n = (n).

3.96. . , p = 3 2k1 1, q = 3 2k 1 r = 9 22k1 1 , m = 2k p q n = 2k r. - .

3.97. , ) (n) > 3n; )* (n) > 100n?3.98. . n = 2p1p2,

p1 p2 , , (n) = 3n.

36 3.

3.99. , d-. , , d,

[

d

].

3.100. , -

d [

d

]=[[]

d

].

3.101. . n! n! = p11 . . . pss .

k =[n

pk

]+[n

p2k

]+[n

p3k

]+ . . .

3.102. , p n! ,

[n

p 1

].

3.103. n :

n = 2e1 + 2e2 + . . .+ 2er (e1 > e2 > . . . > er > 0).

, n! 2nr, 2nr+1.3.104. .

p n p- :

n = akpk + ak1p

k1 + . . .+ a1p1 + a0.

, p, p n!, n, p ak.

3.105. 3.101 , 1

n+ 1Cn2n (n > 0) . (. 2.115.)

3.106. , (2m)! (2n)!m!n! (m+ n)!

(m, n > 0) .

3.107. r, n!2nr

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, Fn+2 = Fn+1 + Fn (n > 0).

4. , 37

(1202 .) - ().

3.108. . - . , , , ?

3.109. , . , - , . . , . - n- ? (. 3.114.)

3.110. 6 , :

, , , - 12 . ?

3.111. F1, F2, . . . , Fn, . . . ?

3.112. .

Fn+1Fn1 F2n = (1)

n (n > 0).

n? (. 12.13.)

3.113. :) F1 + F2 + . . .+ Fn = Fn+2 1; ) F2 + F4 + . . .+ F2n = F2n+1 1;) F1 + F3 + . . .+ F2n1 = F2n; ) F21 + F22 + . . .+ F2n = FnFn+1.3.114. , n > 1 m > 0

Fn+m = Fn1Fm + FnFm+1.

: - 3.109. , .

38 3.

3.115. ) F2n+1 = F2n + F2n+1;) Fn+1Fn+2 FnFn+3 = (1)n+1;) F3n = F3n + F3n+1 F3n1.

3.116. F4n+2 FnFn+1Fn+3Fn+4.

3.117.

1

1 2 +2

1 3 + . . .+Fn

Fn1 Fn+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.

3.121. , Fn1 Fn(n > 1) .

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

3.123. 8 , . , -.

3.124. n, - 0 1, 1 . , Fn+2. - 3.109.

3.125. . , - n, Fm,

n =

mk=2

bkFk,

b2, . . . , bm 0 1, , bkbk+1 = 0 (2 6 k 6 m 1).

4. , 39

:

n = (bk . . . b2)F.

(. 12.14, 4.193 .)3.126. . :

Fn =n n

5,

= 1+5

2 , = 1

5

2( ) . (. 11.43, 11.75.)

3.127. :

2n1Fn =

[(n1)/2]k=0

Cn2k+15k.

(. 4.129.)3.128. , Fn

n

5,

Fn =

[n5

+1

2

].

3.129. . :

C0n + C1n1 + C

2n2 + . . . = Fn+1.

, , -, , (. , II 2.67, 3.124, 11.44 11.45.)

3.130. :

Sn = C0n C

1n1 + C

2n2 . . .

(. 11.44, 11.45.)3.131. 1 2,

n? , n = 4, :

11111, 112, 121, 211, 22.

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 = Fn1 + Fn+1;) 5 Fn = Ln1 + Ln+1;) F2n = Ln Fn;) L2n+1 + L2n = 5F2n+1;) Fn+2 + Fn2 = 3Fn.

(. 9.79, 11.41.)3.134. -

1 2. , n (n > 3)? , .

3.135. Ln . (. 11.77.)

3.136.

) 47+ 3

5

2

4

7 3

5

2= 1;

) 511+ 5

5

2+

9

76 34

5

2= 1.

, .

3.137*. :) x2 xy y2 = 1;) x2 xy y2 = 1.

3.138. ) , m > 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

5. 41

k m0 m1 k 6 5t.

3.141. .

1

1 1

1 1 1

1 2 2 1

1 3 6 3 1

1 5 15 15 5 1

1 8 40 60 40 8 1

1 13 104 260 260 104 13 1

- Fkn,

Fkn = Fn Fn1 . . . Fnk+1Fk Fk1 . . . F1 (0 6 k 6 n).

) , - Fkn = Fknk.

) , Fkn Fk1n1 Fkn1 ( ) 2.77).

) , .

3.142*. . A1, A2, . . . , ,

(Am, An) = A(m,n) (m, n > 1).

,

Akn = An An1 . . . Ank+1Ak Ak1 . . . A1 . (. 8.89.)

5. . a0 , a1, a2, . . . , an

an > 1. n- ()

a0 +1

a1 +1

a2 + . . . + 1an

(3.1)

42 3.

( [a0; a1, a2, . . . , an]). a1, a2, . . . , an (3.1).

3.143. 14713

129111

.

3.144. PnQn

= [1; 1, . . . , 1 n

]. Pn Qn?

3.145. ?

3.146. . (. 2) - . m0 m1 (m1 6 m0) - a0 m1 m1, m1 m2 (m2 6 m1) a1 m2 m2, . . , . (. 3.157.)

- m1/m2.

3.147. n -, n .

3.148. . - a/b , a/b. a b 3.146?

3.149. a0 , a1, . . . , an . -

P1 = 1, P0 = a0, Pk = akPk1 + Pk2 (1 6 k 6 n);Q1 = 0, Q0 = 1, Qk = akQk1 +Qk2 (1 6 k 6 n).

, k = 0, 1, . . . , n :

) PkQk

= [a0; a1, a2, . . . , ak];

) PkQk1 Pk1Qk = (1)k+1;) (Pk, Qk) = 1.

.

Pk

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

5. 43

(3.1).

3.150. :

) PkQk2 Pk2Qk = (1)kak (k > 2);) PkQk

Pk1

Qk1=

(1)k+1

QkQk1(k > 1);

) Q1 < Q2 < . . . < Qn;

) P0Q0

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. , 11999 + 21999 + . . .+ 161999 17.

4.35. , - . , 13.

52 4.

4.36. , 1 2001 - , .

4.37. , 77777

7777

... 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 6 k 6 p 1, Ckp ..

. p.4.43. , 4.42:

Ckn ... n 1 6 k 6 n 1, n .

4.44. , p 1 6 k 6 p 2, Ckpk+1 C

k2pk1 ..

. p. ?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

a m. a.

4.52. , a mt+ a, t .

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

0 0 1 2 3 4 5

1 1 2 3 4 5 0

2 2 3 4 5 0 1

3 3 4 5 0 1 2

4 4 5 0 1 2 3

5 5 0 1 2 3 4

0 1 2 3 4 50 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 1

m = 7, 8, . . . , 13.

4.57. a b (mod m) ac bc (mod m) -?

4.58. ab (mod m) acbc (mod mc)?4.59. 100 . 1 5

. , . - . (. 5.81.)

4.60. - 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. ,

p3 + 2 .4.72. 6

, .

4.73. 7777

.

4.74. n2 + 1

3 n?

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) = anxn + . . .+ a1x+ 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. 57

4.97. . , p

(p 1)! 1 (mod p).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

a1a2 + a2a3 + . . .+ an1an + ana1 = 0.

, n ... 4. F(x1, . . . , xn) -

x1, . . . , xn. ,

F(x1, . . . , xn) = 0 (4.1)

F(x1, . . . , xn) 0 (mod m) (m > 1). (4.2), m (4.2) , (4.1) .

4.102. , :

) x2 + y2 = 2003; ) 15x2 7y2 = 9;) 12x+ 5 = y2; ) x2 5y+ 3 = 0;) x2 + 7y3 + 6 = 0; ) x41 + . . .+ x414 = 1999;) x2 + y2 + z2 = 1999; ) 8x3 13y3 = 17.

58 4.

4.103. , .

4.104. . ,

Hn = 1+1

2+1

3+ . . .+

1

n

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).

4.

4.108. n, 10n 1 ) 7; ) 13; ) 91; ) 819.4.109. , ) 111 . . . 1

12

... 13; ) 111 . . . 1 16

... 17.

. p p - a.

ap1 1 (mod p).4.110. , (1+ 1+ . . .+ 1)p -

.4.111. p , p 6= 2, 5. ,

111 . . . 11, p. : ,

, .4.112. n n2001 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. ,

a12 + b12 + c12 + d12 + e12 + f12 ... 13.

, abcdef ... 136.

4.117. -. p > 2 . - p- a ? (, , .) - .

4.118. 103 ) 5102; ) 3104.

4.119. , 30239 + 23930 .

4.120. 2571092 + 1092?

4.121. , p , p 6= 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 p

2 . . . 2 p

3 . . . 3 p

. . . 9 . . . 9 p

123 . . . 9 ... p.

4.125. p > 2 a, p, x2 a (mod p). ,

a(p1)/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, 2, 3, . . . , b

b+ 1, b+ 2, b+ 3, . . . , 2b

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

(a 1)b+ 1, (a 1)b+ 2, (a 1)b+ 3, . . . , ab.

b? a? , .

. -m , ,

4. 61

. (, 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 = p11 . . . pss .

(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) = nk1(n)?4.143. ) (5x) = 100; ) (7x) = 294; ) (3x 5y) = 600.4.144. , (m, n) > 1. (m n) (m)

(n)? (. 3.93.)4.145. a = 2(a).4.146. , n > 2, -

n.4.147.

n.4.148. n

. , n = 12 :

0

1,1

12,1

6,1

4,1

3,5

12,1

2,7

12,2

3,3

4,5

6,11

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:a) a5 a ... 30; ) a11 a ... 66;) a17 a ... 510; ) a73 a ... 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 = p11 . . . pss m . (m) (p11 ), . . . , (pss ):

(m) = [(p11 ), . . . , (p

ss )].

, a , (a, m) = 1,

a(m) 1 (mod m).

5.

4.163. 3, 9 11. N

N = anan1 . . . a1a0.

:) N ... 3 an + an1 + . . .+ a1 + a0 ... 3;) N ... 9 an + an1 + . . .+ a1 + a0 ... 9;) N ... 11 an an1 . . . 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. 65

4) 19, 19 | N, 19 - N. .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 = anan1 . . . a1a0, ri- 10i m (i = 0, . . . , n). , N m , M = anrn + an1 + . . .. . . + a1r1 + a0 m.

4.190. - 3, 9, 6, 8, 12, 15, 11, 7, 27, 37.

ri , , , 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 6= j, m = m1 . . .mn, a1, . . . , an, A . x ,

x a1 (mod m1),. . . . . . . . . . . . . .x an (mod mn)

(4.3)

A 6 x < A+m. (. 6.51.)

. - ( ) -. - .

4.194. n an = n2 + 3n + 1 55?

4.195. :) 1910 66; ) 1914 70; ) 179 48; ) 141414 100.

4.196. m1, . . . , mn .,

a b (mod m1 m2 . . . mn)

a b (mod m1),a b (mod m2),. . . . . . . . . . . . .a b (mod mn).

6. 67

4.197. m1, . . . ,mn . -, x = (m2m3 . . .mn)(m1)

x 1 (mod m1),x 0 (mod m2),. . . . . . . . . . . . .x 0 (mod mn).

4.198. , x, (4.3).

4.199. .

4.200. x, :

){x 3 (mod 5),x 7 (mod 17); )

{x 2 (mod 13),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 = x1m2 . . .mn +m1x2m3 . . .mn + . . .+m1m2 . . .mn1xn

m1m2 . . .mn. .

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

68 4.

4.207. , m1, . . . , mn -. , c

m1 . . .mn

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 3, , 2 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,a1a2 . . . akb1b2 . . . bnb1b2 . . . bnb1b2 . . . bn . . . ,

b1b2 . . . bn , - , . b1b2 . . . bn , a1a2 . . . ak , n

= 0,a1a2 . . . ak(b1b2 . . . bn).

5.1. :

) 17; ) 2

7; ) 1

14; ) 1

17.

5.2. a b , 0,aaaaa . . . = 0,bbbbb . . .

5.3. 1

49= 0,0204081632 . . .

, 2?

5.4. . - :

1

243= 0,004115226337448 . . .

1. 71

5.5. - :

) 0,(12) + 0,(122); ) 0,(3) 0,(4); ) 0,(9) 0,(85).5.6. , ,

.

5.7. n 1n

?

5.8. ) = 0,101001000100001000001 . . . ;) = 0,123456789101112131415 . . . ?

5.9. , , .

5.10. ,

2. , ,

, .

5.11. n, 1n

1n+ 1

.

5.12. , [2k2] (k = 0, 1, . . . )

.

5.13. :) 317; ) 3

3

2; ) sin 1;

)2+

3; ) cos 10; ) log2 3.

)2+

3+

5; ) tg 10;

5.14. . , ) 8x4 + 4y4 + 2z4 = t4; ) x2 + y2 + z2 + u2 = 2xyzu;) x2 + y2 + z2 = 2xyz; ) 3n = x2 + y2

.

5.15. , x3 + x2y + 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

13+

48.

5.20.

3

6+

847

27+

3

6

847

27= 3.

5.21. 17 :

) 11+

2+

12+

3+ . . .+

199+

100

;

)2+

3/2

2+2+

3+

2

3/2

22

3;

)

|402 57|

402+ 57.

5.22. :) 320+

392+

320

392;

) 352+ 7

352 7;

)x+ 6

x 9+

x 6

x 9 (9 6 x 6 18).

5.23. . 10+

24+

40+

60.

5.24. . :a

b =

a+

a2 b

2a

a2 b

2.

(. 7.15.)5.25*. ,

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 6= 4 n- .

5.31. , (2;3)

.5.32. :) 11+

a; ) 1

2+2+

3; ) 1

3a+

3b+ 3

c.

) 1a+

b+

c; ) 1

a+4ab+

b;

) 11 3

a+

3a2; ) 1

42+

44+

48+ 2

;

5.33. n (2 + 1)n (

2 1)n

?5.34. :

)

2+

2+ . . .+

2+

6

10

=10242+

3+

10242

3;

)

2+

2+ . . .+

2+

2

n

= 2 cos pi2n+1

.

5.35. e. e - e = lim

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

n(2+ 1/2! + 1/3! + . . .+ 1/n!);) e = 2+ 1/2! + 1/3! + . . .+ 1/n! + rn, 0 < rn 6 1/(n!n);) e .

(. 11.73, 7.51).5.36*. e . N ,

. , k .

74 5. , ,

, N > [k! e], , - . (. 2.33.)

5.37*. {xn} {dn}

x1 = 1, xn+1 = [2xn(xn + 1) ], dn = x2n+1 2x2n1 (n > 1).

, 2

2 = (d1, d2d3 . . . )2. (

2

.)

2. -

, .. n , -

n En = 11 . . . 1 n

.

. - .

5.38. , 10n 1

m= a1a2 . . . an

, 1/m 1/m = 0, (a1a2 . . . an).

5.39. , (m, 10) = 1, En, m. ?

5.40. {p/q} {10kp/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 =N1N2 , 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(m1m2) = [L(m1), L(m2)]. 1/m1 + 1/m2?

5.51. , .

5.52. , .

5.53. , , - 5, 6 8 .

5.54. m m = 2a5bm1, (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. : 1. 0

) 100; ) n? (. 6.77.)5.64. . ,

x n. , , n = 16,

3. 77

15 x16 = x x . . . x, :

x1 = x x = x2, x2 = x1 x1 = x4, x3 = x2 x2 = x8, x4 = x3 x3 = x16.

n = 2e1 + 2e2 + . . .+ 2er (e1 > e2 > . . . > er > 0).

, 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 3 5 7

9 11 13 15

17 19 21 23

25 27 29 31

2 3 6 7

10 11 14 15

18 19 22 23

26 27 30 31

4 5 6 7

12 13 14 15

20 21 22 23

28 29 30 31

8 9 10 11

12 13 14 15

24 25 26 27

28 29 30 31

16 17 18 19

20 21 22 23

24 25 26 27

28 29 30 31

, . , , - ? , 1 63?

5.67. . ) 27 ( ). . - , ( , , , . .). , . , - . . , , ?

78 5. , ,

) , 3n (n < 9) ?

5.68. : 1, 2 3. , , . , ?

5.69. 1 200. ,

) ;) , ?

5.70*. 1 200, . ( ) . , ?

5.71. , A

A = a0 + 2a1 + 22a2 + . . .+ 2

nan,

ak = 0, 1 1 akak+1 = 0 0 6 k 6 n1, .

5.72. . 0 1 . (1/3; 2/3) , , . , , .

) .) , 1/4 .)

2

3+2

9+2

27+2

81+ . . .

. , .

) , x [0, 2] x = + , .

5.73. . -

01101001100101101001 . . .

3. 79

. . - . - , , .

) 2001 ?) , ,

?) ,

01, - 10.

) , - .

) , n , n- ? (. 11.88.)

5.74. . - , 28 1 . (.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 = (1bm1bm2 . . . b1b0)2, J(n) = (bm1bm2 . . . b1b01)2.5.76. -. , n --

m k (mk = n), m k .

1) m k

m = (ms . . .m1m0)2, k = (ks . . . k1k0)2

( ).2) -

2:

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

80 5. , ,

3) (ns, . . . , n1, n0) n:

(ns . . . n1n0)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.

, - :) mm = 0; ) m k = km; ) (m t) k = m (t k);) n 6= 0

m1 m2 . . .ml = n, (5.1)

j (1 6 j 6 l), mj n < mj.5.77. . .

. () , . , .

m1, m2, . . . , ml (5.1).

) , -, - n 6= 0.

) , - - n = 0.

) .) , : 3, 4 5

?

5.78. II. - - . , 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. II. - 4.21.

5.80. . , 6 8 = 48 . :

3. 81

- , , , . , , .

) . , , ?

) ?

) ?

5.81. . . 1 5 . , , ) ; ) . (. 4.59.)

5.82*. . 3n 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, x1x2 = q.

6.1. x1, x2 x2+px+q = 0. p q

) 1x1

+1

x2; ) 1

x21+1

x22; ) x31 + x32; )

1

(x1 + p)2+

1

(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 :

) x31, x32; )1

x21, 1x22; ) x1 +

1

x2, x2 +

1

x1; ) x2

x1, x1x2.

6.4. x1, x2 ax2 + bx+ c = 0 Sn = x

n1 + x

n2 (n > 0).

aSm + bSm1 + cSm2 = 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

4x+ a3 = 0 ?

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 6= x2, y2+2x1y+2x2 = 0 z2 + 2x2z+ 2x1 = 0. b.

6.14. , ax2 + bx+ c bx2 + cx+ a (a 6= 0) . .

6.15. a x2 + ax+ 1 = 0 x2 + x+ a = 0 ?

6.16. x2 + px + q = 0, x2 px q = 0. , x2 2px 2q = 0.

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 6 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 6 0, x2 > 2; ) 1 6 x1 6 0, 1 6 x2 6 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

x2 + cx+ d = 0 1. ,

x2 +a+ c

2x+

b+ d

2= 0

86 6.

1.

6.30. x2 + px + q = 0 p q 1 1. , .

6.31. a (2 a)x2 3ax+ 2a = 0 1

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, (r4)x22(r3)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),

degR(x) < degQ(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 anxn+. . .+a1x+a0 = 0.

) a0xn + . . .+ an1x+ an = 0; ) anx2n + . . .+ a1x2 + a0 = 0?

6.42.

P(x) = xn + an1xn1 + . . .+ a1x+ a0

x1, x2, . . . , xn,

P(x) = (x x1)(x x2) . . . (x xn).

Q(x) = P(x)P(x). , ) Q(x) 2n -

x;) Q(

x) x21, x22, . . . , x2n.

(. 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) = x1000+ax2+9 x+ 1?

6.46.

P(x) = x81 + x27 + x9 + x3 + 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 x1, 1 x2. P(x) (x 1)(x 2)?

88 6.

6.49. ,

x3 + y3 + z3 + k xyz

x+ y+ z.6.50. n 1 + x2 + x4 + . . . + x2n2

1+ x+ x2 + . . .+ xn1?. 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 6= j, a1(x), . . . , an(x) . , p(x) ,

p(x) a1(x) (mod m1(x)),. . . . . . . . . . . . . . . . . . . .p(x) an(x) (mod mn(x))

deg p(x) < degm1(x) + . . .+ degmn(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 x2 1.

6.56. x36x2+ax6 = 0 3. .

6.57. a P(x) = xn+axn2(n > 2) x 2?

6.58. p q x4 + 1 x2 + px+ q?

2. 89

6.59. a

P(x) = a3x5 + (1 a)x4 + (1+ a3)x2 + (1 3a)x a3

x 1?

6.60. ,

x P(x 1) = (x 26)P(x).

6.61. xnan1xn1 . . .a1xa0 = 0, an1, . . .. . . , a1, a0 > 0. , .

6.62. . , -

f(x) = anxn + . . .+ a1x+ a0

an, . . .. . . , a1, a0.

6.63. -

f(x) = anxn + . . .+ a1x+ 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 6.

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

degQ(x) > degR1(x) > degR2(x) > . . . > degRs(x) > 0,

P(x) = Q(x) A0(x) + R1(x),Q(x) = R1(x) A1(x) + R2(x),R1(x) = R2(x) A2(x) + R3(x),. . . . . . . . . . . . . . . . . . . . . . .Rs2(x) = Rs1(x) As1(x) + Rs(x),Rs1(x) = Rs(x) As(x),

(P(x), Q(x)) = Rs(x). ( 3.36.)6.67. (P(x), Q(x)) = D(x). , -

U(x) V(x) , degU(x) 0 x > 0., m k (am, ak) = a(m,k).

6.71. {x6 x5 + x4 x3 + 5x2 = 5,

x6 2x5 + 3x4 4x3 + 2x = 0.

6.72. p 3x24px+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. 2n+ 1n(n+ 1)

n n+ 1?

6.77. .

Pn(x) = anxn + an1x

n1 + . . .+ a1x+ a0 (an 6= 0) x = c , n . Pn(x)

Pn(x) = (. . . (anx+ an1)x+ . . .+ a1)x+ a0.

(. 5.63.) bn, bn1, . . . , b0 , -

Pn(c),

bn = an, bk = c bk+1 + ak (k = n 1, . . . , 0)., Pn(x) (x c) -

, bn1, . . . , b1, b0. , - :

Pn(x) = (x c)(bnxn1 + . . .+ b2x+ b1) + b0.

6.78. . - :

an+1 bn+1 = (a b)(an + an1b+ . . .+ bn);

a2n+1 + b2n+1 = (a+ b)(a2n a2n1b+ a2n2b2 . . .+ b2n).

6.78 . , n > 2

nn1 1 ... (n 1)2

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

Pn(x) =

nk=0

ck (x c)k,

92 6.

ck

ck =P(k)(x)

k!

x=c

(0 6 k 6 n).

(. 11.21.)6.80. , x4 + 2x3 3x2 4x + 1

x+ 1.6.81. P(x+ 3) x, P(x) = x4 x3 + 1.

3. . -

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

-, .

6.82. - :

) x4 + 4; ) (a+b+ c)3 a3 b3 c3;) 2x3 + x2 + x 1; ) (xy)5 +(y z)5 +(z x)5;) x10 + x5 + 1; ) a8 +a6b2 +a4b4 +a2b6 +b8;) a3 +b3 + c3 3abc; ) (x2 + x+ 1)2 + 3x(x2 + x+ 1)+ 2x2;) x3 + 3xy+y3 1; ) a4 +b4 + c4 2a2b2 2a2c2 2b2c2;) x2y2 x2 + 4xyy2 + 1; ) (x+ 1)(x+ 3)(x+ 5)(x+ 7)+ 15.

(. 9.8.)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. 93

6.86. , m

(x+ y+ z)m xm ym zm

(x+ y+ z)3 x3 y3 z3.

6.87. a, b, c . , -

a2(c b) + b2(a c) + c2(b a)

.

6.88. , a, b, c

1

a+1

b+1

c=

1

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) = anxn + . . .+ a1x+ 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 3a2x2 2a2x+ 2a4 = 0; ) x3 3x = a3 + a3.

4. . P(x) = (x a)kQ(x), k > 1 Q(a) 6= 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)Q1(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+x2

2!+ . . .+

xn

n!

.

6.99. A B Axn+1+Bxn+1 x = 1 ?

6.100. , x2n nxn+1 + nxn1 1 n > 1 x = 1.

6.101. , P(x) , P(x) = an(x x0)n.

6.102. , n > 0 nxn+1 (n + 1)xn + 1 (x 1)2.

6.103. , n > 0

n2xn+2 (2n2 + 2n 1)xn+1 + (n+ 1)2xn x 1

(x 1)3.

6.104. , n > 0

x2n+1 (2n+ 1)xn+1 + (2n+ 1)xn 1

(x 1)3.

6.105. ,

P(x) = a0 + a1x+ . . .+ anxn

5. 95

1 m , - :

a0 a1 + a2 a3 + . . .+ (1)nan = 0,

a1 + 2a2 3a3 + . . .+ (1)nnan = 0,

. . . . . . . . . . . . . . . . . . . . . . . . . . . a1 + 2

ma2 3ma3 + . . .+ (1)

nnman = 0.

(. 11.12.)6.106. ,

P(x) = (xn+1 1)(xn+2 1) . . . (xn+m 1)

Q(x) = (x1 1)(x2 1) . . . (xm 1).

(. 11.95.)

5. . x1, x2,. . . , xn

anxn + an1x

n1 + an2xn2 + . . .+ a1x+ a0

(an 6= 0). x1 + x2 + . . .+ xn = an1/an,

x1x2 + x2x3 + . . .+ xn1xn = an2/an,

. . . . . . . . . . . . . . . . . . . . . .x1x2 . . . xn = (1)

na0/an.

. , - , .

1(x1, x2, . . . , xn) = x1 + x2 + . . .+ xn,

2(x1, x2, . . . , xn) = x1x2 + x2x3 + . . .+ xn1xn,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .n(x1, x2, . . . , xn) = x1x2 . . . xn,

.. F(x1, . . . , xn) -

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

96 6.

G F , , , F , G .

- - (. . 92). - F(x1, . . . , xn) m - a11 . . . ann , (a1 + 2a2 + . . .+nan) m.

6.107. :

) (x+ y)(y+ z)(x+ z); ) (x2 + y2)(y2 + z2)(x2 + z2);) x3 + y3 + z3 3xyz; ) x21 + x22 + . . .+ x2n;) x3 + y3; ) 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,1x

+1

y+1

z=1

a.

, a.

6.110. :x+ y+ z = a,

x2 + y2 + z2 = a2,

x3 + 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 = x2x3, y2 = x1x3, y3 = x1x2.

5. 97

6.114. c

x3 + ax2 + bx+ c = 0

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

x+1

y+1

z=11

6,

xy+ yz+ xz = 11;

)

x2

y+y2

x=7

2,

1

y+1

x=1

2;

)

x(y+ z) = 2,y(z+ x) = 2,z(x+ y) = 3;

)

x+ y+ z = 1,xy+ xz+ yz = 4,x3 + y3 + z3 = 1;

){x2 + y2 + x+ y = 32,

12(x+ y) = 7xy;){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

ab+1

bc+1

ac=

1

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)

(c a)(c b)+ b

(x a)(x c)

(b a)(b c)+ a

(x b)(x c)

(a b)(a c)= x.

6.126.

c2(x a)(x b)

(c a)(c b)+ b2

(x a)(x c)

(b a)(b c)+ a2

(x b)(x c)

(a b)(a c)= x2.

6.127. x1 < x2 < . . . < xn . f1(x), f2(x), . . . , fn(x) n1, fi(xi) = 1 fi(xj) = 0 i 6= j (i, j = 1, 2, . . . , n).

6.128.

f(x) = f1(x) + f2(x) + . . .+ fn(x),

6. 99

fi(x) .

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) xa, x b x c. (x a)(x b)(x c).

. n1, 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+ a2x+ a3 = 0,

z+ by+ b2x+ b3 = 0,

z+ cy+ c2x+ c3 = 0.

100 6.

6.137. a, b c . ,

x+ ay+ a2z = 0,

x+ by+ b2z = 0,

x+ cy+ c2z = 0,

x = y = z = 0.

6.138. f(x) = x10 + a9x9 + . . .+ a0 ,

f(1) = f(1), . . . , f(5) = f(5).

, f(x) = f(x) x.

6.139. P(x) = anxn + . . . + a1x + a0 . ,

|3n+1 P(n+ 1)|, . . . , |31 P(1)|, |1 P(0)|

1.

6.140. , f(x) , n,

f(x)

(x x1)(x x2) . . . (x xn)

(x1, x2, . . . , xn ) n :

A1

x x1+

A2

x x2+ . . .+

An

x xn,

A1, A2, . . . , An . (. 6.51.)

6.141.

x1

a1 b1+

x2

a1 b2+ . . .+

xn

a1 bn= 1,

x1

a2 b1+

x2

a2 b2+ . . .+

xn

a2 bn= 1,

. . . . . . . . . . . . . . . . . . . . . .x1

an b1+

x2

an b2+ . . .+

xn

an bn= 1.

7

1. . z =

= x + iy, x y , i - , , 1; x z, y ( x = Re z, y = Im z). z x = 0, y 6= 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 pi pi.

|z| = r, arg z = , z z = r(cos+ i sin). - z. z = x+ iy - z.

7.3. :) z+ z= 2Re z; ) z z= 2i Im z; ) z z = |z|2.7.4. :) |z1 + z2| 6 |z1| + |z2|; ) |z1 z2| >

|z1| |z2|; ) |z 1| 6 | arg z|, |z| = 1.

102 7.

7.5. :

) 1+ i; ) sin pi6

+ i sin pi6;

) 2+3+ i; ) cos+ i sin

cos i sin.

) 1+ cos+ i sin;7.6. -

:

) |z| 6 1; ) argz iz+ i

= pi4; ) |z i| + |z+ i| = 2;

) |z i| 6 1; ) Re(z2) 6 1; ) Im 1z<

1

2;

) |z| = z; ) |iz+ 1| = 3; ) pi6< arg(z i) < pi

3?

)z 1z+ 1

< 1;7.7. min |3+ 2i z| |z| 6 1.7.8.

:) , ;) , ;) , , -

;) 1 ( ) O,

.

7.9. z, - |z 1 i| = 2|z+ 1 i|.

7.10. . , |z a| = k|z b| k 6= 1 (a b ).

7.11. , z1 z2

|z1 + z2|2 + |z1 z2|

2 = 2(|z1|2 + |z2|

2).

?

7.12. , aj, bj (1 6 j 6 n)

(a1 + a2 + . . .+ an)2 + (b1 + b2 + . . .+ bn)2 66a21 + b

21 +

a22 + b

22 + . . .+

a2n + b

2n.

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

2 i

a2 + b2 a

2

).

, , ? (. 5.24.)

7.16. )3 4i; )

24+ 70i; )

7 24i;

)

2+ i2; )

1+ i

3; )

12 5i.

7.17. -:

) z2 + z+ 1 = 0; ) z2 (3+ 2i)z+ 6i = 0;) z2 + 4z+ 29 = 0; ) z2 (3 2i)z+ 5 5i = 0;) z2 (2+ i)z+ 2i = 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 6= 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).) z+ 3z2; ) z3; ) (z 2)1;7.23. z

1 i, 2 i, 2+ 2i, 1+ 2i.

104 7.

a) z2; ) z3; ) z1?

7.24. . . , z = r(cos+ i sin):

zn = rn (cosn+ i sinn) (n > 1).

n n- :

wk = r1/n

(cos + 2kpi

n+ i sin + 2kpi

n

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

(. 12.11.)

7.25. :a)i; ) 4

1; )

8i; ) 3

1 i; ) 6

1; ) 8

i3 1.

7.26. , wk (k = 0, . . . , n 1), - wn = z z - n-. (. 8.2.)

7.27. , zn = 1 - 1, , 2, . . . , n1.

7.28. :) z4 = z4; ) z2 + |z|2 = 0;) z2 + |z| = 0; ) (z+ i)4 = (z i)4;) z2 + z= 0; ) z3 z= 0.7.29. s

zn = 1, s .

7.30. :

) cosncosn

= 1 C2n tg2+ C4n tg

2 . . . ;

) sinncosn

= C1n tg C3n tg

3+ C5n tg5 . . . .

7.31. a) (1+ i)n; ) (1+ cos+ i sin)n;) (1+ i

3)n; ) (

3+ i)n;

)(1+ i

3

1 i

)20; )

(cos+ i sin

cos+ i sin

)n.

)(1

3 i

2

)20;

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

1. 105

7.33. , x44 + x33 + x22 + x11 + 1 = 0 x4 + x3 + x2 + x+ 1 = 0.

7.34. :) cos 2pi

7+ cos 4pi

7+ cos 6pi

7; ) cos 2pi

7 cos 4pi

7 cos 6pi

7.

7.35. ) ,

P(x) = (cos+ x sin)n cosn x sinn

x2 + 1.) ,

Q(x) = xn sin n1x sinn+ n sin(n 1)

x2 2x cos+ 2.

7.36.

) x2n 1 = (x2 1)n1k=1

(x2 2x cos kpi

n+ 1

);

) x2n+1 1 = (x 1)n

k=1

(x2 2x cos 2kpi

2n+ 1+ 1

);

) x2n+1 + 1 = (x+ 1)n

k=1

(x2 + 2x cos 2kpi

2n+ 1+ 1

);

) x2n + 1 =n1k=0

(x2 2x cos

(2k+ 1)pi

2n+ 1

).

7.37. , ,

cosnx = Tn(cos x), sinnx = sin xUn1(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) Tn1(x), Un+1(x) = 2xUn(x) Un1(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) 6= 0, 1/2, 1, cos(p/q) .

7.42. ,

cosn x =n

k=0

ak cos kx, sinn x = sin xn1k=0

bk sinkx,

a0, . . . , an, b0,. . . , bn1 . - n = 2, 3, 4, 5. sinn x n

sinn x =n

k=0

ck cos kx, sinn x =n

k=0

dk sinkx.

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

525, n, 5.

7.44. P0(x) = 1, P1(x) = x, P2(x) == x2 1, . . .

Pn+1(x) = x Pn(x) Pn1(x).

, P100(x) = 0 100 - [2; 2]. ?

7.45. :(1+ i tg

1 i tg

)n=1+ i tgn

1 i tgn.

7.46. , z + z1 = 2 cos, zn + zn = 2 cosn. zn + zn y = z+ z1? (. 1.5.)

7.47. x = cos

Tn(cos) = cosn, Un1(cos) =sinn

sin.

, x = sin?7.48. a, b (a, b) = 1. ,

(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. , aib f(x).(. 7.82.)

7.50. , , -.

7.51. . a b .

ea+ib = limn

(1+

a+ ib

n

)n.

:

ea+ib = ea(cosb+ i sinb).

, sin x cos x - :

cos x = eix + eix

2, sin x = e

ix eix

2i.

(. 5.35, 11.73 12.12.)7.52. , z1, z2 -

ez1ez2 = ez1+z2 . (. 11.73.)7.53. , -

.7.54. ln z z?7.55.

az? (. 12.12.)

7.56. i

1 = (1)1/i 2317.

7.57. z = e2pii/n = cos 2pin

+ i sin 2pin.

a ) 1+ za + z2a + . . .+ z(n1)a; ) 1+ 2za + 3z2a + . . .+ nz(n1)a.7.58. ) :

cos+ . . .+ cosn =sin(n/2) cos((n+ 1)/2)

sin(/2);

108 7.

) :sin+ . . .+ sinn.

(. 8.11.)7.59. :

sin+ sin 3+ . . .+ sin(2n 1)

cos+ cos 3+ . . .+ cos(2n 1)= tgn.

7.60. :) cos2 x+ cos2 2x+ . . .+ cos2 2nx; ) sin2 x+ sin2 2x+ . . .+ sin2 2nx.7.61. (1 + i)n ,

:) C0100 C2100 + C4100 . . . + C100100; ) C199 C399 + C599 . . . C9999.7.62. ) :

C0n C2n + C

4n . . . = 2

n/2 cos npi4.

) :C1n C

3n + C

5n . . .

7.63. ) :

1+ C3n + C6n + . . . =

1

3

(2n + 2 cos npi

3

).

) :

C1n + C4n + C

7n + . . . ; C

2n + C

5n + C

8n + . . .

7.64. :

C1n 1

3C3n +

1

9C5n . . . =

2n

3(n1)/2sin npi

6.

7.65. :) 1+ a cos+ . . .+ ak cos k+ . . . (|a| < 1);) a sin+ . . .+ ak sink+ . . . (|a| < 1);) cos+ C1n cos 2+ . . .+ Cnn cos(n+ 1);) sin+ C1n sin 2+ . . .+ Cnn sin(n+ 1).7.66.

limk

(1+

1

2cos x+ . . .+ 1

2kcos kx

).

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

1. 109

) z1 + . . .+ zn 6= 0; ) z11 + . . .+ z1n 6= 0.7.68. z1, z2, . . . , zn .

z = 1z1 + 2z2 + . . .+ nzn,

1, 2, . . . , n , 1 + 2 + . . .+ n = 1.

7.69. ,

1

z a+

1

z b+

1

z c= 0,

a, b, c , a, b, c, ( ).

7.70. f(x) = (xa)(xb)(xc) a, b, c. , a, b, c.

7.71. . f(x) n 1, . . . , n. M 1, . . . , 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) = x6n + x5n + x4n + x3n + x2n + xn + 1

Q(x) = x6 + x5 + x4 + x3 + x2 + x+ 1,

, n 7.

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

n1m=1

1

sin2(pim/n)=n2 1

3.

7.81*. . ) , n > 1

(n1)/2m=1

1

m2=pi2

6pi2

2n (0 < < 1).

) :

m=1

1

m2=pi2

6.

7.82*. . P(x) x . -, a(x) b(x), P(x) == a2(x) + b2(x).

2. :Ta a;Sl l ( l);RA A ;HkA A k.

7.83. : 0, 1 i,1+ i

w =(12+

i2

)z?

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) -:

) H2O T3+4i; ) Rpi/4i ; ) H21 H1/21 ;) T3+4i H2O; ) HkA; ) Rpi/4i Rpi/41 Rpi/4i Rpi/41 . O = (0; 0) . -

: (f g)(z) = f(g(z)).7.88. H2i

O.

7.89. . , - :

Hk2A2Hk1A1 =

{Ta, k1k2 = 1,

HkA, k1k2 6= 1,

a A1A2, A A1A2 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 = z1.

7.91. 2 < Re z < 3 :) w = z1; ) w = (z 2)1; ) w = (z 5/2)1?

7.92. ) |zabi|=

a2 +b2 w= 1/z;

) |z a| = R w = 2aRz2 a2 + R2

.

7.93*. n- .,

) n2;) n ctg pi

2n;

) nn/2.

112 7.

. - ,

w =az+ b

cz+ d, (7.1)

w =az+ b

cz+ d, (7.2)

= ad bc 6= 0.7.94. (7.1) (7.2) , =

= ad bc = 0?

. C - C, = 1

0, C= C {}.

7.95. , - - -.

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

8 +

1.

8.1. , , - n- , .

8.2. :

a) cos pi5

cos 2pi5

=1

2;

) 1sin(pi/7)

=1

sin(2pi/7)+

1

sin(3pi/7);

) sin 9 + sin 49 + sin 89 + . . .+ sin 329 = 0.(. 7.26.)

8.3. ) cos pi

9cos 4pi

9cos 7pi

9; ) cos pi

7+ cos 3pi

7+ cos 5pi

7.

8.4. cos 36 cos 72.8.5. ) , , -

36 (. . ).

) -

2.

8.6. 0 < x < 90:

a)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.

8.7. :

arctg 1+ arctg 12+ arctg 1

3=pi

2.

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 6 x, y, z 6 8. -

S = 2x2y2 + 2x2z2 + 2y2z2 x4 y4 z4 ?

8.11. xk

cos x+ cos 2x+ cos 3x+ 12

= 0.

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, - :

) arg z z1z z2

= 0; ) arg z1 zz z2

= 0.

.

V(z2, z1, z0) =z2 z0z1 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) ,

z0 z2z1 z2

=z0 z2z1 z2

.

8.17. , , z1 z2 z,

z z2z1 z2

=z z2z1 z2

.

8.18. ,

BzBz+ C = 0,

C .

8.19. , , z0, z1, z2,z3 ( )

V(z0, z1, z2)

V(z0, z1, z3)=z0 z2z1 z2

:z0 z3z1 z3

.

.

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

V(z0, z1, z3)

( - ) z0, z1, z2, z3.

8.20. . z 1, z 2, z 3,z 4 , - - (7.1) z1, z2, z3, z4. ,

W(z 1, z2, z

3, z

4) =W(z1, z2, z3, z4).

8.21. W(z1, z2, z3, z4) - (7.2)?

116 8. +

8.22. - ., - - .

8.23. , ( ) -

Azz+ BzBz+ C = 0, (8.1)

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+ bcz+ d

w = z -

) i R = 1; ) Rei R;) z0 R.

8.27. . , - .

8.28. (8.1). (7.1) -

A zz+ B zBz+ C = 0,

A C . A , B C A,B C.

2. 117

. A - R O |OA|2 R2.

8.29. , w

Azz+ BzBz+ C = 0

ww+B

Aw

B

Aw+

C

A.

8.30. . , w, S1 S2 , .

S1 S2.

8.31. . S1, S2 S3. , Q, .

Q S1, S2 S3.

8.32. . a1, a2 a3 zz= 1. , h = a1+a2+a3 a1, a2 a3.

8.33. . a1, a2 a3 zz = 1. , e = h/2 1/2 - a1a2a3, , a1, a2, a3 h.

8.34. . , m = (a1++a2+a3)/3 a1a2a3.

8.35. . , - , .

8.36. . u - zz = 1 u1, u2, u3 , u a2a3, a1a3, a1a2 a1a2a3.

) , u1, u2, u3

u1 = (a2 + a3 + u a2a3/u)/2,

u2 = (a1 + a3 + u a1a3/u)/2,

u3 = (a1 + a2 + u a1a2/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 pi15

cos 2pi15

cos 3pi15

cos 4pi15

cos 5pi15

cos 6pi15

cos 7pi15

=(1

2

)7.

8.40. :

cosa cos 2a cos 4a . . . cos 2n1a.8.41. :

) sin pi2n+ 1

sin 2pi2n+ 1

sin 3pi2n+ 1

. . . sin npi2n+ 1

;

) sin pi2n

sin 2pi2n

sin 3pi2n

. . . sin (n 1)pi2n

;

) cos pi2n+ 1

cos 2pi2n+ 1

cos 3pi2n+ 1

. . . cos npi2n+ 1

;

) cos pi2n

cos 2pi2n

cos 3pi2n

. . . cos (n 1)pi2n

.

8.42. :

tg 20 tg 40 tg 80 =3.

8.43. :

cospi x31

cos 2pi x31

cos 4pi x31

cos 8pi x31

cos 16pi x31

=1

32.

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. 119

, + 2 = pi/2.

8.46. :

) sin 15 =6

2

4, cos 15 =

6+

2

4;

) sin 18 = 1+5

4, cos 18 =

10+ 2

5

4.

8.47. :

sin 6 =30 6

5

6+ 2

5

8, cos 6 =

18+ 6

5+

10 2

5

8.

8.48. :) sin+ sin+ sin sin(++) = 4 sin +

2sin +

2sin +

2;

) cos+cos+cos+cos(++) = 4 cos + 2

cos + 2

cos + 2

.

8.49. :

tg+ tg+ tgsin(+ + )

cos cos cos= tg tg tg.

8.50. , , ,

tg+ tg+ tg = tg tg tg.8.51. , + + = pi,

sin+ sin+ sin = 4 cos 2cos

2cos

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 + cosy = a, sin x + siny = b. cos(x + y) sin(x+ y).

8.54. , cosx .

8.55. n

y = cosnx sin 5nx

3pi?

8.56. f(x) = A cos x + B sin x, A B . , f(x) x1 x2 , x1 x2 6= kpi (k ), f(x) .

120 8. +

8.57. ,

a1 cos(1 + x) + a2 cos(2 + x) + . . .+ an cos(n + x)

x = 0 x = x1 6= kpi (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.

, cos2

2=

c2

a2 + b2.

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(pi

7

)]; ) arcsin

(cos 33pi

5

).

8.65. , :

) cos arcsin x =1 x2; ) sin arccos x =

1 x2;

) tg arcctg x = 1x; ) ctg arctg x = 1

x;

) cos arctg x = 11+ x2

; ) sin arctg x = x1+ x2

;

) cos arcctg x = x1+ x2

; ) sin arcctg x = 11+ x2

.

8.66. :) arctg x+ arcctg x = pi

2; ) arcsin x+ arccos x = pi

2.

8.67. :) arcsin(x) = arcsin x, ) arccos(x) = pi arccos x.

8.68. arctg x+ arctg 1x?

8.69. :

arctg x+ arctg y = arctg x+ y1 xy

+ pi,

3. 121

= 0, xy < 1, = 1 , xy > 1 x < 0, = +1, xy > 1 x > 0.

8.70. :

4 arctg 15 arctg 1

239=pi

4.

8.71. :

arctg 13+ arctg 1

5+ arctg 1

7+ arctg 1

8=pi

4.

8.72. :

arctg x1+ 1 2x2 + arctg

x

1+ 2 3x2 + . . .+ arctgx

1+ n (n+ 1)x2 (x > 0).

8.73. :

arctg r1+ a1 a2 + arctg

r

1+ a2 a3 + . . .+ arctgr

1+ an an+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 F2n+1 + . . . =pi

4.

8.75. , x > 1 :

2 arctg x+ arcsin 2x1+ x2

= pi.

8.76.

arcsin x2 8

8= 2 arcsin x

4pi

2.

8.77. :

arccos x ={ arcsin1 x2, 0 6 x 6 1;pi arcsin

1 x2, 1 6 x 6 0.

8.78. :

arcsin x+ arcsiny = arcsin(x1 y2 + y

1 x2) + pi,

122 8. +

= 1, = 0, xy < 0 x2 + y2 6 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 1+ x1 x

, = arctg 1 x2

1+ x2,

+ = pi.8.80.

arcsin cos arcsin x arccos sin arccos x.

8.81. , 0 6 6 pi2

cos sin > sin cos.

8.82.

sin(2 arctg 1

5 arctg 5

12

).

8.83. . ,

a

sin=

b

sin=

c

sin, + + = pi (8.3)

:a = b cos+ c cos,b = c cos+ a cos,c = a cos+ b cos.

(8.4)

(. 8.12.)8.84. , (8.4) -

0 < < pi, 0 < < pi, 0 < < pi, a > 0, b > 0, c > 0 (8.3).

8.85. . , (8.4) -

a2 = b2 + c2 2bc cos,b2 = a2 + c2 2ac cos,c2 = a2 + b2 2ab cos,

(8.5)

(8.4) (8.5) .8.86. -

. , , A, B, C.

3. 123

(8.7) (8.6),(8.8) ( ). , , . ,

cos = cos cos+ sin sin cosA,cos = cos cos+ sin sin cosB,cos = cos cos+ sin sin cosC,

(8.6)

, , , , A, B, C 0 pi.,

sinA

sin=

sinB

sin=

sinC

sin. (8.7)

8.87. . , (8.6)

cosA = cosB cosC+ sinB sinC cos,cosB = cosA cosC+ sinA sinC cos,cosC = cosA cosB+ sinA sinB cos,

tg A+ B+ C pi4

=

tg p2tg p

2tg p

2tg p

2,

(8.8)

2p = + + .8.88. . :

) 3

cos 2pi7

+ 3

cos 4pi7

+ 3

cos 8pi7

=3

5 3

37

2;

) 3

cos 2pi9

+ 3

cos 4pi9

+ 3

cos 8pi9

=3

339 6

2.

(. 8.11.)8.89.

uk =sin 2nx sin(2n 1)x . . . sin(2n k+ 1)x

sinkx sin(k 1)x . . . sin x .

, uk cos x.(. 3.142.)

8.90. uk . - :

) 1u1+u2. . .+u2n = 2n(1cos x)(1cos 3x). . .(1cos(2n1)x);) 1u21+u22 . . .+u22n = (1)n

sin(2n+ 2)x sin(2n+ 4)x . . . sin 4nxsin 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

.

9.4. 32+

5+

32

5 = 1.

9.5.

x3 + x2 + x = 1

3.

9.6. ,

x3 + ax2 b = 0,

a b b > 0, - .

9.7. a b, -

x3 + px+ q = x3 a3 b3 3abx?

1. 125

9.8.

a3 + b3 + c3 3abc

. (. 11.74.)

9.9. a b

x3 a3 b3 3abx = 0.

.

9.10. ,

(a2 + b2 + c2 ab bc ac)(x2 + y2 + z2 xy yz xz) =

= X2 + Y2 + 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+

q2

4+p3

27+

3

q

2

q2

4+p3

27.

9.12. x3 + x 2 = 0 .

9.13. ,

=1

2

(352+ 7

352 7

).

.

9.14. a - x3 x a = 0.

9.15. x3 x 233

= 0.

?

9.16. , x1, x2, x3 x3+px+q = 0,

x22 + x2x3 + x23 = x

21 + x1x3 + x

23 = x

21 + x1x2 + x

22 = p.

(. 8.13.)

126 9.

. f(x) n > 2, f(x) = an(x 1) . . . (x n) f(x) . D(f) f(x) :

D(f) = a2n2n

16j

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 + 27q2 6 0 (9.2)

t1 = cos

3, t2 = cos

+ 2pi

3, t3 = cos

+ 4pi

3,

= 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 + 27q2 6 0 9.1 x = y+

ay3 3by2 3ay+ b = 0 (9.3)

y.) , (9.3)

y1 = tg

3, y2 = tg

+ 2pi

3, y1 = tg

+ 4pi

3,

128 9.

:

sin = ba2 + b2

, cos = aa2 + b2

.

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 0).

, limn xn =

2. (. 9.65.)

3. 131

9.47. , x0 = 1?

9.48. . , - {xn},

x0 = 1, xn+1 =1

2

(xn +

k

xn

), (n > 0),

. .

9.49. a k > 0 . - {an}

a0 = a, an+1 =1

2

(an +

k

an

)(n > 0).

, n

an k

an +k

=(a

k

a+k

)2n.

9.50. a0 a1. {an}

an+1 =an + an1

2(n > 1).

an a0, a1 n.

9.51. I. ) , 3

x (x > 0) ,

x.

. {yn}, y0- , , y0 =

x,

yn+1 =

xyn (n > 0).

, limnyn = 3

x.

) .

9.52. II. ln x x = 1 1.

limx0 ln(1+ x)x = limx0 ln(1+ x) ln 1(1+ x) 1 = 1.

132 9.

- N. 9.51, - .

9.53. . , , f(x) = x, . x0, - {xn} xn+1 = f(xn) (n > 0). , x = lim

n 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), . . . B0A0B1A1 . . . BnAn . . . -.

9.54. :

) f(x) = 1+ x2, x0 = 0, x0 = 8;

) f(x) = 1x, x0 = 2;

) f(x) = 2x 1, x0 = 0, x0