-
. , .
2014
(e-book) CD-ROM (ISBN 978-960-16-4244-4) . Copyright . ( ), , ,
2004 2014. , .
http://www.patakis.gr/viewshopproduct.aspx?id=601807
-
(. 2121/1993 )
.
(, ) ,
, , , ,
.
-
. , . ,
3
2o
474 8 477 9 478 10 480 11
481 12 483 13 484 14 485 15
486 16 487 17 488 18 489 19
490 20 491 21 492 22 493 23
495 24 496 25 497 26 498 27
499 28 503 29 504 30 505 31
506 32 507 33 508 34 509 35
510 36 936 37 938 38 944 39
947 40 950 41 952 42 955 43
991 44 996 45 999 46 1003 47
1005 48 1007 49 1009 50 1015 51
1024 52 1032 53 1039 54 1042 55
1050 56 1055 57 1057 58 1062 59
1064 60 1067 61 1070 62 1074 63
1077 64 1080 65 1082 66 1086 67
1088 68 1089 69 1090 70 1091 71
1092 72 1093 73 1096 74 1097 75
1100 76 1101 77 1102 78 1273 79
1275 80 1276 81 1277 82 1278 83
1281 84 1282 85 1283 86 1287 87
1288 88 1293 89 1297 90 1298 91
1300 92 1301 93 1302 94 1305 95
1506 96 1509 97 1512 98 1513 99
1520 100 1529 101 1532 102 1533 103
1537 104 1541 105 1542 106 1544 107
1553 108 2212 109 2702 110 3378 111
3379 113 3380 115 3381 116 3382 117
3383 118 3384 119 3828 120 3839 121
3847 122 3852 123 3857 124 3863 125
3870 126 3874 127 3878 128 3884 129
4288 130 4290 131 4295 132 4299 133
4300 134 4301 135 4302 136 4303 137
4304 138 4305 139 4306 140 4308 141
4309 142 4310 143 4311 144 4312 145
4313 146 4314 147 4315 148 4316 149
4317 150 4318 151 4319 152 7518 153
7519 154 7520 155 7521 156 8173 157
-
. , . ,
4
4o
1868 159 1874 160 1880 162 1890 163
1936 164 1955 166 1963 168 2046 169
2047 171 2052 172 2055 173 2064 174
2073 175 2080 176 2081 177 2083 178
2084 179 2220 180 2226 182 2229 184
2234 186 2238 187 2244 188 2255 189
2273 190 2287 191 2301 192 2302 193
2323 194 2332 195 2336 196 2338 198
2339 199 2340 201 4542 203 4545 204
4548 205 4551 206 4558 207 4575 208
4607 209 4629 211 4647 213 4654 215
4656 216 4657 217 4659 219 4660 220
4663 221 4665 222 4667 223 4671 224
4679 225 4680 226 4681 227 4682 228
4819 229 4833 230 4835 231 4836 232
4853 233 4857 235 4858 236 4859 237
4861 238 4862 240 4886 241 4903 243
4912 244 4925 245 4946 246 4952 247
4957 248 4962 249 4970 250 4975 251
4992 252 5275 254 5285 255 5316 256
5317 257 5322 258 5879 259 5882 261
5884 263 5885 264 6143 265 6144 267
6146 269 6223 271 6224 272 6226 274
6227 276 6228 277 6229 278 6231 280
6678 282 6859 283 7263 284 7502 285
7503 287 7504 288 7506 289 7510 290
7511 292 7512 294 7514 295 7515 296
7516 297 7517 298 7677 299 7684 300
7745 301 7784 302 7791 304 7940 305
7958 306 7974 308 8170 309 8217 311
8443 313 8445 314 8448 315 8451 317
8453 318 8455 319 8458 320 10774 321
10775 323
-
. , . ,
5
2
2
497 499 1003 1102 1287 1.2 4
1506 1520 3383 3384 3878
2.1 5 1070 1080 3874
486 487 506 1092 1541 2.2 6
3852 3870 4299 7519 7520
504 509 996 1009 1089 2.3 7
1091 2702 3884
936 938 944 947 950
952 955 1276 1300 4311 2.4 8
4314 4316 8173
3.1 9 485 507 1055 3382 4302
481 483 493 496 1005
1007 1093 1097 1275 1281
1282 1298 1509 1533 3839
3847 3857 3863 4308 4309
3.3 11
4310 4313 4317 7518
489 491 503 505 991
1039 1062 1074 1077 1273
1305 4290 4295 4305 4306 4.1 12
4318 7521
478 484 490 498 1067
1277 1278 1288 1297 1512 4.2 13
1544 3380
474 480 508 1015 1050
1057 1064 1086 1101 1301
1513 4300 4301 4303 4304 5.2 16
4312 4319
495 1032 1088 1100 3828 5.3 17
4288 4315
488 510 999 1042 1082 6.1 19
1302 1532 1537
477 492 1024 1090 1542 6.2 20
1553 3381
1096 1283 1293 1529 2212 6.3 21
3378 3379
-
. , . ,
6
4
4
1868 1936 2064 2073 2080 1.2 4
6144
2.3 2301
3.1 2302
1955 2332 4551 4558 4654
4659 4665 4667 4857 4903
4957 4962 4970 4975 4992
5317 6223 6224 6231 7510
3.3 11
7515 7516 7940
1890 2081 2238 2287 4833
4835 4946 4952 7263 7791 4.1 12
8443 8453
1874 2055 2244 2255 2273
2336 4542 4548 4607 4663
4680 4681 4682 4819 4834
4836 4853 4859 5285 5322
5884 5885 6226 6227 7677
4.2 13
7684 7958 7974 8445 8455
2047 2083 2323 4671 4858
4925 6143 7503 7504 7514 5.2 16
8458 10775
5.3 17 2340 4629 6678 6859 8170
2052 2084 2220 2226 2229
2234 4545 4575 4861 4862
5879 6228 7506 7511 7512 6.1 19
7517 7745 8217
1963 2338 4656 4660 4679
4886 4912 5275 5882 6146 6.2 20
8448 8451
1880 2046 2339 4647 4657 6.3 21
6229 7502 7784 10774
1.2 326 2.1 330
2.2 331 2.3 333
2.4 335 3.1 337
3.3 338 4.1 347
4.2 352 5.2 362
5.3 367 6.1 370
6.2 377 6.3 383
-
2014
2
-
. , . ,
8 8
, 16
. . .
GI_A_ALG_2_474 ( ) : 1, 3, 5, 7,
) ( ) -
. ( 15)
)
. ( 10)
: 5.2 :
) 2 1 3 2 4 3 2 = = = ,
*
, ,N
= 2 1 1= .
* 1 ( 1) 1 ( 1) 2 1 2 2 2 1, = + = + = + = N .
100 2 100 1 200 1 199= = = ,
199.
) *
, ,N :
( ) ( )1 2 *
2 1 2 1 1 2 2 2 2S ,
2 2 2
+ + + = = = = N ,
.
-
. , . ,
9 9
, 11, 19,
20
. . .
GI_A_ALG_2_477
f, 2x 5x 6
f (x)x 3
+=
.
) f. ( 7)
) f. ( 9)
) f xx
yy . ( 9)
: 3.3, 6.1, 6.2 :
) x 3 0 x 3, fA {3} ( , 3) (3, )= = +R .
) 2x 5x 6 + ( )2 5 4 1 6 1= =
( )5 1 6
35 1 5 1 2 2
x5 1 42 2
22 2
+ = = = = =
= =
, 2x 5x 6 (x 2)(x 3) + = .
x {3} R 2x 5x 6 (x 2)(x 3)
f (x) x 2x 3 x 3
+ = = =
.
) f
C yy x = 0, :
f(0) = 0 2 = 2, (0, 2).
fC xx y = 0, x 3:
f(x) = 0 x 2 = 0 x = 2 (), (2, 0).
-
. , . ,
10 10
, 11, 13
. . .
GI_A_ALG_2_478
: ( )2 2x x 1 0 + + = (1), . ) , (1)
. ( 12)
) : 2S P 2 0 , S P - (1). ( 13)
: 3.3, 4.2 :
) ( )2 2x x 1 + + ( ) ( )2 2 2 2 2 4 1 4 4 4 3 4 4= + = + = +
.
, 0,
23 4 4 0 + ().
23 4 4 + ( ) ( )2 4 4 3 4 64= = ,
( )
( )
4 8 122
4 64 4 8 6 6
4 8 4 22 3 6
6 6 3
+ = = = = = = =
:
+2
2
3
23 4 4 + +
2
() 2 3
.
) 2
2,3
, Vieta 1 2x , x
(1), :
1 2
S x x
1
= + = =
22
1 2
1P x x 1
1
+ = = = + .
: 2S P 2 0 2 2 ( 1) 2 0 + 2 2 1 2 0 +
2 1 1 1 .
1 2
2,3
, [2, 1].
+2 1
2
3
-
. , . ,
11 11
, 16
. . .
GI_A_ALG_2_480
. 7 36
300.
) ; -
. ( 12)
) ; ( 13)
: 5.2 :
)
,
*
, ,N 10, =
1 .
) ( ) ( ) * 1 1 1 1 , ,= + = + N 10. 7 = 36, :
( )1 7 1 36+ = 1 6 36+ = ().
*
, ,N 10, :
( ) ( )1 1 *
2 1 2 1 S , ,
2 2
+ + = = N 10.
:
10S 300= ( )12 10 1
10 3002
+ = 15(2 9) 300+ =
12 9 60+ = (). (), () :
1
1
6 36
2 9 60
+ =
+ = 1
1
2 12 72
2 9 60
=
+ =
1
12 9 72 60
2 9 60
+ = +
+ =
1
3 12
2 60 9
=
=
1
4
2 60 9 4
=
=
1
4
2 24
=
=
1
4
12
=
=.
:
12, 16, 20, 24, 28, 32, 36, 40, 44, 48.
-
. , . ,
12 12
, 11
. . .
GI_A_ALG_2_481
( )2x 2x 4 1 0 + = , . ) . ( 8)
) . ( 8)
) 1 2x , x ,
: 1 2 1 2x x x x+ = . ( 9)
: 3.3 :
) ( )2x 2x 4 1 +
( ) ( ) ( ) ( )2 22 2 2 4 1 4 1 4 16 16 4 4 4 4 2= = + = + = . )
( )2 2 0 0 ,
( )2x 2x 4 1 0 + = . ) Vieta 1 2x , x
( )2x 2x 4 1 0 + = , :
1 2
2S x x 2
1
= + = = 1 2
4( 1)P x x 4 4
1
--= = = .
:
x x x x1 2 1 2+ =+ =+ =+ = 2 = 4 4 2 4 = 4 2 = 4 = 2.
-
. , . ,
13 13
, 7, 9, 11
. . .
GI_A_ALG_2_483 ) 2x 1 3 = . ( 12)
) , < (),
2x x 3 0+ + = . ( 13)
: 2.3, 3.1, 3.3 :
) 2x 1 3 = 2x 1 = 3 2x 1 = 3 2x = 1 + 3 2x = 1 3
2x = 4 2x = 2 x = 2 x = 1. ) , 2x 1 3 = , 2 1.
, < , = 1 = 2. 2x x 3 0+ + = 2x 2x 3 0 + + = ().
2x 2x 3 + + ( )2 2 4 1 3 16= =
2 4 21
2 16 2 4 2 2x
2 4 62 ( 1) 23
2 2
+ = = = = = = =
.
() x = 1 x = 3.
-
. , . ,
14 14
, 12, 13
. . .
GI_A_ALG_2_484 ) : 2x 5 3 22x x 1 0 . ( 16)
) ). ( 9)
: 4.1, 4.2 :
) 2x 5 3 3 2x 5 3 5 3 2x 3 + 5 2 2x 8
1 x 4. 22x x 1 2 ( 1) 4 2 ( 1) 9= = ,
1 3 2 1
( 1) 9 1 3 4 4 2x
1 3 42 2 41
4 4
= = = = =
+ = =
:
+1
2 1
x
22x x 1 + +
22x x 1 0 [ )1x , 1,2
+ .
) :
+1
2 1 4
x[1, 4].
-
. , . ,
15 15
, 9
. . .
GI_A_ALG_2_485
2x x 1 = + , . ) :
( )( ) ( 1 x 1 , 1) = + . ( 8) )
, . ( 8)
)
; . ( 9)
: 3.1 :
) : 2x x 1 = + 2x x 1 = ( )( ) ( 1 x 1 , 1) = + .
1 0 1 ( )( ) ( 1 x 1)1 = +
( )( ) 1 1
x 1 1
+= = +
.
= 1 ( )( ) ( 1 x 1)1 = + : (1 1)x = (1 1)(1 + 1) 0x = 02 0x = 0,
.
) , ( ) 1 x 1 ( ) ( ) ,1 , = + - 1 x 1= + .
) , ( ) 1 x 1 ( ) ( ) ,1 , = + - = 1.
-
. , . ,
16 16
, 6
. . .
GI_A_ALG_2_486 0 < < 1, :
) : 3 < . ( 13) ) :
3 10, ,1, ,
. ( 12)
: 2.2 :
) 0 < < 1, > 0
+ 1 > 0 + 1 + 1 > 1, + 1 > 0. , 0 < < 1 0 1 <
1 < 1 1 1 < 1 < 0, 1 < 0.
3 2 ( 1) ( 1)( 1) 0 = = + < , 3 < .
) 3 10 1
< < < < . :
30 < , > 0, 3 < , () , < 1, ,
1
1
< , < 1 1 1
1>
11
> .
-
. , . ,
17 17
, 5, 6
. . .
GI_A_ALG_2_487 ) x, y :
( ) ( )2 2 2 2x 1 y 3 x y 2x 6y 10 + + = + + + . ( 12) ) x, y :
2 2x y 2x 6y 10 0+ + + = . ( 13)
: 2.1, 2.2 :
) x, y :
( ) ( )2 2 2 2 2 2x 1 y 3 x 2x 1 y 6y 9 x y 2x 6y 10 + + = + + +
+ = + + + . ) () :
2 2x y 2x 6y 10 0+ + + = ( ) ( )2 2x 1 y 3 0 + + = x 1 = 0 y + 3
= 0 x = 1 y = 3.
-
. , . ,
18 18
, 11, 19
. . .
GI_A_ALG_2_488
f, 2
2
2x 5x 3f (x)
x 1
+=
.
) A. ( 5)
) 22x 5x 3 + . ( 10)
) xA : ( ) 2x 3f xx 1
=
+. ( 10)
: 3.3, 6.1 :
) 2 2x 1 0 x 1 x 1 x 1 . { }A 1,1= R .
) 22x 5x 3 + ( )2 5 4 2 3 25 24 1= = =
( )
5 1 6 3
5 1 5 1 4 4 2x
5 1 42 2 41
4 4
+ = = = = =
= =
.
23 3
2x 5x 3 2 x (x 1) 2 x 2 (x 1) (2x 3)(x 1)2 2
+ = = =
.
) { }x A 1,1 = R : ( )
( )( )
2
2
(2x 3) x 12x 5x 3 2x 3f (x)
x 1 x 1 x 1 x 1
+ = = =
+ +.
-
. , . ,
19 19
, 12
. . .
GI_A_ALG_2_489 ) x 5 2 < . ( 8)
) 2 3x 5 > . ( 8)
)
. ,
. ( 9)
: 4.1 :
) x 5 2 < 2 < x 5 < 2 5 2 < x < 2 + 5 3 < x
< 7.
) 2 3x 5 > 2 3x < 5 2 3x > 5 3x < 5 2 3x > 5
2
3x < 7 3x > 3 7
x3
> x < 1.
) :
+1
7
33 7
x(3, 7) 3 < x < 7.
-
. , . ,
20 20
, 6, 8, 11,
13
. . .
GI_A_ALG_2_490 22x 3x 1 + . ) . ( 10)
) x : 22x 3x 1 0 + < . ( 5)
) 3
2
1
2 :
22x 3x 1 0 + < . ( 10)
: 2.2, 2.4, 3.3, 4.2 :
) 22x 3x 1 + ( )2 3 4 2 1 1= =
( )3 1 4
13 1 3 1 4 4
x3 1 2 12 2 4
4 4 2
+ = = = = =
= =
.
) :
+1
2 1x
22x 3x 1 + + +
22x 3x 1 0 + < 1
x ,12
.
) 1 3
12 2< < 1 3 2< <
22 21 3 2< < 1 < 3 < 4,
, .
, 1 1 2 2
22 2 2
= =
1 11
2 2< <
1 21
2 2< < 1 2 2< <
22 21 2 2< < 1 < 2 < 4,
, .
3
2
1
2 22x 3x 1 0 + < .
-
. , . ,
21 21
, 12
. . .
GI_A_ALG_2_491
: 3x 1 < x + 9 x 1
2 x2 2
+ .
) . ( 15)
) . ( 10)
: 4.1 :
) :
3x 1 < x + 9 3x x < 1 + 9 2x < 10 x < 5. :
x 12 x
2 2 +
x 12 2 2 2 x 2
2 2 + 4 x 2x + 1
x 2x 1 4 3x 3 x 1. ) :
+1 5
x[1, 5) 1 x < 5.
-
. , . ,
22 22
, 19, 20
. . .
GI_A_ALG_2_492
( ) 2f x x 2x 15, x= + . ) f(1) + f(0) + f(1). ( 10) ) f .
( 15)
: 6.1, 6.2 :
) :
( ) 2f 1 2 1 15 1 2 15 121 = + = + = , ( ) 2f 0 0 2 0 15 15= + =
( ) 2f 1 ( 1) 2 ( 1) 15 1 2 15 16 = + = = .
:
f(1) + f(0) + f(1) = 16 + (15) + (12) = 43. )
fC yy x = 0, :
f(0) = 15, (0, 15).
fC xx y = 0, :
f(x) = 0 2x 2x 15 0+ = ().
2x 2x 15+ ( )2 = 2 4 1 15 64 =
2 8 63
2 64 2 8 2 2x
2 8 102 1 25
2 2
+ = = = = =
= =
.
() x = 3 x = 5,
fC xx (3, 0) (5, 0).
-
. , . ,
23 23
, 9, 11
. . .
GI_A_ALG_2_493
) | x 2 3| = . ( 10) )
) . ( 15)
: 3.1, 3.3 :
) x 2 3 x 2 3 x 2 3 x 2 3 x 2 3| | = = = = + = .
) 1 2x 2 3, x 2 3= + = () .
1 2x , x 2x Sx P 0 + = ,
:
1 2 2 3S x 2 4x 3+ += + ==
( )( )222
1 2 3 2 3 2 3 4 1P 3x x + = = = == .
1 2x , x 2x 4x 1 0 + = .
-
. , . ,
24 24
, 17
. . .
GI_A_ALG_2_495 ( ) , : 3 1= 5 4= . ) . ( 13)
) - : 3 2
= . ( 12)
: 5.3 :
) *
, ,N ,
1 * 1 ,
= N . : 4
2 1 2
3 1 2
1 2245 1111 2
1
4 2 2 1 1 4
1 1 4 2 1 1 4
4 1
== = = = =
= = === =
,
= 2 . ) :
1 1 2 1 2 1 3
1
*
1 2 2 2 2 2
4, + = = = = = N .
-
. , . ,
25 25
, 11
. . .
GI_A_ALG_2_496
( )2x 2x 4 1 0+ + = . ) . ( 8)
) . ( 8)
) 1 2x , x ,
: ( )21 2 1 2xx x x 5 0+ + =+ . ( 9)
: 3.3 :
) ( )2x 2x 4 1+ + ( ) ( )2 2 2 2 2 4 4 1 4 16 16 4( 4 4) 4( 2)=
= + = + = .
) 2( 2) 0 0 , ( )2x 2x 4 1 0+ + = .
) 1 2x , x ( )2x 2x 4 1 0+ + = , Vieta :
1 2S x x2
21
= == + 1 24( 1)
P x x 4 41
= = = .
:
( )21 2 1 2xx x x 5 0+ + =+ ( )2
4 42 5 0 + + = 2 44 1 0+ + =
2(2 +1) 0= 2 +1 0= 2 = 1- 1
=2
- .
-
. , . ,
26 26
, 3, 4
. . .
GI_A_ALG_2_497 .
3 : (), (), (), 2 -
: () ().
.
) . ( 10)
) :
: .
: .
: . ( 15)
: 1.1, 1.2 :
) :
{ } = , , , , , () = 6.
) :
= {, , , }, () = 4 N(A) 4 2
P(A)N() 6 3
= = = ,
= {, , }, () = 3 N() 3 1
P()N() 6 2
= = = .
, , -
, :
= {, }, () = 2 N() 2 1
P()N() 6 3
= = = .
-
. , . ,
27 27
, 9, 13
. . .
GI_A_ALG_2_498
) : | x 1| | x 1| 4 2
3 5 3
+ + + = . ( 9)
) N : 2x 2x 3 0 + + . ( 9) ) ()
() . ( 7)
: 3.1, 4.2 :
) | x 1| | x 1| 4 2
3 5 3
+ + + =
| x 1| | x 1| 4 215 15 15
3 5 3
+ + + =
5 | x 1| 3(| x 1| 4) 5 2+ + + = 5 | x 1| 3 | x 1| 12 10+ + = 5 |
x 1| 3 | x 1| 12 10+ + = + 2 | x 1| 22+ = | x 1| 11+ = x 1 11+ = x
1 11+ = x 11 1= x 11 1= x 10= x 12= .
) 2x 2x 3 + + 2 2 4 ( 1) 3 16= = ,
2 4 21
2 16 2 4 2 2x
2 4 62( 1) 23
2 2
+ = = = = = = =
:
+13x
2x 2x 3 + + +
:
2x 2x 3 0 + + x ( , 1] [3, ) + . ) 10 [3, ) + 12 ( , 1] , -
() () .
-
. , . ,
28 28
, 3, 4
. . .
GI_A_ALG_2_499 , 25% , 30%
15%
. . :
:
: ,
) :
i) ii) iii) B A iv) ( 12) ) :
i)
ii) . ( 13)
: 1.1, 1.2 :
) i) = . ii) = . iii) B A = .
iv) = .
) :
( ) ( ) ( ) A 25% 0, 25, B 30% 0,3, A B 15% 0,15= = = = = = . i)
( ) ( ) ( ) B A B B A 0,3 0,15 0,15 15% = = = = . ii) , ( ) ( ) ( )
( ) A B A B 0, 25 0,3 0,15 0, 4 40% = + = + = = .
:
( ) = , ( )( ) ( ) 1 1 0,4 0,6 60% = = = = .
-
. , . ,
29 29
, 12
. . .
GI_A_ALG_2_503
) : 1
x 42
< . ( 9)
) : | x 5 | 3+ . ( 9) ) () ()
. ( 7)
: 4.1 :
) 1 1 1 1 1 8 1 8
x 4 4 x 4 4 x 4 x2 2 2 2 2 2 2 2
< < < < < + < < +
7 9x
2 2 < < .
) | x 5 | 3+ x + 5 3 x + 5 3 x 3 5 x 3 5 x 8 x 2.
) :
+8
7
2 2
9
2
9
x 2,2
92 x
2 < .
-
. , . ,
30 30
, 6, 7
. . .
GI_A_ALG_2_504
) < 0, : 1
2
+ . ( 15)
) < 0, : 1
2
+ . ( 10)
: 2.2, 2.3 :
) < 0 : 0
2 21 1 2 ( 2) 1 2 2 1 0
. ( 9)
) ()
() ; . ( 7)
: 3.1, 4.1 :
) | 2x 4 | 3 | x 1| = ( )| 2x 4 | | 3 x 1 | = 2x 4 = 3(x 1) 2x 4
= 3(x 1) 2x 4 = 3x 3 2x 4 = 3x + 3 2x 3x = 4 3 2x + 3x = 4 + 3
x = 1 5x = 7 x = 1 7
x5
= .
) 3x 5 1 > 3x 5 < 1 3x 5 > 1 3x < 5 1 3x > 5 +
1
3x < 4 3x > 6 4
x3
< x > 2 ( )4x , 2,3
+
.
) :
4 72
3 5< <
4 715 15 15 2
3 5 < < 5 4 3 7 15 2 < < 20 < 21 < 30,
, .
( )7 4, 2,5 3
+
, 4
1 ,3
,
x 1= | 2x 4 | 3 | x 1| =
3x 5 1 > .
-
. , . ,
32 32
, 6
. . .
GI_A_ALG_2_506 2 x 3 1 y 2 , :
) x + y ( 5)
) 2x 3y ( 10)
) x
y ( 10)
: 2.2 :
) 2 x 3 1 y 2 , : 2 1 x y 3 2+ + + 3 x y 5 + . x + y 3 5.
) :
2 x 3 2 2 2 x 2 3 4 2x 6 , 1 y 2 ( 3) 1 ( 3) y ( 3) 2 3 3y 6
6 3y 3 ,
4 6 2x 3y 6 3 2 2x 3y 3 . 2x 3y 2 3.
) :
2 x 3
1 y 2 1 1 1
1 y 2
1 11
2 y ,
( )
1 12 x 3 1
2 y
x1 3
y .
x
y 1 3.
-
. , . ,
33 33
, 9
. . .
GI_A_ALG_2_507
: ( )2 2 9 x 3 = , (1). ) ,
. ( 6)
) , (1) . ( 9)
) , (1) 4. ( 10)
: 3.1 :
) :
= 0 (1) ( )2 20 9 x 0 3 0 = 9x = 0. = 1 (1) ( )2 21 9 x 1 3 1 =
8x = 2. = 3 (1) ( )2 23 9 x 3 3 3 = 0x = 0.
) ( )2 2 9 x 3 = ( )( )2 9 0 3 3 0 3 0 3 0 3 + + .
) 3 (1) ,
( )2 2 ( 3) 9 x 3 ( 3)( 3)x ( 3) x x( 3)( 3) 3
= + = = =
+ +.
4, :
4 4( 3) 4 12 4 12 3 12 4
3= + = + = = = =
+,
.
-
. , . ,
34 34
, 11, 16
. . .
GI_A_ALG_2_508 )
1, 2, 3, , . ( 12)
)
45. ( 13)
: 3.3, 5.2 :
) *
, ,N
1= 1 = 1, :
( ) ( ) 21 *
2 1 2 1 1 1 2 1 ( 1) S ,
2 2 2 2 2
+ + + + += = = = = N .
) *
N
S 45= , : 2
= 452
+ 2 = 90+ 2 90 = 0+ ().
2
90+ 2 1 4 1 ( 90) = 361=
1 19 189
1 361 1 19 2 2
1 19 202 1 210
2 2
+ = = = = =
= =
.
() = 9 = 10 (), 45.
-
. , . ,
35 35
, 6, 7
. . .
GI_A_ALG_2_509
) { }, 0 , : 2 + (1). ( 15)
) (1); .
( 10)
: 2.2, 2.3 :
) { }, 0 : |||| 0 | | | | | | | |
2 2 | || | | || | 2 | || | | | | | | | | |
>
+ + +
( )22 2 2 2| | | | 2 | || | | | | | 2 | || | 0 | | | | 0 + + , ,
.
) { }, 0 : |||| 0 | | | | | | | |
2 2 | || | | || | 2 | || | | | | | | | | |
>
+ = + = + =
( )22 2 2 2| | | | 2 | || | | | | | 2 | || | 0 | | | | 0 + = + =
= | | | | 0 | | | | = = = ,
= .
-
. , . ,
36 36
, 10, 19
. . .
GI_A_ALG_2_510
f, : 2
2x 5, x 3f (x)
x , 3 x 10
=
<
-
. , . ,
37 37
, 8
. . .
GI_A_ALG_2_936
: ( )( )A x 4 x 1 x 4 x 1= + + + . ) x A;
. ( 12)
) A , x.
( 13)
: 2.4 :
) x 4 0 x 4
x 4x 1 0 x 1
+ ,
x [4, ) + . ) x [4, ) + :
( )( ) ( ) ( )2 2A x 4 x 1 x 4 x 1 x 4 x 1= + + + = + (x 4) (x
1) x 4 x 1 5= + = = ,
A ( x).
-
. , . ,
38 38
, 8
. . .
GI_A_ALG_2_938
) : 33 30 4< < . ( 12)
) 3 30 36 30 . ( 13)
: 2.4 :
) : 3
3 33 33 30 4 3 30 4 27 30 64< < < < < < , ,
.
) :
3 36 30 30 < 3 36 30 30< + 36 2 30< ( )33 36 2
30<
33 3 36 2 30< 216 8 30< 216 < 240,
, .
3 36 30 30 < .
-
. , . ,
39 39
, 8
. . .
GI_A_ALG_2_944
: A x 4 6 x= + . ) x A;
x .
( 13)
) x = 5, : 2A A 6 0+ = . ( 12)
: 2.4 :
) x 4 0 x 4
4 x 66 x 0 x 6
,
x [4, 6] . ) x = 5 :
A 5 4 6 5 1 1 1 1 2= + = + = + = , 2 2A A 6 2 2 6 4 2 6 0+ = + =
+ = .
-
. , . ,
40 40
, 8
. . .
GI_A_ALG_2_947
: 2A x 4 x 4= + . ) x A;
x .
( 12)
) x = 4, : ( )2A A 2 10 5 = . ( 13)
: 2.4 :
) 2 2 xx 4 0 x 4
x 4x 4x 4 0 x 4
+
R,
x [4, ) + . ) x = 4 :
2A 4 4 4 4 16 4 0 20 4 5 4 5 2 5= + = + = = = = , :
( ) ( )2 22 2A A 2 5 2 5 2 5 2 5 4 5 2 5 20 2 5 2 10 5 = = = = =
.
-
. , . ,
41 41
, 8
. . .
GI_A_ALG_2_950
: 4 4A 1 x x= . ) x A;
x .
( 13)
) x = 3, : 3 2A A A 1 0+ + + = . ( 12)
: 2.4 :
) 4 4
1 x 0 x 1 x 1x 1
xx 0 x 0
R,
x ( ,1] . ) x = 3 :
44A 1 ( 3) ( 3) 1 3 | 3 | 4 3 2 3 1= = + = = = ,
3 2 3 2A A A 1 ( 1) ( 1) ( 1) 1 1 1 1 1 0+ + + = + + + = + + =
.
-
. , . ,
42 42
, 8
. . .
GI_A_ALG_2_952
: ( )55B x 2= . ) x B;
x -
. ( 13)
) x = 4, 2 4B + 6B = B . ( 12)
: 2.4 :
) : 5(x 2) 0 x 2 0 x 2 , x [2, ) + .
) x [2, ) x 2 x 2 0 + = |x 2| = x 2. x = 4 = 4 2 = 2,
2 2 4 4B + 6B 2 + 6 2 4 12 16 2 B= = + = = = .
-
. , . ,
43 43
, 8
. . .
GI_A_ALG_2_955
: ( )6A 2= ( )63B 2= . ) : = 4. ( 13)
) : 32, 1, 2 .
( 12)
: 2.4 :
) :
( ) ( )36 2
3A 2 2 2 8= = = = ( ) ( )26 3
23 3B 2 2 2 4= = = = ,
= 8 4 = 4.
) :
3
33 31 < 2 1 < 2 1 2 < , ,
6 6
2 33 32 2 2 2 2 2 4 8< < < < , , .
31 < 2 2< .
-
. , . ,
44 44
, 7, 12
. . .
GI_A_ALG_2_991 x : x 1 2+ < ,
) x(3, 1). ( 12)
) : x 3 x 1
K4
+ + = -
x. ( 13)
: 2.3, 4.1 :
) x 1 2+ < 2 < x + 1 < 2 2 1 < x < 2 1 3 < x
< 1
x(3, 1). ) 3 < x < 1 3 3 < x + 3 < 1 + 3 0 < x +
3 < 4,
x + 3 > 0 |x + 3| = x + 3.
, 3 < x < 1 3 1 < x 1 < 1 1 4 < x 1 < 0, x 1
< 0 |x 1| = (x 1) = x + 1. :
x 3 x 1 x 3 ( x 1) x 3 x 1 4K 1
4 4 4 4
+ + + + + + += = = = = ,
x.
-
. , . ,
45 45
, 6, 7
. . .
GI_A_ALG_2_996 : A = |x 1| + |y 3|, x, y ,
: 1 < x < 4 2 < y < 3.
:
) A = x y + 2 ( 12)
) 0 < A < 4 ( 13)
: 2.2, 2.3 :
) :
1 < x < 4 1 1 < x 1 < 4 1 0 < x 1 < 3, x 1
> 0 |x 1| = x 1.
2 < y < 3 2 3 < y 3 < 3 3 1 < y 3 < 0, y 3
< 0 |y 3| = (y 3) = y + 3.
= x 1 + (y + 3) = x 1 y + 3 = x y + 2. ) :
1 < x < 4 1 + 2 < x + 2 < 4 + 2 3 < x + 2 < 6,
2 < y < 3 (1)2 > (1)y > (1)3 2 > y > 3 3 < y
< 2,
3 3 < x + 2 y < 6 2 0 < x + 2 y < 4 0 < <
4.
-
. , . ,
46 46
, 11, 19
. . .
GI_A_ALG_2_999
) 2x 5x 6 + . ( 12)
) ( )2
x 2f x
x 5x 6
=
+.
i) A . ( 5)
ii) xA : ( ) 1f xx 3
=
. ( 8)
: 3.3, 6.1 :
) 2x 5x 6 + ( )2 5 4 1 6 1= =
( )
5 1 63
5 1 5 1 2 2x
5 1 42 1 22
2 2
+ = = = = =
= =
,
2x 5x 6 (x 2)(x 3) + = .
) i) f 2x 5x 6 0 + x 2 x 3, { }A 2, 3 ( , 2) (2, 3) (3, )= = +
.
ii) x A ( )( )( )2
x 2 x 2 1f x
x 5x 6 x 2 x 3 x 3
= = =
+ .
-
. , . ,
47 47
, 3, 4
. . .
GI_A_ALG_2_1003 , , .
5, 9, 16.
. :
:
K: KOKKINH
:
) , -
:
i) ,
ii) . ( 13)
)
(). ( 12)
: 1.1, 1.2 :
) i) A = .
ii) K = . ) () = 5, N(K ) 16 = () = 5 + 9 + 16 = 30.
:
( ) ( ) ( )( )
N A 5 30 5 25 5P 1 P A 1 1
N 30 30 30 30 6= = = = = =
( ) ( )( )
16 8P K
30 15
= = = .
-
. , . ,
48 48
, 5, 11
. . .
GI_A_ALG_2_1005
1 x
x 1
+=
2
2B
x x=
, x
.
) , :
x 1 x 0. ( 12)
) x A = B. ( 13)
: 2.1, 3.3 :
) x 1 0 x 1. :
( )2x x 0 x x 1 0 x 0 x 1 0 x 0 x 1 . , , :
x 0 x 1.
) x 0 x 1 :
( )( ) ( )
( )21 x 2 1 x 2 1 x 2
A B x x 1 x x 1x 1 x x x 1 x x 1 x 1 x x 1
+ + += = = =
( ) 2 2x 1 x 2 x x 2 x x 2 0 + = + = + = (). 2x x 2+ 2 1 4 1 (
2) 9= =
1 3 21
1 9 1 3 2 2x
1 3 42 1 22
2 2
+ = = = = =
= =
.
:
() x = 1 () x = 2 () x = 2.
-
. , . ,
49 49
, 11
. . .
GI_A_ALG_2_1007 ) : 22x 10x 12 + = . ( 15)
) : 22x 10x 12
= 0x 2
+
. ( 10)
: 3.3 :
) 22x 10x 12 + = 22x 10x 12 = 0 + 2x 5x 6 = 0+- ().
2x 5x 6+- 2 ( 5) 4 1 6 25 24 1= = =
5 1 63
( 5) 1 5 1 2 2x
5 1 42 1 22
2 2
+ = = = = =
= =
.
() x = 2 x = 3. ) x 2 0 x 2 :
22x 10x 12= 0
x 2
+
22x 10x 12 = 0 + 2x 5x 6 = 0+-
x = 2 () x = 3 () x = 3.
-
. , . ,
50 50
, 7
. . .
GI_A_ALG_2_1009 : = |3x 6| + 2, x .
) :
i) x 2, A 3x 4 = ii) x 2, A 8 3x< = . ( 12)
) x x 2, : 29x 16
3x 43x 6 2
= +
+.
( 13)
: 2.3 :
) :
3x 6, 3x 6 0 3x 6, 3x 6 3x 6, x 2| 3x 6 |
(3x 6), 3x 6 0 3x 6, 3x 6 3x 6, x 2
= = =
< + < +
-
. , . ,
51 51
, 16
. . .
GI_A_ALG_2_1015 ( ) 2 4 0, 4= = .
) = 2 1 2= ,
1 . ( 10)
) - 2 4,
= N 98. ( 15)
: 5.2 :
) *
, ,N 1 ,
( ) * 1 1 , = + N . :
( )( )
12 1 1
14 1
2 1 0 0 0
4 1 4 4 3 4 3 4
+ == + = =
+ == + = + =
1 1 1 2
2 4 2 2
= = =
= = = .
) :
( ) ( ) * 1 1 2 1 2 2 2 2 2 4, = + = + = + = N . 98,
*N :
98 2 4 98 2 4 98 2 102 51= = = + = = .
98, 51 98= .
-
. , . ,
52 52
, 20
. . .
GI_A_ALG_2_1024 f(x) = x + , , .
) f A(1, 6),
B(1, 4), , . ( 13) ) = 1 = 5,
f xx yy . ( 12)
: 6.2 :
) fC (1, 6), f(1) = 6,
(1, 4), f(1) = 4. : 1 6
( 1) 4
+ =
+ =
6
4
+ = + =
6 4
4
+ + = +
+ =
2 10
4
= + =
5
4
= + =
5
5 4
= + =
5
4 5
= =
5
1
= =
5
1
=
=.
) = 1 = 5 f(x) = x + 5.
fC yy x = 0, :
f(0) = 0 + 5 = 5, (0, 5).
fC xx y = 0, :
f(x) = 0 x + 5 = 0 x = 5, (5, 0).
-
. , . ,
53 53
, 17
. . .
GI_A_ALG_2_1032 ) x : x, 2x + 1, 5x + 4,
, .
( 13)
) , :
i) x = 1 ii) x = 1 ( 12)
: 5.3 :
) x, 2x + 1, 5x + 4 ,
:
( ) ( )2 2 2 2 22x 1 x 5x 4 4x 4x 1 5x 4x 4x 4x 1 5x 4x 0+ = + +
+ = + + + = 2 2x 1 0 x 1 x 1 + = = = .
) i) x = 1 1, 3, 9,
3 9
31 3= = .
ii) x = 1 1, 1, 1, -
1 1
11 1
= =
.
-
. , . ,
54
, 7, 12
. . .
GI_A_ALG_2_1039
) x 1 5 . ( 8)
) x 5 3.
( 9)
) () (). ( 8)
: 2.3, 4.1 :
) x 1 5 x 1 5 x 1 5 x 1 5 x 1 + 5 x 4 x 6.
) H x 5 d(x, 5) | x 5 |= . :
x 5 3 3 x 5 3 3 5 x 3 5 2 x 8 < < < + < < + <
< .
) :
+4 2 6 8
6 x 8 < .
-
. , . ,
55
, 19
. . .
GI_A_ALG_2_1042
: 2x 4, x 0
f (x)x 1, x 0
+
-
. , . ,
56
, 10, 17
. . .
GI_A_ALG_2_1050
) x : ( )2x 2, x 1 , 3x 2+ + +
.
( 13)
) , :
i) x 1= ii) x = 1 ( 12)
: 3.2, 5.2 :
) ( )2x 2, x 1 , 3x 2+ + +
, :
( )22 x 1 x 2 + 3x 2+ = + + ( )22 x 2x 1 4x 4+ + = + 22x + 4x +
2 4x 4 = 0- - 22x 2 = 0- 22x 2= 2x 1= x 1= x 1= .
) i) x = 1, 3, 4, 5, -
4 3 = 5 4 = 1.
ii) x = 1, 1, 0, 1, - 0 1 = 1 0 = 1.
-
. , . ,
57
, 9
. . .
GI_A_ALG_2_1055
: ( )( )2( 1)x 1 2 = + + , . ) = 1 = 1. ( 12) ) ;
. ( 13)
: 3.1 :
) = 1 , :
( ) ( )2(1 1)x 1 1 1 2 0x 6 = + + = , . = 1 , :
( )( )2( 1) 1 x 1 1 1 2 0x 0 = + + = , . ) 2 21 0 1 1 1 ,
( )( )2( 1)x 1 2 = + +
( )( ) ( ) ( )2
1 2 1 2 2x
1 ( 1)( 1) 1
+ + + + += = =
+ .
-
. , . ,
58
, 16
. . .
GI_A_ALG_2_1057 10 , 120
20 .
)
- . ( 9)
) ; ( 8)
) ; ( 8)
: 5.2 :
) 20 ,
*, N 10 = 20 1 120 = .
: *120 ( 1) 20 120 20 20 20 100, = + = + = + N 10.
) 10, 10 , :
10 20 10 100 200 100 300 = + = + = . 300 .
)
10 , :
1 1010
120 300 420S 10 10 10 210 10 2100
2 2 2
+ += = = = = .
2100 .
-
. , . ,
59
, 6, 12
. . .
GI_A_ALG_2_1062 ) y : y 3 1 < . ( 12)
) x, y ,
1 x 3< < 2 y 4< < , E . ( 13)
: 2.2, 4.1 :
) y 3 1 1 y 3 1 3 1 y 3 1 2 y 4 < < < < < + <
< .
) ( )
1 x 3< < 2 y 4< < , : 1 2 x y 3 4 < < 2 E
12< < , 2 12.
-
. , . ,
60
, 16
. . .
GI_A_ALG_2_1064 ( ) :
1 19= 10 6 24 = . ) = 6. ( 9)
) 20 . ( 8)
) 20 . ( 8)
: 5.2 :
) *, , N
*1 ( 1) 19 ( 1) , = + = + N . :
10 6 24 19 (10 1) [19 (6 1)] 24 19 9 (19 5) 24 = + + = + + =
19 9 19 5 24 4 24 6 + = = = . ) *19 ( 1) 6 19 6 6 6 13, = + = +
= + N .
20 6 20 13 120 13 133 = + = + = . )
( ) ( )1
2 1 2 19 1 6 2 [19 3( 1)]S
2 2 2
+ + + = = =
2 *2(19 3 3) (3 16) 3 16, 2
+ = = + = + N .
220S 3 20 16 20 3 400 320 1200 320 1520= + = + = + = .
-
. , . ,
61
, 11, 13
. . .
GI_A_ALG_2_1067
: 2
2
x 4x 4K
2x 3x 2
+=
.
) 22x 3x 2 . ( 10) ) x ; -
. ( 7)
) . ( 8)
: 3.3, 4.2 :
) 22x 3x 2 ( ) ( )2 3 4 2 2 9 16 25= = + =
( )
3 5 82
3 25 3 5 4 4x
3 5 2 12 2 4
4 4 2
+ = = = = =
= =
.
:
2 1 12x 3x 2 2 x (x 2) 2 x (x 2) (2x 1)(x 2)2 2
= = + = +
.
) : 22x 3x 2 0 (2x 1)(x 2) 0 2x 1 0 x 2 0 + +
12x 1 x 2 x x 2
2 .
1
x , 22
.
) 1
x x 22
:
( )( ) ( )
22
2
x 2x 4x 4 x 2K
2x 3x 2 x 2 2x 1 2x 1
+ = = =
+ +.
-
. , . ,
62
, 5
. . .
GI_A_ALG_2_1070 , , , 0 :
+ = 4
1
4=
.
) = 3 = 5. ( 10)
) :
+=
. ( 15)
: 2.1 :
) 0
4 4 4 3
+= + = = = .
, 1
4 4 5 4
= = + = =
.
) 0 :
( )( )
( )( )
3 4 1
5 4
+ ++= = = = =
.
-
. , . ,
63
, 6, 12
. . .
GI_A_ALG_2_1074 ) y : | y 3 | 1 < . ( 12) ) x, y ,
1 x 3< < 2 y 4< < , 6 14< < , . ( 13)
: 2.2, 4.1 :
) y 3 1 1 y 3 1 3 1 y 3 1 2 y 4 < < < < < + <
< .
) = 2x + 2y,
x, y > 0.
:
1 x 3< < 2 1 2 x 2 3 < < 2 2x 6< < , 2 y 4<
< 2 2 2 y 2 4 < < 4 2y 8< < ,
2 4 2x 2y 6 8+ < + < + 6 14< < .
x
y
-
. , . ,
64
, 6, 12
. . .
GI_A_ALG_2_1077 ) : |x 5| < 4. ( 10)
) , :
1 11
9 < < . ( 15)
: 2.2, 4.1 :
) |x 5| < 4 4 < x 5 < 4 5 4 < x < 5 + 4 1 < x
< 9. ) () , :
1 1 1 1 11 9 1
1 9 9 < < > > < < .
-
. , . ,
65
, 5
. . .
GI_A_ALG_2_1080
x, y : 4x 5y
2x 4y
+=
.
) : y = 2x. ( 12)
) : 2 22x 3y xy
Axy
+ += . ( 13)
: 2.1 :
) x 4y 0 x 4y :
( )4x 5y 2 4x 5y 2 x 4y 4x 5y 2x 8yx 4y
+= + = + = +
4x 2x 8y 5y 6x 3y 2x y y 2x + = = = = . ) x 4y, x 0, y 0 :
2 2 2 2 2 2 2
2
2x 3y xy 2x 3 (2x) x (2x) 2x 3 4x 2xA
xy x (2x) 2x
+ + + + + += = =
2 2 2 2
2 2
2x 12x 2x 16x8
2x 2x
+ += = = .
-
. , . ,
66
, 19
. . .
GI_A_ALG_2_1082
2
x 2f (x)
x x 6
+=
.
) f. ( 15)
) : f(2) + f(4) = 0. ( 10)
: 6.1 :
) 2x x 6 0 ().
2x x 6 ( ) ( )2
1 4 1 6 25= =
( )
1 5 63
1 25 1 5 2 2x
1 5 42 22
2 2
+ = = = = =
= =
.
() x 3 x 2, fA { 2, 3}= R .
) x 3 x 2 ( ) 2
x 2 x 2 1f x
x x 6 (x 3)(x 2) x 3
+ += = =
+ .
:
1 1 1 1f (2) f (4) 1 1 0
2 3 4 3 1 1+ = + = + = + =
.
-
. , . ,
67
, 9, 16
. . .
GI_A_ALG_2_1086 = 1, = x + 4, = x + 8 , ,
( ) .
) x. ( 10)
) x = 1 A ( ) ,
i) . ( 7)
ii) . ( 8)
: 3.1, 5.2 :
) A, B, , :
2 = + 2(x + 4) = 1 + x + 8 2x + 8 = x + 9 2x x = 9 8 x = 1.
) x = 1 A = 1, B = 5, = 9.
i) = = = 5 1 = 9 5 = 4.
ii) *
, ,N = 4 1 A 1 = = , :
*
1 ( 1) 1 ( 1) 4 1 4 4 4 3, = + = + = + = N . :
20 4 20 3 80 3 77 = = = .
-
. , . ,
68
, 9, 11,
15, 16
. . .
GI_A_ALG_2_1088 ) 4 x, x, 2 , -
x. ( 9)
) 4 x, x, 2 , - x. ( 9)
) x 4 x, x, 2 - . ( 7)
: 5.2, 5.3 :
) 4 x, x, 2 , : 2x = 4 x + 2 2x + x = 4 + 2 3x = 6 x = 2.
) 4 x, x, 2 , :
( )2 2 2x 4 x 2 x 8 2x x 2x 8 0= = + = ().
2x 2x 8+
( )2 2 4 1 8 36= =
2 6 42
2 36 2 6 2 2x
2 6 82 1 24
2 2
+ = = = = =
= =
.
() x = 2 x = 4. ) () () , x = 2.
-
. , . ,
69
, 7
. . .
GI_A_ALG_2_1089 x 5 < x < 10,
) |x 5| |x 10| .
( 10)
) : x 5 x 10
A =x 5 x 10
+
. ( 15)
: 2.3 :
) 5 < x < 10 5 5 < x 5 < 10 5 0 < x 5 < 5, x 5
> 0 |x 5| = x 5.
5 < x < 10 5 10 < x 10 < 10 10 5 < x 10 < 0, x
10 < 0 |x 10| = (x 10).
) 5 < x < 10 :
x 5 x 10 x 5 (x 10)A = = 1 ( 1) 0
x 5 x 10 x 5 x 10
+ + = + =
.
-
. , . ,
70
, 19, 20
. . .
GI_A_ALG_2_1090
f, 2
1f (x)
x 1=
.
) . ( 13)
) ,
1M ,
8
f. ( 12)
: 6.1, 6.2 :
) 2 2x 1 0 x 1 x 1 x 1 , { }fA 1,1= .
) 1
M ,8
f,
1
f ( )8
= .
1 1 :
2 2 2
2
1 1 1f ( ) 1 8 8 1 9 3
8 1 8 = = = = + = =
.
-
. , . ,
71
, 7
. . .
GI_A_ALG_2_1091 : = |x 1| |x 2|. ) 1 < x < 2, : A = 2x 3.
( 13)
) x < 1, A ( x),
. ( 12)
: 2.3 :
) 1 < x < 2 1 1 < x 1 < 2 1 0 < x 1 < 1, x 1
> 0 |x 1| = x 1.
, 1 < x < 2 1 2 < x 2 < 2 2 1 < x 2 < 0, x 2
< 0 |x 2| = (x 2) = x + 2. 1 < x < 2 :
= |x 1| |x 2| = x 1 (x + 2) = x 1 + x 2 = 2x 3. ) x < 1 x 1
< 0, |x 1| = (x 1) = x + 1.
, x < 1 x 2 < 1 2 x 2 < 1, x 2 < 0 |x 2| = (x 2) = x
+ 2. x < 1 :
= |x 1| |x 2| = x + 1 (x + 2) = x + 1 + x 2 = 1, ( x).
-
. , . ,
72
, 5, 6
. . .
GI_A_ALG_2_1092 y.
Z
x
y
y
y
EH
)
: = 2x + 4y. ( 10)
) 5 < x < 8 1 < y < 2,
. ( 15)
: 2.1, 2.2 :
) x > y > 0 = = + = + = y + y = 2y
EZ = HZ HE = AB = x y.
x > y > 0
= + + + + + = x + 2y + x y + y + y + y = 2x + 4y.
) :
5 < x < 8 25 < 2x < 28 10 < 2x < 16, 1 < y
< 2 41 < 4y < 42 4 < 4y < 8,
10 + 4 < 2x + 4y < 16 + 8 14 < < 24, 14 24.
-
. , . ,
73
, 8, 11
. . .
GI_A_ALG_2_1093
: 1
5 5 =
+ ,
1
5 5 =
.
) :
i)
1
2+ = ( 8)
ii) 1
20 = ( 8)
) 2 .
( 9)
: 2.4, 3.3 :
) :
( ) ( ) ( ) ( )1 1 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5 5
++ = + = +
+ + +
( ) ( ) ( )225 5 5 5 10 10 10 1
25 5 20 25 5 5 5 5 5
+ += = = = =
+ .
:
( ) ( ) ( )221 1 1 1 1 1 1
25 5 205 5 5 5 5 5 5 5 5 5
= = = = = + +
.
) 1 2x , x 2x Sx P 0 + = , 1 2S x x= + 1 2P x x= .
:
1S A+B=
2=
1P A B=
20= ,
2 1 1x x 02 20
+ = 220x 10x 1 0 + = .
-
. , . ,
74
, 21
. . .
GI_A_ALG_2_1096 y ( ) , x
, : y = 35 + 0,8x.
) 25 ;
( 12)
) , 75
; ( 13)
: 6.3 :
y = 35 + 0,8x x 0.
) x = 25 y 35 0,8 25 35 20 55= + = + = , 25
55 .
) x 0 y = 75.
:
75 35 0,8x 75 35 0,8x 40 0,8x x 50= + = = = , 75 50 .
-
. , . ,
75
, 11
. . .
GI_A_ALG_2_1097
22x x 5+ , . ) 0x 1= ,
. ( 12)
) = 3, . ( 13)
: 3.3 :
) 1 , : 22 1 1 5 0 2 5 0 3 0 3 + = + = = = .
) = 3 22x 3x 5+ , ( )23 4 2 5 9 40 49 = = + =
3 7 41
3 49 3 7 4 4x
3 7 10 52 2 4
4 4 2
+ = = = = =
= =
.
:
( )2 5 52x 3x 5 2 x x 1 2 x (x 1) (2x 5)(x 1)2 2
+ = = + = +
.
-
. , . ,
76
, 11, 17
. . .
GI_A_ALG_2_1100 : 2 22x 5 x 2 0 + = (1), > 0.
) (1) : 1 2x 2 x2
= = . ( 12)
) 1 2x , x (1), 1 2x , , x , ,
. ( 13 )
: 3.3, 5.3 :
) 2 22x 5 x 2 0 + =
( )2 2 2 2 25 4 2 2 25 16 9 0 = = = >
( ) 2
5 3 82
5 9 5 3 4 4x
5 3 22 2 4
4 4 2
+ = = = = =
= =
.
(1) 1 2x 2 x2
= = .
) 1 2x , , x :
2 2 2 2
1 2x x 22
= = = , , .
1 2x , , x .
-
. , . ,
77
, 11, 16
. . .
GI_A_ALG_2_1101 : 2 2(2 x 0)x 4 + = (1) .
) (1) : 1 2x 2 x 2= = + . ( 12)
) 1 2x , x (1), 1 2x , , x ,
,
. ( 13)
: 3.3, 5.2 :
) 2 2x 2 x 4( ) +
( ) ( )2 2 2 22 4 1 4 4 4 16 16 = = + =
( )2 16 2 4 2( 2)
x 22 1 2 2
= = = =
.
(1) 1 2x 2 x 2= = + .
) 1 2x , , x
1 22 x x = + 2 2 2 = ++ 2 = 2, , .
1 2x , , x .
-
. , . ,
78
, 3, 4
. . .
GI_A_ALG_2_1102 ,
3( )
4 = ,
5( )
8 =
1( )
4 = .
) ( ) . ( 9) ) i) Venn -
: . ( 7)
ii) -
. ( 9)
: 1.1, 1.2 :
) ( ) ( ) ( ) = ( ) ( ) ( ) = ,
3 5 6 5 1
( )4 8 8 8 8
= = = .
) i)
, , Venn -
.
ii) -
A B , :
3 1 1 8 1 7( ) ( ) ( ) ( )
4 4 8 8 8 8 = + = + = = .
B
-
. , . ,
79
, 6, 12
. . .
GI_A_ALG_2_1273 x y, :
|x 3| 2 |y 6| 4.
) : 1 x 5 2 y 10. ( 12)
)
2x y. ( 13)
: 2.2, 4.1 :
) |x 3| 2 2 x 3 2 3 2 x 3 + 2 1 x 5. |y 6| 4 4 y 6 4 6 4 y 6 + 4
2 y 10. ) :
= 22x + 2y = 4x + 2y x, y > 0. :
1 x 5 41 4x 45 4 4x 20, 2 y 10 22 2y 210 4 2y 20,
4 + 4 4x + 2y 20 + 20 8 4x + 2y 40.
8 (x, y) (1, 2)= , 40
(x, y) (5,10)= . 2x
y
-
. , . ,
80
, 11
. . .
GI_A_ALG_2_1275
22x 5x 1+ . ) , 1x 2x .
( 6)
) : 1 2 1 2x x , x x+ 1 2
1 1
x x+ . ( 9)
) 2 1
1
x
2
1
x. ( 10)
: 3.3 :
) 22x 5x 1+ 25 4 2 ( 1) 25 8 33 0 = = + = > , , 1x 2x .
) 1 2x , x , Vieta :
1 2
5S x x
2= + = 1 2
1 1P x x
2 2
= = = .
, 2 1 2 1
1 2 1 2 1 2 1 2
5x x x x1 1 S 2 5
1x x x x x x x x P
2
++ = + = = = =
.
) 2 1
1
x
2
1
x 2x Sx P 0 + = ,
1 2
1 1S 5
x x= + =
1 2
1 1 1 1P 2
1x x P
2
= = = =
.
2 1
1
x
2
1
x 2x 5x 2 0 = .
-
. , . ,
81
, 6, 8
. . .
GI_A_ALG_2_1276
2 2x 4x 4 x 6x 9
Kx 2 x 3
+ + +=
+ .
) x, K
. ( 12)
) 2 < x < 3, K , - x. ( 13)
: 2.2, 2.4 :
) H K : 2 2
2 2
xx 4x 4 0 (x 2) 0
x 2 x 2x 2 0 x 2
x x 3x 6x 9 0 (x 3) 0
x 3x 3 0 x 3
+ + + +
+
R
R.
x { 2,3} R . ) 2 < x < 3 2 2 < x + 2 < 3 + 2 0 <
x + 2 < 5,
x + 2 > 0 |x + 2| = x + 2.
, 2 < x < 3 3 2 < x 3 < 3 3 5 < x 3 < 0, x 3
< 0 |x 3| = (x 3). x ( 2, 3) :
2 22 2 (x 2) (x 3)x 4x 4 x 6x 9 | x 2 | | x 3 |K
x 2 x 3 x 2 x 3 x 2 x 3
+ + + + + = = =
+ + +
x 2 (x 3)1 ( 1) 1 1 2
x 2 x 3
+ = = = + =
+ ,
K ( x).
-
. , . ,
82
, 13
. . .
GI_A_ALG_2_1277
2x 5x 6 0 + < (1) 2x 16 0 (2). ) (1) (2). ( 12)
) (1) (2)
. ( 13)
: 4.2 :
) 2x 5x 6 + 25 4 ( 1) ( 6) 25 24 1 = = = ,
5 1 42
5 1 5 1 2 2x
5 1 62 ( 1) 23
2 2
+ = = = = = = =
:
+2 3x
2x 5x 6 + +
2x 5x 6 0 + < x < 2 x > 3. 2x 16 0 2x 16 |x| 4 4 x
4.
) (1) (2) :
+4 2 3 4
(1) (2) x [ 4,2) (3, 4] .
-
. , . ,
83
, 7, 13
. . .
GI_A_ALG_2_1278 x d(x, 2) < 1. :
) 3 < x < 1 ( 10) ) 2x 4x 3 0+ + < ( 15)
: 2.3, 4.2 :
) d(x, 2) < 1 |x (2)| < 1 |x + 2| < 1 1 < x + 2 <
1 1 2 < x < 1 2 3 < x < 1.
) To 2x 4x 3+ + 24 4 1 3 16 12 4 = = = ,
4 2 21
4 4 4 2 2 2x
4 2 62 1 23
2 2
+ = = = = =
= =
:
+3 1
x
2x 4x 3+ + + +
2x 4x 3 0+ + < 3 < x < 1 d(x, 2) < 1 |x + 2| <
1.
-
. , . ,
84
, 11
. . .
GI_A_ALG_2_1281
( )2x 3 1 x 3 + + . ) ( )23 1 = + .
( 12)
) . ( 13)
: 3.3 :
) ( ) ( )2 22 22 23 1 4 ( 1) 3 3 2 3 1 4 3 3 2 3 1 3 1 = = + + =
+ + = + . ) :
( )2 2x 3 1 x 3 x 3x x 3 + + = + + ( )x(x 1) 3(x 1) (x 1) 3 x= +
+ + = + .
2 () .
( )2x 3 1 x 3 + + :
( ) ( ) ( )2
3 1 3 1 3 1 3 1x
2 1 2
+ + += =
( )
( )
3 1 3 1 3 1 3 1 21
2 2 2
3 1 3 1 3 1 3 1 2 33
2 2 2
+ + + + + + = = =
= + + +
= = =
.
( ) ( ) ( )2x 3 1 x 3 1 (x 1) x 3 (x 1) x 3 + + = = + .
-
. , . ,
85
, 9, 11, 19
. . .
GI_A_ALG_2_1282
) 23x 2x 1 . ( 8) ) x :
2
x 1A(x)
3x 2x 1
=
. ( 9)
) : |A(x)| = 1. ( 8)
: 3.3 :
) 23x 2x 1 2( 2) 4 3 ( 1) 4 12 16 = = + =
2 4 61
( 2) 16 2 4 6 6x
2 4 2 12 3 6
6 6 3
+ = = = = =
= =
.
21 1
3x 2x 1 3 x (x 1) 3 x (x 1) (3x 1)(x 1)3 3
= = + = + .
) 21
3x 2x 1 0 x x 13
,
A(x) 1
,13
R .
) 1
x 1,3
R 2
x 1 x 1 1A(x)
3x 2x 1 (3x 1)(x 1) 3x 1
= = =
+ +,
:
1 |1|A(x) 1 1 1 | 3x 1| 1
3x 1 | 3x 1|= = = + =
+ +
3x + 1 = 1 3x + 1 = 1 3x = 1 1 3x = 1 1
3x = 0 3x = 2 x = 0 2
x3
= , .
-
. , . ,
86
, 11, 19,
20, 21
. . .
GI_A_ALG_2_1283
) 2x 2x 3+ . ( 8)
) 2x 2x 3
f (x)x 1
+ =
. ( 9)
) . ( 8)
: 3.3, 6.1, 6.2, 6.3 :
) 2x 2x 3+ 22 4 1 ( 3) 4 12 16 = = + =
2 4 21
2 16 2 4 2 2x
2 4 62 1 23
2 2
+ = = = = =
= =
.
[ ]2x 2x 3 (x 1) x ( 3) (x 1)(x 3)+ = = + . ) x 1 0 x 1, fA {1}=
R .
x 1 : 2x 2x 3 (x 1)(x 3)
f (x) x 3x 1 x 1
+ += = = +
.
) f y = x + 3,
(1, 4), x 1.
x = 3 y = 0, B(3, 0), x = 0 y = 3, (0, 3).
f :
A
B
-
. , . ,
87
, 3, 4
. . .
GI_A_ALG_2_1287 :
.
.
: ( 7)
: 3 ( 9)
: 3 ( 9)
: 1.1, 1.2 :
= {11, 12, 13, 21, 22, 23, 31, 32, 33}, () = 9.
= {12, 22, 32}, () = 3 N(A) 3 1
P(A)( ) 9 3
= = =
.
, = {12}, () = 1 N( ) 1
P( )( ) 9
= =
.
, = {12, 21, 22, 32, 33}, () = 5 N( ) 5
P( )( ) 9
= =
.
-
. , . ,
88
, 6, 7, 11,
13
. . .
GI_A_ALG_2_1288 ) : 2x 10x 21 0 + < . (M 12) ) : 2A | x 3 | |
x 10x 21|= + + .
i) 3 x 7< < , : 2A x 11x 24= + . ( 8) ii) x (3, 7), A 6= .
( 5)
: 2.2, 2.3, 3.3, 4.2 :
) 2x 10x 21 + 2( 10) 4 1 21 16 = = ,
10 4 147
( 10) 16 10 4 2 2x
10 4 62 1 23
2 2
+ = = = = =
= =
:
+3 7x
2x 10x 21 + + +
2x 10x 21 0 3 x 7 + < < < .
) i) 3 x 7< < , 2x 10x 21 0 + < , 2 2 2| x 10x 21| (x
10x 21) x 10x 21 + = + = + . , 3 x 7< < 3 3 x 3 7 3 < <
0 x 3 4< < , x 3 > 0 |x 3| = x 3.
: 2 2 2A | x 3 | | x 10x 21| x 3 x 10x 21 x 11x 24= + + = + = +
.
ii) x(3, 7) : 2 2 2A 6 x 11x 24 6 0 x 11x 24 6 x 11x 30 0= + = =
+ + + = ().
2x 11x 30 + 2( 11) 4 1 30 1 = =
11 1 126
( 11) 1 11 1 2 2x
11 1 102 1 25
2 2
+ = = = = =
= =
.
() x = 6 x = 5, .
-
. , . ,
89
, 21
. . .
GI_A_ALG_2_1293
T ( )C , x , : 15 25x = + , 0 x 200 . ) 30
. . ( 7)
) 290 C
. -
. ( 10)
) ,
440 C
; . ( 8)
: 6.3 :
) x = 30 :
= 15 + 2530 = 15 + 750 = 765, 30
765 C
.
) x[0, 200] : 290 15 25x 290 25x 290 15 25x 275 x 11 = + = = = =
,
290 C
11 km.
) x[0, 200] : 440 15 25x 440 25x 440 15 25x 425 x 17 > + >
> > >
x[0, 200] x(17, 200]. 440 C
17 km
200 km.
-
. , . ,
90
, 6, 13
. . .
GI_A_ALG_2_1297
) : 23x 4x 1 0 + . ( 12) ) , ,
3 6
9
+ . ( 13)
: 2.2, 4.2 :
) 23x 4x 1 + 2( 4) 4 3 1 16 12 4 = = = ,
4 2 61
( 4) 4 4 2 6 6x
4 2 2 12 3 6
6 6 3
+ = = = = =
= =
:
+1
31
x
23x 4x 1 + + +
21
3x 4x 1 0 x 13
+ .
) , , :
1 11 3 3 3 1 1 3 3
3 3
1 1
1 6 6 6 1 2 6 63 3 ,
1 2 3 6 3 6+ + + 3 3 6 9 + 3 3 6 9
9 9 9
+
1 3 6
13 9
+ ,
3 6
9
+ 23x 4x 1 0 + .
-
. , . ,
91
, 5, 11
. . .
GI_A_ALG_2_1298 , :
2+ = 2 2 30 + = ) : 15 = . ( 10) ) ,
. ( 15)
: 2.1, 3.3 :
) :
2 2
2 2 2 2
30 ( ) 30 2 30 15
+ = + = + = + =
+ = + = = = .
) 2x Sx P 0 + = , S 2= + = P 15= = .
2x 2x 15 0 = . 2x 2x 15 2( 2) 4 1 ( 15) 4 60 64 = = + =
2 8 105
( 2) 64 2 8 2 2x
2 8 62 1 23
2 2
+ = = = = =
= =
.
2x 2x 15 0 = x = 5 x = 3, ( , ) (5, 3) ( , ) ( 3, 5) = = .
-
. , . ,
92
, 8
. . .
GI_A_ALG_2_1300
: ( ) ( ) ( )6 6 63 62 , 3 , 6 = = = . ) : 23 + + = . ( 13)
) : 3 63 , 6 . .
( 12)
: 2.4 :
) :
( ) ( ) ( )36 2 3 2
32 2 2 2 8
= = = = = , ( ) ( ) ( )26 3 2 3
23 3 33 3 3 3 9
= = = = =
( )66 6 6 = = . 8 9 6 23+ + = + + = .
) :
( )26 6 3
23 6 3 6 33 6 3 6 3 6 3 6 9 6> > > > > ,
, .
3 63 6> .
-
. , . ,
93
, 16
. . .
GI_A_ALG_2_1301 ( ) : 4 2 10 = . ) = 5. ( 12)
) 33,
. ( 13)
: 5.2 :
) *, , N
*1
( 1) , = + N . :
4 1 1(4 1) 3 = + = + ,
2 1 1(2 1) = + = +
4 2 1 1 1 110 3 ( ) 10 3 10 = + + = + =
2 10 5 = = .
) :
1 2 3 1 1 1 133 2 33 3 3 33 + + = + ++ + = + =
1 1 1 1 13 3 5 33 3 15 33 3 33 15 3 18 6 + = + = = = = .
-
. , . ,
94
, 19
. . .
GI_A_ALG_2_1302
f, 8 x x 0
f (x)2x 5 x 0
-
. , . ,
95
, 7, 12
. . .
GI_A_ALG_2_1305 ) | x 4 | 3+ . ( 12)
) 1 - , | 4 | 3= + .
. ( 13)
: 2.3, 4.1 :
) | x 4 | 3 x 4 3 x +4 3 x 3 4 x 3 4 x 7 x 1+ + . ) :
1 4 1 4 4 3 + + + - , + 4 > 0 | + 4| = + 4. , :
1 1 1 1 1 0 + + + - , | + 1| = + 1. :
| 4 | 3 4 3 1 1= + = + = + = + .
-
. , . ,
96
, 3, 4
. . .
GI_A_ALG_2_1506 { }1,2,3,4,5,6 =
{ }1,2,4,5 =
{ }2,4,6 = . ) N Venn , ,
. , , , . ( 13)
) . -
:
(i) . ( 4)
(ii) . ( 4)
(iii) , . ( 4)
: 1.1, 1.2 :
) :
{1, 2, 4, 5, 6} = , {2, 4} = ,
{3, 6} = {1, 3, 5} = .
) (i) ( ) 2 1
( )( ) 6 3
= = =
.
(ii) ( ) 2 1
( )( ) 6 3
= = =
.
(iii) ( ) 5
( )( ) 6
= =
.
1
2
4
5
3
6
-
. , . ,
97
, 9, 11
. . .
GI_A_ALG_2_1509
2x ( 1)x 6 0 + = (1) R . ) 1, . ( 13) ) = 2 (1). ( 12)
: 3.1, 3.3 :
) 2x ( 1)x 6 0 + = 1, : 21 ( 1) 1 6 0 1 1 6 0 1 1 6 8 8 + = + +
= = = = .
) = 2 2 2x (2 1)x 6 0 x x 6 0 + = + = ().
2x x 6 + 2( 1) 4 1 6 1 24 23 0 = = = < , () .
-
. , . ,
98
, 6, 11, 13
. . .
GI_A_ALG_2_1512
) : 2x x 2 0 = . ( 8) ) : 2x x 2 0 >
. ( 12)
) 4
3 .
4
3
(); .
( 5)
: 2.2, 3.3, 4.2 :
) To 2x x 2 2( 1) 4 1 ( 2) 9 = =
1 3 42
( 1) 9 1 3 2 2x
1 3 22 1 21
2 2
+ = = = = =
= =
.
2x x 2 0 = x = 2 x = 1. ) :
+1 2
x
2x x 2 + +
2x x 2 0 > x < 1 x > 2.
) 4
3 ,
41
3 < .
+1 2
4
3
-
. , . ,
99
, 6, 12, 16
. . .
GI_A_ALG_2_1513 ( ) 1 1 = 3 9 = . ) . ( 12)
) , 30 > . ( 13)
: 2.2, 4.1, 5.2 :
) *, , N 1 1 = ,
*1
( 1) 1 ( 1) , = + = + N .
3
1 (3 1) 1 2 = + = + ,
1 + 2 = 9 2 = 9 1 2 = 8 = 4.
) *1 ( 1) 4 1 4 4 4 3, = + = + = N .
*N 33
30 4 3 30 4 3 30 4 33 8,254
> > > + > > > ,
9.
-
. , . ,
100
, 3, 4
. . .
GI_A_ALG_2_1520 , 50% , 40%
, 10% . -
. :
:
:
:
)
. ( 12)
) .
( 13)
: 1.1, 1.2 :
( ) 50% 0,5 = = , ( ) 40% 0,4 = = ( ) 10% 0,1 = = . ) :
= , ( ) ( ) ( ) ( ) 0,5 0,4 0,1 0,8 80% = + = + = = .
) :
( ) = ,
( )( ) 1 ( ) 1 0,8 0,2 20% = = = = .
-
. , . ,
101
, 19, 20,
21
. . .
GI_A_ALG_2_1529 f (x) x , ,= + R , : f (0) 5= f (1) 3= . ) = 2 =
5. ( 12) ) f
xx yy . ( 7) ) f. ( 8)
: 6.1, 6.2, 6.3 :
) :
f (0) 5 0 5 5 5 5 5
f (1) 3 1 3 3 5 3 3 5 2
= + = = = = =
= + = + = + = = = .
) = 2 = 5 f (x) 2x 5= + .
f
C yy x = 0, :
f(0) = 20 + 5 = 5, (0, 5).
fC xx y = 0, :
5 5f (x) 0 2x 5 0 2x 5 x x
2 2
= + = = = =
,
5
B , 02
.
) f
:
-
. , . ,
102
, 19
. . .
GI_A_ALG_2_1532
3x 16x
f (x)x 4
=
.
) f , x
, 2f (x) x 4x= + . ( 15) ) x f(x) = 32. ( 10)
: 6.1 :
) x 4 0 x 4, f
A {4}= R . , x 4 :
3 22x 16x x(x 16) x(x 4)(x 4)f (x) x(x 4) x 4x
x 4 x 4 x 4
+= = = = + = +
.
) x 4 : 2 2f (x) 32 x 4x 32 x 4x 32 0= + = + = ().
2x 4x 32+ 24 4 1 ( 32) 144 = =
4 12 84
4 144 4 12 2 2x
4 12 162 1 28
2 2
+ = = = = =
= =
.
() x = 4 () x = 8 x = 8.
-
. , . ,
103
, 11
. . .
GI_A_ALG_2_1533
2x 2x 2 0+ + = R . ) . ( 10)
) 1 2
x , x
: 1 2 1 2
x x 2(x x ) 1 + = . ( 15)
: 3.3 :
) To 2x 2x 2+ + 22 4 1 ( 2) 4 4 8 12 4 = = + = .
2x 2x 2+ + 12
0 12 4 0 4 12 34
.
) 3 1 2
x , x 2x 2x 2 0+ + = , Vieta :
1 2
2S x x 2
1= + = =
1 2
2P x x 2
1
= = = .
3 :
1 2 1 2x x 2(x x ) 1 2 2 ( 2) 1 2 4 1 + = = + =
2 4 1 1 = + = , .
-
. , . ,
104
, 11, 19
. . .
GI_A_ALG_2_1537
1
f (x) x , x 0x
= + .
) : 1
A f f (1) f (2)2
= +
. ( 10)
) : 5
f (x)2
= . ( 15)
: 3.3, 6.1 :
) :
1 1 1 1 1 4 5f 2
12 2 2 2 2 2
2
= + = + = + =
, 1
f (1) 1 21
= + = , 1 4 1 5
f (2) 22 2 2 2
= + = + = ,
1 5 5
A f f (1) f (2) 2 22 2 2
= + = + =
.
) x 0 :
5
f (x)2
= 1 5
xx 2
+ = 1 5
2x x 2x 2xx 2
+ = 22x 2 5x+ =
22x 5x 2 0 + = ().
22x 5x 2 + ( )25 4 2 2 9 = =
5 3 82
( 5) 9 5 3 4 4x
5 3 2 12 2 4
4 4 2
+ = = = = =
= =
.
() x = 2 1
x2
= , .
-
. , . ,
105
, 6
. . .
GI_A_ALG_2_1541 x y ,
. x y : 4 x 7 2 y 3, :
)
. ( 10)
) x 1 y ,
-
. ( 15)
: 2.2 :
) :
4 x 7 24 2x 27 8 2x 14, 2 y 3 22 2y 23 4 2y 6,
8 + 4 2x + 2y 14 + 6 12 20, = 2x + 2y .
12 20.
) :
4 x 7 4 1 x 1 7 1 3 x 1 6 23 2(x 1) 26 6 2(x 1) 12,
2 y 3 32 3y 33 6 3y 9 26 23y 29 12 6y 18,
6 + 12 2(x 1) + 6y 12 + 18 18 30, .
18 30.
x
y
x
3y
-
. , . ,
106
, 5, 20
. . .
GI_A_ALG_2_1542
) : 3 2A x x 3x 3= + . ( 13)
) 3
f (x)x
=
2g(x) x x 3= + , (1, 3). ( 12)
: 2.1, 6.2 :
) 3 2 2 2A x x 3x 3 x (x 1) 3(x 1) (x 1)(x 3)= + = + = + .
) *fA = R gA = R .
f, g
f (x) g(x)= , x 0. x 0 :
23 x x 3x= + 3 23 x x 3x= + 3 2x x 3x 3 0 + =
2(x 1)(x 3) 0 + = x 1 0 = 2x 3 0+ = () x = 1. f, g
1, A(1, f (1)) (1, 3), f(1) = 3.
-
. , . ,
107
, 7, 13
. . .
GI_A_ALG_2_1544
) 2x 4x 5 0+ + > , x. ( 10) ) :
2 2B x 4x 5 x 4x 4= + + + + . ( 15)
: 2.3, 4.2 :
) 2x 4x 5+ + 24 4 1 5 16 20 4 = = = ,
2x 4x 5 0+ + > xR.
) 2x 4x 5 0+ + > xR, 2 2x 4x 5 x 4x 5+ + = + +
2 2x 4x 4 (x 2) 0+ + = + xR, 2 2x 4x 4 x 4x 4+ + = + + .
:
( )2 2 2 2B x 4x 5 x 4x 4 x 4x 5 x 4x 4= + + + + = + + + + 2 2x
4x 5 x 4x 4 1= + + = .
-
. , . ,
108
, 20
. . .
GI_A_ALG_2_1553
3f (x) x= g(x) x, x= R . ) f, g
, . ( 13)
) , , ,
(0, 0), , . ( 12)
: 6.2 :
) f, g
f (x) g(x)= . xR :
3x x= 3x x 0 = 2x(x 1) 0 = x(x 1)(x 1) 0 + = x = 0 x 1 0 = x 1
0+ = x = 0 x = 1 x = 1. x = 0, x = 1, x = 1 f(0) = 0, f(1) = 1,
f(1) = 1, (0, 0), A(1, 1), B(1, 1).
) , (0, 0),
.
-
. , . ,
109
, 6, 19, 21
. . .
GI_A_ALG_2_2212
f ( )22x 6 x
f x2 x 6
=
.
) f . ( 10)
) ( )f x x=
x . ( 10)
) f x 0> . ( 5)
: 2.3, 6.1, 6.3 :
) 2 x 6 0 2 x 6 x 3 x 3 x 3 ,
{ }3, 3 ( , 3) ( 3, 3) (3, ) = = + . ) x :
( )( )( )
22 2 x x 32x 6 x 2 x 6 xf x x
2 x 6 2 x 6 2 x 3
= = = =
.
) x > 0 f(x) = x,
, (0, 0), (3, 3) .
:
B
O
-
. , . ,
110
, 7
. . .
GI_A_ALG_2_2702 2x 4 =
x 3 = , x .
) 2 x 3 < x 1 + = . ( 16) ) [ )x 2, 3
2+ = ;
. ( 9)
: 2.3 :
) :
2 x < 3 22 2x < 23 4 2x < 6 4 4 2x 4 < 6 4 0 2x 4
< 2, 2x 4 0 |2x 4| = 2x 4,
2 x < 3 2 3 x 3 < 3 3 1 x 3 < 0, x 3 < 0 |x 3| = x +
3.
:
2x 4 x 3 (2x 4) ( x 3) 2x 4 x 3 x 1 + = + = + + = + = .
) ( ) [ )x 2, 3
2+ = .
x 1 = 2 x = 2 + 1 x = 3 (), [ )x 2, 3
2+ = .
-
. , . ,
111
, 19, 20,
21
. . .
GI_A_ALG_2_3378 -
f.
) . ( 6)
) :
x 2 1 1 2
y 1 3 ( 6) ) .
( 6)
)
. ( 7)
: 6.1, 6.2, 6.3 :
) f
xx , [ ]fA = 2, 6- . )
fC 2 (2, 3).
f
C 1 (1, 0).
f
C 1 (0, 1).
f
C 1 (1, 0).
f
C 2 (2, 3).
-
. , . ,
112
f
C 3 (6, 3).
:
x 2 1 0 1 2 6 y 3 0 1 0 3 3
) f
C xx (1, 0), (1, 0), (4, 0).
f
C yy (0, 1).
) [ ]fx A = 2, 6 - fC xx , x ( 1,1) (4, 6] .
-
. , . ,
113
, 19, 20,
21
. . .
GI_A_ALG_2_3379
f.
) . ( 6)
) :
x 3 1 0 3
y 2 4 ( 6) ) .
( 6)
)
. ( 7)
: 6.1, 6.2, 6.3 :
) f
xx , [ ]fA = 3, 8- . )
fC 3 (3, 0).
f
C 1 (1, 2).
-
. , . ,
114
f
C 0 (0, 3).
f
C 3 (3, 6).
f
C 2 (7, 2).
f
C 4 (8, 4).
:
x 3 1 0 3 7 8 y 0 2 3 6 2 4
) f
C xx (3, 0) (6, 0).
f
C yy (0, 3).
) [ ]fx A = 3, 8 - fC xx , x ( 3, 6) .
-
. , . ,
115
, 6, 8, 13
. . .
GI_A_ALG_2_3380 ( ) 2f x = 3x + 9x 12, x . ) ( )f x 0
. ( 13)
) 3 2 ().
. ( 12)
: 2.2, 2.4, 4.2 :
) ( ) 2 2f x = 3x + 9x 12 = 3(x + 3x 4), x . 2x + 3x 4
23 4 1 ( 4) 25 = = ,
3 5 21
3 25 3 5 2 2x
3 5 82 1 24
2 2
+ = = = = =
= =
:
+4 1
x
2x 3x 4+ + +
x : ( ) 2 2f x 0 3(x + 3x 4) 0 x + 3x 4 0 4 x 1 .
) : 3
33 32 1 2 1 2 1> > > , , , 3 2 .
-
. , . ,
116
, 11, 19,
20
. . .
GI_A_ALG_2_3381
g, ( )22x 4x
g xx 1
+=
+.
g ( )1, 4 , ) 6= . ( 9) ) . ( 9)
) = 6 . ( 7)
: 3.3, 6.1, 6.2 :
: , ,
.
(), () .
) x 1 0+ x 1 , { }g 1 = . g
( )A 1, 4 , :
( )22 1 4 1 2 4
g 1 4 4 4 2 8 61 1 2
+ += = = + = =
+.
) { }g 1 = .
) = 6 ( )2 22x 4x 6 2(x 2x 3)
g xx 1 x 1
= =
+ + ().
2x 2x 3 ( ) ( )
22 4 1 3 16 = =
2 4 63
( 2) 16 2 4 2 2x
2 4 22 1 21
2 2
+ = = = = =
= =
.
( )( ) ( )( )2x 2x 3 x ( 1) x 3 x 1 x 3 = = + ,
x 1 ( ) 2(x 1)(x 3)g x 2(x 3) 2x 6x 1
+ = = =
+.
-
. , . ,
117
, 8, 9
. . .
GI_A_ALG_2_3382
3 5
5 3 5 3 = +
+.
) = 4. ( 12)
) x + = 1. ( 13)
: 2.4, 3.1 :
) ( )
( )( )( )
( )( )3 5 3 5 5 33 5
5 3 5 3 5 3 5 3 5 3 5 3
+ = + = +
+ + +
( ) ( )( )( ) 2 2
3 5 3 5 5 3 3 5 3 3 5 5 5 3
5 3 5 3 5 3
+ + + + = =
+
2 2
3 5 3 5 84
5 3 2 2
+ += = = =
.
) x 1 x 4 1 x 4 1 x 4 1+ = + = + = + =
x 1 4 x 1 4 x 3 x 5 = = = = .
-
. , . ,
118
, 3, 4
. . .
GI_A_ALG_2_3383 70% , 40%
20% .
. :
A:
: .
) :
i)
ii)
iii) ( 9) ) :
i) ( 7)
ii) . ( 9)
: 1.1, 1.2 :
) :
i) = ii) = . iii) = .
) ( )P 70% 0,7 = = , ( )P 40% 0,4 = = ( )P 20% 0,2 = = .
( ) ( ) ( ) ( )P P P P 0,7 0, 4 0,2 0,9 = + = + = .
i) ( ) ( )P 1 P 1 0, 4 0,6 = = = .
ii) ( ) = , ( )( ) ( )P 1 P 1 0,9 0,1 = = = .
-
. , . ,
119
, 3, 4
. . .
GI_A_ALG_2_3384 180 , 20 ,
30 , 10
. . :
A:
:
) :
i)
ii)
iii) ( 9) ) :
i) . ( 9)
ii) . ( 7)
: 1.1, 1.2 :
) :
i)
= ii)
= iii) = .
) :
( ) ( ) 20 1P = =( ) 180 9
=
,
( ) ( ) 30 1P = = =( ) 180 6
( ) ( ) 10 1
P = =( ) 180 18
=
.
20 30 10 40 2( ) ( ) ( ) ( )
180 180 180 180 9 = + = + = = .
i) ( ) = ,
( ) 2 7( ) 1 ( ) 1
9 9 = = = .
ii) 30 10 20 1
( ) ( ) ( )180 180 180 9
= = = = .
-
. , . ,
120
, 17
. . .
GI_A_ALG_2_3828 2, 2
7 4, , + , , ( ) .
) = 4 . ( 12)
) i) 2 , 5 4
1
. ( 6)
ii) ( )2 5 1 44 + = + . ( 7)
: 5.3 :
) 2, 2 , 7 4 + , :
( ) ( )( )2 2 22 7 4 2 4 7 14 4 8 = + = +
2 2 27 4 14 4 8 0 3 10 8 0 + = = ().
23 10 8 ( ) ( )210 4 3 8 196 = =
10 14 244
( 10) 196 10 14 6 6
10 14 4 22 3 6
6 6 3
+ = = = = =
= =
.
() = 4 2
3
= (, )
2, 8, 32 8 32
42 8
= = = .
) i) *, , N = 4, 1
1 1 *1 1 4 ,
= = N .
2 14 = , 4
5 1 14 256 = = 3
4 1 14 64 = = . ii) , ( ) ( )2 5 1 1 1 1 1 44 256 4 64 4 + = + =
+ = + .
-
. , . ,
121
, 11
. . .
GI_A_ALG_2_3839
: ( )2x 1 x 1 0 = , 0. ) 2. ( 12)
) 0.
( 13)
: 3.3 :
) 2 ( )2x 1 x 1 0 = , :
( ) ( ) ( )2 2 1 2 1 0 4 2 2 1 0 6 3 0 = + = = 3 1
6 3 6 2
= = = .
) 0, ( )2x 1 x 1 0 = . ( )2x 1 x 1
( ) ( )2 2 2 2 2 1 4( 1) 1 4 2 1 4 2 1 ( 1)= = + = + + = + + = +
. 0 2( 1) 0 0+ , 0 .
-
. , . ,
122
, 11
. . .
GI_A_ALG_2_3847
( ) 2 2 x 2x 1 0+ + + = , 2 . :
) . ( 13)
) 2. ( 12)
: 3.3 :
) 2 2 0 + , ( ) 2 2 x 2x 1 0+ + + = .
( ) 2 2 x 2x 1 0+ + + =
( ) ( )( )2 2 2 2 2 2 4 2 1 4 4( 2 2) 4 4 4 8 8= + = + = + + 8 4
4(2 )= = .
( ) 2 2 x 2x 1 0+ + + = :
> 0 4(2 ) > 0 2 > 0 < 2. 2, ( ) ( ) , 2 2, 2 .
) 1 2x , x ( ) 2 2 x 2x 1 0+ + + = ,
Vieta 1 22
S x x2
= + =
+.
( ) ( ) , 2 2, 2 :
( )2S 2 2 2 2 2 2 2 4 2 2 4 2
= = = + = + = +
4 4 1 = = , .
-
. , . ,
123
, 6
. . .
GI_A_ALG_2_3852 , : 2 4 4 3 . :
) 2 ( 12)
) 2 2 ( 13)
: 2.2 :
) :
2 4 , ( ) ( )4 3 2 4 2 2 3 8 2 6 6 2 8 ,
2 6 2 4 8 8 2 12+ + , 2 8 12.
) 2 2 4 8 2 12 . ( )
22 8 ( 2) 4 12 16 2 48 ,
2 2 16 48.
-
. , . ,
124
, 5, 11
. . .
GI_A_ALG_2_3857 , :
4 = 2 2 20+ = ) : 5+ = . ( 10) ) 2 ,
. ( 15)
: 2.1, 3.3 :
) : 2 2 ( ) 20 4( ) 20 520
4 4 44
+ = + = + = + =
= = = = .
) 2 2x Sx P 0 + = , S 5= + = P 4= = .
2 2x 5x 4 0 + = (). 2x 5x 4 + 2( 5) 4 1 4 9 = =
5 3 84
( 5) 9 5 3 2 2x
5 3 22 1 21
2 2
+ = = = = =
= =
.
() x = 1 x = 4, (, ) = (1, 4) (, ) = (4, 1).
-
. , . ,
125
, 5, 11
. . .
GI_A_ALG_2_3863 , :
1+ = 3 2 2 3 2 12+ + = ) : 12 = . ( 10) ) 2 ,
. ( 15)
: 2.1, 3.3 :
) :
3 2 2 3 2 2 2
1 1 1
2 12 ( 2 ) 12 ( ) 12
+ = + = + =
+ + = + + = + =
2
1 1
12( 1) 12
+ = + =
= = .
) 2 2x Sx P 0 + = , S 1= + = P 12= = .
2 2x x 12 0+ = (). 2x x 12+ 21 4 1 ( 12) 49 = =
1 7 63
1 49 1 7 2 2x
1 7 82 1 24
2 2
+ = = = = =
= =
.
() x = 3 x = 4, (, ) = (3, 4) (, ) = (4, 3).
-
. , . ,
126
, 5, 6
. . .
GI_A_ALG_2_3870
: 2 2 2 9= + + ( ) 2 3 = , , . ) : = ( ) ( )2 2 2 2 6 9+ + + + .
( 3) ) : , , . ( 10)
) , = ; - . ( 12)
: 2.1, 3.3 :
) , :
( ) ( )2 2 2 2 2 2 9 2 3 9 6 2 = + + = + + + + ( ) ( )2 2 2 2 6
9= + + + + .
) () , , :
( ) ( )2 2 2 2 2 2 6 9 ( ) ( 3) = + + + + = + + . ( )2 0+ (
)
