!(www.legionr.ru) - . . -2011. II. . / , , . .1. . -. -2011. II. -()(). , - , 4, 70, 71, , . 146, , . 7.12.2. (, ) - (www.legionr.ru) / - .3. (, ) - -, - .4. (, ) - . - .: . . . ,. . . : . ., . ., . ., . ., . ., . ., . ., . ., . .. . -2011. II. - : - / - . . , . . . --:-, 2010. ( ) - . - . -2011 .. , .. . . I , - . - - . -2011 . . , . . , . II , , www.legionr.ru ()., . -. .c ( , 2010. www.legionr.ru 2. 23 + 3 =

(23 + 3)2=

21 + 123 6

21 123 3

23 + 3 =6

21 123 6

21 + 123 ==6

212(123)2=6441 432 =69 =33,33 372 =3398 ==333 23= 32 = 6.: 6.3.

1 +6

150 +25

6 1 + 1 5 ==

1 +6

1 5

6 1 + 1 5 ==

6 16 + 1

1 5

+ 1 5 ==5

1 5

+ 1 5 =5 5 + 1 5 = 4.: 4.4.6

27 + 8 6

8 27 336 =6

82(27)2

336 =636 336 =3636 = 6.: 6.5. , :m1,5+ 22(m+ 2) 2m +2 2m1

==

m0,5+2

m2m+ 2

(m+ 2) 2m+22m1

= m0,5+2+2m0,52 == 3m0,5. m = 9: 3m0,5= 39 = 33 = 9.: 9.6. , :n1,233(n0,8+ 3) +3n0,8 3

3n0,81

==

n0,43

n0,8+3n0,4+ 3

(n0,8+ 3) +3n0,83

3n0,81

== n0,43 3n0,4+3 = 2n0,4. n = 32:2n0,4= 23225= 24 = 8.: 8. www.legionr.ru4 87. .90,521 90,5+ 90,5=13 21 3 + 13= 1;91 = 1.: 1.8.3

75a23a53

25 a3=33a2. a = 3, 3.: 3.9.

35a67a5

15a23a7a53a=7a5

5a 1 3a 5a 17a53a==5a 1(7a53a)7a53a=5a 1. a = 2, 3.: 3.10.4

7 43

2+3 =4

(3)222 3 + 4

2+3 ==4

3 2

4+3 =

32

+3. 32 < 0,

32

= 23.: 2 3 +3 = 2.: 2.11. :6 25 = (1 5)2 9 + 45 = (2 +5)2. ,

6 25

9 45 =

(1 5)2

(2 +5)2= [15[[2+5[. 5>1 2 +5>0, 5 1 (2 +5) = 3.: 3.12.

3 + 22 +

11 62 =

1 +2

2+

3 2

2==

1 +2

+

3 2

. 3>2, :1 +2 + 3 2 = 4.: 4.13. :2947 =

127

2 1147 =

27

2. 2 0 2 +a 4 > 0. :2 a 4 + 2 +a 4 = 4.: 4.15. :a + 6a 9 = a 9 + 3

2a 6a 9 = a 9 3

2.

a 9 + 3

2+

a 9 3

2==

a 9 + 3

+

a 9 3

. a=9,75, a 9 + 3>0 a 9 3 < 0. :a 9 + 3 + 3 a 9 = 6.: 6.16.

4x25(4x 5) + 2

9 +x(x + 6) =

(2x 5)2+ 2

(x + 3)2== [2x 5[ + 2[x + 3[. , x = 2,1283, [2x 5[ + 2[x + 3[ = 5 2x + 2(x + 3) = 11.: 11.17. a = 1 .

2 3

2 +3 =4 3 = 1.: 1.18.1414 +6 +614 6=14 146 +146 + 614 6=208= 2,5.: 2,5.20.

x + 1 4x 3 +

x + 1 + 4x 3 ==

x 3 2

2+

x 3 + 2

2=

x 3 2

+

x 3 +2

. x = 3,185,

x 32

+

x 3+2

= 20,185+0,185+2 = 4.: 4.21.3337 33+3737 +33=33(37 +33) +37(37 33)37 33==33 + 374= 17,5.: 17,5.22. (3x)1

(x 3)2(x + 1) =13 x [x3[ x + 1. x = 2,61, [x3[ = 3x 13 x (3x)x + 1 =x + 1 ==3,61 = 1,9.: 1,9. www.legionr.ru6 2423.4565 45+6565 +45=45(65 +45) +65(65 45)65 45==45 + 6520= 5,5.: 5,5.24. 232 23+2100 = 232+3+2100 = 26100 = 64100 = 36.: 36.25. 32+3 323+ 32= 3(2+3)+(23)+ 32= 34+ 32= 81 + 9 = 90.: 90.26.

9 45(2 +5) =

(2 5)2 (2 +5) == [2 5[(2 +5) = (5 2)(5 + 2) = 5 4 = 1.: 1.27.

16 67 (7+3) =

(3 7)2 (7+3) = [37[(7+3) == (3 7)(3 +7) = 9 7 = 2.: 2.28.

a 5

a +5 =

(a 5)(a +5) =

a2(5)2==a25. a = 3:a25 =9 5 = 2.: 2.29.3

33 a 3

a +33 =3

(33 a)(33 +a) =3

(33)2a2==333 a2. a = 5:333 a2=333 25 = 2.: 2.30. 3 3216

49

=3649= 8.: 8.31.2051024 3252=5432452=95= 1,8.: 1,8.32.3+2n49n 733n1

27n3=3+2n72n+33n133n1= 7 3 = 4.: 4.33.n92n2n25n+ 1 =n

(92)n2n52n+ 1 = 925 + 1 = 77.: 77.34.log5 50log2 5= log5(252)log5 2 = (log5 25 + log5 2)log5 2 == (2 + log5 2)log5 2 = log25 2 + 2 log5 2; www.legionr.ru 35 7log5 10log10 5= log5 10log5 10 == log25 10 = log25(25) = (log5 2 + log5 5)2= (1 + log5 2)2== 1 + 2 log5 2 + log25 2; 5,5(log25 2 + 2 log5 2 1 2 log5 2 log25 2) == 5,5(1) = 5,5.: 5,5.35. log7 21log7 21 log7(721)log7 3 == (log7(73))2log7(72 3)log7 3 == (log7 7 + log7 3)2(log7 72+ log7 3)log7 3 == (1 + log7 3)2(2 + log7 3)log7 3 == 1 + 2 log7 3 + log27 3 2 log7 3 log27 3 = 1.: 1.36. log223 + log2 12log12 2 log2 144log3 2= log223 + log2212 log2 144log2 3 == log223+log22122 log2 12log2 3 = (log2 12log2 3)2= log224 = 22= 4.: 4.37.12log2

3+5

+log2

45

log2 235 ==12log2

12+55

log2 235 =12log2(7+5)log12 25 ==

12log12 2

log2(7+5)5 = 2log2(7+5)5 = 7 +5 5 = 7.: 7.38. 7,15

log7 21log21 7 log7 147log3 7

== 7,15

log7(37)log7(37) log7

72 3

log7 3

== 7,15

log7 3 + 1

22 log7 3 log27 3

== 7,15

log27 3 + 2 log7 3 + 1 2 log7 3 log27 3

= 7,15.: 7,15.39. 2,3

log5 45log45 5 log5 225log9 5

= 2,3(log25 45 log5 225log5 9) == 2,3((log5 5 + log5 9)2(log5 25 + log5 9)log5 9) == 2,3((1 + log5 9)2(2 + log5 9)log5 9) == 2,3(1 + 2 log5 9 + log25 9 2 log5 9 log25 9) = 2,31 = 2,3.: 2,3.40. (3 log27 3,5 log3 10,5 1)53 log5 2= www.legionr.ru8 41= (log3 3,5 log3 10,5 1)8 =

log313 1

8 = 28 = 16.: 16.41. 5 log3 49log7 81 + 17log17 8= 10 log3 74 log7 3 + 8 == 40 log3 7log3 7 + 8 = 48.: 48.42. 12 ln a = 1; ln a = 2.logba2e5=1loge5ba2=5lnb ln a2=512 ln b 2 ln a=53 4= 5.: 5.43. loga6bb =logabloga(a6

b)=logab6 logaa + 12 logab=126 + 6= 1.: 1.44. loga3ba2=2 logba12 logba 13 logbb=22122 131=122= 6.: 6.45. 7 logabba=7(logab logaa)logaa logab=7

124 1

12 4=727= 2.: 2.46. 5 logb2aa2b=5(logaa2logab)logab2logaa=5(2 3)23 1=55= 1.: 1.48. :logab logbc = logablogbc= logac. :log3 5log2 7log5 8log7 9 = log3 5log2 7log5 23 log7 32=6 log3 5log5 2log2 7log7 3 = 6 log3 2log2 3 = 6.: 6.49. :logab logbc = logablogbc= logac. :log36 5log8 3log25 2log33 6 = log62 5log23 3log52 2log3136 = www.legionr.ru 50 90, 25 log6 5log5 2log2 3log3 6 = 0, 25 log6 2log2 6 = 0, 25.: 0,25.50. logp2(ab) = 12; ab = p24; logp2ab= 4;ab= p8

ab = p24,ab= p8;b = p8; a = p16;logpp16logpp8=168= 2.: 2.51. logt(ab) = 7; ab = t7; logtab= 1;ab= t.

ab = t7,ab= t;a = t4; b = t3; logtt4 logtt3= 43 = 12.: 12.52. a :251+ log21a51log4a3a0,5=251+ log2116510,5 log2 16 3160,5==

15

22215 212 =

15 2

15 + 2

15 212 = 9,8.: 9,8.53. 74 loga 7log49a= 72 loga 7log7a. loga 7log7a = 1, 72 loga 7log7a== 72= 49.: 49.54. 32(log0,2a+log5a)= 32(log5a+log5a)= 30= 1.: 1.55. lg 2log5 10log2 5=

lg 21lg 5

log2 5=lg 2lg 5log2 5= log5 2log2 5 = 1.: 1.56. m = 9 : (72)2= 722= 72= 49.: 49.57. a = 5 :5 2 log12 3 +1log10 12 + log12 1,6log10 2 + lg 5. www.legionr.ru10 58 :2 log12 3 = log12 32= log12 9,1log10 12= log12 10,log12 9 + log12 10 + log12 1,6 = log12(9101,6) = 2,log10 2 = lg 2, lg 2 + lg 5 = lg(25) = 1. 5 2 log12 3 +1log10 12 + log12 1,6log10 2 + lg 5= 5 21= 3.: 3.58. 1) a = 7 a +2 log13+a(a 2) +1log32(3a 1) log20 (a 5)loga+3 20 + lg(12 a)== 7 +2 log20 5 +1log32 20 log20 2log10 20 + lg 5.2) :2 log20 5 = log20 52= log20 25,1log32 20= log20 32, log20 25 + log20 32 log20 2 = log2025322= 2,log10 20 = lg 20, lg 20 + lg 5 = lg(205) = 2. 7 +2 log20 5 +1log32 20 log20 2log10 20 + lg 5= 7 + 22= 8.: 8.59. , :49x107x+ 25 = (7x5)2. :49x107x+ 25+7x+2,5 =

(7x5)2+7x+2,5 = [7x5[+7x+2,5. 6x= 0,25< 1, x< 0. , 7x< 1 [7x 5[ = 5 7x.: [7x5[ + 7x+ 2,5 = 5 7x+ 7x+ 2,5 = 7,5.: 7,5.60. ln((x + 3)(x 2))ln(x + 3)logx+3(x 2) == logx+3(x 2) + 1 logx+3(x 2) = 1.: 1. www.legionr.ru 62 1161. lg((x 1)(x + 4))lg(x 1)logx1(x + 4) == logx1(x + 4) + 1 logx1(x + 4) = 1.: 1.62. 3 lg 2 lg 0,25lg 14 lg 7=3 lg 2 lg 22lg 147=3 lg 2 + 2 lg 2lg 2=5 lg 2lg 2= 5.: 5.63. 3 log7 2 log7 24log7 3 + log7 27=log72324log7 81=log713log7 81= log8113= 0,25.: 0,25.64.72p5p72p+ 27p 5p+ 52p=(7p5p)(7p+ 5p)(7p+ 5p)2==7p5p7p+ 5p=5p5p

75

p1

75

p+ 1=3 13 + 1=12, log1,4 3 = log753.: 0,5.65. 987log7149= 98 149= 2.: 2.66. 2564log4164= 256 164= 28 26= 22= 4.: 4.67. log7x 5log7 2= 5 log7xlog7 2= 5log2x = 55 = 25, x = 5.: 25.68. log6y10log6 5= 10log6ylog6 5= 10 log5y = 10 5 = 50, log5y = 5.: 50.69. log2t73 = 7 log2t 3 = 75 3 = 32, log2t = 5.: 32. www.legionr.ru12 7170. tgt2=13, tg2t =

2 tgt21 tg2t2

2=

2 131 19

2=916. tg2t + 35=916 + 35==5780= 0, 7125.: 0, 7125.71. 210lg 14lg 4+lg 5= 210lg1454= 2 752= 35.: 35.72. 3log3x2+1 =32 log3x+1. , log3x =12, 3212+ 1 = 4.: 4.73.

12

log2x34 =

12

3 log12x4. , log12x =13,

12

334 = 2 4 = 2.: 2.74. logab3 logbaab=3ab logablogba =3ab. , a = 3, ab = 5,335=15.: 0,2.75. cd2

logcd3logcd=3cd2. , c =16, d =4, 3 1616 = 18.: 8.76. log5 (7x4) log25 (49x2) = log5 (7x4) 12 log5 (49x2) == log5 (7x4) log5 (49x2)12= log57x47x= log5x3= 3 log5x = 3 log15x. www.legionr.ru 77 13 log15x = 1, 3 log15x = 3.: 3.77. log123x2+ log2 3 = log23x2+ log2 3 = log2x23+ log2 3 == log2

x23 3

= log2x2= 2 log2x. log2x = 3, 2 log2x = 6.: 6.78. 11 3 log33 = 11 3 log3 312= 11 3 12= 11 1,5 = 9,5.: 9,5.79. 13 3log28 = 13 3log2 812= 13 32log2 23= 13 4,5 = 8,5.: 8,5.80. 49log7 4= (72)log7 4= (7log7 4)2= 42= 16.: 16.82. 3(sin 10 + sin80)(cos 80cos 10)2 sin 70==32 sin 45 cos 35 2 sin 45 sin 352 sin 70=3 sin 70 2

22

22 sin 70==3 sin 702 sin 70= 1,5.: 1,5.83.sin 91sin 192cos 46 +2sin 44=2 sin 45 cos 462 cos 46(9 + 1)=2 22102= 0,1.: 0,1.84. (tg 20 + tg 5)ctg 25 + (tg 20tg 5)ctg 15 ==(sin 20 cos 5 + sin5 cos 20) cos 25cos 20 cos 5 sin25++ (sin 20 cos 5sin 5 cos 20) cos 15cos 20 cos 5 sin 15==sin 25 cos 25cos 20 cos 5 sin 25+sin15 cos 15cos 20 cos 5 sin 15=cos 25 + cos 15cos 20 cos 5==2 cos 20 cos 5cos 20 cos 5= 2: 2. www.legionr.ru14 8685. sin 30(sin 12 cos 18 + cos 12 sin18) = sin30 sin 30 =14= 0,25.: 0,25.86. cos 60(cos 25 cos 35sin25 sin35) = cos 60 cos 60 ==14= 0,25.: 0,25.87. sin 10 sin 50 sin 70 = sin10 cos 20 cos 40 ==2 sin 10 cos 20 cos 40 cos 102 cos 10=2 sin 20 cos 20 cos 404 cos 10==2 sin 40 cos 408 cos 10=sin808 sin 80=18= 0,125.: 0,125.88. cos2135(sin 80 cos 55 + cos 80 sin55)2== cos2135sin2135 = cos 270 = 0.: 0.89.cos 70 cos 10 + cos 80 cos 20cos 68 cos 8 + cos 82 cos 22==cos 70 cos 10 + sin 10 sin 70cos 68 cos 8 + sin 8 sin68=cos 60cos 60= 1.: 1.90. tg

2arcctg

33

3

= tg

2 233

= tg = 0.: 0.91. (sin sin)2+ (cos cos )2sin2 2==sin2 2 sin sin + sin2 + cos2 2 cos cos + cos2sin2 2==2 2 cos( )1 cos( )2=22(1 cos( ))1 cos( )= 4.: 4.92. cos 75 + cos 75 cos 15cos 15 + sin75 sin15 +22= www.legionr.ru 93 15= (cos 75cos 15) + cos 60 +22= 2sin 45 sin 30 + 12 +22== 2 22

12 + 12 +22= 0,5.: 0,5.93. (sin 35 cos 55 + cos 35 sin 55) + 6tg2(5525) == sin 90 + 6tg230 = 1 + 6

13

2= 3.: 3.94.

cos3sin sin3cos

tg 2 =cos4 sin4sin cos tg 2 ==(sin2 + cos2)(cos2 sin2)sincos tg 2 =2 cos 2sin 2 tg 2 == 2 ctg 2tg 2 = 2.: 2.95.

sin3cos +cos3()sin()+ 4 ctg 2

tg 2 ==

sin4 cos4sin cos + 4 ctg 2

tg 2 ==

2 cos 2sin2+ 4 ctg 2

tg 2 = 2 ctg 2tg 2 = 2.: 2.96. 8 sin 36(cos 36sin18)cos 54=4(sin 722 sin 36 sin 18)cos 54==4(cos 18cos 18 + cos 54)cos 54= 4.: 4.97.3sin 40+1cos 40=3 cos 40 + sin 40sin 40 cos 40==232cos 40 + 12 sin40

12 sin 80=2(cos(4030))12 sin 80=4 cos 10cos 10= 4.: 4.98. 1 + tg2 =1cos2, www.legionr.ru16 991 + ctg2 =1sin2 sin 75

1 + tg2105+cos 75

1 + ctg2105== sin75 [ cos 105[ + cos 75 [ sin 105[ == sin 75 cos 105 + cos 75 sin105 == sin(10575) = sin30 = 0,5, cos 105< 0, sin105> 0.: 0,5.99.

1 1sin263

1 1cos263

= (1 1 ctg263)(1 1 tg263) == ctg263 tg263 = 1: 1.100. cos 15 cos 45cos 45 cos 75 = cos 15 cos 45sin 45 sin 15 == cos 60 = 0,5.: 0,5.101.3 cos275 +3 cos 165 sin75 =3 cos2753 sin275 ==3(cos275sin275) =3 cos 150 = 3 cos 30 == 3 32= 1,5: 1,5.102. 16

cos412 + 14 sin26

sin215 == 16

1 + cos 6

24+ 14 sin26

sin215 == 4

1 + 2 cos 6+ cos26+ sin26

1 cos 302== 4

2 + 2 32

1 322

= 4(2 +3)

2 34

= 4 3 = 1.: 1.103.72 cos 75 cos 15cos 15 cos 105sin18 sin63 + sin 108 sin 27= www.legionr.ru 104 17=72 cos 75 cos 15 + sin 75 sin 15sin 18 sin 63 + cos 18 cos 63==72cos 60cos 45=712222=72= 3,5.: 3,5.104. cos 2 =1 tg21 + tg2. tg = 0,2, cos 2 =1 0,041 + 0,04=0,961,04=1213;873 + 41213=871387= 13.: 13.105. cos

6

=12 0 < < : 6=3; =2. ctg

2+4

= ctg

34

= 1.: 1.106. sin 2 = 2 sin cos = 2 sin

+4 4

cos

+4 4

== 2

sin

+4

cos 4 cos

+4

sin 4

cos

+ 4

cos 4sin

+ 4

sin 4

= 2

12

2222cos

+ 4

22cos

+4

+ 12 22

= 2

216 24cos2

+4

==18

2 8

1 sin2

+4

=18

2 8

1 14

= 0,5, sin

+4

=12.: 0,5.107. sin2x = 1cos2x = 1413=913. < x < 0, sinx < 0 sin x =

913=313, tg x =sinxcos x=313:213= 1,5,tg 2x =2 tg x1 tg2x=21,51 (1,5)2=31,25= 2,4.: 2,4. www.legionr.ru18 109108. 1,2 tg cos

2

= 1,2 tg sin = 1,2 sin2cos =1,2(1 cos2)cos =1,2

1 125

15=6524525= 5,76, cos =15.: 5,76.109. sin x =15 0 < x 0. : 4log2x+ 1x= 2log2x+ 1x== x +1x=x2+ 1x. x = 2 3:

2 3

2+ 12 3==4

2 3

2 3= 4.: 4.137.sinx cos xsin3x cos3x=sinx cos x(sin x cos x)(sin2x + sin xcos x + cos2x)= www.legionr.ru 139 23=11 + sin xcos x=22 + sin 2x=22 + sin 6= 0,8.: 0,8.139. 1 (cos 45 cos 15sin 15 sin 45)2= 1 (cos(45 + 15))2== 1 cos260 = 1 0,25 = 0,75.: 0,75.140. (sin 75 cos 15sin 15 cos 75)2+ 0,5 =

sin(7515)

2++ 0,5 = (sin 60)2+ 0,5 =

32

2+ 0,5 = 0,75 + 0,5 = 1,25.: 1,25.141. 9 cos 2x = 9(2 cos2x 1) = 18 cos2x 9 == 18 19 9 = 2 9 = 7, cos x =13.: 7.142. 3 tg2x = 3 sin2xcos2x= 3 sin2x1 sin2x= 3 49:59=125= 2,4, sin x =23.: 2,4.143.2(sin 70 + sin20)cos 25=22 sin 70 + 202cos 70202cos 25==22 sin 45 cos 25cos 25=2222= 2.: 2.144. cos 50 + cos 402 cos 5=2 cos 50 + 402 cos 504022 cos 5==2 cos 45 cos 52 cos 5=22 22= 1.: 1.145. sin2t + cos2t 3 sin + 7 cos = 1 30 + 7(1) = 6.: 6.146. cos

3

cos

23

+ 4 = cos

3

cos

3

+ 4 = www.legionr.ru24 147= cos

3

cos 3

+ 4 = 2 12 + 4 = 5.: 5.147. 2

3 sin12 cos12

= 2

3 12 sin 6

= 2

3 14

= 5, 5.: 5, 5.148. 6 tg2x 5 = 6

1cos2x 1

5 = 6 5 = 1, cos2x = 0,5.: 1.149. 5 3 cos2x = 5 31 + tg2x= 5 31 + 2= 4, tg2x = 2.: 4.150. 6

1+sin2

=6

1+

1cos2

=6

2cos2

=6

2

13

2

== 6

2 13

= 6 53= 10.: 10.151. 9(1 + cos2) = 9

1 +

1 sin2

= 9

2 sin2

== 9

2 49

= 18 4 = 14.: 14.152. log2sin 25 sin 65tg 45 cos 40= log2cos 40cos 902 cos 40= log212= 1.: 1.154.0,04+log4 22+2log2 3=

(0,2)2+12 log2 232+3 = 0,2+34+3 = 3,95: 3,95.155. 3log3 20,09 + 3 log91333= 2

(0,3)23 12log3 333 == 2 0,3 32log3 3113= 1,7 32 43= 1,7 2 = 0,3.: 0,3.156. 53x1 2575x= 0,2; 53x1+1410x= 51; 7x + 13 = 1; x = 2.: 2.157. 82x+343x+2= 62; 26x+926x+4= 62; 26x+4(251) = 62;26x+4 31 = 62; 26x+4= 2; 6x + 4 = 1; x = 0,5.: x = 0,5. www.legionr.ru 159 25158.3x+323x= 675; 3x

332

= 675; 3x 25 = 675; 3x= 27; x = 3.: 3.159. 725 45x= 5x+2; 5x+2+ 45x= 725; 5x(52+ 4) = 725;5x 29 = 725; 5x= 25; x = 2.: 2.160.3log5 6= 6log5x+log5 4. : x > 0. 3log5 6= 6log5 3, 6log5 3= 6log5 4x; 4x = 3; x =34.: 0,75.161. 8ln =ln xln 2, : x> 0. 8ln =lnx2; ln 8=lnx2;x2= 8;x = 16.: 16.162. 2log3 6 xlog3 2=8; x>0, 2log3 6 2log3x=23; 2log3 6+log3x=23;2log3 6x= 23; log3 6x = 3; 6x = 27; x =276; x = 4,5.: 4,5.163. 4x2+ 24x1= 9; 4x2 (1 + 24) = 9; 4x2 9 = 9; 4x2= 1;4x2= 40x 2 = 0, x = 2.: 2.165. 9x= 83x+1+81;

32

x833x81 = 0;

3x

2243x81 = 0. t = 3x> 0. t224t 81 = 0; t1,2 = 12 144 + 81 = 12 15 t1 = 27, t2 = 3 < 0t = 27. , 3x= 27; 3x= 33x = 3.: 3.166. 0,72x 0,725x4 0,713x= 33049; 0,72x+25x4 0,713x= 33049;0,723x40,713x= 33049; 0,713x 0,7 40,713x= 33049;0,713x

710 4

= 33049; 0,713x

3310

= 33049; 0,713x=10049;0,713x=

107

2; 0,713x= 0,721 3x = 2; 3x = 3; x = 1.: x = 1.167. 22x+1. , 22x+1(1 + 21+ 22+ 23) = 240;22x+1(1 + 2 + 4 + 8) = 240; 22x+1= 16; 2x + 1 = 4; x = 1,5.: 1,5. www.legionr.ru26 169168. 5. -, 58x4+ 58x3+ 58x2+ 58x1= 6,24. 58x4. , 58x4(1 + 51+ 52+ 53) = 6,24;58x4(1 + 5 + 25 + 125) = 6,24; 58x4=125; 8x 4 = 2; x = 0,25.: 0,25.169.: 7 4x > 0,4x < 7, x 0; x2< 1; 1 < x < 1.1) 2455x1+5252x+1= 0; 1052x495x5 = 0; 52x4,95x0,5 = 0.: 5x= t, t > 0; t2 4;9t 0,5 = 0. , t1 = 5, t2 = 0,1. t2 = 0,1 t > 0. 5x= 5 x = 1, x = 1 , .2) log0,5

2 2x2

= 0; 2 2x2= 1; x2= 0,5; x1 =0,5, x2 = 0,5. 0,5 0,5, , . 0,5 0,5 = 0,5.: 0,5.172. lg (x + 3) = 2 lg 2 lg x lg (x + 3) + lg x = lg 22lg

x(x + 3)

= lg 4 x2+3x = 4x2+3x4 = 0x1= 4, x2= 1.: x = 1, lg(3 + 1) = lg 4 = 2 lg 2 = 2 lg 2 0 = 2 lg 2 lg 1.x = 4 , lg(4) -.,x = 1.: 1.173.log7 36 log7(3x 12) = log7 4; log7(3x 12) = log7 36 log7 4;log7(3x 12) =log7 36 log7 4; log7(3x 12) =log7 9; 3x 12=9;3x = 21; x = 7. www.legionr.ru 174 27: log7 36 log7(37 12) = log7 4; log7 36 log7 9 = log7 4;log7 4 = log7 4. x = 7 .: 7.174.log5(8 24x) log5 8 = log5 7; log5(8 24x) = log5 56; 8 24x = 56;24x = 48; x = 2.: log5(8 + 242) log5 8 = log5 7; log5 56 log5 8 = log5 7;log5 56 log5 8 = log5 7; log5 7 = log5 7. x = 2 .: 2.175. 216x 34x=2; 2(4x)2 34x=2. 4x=t,t >0. 2t23t 2 = 0; t1,2 =3 54; t1 = 2, t2 = 0,5. , t2 t > 0. 4x= 2; x = 0,5.: 0,5.176. :

3 log2x log2y = 1,log2x + 2 log2y = 3.-, log2x = a, log2y = b:

3a b = 1,a + 2b = 3. : a = 1,b=2. log2x=1, log2y=2. x=2, y=4. x +y = 2 + 4 = 6.: 6.177. 32x+1= 28 3x9. 3x= t, t > 0. 3t2= 28t 9;3t2 28t + 9=0; t1,2=14 133; t1=9, t2=13 x1=2, x2= 1. : x1 x2 = 2(1) = 2.: 2.178.

log3x + 3 log3y = 1,log3x log3y = 1;log3x = a, log3y = b,

a + 3b = 1,a b = 1;a = 2, b = 1; x = 9, y = 3. - x 2y = 3.: 3.179. y = 39x. : 525x815x+39x= 0. 9x= 0. 5

53

2x8

53

x+ 3 = 0.

53

x= t > 0. 5t2 8t + 3=0. t1=1,t2=35. , www.legionr.ru28 180

53

x= 1,

53

x=35; x1= 0 x2= 1. y1= 3 y2=13. (0; 3)

1; 13

. x +y 0 + 3 = 3.: 3.180. : x > 0.9 log3x x2log x = 0; (9 x2) log3x = 0;) 9 x2=0 x=1= 3,x2= 3. , x=1= 3 .) log3x = 0; x3 = 1. : x2 +x3 = 4.: 4.181. y = 4x+1 : 1449x1814x+ 4x+1= 0; 1472x187x 2x+ 422x= 0. 22x= 0. 14

72

2x18

72

x+4 = 0.

72

x=t> 0. 14t2 18t + 4 = 0. t1= 1,t2 =27. :

72

x= 1, x1 = 0;

72

x=27; x2 = 1. y1 = 4 y2 = 1. , (0; 4) (1; 1). x +y 2.: 2.182. 3y=z

9x2+z2= 6zx,7x +z = 30. z = 30 7x -, x26x+9 = 0. x = 3 z = 30 73 = 9. , x + 3y= x +z = 12.: 12.183.: x > 3.2xlog2(x 3) 3 log2

x26x + 9

= 4x 12;2xlog2(x3) 6 log2(x3) 4(x3) = 0; 2(x3)

log2(x3) 2

= 0;x1 = 3, x2 = 7. x1 .: 7.184. : x > 2.x2log3(x + 2) 9 log3(x + 2) = 0; (x29) log3(x + 2) = 0.) x29 = 0 x1 = 3, x2 = 3. , x1 = 3 - www.legionr.ru 185 29 .) log3(x + 2) = 0 x3 = 1. : x2 +x3 = 2.: 2.185. : x < 3.25xlog2(3 x) x3log2(3 x) = 0; (25x x3) log2(3 x) = 0;) 25x x3= 0; x(25 x2) = 0 x1 = 0, x2 = 5, x3 = 5. , x3 .) log2(3 x) = 0; 3 x = 1; x4 = 2. : x1 +x2 +x4 = 0 5 + 2 = 3.: 3.186.x25x + 65x125= 0

x25x + 6 = 0,5x1= 25;

x1 = 2,x2 = 3,x = 3;x = 2.: 2.187. : x = 2.y1=12, y2= 3.y =12: 52x 265x+ 25 = 0. 5x=t, t2 26t + 25 = 0;t1= 25, t2= 1. x1= 2 , x2= 0. y = 3, , . y x =12 0 = 0,5.: 0,5.188. :

3x + 1 > 0,3x + 1 = 1;x

13; 0

(0; +). 4 1log2(3x + 1)log2(3x+1) +3 = 0. log2(3x+1) = t, t2 3t 4=0; t1= 1, t2=4. log2(3x + 1) = 1,log2(3x + 1) = 4;x1= 16,x2=5. , x = 5.: 5.189. log4 log2

1x

= 1; log2

1x

= 4;1x= 16; x =116. www.legionr.ru30 191: log4 log2 16 = log4 4 = 1. , x =116= 0,0625.: 0, 0625.191. lg(x23x + 1) lg(x 1) = 0;) lg(x23x + 1) = 0; x23x + 1 = 1; x1 = 0, x2 = 3.) lg(x 1) = 0; x 1 = 1; x3 = 2.: x1 = 0 ; x2 = 3: lg(99+1) lg(31) = lg 1lg 2 = 0; x3 = 2: lg(46+1) lg(21) = lg(1)lg 1 . - x2 = 3.: 3.192. 2x log2 7 7x2+x= 1; 7x 7x2+x= 1; 7x2+2x= 70; x2+ 2x = 0; x1 = 0,x2 = 2. x1 = 0.: 0.193.

3x3y= 6,x +y = 3.x = 3 y . :33y3y= 6. 3y= z, z> 0. 27z z = 6, : z1 = 9 ( z> 0), z2=3. y=1, x =2., 3y2x = 1.: 1.194. log3(x 1) + log9(x 1) =3; log3(x 1) +12 log3(x 1) =3;log3(x 1) = 2; x 1 = 9; x = 10.: 10.195. , : 212x= 21, x =12. 3x+y=71x; x=12 3x+y= 712; 9x+y=

3x+y

2= 7.: 7.196. 23x 503x= 0,110x2+3; 1003x= 10x2+2; 106x= 10x2+2; 6x = x2+2;x26x + 2 = 0; x1,2 = 3 7; x1 +x2 = 3 +7 + 3 7 = 6;: 6.197. 2x= t, t > 0.

3y = 10 t,3t212y = 24. 3y = 10 t : www.legionr.ru 198 313t2+ 4t 64 = 0; t1 = 163 t> 0; t2 = 4. x = 2, y = 2. 41,5x6y= 4362= 28.: 28.198. :

2x + 6 > 0,x 1x + 3> 0;

x + 3 > 0,x 1 > 0;x > 1.log2x 1x + 3 + log2(2x 6) = 0; log2(x 1)(2x + 6)x + 3= 0; 2(x 1) = 1;x = 1,5. , x = 1,5 .: 1,5.199.

12xy 6y= 10,2xy=53;

12x10= 2y,2y=352x. 2y=352x :12x10=352x, x = 1. : 122xy 6y= 12xy 6y 12x= 120.: 120.201. log3(x + 5) = 3. x + 5 = 33, x + 5 = 27, x = 22. , x = 22 - .: 22.202. lg(10 x) =2, 10 x=102,x = 10 100, x = 90.: 90.203. 0, 23x+5>0, 04; 0, 23x+5>0, 22; 3x + 5 125; 52x1> 53; 2x 1 > 3; 2x > 4; x > 2., x, , - 3.: 3.205.

14

x+1= 64.

14

x+1=

14

3; x + 1 = 3; x = 4.: 4. www.legionr.ru32 207206. 73x= 343. 73x= 73,3x= 3, x = 1.: 1.207. 4x4= 64; 4x4= 43; x 4 = 3; x = 7.: 7.208. 5x+2= 125; 5x+2= 53; x + 2 = 3; x = 1.: 1.209. log3(4 x) = 4; 4 x = 81; x = 77.: 77.210. cos 3x + ctg xsin3x = sin 4x [0; 2].: sinx = 0, x = n.cos 3x +sin 3xcos xsinx= sin 4x; sin xcos 3x + sin 3xcos x = sin4xsin x;sin 4x(1 sinx) = 0;1) sin 4x = 0; 4x = n; x =n4, n Z.2) sin x = 1; x =2+n, n Z.x=n4, n Z. [0; 2]:0n42; 0n42; 0n8. n Z, n=0, 1,2, 3, 4, 5, 6, 7, 8, x =n, n =0, 4, 8. , n = 1,2,3,5,6,7 [0; 2]. 6 .: 6.211. tg 2xcos x + sinx = sin 3x (0; 2).: cos 2x = 0;2x =2+n, n Z; x =4+n2, n Z.sin2xcos xcos 2x+ sinx = sin 3x; sin2xcos x + sin xcos 2x = sin 3xcos 2x;sin 3x(1 cos 2x) = 0;1) sin 3x = 0; 3x = n; x =n3, n Z.2) cos 2x = 1; 2x = 2n, n Z; x = n, n Z. x =n3, n Z. www.legionr.ru 212 33. 1. (0; 2): 0 0,25 6x0;

x < 0,5,x 256l;x 0,x > 0, x = 1. logx 5 (logx 5 =0), - log53x + 4=1logx 5; log53x + 4=log5x 3x + 4 =x; 3x + 4 =x2; x2 3x 4 = 0; x1,2 =3 9 + 162=3 52x = 4 .: 4.290. :

0, 5x + 0, 5 > 0,x > 0, x = 11 < x < 0.log110, 5x + 0, 5logx 11 = 1. logx 11(logx 11 = 0), log110, 5x + 0, 5 ==1logx 11;log110,5x + 0,5 = log11(x)

0, 5x + 0, 5 = x,x < 0;

0, 5x + 0, 5 = x2,x < 0;

2x2x 1 = 0,x < 0;

x1,2 =1 1 + 84=1 34,x < 0;x = 0,5 .: 0,5.291. :

x > 0,log29x + log3x2 0. x , log3x2= 2 log3[x[ = 2 log3x, x > 0. - , , -.

(log32 x)2+ 2 log3x = log3(93)log3x;

14 log23x + 2 log3x =52log3x. t = log2x,

14t2+ 2t =52 t

t2+ 8t = (5 2t)2,52 t0; www.legionr.ru 292 55

3t228t + 25 = 0,t 52;

t = 1,t =253,t 52;t = 1. : log3x = 1 x = 3. , .: 3.292. :

10,5x1,2x2.1) arcsin(0, 5x) = 0; 0,5x = 0; x = 0 , .2) sin6xcos6x = 0; sin2xcos2x

sin4x+sin2xcos2x+cos4x

= 0;sin4x + sin2xcos2x + cos4x> 0, sin2x cos2x = 0; cos 2x = 0;2x =2 +k, k Z; x =4 + k2, k Z. 4,4. 3 .: 3.293.4 x2= 0; x2= 4; x = 2. sin2 = 0 sin(2) = 0, x = 2 . 4 x2 0; x2 4; [x[2; 2x2.1sin x 1 = 0; sin x = 1; x =2+ 2k, k Z. k = 0: x0 =2, 2 22; x0 =2 . k = 1: x1 =2+ 2 =92/ [2; 2]. k = 1k > 1: xk .k= 1: x1= 32, 32/ [2; 2]; x1= 32 . , k< 1, xk/ [2; 2]. : 2;2.: 3.294. :

5 x2 0,x = 0,x =2+k, k Z.

5x 5,x = 0,x =2+k, k Z; www.legionr.ru56 295

5; 2

2; 0

0; 2

2;5

.1)

5 x2x= 0; 5 x2; x = 5 .2) tg x = 0; x = k, k Z. .: 2.295. :

9 x2 0,3x3.(2 sin 2x 1)9 x2= 0.1. 2 sin 2x 1 = 0; sin2x =12; 2x = (1)n6+n;x = (1)n12 +n2,n Z. [-3;3].1) n = 2: x1 =12 = 1112; 1112 [3; 3];2) n = 1: x2 = 12 2= 712; 712 [3; 3];3) n = 0: x3 =12;12 [3; 3];4) n = 1: x4 = 12 +2=512; 512 [3; 3]. n Z x/ [3; 3].2.9 x2= 0; 9 x2= 0; x5,6 = 3. 6 .: 6.296. :

sin2x = 0,9 x2 0;

x = n, n Z,3x3.

1 1sin2x

9 x2= 0.1. 1 1sin2x= 0; sin2x = 1; sin x = 1; x =2+ n, n Z. , :1) n = 1: x =2 = 2, 2 [3; 3].2) n = 0: x =2,2 [3; 3]. n Z x/ [3; 3].2.9 x2= 0; 9 x2= 0; x = 3. 4 .: 4. www.legionr.ru 298 57297. : x74.)4x + 7; x1 = 74 .)

23

5x229

32

x2+5= 0;

32

295x2

32

x2+5; 29 5x2= x2+ 5;6x2=24; x2=4; x2=2, x3=2 . x1= 1,75 x2=2 . :1,75 + 2 = 0,25.: 0,25.298. : 0 < x3.sin x log3x3 x = 0;3 x = 0; x1 = 3; log3x = 0; x2 = 1; sin x = 0;x =n,n Z, 0 0. x y :lg x = 2 lg y; x = y2. y2+ 8 = 2y 1

y2+ 8 = 4y24y + 1,2y 10;

3y24y 7 = 0,y 12;

y1 = 1, y2 =73,y 12;y =73. x =

73

2=499. 499 73 .

499 ; 73

.3x0y0=349397= 7.: 7.304. : x1,8.1)

25

4x223

52

5x213= 0;

25

4x223=

25

135x2; 4x223 = www.legionr.ru 305 59= 13 5x2; 9x2= 36; x2= 4, x1= 2, x2= 2. x1 = 2, , .2)65x + 9=0; x3= 1,8 , , . : 2(1,8) = 3,6.: 3,6.305. : 4 x2> 0; 2 < x < 2.1) sin2x 1 = 0;1 cos 2x21 = 0; cos 2x = 1; 2x = + 2n, n Zx=2+ n,n Z. (2; 2) x1=2,x2 = 2.2) log0,5

4 x2

= 0; 4x2= 1; x2= 3; x3 =3, x4 = 3. , , . 4 .: 4.306. : 7 < x 2 - .2) log2

7 x2

1=0; 7 x2=2; x2=5; x11= 5, x12= 5. 5 5 , , www.legionr.ru60 307. 12 .: 12.307. :

x23x > 0,1 x5 561;

x(x 3) > 0,16 x5 116 ;

x < 0, x > 3,56x 556 ;

56x < 0,3 < x 556.

log2

x23x

2 = 0,arcsin

x5 56

= 0,

56x < 0,3 < x 556 ;

x23x = 4,x5 56= 0,

56x < 0,3 < x 556 ;

x23x 4 = 0,x5=56,

56x < 0,3 < x 556 ;

x1 = 1,x2 = 4,x3 =256,

56x < 0,3 < x 556 ;x2 = 4, x3 =256. xmin = 4, 3xmin = 34 = 12.: 12.309. :

5x 2x21 > 0,13x 21;

2x25x + 1 < 0,13x3;

5 174< x 0; 0 < x < 1,5. :

cos x = 12,3x 2x2= 1;

x = 23+ 2k, k Z,2x23x + 1 = 0;

x = 23 + 2k, k Z,x1,2 =3 14. , - 12, 23, 1, 43. 12+23+1+43= 3,5.: 3,5.317. :

sin 2x = 0,3 2x 5x2 0;

2x = n, n Z,5x2+ 2x 30;

x =n2, n Z,(x + 1)

x 35

0;

x =n2, n Z,1x0,6.1) ctg 2x = 0; 2x =2+ k; k Z; x =14 +k4, k Z. , 34, 14, 14.2)43 2x 5x2= 0, x1 = 1, x2 = 0,6. , 0,6. :0,75 0,25 + 0,25 + 0,6 = 0,15.: 0,15.318. :

cos x = 0,2 + 3x 2x2 0;

x =2+n, n Z,x21,5x 10;

x =12 +n, n Z,(x 2)(x + 0,5)0;

x =12 +n, n Z,0,5x2. www.legionr.ru 319 651) tg x + 1 = 0; x = 14 +k, k Z. 14, 34, 134.2)2 + 3x 2x2= 0; x = 0,5, x = 2. 2. : 14 + 34 + 2 + 134= 4,25.: 4,25.319. :

sin x2= 0,13x x2400;

x2= n, n Z,x213x + 400;

x2 = n, n Z,(x 5)(x 8)0;

x = 2n, n Z,5x8.1) ctg x21 = 0; ctg x2= 1;x2=4+k, k Z; x =12 + 2k, k Z. 6,5.2)13x x240=0; x1=5, x2=8. 5. : 5 + 6,5 = 11,5.: 11,5.320. : 4 x2 0; x2 4; 2x2.1) 52x7= 52; 2x 7 = 2; x = 4,5 .2) 4 x2=0; x1=2; x2=2., . : 2 2 = 0.: 0.321. : 9 x2 0.

792x49 = 0,9 x2 0,9 x2= 0;

9 2x = 2,3x3,x1 = 3, x2 = 3;

x = 3,5,3x3,x1 = 3, x2 = 3. , : x1 = 3, x2 = 3. : 3 3 = 0.: 0.322. : 7x262 > 0; x <

627, x >

627. x27x + 15 = 5,7x262 = 1;x27x + 10 = 0,x2= 9;x1 = 2, x2 = 5,x3 = 3, x4 = 3. www.legionr.ru66 323 , - 3, 3, 5. 3 + 3 + 5 = 5.: 5.323. :

9 x2> 0,cos x = 0;

3 < x < 3,x =2+n, n Z; 9 x2= 1tg x = 0;x1 = 22, x2 = 22x = n, n Z. , - 22, 22, 0, , 3.: 3.324. : x22x2 > 0; x(13)

x(1+3)

> 0; x < 13,x > 1 +3.1) 93x 10836x27+ 1 = 0; 36x 436x+ 1 = 0; 336x= 1; 36x=13;6x = 1; x = 16 .2) x2 2x 2 = 1; x2 2x 3 = 0; x1= 1, x2= 3. - , . 1 + 3 = 2.: 2.325. :

9 x2 0,cos x = 0;

3x3,x =2+k, k Z.1)9 x2= 0; x1 = 3, x2 = 3. , - , .2)tg x =0; x =n, n Z. 0, x=0 , . .: 0.326. :

4 x2 0,sin x = 0;

2x2,x = n, n Z.)4 x2= 0; x1 = 2, x2 = 2. , .) ctg x = 0; x =2 +k, k Z. x1 = 2, x2 =2 www.legionr.ru 327 67 . 2 .: 2.327. :

1x241,9 x2> 0;

3x2 5,3 < x < 3.1) arcsin

x2 4

= 0; x2 4 = 0; x1= 2, x2= 2. , x1 = 2, x2 = 2 .2) ln

9x2

= 0; 9x2= 1; x3 = 22, x4 = 22. . : 22 = 4.: 4.328. : 49 4x2> 0; 72< x 0;1)

1 2 sin2x = 0,18 +x 4x2> 0;

cos 2x = 0,4x2x 18 < 0; www.legionr.ru68 331

2x =2+k, k Z,(4x 9)(x + 2) < 0;

x =4+k2, k Z,2 < x 0,6 5 log5x0;0 < x6. ,x2+x6 = 0;x1 = 3,x2 = 2. , , x = 2.: 2.350. 42x= t, 0 < t8,5

3 14t

2=17 2t; 9 32t +116t2+ 2t 17=0; t2+ 8t 128=0;t1 = 8, t2 = 16 t (0; 8,5]. , 42x= 8;24x= 23; x = 0,75.: 0,75.351. t =22x2,t >0. 22x=4t. -: 3 t = 9 4t. , 9 6t + t2= 9 4t; t2 2t = 0; t1 = 0, t2 = 2. T t> 0, t = 2. 22x2= 2; 2x 2 = 1; 2x = 3; x = 1,5. , x = 1,5 .: 1,5. www.legionr.ru 353 73352. t =52x1, t >0. - 1 +t =9 5t.,1 + 2t + t2= 9 5t; t2+ 7t 8 = 0; t1= 1, t2= 8. T t> 0, t = 1. 52x1= 1; 2x 1 = 0; x = 0,5. , x = 0,5 .: 0,5.353. : x>2. x(x2 4) log3 (x 2)=0, , x1=0, x2= 2, x3=2, x4=3. C x = 3.: 3.354. 9x+50 = 5+14 3x. 3x= t, t214t+45 = 0; t1,2 = 72;x1 = 2, x2 = log3 5; x1> x2.: 2.355. 54|x|(53x+74|x| 1)=0, , x1= 1,x2 = 7. 7.: 7..356. x28x+16 = (x4)2, :

y = [x 4[,y = [x + 3[. [x4[ = [x+3[. -: x28x+16 = x2+6x+9. x =12, y =

12 +3

=72.

12; 72

. x0 +y0 =12 + 72= 4.: 4.357. 4x24x + 1 y = 0,y [2x + 3[ = 0;

[2x 1[ = y,y = [2x + 3[;[2x 1[ = [2x + 3[;x = 12, y = 2; x y = 12 2 = 2,5.: 2,5.361. x + 3x + 4, 96 = [x[. x0 :3x + 4, 96=0. x= 4, 96 - x 0. x0.t2 49t 50 0,(t + 1)(t 50)0. t> 0, t + 1> 0, t 500, t50. 7x50, 72=49 8,3x15;

2x> 23,x5;

x > 3,x5;3 < x5. (3; 5], 4 5.: 2.376.

3x 9,4x > 1;

3x 32,x > 14;

x2,x > 0,25;0,25 < x2. (0,25; 2], www.legionr.ru 377 773 : 0, 1, 2.: 3.377. , , -

2x28x + 9 [x 2[ > 0,2x28x + 9 [x 2[ = 1,x > 0. .1)

0 < x2,2x28x + 9 +x 2 > 0,2x28x + 9 +x 2 = 1;

0 < x2,2x27x + 7 > 0,2x27x + 6 = 0;

0 < x2,x ,(2x 3)(x 2) = 0;0 < x < 1,5, 1,5 < x < 2.2)

x > 2,2x28x + 9 x + 2 > 0,2x28x + 9 x + 2 = 1;

x > 2,2x29x + 11 > 0,2x29x + 10 = 0;

x > 2,x ,(2x 5)(x 2) = 0;2 < x < 2,5, x > 2,5. (0; 1,5)(1,5; 2)(2; 2,5)(2,5; +). - , : 1,5, 2, 2,5. 1,5 + 2 + 2,5 = 6.: 6.380. y=t14 t 0, - (6; 0)

6 < x < 0,[3 x[ +x0;

6 < x < 0,3 x +x0;

6 < x < 0,0x3;6 < x < 0. (6; 0) 5 .: 5.381. - log0,4[0,2x 1[ 10, log0,4[0,2x 1[1,log0,4[0,2x 1[log0,4 0,4,

[0,2x 1[0,4,0,2x 1 = 0; y = log0,4t, t > 0, 0 < 0,4 < 1 . www.legionr.ru78 382

0,40,2x 10,4,x = 5;

0,60,2x1,4,x = 5;

3x7,x = 5;3x < 5, 5 < x7. [3; 5) (5; 7] 4 .: 4.382. - 2x + 5[x + 1[ 1> 0,2x + 5 [x + 1[[x + 1[> 0,[x + 1[> 0, x = 1, , 2x + 5 [x + 1[ > 0, x = 1.1)

x < 1,2x + 5 +x + 1 > 0;

x < 1,3x > 6;

x < 1,x > 2;2 < x < 1.2)

x > 1,2x + 5 x 1 > 0;

x > 1,x > 4;x > 1. (2; 1)(1; +). - 0.: 0.383. y =16 133|x|. E(y) -. [x[ 0, [x[ 0, 0 0.1) x0, f(x) +g(x) = 2(x2+8x) = 18, x2+8x9 = 0, x1 = 9,x2= 1 x0. x0 f(x) +g(x) = 18 , x = 9.2) x > 0, f(x) +g(x) = x28x +x2+ 8x = 18, x2= 9, x1 = 3,x2=3. x1= 3 x>0, x>0 f(x) +g(x) = 18 , x = 3. (9)3 = 27.: 27.421. y = (x + 1)(x 4)(x 3)(2x 1) - [0; 4).x + 1 = 0, x = 1, 1/ [0; 4); x 4 = 0, x = 4, 4/ [0; 4);x 3 = 0, x = 3, 3 [0; 4); 2x 1 = 0, x =12, 12 [0; 4). [0;4) 2 . - f(x) = 0 [6; 0], , T= 4.x = 3. 3 4 = 1, 1 [6; 0]; 1 4 = 5, 5 [6; 0];5 4 = 9, 9/ [6; 0].x=12.12 4= 3,5, 3,5 [6; 0]; 3,5 4= 7,5, 7,5 / [6; 0]. [6; 0] f(x) = 0 3 .: 3.422. y=f(x) x0 , x0 - . h(x)x1= 23; x2= 8., y = f(x) x1 = 23; x2 =23; x3 = 8; x4 = 8. www.legionr.ru 423 87 y =f(x) , f(0) =f(0) = f(0), f(0) = 0. , x5 = 0.: 5.423. . . y=f(x) x0 , x0 .h(x): x1=27; x2=2811; x3=310. y = f(x): x1 =27; x2 = 27; x3 =2811; x4 = 2811; x5 =310; x6 = 310.: 6.424. y = g(x)g(x)f(x) f(x)g(x)f(x); x0 = 0.f(x0) = 5, g(x0) = 2.f(x) = f(x), g(x) = g(x), f(x) g(x) ;y = g2(x)f(x) +f2(x)g(x) = f(x)g(x)(f(x) g(x)) = 523 = 30.: 30.425. y = g(x)f(x)f(x)+f(x)g(x)g(x), x0 = 0, f(x0) = 2, g(x0) = 3,f(x) = f(x) g(x) = g(x) f(x) g(x) .y = g(x)f2(x) +f(x)g2(x) = f(x)g(x)

f(x) +g(x)

= 65 = 30.: 30.426. h(x) = x(x + 2)(3x 4)(5x 2); h(x) = 0,x(x + 2)(3x 4)(5x 2) = 0; x1= 0; x2= 2; x3=43; x4=25. y = f(x) , f(x) = f(x), , x0 f(x) = 0 0 2, 2.: 3.427. y = f(x) - , x < 0 f(x) =f(x) =g(x), x>0 - y=f(x) y=g(x). , - g(x) = g(x) . , x(x 2)(x + 2)(x 4)(x + 9) =x(x 2)(x + 2)(x 4)(x + 9),x(x + 2)(x 2)(x + 4)(x 9) = x(x 2)(x + 2)(x 4)(x + 9),x(x24)

(x 4)(x + 9) + (x + 4)(x 9)

= 0,x(x24)(x2+5x36+x25x36) = 0, 2x(x2)(x+2)(x6)(x+6) = 0. 2 6. 8.: 8. www.legionr.ru88 430428. f(x) g(x) , - , , - . , f(x) =x , f(x) = x. x f(x)=x -g(x) =x x(x + 1)(x 2) =x,x(x2x 3) =0 x1=0,x2,3=1 1 + 122=1 132. g(x) =x

0, 1 +132. f(x) = x

1 +132, 0, 1 +132. , 3 .: 3.430. y =f(x) . x g(x) = (x2+ 4x + 4)(x21)(2x + 3)(x 2). - f(x). , f(x) > 0.. f(x) >0 -. f(x) , - . ,f(x) = 0 x = 1,x = 1,5, x = 2. g(x)> 0 (. 14).. 14.: 5.431. , B8 51. y = f(x) x1. ,(x2+ 5x + 6)(x2x 12) = 0, x1 x = 3, x = 2. -, y=f(x) x=2, x=3. www.legionr.ru 432 89, x = 0 .: 5.432. g(x), :g(5) = 7 + (2)3 f(2) + 5 = 12 8f(2), g(7) = 7 + 0f(0) + 7 = 14,g(9) = 7 + 23 f(2) + 9 = 16 + 8f(2). , g(5) +g(7) +g(9) ==428f(2)+8f(2)=42, f(x): f(2)=f(2).: 42.433. g(1) +g(2) = 1,4 +f(0,5)(0,5)5+ 1,4 +f(0,5)0,55= 2,8++f(0,5) +f(0,5)0,55= 2,8, f(x):f(0,5) = f(0,5).: 2,8.434. f(x) , f(x)= f(x) f(0)=0. g(x) 0, 1, 2 4:g(0) = (1)f(3) f(1) = f(3) +f(1),g(1) = (1 1)f(1 3) f(1 1) 0,5 = 0,5,g(2) = (2 1)f(1) f(1) 0,52 = 2f(1) 1,g(4) = 3f(1) f(3) 0,54 = 3f(1) f(3) 2,g(0)4g(1)+2g(2)+g(4) = f(3)+f(1)+24f(1)2+3f(1)f(3)2 = 2.: 2.435. x 0, g(x) =f(x) +f(x)2=f(x), x 0 2) a 2x = 2 + 6 > 0.: a = 2.606. y = arcctg t . y , t . t = 13 +10ax 2x2x=10a22.x=15.a, 10a4= 15; a = 6.: 6.607. :y =165x21 8ax2= 24

5x21

23

ax2=220x2+3ax10.y =2t. , t, . y = 220x2+3ax10 , - t = 20x2+3ax10 . - www.legionr.ru122 608 x= B2A, , A , B -. , x = 3a40. , x= 3a40. , x = 0,15. 3a40= 0,15 , a = 2.: 2.608. y = 9xx2 3x25ax+3= 32x2x2+x25ax+3= 32xx25ax+3== 3x2(5a2)x+3. y = 3t, 3 > 1 . y , t . t = x2(5a2)x+3 x =5a 22. x = 3. a 5a 22= 3.5a = 4, a = 0,8.: 0,8.609.1) 10%-x, :x10 .2) 600 15%- : 600 15100= 90 .3) 30%-:

90 x10

10030=

300 x3

.4) 10%- :600

300 x3

=

300 +x3

. .x = 300 +x3,23x = 300, x = 450 .: 450.610. 24 x , - (24+x) 40%- , 0,4(24+x) . 45%- 24 - 240,45=10,8 ; www.legionr.ru 611 123. : 0,4(24 +x) = 10,8; 24 +x = 27, x = 3., 3 .: 3.611. 1- x% , 2- (x 40)%. 4 x% , 1- 4 :x100=400x(); 2- 6 , (x 40)% , , :6:x 40100=600x 40().

400x+600x 40

, - 20 . :400x+600x 40= 20;x(x 40) =0;20x+30x 40=1; 20(x 40) + 30x =x(x 40);20x800 +30x = x240x; x290x+800 = 0; x1,2 = 45 2025 800;x1,2 = 45 1225; x1 = 45 35; x2 = 45 + 35; x1 = 10; x2 = 80. . x> 40,, x = 80. 80% 1- .: 80.612. x . x 5 ;x 5x 100 ,x 5x + 15100 . :(x 5)100x(x 5)100x + 15=30, 10(x 5)(x + 15 x)=3x(x + 15),50(x 5) =x(x + 15), x2 35x + 250 = 0, x1,2=35

35225042,x1,2 =35 549 402=35 152, x1 = 25, x2 = 10. , x < 20, x = 10 .: 10.613. x . (x 80) www.legionr.ru124 614 , 80x 100% . 100 (x+100) ; 80 + 100x + 100 100% , 20% -. :180x + 100100 80x 100 = 20;1805x + 100805x= 1; x(x+100) = 0;1805x805(x+100) = x(x+100);x2+100x = 900x 400x 40000; x2400x +40000 = 0; (x 200)2= 0;x = 200 ( x(x + 100) = 0).200 , 200 80 = 120 ().: 120.614. x 1- , 40%, , 2- 60% x, 0,6x -; 3- 60% 0,6x, 0,36x . . , : 1- b1 = x, q = 0,6, n = 5, b5 = xq4, b5 = 324, - x0,64= 324, x =3240,64; x = 2500, b1 = 2500.b1(1 q5)1 q , .2500

1

35

5

1 0,6= 6250

1 2433125

==625028823125= 5764.: 5764.615. x 6%- , x =26100= 0,12 . y 1%- , y =31100= 0,03 .(x +y) , - www.legionr.ru 616 125x +y2 + 3100% =0,12 + 0,035 100% = 3%.: 3.616. x , y . (x +y) , 0,6(x +y) . , - x + 0,2y. : 0,6(x + y)=x + 0,2y;0,4y = 0,4x;yx100% = 100%.: 100.617. x . 1,3x , 0,91,3x - . , (0,9x1,3x x)x 100% = 17%.: 17.618. v ,c1,c2 - , c1, c2 . 1,1v, s=v(c2 c1) .., - s = 1,1v(c2c1) .. , c2 = 1,15c2, c1 = 0,75c2, c1 = 1,2c1 s = v(c20,75c2) = 0,25vc2, c2c1 = 1,15c21,2c1 == 1,15c21,20,75c2 = 1,15c20,9c2 = 0,25c2, s = 1,1v(c2c1) == 1,1v0,25c2 ss= 1,1, 10%.: 10.619. v ,c , vc . 3v, 0,5c, 3v0,5c = 1,5vc. 1,5vc vc = 0,5vc. , 6vc, - (6vc) : (0,5vc) = 12 .: 12.620. a , b , 1,2a , www.legionr.ru126 6221,05b . , - 15%, :1,2a + 1, 05b=1,15(a + b), 1,2a 1,15a=1,15b 1,05b, 0,05a=0,1b,b = 0,5a. x , y , ax 1 , by 1 . 1 18 1 , - :ax=18by,ax=180,5ay, y= 9x. , :xx + 9x=110= 0,1.: 0,1.622. 100 , x , 25 , 35x . - 79, - 7009. 27% . :35x7009=27100, x = 35.: 35.623. a1=100 ,x% , a2 , 2 a3 = 200.a2 = 100 + 100x100= 100 +x; a3 = a2 +a2

x100= 100 +x + 100 +x100 x;a3 =10000 +x2+ 200x100; a3 = 200; 10000 +x2+ 200x = 20000;x2+200x10000 = 0; x1,2 = 10010000 + 10000. x > 0:x = 100 +20000; x 100 + 141,42 41%.: 41%.624. ai , i- - , a1=42, a2=38, an www.legionr.ru 625 127, n = 10, a1 = 42, d = 4, S10 =2a1 + 9d2 10 ==242 362 10 = 2410 = 240. 240 .: 240.625. , , , , 90 . ., . s . 1000 . , , 90. . , 1,2s 90. 1,2(1,2s 90), , , - 90 . . : 1,2(1,2s 90)=90,1, 44s 108 = 90, s =19800144=220016=2752= 137,5. ,,, 137500 .: 137500.626. p100% . , -30.., (1 +p)20 +30 . . , - , (1+p)((1+p)20+30). ., , , 60,95 . . : (1 +p)2 20 + (1 +p)30 = 60,95, 20p2+ 70p 10,95 = 0,D = 702+ 42010,95 = 5776 = 762, p > 0, p = 70 + 7640= 0,15. , , , 15% .: 15.627. p . - , (10,2)p = 0,8p. 0,850,8p=0,68p, , 0,90,68p = 0,612p. - , 0,388p, 0,388pp 100% = 38, 8% p.: 38, 8. www.legionr.ru128 629628. S -, v , . - Sv. , 20% 10% , 1,11,2S = 1,32S. , 30%, 1, 3v. , 1,32S1, 3v = 1,716Sv. , 0,716Sv, 0,716SvSv 100% = 71,6% .: 71,6.629. x, - 1,1x. y (1,1x y), 1,25y0,85(1,1x y).

1,25y + 0,85(1,1x y)

. :1,25y + 0,85(1,1x y)=x; 1,25y + 0,935x 0,85y=x; 0,4y=0,065x;y = 0,1625x; y = (0,1625 : 1,1)(1,1x).y 0,15(1,1x), 15% - .: 15.630. x , 1,2x. y (1,2xy) , 1,1y 0,75(1,2xy) .

1,1y +0,75(1,2xy)

. : 1,1y + 0,75(1,2x y) = x; 1,1y + 0,9x 0,75y = x;0,35y = 0,1x; x = 3,5y.1,1yx 100% 31%, 31% .: 31.631. 20 y . 0,420 +0,2y . - 30% 20 + y , 0,4 20+0,2 y = 0,3(20+y), 8+0,2y = 6+0,3y,0,1y = 2 y = 20.: 20. www.legionr.ru 633 129632.x-. 150 150 , x100 150+x100 250 , 30% (400 ) . -,x100150 +x100250=0,3400, 4x=120 x=30. 100 26 30 = 44% . ,150 0,4150 = 60 , 250 0,44250 = 110 . , 60 + 110 = 170 .: 170.633. x , 2x. 300400, , 2x100300 +x100400 , - 30% 700. ,2x100300 +x100400 = 0,3700, 10x = 210 x = 21. 1001042 = 48%. , 300 0,48300 = 144 , 400 - 0,4400 = 160 . , 144 + 160 = 304 .: 304.634. x , y. 0,5(0,05x + 0,1y)0,5(x +y)=0,07; 0,05x + 0,1y=0,07(x + y); 0,02x=0,03y;x : y = 3 : 2.: 1,5.635. x , y . 0,01x , 0,05 .

x +y2

, 0,01x + 0,05y2 ., : 0,01x + 0,05yx +y100%, 2%. www.legionr.ru130 6360,01 + 0,05y = 0,02(x +y); 0,01x = 0,03y;xy=31., 3 , -.: 3.636.x, y, (x+y). 1,15x, 1,05(x + y). : 1,15x + y=1,05(x + y), 0,1x=0,05y,y=2x,xx +y=xx + 2x=13,13100%=33,(3)%. , , 33% .: 33.637. x, y, (x +y).- 1,5x,- 1,01(x +y).: 1,5x +y =1,01(x +y),0,49x=0,01y, y=49x,xx +y=xx + 49x=0,02. , - 2% .: 2.638. 4% , - 96%. 15 150,96=14,4 . 40% , - 60%. , - 15 , 14,4 - , 60%. , 100% 14,4 10060= 24 .: 24.639. x y - , P . P x100 , P y100 . 12 , www.legionr.ru 640 131 P x100 +P y100= 12. - 2P , 2P x100 +P y100 ,, , 16 . , 5% , -

P x100 +P y100= 12,2P x100 +P y100= 16,y x = 5. , - , y = x + 5. - :x + (x + 5)2x + (x + 5)=1216, 2x + 53x + 5=34, 4(2x+5) = 3(3x+5),x = 5. , y = 10. P :P5100+P10100= 12 P=1210015= 80. , - 160 12 -. , 12160100 = 7,5%.: 7,5.640. P, -x% , y% . -P x100, P y100. 3, P x100+ P y100=3. 2P, - P 100 x100+2P 100 y100 , 11 . , , 20% , www.legionr.ru132 641

P x100 +P y100= 3,P 100 x100+ 2P 100 y100= 11,x y = 20. x y: x = y+20, - . - P100(y + 20 +y)P100(100 y 20 + 2(100 y))=311,2y + 20280 3y=311 11(2y + 20) = 3(280 3y) y= 20. , x = 40. P P 40100 +P 20100= 3 P=310060= 5. , , 5 , 10 3 . , 10 3 = 7 .: 7.641. 1- :10561056 + 44100% =10561100100% =105611% = 96%. 2- :756756 + 1344100% =7562100100% =75621 % = 36%. x (1500 x) -. : 96x100; 36(1500 x)100; 401500100. : www.legionr.ru 642 13396x100+36(1500 x)100=401500100, 96x+36(1500 x) =40 1500,8x + 3(1500 x) = 5000, 5x = 500, x = 100.: 100.642. 600 85100 .1 2 x (600 x) x 92100 (600 x) 80100 :x92100+ (60 x) 80100 .x 92100 + (600 x) 80100= 600 85100; x = 250.: 250.643. x , 0,8x0,75 ==0,6x 0,75 . 0,25 1,8x 0,4, - 0,25x (0,721) = 0,07x( ). : 0,6x 0,07x = 0,53x, - 53% .: 53.644. x , 2,4x , - . - . 0,85 2,4x0,85 = 2,04x, 2,04x 0,85x = 1,19x. 1 0,85=0,15 . k% , 2,4x (10,01k)0,15 0,15 , 2,4x(1 0,01k)0,15 0,15x = 0,21x 0,0036xk. 1,13x. - 1,19x + 0,21x 0,0036xk = 1,13x; k = 75.: 75.645. x, y, 1,4(x+y) . 1,35x , , 1,6y , -. : 1,35 + 1,6y = 1,4(x +y),xy= 4. www.legionr.ru134 646, 4 -.: 4.646. x , , 0,25x1,1+0,4x+0,35x1,2 = 1,095x, 1,095x xx 100% = 9,5%.: 9,5.647. x . - 0,6x. y% - , (x 0,01xy) -.(x 0,01xy) 2,5y100. ,xy100+xy40

1 y100

- , 0,6x.y100 +y40y24000= 0,6, y2140y +2400 = 0, y1 = 120, y2 = 20. y = 20.: 20.648. 1 , x% ,1,5x% . :1 +0,01x +(1 +0,01x)0,015x = 1,56; 3x2+500x 11200 = 0; x1 = 20,x2< 0 .: 20.649. p, t- , pt I, 2ptIII,IIIIV3pt ( 50% ). 5pt . - 0,5pt, , 0,5pt5pt=110= 10%.: 10.650.p, t- , pt 1 -, 2pt I II . - 1,5p. III IV , , 3pt. 5pt . www.legionr.ru 651 135 pt, I , pt5pt=15= 20%.: 20.651. x , (20x) - . - 20 x20.

20 x x(20 x)20

, :20 x x(20 x)2020=(20 x)2400, 0,36. :(20 x)2400= 0,36 x1 = 8, x2 = 32, x < 20 x = 8.: 8.652. x , (10x) . 10 x10. -

10 x x(10 x)10

=(10 x2)100, 0,81. :(10 x)2100= 0,81 x1 = 1, x2 = 19, x < 10 x = 1.: 1.653.1210045 = 5,4 . 12 .5,440100 = 13,5 () .13,5 12 = 1,5 . .: 1,5.654. x , y 30 ,x +y2= 1,25x; y = 1,5x;1,5x 1,25x1,25x= 0,2100% = 20%.: 20. www.legionr.ru136 658655. x , :0,61,45x + 0,41,45x0,6 = 1,218x. , 1,218x x = 0,218x.: 21,8.658. 12,50,4 = 5 () .12,5 5 = 7,5 () .7,5 5 = 2,5 () , .: 2,5.659. 6000,18 = 108 () , 108 100 = 8 () .: 8.660. 2 , 120 , 0,25 /120 == 30 . 30 3075100% = 40% 75.: 40.661. 1,527/=118 /. - 20% 1181,2 / =115 /. 80 80:115=1200 , 20.: 20.662. 900,9= 100 , 100 + 30 = 190 .: 190.663. x , (19 x) . 5x100%, 219 x100%. , 5x100% 32=219 x100%,15x=419 x,15(19 x) = 4x, 1519 = 19x, x = 15.: 15. www.legionr.ru 665 137664. 32 12001,6=1920 , 1200 + 1920 = 3120 .: 3120.665. x , (44x) . 7x100%, - 544 x100%. , 7x100% = 3 544 x100%,7x=1544 x, 7(44 x) = 15x, 744 = 22x,x = 14.: 14.666. 30 0, 430 = 18() = 1800 (). 1800:2=900 (), 900:40=22, 5 (). , , 23 .: 23.667. 15% 4201, 15 = 483 (.). , 5000 : 5000: 483 10, 35., 5000 10 .: 10.668. 900: 20=45 (.). 450, 9 = 40, 5 (.)., 900 . 900: 40, 5 22, 2. , 900 22- .: 22.669. 87 : (200,9) = 4,8. 5 .: 5.670. 2 26010= 12 , - 1230100% = 40% .: 40.671. , - , 23 + 56 www.legionr.ru138 672 .

23 + 56

1 =12 - , .12100% = 50%.: 50.672. , , ,,35+710 .

35 +710

1=0,3 .0,3100% = 30%.: 30.673. 201,25=25 (.) . 200: 25=8, , 8 .: 8.674. 10 0,9 = 9 (.) . 300 : 9 = 3313,, 33 .: 33.675. 81,25 = 10 , , 120 12010= 12 .: 12.676. 120,75=9 .50 : 9 = 559, , 5 .: 5.677. , 1- - a1 = 720, d = 40, n , i -, i- . Sn = 5040 . n : 2720 40(n 1)2n = 5040. :(720 20(n 1))n = 5040;720n 20n2+ 20n 5040 = 0; n237n + 252 = 0; www.legionr.ru 678 139D=372 4252=361; n1,2=37 3612; n1,2=37 192;n1=9;n2 = 28 ( ).9 .: 9.678. -: 90 54 =36(); x /, 36x. - 5 . =112 , (x+6) / 36 - 36x + 6 . , :36x + 6 +112=36x . : 3612x+x(x+6) = 3612(x+6);x2+6x = 36126; x2+6x2592 = 0; x1,2 = 3 9 + 2592; x1 = 54;x2 = 48. x(x+6) = 0, x = 54 .48 / .: 48.679. t , C -, v1, v2 1- 2- -. 1- A C( AC) t , C B ( BC) 15 t, v1 t =AC, v1(15 t) =BC. 2- : v2 t = BC, v2 4 = AC. , v1, v2, t - :

v1 t = v2 4,v2 t = v1 (15 t). v1 v2,, t2= (15t)4, t2+4t60 = 0, t1 = 10, t2 = 6. t , t = 6 .: 6.680. t , - , , C ,v1 ,v2 . , B 45 , www.legionr.ru140 68134, , C B 34 t . -: AC=v1t, BC=v1

34 t

. , , : BC=v2 t, AC=v2 1. , v1, v2, t :

v1

34 t

= v2 t,v2 = v1 t. v1 v2,:34 t=t2, 4t2+ 4t 3=0,t1= 32,t2=12. - t , t=12, 30 .: 30.681. t(), 1- 2- . 2(t + 25) , 1-. v1, v2 1- 2- , A , B C . : v1t=BC, v125=AC; v2t=AC,v2 (36 t) = BC. , v1, v2, t :

v1 t = v2 (36 t),v2 t = v1 25. v1v2, : t2= (36t)25,t2+25t900 = 0, t1 = 45, t2 = 20. t - . t = 20, , 1- 2(25 + 20) = 90 , 1,5 .: 1,5.682. v/. - (v 5)/,(v + 5) /. 10v 5, 45v + 5. :10v 5+45v + 5=2. :10v 5 +45v + 5= 2 www.legionr.ru 683 14110(v +5) +45(v 5) = 2(v225) 2v255v +125 = 0, v1 =52, v2 = 25.v1 , . , v = 25 /.: 25.683. 20 260=23. x , 20 /, , . 5 /, -, x5 x20=160, x =19. , , 23+19=79/. 79/ 20 /, 7180 140 .: 140.684. 5 , , 6 / , - 2 / 5 -. 6 /2 /=3, 5 23, , 7,5 .7,5 + 5 = 12,5.: 12,5.686. 40 =4060 =23 . x / , x > 2.v (/) t () s () x + 216x + 216 x 216x 216 23 . - : www.legionr.ru142 68716x 2 16x + 2=23; 24(x + 2) 24(x 2)=x2 4;x2 4=96,x2= 100; x1 = 10, x2 = 10 x > 2.v=10 / , v=2 / - .vv=10 /2 /= 5 5 .: 5.687. x / , (x+4) /, (x4) /. 30 304= 7,5 . , , -42x + 4 + 1 +12x 4. :42x + 4 + 1 +12x 4= 7,5;42(x 4) + 12(x + 4)=6,5(x2 16); 13x2 108x + 32=0; x1=413,x2 = 8. , 8, 2 .: 2.688. x / , y / , S A B. S = 3(x +y), S = 4(x y), Sy.3(x +y) = 4(x y), x = 7y, S = 3(x +y) = 24y, Sy= 24.: 24.690. x / , y / - . 180 180x, -180y., 27. , 180x 180y=2760=920. www.legionr.ru 691 143 180 180x + 10, 180y 10. 180x + 10=180y 10. :

180x 180y=920,1x + 10=1y 10;

y = x + 20,20x20x + 20=120.: x + 20 x=x2+ 20x400; x2+ 20x 8000=0; x1= 100;x2 = 80. x> 0, 80 /.: 80.691. V1 / , V2 / - , S (. . 21). , . SV1, SV2.. 21.30-,SV2SV1=3060. , -, S120, 1 . , V1160=S120; S=2V1. 1 , , www.legionr.ru144 692

S120 0,146

. ,S120 0,146V2=S120V1;S 1200,146V2=SV1. , S= 2V1, -:

2V1V2 2V1V1=12,2V11200,146V2=2V1V1;

V2 =2V12,5 ,V2 =2V11200,1462;2V12,5= V18,76.0,5V1 = 21,9; V1 = 43,8 /.: 43,8.692. sAB, v/-.sv-,

0,5s30+0,5sv + 20

.B-, sv=0,5s30+0,5sv + 20, v >0.2v=130+1v + 20; 60(v+ 20) == v(v + 20) + 30v; v2 10v 1200 = 0; v1,2 = 5 35. v> 0, v = 5 + 35 = 40 /.: 40.693. 1. : 1- x ; 2- 4x ; 3- (x9) . : 1- 1x ; 2-14x ; 3- 1x 9 . ( ):

1x +14x +1x 9

.1x +14x +1x 9=4(x 9) + (x 9) + 4x4x(x 9)=9x 454x(x 9). ( ) www.legionr.ru 694 145

1 :9x 454x(x 9)=4x(x 9)9(x 5)

, 18 .4x(x 9)9(x 5)= 18, 4x(x 9) = 918(x 5),2x(x 9) = 81(x 5),2x218x 81x + 815 = 0, 2x299x + 815 = 0.x1,2 =99

992428154=99

92(112425)4==99 9121 404=99 994. x1 =1804= 45, x2 =184=92.x2 =92 , - , . , x = 45 ., 1- 3- , 2- . , . : 1- 45 , 3- 36. : 1- 145, 3- 136. 1-3-:145+136=4 + 5180=9180=120. 1- 3- - 1 :120= 20.: 20.694.1. -30, , 1130, 6630=15. 61 15=45,2- 40.45: 40 =150 , 1- 130 150=175 . 1- - www.legionr.ru146 695 1 :175= 75 .: 75.695. 1. 1- - x , 1x , 127 , 2-

127 1x

. 13:1x=x3. 1- 13,2-

127 1x

x3, 13 +

127 1x

x3=x81 .

1 x81

1- 8 1x , , -8x . : 1 x81=8x, x> 0; 81x x2= 881;x281x+648 = 0; D = 8124648 = 65612592 = 3969; x1,2 =81 632;x1 = 9; x2 = 72. x > 0. x = 72. 72 1- .: 72.696. : a1= 18, d = 1, Sn= 100, n .218 + (n 1)2 n = 100; n2+ 35n 200 = 0.,, n1=40( : n N.), n2 = 5. 5 .: 5.697. , , - : a1 = 1,8, d = 0,2,Sn1 = 7,8, n . www.legionr.ru 698 147Sn1 =2a1 +d(n 2)2 (n 1); 21,8 0,2(n 2)2 (n 1) = 7,8;(1,8 0,1(n 2))(n 1) = 7,8; (2 0,1n)(n 1) = 7,8;2n 2 0,1n2+ 0,1n = 7,8; 0,1n2 2,1n + 9,8 = 0; n2 21n + 98 = 0;D=441 392=49;n1,2=21 72;n1=7,n2=14. , an> 0. - : a1 + d(n 1)> 0, 1,8 0,2(n 1)> 0,0,2(n 1)> 1,8, n 1< 9, n< 10. 7< 10, , 7- - ; 14 > 10, 14- . 7 .: 7.698. , an, n-. , d . 20 20 a1 +a202 20, - (a1 +a20)10. (1) , 10 , 10 (bn) :b1=a21, b10=a30: b1+b105 (2) (1)(2): (a1 + a20)10=(a21 + a30)5. , an=a1 + d(n 1), (2a1 + 19d)2 = 2a1 + 49d; 4a1 + 38d = 2a1 + 49d; 2a1= 11d;a1=112d. , 15: a1 +a15215 = (a1+7d)15; a16 +a30215 = (a1+22d)15. , 15 15- : (a1 + 22d)15(a1 + 7d)15=a1 + 22da1 + 7d. a1 =112d, 112d + 22d112d + 7d=11d + 44d11d + 14d=55d25d= 2,2.: 2,2.699. 200% , . 1- x , 2- www.legionr.ru148 700: x+2x = 3x, 12, x = 4; 3- 3x+6x = 9x; 4- 9x + 18x = 27x, . . . , , n , - : b1= 4, q= 3, bn= 43n1, 2916. : 4 3n1= 2916, 4 3n1= 4 36, 3n1= 36,n 1 = 6, n = 7.7 .: 7.700. x , (x 5). 4040x, 3030x 5, 2.:30x 5 40x=2, x(x 5)=015x 5 20x=1;15x20(x5) = x(x5); x25x = 15x20x+100; x2= 100; x1 = 10,x2=10. x(x 5) =0, , (1).x >5x=10. 10 , : 10 5=5 ( ), : 10 + 5 = 15 , , 120 120 : 15 = 8 .: 8.701. x 1 , - (x + 7). 45 45x, 20 20x + 7, 5 , 45x.:20x + 7 + 5 =45x , x(x 7) = 0. (1)4x + 7+ 1=9x; 4x +x(x + 7) =9(x + 7); 4x +x2+ 7x=9x + 63;x2+ 2x 63 = 0; x1,2 = 1 1 + 63; x1,2 = 1 8; x1 = 9, x2 = 7. 9 7 x(x + 7) = 0, , (1), x = 9 , x = 7. www.legionr.ru 702 149 105 1057= 15 ().: 15.702. x , 360x. 360(x + 1), 360x + 1, 4 , 360x. :360x + 1+ 4=360x,x(x + 1) =0;90x + 1+1 =90x ; 90x+x2+x = 90x+90; x2+x90 = 0; x1,2 = 1 1 + 3602;x1,2= 1 192; x1= 10, x2= 9. x1= 10, x2= 9, x(x + 1) = 0.x = 9 .9 .: 9.703. , an n- ; - 6 . n , Sn n , :260 6(n 1)2 n = 324, Sn =2a1 +d(n 1)2 n;(60 3(n 1))n=324; 60n 3n2+ 3n=324; 3n2+ 63n 324=0;n221n + 108 = 0. , ,n1=12,n2=9. , n . an>0, a1 + d(n 1)>0 1)>0; (an=a1 + d(n 1)) 60 6n + 6>0;6n > 66; n < 11. , n = 9. 9 .: 9.704. x, yz, -. xy=35, x +y =4z6(x + y + z)=V , V . : x=3y5, www.legionr.ru150 705z =x +y4, x +y+z =54(x +y) =54

35+ 1

y =2y. ,V= 6(x +y +z) = 12y y =V12. x =3y5=V20, z =14(x +y) ==14

V20+V12

=V30. 3 36 3,6 , 3,6(x+z) .3,6(x + z) = 3,6

V20 +V30

= 0,36 5V6= 0,3V . 3 36 30% .: 30.705. x, yz, -. yz=23, 3x =y+z5(x +y +z) = V , V . , y =2z3, x =y +z3, x +y +z =43(y +z) =43

23 + 1

z =20z9. , V= 5(x +y +z) ==100z9z = 0,09V . y =2z3= 0,06V , x =13(y +z) ==13(0,06V+ 0,09V ) = 0,05V . 6 , 5 , -, , : 6x+5y = 60,05V +50,06V= 0,6V . 60% .: 60.706. , B9 3.xy-,V , t , - , (5x + 2y)t = V .xy=29 x =2y9 V=

5 29 + 2

yt =289yt yt =9V28 .(6x +y)t =

6 29 + 1

yt =219yt =219

9V28= 0,75V .. . t - www.legionr.ru 707 151 75% .: 75.707. 3x, 4x, 5x . , - (3x +4x +5x)5 = 60x . 3x , 12x . 45x , 9x . , 5 . 7 .: 7.708. . x - , y , (x + y) . :

6(x +y) = 1,3(x +y) + 4x = 1;x =18, y =124. , 24 .: 24.709. . -2x, x. 10(2x +x) =1, x =130. y . :y230 + (11 y) 330= 1, 2y + 33 3y = 30, y = 3.: 3.710. , 1. a 4 , b 4 . , , 35 , 4 ,

a +b =35,ab=45;

4b5+b =35,a =4b5., b =13, a =415. V1 ,Vn . www.legionr.ru152 711V1 =a4=115, Vn =V12=130. . 1, - 20% , 80%, 45. , 45 35=15. t, : t =0,2Vn=1305= 6 .: 6.711. x ,y -.S, S1, S2.S =10x, S1=2(x +y), S2=6x. S =S1+S2, S=2(x + y) + 6x.

S = 10x,S = 2(x +y) + 6x., 10x = 2(x +y) + 6x, 10x 2x 6x = 2y, x : y = 1.: 1.689.,- 13 , 15 . 13 + 15=815 , , 2 :815=154= 3,75 .: 3,75.712. pi i- (i 1, 2), , ;ti , i- . :

t1p1 =12,t2p2 =12,t1 +t2 = 860,(p1 +p2)(360 + 45) = 1. . www.legionr.ru 713 153 : p1=1360, t1=180, p2=1600, t2=300; - . 1 2 . , 360 .: 360.713. x 3 , (x 5) 3 . 45 3 - 45x, 50 3

50x 5

-. :50x 5 45x=12, x > 5.x215x450 = 0, x1 = 30, x2 = 15, x2 x > 5.30 3 .: 30.714. . x - , (x > 0), (x + 6) . 1x 1x + 6 . 1x +1x + 6=14, x > 0,4x + 24 + 4x = x2+ 6x; x22x 24 = 0,x1 = 6, x2 = 4.x2 x > 0. 6 , -.: 6.715. , :

5x + 5y = 1,3x + 7,5y = 1; x , y -. , : x=545, y =445, www.legionr.ru154 716 454.: 11,25.716. , :

15x1 + 15x2 = 1,7x1 + 21x2 = 1, x1 , x2 -. , : x1=135, x2=4105, 35 .: 35.717. x, y, z , . 4 , 1 14 , x+y +z =14. - : x + y =16. z =14 (x + y) x + y =16, : z=14 16=112. , 12 .: 12.718. x, y, z , - . , 1x +y=4,1y +z=6,1x +z= 12. 1x +y +z. :

x +y =14,y +z =16,x +z =112. , :2(x +y +z) =14 + 16 +112; x +y +z =14;1x +y +z= 4. 4 .: 4.722. - . - www.legionr.ru 723 155 . : 5(5003 + 3502) = 11000 (.); : 5(5002 + 3504) = 12000 (.); : 5(5004 + 3501) = 11750 (.).11000 < 11750 < 12000. 12 . .: 12.723. y(t) = x(t), 2 + 11t 5t2= 2t, t > 0, 5t29t 2 = 0, t1 = 2, t2 = 0,2. t2 t > 0.: 2.724. , .1- : 1020 + 57 = 235 ;2- : 720 + 137 = 231 ;3- : 1120 + 27 = 234 ., 1-, 235500 = 117500 (.) = 117,5 . .: 117,5.725. y(t) = 2x(t), 8 + 24t 5t2= 6t, t > 0.5t2 18t 8=0; t1= 0,4, t2=4. t1= 0,4 -t >0,t =4y(4) = 8 + 244 542= 24.: 24.726. 23,6-, 201..2050: 201=10 40201, 2050 .. 10 .: 10.727. a , b , 4a + 4b = 52 a2+b2= 89. a b.

4(a +b) = 52,a2+b2= 89;

a +b = 13,a2+b2= 89;

b = 13 a,a2+ (13 a)2= 89;

b = 13 a,a213a + 40 = 0. : a = 5, b = 8 a = 8, b = 5. www.legionr.ru156 728 8 .: 8.728. 24-, 266.,1500 : 266 = 5 85133, 1500 .. 5 - .: 5.729. a , b , ab = 24, 2a + 2b = 22. a b.

2(a +b) = 22,ab = 24;

a +b = 11,ab = 24;

b = 11 a,a(11 a) = 24;

b = 11 a,a211a + 24 = 0. : a = 3, b = 8 a = 8, b = 3. 3.: 3.730. , .: t =8080= 1 ; C: t =AC60+BC100=6060 +50100= 1,5 ., 1 .: 1.731. 2t2+ 86 t0.2t2+ 86, 2t2 2, t2 1, 1t1. t0, - t [0, 1], 1 .: 1.732. , .: t =AB60=6060= 1 ; C: t =AC100 +BC80=50100 + 2080= 0,75 . 1 .: 1.733. 12t2+ 5 3 t 0. www.legionr.ru 734 15712t2+ 53, t2 4, t2 4, 2 t2. t0, t [0, 2], 2 .: 2.734. 20 , 0, 420=8 , 20 8=12 . 3 , 14 3 = 11 . 30 , 1330 = 10 . , - 8 .: 8.735. 22 , = 20 + 5 cos t - = 18 t [0; 12].. 22.: 4.736. x , 23x. 23x = 2 + 3 + 3, x = 12.: 12. www.legionr.ru158 738737. (. . 23) , -, .. 23., 8- ADC, 38 .: 38.738. h=0, d - : 3 0, 48d2= 0. d = 2, 5.: 2, 5.739. x y -. :

3x 7 = y,3y 11 = x. x = 4, y = 5., 5 5 5x + 5y = 45 .: 45.740. , , 3000 + 2700 + 1500 = 7200 ()., , 480 + 450 + 270 = 1200 ()., : 7200601200= 360 (/).: 360.741. :v(t)=S

(t)=30 10t. 0 /, www.legionr.ru 742 159 30 10t = 0; t = 3 c.: 3.742. 11 x , y z . 11 x 3 ,y2 2z . 11 28 . , x 3 +y2 + 2z = 28., 11x 3,3y2 2z 16 . 30 . , x 3 +3y2+ 2z 16 = 30.

x 3 +y2 + 2z = 28,x 3 + 3y2+ 2z 16 = 30;3y2y2 16 = 2; y = 18.: 18.743. 1 6 I - :20 + 1, 5 + 20 + 1, 5 + 25 + 1, 5 + 26 + 1, 5 + 20 = 117 (). II :20 + 1, 5 + 25 + 1, 5 + 24 + 1, 5 + 25 + 1, 5 + 20 = 120 ()., II , - I. 1 6 117 , - II 11760120= 58, 5 (/).: 58, 5.744. , 3 , 4t t2 3. 1t3. , 2 .: 2.745. : x , y , z . : 3x , 2y , 6z . 200 . , 3x+2y +6z = www.legionr.ru160 746= 200. , : 3x , 5y ,(6z 9) . 260 , 3x + 5y + 6z 9 = 260. :

3x + 2y + 6z = 200,3x + 5y + 6z 9 = 260;3y 9 = 60; y = 23.: 23.746. A:(42 + 13 + 87 + 23)103= 165103= 1,65105(..) B:(63 + 47 + 22 + 37)103= 169103= 1,69105(..): 4000.747. a(t0) = s

(t0),s

(t) = t324t 3,s

(t) = 3t224,s

(t) = 3, 3t224 = 3, t2= 9, t = 3, t > 0.: 3.748. : 19 + 23 + 26 + 18 + 20 + 20++20 + 20 + 32 + 27 + 35 + 40 = 300 .120000250 300 = 144000 (..) .: 144000.749. a(t0) = S

(t0), S

(t) = t24t + 3, S

(t) = 2t 4,S

(3) = 23 4 = 2 (/2).: 2.750. , 19. , . 278.: 278.751. , .S1(t) =S2(t); t2+ 6t = 4t; t2 2t = 0; t = 0, t = 2. t =0 , t = 2 S1(2) = 8 .: 8.752. a , - 2a. b, www.legionr.ru 753 1611,5b. , a +2a = b +1,5b, b = 1,2a, 1,2 .: 1,2.753. , 31. - , , 87.: 87.754. , .S1(t) =S2(t); 0,25t = 1,25t 0,25t2; t2 4t = 0; t = 0, t = 4. t = 0 , t = 4 S1(t) = 1 .: 1.755. a, - 23a. b , 56b. , 23a + a=56b + b,5a3=11b6, a = 1,1b, 1,1 , .: 1,1.756. , , 1 , : AB 45012225 675ACDB 645434520 665ADB 525710135 660 AB, - 450 .: 450.757. : 2+4tt2 5; t24t+30; www.legionr.ru162 7581t3. , 5 - 2 .: 2.758. , -: (.) () (.)I 1200 3,512003,5 342,8II 1375 3,913753,9 352,5III 1500 4,515004,5 333,3 -. 333 .: 333.759. 6t t2> 5;t26t + 5 < 0;1 < t < 5, , 5 - 4 .: 4.760. 2 x , - (3x + 106,5) . , 270 100 = 170 (). 3x + 65 = 170 x = 35., : 3510, 235 + 510 = 120 (.).: 120.761. : , -, 6 + 16 + 19 = 41 (.), : 23 + 21 + 28 = 72 (.). x - , . :41 100%72 x%, x =72100%41 175,61%., 175,61 100% = 75,61%. , 76%.: 76. www.legionr.ru 763 163762. x , , 0,4 34x . , x 100,422= x 0,8. : 34x = x 0,8;x4= 0,8; x = 3,2.3,2 , .: 3,2.763. x . 1 0,2x, 20,8x. -, , 0,2x2=0,4x (), 0,8x3 = 2,4x (). 0,4x+2,4x = 254100; 2,8x = 154,x = 55. , 55 . 0,255 ==11 (.), 55 11=44 (.). , 111 + 244 = 99 (.).: 99.764. : , -, 3+5+3+3+1+5+2+5+2+1+3 = 33 (.). 3 + 3 + 5 = 11. :33 100%11 x%, x =11100%33 33,33% 33%.: 33.765. : t =xv0. - , : h=y=gt22=gx22v02(). h 60, ABC (. . 24) : x = CB =ACtg ABC=htg 60=h3. x (), : h =gh26v20; h =6v02g;g 10 /c2=10360021000/2; v0 = 180 /. h =61802 10036002== 1,5 ().: 1,5. www.legionr.ru164 767. 24.766. , . : t1 =8,11,8= 4,5 ; t2 =6,61,5= 4,4 ; t3 =7,42= 3,7 . : t1 =6,31,4= 4,5 ; t2 =7,21,5= 4,8 . , , 3,7 + 4,5 = 8,2 .: 8,2.767. a , b , - S = ab. - P= 34, 2a + 2b = 34, a +b = 17. b = 17 a , S=a(17 a). a< 17, S66.a(17a)66, a217a+660, (a6)(a11)0, 6a11. a 11.: 11.768. , - . t1=2888=36 ; t2=2826=47 ; t3=2947=42 ;t4 =3519= 39 ; t5 =38511= 35 . , - 1, 2, 3 4, 5. 36 + 35 = 71 .: 71.769. a , b , -S =ab. , -P =32, 2a + 2b =32, a +b =16. a=16 b www.legionr.ru 770 165 , S =b(16 b). -b, 0 0;d2 = 6 (6) = 12, 12 > 0. , a1 = 6, q =66 + 8=62= 3.: 3. www.legionr.ru 781 167780. 1. SABC =12PABC r, SABC =12ACBCsin C,12PABC r =12ACBCsin C, r = 1, AC = 5, sin C = 0,6, PABC = 50,6BC, PABC = 3BC (. . 25).2. AB< AC, C< B. , C . -. 25., cos C> 0.cos C =1 sin2C =1 925=1625=45= 0,8.3. AB BC. AB2== BC2+AC22BCACcos C, AB =BC2+ 25 8BC.4. PABC=AB + BC + AC, PABC= BC2+ 25 8BC + BC + 5, PABC=3BC. a BC, a>0, a :(1)a2+ 25 8a +a + 5 = 3a a2+ 25 8a = 2a 5 a2+ 25 8a = (2a 5)2,2a 50; 3a212a = 0,a2,5;a = 4.a = 4 (1). BC = 4.: 4.781. : ABC , B = 30, O - ABC, AD , r = AO == 72.: , AEC.. (. 26).1. ABC , B = 30, A = C = 75.2. ABC ADC AC, ADC = ABC = 30. www.legionr.ru168 782. 26.3. ACD = 90.ACD , AC=12AD = 72, ADC = 30.4. AEC: EAC = 60, ECA = 75, AEC = 45. AE EC .AEsin C=ACsin E=ECsin A.AE =72sin 75sin 45= 72sin752 = 14 sin 75.EC =72sin 60sin 45=72 3 22= 73.SAEC =12AEACsin EAC, SAEC =1214 sin 75 72 32==4962sin75. d , AEC.d =abc2SAEC, a,b,c AEC.d =7214 sin 75 732 4962sin75= 14.: 14.782.OABC,OA=OB=OC=R, R (. . 27). AOB ACB-AB, AOB =2ACB =150. - AOC ABC: AC AOC=2ABC=120. OAOC = OA OCcos AOC, R2 cos 120 =3 22 www.legionr.ru 783 169. 27.R2=3 22 (2) = 2 3. AB - AOB:AB2= OA2+OB22OA OB cos AOB = R2+R22R R cos 150 == 2R21 32== 2(2 3) 2 +32= 4 3 = 1.: 1.783.. 28. ABC (. . 28) O - ABC, OA=OB=OC=R, R . AOB ACB AB, AOB = 2ACB = 90. OB OC = OBOC cos BOC, R2cos BOC =9,cos BOC =9R2(BOC ). www.legionr.ru170 785 BOC. BC2= OB2+OC22OBOCcos BOC == R2+R22R29R2= 2R218. 2R218 = (32)2, 2R2= 36, R2= 18,R =18.AOB: AB =OBsin 45, AB =18 2 = 6.: 6.785. ABC-AC, O, T, K, H- AB,BC,AC- (. . 29), ABC-r.,HC=KC, HA=AT,BT =BK. KC=AT =AH=HC=2x,BK=BT =5x,AB=BC=7x. BH= (7x)2(2x2)2=35x. -ABC12BHAC=1235x4x=65x2. ,SABC=12r(AB + BC + CA)=122518x=185x,185x = 65x2, x = 3, AB = BC = 7x = 21.. 29.: 21.786. (. . 30), - 1 : 2 : 3, ABCB= 90, A = 60, C= 30. OA =2, AC = 22; AB =2; BC =6.SABC =ABBC2=232=3. a, S=a234. SABC= 3, a2= 4, a = 2. www.legionr.ru 787 171BACO. 30. a2= 4; a = 2.: 2.787. R =12 2.AB :BC :CA = 3 : 4 : 5.BACO. 31.O (. . 31)., :3 + 4 + 5=12, 360:12=30. AOB=90, BOC=120,COA = 150. SABC = SAOB +SBOC +SAOC,SABC =12R2(sin 90 + sin150 + sin 120) ==12R21 + 12 +32=14R2(3 +3) =14(12 2)(3 + 3) =3.a. , www.legionr.ru172 789S=a234 . a234= 3 a2= 4 a = 2.: 2.789. ABD ABC ( 2 ) (. . 32).ABAD=ACAB,AB2= ADAC = 49, AB = 6. SABD =12ABADsin 30 =126412= 6.. 32.: 6.790. MN= MP, M B(. . 33).ABC , (B, M, K. 33. BK). BMP: BM=MPsin 60=323= 23,BK = 23 +33=733. BKC: KC = BKtg 60,KC =7333 = 7, AC = 2KC = 14.: 14. www.legionr.ru 792 173791. BOK (. . 34) : BO =OKsin60=2 3 23=. 34.=232 33=43 63; BM= BO +OM=43 63+ 2 3 ==43 6 + 6 333=33. BMC : MC = BMtg 60 ==333 =33= 1. AC = 2.: 2.792. , (. . 35). -. 35., SABC = 96, SBMC = 48, SBMC =12BMBCsinMBC,48 = 6297sin MBC, sin MBC =46297497;cos MBC=8197=997. BMC:MC2= BM2+BC22BMBCcos MBC, www.legionr.ru174 793MC2=144 + 388 212297 997, MC2=100, MC =10;AC = 20. ABC : SABC =12BCACsin C, sin C =24597;cos C = 1 5762597= 43597. , BC> BM. BMC = BCM. , BCM< 90; cos C =43597. BC>MC, MBC< BMC. ,MBC= 90; cos MBC=927. ABC -: AB2= BC2+AC22ACBCcos C,AB2= 497 + 400 229720 43597= 100; AB = 10.: 10.793. 1. ABC (. . 35).. 36.SABC =12ABACsin A; 96 =121020sin A, sin A =9621020= 0,96;cos A = 1 0,962= (1 0,96)(1 + 0,96) == 0,041,96 = 0,21,4 = 0,28. A, cos A = 0,28.2. ABM : BM2= AB2+AM22ABAMcos A,BM2= 100 + 100 210100,28; BM2= 144; BM= 12.: 12.794. 17AB=x;18BC =y, AB=7x, MB=4x,BN= 3y, BC = 8y (. . 37). www.legionr.ru 795 175. 37.1.SABNSMBN=12ABBNsinB12MBBNsinB=74. SMBN= SABN SAMN= SABN 9,SABNSABN 9=74; 4SABN= 7SABN 63; SABN= 21, SBMN= 12.2.SBMNSABC=12BMBNsinB12ABBCsinB;12SABC=4378; SABC = 56.: 56.795. (. 38).. 38.15AB=x,113BC =y, AB=5x, BM=3x,BN= 2y, BC = 13y. SABC =12AB BC sin B =12 5x 13ysin B ==65xy2sinB, www.legionr.ru176 796SBNM=12BM BNsin B=123x4ysinB=12 xy sinB,SABCSMBN=65xy sin B12xy sin B, SMBN= SABC 1265=1301265= 24.SAMNC = 130 24 = 106.: 106.796. (. . 39).. 39. ABC+ ADB=180,ADBBDC-, BDC + ADB=. , ABC= BDC. ABC BDC (ABC = BDC, C ). , - BCDC=ACBC, BC2=ACDC. DC=x> 0. 4 = (3 + x)x, x2+ 3x 4 = 0; x1= 4 ( y> 0),x2= 1. , DC= 1; AC=AD + DC= 4, - ABC , :AC2= AB2+BC22ABBC cos ABC. ABC=BDC, cos ABC=cos BDC=1320. -, AB=y, (y >0).: 16 =y2+ 4 4y 1320,5y213y 60 = 0, y1 = 5, y2 = 2,4 ( y> 0)., AB = 5, ABC 2 + 4 + 5 = 11.: 11.797. KOH MOP(. . 40) (KOH= MOP -;KHO = OMP ).HOOM=KOOP , 45=KOOP , MH = KP= 9; OK =45OP; 9 = PK = OK+OP=45OP+OP=95OP; www.legionr.ru 798 177. 40.OP= 5, KO = 4. , MKHP ,POKMPOHP= 1.: 1.798. ABC(. . 41)OD=12AO;. 41.BD =12BC = 65. BOD: BD2= BO2+OD2= BO2+14AO2.OE =12BO; AE =12AC = 15. AOE : AE2= AO2+OE2= AO2+ 14BO2.BD2= BO2+ 14AO2,AE2= AO2+ 14BO2;180 = BO2+ 14AO2,225 = AO2+ 14BO2;405 = (AO2+BO2)+14(AO2+BO2). ABOAO2+BO2= AB2, www.legionr.ru178 799 405 =54AB2, AB2= 324; AB = 18.: 18.799. CBA = (. . 42), CAB= 2. . 42. ABC:ACsin CBA=BCsin CAB10sin =12sin 2 sin 2sin =1210;2 sin cos sin=65 cos =35. : AC2= AB2+BC22AB BC cos . - AB = x, x > 0, - , : 100 = x2+14421235x;x2 725x + 44 = 0; 5x272x + 220 = 0; x1 = 10, x2 =225. AB = 10, ABC (AB = AC), - ACB = CBA = . , CAB+ACB+CBA = 180 4 = 180; = 45; cos =22=35. ., AB = 10. AB =225= 4,4.: 4,4.801. BO AC B1 (. . 43), BB1 - ( O ). AB=BC, BB1 -, SB1OC=SBB1C SBOC=12SABC 13SABC=16SABC.,BOB1O=SBOCSB1OC=SABC3:SABC6=2 O www.legionr.ru 802 179. 43. (BB1 , O - 2:1). ABC ABC . a ABC, r =43 , , - ABC S =a234=(23r)234= 33r2= 33 3 = 9.: 9.802. ABC AB = BC; AC = 6 (. . 44). r ; R .. 44. BC = x, x > 3, BD =x29, SABC =12AC BD. SABC =12PABCr. ACBD = PABCr; www.legionr.ru180 8036x29 = (2x+6)1;36x29= 4x2+24x+36;32x224x360 = 0;4x23x45 = 0; x1 = 3,75, x2 = 3 x > 3.SABC =12 3,75 + 3,75 + 6 1 = 6,75;R =ABBCAC4SABC=3,753,75646,75= 3,125.: 3,125.803. 1) ABB1 (. . 45): B1 = 90; AB =AB21 +BB21;AB =322+ 242= 40.. 45.2) AB = AC. , B1C = AC AB1;B1C = 40 24 = 16.3) BB1C: B1 = 90; BC =BB21 +B1C2;BC =322+ 162= 165.4) AOB: O = 90; AO =AB2BO2;AO =402852= 165.5) AOC AB1M (A , O = B1 = 90), AOAB1=OCMB1; 16524=85MB1; MB1 = 12, MC1 = 12.6) SABM=12ABMC1; SABM=124012 = 240.: 240.804.1) LP= LO +OP(. . 46). , LP= 5 + 4 = 9.2) MLP OLB (L , P= B = 90), LPLB=PMBO , PM= KP;9LB=KPBO, LB =9BOKP.3) KOP LOB (1=2 , P =B=90), www.legionr.ru 805 181,OPOB=KPLB ;4OB=KPLB , LB =OBKP4.4) 2) 3) , 9BOKP=OBKP4, KP2= 36; KP= 6.5) SKLO = SKLP SKOP=12KPLP 12KPOP==1269 1264 = 15.. 46.: 15.805. O ABC , (. . 47) AM BN A B ABC.. 47. A, B C, , -. , CAN= CBN= /2, , . , BNA= BCA=, MNB= MAB=/2. AM=MN MAN=MNA. MAN www.legionr.ru182 806MNA , , : MAN= MAC +CAN= /2 +/2,MNA = MNB +BNA = /2 +. :/2+/2 = /2+; = 2. , = 30, - + + = 180 = 2, : 3 = 150, = 50.: 50.806. ABC (. . 48).. 48.ABAC=BDDC=58AB =58AC. AC = x, x > 0, AB = BC =58x. r =SABCp, p ABC, p=AB +BC +AC2=58x + 58x +x2=98x. H AC. AB=BC, BH AC. ABH BH= AB2AH2=2564x2x24=38x. ,SABC=12BHAC=316x2. , r =SABCp=316x298x=x6. r = 2, x6= 2; x = 12.: 12.807. SANMB =12AMBNsinAON(. 49). ACB C=120, A=B=30. AM www.legionr.ru 808 183. 49.BN MAB=NBA=15, AON=30.SANMB =1255 12= 6,25.: 6,25.808. 1. ABC C = 120, BAC = ABC = 30.2. AM BN, , MAB= NBA=15, AON= 30, AOB (. . 49).3. SANMB =12AMBNsin AON AMBN=2SANMBsin AON==212,250,5= 49. , AM= BN= 7.: 7.809. CD ACB (. . 50), CBBD=ACAD=63=21., AC= 2x; AD =x, x> 0. ABC :62+ (3 +x)2= (2x)2; x22x 15 = 0. : x1 = 5; x2 = 3. , x > 0, x = 5. SACD =12x6 = 15.: 15.810. AD BAC (. . 51), ACAB=DCDB=24=12.AB =2x, AC =x, x >0. ABC: BC2= AB2+AC22ABAC cos BAC;36=4x2+x2 22x x 12.x=23.ABC. , , www.legionr.ru184 811. 50.. 51. (23)2+ 62=(43)2 , ABC- AC AB. ABC = 30.: 30.811. ABC (. . 52), BH, AC, , . BH.SABC =12ACBH = 300 =1225BH. BH = 24. BCH - : CH =BC2BH2=252242= 7., AH=AC CH= 25 7 = 18. ABH :AB =BH2+AH2=242+ 182= 30.: 30. www.legionr.ru 814 185A BCH. 52.813. OxyABC(.. 53).. 53. BH AC. SABC =12ACBH. AC , AC = 3 (4) = 7. BH , BH = 2 (1) = 3.SABC =1273 = 10,5.: 10,5.814. Oxy ABC(. . 54). AH BC, SABC =12AHBC. BC , BC = 5 2 = 3. AH , AH = 3 0 = 3.SABC =1233 = 4,5.: 4,5.815. KH . www.legionr.ru186 816xyO11C(3;5)HB(3;2)A(0;1). 54.KH = PK sinP= PK1 cos2P= 131 105169= 13 813= 8.: 8.816. 1) CLD= BCL, , (. . 55). CL C, DCL = BCL.. 55. CLD=DCL; DL=CD., AL=AB.-AB=CD, ,AL=DL, , , AL=12AD. h-ABCD, AD. SABCD=hAD,SABL=12hAL=14hAD. ,SABCD=4SABL=60. , p ABCD -: p = 2(AB +AD) = 2 32AD = 3AD = 3BC.2) BCL =12C, CBL =12B, B +C = 180. ,BCL+CBL = 90, BLC = 180BCLCBL = 90. -, BCL . , 2SBCL = BL CL = h BC == SABCD = 60, BL =60CL= 5. www.legionr.ru 817 187 BCL : BC = BL2+CL2= 13. ,p = 3BC = 39.: 39.817. 1) CLD = BCL DCL = BCL, CLD = DCL,(. . 56). , DL=CD, , , AL=AB. -. 56. CD = AB, DL = AL =12AD.2) BCL =12C, CBL =12B, BCL +CBL ==12C +12B=90 BLC=90. CL=x, (x>0), BLC : BC= 36 +x2. ,DL =CD=1236 +x2 CL + CD + DL =x +36 +x2. BCL = 18, x+36 +x2= 18,36 +x2= 18x. x = 8. , SBCL =12BLCL = 24, SABCD = 2SBCL = 48.: 48.818. 1) ABL= CBL= BLA, AB=AL. ,CD = DK. , , :LK = ADALDK = 3AB2AB = AB, (. . 57). P -. 57. BL CK. AL+DK = 2AB =23AD < AD, P . LK | BC, www.legionr.ru188 819LKP BCPk =LKBC=13. :PL=13BP,BL=23BP,BP =32BL=9. , CP=32CK = 12.2) PBC =12B, PCB =12C, PBC +PCB ==12(B +C) = 90, BPC = 90.3)SBCP=12BPCP=54,SLKP=k2 SBPC=1954=6. ,SBCKL = SBCP SLKP= 48, SBCKL =h2(BC +LK) =h2 43BC ==23SABCD ( h ). -, SABCD =32SBCKL = 72. : 72.819. ABL=CBL=BLA, AB=AL. ,CD=DK. AL +DK=2AB=43AD>AD, BL CK .P= BLCK (. .58). KL = AL+DKAD = 2ABAD == 2 23AD AD=13AD. KL =AL + DK AD=13AD. h . 58. ABCD AD. SBCLK=12h(BC + KL) =12h(BC + 13AD) =12h 43BC =23SABCD, www.legionr.ru 820 189SABCD =32SBCLK.2) , BPC = 90, PBC +PCB ==12(B + C) =90. SBCLK=12BL CKsin 90=30. SABCD =32SBCLK =3230 = 45.: 45.820. ML MEC. KL - FECD (. . 59).. 59.MECMFD. ECM FDM-, CEM DFM , CME DMF . , CME FMF -.,, EC: FD=2,ML: KM=2.-, EC=2FD, ML=2KM. C , LK=ML KM=2KM KM= KM, SFECD =EC +FD2KL = 3FDKM2.SECM=12ECML =2FD2KM2= 2FDKM. ,SFECDSECM=3FDKM4FDKM=34= 0,75.: 0,75.821. PABD = AB + AD + BD. AB,AD BD (. . 60).AD=AF+ FD. FD , BOEFOD. EBO ODF , BOE FOD , BO =OD O -. , BOE = FOD www.legionr.ru190 822. 60.. FD = BE = 1,6. , AD = 6,4 + 1,6 = 8. AB. BME AMF.MBE MAF , BME AMF ., BME AMF . - MBAM=BEAF , AM=MBAFBE=16,41,6=4., AB = AM BM= 4 1 = 3. BD. BD2= AB2+AD22AB ADcos BAD = 32+822 3 8 cos 60 = 49,BD = 7., PABD = 3 + 8 + 7 = 18.: 18.822. 1) BKP CDP( )., BKCD=PKPD=PKPK +DK=66 + 9=25. BK =25CD = 4.2) , BKP (. . 61).BPK= PDA ( AD,PC PD), BKP= CDP( - AB, CD PD), PDA = CDP(DP - D). , BPK = BKP BP= BK = 4.3) PBKP= BP +BK +PK = 4 + 4 + 6 = 14.: 14.823. 1) ABN DMN ( ) (. . 62). ,ABMD=BNMN=BM +MNMN=6 + 44=52. AB =52MD = 12,5. www.legionr.ru 824 191. 61.2) ANB =NBC(),ABN =NBC,BN B. , ANB = ABN, ,ABN , AN= AB = 12,5.3) PABN= AN +AB +BN= 12,5 + 12,5 + 10 = 35.. 62.: 35.824. cos B = cos(180A) = cos A =55(. . 63). AH1 , CB.BH1= 25cos B= 2; AH1=CH= 4. AHK KBC,AKKC=AHBC=35. AH=3x, BC=5x. - AHCH1 , AH=CH1=3x BH1 = BCCH1 = 2x = 2 x = 1, BC = 5. SABCD = BCAH1 = 54 = 20.: 20.825. CDN= DNA ( ), CDN= NDA (DN www.legionr.ru192 826. 63. D). ADN=DNA. ADN (. . 64). AD=AN= AB BN== DC BN= 33 3 =23, ADN= BMN( -), BNM= DNA (). NDA= DNA, BNA = BMN. , NBM . BN= BM=3.DCM NBM ( : M , CDM= BNM ),DCBN=DMNM . NM=BNDMDC, NM= 3. BNM MN2= BN2+BM22BNBMcos NBM,cos NBM= 12, NBM=120. NBC=180 120=60. CBN CN2= BC2+BN22BCBN cos NBC, CN2= 9, CN= 3.. 64.: 3.826. h1 = 5, h2 = 7, AB = b, AD = a (. . 65).SABCD = ah1 = bh25a = 7b, - 2(a +b) = 48. : 5a 7b = 0,a +b = 24;a = 14,b = 10. www.legionr.ru 827 193 sin =h1b=12.. 65.: 0,5.827. h1=32, h2=52 (. . 65). SABCD=ah1=bh2, 3a2=5b2, 2(a + b)=32. 3a2 = 5b2,a +b = 16. , a = 10, b = 6. sin =h1b=22tg = 1.: 1.829. x, SMBCN= 3x, SAMND= 5x, SABCD= 8x. AD=a, a> 10. . 66. MN=BC +AD2( ), BC == 2MNAD = 20a. BK = h1+h2, h1 MBCN, h2 AMND (. . 66).SMBCN=MN +BC2 h1; 3x =10 + 20 a2 h1; h1 =6x30 a.SAMND =MN +AD2 h2; 5x =10 +a2 h2; h2 =10x10 +a. www.legionr.ru194 830SABCD=10(h1 + h2); h1 + h2=8x10=4x5,6x30 a+10x10 +a=4x5;330 a +510 +a=25; a2 25a + 150 = 0; a1 = 15, a2 = 10 - a > 10. AD = 15.: 15.830. : ABCD , ,AD = 21, BC = 9, BH , BH = 8 (. . 67).: .. 67. O (. . 67). MN , O. OC =OB(), OBC-. OM BOC, . BM= MC; MC =BC2=92. , ND = AN. ND =AD2=212. MO = x, x > 0, ON= 8 x. MN -, CMO = 90, OND = 90. , CMO OND . MOC :OC2= MC2+MO2. R . R2= OC2=922+x2(1). NOD :OD2=ON2+ ND2, R2=OD2= (8 x)2+2122(2). (1) (2):922+ x2= (8 x)2+2122; 814+ x2=4414+ 64 16x + x2; www.legionr.ru 831 19516x=154; x=778=958. (1) : R2=922+7782=722564;R =858. D = 2R =854= 21,25.: 21,25.831. C CE |BD (. . 68). - AD CE. DBCE -. 68.,CE=BD=12,DE=BC=5.SABCD=SACE. SACE .AC=9,CE=12,AE=15. ACEp =9 + 12 + 152= 18. SACEp(p AC)(p CE)(p AE) ==18(18 9)(18 12)(18 15) = 318 = 54. SABCD = 54.: 54.832. O ABCD . O (. . 69) MN, - AD BC. BAD = 90, OMA. , M . , MO = 5. H - CD. NHO=90, OH=5. OHNON=2OH=10, MN= MO +ON= 15.: 15.833. 1. O AC.AO=KO, AOK.OM,OM KOA AK=2AM=2AOcos =18 cos (.. 70).2. ABCD DF CK (CKA = 90 AC). 2 = 90, DB - DF= FK. , , CK = DB. CK = 18 cos = 18 cos 45 = www.legionr.ru196 834. 69.. 70.= 92 = 16 = DB. , 2 = 90. AKC, CK = AC sin == 18 sin . BFD: 1) 2 < 90, DF= BDsin 2 = 16 sin 2;2) 2 > 90, DF= BDsin(1802) = BDsin 2 == 16 sin 2. CK = DF; 18 sin = 32 sin cos ; cos =916.3. AK = 18 916=818= 10,125.: 10,125.834. AC=12;DB=10; OAB=x, OBA=2x (.. 71).1. ACsinx. 2x=90, . 71. www.legionr.ru 835 197DB . , , ACsin x = 1222= 62 = 10 = DB., 2x = 90.2. 2x > 90, DBsin(1802x) ==DBsin 2x. 2x< 90, DBsin 2x., . 1, AC sinx =DBsin2x; 12 sin x = 10 sin 2x,sinx2 sin xcos x=1012,12 cos x=56, cos x =35, sin x =45.3. SADCB =12ACDBsinAOB =121210 sin (180(x + 2x)) = 60 sin 3x. sin3x = sin(2x +x) = sin 2xcos x + cos 2xsin x == 2 sin xcos2x+cos2xsin xsin3x = 3cos2xsinxsin3x =108125 64125=44125. SADCB = 60 44125= 21,12.: 21,12.835.. 72. MN AD (. . 72). M N BC AD .MC = CK = 2 DK = DN= 8 .BM= BP= PA = AN= x (). CH.HD = DNNH = 82 = 6. HCD: CH2= 10262= 82; CH = 8.BA |CH, BAD = 90. , BA =CH( , ). ,2x = 8; x = 4. PABCD = 4x + 2 + 10 + 8 = 20 + 16 = 36.: 36.836. 1. 73, SABD = SABO +SAOD; www.legionr.ru198 837SACD = SCOD +SAOD. SABD=SACD=HAD2, SABO + SAOD=SCOD + SAOD;. 73.SABO = SCOD.2. BOC AOD,SBOCSAOD= k2=19, k ; SAOD = 9SBOC.3. OC =13AO, SBOC =12hOC =12h 13AO ==13SAOB =136 = 2. SAOB = 3SBOC. :SABCD = SAOB +SBOC +SCOD +SAOD = 6 + 2 + 6 + 18 = 32.: 32.837. BC=x, AD=2x (. .74). BOC AOD( ), OH1OH2=BCAD=12. BC =12AD, BC APD. BCC PH = H1H2 = 3OH1. xOH12= 3, SBOC =x3OH12= 9 = SBPC.SPCOB = SBPC +SBOC = 9 + 3 = 12.: 12.839. 1) A1A2A3 A3A4A5 (. . 75) A1A3 = A3A5.A1A3A5 = A2A3A4A2A3A1A4A3A5. A2A3A4 ==180(8 2)8= 135 A2A3A1 + A4A3A5= 180 135= 45, A1A3A5 = 90. A3A5 A1A3 = A3A5 = R2. www.legionr.ru 839 199. 74.SA1A3A5 =12A1A3 A3A5 = R2; R2= 9; R = 3.. 75.2) A1OA2 =3608= 45, A2OA5 = 18045 = 135;SA2OA5 =12A2OA5Osin 135 =12R222=R224=924.3) A1A8A7 A1A2A3 A1A7 = A1A3 = R2. -, SA1A7B =12A1A7BH =12R2BH =322BH. , A1A7B A2OA5, 322BH=924; www.legionr.ru200 840BH = 1,5.: 1,5.840.. 76.1)(n 2)180n=135; 180n 360=135n; 45n=360, n=8, , (. . 76).2) A1A5A4 = A1A5A6 ( , A1A2A4A1A8A6). , A1A5A4 =12A5 =12135 = 67,5.3) A1A2A3A4 A3A4A1 +A2A1A4 == 360A1A2A3A2A3A4 = 3602135 = 90. A3A4A1 == A2A1A4 = 45. A1A4A5 = A3A4A5A3A4A1 == 13545 = 90. , A1A5 -. a8 = 2Rsin 1808, 2R = A1A5 = 4042;a8 = 4042sin 22,5.4) SA1A4A5 =12A1A5 A4A5 sin A4A5A1 ==1240424042sin22,5 sin 67,5 = 8002 sin22,5 cos 22,5 == 4002sin45 = 4002 22= 400.: 400.841. , A6OA9 3 36012= 90 (. . 77). r , . , www.legionr.ru 842 201. 77.SA6OA9 =12A6OA9O =12r2.SA1OA9 =12A1OA9Osin 120 =12r232=3r24;SA7OA9 =12A7OA9Osin 60 =12r232=3r24., SA1A7A9=SA1OA9+SA7OA9=3r22,,SA1A7A9 = 63, 3r22= 63, r2= 12 SA6OA9 = 6.: 6.842. A1A2=a, a> 0, OA1A2 ,( ) OB1 =a32(. . 78).SA1A2A3A6 = 3SA1OA2 = 3 a234;SB1B2B3B4B5B6 = 6 12SB1OB2 = 3OB1 OB2 sinB1OB2 == 3 a32a32sin60 =9a238;SB1B2B3B4B5B6SA1A2A3A6=9 a2383 a234= 1,5.: 1,5.844. APHADP, www.legionr.ru202 844. 78.PAH 90 (. . 79).. 79. APH = 30. AP=AHsin30= 4.PD=APctg 30= 43 (. ). DAC DBC , . BPC = APD ( ). -, ADP BCP.AP: PD = BP: PC. BP=APPCDP= 23. www.legionr.ru 845 203SADCSABC=12DPAC12PBAC=DPBP= 2.: 2.845. O1A = 3, O2A = 4, O1O2 = 5, O1O2A - , (. . 80).SO1O2A =12 O1O2 AH AH =2SO1O2AO1O2=2 12O1A O2AO1O2=125.. 80.AB = 2AH =245.: 4,8.846. BCBC(. . 81);BAC = . . 81.BC = AC == 9 = 9.P. = AB +BC +AC = 29 + 9 = 18 + 9. www.legionr.ru204 847 P. = 18 + 3, 9 = 3, =3. CAO == BAO =2=6. r , . AOM: sin OAM=OMAO ; sin 6=r9 r;r9 r=12; r = 3.: 3.847. 3 6 , , -,610/. 6, 6 :610=10(.). , - 3.: 3.848. 2 3 6 2, , - 1 2 100 . 5 - 25 2, , 2500 = 2,5 . , 8.: 8.849. 82, AB .. 82.ABC , AB2= AC2++BC2, AB2= 32+ 42, AB2= 25, AB = 5.: 5.850. ABC , AB2= AC2+BC2,AB2= 82+ 62= 100, AB = 10.: 10.851. : A(1; 1), B(3; 5), C(3; 2).AB(31; 51),AB(2; 4);AC(31; 21),AC(2; 1). [AB[ ==22+ 42=20, [AC[ =22+ 12=5. , [AB[[AC[ ==20 5 =100 = 10. www.legionr.ru 852 205. 83.: 10.852. M BC, :xM=xB +xC2=2 + 32=52; yM=yB +yC2=4 + 22= 3. A: xA = 0; yA = 3. AM=(xAxM)2+ (yAyM)2=0 522+ (3 3)2= 6, 5.: 6, 5.853. , 3y +x = 5, 2y x = 5 x = 1 - ABC (. . 84). . 84., .1) 3y +x = 5,2y x = 5;x = 5,y = 0;A(5; 0). www.legionr.ru206 8542)x = 1,3y +x = 5;x = 1,y = 2;C(1; 2).3)x = 1,2y x = 5;x = 1,y = 3;B(1; 3).A(5; 0)OxBCOx,SABC =12AKBC.BC = 5; OK = 6. , SABC =562= 15.: 15.854. , 3y + 8x = 6, y x = 2 y = 6, ABC (. . 85).. 85. , .1) 3y + 8x = 6,y x = 2;x = 0,y = 2;A(0; 2).2) 3y + 8x = 6,y = 6;x = 3,y = 6;B(3; 6). www.legionr.ru 855 2073)y x = 2,y = 6;x = 8,y = 6;C(8; 6). A(0; 2) Oy BC Oy, SABC =12NA BC =12811 = 44.: 44.855. . 14 , 140000 . :14000028=5000. 5 , 50000 . :50000 : 5000 = 10 (.): 10.856. n =2r A1A2, n =2324= 16.: 16.857. ABC, AB2= (1 2)2++ (3 (1))2= 25, AB = 5;AC2= (1 2)2+ (3 2)2= 10, AC =10;BC2= (2 2)2+ (1 2)2= 9, BC = 3.. 86. BK , AKKC=ABBC=53, AK =58AC.16AK2= 16 256410 =25104=1252= 62,5.: 62,5.858. ABC. AB2= (12 6)2+ www.legionr.ru208 859+ (1 7)2= 100, AB = 10;AC2= (12 12)2+ (1 5)2= 36, AC = 6;BC2= (6 12)2+ (7 5)2= 40, BC = 210.. 87. AK , BKCK=ABAC=53, CK =38BC.CK2=96440 =958=458= 5,625.: 5,625.859. 4 , 4 , 8 . 1 , 4(4 + 8) 12= 24 (2).: 24.860. 4 , 1 -, 7 . 1 , 4(1 + 7) 12= 16 (2).: 16.861. 20 = 0,2 , , (. . 88), - 2 6 : 0,2 = 60 . ( .). 88. , www.legionr.ru 862 209 0,2 32=0,13 (). - (. . 88). 53 53 : 0,13 = 50 . , 50 - 60 : 5060 = 3000 .: 3000.862. , - A(0; 0), B(3; 6) C(7; 2) (. . 89).xy623 70ABC. 89.AB2= (3 0)2+ (6 0)2= 45, AB = 35;BC2= (7 3)2+ (2 6)2= 32, BC = 42;AC2= (7 0)2+ (2 0)2= 53, AC =53. cos =AB2+AC2BC22ABAC=3335 53.sin =1 cos2 =1 10892385=12962385=3635 53, SABC =12ABACsin=1235 53 3635 53 = 18.: 18.863. , .SABCD = SABM +SBNC +SMNCD; SMNCD = MD CD = 42 = 8;SABM=12BM AM=12 6 3 = 9; SBNC =12BN NC =12 4 4 = 8;SABCD = 8 + 9 + 8 = 25.: 25.864. 9 , - www.legionr.ru210 865ANBMDC. 90. 5 . 952= 22,5 (2).: 22,5.865. (. . 91), . 91. , ABC. ABC -, O O . , SBA = 30.SABC =12663 = 183, AB =36 + 336 = 12, OB =12AB = 6.SO = OBtg 30 =63. VSABC =13SABCSO =13183 63= 36.: 36.866. , (. . 92).1) V. =13S.SO; S. =12ABBCsin ABC =1263sin 60 ==932.2) R ABC. , www.legionr.ru 867 211. 92. ACsin 60= 2R.3) ABC AC2= AB2+BC22ABBC cos 60;AC2= 36 + 9 263 12= 27; AC2= 27; AC = 33. :33sin 60=2R;3323=2R;R=3. ,- , ABC., OC = R = 3.4) SOC : SO =SC2CO2; SO =21 9 = 23.5) V. =13 932 23 = 9.: 9.867. , (. . 93).. 93. www.legionr.ru212 868V. =13SABCSO.1) SABC =12ABBCsin120 =1266 32= 93.2) BAC = BCA = 30.3) R ABC. , ABsin 30=2R;612=2R; R=6. , , ABC.4) SOC : SO =SC2CO2;SO =163 36 = 23; V. =139323 = 18.: 18.868. V. =13S. SO (. . 94).. 94.1) S. =12ACBC =1268 = 24.2)SABC=12PABC r., PABC ABC,r. , ABC.3) ABC : AB =AC2+BC2=62+ 82= 10;r. =2SABCPABC=2246 + 8 + 10= 2.4) - www.legionr.ru 869 213,., OK = r. = 2. SOK : SO = OKtg ;SO = 23 = 6.5) V . =13246 = 48.: 48.869., (.. 95), MEN.SMEN=12MNEK, EK SC ABOMPKEN. 95.MEN. , MN ABC. - MN=12AC ==122 = 1. P , - B AC AC. BCP: BP=BCsin 60;BP= 2 32= 3. SABC , O , O BP,BOOP=12. - BO =23BP=233; BK=12BP; BK=32. OSB BKE( : EKB = SOB = 90; EBK = SBO), :SOEK=BOBK; EK =SOBKBO=24 33223= 18; SMEN=12 1 18 = 9.: 9.870. S. =S.cos 60; S. =12AC BC (. . 96); S. =12 8 6 = 24; www.legionr.ru214 871. 96.S. =2412= 48.: 48.871. CM AS; BM AS (. . 97), BMC=120.,SCAB,-. 97., BMC; CBM=BCM=30. NCB. BN =x. MNBMN= BNtg 30; MN=x33. ABN AN= AB sin 60 =2x 32; SANS =12AS MN. SO . SABC , SO NSA, SANS =12ANSO; ASMN= ANSO =ANSOMN=x336x33= 96. www.legionr.ru 872 215 ASO : AO =AS2SO2== 816 96 = 123. O CAB, NA =32AO =32123 = 183. AB =ANsin 60=18323= 36.: 36.872. L SB (. . 98). LK |AS,. 98. AS LC KLC = 60.1)KL SAB,CK- ABC. ,AKKCAKKL( ). , AK KLC, AK LC. - AK AS CL. AK =3. CK ABC, AB = 2AK = 23. CK ABC, CK =AC2AK2=12 3 = 3. - SABC =12CKAB =12323 = 33.2) KLC KL = CKtg KLC = 3 tg 60 == 3 33=3. KL ASB, AS = 23. V. ==13SABCAS =133323 = 6.: 6.874. SAB (. . 99). SB AB[[CD. . SAB[[CD, , CD SB www.legionr.ru216 875. 99.CD SAB, CD SAB. N CD SAB. M AB NH - SMN, NH. . OB =12DB =12 2AB = 42. SO , SOB:SB= SO2+OB2= 9 + 32= 41. SMB: SM= SB2MB2= 41 16=5. SMN. SSMN=12SO MN=1238 = 12, SSMN=12SMNH =125NH =52NH. , 12 =52NH, NH =245= 4,8.: 4,8.875. PN | BD(. . 100), (PSN) | BD; OK (PSN);OK . AO =12AC; AC = AB2 = 22; AO =2,MO =22, PN ABD. SMO : www.legionr.ru 876 217. 100.SM=S02+MO2; SM=4924+ 24= 5; SSMO =12SMKO SSMO =12MOSO, SMKO == MOSO =MOSOSM; KO =272225=710= 0,7.: 0,7.876. SMO=SNO=SPO(. . 101), O . SABC=12ACBC; SABC=12PABCr.,. 101.r.. ACBC = PABCr.;r. =ACBCPABC. ABC : AB =AC2+BC2; AB ==144 + 25 =13; r.=12530=2; MO=2. SMO: www.legionr.ru218 877SO = MOtg 30; SO =233; 23h = 23 233= 4.: 4.877. AS = BS = CS (. . 102), AO = BO = CO = R, (R. 102. ABC). SBO :BO= BS2SO2= 352152 5=10. :ABsin ACB= 2R;AB = 2Rsin30; AB = 210 12= 10.: 10.878. SABCD (. . 103), O . 103. , SO . M - www.legionr.ru 879 219BC, OMBC. OH SOM.OMBCSOBC,BC. SOM BCOH.OHBCOHSM, OH. SBC. -,OHOSBC,OH= 5.ABCD,10, O,M,OM=5. OHM:HM=OM2OH2=25 5 = 25; cos HMO =HMOM=255. SOM: SM=OMcos OMS= 5 525=552. SMB: SB =BM2+SM2=25 + 1254=2254=152.: 7,5.879. , H. 104. DBC (. . 104). H - MN | BC. M MF | CD. MFN . d D MFN.1) ABC AK; AK = ACsin 60 =63 32= 9. DABC , H ABC . , AH =23AK =239 = 6.2) CAB MAN k =23. MC =13AC,MN=23BC=2363=43; CAD MAFc- www.legionr.ru220 880 k=23. , FD=13AD. FO |DH,O ABC. HAD OAFc k=23,, OH =13AH = 6 13= 2, OF=23DH =83.3) DH .ABC, OF .ABC, OF AH. FOH: FH =FO2+OH2=832+ 22=103.4) FP |AH, AH (MDN), FP (MDN), -, FP- FMND.5) DFMN: V=13SFMNd, - V=13SMDNFP, FP= OH = 2,d =SMDNFPSFMN=434234310= 2,4.: 2,4.880. AB = x, x > 0. , AC = x2, HC =x22. SHC :SC =SH2+HC2=100 +x22(. . 105).NL = MK =12100 +x22( NL | SC LC =12BC). KL = AB = x,MN=12KL =12x. MNLK (. . 106)., KT=KL MN2=x4. KMT:MT=KM2KT2=200 +x28x216=400 +x24.SKMNL =12(MN +KL)MT=12 32x 400 +x24. www.legionr.ru 881 221. 105.. 106. SKMNL = 629, 3x400 +x216= 629;x400 +x2= 2529; 400x2+x4= 210 29; x4+ 24 52x2210 29 = 0.x2=t, t >0, t2+ 24 52t 210 29=0, , t > 0, t = 400 + 24 3112= 64. x2= 64, x = 8, , 8.: 8.881. OABC - ABC , (. . 107). , ABC . - a ABC, AB = AC = BC = a, a > 0.SABC =12ABAC sin ABC =12a2sin 60 =3a24. M AC, BM ABC BM= AC sin 60 =3a2. H ABC, BH=23BM=3a3,DH, DBH-DB DBH=. DHB:DHBH=tg;DH=tg3a3. sin = 1 cos2=223, tg = 22. www.legionr.ru222 882. 107. V=13DHSABC =a312tg =a326= 163 a = 26. K , M DB MKB MKBM= sin MK = BM sin = sin 3a2=22332 26 = 4. , - 4.: 4.882. ABCD - ABC , (. . 108). , ABC . - a ABC, AB = AC = BC = a, (a > 0).SABC =12ABAC sin ABC =12a2sin 60 =3a24. M AC, BM ABC BM=AC sin60=3a2. H ABC, DH , MH=13BM=3a6. DM - www.legionr.ru 883 223. 108. , HDM= . HDM :DH = ctgMH = ctg3a6=a36ctg . ctg. ,sin =1 cos2 =56, ctg =15. , V =13DHSABC=a324ctg=a3245, v=403,a3245= 403, a3= 96015, a = 415., DH =36 415 15= 2.: 2.883. ABCD, - (. . 109). A1,B1C1 -. 109. AD, BDCD,. www.legionr.ru224 884A1DA1A=13,B1DB1B=14,C1DC1C=23. HH1 , A A1 - BCD. , ADH A1DH1. -,AHA1H1=ADA1D=3 + 11= 4. BDC B1DC1 D, -, , :SBDCSB1DC1=BDCDB1DC1D=(1 + 4)(2 + 3)12=252.,VABCD=13AHSBCDVA1B1C1D=13A1H1SB1C1D. ,VABCDVA1B1C1D= 4 252= 50. , VABCDVA1B1C1DVA1B1C1D==VABCDVA1B1C1D11 = 49.: 49.884. 1) L BC (. . 110). -ASLBC,ALSL.. 110.,KLBC.-, K BC KL.2) ABC , AL =32 AC = 33. www.legionr.ru 885 225O ABC,H , K ABC. - AO=23AL=23. HAO , AH =3. , HL = AL AH = 23.3) KAH. , KH=AHtg . cos =13, tg =1cos2 1 = 22. , KH = 26.4) KHL. - KL =KH2+HL2= 6.: 6.885. 1) L AC (. . 111). -BSLAC,BLSL., KL AC. , K AC KL.. 111.2) x,x>0 ABC. -ABC,BL=32BC=3x2.O ABC, H , - K ABC. BO =23BL =3x3. H BO , BH =3x6. HL = BL BH =3x3. www.legionr.ru226 8863) KBH. , KH=BHtg . cos =13, tg =1cos2 1 = 22. , KH =6x3.4) KHL. - KL =KH2+HL2=6x32+3x32= x = 12.: 12.886. 1) L BC (. . 112). -ASLBC,ALSL.,KLBC.-, KBC KL.. 112.2) ABC , AL =32 AC =732.O ABC,H ,KABC.-AO=23 AL=733.HAO 1: 3, O, AH=34AO=734. HL = AL AH =732.3) KAH. , KH = AHtg 30 =74.4) KHL. - www.legionr.ru 887 227 KL =KH2+HL2=72. SKBC =12KLBC ==494= 12,25.: 12,25.887. 1) L BC (. . 113). -ASLBC,ALSL.,KLBC.-, KBC K BC KL.. 113.2) AB = AC = BC = x, x > 0. ABC -, AL =32 AC=3x2. O ABC,H , K ABC. AO =23AL =3x3. H AO 1: 3, O, AH =34AO =3x4; HL = AL AH =3x4.3) KAH. , SAL = SBO = 30. , KH = AHtg 30 =x4.4) KHL. - KL =KH2+HL2=x2. SKBC=x24= 36. - www.legionr.ru228 888. 114. , x = 12. , KL = 6.: 6.888. AH(. . 114) BC. SA ABC, AH SA. AH AS BC. AH . SAB :AB =SB2SA2= 3; AH = AB sinABH = AB sin(180150) ==AB2= 1,5.: 1,5.889. 1) . A ACD CD, K B , L K CD, (. . 115).=KAB, sin =BKAB .,KL | AD,,,. 115.KLBCD, KLB=90. DL | AK AK BK, DL BK, , - www.legionr.ru 890 229, DL BL.2) , KL=AD=13, BKL ABD :BK =KL2+BL2=1 + 3BL23, AB =AD2+BD2=73. BL BDC BDC :2SBDC = BLCD = BCDH, DH BC. BD = CD, DH BDC, - DH =BD2BH2= 1. , :BL =BCDHCD=152, BK =1 + 3BL23=723,sin =BKAB=12, =6, 30.: 30.890. SB (. . 116), SD BD, , SDB SD. -1cos2SDB=1 + tg2SDB=2516,- cos SDB=45. SBD:BD=SDcos SDB=12,SB= 152122=9. CBSABS:BC =SC2SB2=15292= 12; AB =SA2SB2==15292=12. AB=BC. , - 12. 48.: 48.891. SD(. . 116) SDB, SB . SB SAB: SB =AS2AB2== 152122= 9. SC AB SCD, AB | CD, SCD : SC CD, . SD:SD2= SC2+DC22SCDCcos SCD = 152+122215120,4 = 152,SD = 15. SBD www.legionr.ru230 892. 116.BD =SD2SB2=15292= 12, , SDB.tg SDB =SBDB=912= 0,75.: 0,75.892. AB| CD(..116),SCAB-SCD.ABCD,CD=AB=16. ABSAS=20,AB=16, SB=AS2AB2=202162=12. SD-SDB, SB-. tg SDB =0,75, SB =12, BD=SBtg SDB=16,SD=SB2+DB2=122+ 162=20. SCD SCD. :SD2= SC2+DC22SCDCcos SCD;202+ 16222016cos SCD = 202;cos SCD =25= 0,4.: 0,4.893. , - B ABC SABC. KSPASC(.. 117). , BK SP, tg KBP=233. SO www.legionr.ru 894 231 SABC, S ABC. VSABC=13SABCSO. SABC , BP ABC BP=BAsin60=6. -SABC=12BP AC=123. BP ABC, OP=13BP= 2. BKPSOP( ). ,OSP= KBP. tg OSP=OPSO=233. SO = 3,, , VSABC =13123 3 = 12.. 117.: 12.894. , -BABCSABC. K SPASC(.. 117). ABC , 33, BP=92,OP=13BP=32(BP ; O ABC),BO=BP OP =3. KBP =. - tg =38. sin =373; cos =873. BKP OSPKPO PKB=POS=90. www.legionr.ru232 895 . SOBK=OPKP ,SO=BKOPKP. BKP PK = BP sin =92 373=27273; BK = BP cos ==92 873=3673. SO=36327373227=4. ,SO=4;BO = 3. BOS SB =BO2+SO2= 5.: 5.895. 1) SM SAB. O - SAB, SO: OM=2: 1 (. 118). SOK SMB,. 118.SO: OM=SK: SBS.LKAB=SKSB=23 LK =23AB = 2.2) ABC , CM ABC. -, CM=AB 32=332. SMB SM=SB2BM2==3215; OM=13SM=152. SABCD -, H ABC. ABC MH =13CM==32, cos SMH=MHSM=135. COM www.legionr.ru 896 233 CO =OM2+CM22OMCMcos OMC = 3.3) SCLK =12COLK = 3.: 3.89