..

,

2007

512.1+517.1+511.1 22.141+22.161

70

..70 , :

..: , 2007.608 .: .

ISBN 978-5-94057-263-3

, , , , , -. , . 1000 .

, , , .

22.141+22.161

ISBN 978-5-94057-263-3 .., 2007 , 2007

12

14

1. 16

1.1. (16).1.2. (16). 1.3. (17). 1.4. - (18). 1.5. - (19). 1.6. (19). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2. 26

2.1. (26). 2.2. (26).2.3. (26). 2.4. - (27). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3. 31

3.1. (31). 3.2. - (31). 3.3. - (32). 3.4. (32).3.5. (33). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4. 42

4.1. (42). 4.2. (43). 4.3. - (44). 4.4. (44). 4.5. - (45).

4

4.6. (46). 4.7. - (47). 4.8. (48). 4.9. (49). 4.10. (49). 4.11. - (50). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5. 65

5.1. (65). 5.2. (65). 5.3. (65). 5.4. - (66). 5.5. - (67). 5.6. - (67). 5.7. (67). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6. 74

6.1. (74). 6.2. - (74). 6.3. (75). 6.4. - (76).6.5. (76). 6.6. - (77). 6.7. (78). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7. 88

7.1. (88). 7.2. (88).7.3. (89). 7.4. - (90). 7.5. (91). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8. 96

8.1. x + 1/x > 2 (96). 8.2. - (96). 8.3. (97). 8.4. - - (98). 8.5. , (99). 8.6. (99).8.7. (100). 8.8. (101).8.9. (102). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5

9. 116

9.1. (116).9.2. (117). 9.3. Sk(n)=1

k+2k+ . . .+nk (117). 9.4. (118).9.5. (119). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10. I 125

10.1. (125). 10.2. (125). 10.3. (125).10.4. (126). 10.5. (126).10.6. (128). 10.7. (128). 10.8. (128). 10.9. (129).10.10. (130). 10.11. (130). 10.12. (131). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11. 142

11.1. (142). 11.2. - (143). 11.3. (143).11.4. , - (144). 11.5. (144). 11.6. cosnf . . (145). 11.7. - (146). 11.8. (147). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

12. 159

12.1. (159). 12.2. - (159). 12.3. (160).12.4. (161).12.5. (161). 12.6. - (162). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6

13. 171

13.1. (171). 13.2. (171).13.3. (172). 13.4. - (172). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

14. 178

14.1. (178). 14.2. (179). 14.3. - (180). 14.4. - (180). 14.5. - (181). 14.6. - (181). 14.7. - (181). 14.8. - (182). 14.9. (182). 14.10. - (184). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

15. 201

15.1. (201). 15.2. (201).15.3. (203).15.4. (203). 15.5. - (204). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

16. 210

16.1. (210). 16.2. - (211). 16.3. (211).16.4. (211).16.5. (212). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

17. . 218

17.1. (218). 17.2. (219).17.3. - (220). 17.4. (220). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

7

18. 228

18.1. (228). 18.2. - (229). 18.3. (229). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

19. 236

19.1. (236). 19.2. (237). 19.3. (238). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

20. . . 242

20.1. (242). 20.2. (243).20.3. (243). 20.4. (244). 20.5. - (245). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

21. 256

21.1. (256). 21.2. (256).21.3. (257). 21.4. (257).21.5. (257).21.6. (258).21.7. d- (259). 21.8. - (259). 21.9. (260).21.10. (261). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

22. 269

22.1. (270). 22.2. - (270). 22.3. (270). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

23. 275

23.1. - (276). 23.2. (276). 23.3. - (277). 23.4. (279). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

8

24. , 28524.1. (286). 24.2. - (286). 24.3. - 4- (287). 24.4. , - (287). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

25. 29325.1. (293). 25.2. - (294). 25.3. (295). 25.4. - e (296). 25.5. (297). 25.6. (297). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

26. 31026.1. (310). 26.2. (310). 26.3. (310). 26.4. - (311). 26.5. - (312). 26.6. , - (312). 26.7. (313). 26.8. - (314). 26.9. (314). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

27. 32227.1. - (322). 27.2. (323). 27.3. - (323). 27.4. - (324). 27.5. (324).27.6. (324). 27.7. - (324). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

28. 33128.1. (331). 28.2. (332). 28.3. (333). 28.4. (333).28.5. (334). 28.6. (335).28.7. , (335).28.8. (337). 28.9. (338).

9

28.10. (338). 28.11. - (339). 28.12. (339). 28.13. (340). 28.14. (340). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

29. 361

29.1. (361). 29.2. - (362). 29.3. (364).29.4. (365). 29.5. - (365). 29.6. (366). 29.7. - (367). 29.8. (367). 29.9. - (368). 29.10. (369). 29.11. - (369). 29.12. - (369). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

30. 384

30.1. (384). 30.2. - (384). 30.3. - (384). 30.4. (386). 30.5. p (387). 30.6. - (387). 30.7. (388). 30.8. - (388). 30.9. - (388). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

31. 399

31.1. (399). 31.2. (399). 31.3. (400). 31.4. - (400). 31.5. (401).31.6. sk(n). (402). 31.7. - (403). 31.8. - (404). 31.9. (406). 31.10. (406). 31.11. (407).31.12. (408).31.13. (409).31.14. (410). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

10

32. II 434

32.1. (434). 32.2. - (436). 32.3. (439).32.4. (442). 32.5. (444). 32.6. (445). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

33. 460

33.1. (460). 33.2. - . (460). 33.3. - (461). 33.4. (463). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

34. 470

34.1. (470). 34.2. (470). 34.3. - (471).34.4. (472). 34.5. - (472). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

35. 484

35.1. (484). 35.2. - (486). 35.3. - (486). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

36. - 493

36.1. (493). 36.2. (494). 36.3. (494).36.4. (494). 36.5. (495). 36.6. (496). 36.7. - (497). 36.8. (497). 36.9. (498). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

11

37. 51037.1. (510). 37.2. - (511). 37.3. - (511). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

38. 51538.1. (516). 38.2. - (516). 38.3. (519). 38.4. - (520). 38.5. . - (521). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

39. 53139.1. (531). 39.2. - (531). 39.3. (532).39.4. (533). 39.5. - (533). 39.6. (534). 39.7. (534). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

5391. (539).2. (543). 3. (546). 4. 17- (549). 5. - (553). 6. (561). 7. e p (565).8. (570). 9. - (584). 10. (589).11. (594).

597

598

39 , - , , . - . - , - .

, , - . , , - . -, . , : , , , , - .

-. 39 , . - , - (, , , ..).

13

, - : , . .

, - , : - ; ; -; 17-; ; ; e p; - ; - ; - . , - .

. , . , - . 1415.

[x] x, , - x

{x} x, x

n!=1 2 3 . . . n

(2n)!!=2 4 6 . . . 2n

(2n+1)!!=1 3 5 . . . (2n+1)

Ckn=n!

k! (nk)! d |n d nlim

min{a1, . . . , an} min(a1, . . . , an) a1, . . . , an

max{a1, . . . , an} max(a1, . . . , an) a1, . . . , an

lg 10

ln , e (- )

sgn

15

sh

ch

th

cth

Arsh

Arch

Arth

Arcth

f(x) f(x)]

1

1.1.

1.1. , x>0, x+1/x>2.1.2. ) , x(1 x) 6 1/4. ) ,

x(ax)6a2/4.1.3. , a, b, c, -

0 1, a(1 b)>1/4, b(1 c)>1/4 c(1a)>1/4.

1.4. x f(x)= (xa1)2+ . . .+ (xan)2 ?

1.5. x, y, z , - 1. , 1/x+1/y+1/z>9.

1.6. , (x0, y0) -

ax+ by+ c=0 |ax0+ by0+ c|p

a2+ b2.

1.7. a1, . . ., an , a1+ . . .+an=a. ,

a1a2+a2a3+ . . .+an1an6a2/4.

1.2.

1.8. ) a, b, c . ,

(xa)(x b)+ (x b)(x c)+ (x c)(xa)=0 .

) , (a+ b+ c)2>3(ab+ bc+ ca).

17

1.9. a1, . . ., an, b1, . . ., bn .

(a1b1+ . . .+anbn)26 (a21+ . . .+a

2n)(b

21+ . . .+ b

2n).

1.10. , (a+ b+ c)c4ac.

1.3.

1.11. ) , , . ?

) a 72, b . -, a b a + b .

1.12. , ax2 + bx + c x , 2a, a+ b c .

1.13. x2+ax+b=0 x2+ cx+d=0 -

, x0. , x2 +a+ c

2x+

+b+d

2=0 , x0.

1.14. , -

x2+ p1x+ q1=0,

x2+ p2x+ q2=0

, p1= p2 q1= q2.1.15. , p

ax2+bx+c+p=0 -, a .

1.16. ) ax2+ bx+ c x . , a= b= 0.

) ax2+ bx+ c x - . , ax2+bx+c=(dx+e)2.

18 1.

1.17. x1 x2 x2++ax+ b=0, sn=xn1 +x

n2. ,

sn

n=m

(1)n+m (nm1)!m! (n 2m)! a

n2mbm,

m, 06m6n/2 ( ).

1.4.

- , . f(x)= ax2 + bx + c, a 6= 0. f(a)f(b) 6 0, [a, b] - ax2+ bx+ c= 0.

f(x) = f(x) y(x) , , ,f(a)> y(a) f(b)6 y(b), [a, b] x0, f(x0)=y(x0).

1.18. .

1.19. a x2+ax+ b=0, b- x2axb=0. , - a b x22ax2b=0.

1.20. f(x) , n>>2. , f(x)+ f(x+1)+ . . .. . .+ f(x+n) .

1.21. ax2+ bx+ c=0 ax2+ bx+ c=0., x1 , x2

, x3 a

2x2 + bx+ c= 0,

x16x36x2, x1>x3>x2.1.22. , [1, 1] -

f(x) = x2 + ax + b , 1/2.

1.23. , |ax2 + bx + c| 6 1/2 |x|61, |ax2+ bx+ c|6x21/2 |x|>1.

19

1.24. , |ax2+ bx+ c|6 1 |x|6 1, |cx2+ bx+a|62 |x|61.

1.5.

1.25. ax2 ++ bxy+ cy2+dx+ ey+ f=0 (x0, y0).

1.26. ) , (x0, y0) - ( ) ax2 + by2 = 1 ax0x+ by0y=1.

) , (x0, y0) - xy=a x0y+ y0x=2a.

1.6.

1.27. , x2+p1x+q1== 0 x2 + p2x + q2 = 0 () ,

(q2 q1)2+ (p1 p2)(p1q2 q1p2)=0. (q2 q1)2 + (p1 p2)(p1q2 q1p2)

x2 + p1x+ q1 x2 ++ p2x+ q2.

1.28. , p1, p2, q1, q2 - (q2 q1)2 + (p1 p2)(p1q2 q1p2) < 0, x2 + p1x + q1 x2 + p2x+ q2 - .

1.1. x > 0 x2 2x+1> 0, . . (x1)2 > 0.

1.2. ) x2x - x= 1/2; 1/4.

) x2 ax - x= a/2; a2/4.

20 1.

1.3. 1.2 ) a(1a)61/4, b(1b)61/4, c(1c)66 1/4.

a(1 b)b(1 c)c(1a)=a(1 a)b(1 b)c(1 c)6 (1/4)3.1.4. , f(x)= nx2 2(a1 + . . .+ an)x+ a21 + . . .+ a2n.

x =

=a1+ . . .+ an

n.

1.5. 1

x=

x+ y+ z

x= 1+

y

x+

z

x.

1/y 1/z. 1.2 y/x+ x/y> 2. , .

1.6. (x, y) ax+ by+ c= 0.

y=ax+ cb

,

(xx0)2+ (yy0)2= (xx0)2+ax+ c

b+ y0

2=

=a2+ b2

b2x2+ 2

x0+ ac

b2+

ay0

b

x+x20+

c2

b2+2cy0b

+ y20.

b2

a2+ b2

x0+ ac

b2+

ay0

b

2+x20+

c2

b2+2cy0b

+ y20=(ax0+ by0+ c)

2

a2+ b2.

1.7. x = a1 + a3 + a5 + . . . a x = a2 + a4 + a6 + . . . 1.2 )

a2/4> x(ax)= (a1+ a3+ a5+ . . .)(a2+ a4+a6+ . . .)>> a1a2+ a2a3+ . . .+an1an.

1.8. ) a6b6 c. x=b (bc)(ba)60. x , .

) , ) .

1.9. (a1x+ b1)2 + . . .+ (anx+ bn)

2 - x, . , .

1.10. f(x)= x2 + bx+ ac. f(c)< 0. x2 , -

21

-. , D= b2 4ac .

1.11. ) :

512

. -

y z, y < z.

, y

z=

z

y+ z= x, x . x

x(x+1)=y

z

y+ z

z=1.

, x=12+

5

2.

) ABC A=36, B=C=72., , a=BC b=AC. - BK. KBC= 36, BCK 72. , AK=KB, ABK= 36 = BAK. , KC= b a, -

AB

BC=

BC

KC

b

a=

a

b a , . . b2 ba= a2.

,a+ b

b=

b

a, .

1.12. , f(x)==ax2+ bx+ c x. , , f(0) = c . f(1) c = a b . , 2a= (a+ b)+ (a b) .

, 2a, a+b c . x(ax+ b) x. x= 2k+ 1, ax+ b==2ka+ a+ b .

1.13. , x > x0, x2 + ax + b > 0

x2+ cx+d>0. x>x0, 2x2+ (a+ c)x+ (b+d)>0.

1.14. x2+p1x+q1== 0 x1, ., x1=m/n . m

2++p1mn+q1n

2=0, m2 n. m n . , x1 . , , x1 .

, x1=a++b, a b ,

b .

p1a

b q1= (a+

b)(p1a

b). ,

q1 = (p1 2a)b + r, r = ap1 a2 b .

p1=2a q1= (a+b)(ab)= a2 b.

p2 q2 .1.15. , a > 0. -

p D= b2 4ac 4ap ,

22 1.

. -, a< 0. p -

, c+ p

a, .

1.16. ) , a> 0 c> 0. x, - 1, 2, . . . , n. a b , ax2 + bx + c x n/2 . - 0 an2 + |b|n + c. , 0 an2 + |b|n+ c, 4pan2+ |b|n+ c+ 1. 4pan2+ |b|n+ c + 1 > n/2, . . an2 +

+ |b|n+ c > (n/2 1)4. n , (n/2)4 , an2.

) f(x)=ax2+ bx+ c.

f(x+ 1) f(x)= (f(x+1))2 (f(x))2

f(x+1)+ f(x)=

a(2x+1)+ b

f(x+1)+ f(x),

limx

(f(x+1)f(x))=a. x f(x+1)f(x) ,

a = d, d .

, x> x0 f(x+ 1) f(x) - d. y = x0 + n. f(y)= f(x0)+nd n. ,

ay2+ by+ c= (f(x0)+nd)2= (dydx0+ f(x0))2

y=x0+n, n. - y. , d=

a e= f(x0)dx0, x0

.1.17. x1+x2=a x1x2= b. n= 1

n= 2 x1 + x2 =a 12(x21 + x

22)=

=1

2a2 b. . ,

, - n 1, n> 3. sn

n=1

n(asn1 bsn2)= n1

n

Xm

(1)n+m (nm2)!m! (n2m 1)!a

n2mbm+

+n2n

Xm

(1)n+m1 (nm 3)!m! (n 2m2)!a

n2m2bm+1.

23

m+1 m,

n2n

Xm

(1)n+m (nm2)!(m1)! (n2m)!a

n2mbm.

,

(nm 2)! (n1)m! (n2m 1)!n +

(nm2)! (n2)(m1)! (n2m)!n =

=(nm 2)!

(m1)! (n 2m1)!nn 1m

+n 2n 2m

=

(nm1)!m! (n 2m)! .

.1.18. , -

. , . x1 x2, x1 < x2, x< x1 x> x2 , x1 < x< x2 . , x1 x2.

1.19. f(x) = x2 2ax 2b. a2 = aa b b2=ab+b, f(a)=a2+2a2=3a2 f(b)=b22b2=b2., f(a)f(b)6 0. [a, b] x2 2ax2b=0.

1.20. f(x) - x1 x2=x1+n+a, a>0. , x2 . x0=x1+a/2. - x0>x1 x0+n6x2. f(x0)+f(x0+1)+ . . .+f(x0+n) g(0). , f g : [1, 0], [0, 1].

, f g . x2 + ax + b = x2 1/2 ,

24 1.

ax + b = 1/2. f(0) > g(0) , , b 6= 1/2. ax + b = 1/2 .

1.23. , ax2+ bx+ c6 x2 1/2 ( - ax2bxc ).

f(x)=ax2+bx+c g(x)=x21/2. f(0)>g(0)=1/2 f(1)6 g(1)= 1/2. f g [1, 1], ( f g -).

f(1) < g(1), f(x) < g(x) - |x|> 1. , f(1)= g(1) f(1)= g(1). , , f(1)=g(1) f(x)>g(x) x, 1. f(x)g(x) x 6=1. , f(1)>g(1), .

1.24. 1.23 |ay2 + by + c| 6 2y2 1 |y|> 1. y=1/x.

|cx2+ bx+ a|= 1y2|ay2+ by+ c|6 1

y2(2y2 1)6 2

|y| > 1, . . 0 < |x| 6 1. x = 0 , x, .

1.25. , (x0, y0)

ax2+ bxy+ cy2+dx+ ey+ f= 0.

, (x0, y0), - y y0 = k(x x0) ( x= x0, k=). . (. . - y k(xx0)+ y0) - A B x2 x ( C ):

A= a+ bk+ ck2;

B=bkx0+ by0 2ck2x0+ 2cky0+d+ ke. -

,

25

(x0, y0).* : 2x0=B/A, . .k(bx0+ 2cy0+ e)=(2ax0+ by0+d).

,

(xx0)(2ax0+ by0+d)+ (yy0)(bx0+ 2cy0+ e)= 0. (x0, y0) , . . ax

20+ bx0y0+

+ cy20+dx0+ ey0+ f= 0. , - :

(2ax0+ by0+d)x+ (bx0+2cy0+ e)y+dx0+ ey0+ 2f=0.

1.26. 1.25.1.27. x1 .

, (p1 p2)x1 = q2 q1. p1 = p2, q1 = q2. p1 6= p2, x1 = q2 q1

p1 p2.

x1 , - . p1= p2 q1= q2 .

, (q2q1)2+(p1p2)(p1q2q1p2)=0. p1=p2, q1=q2; ,

. p1 6= p2, x1= q2 q1p1 p2

. -

, x21+ p1x1+ q1= 0 x21+ p2x1+ q2= 0.

1.28. , , , p1 6= p2. x1=

q2 q1p1 p2

.

x21+ p1x1+ q1=x21+ p2x1+ q2=

(q2 q1)2+ (p1 p2)(p1q2 q1p2)(p1 p2)2

< 0.

, f1(x)= x2 + p1x+ q1 f2(x)=

= x2 + p2x+ q2 (x1, y1), y1 < 0. x2 + p1x+ q1 x

2 + p2x+ q2 .

* . , x= x0 y= y0 xy=1 , . - . .

2

2.1.

2.1.

(x2x1)3+ (x23x+2)3= (2x24x+1)3.2.2. x4+ax3+ bx2+ax+1=0.2.3. x4+ax3+ bx2ax+1=0.2.4. x4+ax3+ (a+ b)x2+2bx+ b=0.

2.5. 1

x2 1(x+ 1)2

=1.

2.2.

2.6. (x2x+ 1)3x2(x 1)2 =

(a2 a+1)3a2(a 1)2 , a>1.

2.7.

1 x1+x(x 1)

2! . . .+ (1)nx(x 1) . . . (xn+1)

n!=0.

2.8. n> 1 . - xnnx+n1=0.

2.3.

2.92.15 , . - .

2.9. 2x6+x+4=5.

27

2.10.

m(1+x)2 m

(1x)2= m

1x2.

2.11.

3x29+4

x216+5

x225= 120

x.

2.12. x+3+

x=3.

2.13. aa+x=x.

2.14. x+34x1+

x+86x1=1.

2.15. 31x+ 31+x= p, p-

.

2.4.

2.16.

|x+1| |x|+3|x1| 2|x2|=x+2.2.17. x3 [x] = 3, [x] -

, x.

2.1. u=x2x1 v=x23x+2. - u3+v3=(u+v)3, . . 3uv(u+v)=0. x2x1=0, x23x++2= 0 2x24x+1= 0.

2.2. y=x+ 1/x.2.3. y=x 1/x.2.4. y=1/x+ 1/x2.2.5. x4+2x3+x22x

1 = 0. 2.4 a= 2 b = 1 .

2.6. R(x)=(x2x+1)3x2(x1)2

x 1/x 1x. a, 1/a, 1a, 1/(1a), 11/a a/(1 a).

28 2.

a> 1 . - 6, .

2.7. : x= 1, 2, . . . , n.

1 k1+

k(k1)2!

. . .+ (1)k k(k1) . . . 2 1k!

= (11)k= 0

, k= 1, 2, . . . , n . n , - n.

2.8. : x= 1. ,

xnnx+n1= (1+x+ . . .+xn1n)(x1).

x>1, 1+x+ . . .+xn1n>0, 0 3, x < 1, x +

+p3+

x

29

a . , :

a=x2+x+ 1;

a=x2x. x, - :

x1,2=12ra 3

4;

x3,4=1

2ra+

1

4.

2.14. : 56 x6 10. ,

x+3 4x1= (x 1 2)2,x+8 6x1= (x 1 3)2.

|x 1 2|+ |x1 3|= 1( ). .

1.x 1 2 > 0 x1 3 > 0, . . x > 10.

x=10.2.x 1 2> 0 x 1 36 0, . . 56 x6 10.

, . . 5 6 x 6 10, x .

3.x 1 26 0 x 1 36 0, . . x6 5.

x=5.,

x1 26 0 x 1 3> 0, , -

.2.15. :

2+33p1x2(31x+ 31+x)= p3.

p 31x+ 31+x.

2+3p 31x2= p3,

x=r1

p323p

3;

p=1 0< p6 2. : , .

30 2.

. - u= 3

1x, v= 31+x. u+ v= p

( ). u3 + v3 + 3uv(u+ v)= p3. p u+ v. u3 + v3 + 3uvp= p3, . . (u+ v)3 p3 3uv(u+ v p)= 0. (u+ v)3 p3 : a3 b3 == (a b)(a2 + ab + b2). (u + v p) ,

(u+ v p)(u2+ v2+ p2+up+ vpuv)= 0.

(u2+ v2+ p2+up+ vpuv)=0, . . (u+ p)2+ (v+ p)2+ (u v)2=0., , u= v= p,. . 3

1x= 31+x=p. -

x= 0, p=1. x= 0 p=1 .

2.16. : x=2 x> 2. x> 2, . 1 6 x < 2, 4x= 8,

. 06 x< 1, 2x= 2,

. 16 x

3

3.1.

3.13.8 - .

3.1.

x(y+ z)=35,

y(x+ z)=32,

z(x+ y)=27.

3.2.

x+ y+xy=19,

y+ z+ yz=11,

z+x+ zx=14.

3.3.

{2y=4x2,2x=4 y2. 3.4.

x+ y+ z=a,

x2+ y2+ z2=a2,

x3+ y3+ z3=a3.

3.5.

1x1x2=0,1x2x3=0,1x3x4=0,................

1xn1xn=0,1xnx1=0.

3.6.

{x3 y3=26,x2yxy2=6.

3.7.

3xyzx3 y3 z3= b3,x+ y+ z=2b,

x2+ y2 z2= b2.3.8.

x2+y22z2=2a2,x+y+2z=4(a2+1),

z2xy=a2.

3.2.

3.93.14 .

32 3.

3.9.

{x+ y+xy=2+3

2,

x2+ y2=6.3.10.

{x3+ y3=1,

x4+ y4=1.

3.11.

{x+ y=2,

xy z2=1.3.12.

x+

3xyx2+y2

=3,

y x+3yx2+y2

=0.

3.13.

(x3+x4+x5)5=3x1,

(x4+x5+x1)5=3x2,

(x5+x1+x2)5=3x3,

(x1+x2+x3)5=3x4,

(x2+x3+x4)5=3x5.

3.14.

2x211+x21

=x2,

2x221+x22

=x3,

2x231+x23

=x1.

3.3.

3.15. (x1>0, x2>0,x3 >0, x4 >0, x5 >0)

x1+x2=x23,

x2+x3=x24,

x3+x4=x25,

x4+x5=x21,

x5+x1=x22.

3.4.

3.16. {x2 y2=0,(xa)2+ y2=1

33

, , . - a ?

3.5.

3.17.

x1+2x2+2x3+2x4+2x5=1,

x1+3x2+4x3+4x4+4x5=2,

x1+3x2+5x3+6x4+6x5=3,

x1+3x2+5x3+7x4+8x5=4,

x1+3x2+5x3+7x4+9x5=5.

3.18.

x1+x2+x3=6,

x2+x3+x4=9,

x3+x4+x5=3,

x4+x5+x6=3,x5+x6+x7=9,x6+x7+x8=6,x7+x8+x1=2,x8+x1+x2=2.

3.19. a, b, c .

x+ay+a2z+a3=0,

x+ by+ b2z+ b3=0,

x+ cy+ c2z+ c3=0.

34 3.

3.20. a1, . . ., an . -,

x1+ . . .+xn=0,

a1x1+ . . .+anxn=0,

7a21x1+ . . .+a2nxn=0,

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

an11 x1+ . . .+an1n xn=0

.3.21.

x(1 1

2n

)+ y

(1 1

2n+1

)+ z(1 1

2n+2

)=0,

n=1, 2, 3, 4, . . .3.22. 100 a1, a2, a3, . . ., a100, -

:

a13a2+2a3>0,a23a3+2a4>0,a33a4+2a5>0,......................

a993a100+2a1>0,a1003a1+2a2>0.

, ai .3.23.

10x1+ 3x2+ 4x3+ x4+ x5 =0,

11x2+ 2x3+ 2x4+ 3x5+ x6 =0,

15x3+ 4x4+ 5x5+ 4x6+ x7=0,

2x1+ x2 3x3+12x4 3x5+ x6+ x7=0,6x1 5x2+ 3x3 x4+17x5+ x6 =0,3x1+ 2x2 3x3+ 4x4+ x516x6+ 2x7=0,4x1 8x2+ x3+ x4 3x5 +19x7=0.

35

3.24. , x1x2=a,x3x4= b,x1+x2+x3+x4=1

(x1 > 0, x2 > 0,x3 >0, x4 >0) , |a|+ |b|

36 3.

(x+ y + z)3 (x3 + y3 + z3)= 3(x + y)(y + z)(z+ x) -, (x + y)(y + z)(z + x) = 0. (1), xyz= (xy+ yz+ xz)(x+ y+ z) (x+ y((y+ z)(z+ x)= 0. x= 0, (1) , yz= 0. y= 0 z= a, z= 0 y= a. - . : (0, 0, a),(0, a, 0) (a, 0, 0).

3.5. : x1=x2= . . .=xn=1 n, x1=x3= . . .. . .=xn1= a x2=x4= . . .=xn= 1/a (a 6= 0) n.

n . , x2 6= 0, - x1 = x3. x2=x4 . . , - xn = x2. x1 = x3 = . . .= xn = x2 == x4 = . . .= xn1. x21 = 1,. . x1 = 1. , - .

n x1 = x3 = . . .= xn1, x2 == x4 = . . . = xn2 x2 = xn. , .

3.6. y= kx. , k 6=1. x3 k3x3= 26,kx3yk2x3= 6

x3=26

1 k3 x3=

6

k k2 . ,

26

1 k3 =6

k k2 .

1 k. 26

1+ k+ k2=6

k,

k= 3 1/3. x3 =1 x3 = 27. - : (1, 3),

1 i32

,3

2(1 i3)

, (3, 1),3 3i3

2,1

2(1 i3)

.

3.7. , b = 0. z=xy z2=x2+y2. , xy= 0. , x= 0,z=y y= 0, z=x. .

37

, b 6= 0. -

3xyzx3y3 z3= (x+ y+ z)(xy+ yz+xzx2 y2 z2). , xy+yz+xzx2y2 z2 = b

2

2. x + y + z = 2b,

x2+ y2+ z2+ 2xy+ 2yz+ 2xz= 4b2. , x2+ y2+ z2= b2

xy + yz+ xz =3

2b2. -

, z = 0.

, x2 + y2 = b2 xy =3

2b2. ,

x=

1

r12

b, y=

1

r12

b.

3.8. :8>:x2+y2= 2z2+ 2a2,

x+ y= 4(a2+ 1)2z,xy=a2 z2.

, , 2, . - :

0= 16(a2+1)216(a2+1)z,. . z=a2+1. :(

x+y=2(a2+1),

xy= a4+ a2+1.

-; ,

x=a2 a+1, y= a2 a+ 1.3.9. u=x+ y v=xy. u+ v= 2+3

2 u22v= 6,

u2 + 2u = 6 + 2(2 + 32) = 10 + 6

2. , u = 1

p11+6

2=1(3+2), . . u=2+2 42.

v=2+32u=22 6+42. u=42 v=6+42,

(x y)2 = (x+ y)2 4xy = (4+2)2 4(6+ 42) < 0. x+ y= 2+

2 xy= 2

2. -

: (x, y)= (2,2) (

2, 2).

.

38 3.

3.10. , x= 0 1, y=1 0. , x 6=1 y 6=1. : x= 0, y = 1 x= 1, y = 0. , .

, 0 < |x|, |y|< 1. |x|3+ |y|30 y 2,

x+ y= 2 , x= y= 1. z= 0.

3.12. y, x - . 2xy 1= 3y. , y 6= 0, x= 3

2+

1

2y.

, 4y4 3y2 1=0. , y2> 0, y2= 1, . . y1= 1 y2=1. y x1= 2 x2=1.

3.13. , x1 > xi (i= 2, 3, 4, 5). f(x)= x5 , 3x2= (x4+ x5+x1)

5> (x3+x4+x5)

5= 3x1., x1 = x2 x3 = x1. , 3x4 = (x1 + x2 + x3)

5>

> (x5+x1+x2)5=3x3. , x4=x3 x5=x3.

, x1=x2=x3=x4=x5=x. x (3x)5=3x. : x=0 1/3.

3.14. x1=x2=x3=0. , x1, x2, x3 , . ,

x1x2x3 6= 0. 1+ 1x21

=2

x2,

1+1

x22=

2

x3, 1+

1

x23=

2

x1. ,

1 1x1

2+1 1

x2

2+1 1

x3

2=0.

, x1=x2=x3= 1.3.15. xmin = xi x1, . . . , x5, xmax =

= xj . x2min = xi2 + xi1 > 2xmin ( -

, x0 = x5 x1 = x4) x2max = xj2 + xj1 6 2xmax.

39

xmin xmax , xmin > 2> xmax. -, xmin=xmax=2.

3.16. : a=1, a=2.

y=x. - ,

(x a)2+x2=1. (1) , - (1) . (1) x = 0, a2=1, . . a=1. , (1) (. . ). - (1), a=

2.

3.17. : x1=x3=x5= 1, x2=x4=1. , ,

, , , . .:8>>>>>>>>>:

x1+ 2x2+ 2x3+2x4+ 2x5= 1,

x2+ 2x3+2x4+ 2x5= 1,

x3+2x4+ 2x5= 1,

x4+ 2x5= 1,

x5= 1.

x5, x4, x3, x2, x1.3.18. : x1 = x8 = 1, x2 = x7 = 2, x3 = x6 = 3, x4 =

=x5= 4. , 3(x1 + x2 + . . . + x8) = 0. -

, . 2x1 + x2 + x3 + . . . + x8 = 1, , x1 = 1. .

3.19. P(t)=t3+t2z+ty+x. - a, b, c P(t). P(t) = (t a)(t b)(t c), , x = abc, y = ab + bc + ca,z=(a+ b+ c).

3.20. , a1, . . . , an , P(t), 0 t= a2, . . . , an - 1 t= a1. P(t)= l(t a2) . . . (t an), l(a1 a2) . . . (a1 an)= 1, . . l= 1

(a1 a2) . . . (a1 an).

P(t) P(t)= p0 + p1t+ . . .+ pn1tn1. p0,

40 3.

p1, . . . , pn1. , - P(a1)x1+P(a2)x2+ . . .+P(an)xn=0, . . x1=0. , x2=0, . . . , xn=0.

. 10.35.

3.21. : y=3x, z=2x (x )., -

:

x+ y+ z= 0,

4x+ 2y+ z=0.

-, -

, 1

2n+2, n-

. -, -

x1 1

2

+y1 1

4

+ z1 1

8

= 0,

x1 1

4

+ y1 1

8

+ z1 1

16

= 0

. , - , x/4+y/8+z/16=0. , x+ y+ z=0.

3.22. . ak - 1 3 + 2 = 0. , a13a2+2a3, . . . , 0., 0, . . -, .

(a1a2)+ 2(a3 a2)=0,(a2a3)+ 2(a4 a3)=0,............................

(a100a1)+ 2(a2 a1)= 0. a2a3= (a1a2)/2,a3a4= (a2 a3)/2= (a1 a2)/22, . . . , a1a2= (a100 a1)/2= (a1 a2)/2100. a1 = a2. a2= a3, a3=a4, . . . , a100= a1.

3.23. : x1=x2= . . .=x7= 0.

7Pi=1

aijxi=

= 0. aij : |ajj | >>Pi6=j|aij | j. x1, . . . , x7 -

41

. , x1, . . . , x7 . xk - . |akkxk | >

Pi6=k

aikxi,

7Pi=1

aikxi= 0 . .

3.24. a>0, x1=x2+a, a< 0, x2 = x1 a. x1=x2, x

2=x1. , ,

x1=x2+a a>0. , x3=x4+b b>0. , 1=x1+x2+x3+x4= 2x2+2x4+a+ b> a+ b ( a, b , 1> |a|+ |b|).

, a+ b

4

4.1.

4.1. 1*2*3* . . .*10=0 - , ?

4.2. , - n ( 1 ) , .

4.3. a1, . . ., a2n+1 , b1, . . ., b2n+1 , . -, ak bk, k= 1, 2, . . ., 2n+ 1,.

4.4. a, b, c , a 6=0. , ax2 + bx+ c= 0 - , a, b, c.

4.5. , -

a0xn+a1x

n1+ . . .+an1x+an,

x=0 x=1 , .

4.6. x . - x , 4. , - , .

43

4.7. , - ( , ):

11 1 1

1 2 3 2 11 3 6 7 6 3 1

1 4 10 16 19 16 10 4 1

, , , - .

4.2.

p > 1 , , - 1. n>1 , . n, n= n1n2, n1 < n n2 < n. - n> 1 . - . ( ). , - , - .

a b , a 6 b. - q r, b= qa+ r r < a ( ). - . a0 a1 , a0 > a1. a0 a1 : a0 = q1a1 + a2; a1 a2 : a1= q2a2+ a3 . . ak1 = qkak. a0, a1, . . . , ak ma0 + na1, m n . , , ak a0 a1. -, ak2= qk1ak1+ ak = (qk1qk+ 1)ak . ., a0 a1 ak. , ak - a0 a1. , a0 a1 ma0+ na1, m n

44 4.

. . a b (a, b).

4.8. bc a (a, b) = 1. , c a.

4.9. .

4.10. , 21n+ 4

14n+ 3

n.4.11. a1, a2, . . .

, (am, an)=(amn, an) m>n., (am, an)=ad, d=(m, n).

4.12. , a - m n (am 1, an 1) = ad 1, d=(m, n).

4.3.

4.13. p/q , p q-

. f(p

q

)=

p2q2

q1 . . . qn, q1, . . ., qn-

q. , f - .

4.4.

4.14. anan1 . . . a1a0 - .

) , 3 , a0+a1+ . . .+an 3.

) , 9 , a0+a1+ . . .+an 9.

) , 11 , a0a1+a2a3+ . . .+ (1)nan 11.

4.15. 300 , . - ?

45

4.16. 1, 2, 3, 4, 5, 6, 7., , , .

4.17. anan1 . . . a1a0 - . anan1 . . . a1+2a0. - . . , , 19., 19 , 19.

4.18. anan1 . . . a1a0 - . anan1 . . . a1 2a0. -, 7 , 7.

4.5.

4.19. ) , (a, b)=ab

(a, b).

) ,

(a, b, c)=abc(a, b, c)

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

( 14.26.)) ,

(a, b, c)=abc(a, b, c)

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

4.20. ,

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

4.21. , (a, a+ b)

(a, b)=a+ b

b.

4.22. a b . -, (a+ b, a2+ b2)=1 2.

46 4.

4.23. , - .

4.24. , n - a1 1. , am+bm an+ bn , m=kn, k.

* * *

4.33. ) , 1/2+1/3+ . . .+1/n .

47

) , 1

k+

1

k+1+ . . . +

1

k+n, k

n , .4.34. , :

)(m+n)!m!n!

; )(2m)! (2n)!m!n! (m+n)!

;

)(5m)! (5n)!

m!n! (3m+n)! (3n+m)!;

)(3m+ 3n)! (3n)! (2m)! (2n)!

(2m+ 3n)! (m+ 2n)!m! (n!)2(m+n)!.

4.7.

4.35. 1 100 ?

4.36. p a , n! pa. ,

a=[n

p

]+[n

p2

]+[n

p3

]+ . . .

4.37. , n! 2n.4.38. ,

(n+1)(n+2) . . . 2n.

* * *

4.39. n, 1

n

1

n+1

.4.40. ) , a

m k, ak 1 2m.

) , a - m, am1 2m.

4.41. m, :) 3m1 2m; ) 31m1 2m.

48 4.

4.8.

4.42. : ) 3, ) 4, ) 5, ) 8?

4.43. , :) 5 4, 6

5, 7 6.) 5 4, 6

5, 8 7.4.44. , :) 5 a, 6

b.) 5 a, 6

b, 7 c.4.45. ) n 4 3. -

, n - .

) n 8 7. , n - .

4.46. , - 131 112, 132 98.

4.47. 523 . . . , - 7, 8 9.

4.48. 7

1010+10(102)+10(10

3)+ . . .+10(1010).

4.49. m1, . . ., mk m=m1 . . .mk. , a1, . . .. . ., ak x ai (mod mi), i = 1, . . .. . ., k, , x1 x2 , x1 x2 m ( -).

4.50. n, 2n1 7.

49

4.51. p . , r a1, . . ., ar, p. , r < p, - r + 1 , p ( , ).

4.52. , 2n1 n , n.

4.9.

m n , (m, n)= 1.

4.53. , n n(n+1) .

4.54. ) , - n, n, n + 1, n + 2, n + 3, n + 4 , .

) , n, n, n + 1, n + 2, . . ., n + 9 , .

4.55. n m . -, Fn = 22

n1+ 1 Fm = 22

m1+ 1

.

4.10.

4.56. , p>3 12 1.

4.57. , - ().

4.58. , - 4k1.

4.59. ) , 2n1 , - n .

) , 2n+1 , n=2k.

50 4.

4.60. , n 22

n

+ 22n1

+ 1 n -.

4.11.

4.61. , n a b ab n a b.

4.62. p , a p. , p a,2a, 3a, . . ., (p 1)a , . . 1 p 1 .

4.63. , p , a b p, p a b ab p.

4.1. : . 1 2 3 . . . 10 ; , . 5 , .

4.2. d n n/d. d d n/d (. . n 6= d2), n , . n= d2, , d, , .

4.3. , ak bk . (a1b1)+ . . .+ (a2n+1b2n+1) ( ). 0, a1+ . . .. . .+a2n+1= b1+ . . .+ b2n+1.

4.4. , a, b, c ax2+ bx+ c= 0 x=m/n, m n - . am2+bmn+cn2=0 , m2+mn+n2 . m n ,

51

, . m2 +mn+ n2

.4.5. P(x) = a0x

n + a1xn1 + . . . + an1x+ an.

an = P(0) a0 + a1 + . . .+ an = P(1) . x- , P(x) an (mod 2). x , P(x) a0 + a1 + . . . + an (mod 2). , P(x) , .

4.6. , - 4, , -. , - . , . 2. anx

n++ an1xn1 + . . . + xr bmxm + bm1xm1 + . . . + xs. - - 2, anbmx

n+m+ . . .+xr+s. , - xr+s , -.

4.7. , , - , 0, 1:

1 0 1 01 1 0 1

1 0 0 01 1 1 0

1 0 1 0. . . .

, - . - , .

4.8. m n , ma+nb==(a, b)=1. mac+nbc=c, . . a(mc+n1)=c, n1. , c a.

4.9. , a = p1 . . . pr = q1 . . . qs, p1, . . . , pr,q1, . . . , qs . , (p1, q1) = 1, - (p1, q1) = p1 p1 = q1. (p1, q1) = 1, 4.8 q2 . . . qs p1. (p1, q2)==1, q3 . . . qs p1 . . p1= qi. -.

52 4.

4.10. 21n + 4 14n + 3, . 21n + 4 14n + 3, 7n + 1. 14n + 3 7n + 1, 1. , 21n + 4 14n + 3 .

4.11. {m, n}, m > n, {m, n} {m n, n} {d, d}, d=(m, n). , m n .

4.12. 4.11 , (am1, an1)=(amn1, an1) m>n>1. am1== an(amn 1)+ an 1 a an 1 .

4.13. p = pa11 . . . pamm , q = q

b11 . . . q

bnn . f(p/q) = p

2a11 . . .

. . . p2amm q2b111 . . . q

2bn1n . , -

.4.14. ) 10 3 1. 10k

3 1. , ak10k

3 ak.) ).) 10 11 1. 10k

11 (1)k. , ak10k 11 (1)kak.

4.15. 3, 9, .

4.16. a b , - . , a b a 6= b. a b b. , a b

b6 7. -

, a b 9, b 9. a bb

9. .

4.17. 10a+ b a+ 2b. 10a+b, 19, 9a>b, . . 10a+b>> a + 2b. . , 10a + b 19 - , 20a + 2b 19, . . a + 2b 19.

4.18. 10a + a0. -, 7 , a 2a0 7. , 10a + a0 7 , 20a+ 2a0 7. , 7 20a+ 2a0 , a+ 2a0.

53

4.19. ) a= pa11 . . . pakk b= p

b11 . . . p

bkk .

(a, b)= pmin{a1,b1}1 . . . pmin{ak,bk}k ,

(a, b)= pmax{a1,b1}1 . . . pmax{ak,bk}k .

.

a=pa b=pb, a6b, (a, b)=pa ab

(a, b)=

=papb

pb= pa.

) , a= pa, b= pb, c= pg, a6b6g. (a, b, c)= pa,

abc(a, b, c)

(a, b)(b, c)(c, a)=

papbpgpg

pbpgpg= pa.

) ,

papbpgpa

papbpa= pg.

4.20. p, a, b, c, a, b, g. ,

min(a, min(b, g))=min(min(a, b), g))=min(a, b, g),

min(x, y) x y. - .

4.21. 4.19 )

(a+ b)(a, b)=(a+ b)ab

(a, b)=

(a+ b)ab

(a, a+ b)= b(a, a+ b).

4.22. , a+b a2+b2 d. 2ab= (a+ b)2 (a2 + b2) d. 2a2= 2a(a+ b) 2ab 2b2 = 2b(a+ b) 2ab d. a b , a2 b2 . , d= 1 2.

4.23. a b , a b. - - . . a= pa. . . b= pb. . . , a6 b. a b pa. . . , - pb. . .

54 4.

pa. . . a b .

4.24. a.

a

a1>

a

a2> . . .>

a

an .

a

a1> n, . . a>na1.

4.25. f(a, b)=(a, b) (1)(3); (1) (2) , (3) 4.21. , (2) (3) f(a, b), f(a1, b1) a1 b1, a1 + b1 < a + b. f(a, b) -.

4.26. a1, . . . , an .

1, 2, 3, . . . , N, ak, hN

ak

i. -

a1, . . . , an N, 1, 2, . . . , N , a1, . . . , an. 1, 2, . . . , N, a1, . . . , an, h

N

a1

i+hN

a2

i+ . . .+

hN

an

i.

1, 2, . . . , N N , hN

a1

i+hN

a2

i+ . . .+

hN

an

i6N.

, hN

ak

i>

N

ak 1,

N

a11+N

a2 1+ . . .+

N

an1

55

4.29. : n. , 323 = 17 19, 323 -, 17 19. 20n 3n 20 3 = 17. , 16n (1)n (mod 17), 16n 1 17 , n . 19, 20n 1 20 1= 19 n, n= 2m 16n 3n 162 32 = 13 19, 19.

4.30. xnyn xy ( 5.1 ), bn kn b k. , a bn = (a kn) (bn kn) bk k 6=b. , a= bn.

4.31. 2n 2=nm. 22

n122n 1 =2

22n212n1 = 2

2nm12n 1 =2(2

n(m1)+ 2n(m2)+ . . .+ 2n+ 1).

4.32. m= kn, k , am + bm = (an)k ++ (bn)k an+ bn ( 5.1 ).

m= kn+ r, k 0< r< n. , am+ bm an+ bn. ,

akn+r + bkn+r = ar(akn+ bkn)+ bkn(br ar). ar(akn + bkn) an + bn, bkn(br ar) an+bn, bkn an+bn 0< |brar |

56 4.

p, q . p+1< q p+1 . 2m(p+1) k, k+1, . . . , k+n 2m+1, m.

4.34. Qk

((akm + bkn)!)dk, ak

bk , dk . - p, n!, Pm=1

[n/pm] ( 4.36). x, y > 0

P

dk[akx+ bky]> 0, .

)) ak, bk, dk Pk

dkak=

=Pk

dkbk = 0. f(x, y) =Pk

dk(akx + bky)

: f(x+ 1, y)= f(x, y)= f(x, y+ 1). f(x, y)> 0 x, y, 06 x, y< 1.

) [x+y] [x] [y]>0 06x, y 0 06 x, y< 1. , [x] = [y] = 0, -

0

1

0

1

0

1

x

y

. 4.1. )

f(x, y)=[2x]+ [2y] [x+y]>0. - f(x, y) 06x, y 0 06 x, y< 1. -

57

f(x, y) , [akx + bky] akx + bky = 1, 2, . . . , ak + bk 1; , (. 4.2). -, ).

0 1 0 1

0

2 1

1 0

2

1 0

1

0

2

1

1

0

1

1

0

1 2 1

10

1

0

0

21

1

0

2

1

2

1

2

1

1

1 0 1

2

12

1

1 01

2

1

21

0

2

1 0

12

12 1 2

x

y

. 4.2. )

) f(x, y)= [3x+ 3y] + [3y] + [2x] ++[2y] [2x+3y] [x+2y] [x+y]>0. ,

58 4.

0 1

0 1

0

1

2

10

1

0

10 1

0 1

1 2

1 2

1

2

1

2 1

0

1

2

12

12

x

y

. 4.3. )

), . . 4.3. , x + y = 1 , f(x, y) 1 1,. . .

4.35. : 24. 1 100 20 , 5, , 5, 4 , 25 (, 125, ). 524 525. , 224.

4.36. 1, 2, . . . , n [n/p] , p,[n/p2] , p2, . .

4.37. a , n! - 2a. 4.36

a=hn

2

i+hn

22

i+hn

23

i+ . . .6

n

2+

n

22+

n

23+ . . .=n,

[x]6 x. ,hn

2k

i= 01

2k5l = n 2s5t = n+ 1. n n+ 1 , 2k + 1 = 5t, 5l+ 1= 2s. , 5l+1= 2s. 2s - 6, s= 4m. , 5l= 24m1= (22m 1)(22m+ 1). 22m 1 22m + 1 5.

2k + 1= 5t. t= 2ms, s.

5t 1= (52m 1)(52m(s1)+ 52m(s2)+ . . .+ 1). -, . . . , s= 1. ,

52m 1= (5 1)(5+ 1)(52+ 1) . . . (52m1 + 1).

5+ 1= 6 , m= 0. 22+ 1= 5.

4.40. ) k= 2n.

ak1= (a2n1 1)(a2n1 +1)== (a1)(a+ 1)(a2+ 1)(a4+1) . . . (a2n1 + 1). (1)

a , n+ 1 . , ak 1 2n+1. , k= 2n n>m1, ak1 2m.

) : m=2ns, s -, d(m)=n. , am1= (a2n 1)(a2n(s1)+a2n(s2)+ . . .. . . + 1), . d(am1)=d(a2n 1). , - (1) a2 + 1, . . . , a2

n1+ 1 2

4, 1 - 4. ,

d(a2n 1)=

(d(a 1) n=0;d(a2 1)+n 1 n> 1.

am 1 2m , d(am1)>m. . d(m)=0, . . m , m6 d(a1).

60 4.

. d(m)>1, . . - m , d(a21)+n1>>m= 2ns, n= d(m).

s6d(a21)+n 1

2n( s ). -

, n/2n - n ( 25.18).

4.41. ) : 1, 2 4. - 4.40 ). a= 3, d(a 1)= 1 d(a2 1)= 3. s6

2+n

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

) : 1, 2, 4, 6 8. a = 31, d(a 1) = 1 d(a21)=d(30 32)= 6. s6 5+n

2n

: (s, n) = (1, 1), (3, 1), (1, 2) (1, 3). (, s .)

4.42. ) : 0 1. (3k1)2== 9k26k+ 1.

) : 0 1. (2k+1)2=4k2++ 4k+ 1.

) : 0, 1 4. (5k 1)2 == 25k2 10k+1 (5k 2)2= 25k2 20k+ 4.

) : 0, 1 4. (2k+ 1)2 == 4k(k+ 1)+ 1 (4k+ 2)2 = 16k2 + 16k+ 4 , k(k+ 1) .

4.43. ) : 209. n . n+ 1 5 6 7=210. n=209.

) : 119. n . n+1 (5, 6, 8)= 120. n=119.

4.44. ) : 6a+25b. 6n1(mod 5) 5m1(mod 6).

6na+5mb6na a (mod 5)6na+ 5mb 5mb b (mod 6).

m=5 n= 1.) : 126a+ 175b+ 120c. 42k 1 (mod 5), 35n

1 (mod 6) 30m 1 (mod 7) ( 42= 6 7, 35= 5 7, 30= 5 6).

42ka+ 35nb+30mc 42kaa (mod 5),42ka+ 35nb+30mc35nb b (mod 6),42ka+35nb+ 30mc 30mc c (mod 7).

k= 3, n= 5, m=4.

61

4.45. ) 4.42 ) 4 0, 1 2.

) 4.42 ) - 8 0, 1, 2, 4 5, 0, 1, 2, 3, 4, 5 6.

4.46. : 1946. N . N= 131k + 112 = 132l + 98, k l . -

, N < 10 000, l =N 98132

131. l6 75, k= l l 14= 0. , N= 131 14+112= 132 14+98=1946.

4.47. : 523152 523 656. 7 8 9= 504.

523 000 504 : 523 000= 1037 504+ 352. 504 352= 152, 504 523 152 523152+ 504== 523 656. , 504, 523 000 523 999 .

4.48. : 5. , 106 1 (mod 7), 103 + 1 7, 10k 4 (mod 6) k> 1, - 99 . . . 96 3. , 1010

k 104 (mod 7) k> 1. 10 104=105 7. 5.

4.49. ni = m/mi. ni , mi, (ni, mi)= 1. ri si , rimi + sini = 1( - . 43). ei = sini x = a1e1 + . . . + akek., ei 1 (mod mi) ei 0 (mod mj) j 6= i, xai (mod mi), i= 1, . . . , k.

x1 x2 , x1 x2 0 (mod mi), i = 1, . . . , k. m1, . . . , mk , x1x2 m.

4.50. , 23 = 8 1 (mod 7). 23k 1 (mod 7),23k+12 (mod 7) 23k+24 (mod 7). , 2n1 7 , n 3.

4.51. r. r = 1 :0 a1. , r< p 1 r+1 . 0, s1, . . . , sr, a1, . . . , ar, p, 0+ar+1, s1+ar+1, . . . , sr+ar+1 p - 0, s1, . . . , sr. , ar+1 si (mod p)

62 4.

i. , si + ar+1 sj (mod p) j,. . 2ar+1 sj (mod p). , - p ar+1, 2ar+1, 3ar+1, . . . , (p 1)ar+1 p 0, s1, . . . , sr. p , ar+1 p. p ar+1, 2ar+1, . . . , (p 1)ar+1 . ,p 1> r, .

4.52. , n = a n = b, n = ab. 2ab 1 . n= b, ,2ab 1> 2b 1. 2ab 1 b , b. , 2b 1, b , b. 2ab 1= (2a 1)b+ b 1 , 2a1 2a1 b -, b, . . . - . n = a, 2a 1 a , a. a b . ab ab, .

: n= p.

p. b1 6 b2 6 . . . 6 b2p1 p. p 1 a1 = bp+1 b2,a2=bp+2b3, . . . , ap1=b2p1bp. ai=0, bi=bp+i, ,bi=bi+1= . . .=bi+p. p bi, bi+1, bi+p - p. , a1, . . . , ap1 .

x p b1 + b2 + . . . + bp. x= 0, p . -, x 6= 0. 4.51 a1, . . . , ap1 , p. , ai1 + . . .+ aik, p x. b1 + b2 + . . .+ bp + ai1 + . . .+ aik = b1 + . . .. . .+bp+(bp+i1bi1)+ . . .+(bp+ikbik) p. p .

4.53. , n(n+1)=mk, m k, k > 2. n n + 1 , n = ak

n+ 1= bk, a b . , b> a. (a+1)k >(a+1)ak1=ak+ak1>n+1. bk> (a+1)k >n+1.

63

4.54. ) |k l| 6 4 k 6= l, - k l 4. 4. . 3. .

) 5 . 3, 5 7. 3 , 5 7 . , 3, , 5 7. .

4.55. , n>m. x= 2

(x1)(x+1)(x2+1)(x4+1) . . . (x2n2 +1)=x2n1 + 1. F1F2 . . . Fn1+2=Fn.

, Fn Fm d. 2 ==FnF1F2 . . . Fn1 d. Fn Fm , d 6= 2.

4.56. , p> 3 6. 2 4, - . 3, 3. , p>3 6 1 5, . . 6n 1; 36n212n+1.

4.57. , - , , p1, . . . , pr. p1 . . . pr+ 1. p1, . . . , pr, , p1, . . . , pr.

4.58. , p1, . . . , pr 4k 1. 4p1 . . . pr 1. , 4k 1. , 4k + 1, . , p1, . . . , pr.

4.59. ) xq 1 x 1, 2pq 1== (2p)q 1 2q 1.

) q , xq+1 x+1. n q> 1, 2n + 1 2q+ 1.

64 4.

4.60. x4 + x2 + 1 = (x2 + 1 x)(x2+ 1+ x). x= 22n2 , 22

n1+ 22

n2+ 1 22

n1 22n2 + 1. , ,

22n2+1. n, -

22n1

+22n2

+1 .4.61. a= a1 + a2n b = b1 + b2n. a b = a1 b1 +

+ (a2 b2)n ab= a1b1+ (a2b1+a1b2+ a2b2n)n.4.62. xa ya (mod p), (x y)a p. a

p , x y p. , 16 x, y6 p1, x=y.

4.63. 4.62 p b, 2b, . . . , (p 1)b 1. , - b, 16 b6 p 1, bb 1 (mod p). a b(ab) (mod p).

5

5.1.

5.15.9 , -.

5.1. ) xn yn; ) x2n+1+ y2n+1.5.2. x4+4.5.3. (x+ y+ z)3x3 y3 z3.5.4. x3+ y3+ z33xyz.5.5. (x y)3+ (y z)3+ (zx)3.5.6. a10+a5+1.5.7. a4(b c)+ b4(ca)+ c4(a b).5.8. x4+x3+x2+x+12.5.9. ) x8+x4+1 .) x8+x4+1 ,

- .

5.2.

5.10. , n -

(2n)!

n!=2n (2n1)!!

5.3.

5.11. , m n - ,

66 5.

mn .

5.12. )

(a21+a22+a

23)(b

21+ b

22+ b

23) (a1b1+a2b2+a3b3)2.

)

(a21+ . . .+a2n)(b

21+ . . .+ b

2n) (a1b1+ . . .+anbn)2

( ).

5.4.

5.13. , a+1/a . ) , a2+1/a2 . ) , an+1/an

n.5.14. ,

.5.15. , -

.

5.16. , 1

a+1

b+1

c=

1

a+ b+ c

1

an+

1

bn+

+1

cn=

1

an+ bn+ cn n.

5.17. x, y, z ., (x y)5+ (y z)5+ (zx)5 5(y z)(zx)(x y).

5.18. , a

b c +b

c a +c

a b = 0, a

(b c)2 +b

(c a)2 +c

(a b)2 =0.5.19. , n2+3n+5 n

121.5.20. ,

x5+3x4y5x3y215x2y3+4xy4+12y5

33 x y.

67

5.21. , n>2 24n+2 + 1 .

5.22. , .

5.5.

5.23. 2

x2 1 2x

x2 1 -

a

x1 +b

x+1, a b .

5.24. a1, . . ., an . -, A1, . . ., An ,

1

(x+ a1) . . . (x+ an)=

A1

x+a1+ . . .+

An

x+ an.

5.25. a1

68 5.

5.28. p q ., [

q

p

]+[2qp

]+ . . .+

[(p 1)q

p

]=(p 1)(q1)

2.

5.29. p q - . ,

[q

p

]+[2q

p

]+ . . .+

[p 12

q

p

]+[p

q

]+

+[2pq

]+ . . .+

[q12

p

q

]=(p 1)(q1)

4

().5.30. , [

n+

n+1] = [

4n+2]

n.5.31. , [

n+

n+1+

n+2]= [

9n+8]

n.5.32. , [3

n+ 3

n+1] = [3

8n+3]

n.

5.1. ) (x y)(xn1+xn2y+ . . .+ yn1).) (x+y)(x2nx2n1y+x2n2y2 . . .+y2n).5.2. (x22x+ 2)(x2+2x+ 2).5.3. 3(x+y)(y+ z)(z+x).5.4. (x+ y+ z)(x2+ y2+ z2xy yz zx).5.5. 3(xy)(y z)(zx).5.6. (a2+ a+ 1)(a8 a7+a5 a4+a3 a+1).5.7. (a2+ b2+ c2+ ab+ bc+ ca)(a b)(b c)(a c).5.8. (x22x+ 3)(x2+3x+ 4).5.9. ) (x4+x2+ 1)(x4x2+ 1).)

,

(x2+ax+ 1)(x2 ax+1)=x4+ (2a2)x2+1; a=1 a=

3.

69

5.10. , n! 2n = 2 4 6 . . . 2n. n! 2n(2n 1)!!== (2 4 6 . . . 2n) 1 3 5 . . . (2n 1)= (2n)!.

5.11. (a2+b2)(c2+d2)= (ac+bd)2++ (ad bc)2.

5.12. ) (a1b2 a2b1)2+ (a2b3 a3b2)2+ (a1b3a3b1)2.)P(aibjajbi)2, i< j.

5.13. ) a2+1/a2= (a+1/a)22.)

an+1+1

an+1=an+

1

an

a+

1

a

an1+

1

an1

.

5.14. 2n+1= (n+1)2n2.5.15. ,

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

5.16. 1

a+

1

b+

1

c=

1

a+ b+ c

(bc+ ca+ab)(a+ b+ c)=abc, . . (a+ b)(b+ c)(c+a)=0 (-, abc 6= 0 a+ b+ c 6=0). , a,b, c x, x, y, y 6=x. an,bn, cn n .

5.17. , x2+y2+ z2xyyzxz. , xy=u yz=v. (xy)5++(yz)5+(zx)5=u5+v5(u+v)5=5(u4v+2u3v2+2u2v3+vu4)==5uv(u+ v)(u2 + uv+ v2) 5(y z)(z x)(x y)=5uv(u+ v). , u2+uv+ v2=x2+ y2+ z2xyyzxz.

5.18. ,

a

(b c)2 +b

(c a)2 +c

(a b)2 =

=

a

b c +b

c a +c

a b 1

b c +1

c a +1

a b.

,

a

b c 1c a +

1

a b=

ac ab(a b)(b c)(c a) ,

b

c a 1b c +

1

a b=

ab bc(a b)(b c)(c a) ,

c

a b 1b c +

1

c a=

bc ac(a b)(b c)(c a)

.

70 5.

5.19. , n2 + 3n+ 5= (n+ 7)(n 4)+ 33. - n2 + 3n + 5 121, (n + 7)(n 4) 11. (n+ 7) (n 4)= 11, 11 . , (n+7)(n4) 11, 121. (n+7)(n 4)+33 121.

5.20.

(x+2y)(x y)(x+ y)(x 2y)(x+ 3y). y 6=0 -, 33 ( - -, 1). y= 0 x5. x x5 33.

5.21. 24n+2+ 1 - (22 + 1)(24n 24n2 + . . . 22 + 1), (22n+1+ 2n+1+ 1)(22n+1 2n+1 + 1). n> 2 -.

5.22.

(a b)3+ (b c)3+ c3= 3b2(a c)+ (a3 3b(a2 c2)). t a=12t(t+1), b=(t+1)3

c= 12t(t 1). 72t(t+ 1)6= (a b)3+ (b c)3+ c3.

t 6=1, w= 72t - :

w=

a b(t+ 1)2

3+

b c(t+ 1)2

3+

c

(t+ 1)2

3.

, 72= (4)3+ (2)3+ 03.5.23. ,

a

x1 +b

x+1=

(a+ b)x+ (a b)x2 1 .

2

x2 1 =

=1

x1 1

x+ 1

2x

x21 =1

x 1 +1

x+1.

5.24. n.

n= 2 A1 =1

a2 a1 A2 =

1

a1 a2.

,

1

(x+ a1) . . . (x+ an1)=

B1

x+ a1+

Bn1x+ an1

.

71

Bi

(x+ ai)(x+ an)

cni

x+ ai+

dni

x+ an i= 1, 2, . . . , n 1. .

5.25. n. n=1 -. , n 1.

1

(x+ a1) . . . (x+ an1)=

B1

x+ a1+ . . .+

Bn1x+ an1

,

1

(x+ a2) . . . (x+ an)=

C2

x+ a2+ . . .+

Cn

x+ an,

B1, C2 > 0, B2, C3 0, . . .

1

(x+ a1) . . . (x+ an1) 1

(x+ a2) . . . (x+ an)=

an a1(x+ a1) . . . (x+ an)

,

an a1 > 0. , (ana1)A1=B1, (ana1)A2=B2C2, . . . , (ana1)An1=Bn1Cn1,(an a1)An=Cn.

5.26. x, y, z,

a11= b1c1, 2a12= b1c2+ b2c1,

a22= b2c2, 2a13= b1c3+ b3c1,

a33= b3c3, 2a23= b2c3+ b3c2.

, ,

A11x2+A22y

2+A33z2+2A12xy+ 2A13xz+2A23yz= 0

x, y, z, Aij . x= 1,y= z= 0 A11 = 0. A22 =A33 = 0. x=y=1, z= 0 A12=0. A13=A23= 0.

a11a22a33+2a13a12a23 a223a11+a

213a22+

+a212a33 aij bp cq. - .

5.27.

f(x)= [nx] [x]hx+

1

n

i . . .

hx+

n1n

i.

,

fx+

1

n

= [nx+ 1]

hx+

1

n

ihx+

2

n

i . . .

hx+

n1n

i [x+1] =

= [nx] [x]hx+

1

n

i . . .

hx+

n1n

i= f(x).

72 5.

f(x) = f(y) y, 06 y6 1/n. f(y)= 0.

5.28. 06 x6 p, 06 y6 q y= qx/p. , ( ) . , - , . p q , - ; . , (p1)(q 1) .

5.29. 1 6 x 6p12

, 1 6 y 6q 12

y = qx/p. . , , . , . -

, . . p12

q12

.

5.30. n+

n+ 1

4n+2. -

-, :

n+

n+1

4n+ 2 n+2n(n+1)+n+14n+2 2n(n+1)2n+1 4n2+ 4n4n2+ 4n+ 1. , n+n+ 1 1 -

n+

2

5

2

73

, n+2/5+n+7/10+n+7/5=3n+5/2 n+1/2+n++1+n+3/2=3n+3, 9n+8 0 -

a+ b

2

3n+

3n+1

2>

p3n 3n+1=

= 6n2+n. , 3

8n+4> 3

n+ 3

n+1> 6

64n2+64n> 3

8n+3.

38n+ 3>m, 3

n+ 3

n+ 1>m. , -

m, 3n+ 3

n+ 1>m> 3

8n+ 3,

8n+ 4>m3 > 8n+ 3, 8n+3 8n+ 4 .

6

6.1.

6.1. , : )2 +

3

11; )6+2

7

10+

21; )

11 5 35.

6.2. , :

2+

2+

2+ . . .

( - ) 2?

6.2.

.

6.3. )1

2+3; )

12+

3; )

12+

3+

5.

6.4. )1

3x 3y ; )

13x+

3y.

6.5. )1

nx ny ; )

12n+1

x+2n+1

y.

6.6.1

x 3y.

6.7.1

3x+

3y+

3z.

6.8. , - , .

75

6.3.

6.9. , a>b,

a+pa2 b2

apa2 b2

=ab.

6.10. ab, a

b :

)3+2

2; )

9+4

5; )

7210.

6.11. , 4+

7

47=2.

6.12. , 320+14

2+

32014

2 -

.6.13. , :

)3

3+

242

27+

3

3

242

27;

)3

6+

847

27+

3

6

847

27.

6.14. ,

341+

316 34=

3.

6.15. :

)3

321= 31

9 32

9+ 34

9;

)

35 34=

1

3(32+ 320 325);

)6732019= 3

5

3 32

3;

) 4

3+2

45

3245=

45+ 1

45 1

;

)

328 327= 13(398 3281);

)3

5

32

5 527

5= 5

1

25+ 5

3

25 5

9

25.

76 6.

6.4.

6.16. , :)p, p ;

)p1 . . . pk, p1, . . ., pk ;

)

p1 . . . pk

pk+1 . . . pn, p1, . . ., pn .

6.17. , 2+ 3

3 .

6.18. ,

a=34+

15+

3415.

6.19. ) , a, b,a+

b -

, a

b .

) , a, b, c,a+

b+

c -

, a,b

c .

) , a1, . . ., an,a1 + . . .+

an

, a1, . . .,

an .

6.20. p1, . . ., pk . -,

pk -

pi1 . . . pis, 16 i1 < . . .< is6k1,

.

6.5.

6.21. ) p . , m+n

p, m n .

) - m+ n

p, p -

.

, z m + np, m

n , p - , z=mnp.

77

6.22. , (a+ bp)n=An+Bn

p, p

, a, b, An, Bn, (a bp)n=AnBn

p.

6.23. (2+

3)1000.

6.24. , x, y, z t

(x+ y2)2+ (z+ t

2)2=5+4

2.

6.25. , m n (5+3

2)m= (3+5

2)n.

6.26. ) , (2 1)n =kk1, k

.) m n . ,

(mm1)n=kk1, k

.6.27. , n

[(1+3)2n+1] 2n+1 2n+2.

6.6.

6.28. - , - n. a/b c/d . -, |bcad|=1. 6.28 - Fn.

6.29. a/b

78 6.

6.32. ) , - 2 -.

) , - 2 .

6.33. , -

Fn n

q=2f(q), f .

6.7.

6.34. a b . , :

(1) [a], [b], [2a], [2b], [3a], [3b], . . .;

(2)1

a

+1

b

=1, a b .

6.35. a1, . . ., ak , - : [a], [b], [2a], [2b],[3a], [3b], . . . , k62.

6.1. - , -.

)2+

311 2+26+311 266 69. ,

2+3 3

25 -

.) , (3

5 32)6= 9(7320 19); ,

350, .

) , 3245>0. -, (4

5 1)4(3+ 245)= 10425 22= (45+ 1)4(3 245);

, 45, ; , -

4125, .

) , (398 328 1)2 = 9(328 3) =

= 9(328 327); , 314, .

, 398 > 3

28 + 1, . . 3

14(3

7 32) > 1. -

349+ 3

14+ 3

4,

81

5314> 3

49+ 3

14+ 3

4, . . 43

14> 3

49+ 3

4.

, 4314 > 8, 3

49 < 6

34

82 6.

r2 + c a b 6= 0, (2) , c. r2 + c= a+ b, (1) , 2

ab=2rc, r> 0. , ab= 0 c= 0.

, c .

a

b .) , a1, . . . , an -

. , a1 .

x1=a1, . . . , xn=

an y=

a1+ . . .+

an. -

f(y, x1, . . . , xn)=

Y(yx1x2 . . .xn), (1)

. - y, x1, x

22, . . . , x

2n. x1

,

f(y, x1, . . . , xn)= g(y, x21, . . . , x

2n)x1h(y, x21, . . . , x2n). (2)

(1) y x1 x2 . . . xn = 0. f(y, x1, . . . , xn)= 0. y, x

21, . . . , x

2n -

. h(y, x21, . . . , x2n) 6= 0,

x1= g(y, x21, . . . , x

2n)/h(y, x

21, . . . , x

2n)

. , h(y, x21, . . . , x

2n) = 0. -

(2),

f(y, x1, x2, . . . , xn) f(y, x1, x2, . . . , xn)=2x1h(y, x21, . . . , x2n).,

h(y, x21, . . . , x2n)=

1

2x1f(y, x1, x2, . . . , xn)=

=1

2x1

Y(y+x1x2. . .xn)= 1

2x1

Y(2x1+(x2x2)+. . .+(xnxn));

, y = x1 + . . .+ xn. , 2x1 + (x2 x2) + . . .. . .+ (xn xn) . x1 > 0, xk xk > 0. ,h(y, x21, . . . , x

2n) 6= 0.

6.20. k. k= 2. ,

p2 = a+ b

p1, a b -

. 6.16 ) , ab 6= 0. p2 = a

2 + 2abp1 + b

2p1, , p1

, .

83

, pk+1= a+ b

pk, a b -

p1, . . . ,

pk1. b = 0, ,

pk+1

p1, . . . ,

pk1, -

. a = 0, pk+1 = (a

+ bpk1)

pk.

a= 0 . ., pk+1= r

p1 . . . pk,

r . ( 6.16 ). (, p1, . . . , pk)

pk+1=a+b

pk, ab 6=0. , pk+1=a2+2abpk+b2pk,

. .pk=

pk+1 a2 pkb22ab

. -

( 6.8), pk

p1, . . .

. . . ,pk1, .

6.21. m+ np=m1 + n1

p, m, n, m1 n1

. m1 6=m n1 6= n, p=

mm1n1 n

. -

, p

.6.22. n. , z1z2 =

= z1z2, . . (a + bp)(c + d

p) = A + B

p, (a bp)

(cdp)=ABp. , (a+ bp)(An+Bnp)=

=An+1+Bn+1p, (a bp)(AnBn

p)=An+1Bn+1

p.

6.23. : 9. (2+3)n=An+Bn

3, An Bn-

. 6.22 (2 3)n = An Bn3.

(2+3)n + (23)n . 23

0,2679 < 0,3, (23)1000 0,0 . . . ( ).

6.24. x, y, z t , 6.22

(xy2)2+ (z t

2)2=5 4

2.

5 42 0.6.25. , (5+3

2)m= (3+5

2)n. -

6.22 (5 32)m = (3 52)n. . -, , |5 32|< 1 |3 52|> 1. -, (5+ 3

2)m = (3+ 5

2)n (5 32)m= (3 52)n; -

7m= (41)n.6.26. ) (

2 + 1)n = x

2 + y, 6.22

(2 1)n = (1)n(12)n = (1)n(y+ x2)= (1)n(

py2

2x2).

84 6.

y22x2=(y+x2)(yx2)=(1+2)n(12)n=(1)n. , n , (

2 1)6n=

py2

2x2 y2

2x2=1, n , (21)6n=2x2

py2 2x2y2=1.

)

(am bm 1)(cmdm1)=

= acm+ bd(m 1) (ad+ bc)pm(m1),

(a bpm(m 1))(cmdm 1)=

= (ac+ bd(m 1))m (ad+ bcm)

m 1

, (mm 1)n=am bm 1 n

(mm 1)n=abm(m 1) n (a b-

). (mm 1)n=xy,

x y (x = a2m y = b2(m 1) n, x = a2 y = b2m(m 1) n). xy= (xy)(x+y)= ((mm1)(m+m1))n= 1.

6.27. (1 +3)2n+1 + (1 3)2n+1 , 1 a2 > . . . > a7 , - . a3 > 16, a2 > 17 a1 > 18, a1+a2+a3> 51. a36 15, a46 14, a56 13, a66 12 a76 11, a4+a5+ a6+ a76 50, , a1+ a2+ a3> 50.

7.11. . 2/3 , ., 2/3 , . - , 4/3 ,. . 4/7 .

. x y ; x1 y1 , - x2 y2. 5x1

94 7.

7.13. n , k . 3,56100k/n64,5. , k>1

n>100k

4,5> 22,2k> 22,2. n> 23. -

23 , , .

7.14. 6 7 . 200/5= 40 . - : 3 7 2 9 .

7.15. 8 , , , . , , 8 1,5=12 , 1,5 ; . 4 350 = 1400 < 1500, - 4 . , 8 , , .

7.16. : 3 6 ( -, . . 4 6 3 - 7 ). , 3 6 - 195 . , 15 3 3 20 . 6 , , - . 15+20+140=175 3 + 7 6 = 45 . 5 , 20 ( - 5 ).

, 250 , , 200 .

7.17. , . - a1 a2, a1. , a2 ( a3). a2. , a3 a4, a3 . . n-, an ak, 16k6n1. , , n< 20. , k= 1. ,

95

ak, 2 6 k 6 n 1, k 1 k. n- an. - : . . .

7.18. , . , -, , ; - . .

7.19. a1, a2, . . . , an. - ai k + 1 , k , ai + ai+1 + ai+2 + . . .+ ai+k > 0 ( a1 - ). , ai k+ 1, ai+1, ai+2, . . . , ai+k k, k 1, . . . , 2, 1 . - . , ( K), - K 1 . ,K1 , K2, . . .

7.20. . - . . . , - . , , . - . . , - - .

8

8.1. x + 1/x> 2

, 1.1 x> 0 x+1/x> 2.

8.1. n x1, . . ., xn,

x1+x2+ . . .+xn=3,1

x1+

1

x2+ . . .+

1

xn=3?

8.2. , a1, . . ., an a1 . . . an=1,

(1+a1)(1+a2) . . . (1+an)>2n.

8.3. , x5x3+x=2, 34,

x1

x2+xn+

x2

x3+x1+ . . .+

xn1xn+xn2

+xn

x1+xn1>2.

8.2.

, a, b, c , a+b>c, b+c>a c+a>b.

97

, a,b, c .

8.5. , a, b, c - , (a2+ b2+ c2)2 >2(a4+ b4+ c4).

8.6. , a1,a2, . . ., an (n>3)

(a21+a22+ . . .+a

2n)

2> (n1)(a41+a42+ . . .+a4n),

-.

8.7.

Aa(Bb+Cc)+Bb(Cc+Aa)+Cc(Aa+Bb)>

>1

2(ABc2+BCa2+CAb2),

a>0, b>0, c>0 , A> 0, B> 0, C> 0. a, b, c ?

8.3.

8.8. xy6 (x2+ y2)/2.8.9. ( k

i=1

aibi

)26

( ki=1

a2i

)( ki=1

b2i

).

8.9 . - 1.9 5.12.

8.10.

a21+ . . .+a2n>

(a1+ . . .+ an)2

n.

8.11. x1, . . ., xn .

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

1

xn

)>n2.

98 8.

8.12. S= a1+ . . .+ an, a1, . . . , an- n>2. ,

a1

Sa1 + . . .+an

San >n

n1 .

8.4.

a1, . . . , an . -

A =a1+ . . .+ an

n, -

G = na1 . . . an. 1821 .

A>G. - , : -, n = 2m, n = 2m+1, , n, n 1 (. - 13.10).

8.13. a1, . . ., an .

a1+ . . .+ ann

> na1 . . . an.

( , .)8.14. a, b>0. , 2

a+33

b>55

ab.

8.15. , n> 2 -

n(nn+11)

99

8.17. a1, . . ., an , a1, . . .. . ., an .

x > 0 f(x) =ni=1

aixai. a a ,

y = f(x) y = axa x = x0., f(x) > axa, x=x0.

. 13.10, 25.21.

8.5. ,

8.18. a, b, c . ,

a2ab+ b2+b2 bc+ c2>

a2+ac+ c2,

, 1/a+1/c=1/b.

8.6.

8.19. a1, . . ., an . -,

a1

a2+a2

a3+ . . .+

an

a1>n.

8.20. , a,b, c

a3b+ b3c+ c3a>a2bc+ b2ca+ c2ab.

8.21. 06x1, . . ., xn61. , n>3

x1+ . . .+xnx1x2x2x3 . . .xn1xnxnx16 [n/2].8.22. a, b, c . ,

3+(a+b+c)+(1

a+1

b+1

c

)+(a

b+b

c+c

a

)>3(a+ 1)(b+ 1)(c+1)

abc+ 1.

100 8.

8.7.

8.23. , a, b, c, d -

, a/b< c/d, a

b b>0. , aabb >abba.8.26. , x1, . . ., xn -

, , s1 =

xi, s2 ==i1

2.

8.31. x y .,

x+ y

2x2+y2

2x3+y3

26x6+y6

2.

8.32. , n -

1

23

45

6 . . .

2n 12n

12

2

12n

0

x.8.40. x1, x2, . . . , x100 -

x21+x22+ . . .+x

2100 >10000,

x1+x2+ . . .+x100 b> 0. -,

(a+ b

2

)n1 a< 0. , x> 0, x 6= 1, xaax+a1>0.

) a , 0 1, A1/pB1/q 6 A/p + B/q, pA/p+B/q. ( A 6=B, .)

8.46. xi yi , p q- , 1/p+ 1/q= 1., p>1,

x1y1+ . . .+xnyn6 (xp1+ . . .+x

pn)

1/p(yq1+ . . .+ y

qn)

1/q,

p (xp1+ . . .+x

pn)

1/p(yq1+ . . .+ y

qn)

1/q

( ).8.47. xi yi , p > 1

. , ( ni=1

(xi+ yi)p

)1/p6

( ni=1

xpi

)1/p+

( ni=1

ypi

)1/p.

p< 1, ( ).

103

8.1. xi + 1/xi > 2, 3 + 3 = (x1 + 1/x1) + . . . + (xn + 1/xn) > 2n, . . n 6 3. n = 3 x1 = x2 = x3 = 1. n= 2

x1=3+

5

2, x2=

35

2. n= 1 .

8.2. a1 . . . an= 1 ,

(1+a1)(1+ a2) . . . (1+an)=(1+ a1)(1+ a2) . . . (1+ an)

a1 . . . an=

=1+

1

a1

. . .1+

1

an

.

, (1+ ai)1+

1

ai

=2+ ai+

1

ai> 4.

((1+a1)(1+ a2) . . . (1+an))2=

= (1+ a1)1+

1

a1

. . . (1+ an)

1+

1

an

> 4n.

8.3. 2+x3=x5+x , 2

x3+1=x2+

1

x2>2.

2

x3>1, . . x362. x 6=1, .

x5 x3 + x= 2 x7 x5 + x3 = 2x2, x7+x= 2+ 2x2, . . x6 + 1= 2(x+ 1/x)> 4. , x6 > 3. , x 6=1.

8.4. ) x2+x3=a, x3+x1=b, x1+x2=c, . . x1=b+ca

2,

x2=a+ c b

2, x3=

a+ b c2

.

b+ c a

2a+

a+ c b2b

+a+ b c

2c>3

2,

. . b/a+ c/a+ a/b+ c/b+ a/c+ b/c> 3+ 3= 6. , b/a+ a/b> 2 . .

) n= 4

x1

x2+x4+

x2

x3+ x1+

x3

x4+ x2+

x4

x3+ x1=

x1+ x3

x2+ x4+

x2+ x4

x1+ x3> 2.

n > 4 -. , x1, . . . , xn

104 8.

. x1, . . . , xn+1 -. -

, xn+1. x1

x2+xn+1>

x1

x2+xn,

xn

xn+1+xn1>

xn

x1+xn1

xn+1

x1+xn> 0.

x1

x2+xn+1+ . . .+

xn

xn+1+ xn1+

xn+1

x1+xn>

x1

x2+xn+ . . .+

xn

x1+xn1> 2;

.8.5. a, b, c

,

(a+ b c)(b+ c a)(c+ a b)(a+ b+ c)>0. (1), - . , a + b < c b+ c< a . (1) 2(a2b2+ b2c2 + c2a2) (a4+ b4+ c4)> 0, .

8.6. ,

(a22+ . . .+ a2n)

2> (n 2)(a42+ . . .+a4n).

a41+ 2a21S2+S

22 > (n1)a41+ (n1)S4,

S2= a22+ . . .+ a

2n S4= a

42+ . . .+ a

4n. :

(n 2)a21

S2

n 22+

S22n2 S

22+ (n1)S4 < 0.

, S221 1

n2

> (n 1)S4, . . S22 > (n 2)S4, .

(a2k+ . . .+a2n)

2> (n k)(a4k+ . . .+a4n).

k = n 2 8.5 , an2, an1, an . .

: . a= b= 1, c= 2.

2AB+ 4AC+4BC> 2AB+1

2BC+

1

2AC,

105

. .7

2AC+

7

2BC> 0. -

A, B, C. . a, b, c -

. , A=B= 1,C=e.

a(b+ec)+ b(ec+ a)+ec(a+ b)>1

2(c2+ea2+eb2). (1)

2ab >1

2c2,

(1) e. ,c2 6 4ab6 (a+ b)2. a, b, c , c6 a+ b. a6 b+ c b6 a+ c .

8.8. , (x y)26 0.

8.9. A=

skP

i=1a2i B=

skP

i=1b2i . -

xy6 (x2+ y2)/2,

kXi=1

ai

A

bi

B6

kXi=1

1

2

a2iA2

+b2i

B2

=

1

2A2

kXi=1

a2i +1

2B2

kXi=1

b2i =1.

, kPi=1

aibi

26A2B2=

kPi=1

a2i

kPi=1

b2i

.

8.10. ; b1= . . .= bn=1.

8.11. - : ai=

xi bi=1/

xi.

8.12. bi = S ai. -

S b1b1

+ . . .+S bnbn

>n

n1 , . . S 1b1+ . . .+

1

bn

>

n

n 1 +n=n2

n 1 .

b1+ . . .+ bn= (Sa1)+ . . .+ (San)=nSS= (n1)S. ,

(b1+ . . .+ bn) 1b1+ . . .+

1

bn

>n2

( 8.11).8.13. . n.

n = 1 . , - n . b1 = n+1

a1, . . . , bn+1 = n+1

an+1.

, (bni bnj )(bi bj)> 0.

106 8.

i, j, i> j.

n

n+1Xi=1

bn+1i >

n+1Xi=1

biXj6=i

bnj .

( , .) - n ,

n+1Xi=1

biXj6=i

bnj > n

n+1Xi=1

biYj6=i

bj=n(n+ 1)n+1Yj=1

bj.

,n+1Xi=1

bn+1i > (n+ 1)n+1Yj=1

bj,

. .

: a1 6 a2 6 . . . 6 an. a1 = an, a1 = a2 = . . . = an. A= G. , a1 < an. a1 a1an. , a1+anA> 0.

n. , n 1 . a2, a3, . . . , an1, a1+anA.

a2+ a3+ . . .+ an1 + a1+ anAn1

n1> a2a3 . . . an1(a1+ anA).

a2+ a3+ . . .+ an1 + a1+ anA

n1 =nAAn1 =A.

A:

An> a2a3 . . . an1A(a1+ anA)> a1a2 . . . an; A(a1+anA)>> a1an.

. . 26.22.

107

8.14. a= x10 b= y15.

x5+ x5+ y5+ y5+ y5

5>

5px10y15,

. . 2a+ 33

b> 55

ab.

8.15.

2,3

2,4

3, . . . ,

n+1

n. -

1

n

2+

3

2+4

3+ . . .+

n+1

n

>

nn+ 1,

. .

(1+1)+1+

1

2

+1+

1

3

+ . . .+

1+

1

n

>n

nn+ 1.

-

1,1

2,2

3, . . . ,

n1n

.

1

n

1+

1

2+2

3+ . . .+

n1n

>

n

r1

n,

. .

1+1 1

2

+1 1

3

+ . . .+

1 1

n

>

nnn.

8.16. . n - 8.45. Wn=W Wn1=w1+ . . .+wn1. ,

(aw11 . . . a

wnn )

1/Wn=(aw11 . . . a

wn1n1 )

1/Wnawn/Wnn 6

Wn1Wn

(aw11 . . . a

wn1n1 )

1/Wn1+

+wn

Wnan6

Wn1Wn

w1a1+. . .+wn1an1Wn1

+wn

Wnan=

w1a1+. . .+wnan

Wn.

ab 6ap

p+

bq

q p =

=Wn

Wn1 q =

Wn

wn(, 1/p + 1/q = 1), -

. . 1 +

+ x< ex x 6= 0 ( 28.48). b1, . . . , bn ,

108 8.

ai= (1+ bi)w1a1+ . . .+wnan

W.

wibi=wiai(w1+ . . .+wn)w1a1 . . .wnan

w1a1+ . . .+wnan,

w1b1 + . . .+wnbn = 0. , a1, . . . , an , b1, . . . , bn .

(aw11 . . . a

wnn )

1/W=w1a1+ . . .+wnan

W((1+ b1)

w1 . . . (1+ bn)wn)1/W x

x0

a

f(x0)=axa. , -

x 6=x0, x

x0

ai

, i=1, . . . , n, ,

.8.18. AOC, AO= a, CO= c

O 120. O OB = b. - AB =

a2 ab+ b2, BC = b2 bc+ c2

AC=a2+ ac+ c2.

AC 6 AB + BC. - , B - AC. ,

109

AOB BOC AOB, . . ac sin 120 == (ab+ bc) sin 60. , sin 120 = sin 60, ac==ab+ bc, . . 1/a+1/c=1/b.

8.19.

1

n

a1

a2+

a2

a3+ . . .+

an

a1

>

a1

a2a2

a3 . . .

an

a1

1/n= 1.

8.20. :

a3b+ b3c+ c3aa2bc b2ca c2ab== (a3b 2a2bc+ c2ab)+ (b3c 2b2ca+a2bc)+ (c3a 2c2ab+ b2ca)=

=ab(a c)2+ bc(ba)2+ ca(c b)2> 0.8.21. x1, . . . , xn, xi=x.

ax+b, a b . x=0 x=1( ). , , x1, . . . , xn 0 1.

, xi =xi+1 = 1 ( xn = x1= 1). - , x1=x2=1. , x1 x2, . . x1 + x2 x1x2 x2x3xnx1. x1=1 x2=1 1x3xn, x1=1 x2= 0 1xn > 1x3xn. , x1, x2, . . . , xn, x1 1. [n/2], - .

x1=x3=x5= . . .=0 x2=x4= . . .=1 [n/2].8.22. a2b2c2

(a+ b+ c)2abc(ab+ bc+ac)2abc(a+ b+ c)+ (a2c+ b2a+ c2b)++abc(a2c+b2a+c2b)+ (ab+bc+ac)>0. - ab(b+1)(ac1)2+bc(c+1)(ab1)2+ac(a+1)(bc1)2.

8.23. a

b 1. (a/b)ab > 1,. . aab > bab. .

8.26. : , . , -

f(x)=xns1xn1+s2xn2 . . .+ (1)nsn. x1, . . . , xn . -, x< 0, f(x) 6=0. x> 0,

(1)nf(x)= (x)n+s1(x)n1+s2(x)n2+ . . .+sn >0. . s1>0 s2>0 ,

x1 x2 . , x1= 2+ i x2=2 i, s1=4 s2= 3.

8.27. x6 0, x12, x9, x4, x> 0. 01, x9(x31)+x(x31)+1>0.

8.28. 1 x > 0, 1 + x > 0, 1 y > 0 1 + y > 0. (1x)(1+y)>0 (1+x)(1y)>0, . . 1x+yxy>0 1+ x y xy> 0. , 1 xy> x y 1 xy> y x. , 1xy= |1xy|.

8.29. ,

xnnx+n1= (1+x+ . . .+xn1n)(x1). x > 1, , 0

1

2n, . . . ,

1

2n 1>>

1

2n

1

2n=

1

2n. .

8.31. , m n ,

xm+ ym

2xn+ yn

26

xm+n+ ym+n

2.

(xmym)(xnyn)> 0. m n , x > y xm > ym xn > yn, x 6 y xm 6 ym xn 6 yn. m n , x2> y2 xm> ym xn > yn.

,x+ y

2x3+ y3

26x4+ y4

2x2+ y2

2x4+ y4

26x6+ y6

2.

.

111

8.32. , 123

4 . . .

2n 12n

2=1 3

223 5

42 . . .

(2n 1)(2n+1)(2n)2

12n+1

1, (2k 1)2(2k2)2k =

(2k 1)2(2k1)21 > 1. P

2n >

1

21

2n,

. . Pn >

2

2

12n.

,(2k1)(2k+1)

(2k)2=

(2k21)(2k)2

2n.

8.35. , 1 n, - . n 1 ( n 1), .

8.36. (xi a)(xi b)6 0 xi > 0, abxi+ xi 6

6 a+ b. , abP 1

xi+P

xi 6

6 n(a+ b). x* =1

n

Pxi y

* =1

n

P 1xi.

aby*+x*6a+b, . . y*6a+bx*

ab.

x* ,

112 8.

x(a+bx)6a+ b

2

2 x ( 1.2 ).

x*y*6x*(a+ b x*)

ab6

a+ b2

2ab

=(a+ b)2

4ab.

.8.37. P(x)=4(x+1)(x1/2)2= (x+1)

(4x2 4x+ 1)= 4x3 3x+ 1. , P(xi)> 0. - P(x1)> 0, . . . , P(xn)> 0, 3(x1 + . . .+ xn)+ n> 0,. . x1+ . . .+xn 6 n/3.

8.38. , x2 + 4y2 = 4 xy = 4

1,6.

. :

x2 + 4y2 = 4

2,1

2

2-

xy = 4 (22,2),

1,6. . 8.1, , -

.

x

y

42

22

22

2

2

1(2, 12

2)

(22,2)

. 8.1

113

1.26 x2+4y2=4 2,

1

2

2

2x+ 2

2y= 4, . .

x+2y= 22, (1)

xy=4 (22,2) -

2x+ 2

2y= 8, . .

x+2y= 42. (2)

, (1) (2), . - h -

2 2

2, . ,

h=ab/c, a=2, b=2

2 c=

a2+ b2. , h2=1,6,

.8.39. A(x)

, , x1 > x2 > x3 > x4 > x5.

A1(x)+A2(x)= (x1x2)[(x1x3)(x1x4)(x1x5) (x2x3)(x2x4)(x2x5)]> 0,

x1x2> 0, x1x3> x2x3> 0 . . -, A4(x)+A5(x)> 0. , A3(x) - , A3(x)> 0.

8.40. , x1 > x2 > . . .> x100 > 0. x1 > 100, x1 + x2 + x3 > 100. , x1 < 100. 100x1 > 0, 100x2 >0, x1x3> 0 x2x3> 0, 100(x1+x2+x3)> 100(x1 +x2+x3) (100x1)(x1x3)

(100x2)(x2x3)=x21+x22+x3(300x1x2)>>x21+x

22+x3(x3+x4+ . . .+x100)>

> x21+x22+x

23+ . . .+x

2100 > 10000.

, x1+x2+x3 > 100.8.41. n. n = 2

a2+2ab+b2

114 8.

, anb+abn 0.

8.42. 8.41 n=3, a= 38

b= 37.

38+

37

20 y>0, y 6=1. -, y>1, y1>0 nyn >yn1+ . . .+y+1, 00, y 6=1. y=x1/n, x>0, x 6=1. (xm/n1)/m> (x1)/n, . . xm/n1 m

n(x1)>0.

a=m/n>1. ,

. a > 1 xa = x1b = yb1, b = 1 a < 0 y=x1. xaax+a1>0 yb1 (1 b)y1+1 b 1>0, . . y1(yb by+ b 1)> 0, b1 x>0).

xaax+a1>0 y 1byb+

1

b1>0,

. . yb by+ b11) 28.43.8.46. X = xp1 + . . . + x

pn, Y = y

q1 + . . . + y

qn, A = x

pi /X

B=yqi /Y. p>1, 8.45xi

X1/pyi

Y1/q6

xpi

pX+

yqi

qY.

115

i= 1, . . . , n,

x1y1+ . . .+xnyn

X1/pY1/q6

X

pX+

Y

qY=1

p+1

q=1.

, p< 1, .8.47. p>1. ,

nXi=1

(xi+ yi)p=

nXi=1

xi(xi+ yi)p1+

nXi=1

yi(xi+yi)p1.

( 8.46)nXi=1

xi(xi+ yi)p1

6

nXi=1

xpi

1/p nXi=1

(xi+ yi)p

1/q,

q= p/(p1), . . 1/p+1/q=1. .

nXi=1

(xi+ yi)p6

nXi=1

xpi

1/p+

nXi=1

yqi

1/q nXi=1

(xi+ yi)p

1/q.

. , p

9

9.1.

9.1. a1, . . ., an . -,

1

a1a2+

1

a2a3+ . . .+

1

an1an=n 1a1an

.

9.2. a1, . . ., an - . ,

1a1+

a2+

1a2+

a3+ . . .+

1an1+

an=

n 1a1+

an.

a=0,a1a2a3 . . . , ak+n=ak k>N; n . N=1, . -. 0,a1a2 . . . aN1(aNaN+1 . . . aN+n1).

9.3. , - .

. 17.1.

9.4. 0,1234567891011 . . . ( )?

9.5. , nk, n k- , 1, n .

117

9.6. 1+2x+3x2+ . . .+ (n+1)xn.9.7. a, a+ d, a+ 2d, . . . -

, a, d> 0. , , - , , a/d .

9.8. 4n , , - . , n .

9.9. , - ( 0 1). , .

9.2.

9.10. , n= p+ q1, (a1+a2+ . . .+ap)+(a2+ . . .+ap+1)+ . . .+(anp+1+ . . .+an)=

=(a1+a2+ . . .+aq)+(a2+ . . .+aq+1)+ . . .+(anq+1+ . . .+an).

9.11. a1, a2, . . . , an , - , . n ?

9.3. Sk(n)= 1k + 2k + . . . + nk

1+2+3+ . . .+n . (k+ 1)2= k2+ 2k+ 1 k= 1, 2, . . . , n. - (n+1)2=1+2S1(n)+n,

S1(n) . , S1(n)=n(n+1)

2.

9.12. S2(n)= 12 ++22+ . . .+n2 S3(n)=13+23+ . . .+n3.

118 9.

9.13. ) ,

Ckk+1Sk(n)+Ck1k+1Sk1(n)+ . . .+C

1k+1S1(n)+S0(n)=

= (n+1)k+11.) , k Sk(n) -

k+1 n -

nk+1

k+1.

9.14. ) S=C1kS2k1(n)+C3kS2k3(n)+C

5kS2k5(n)+. . . ,

CkkSk(n) k Ck1k Sk+1(n) k. ,

S=nk(n+ 1)k

2.

) , S2k1(n) k n(n+ 1)

2( k).

9.15. S=C1k+1Sk(n)+C3k+1Sk2(n)+ . . . , -

Ck+1k+1S0(n) -

k Ckk+1S1(n) k. , S =

=(n+ 1)k+1+nk+1 1

2.

9.16.

13+33+53+ . . .+ (2n1)3.

9.4.

9.17. ) , p> 2

m

n=1+

1

2+1

3+ . . .+

1

p1 p.

) , p>3 -

m

n=1+

1

2+1

3+ . . .+

1

p1 p2.

119

9.18. p

q=1 1

2+1

3 14+ . . .+

1

4k 1, 6k1 . , p - 6k1.

9.19. p> 2 , ak -

kp p2. , a1+a2+ . . .+ap1=p3 p22

.

9.5.

9.20. - a b . p , q ., a b= p q.

9.1. , 1

ak 1

ak+1=

ak+1 akakak+1

=d

akak+1, d -

. -

1

d

1a1 1

a2+

1

a2 1

a3+ . . .+

1

an1 1

an

=1

d

1a1 1

an

=

=1

dan a1a1an

=1

d(n1)da1an

=n 1a1an

.

9.2. ,

1ak+

ak+1

=

ak+1

ak

ak+1ak

1

ak+ak+1

=

=

ak+1

ak

ak+1 ak=

ak+1

ak

d,

d . -

1

d(ana1)= 1

d

an a1an+

a1

=n 1an+

a1.

9.3. ,

0,a1a2 . . . aN1(aNaN+1 . . . aN+n1)=a1a2 . . . aN110N+1+

+ 10NaNq+ 10N1aN+1q+ . . .+ 10Nn+1aN+n1q,

120 9.

q = 1 + 10n + 102n + 103n + . . . =1

110n .9.4. , .

17.1 0,a1a2a3 . . . , ak+n = ak k>N (n ). - m , 10m+2n aN. , , , , , . , -.

9.5. a+ (a + 2) + . . .+ (a + 2n 2) = n(a + n 1) nk, a+n1=nk1, . . a=nk1n+1., a .

9.6. , 1+x+

+x2+ . . .+xn=xn+11x 1 , x+x

2+ . . .+xn=xn+1xx 1 , . . . , x

n=xn+1xnx 1 .

(n+1)xn+1 (1+x+ . . .+xn)x1 =

(n+1)xn+1 xn+1 1x1

x1 =

=(n+ 1)xn+2 (n+2)xn+1+ 1

(x 1)2 .

9.7. a/d = m/n, m n . k (1 + n)k 1 n, bk =

a(1+n)k ad

=m

n((1+ n)k 1) . ,

a(1+ n)k = a+ bkd -.

a+ kd, a+ ld, a+md - , k< l c> d> e> . . . a, b, c, d . ab> cd ac> bd, ad= bc, . . d= bc/a. , e= bc/a.

121

9.9. aqn, n > 0. , q = 1, , , q 6= 1. , - k1, k2, . . . , km+1 (m> 2),

aqk1 +aqk2 + . . .+ aqkm = aqkm+1. (1)

l1 < l2 < . . .< lm+1 k1, k2, . . . , km+1, . (1)

aql1 =aql2 . . . aqlm+1. aql1

1= ql2l1(1 ql3l2 . . . qlm+1l2). 1, ql2l1 , 1. - .

9.10. np+1Pi=1

(ai+. . .+ai+p1)=

=np+1Pi=1

p1Pj=0

ai+j=npPi=0

p1Pj=0

ai+j+1=p1Pj=0

npPi=0

ai+j+1=nqPj=0

q1Pi=0

ai+j+1. -

.9.11. 9.10 (a1+a2+ . . .+a7)+ (a2+ . . .+a8)+ . . .

. . . + (a11 + . . . + a17) = (a1 + a2 + . . . + a11) + (a2 + . . . + a12) + . . . ++ (a7+ . . .+a17). n

122 9.

) ) - k. , Ckk+1=k+1.

9.14. ) ,nPj=1

(jk(j+ 1)k jk(j 1)k)= nk(n+ 1)k. ,

jk+C1kj2k1+C2kj

2k2+ . . .+Ck1k jk1+Ckkj

k,

jk+C1kj2k1C2kj2k2+ . . .Ck1k jk1Ckkjk.

2S.) ) S1(n) =

n(n+ 1)

2, C12S3(n) =

n2(n+1)2

2,

C13S5(n)=n3(n+ 1)3

2C33S3(n) . .

9.15. ,nP

j=1((j+ 1)k+1 (j 1)k+1)= (n+ 1)k+1 +

+nk+1 1. , 2S.9.16. 9.12 13 + 23 + 33 + . . .+m3 =

m(m+1)

2

2.

13 + 23 + 33 + . . .+ (2n 1)3 + (2n)3 =2n(2n+1)

2

2, . .

13 + 33 + 53 + . . .+ (2n 1)3 + 23(13 + 23 + . . .+ n3)=2n(2n+1)

2

2.

, , 13+

+ 23+ . . .+ n3=n(n+1)

2

2. 13 + 33+ 53 + . . .

. . .+ (2n 1)3=n2(2n21).9.17. )

1

k+

1

p k , k = 1, 2, . . . ,p12

. 1

k+

1

p k =p

k(p k) . - m

n=

pq

1 2 3 . . . (p1) . , p 2, 3, . . . , p1.

)

1+

1

p1+12+

1

p 2+ . . .+

1

p12

+p+1

2

!=

= p

1

p1 +1

2(p2) + . . .+1p 1

2

p+12

!= p

M

(p1)! ,

123

M=(p1)!p 1 +

(p 1)!2(p2) + . . .+

(p1)!p12

p+ 12

.

, M p.

x (p1)!k(p k) (mod p). xk(p k) (p 1)! (mod p).

( 31.15 ) (p 1)!1 (mod p), xk2 1 (mod p). , , k 1, 2, . . . ,

p 12

, x 12, 22, . . .

. . . ,p12

2. , k2 -

. k k , kk1 (mod p). , k2 k x (mod p).

, M 12 + 22 + . . .+p 12

2(mod p).

9.12 12 + 22 + . . . +p12

2=p12

p+ 1

2

p

6

.

p.9.18. ,

1 12+1

3 14+ . . .+

1

4k1 =

= 1+1

2+1

3+1

4+ . . .+

1

4k1 212+1

4+1

6+ . . .+

1

4k2=

= 1+1

2+1

3+1

4+ . . .+

1

4k1 11

2 13 . . . 1

2k 1 =

=1

2k+

1

2k+1+ . . .+

1

4k 1 .

,

1

2k+ s+

1

4k1 s =6k+1

(2k s)(4k1 s) .

6k1 . - 6k 1 , 6k1.

9.19. , kp+ (p k)p p2. ,(xy)p=yp+pxyp1+. . . , , - x2. (pk)pkp+pkp1pkp (mod p2), kp+ (pk)p p2.

124 9.

p , 1 6 k 6 p 1, kp (p k)p p . , ak + apk = p2. - (p 1)/2 ak + apk, p2(p 1)/2.

9.20. +2, 2, 0. , 2(a b). , 2(p q).

10

I

10.1.

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

10.2.

10.2. ) , f(x) xa f(a) ().

) x0 f(x). , f(x) xx0.

10.3. P(x)= anxn + an1xn1 + . . .+ a1x+ a0 - . , x0=p/q, p/q- . , an q, a0 p.

10.4. ,

2+

3.

10.5. , 3

2+ 3

3.

10.6. xn xn1 xn2 . . .. . .x1, .

10.3.

10.7. , x17 x18 -

126 10. I

(1+x5+x7)20.

10.8. P(x)= anxn + an1xn1 + . . .+ a0 , an>1. -, m |an1 |+ 1, . . .. . ., |a0 |+ 1, Q(x)=P(x+m) - .

10.9. x19511 x4+x3+2x2++ x + 1 . x14.

10.4.

10.10. x1, . . ., xn

xn+an1xn1+an2xn2+ . . .+a0.

, x1+x2+ . . .+xn=an1,16i

127

10.14. P(x) -, P(7)=11 P(11)=13?

10.15. P(x) -, n P(n),P(n+1) P(n+2) 3. , P(m) 3 m.

10.16. , -, , - , .

10.17. P(x), - x2+1, P(x)1 x3+1.

10.18. a0 = 0, an = P(an1) n = 1, 2, . . ., P(x) , P(x)>>0 x>0. , m, n>0, (am, an)==ad, d=(m, n).

10.19. , x15 1 1 14, . . k 6 14 k -, x151.

10.20. , x2n + xn + 1 x2+x+1 , n 3.

10.21. ) , ax3+bx2+cx+d, a, b, c, d- , x 5., a, b, c, d 5.

) , ax4 + bx3 + cx2 + dx + e, a, b, c,d, e , x 7. , a, b, c, d, e 7.

10.22. , p/q - ,

f(x)=a0xn+a1x

n1+ . . .+an , pkq f(k) k.

10.23. , A 3x2n+Axn+2 2x2m+Axm+3.

128 10. I

10.6.

10.24. , x(x+1) . . . (x+n)=1 , 1/n!.

10.7.

10.25. , P(x) n n .

a k (k>1) P(x), P(x) (x a)k (x a)k+1.10.26. , P(x) n

n , . . a1, . . ., am k1, . . . , km, k1+ . . .+km6n.

10.27. ,

xna1xn1a2xn2 . . .an1xan=0, a1 > 0, a2> 0, an> 0, - .

10.28. f1(x)=x22, fn(x)= f1(fn1(x)). , n fn(x)= x 2n .

10.8.

10.29. ,

(1x+x2x3+ . . .x99+x100)(1+x+x2+x3+ . . .+x99+x100) , x .

10.30. x+x3+x9+x27+x81+x243 (x1)?

129

10.31. :

(1x2+x3)1000, (1+x2x3)1000

x20?

10.32. ) a, (x a)(x 10)+1 (x+ b)(x+ c) b c.

) a, b, c, x(xa)(xb)(x c)+ 1 .

10.9.

10.33. x1, . . ., xn+1 ., P(x) - n, xi - ai ( ).

10.34. , n

k=0(1)kkmCkn = 0 m < n (m

) n

k=0(1)kknCkn= (1)nn!.

10.35. a1, . . . , an . -, b0, b1, . . ., bn1

x1+ . . .+xn= b0,

a1x1+ . . .+anxn= b1,

a21x1+ . . .+a2nxn= b2,

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

an11 x1+ . . .+an1n xn= bn1

, .10.36. x0, x1, . . ., xn. , -

f(x) n,

130 10. I

f(x0), . . . , f(xn),

f(x)= f(x0)+ (xx0)f(x0; x1)++ (xx0)(xx1)f(x0; x1; x2)+ . . .

. . .+ (xx0) . . . (xxn1)f(x0; . . . ; xn), f(x0; . . . ; xk) x0, . . ., xk - ( - ).

. - , x0, x1, . . . , xn xn+1 f(x0; . . . ; xn+1); -, , - .

10.10.

10.37. R(x)=P(x)/Q(x), P Q - . , R(x)

R(x)=A(x)+i,k

cik

(x ai)k ,

cik , A(x) .10.38. , -

, 10.37, .

10.11.

p(x) , - x.

10.39. ,

Ckx=x(x1) . . . (x k+ 1)

k!

.10.40. pk(x) k, -

x= n, n+ 1, . . ., n+ k -

131

n.

pk(x)= c0Ckx+ c1C

k1x + c2C

k2x + . . .+ ck,

c0, c1, . . ., ck .

10.12.

n x1, . . . , xn ak1 ...knx

k11 . . . x

knn . k1 + . . . + kn.

- () .

10.41. , x200y200 + 1 - x y.

10.42. ) P= P(x, y, z), Q==Q(x, y, z) R=R(x, y, z) x, y, z, -

(x y+1)3P+ (y z1)3Q+ (z2x+1)3R=1?)

(x y+1)3P+ (y z1)3Q+ (zx+1)3R=1.. 16.10.

10.1. (x+1)(x+2)(x+3)(x+4)+1=x4+10x3+ . . .+25 (x2 + ax + b)2 = x4 + 2ax3 + . . . + b2 , - x2 + 5x 5. , (x2 + 5x+ 5)2 1= (x2 + 5x + 4)(x2 + 5x+ 6) x2+ 5x+ 4= (x+ 1)(x+ 4), x2+5x+6= (x+2)(x+3).

10.2. ) f(x) xa . f(x) = (x a)g(x) + r, r . x = a. f(a)= r.

) ), f(x0)= 0.10.3. anx

n0+an1x

n10 + . . .+a1x0+a0=0 -

qn anpn+an1pn1q+ . . .+a1pqn1+a0qn=0.

anpn=(an1pn1q+ . . .+ a1pqn1+ a0qn) q, -

p q . ,

132 10. I

an q. a0qn = (anpn + an1pn1q + . . . + a1pqn1)

p, p q . -, a0 p.

10.4. x =2+

3. x2 = 5 + 2

6 (x2 5)2 = 24,

. . x4 10x2+1=0.10.5. ,

(32+

33)3=5+3a,

(32+

33)6= 133+ 30a+ 9b,

(32+

33)9= 2555+ 711a+ 135b,

a= 36(32+ 33) b= 3

36(3

4+ 39). x915x687x3

125= 0 x= 32+ 33.10.6. x1, . . . , xn x

n + an1xn1 ++ an2xn2+ . . . , ai=1. x1+ . . .+xn==an1

P16i

npx21 . . . x

2n, . . 3/n > 1. -

, n6 3, n= 3 x21=x

22=x

23=1. (x1)3 ,

n= 3 (x2 1)(x 1). n= 1 2 .

10.7. 18 5 7, x18 .

17 5 7 : 17= 7+ 5+ 5; . 20 - 1+x5+x7 x7, 19 x5.

x17 20 19 18

2= 3420.

10.8. P(x) P(x) = (an 1)xn ++(x+an11)xn1+(x+an21)xn2+. . .+(x+a01)+1. , Q(x)=(an1)(x+m)n+(x+(m+an11))xn1+ . . .. . .+ (x+ (m+a0 1))+ 1. an1> 0, m+ an11, . . .. . . , m + a0 1 . .

133

10.9. :