CIP ) / :,2006.2 ISBN 7-80702 - 113-6..........G.206 CIP (2006)080133 :8501168mm 1/ 32 : 120 :2000 2006 3 1 : 5000 ISBN 7-80702 - 113-6/ G206 : 348.00 (16 )
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,,,,,,,,,,,,,,,!,!,! :! 2006 1
(167) (172) (176) (185) (190) (196) (201) (214) (222) (222) (227) (230) (232) (236) (239) (243) (246) (248) (253) (253) (257) (258) (261) 1
(264) (266) (267) (268) (269) (270) (272) (273) (275) (276) (278) (281) (282) (283) (284) (285) (287) (288) (290) (291) (293) (296) (298) (301) 2
(303) (307) (309) (311) (312) (313) (314) (316) (320) (324) 3
,,, 10, : 1, : 6 X + 6 y = 1 2 8 X - 3 y = 3 10 : 1 X = a, 1 y = b, 6a + 6b = 1 2 8a - 3b = 3 10 a = 1 20, b = 1 30 X = 20 y = 30(, ) 167
,, A, A, : : 3 7 + 5 2 + 3 7-5 2 : A = 3 7 + 5 2 + 3 7-5 2 A 3 = ( 7 + 5 2 ) + ( 7-5 2 ) + 3 3 3 7-5 2 ( 7 + 5 2 + A 3 = 14-3A, A = 2, 3 7 + 5 2 + 3 3 7-5 2 = 2 3 3 7-5 2 ) 7 + 5 2,,,, : 198920002000-200019891989 1989 = X, 2000 = X + 11,, : 1989 = X, 2000 = X + 11. = X10000(X + 11) + (X + 11) - (X + 11) ( 10000X + X) = 10001X(X + 11) - 10001X(X + 11) = 0 4, : 168 ( 3 X + 1-1) 4 + ( 3 X + 1-3) 4 = 16
y = 3 X + 1-2 (y + 1) 4 + ( y - 1) 4 = 16 y 4 + 6y 2-7 = 0 y 2 = 1 y 2 = - 7 () 3 X + 1-2 = 1 X1 = 26, X2 = 0 5,,,, : 2(X + X 2-1 ) = (X - 1 + X + 1) 2 : X1, y = X - 1, X - 1 = y 2, X = y 2 + 1,, 2(y 2 + 1 + y y 2 + 2 ) = (y 2 + y 2 + 2 ) 2 y(y - 1) (y 2 + y + 2 y 2 + 2 ) = 0 y 2 + y + 2 y 2 + 2 > 0, y = 0 y = 1, X1 = 1, X2 = 2 6 AX 2 + Ay 2 + BXy + CX + Cy + E,,,, : X 2 + Xy + y 2 = 3, M = X 2 + y 2 : X = m - n, y = m + n X 2 + Xy + y 2 = 3 3m 2 + m 2 = 3, M = X 2 + y 2 = 2m 2 + 2n 2 = 2m 2 + 2 (3-3m 2 ) = 6-4m 2 169
m1, Mmax = 6, Mmin = 2 7,, : 3 (5 - X) 2 + 3 (4 + X) 2 = 3 20 + X - X 2 + 3 : 3 5 - X = u, 3 4 + X = v u3 + v 3 = 9, u 2 + v 2 = - uv = 3. u = 2 v = 1 X = 4, X 2 = - 3 u = 1 v = 2 8,,, : sinx + cosx + sinxcosx = 1 : sinx + cosx = t( - 2t2 ) sinxcosx = 1 2 ( t2-1),, t 2 + 2t - 3 = 0 t = 1 t = - 3( ) sinx + cosx = 1, X = k( - 1) k 4 - ( kz) 4 9 A B = C D = E F, k, k, : X y z X - 1 = y + 1 2 + y 2 + z 2 = z - 2 3 X y z X 2 170
X - 1 = y + 1 = z - 2 = k 2 3 X = k + 1,y = 2k - 1, z = 3k + 2 X 2 + y 2 + z 2 = 14k 2 + 10k + 6 = 14( k + 5 14 ) 2 + 59 14 59 14, k = - 5 14, X = 9 14, y = - 12 7, z = 13 14, X 2 + y 2 + z 2 59 14 10,,, : : sina + sin3a + sin5a = a cosa + cos3a + cos5a = b (1) b0, tg3a = a b (2) (1 + 2cos2A) 2 = a 2 + b 2 (1986 ) : z = cosa + isina, z + z 3 + z 5 = b + ai z = 1 z 2 + 1 z 2 = 2cos2A z + z 3 + z 5 = z 3 (1 + z 2 + 1 z 2 ) = z 3 (1 + 2cos2A) (1 + 2cos2A)(cos3A + isin3a) = b + ai, b0, tg3a = a b z 3 (1 + 2cos2A) = b + ai, (1 + 2cos2A) 2 = a 2 + b 2,, 171
:, : 1,,,,,,,,, : P, P X Y C, h, PX + PY + PC A :,,,,, : P, P XY, h, PX + PY 1, PX = APtga, PY = BP tga, PX + PY = tga( PA + PB) = tga AB = 2tga AB 2 = 2h ) 172 PX + PY + PC = 3h(a a X Y P C h 1 a B
, M, ( ),, : M - PQ - N A, M AG PQ 45AG N 30, M - PQ - M : 2, AG C, C CBN D, AD, CAD AG N, CAD = 30, C CBPQ B,BD, CBD AC = a, Rt ABC,BC = ACsin45= 2, RtADC, DV = ACsin30= 2a 1 DC a, RtBDC, sincbd = 2 BC = 45, 45 N 1 2 a 2 2 a G P C D B 2 A Q = 2 2, CBD = :, GAB = 45GAD = 30, N GAD, GBD GBD,,, GAB = 45 BG, GAD = 30 GD, DG BG GBD RtGDB, 3,,,, : 3, ABCD - A B C D, a, G 173
B C BG : G B1C EFPQ, E F PQPQ A M, GM, GMB C, MGB BC BG D D E G P Q M B C C MB, GMB, GM = 2 2 a A F B 3 GB = a 2 + ( 2 2 a)2 = 6 2 a BM = PB2 + PM 2 = ( 1 2 a)2 + a 2 + ( 1 2 a)2 = 6a cosmgb = MGB = arccos 4 = 3 6 ( 6 2 a) 2 + ( 2 2 a)2 - ( 6 2 a)2 3 6 2 6 2 a2 2 a,,, 2a, a, A A1 O1 O 4,,,? : C C 1 174
ABCD A1 B1 C1 D1 O1 O ABCD, AA CC AA C 4, AA C C, O1 O, AC AC1 = 2 3a AO = 3a, AS = ( 3-1) a O1 r, AO1 = 3 r, AS = ( 3 + 1) r, ( 3-1 )a = ( 3 + 1) r,, r = 3-1 a(2-3 )a. 3 + 1 5,,,, : 5 10, 20,, :, = 10-5 20 360= 90, B A 5 A B O O B E A B A D A 5, X, 5 10 = x x + 20, X = 20, OA = 40, OB = 30, A B = 30 2 + 40 2 175
ODA B E, ED OD = OA OB = 24, OE = 20, ED = 4 A B,,, : 1,, (1) :, m, (A)m 2 + 1; ( B)m 2 + 2m + 1; ( C) m 2 + m; (D)m + 2 m + 1 : m, (B)(m + 1 ) 2, ( D) ( m ) 2, ( B) (D) 176 ( B) (D)m, m m
(2) B) (D ), (D) : cos = 1 5, 5 << 3, sin 2 2 (A) - ( C) - 10 5 ; (B) 10 5 ; 15 5 ; (D) 15 5 : 5 << 35 2 4 < 2 < 3 sin 2 2, - 2 2 ( C) > sin 2 > - 1 ( C), - (3) :, 85X - 32y = 101 (A)X = 329,y = 86 (B)X = 625,y = 171; ( C)X = 978, y = 256; (D)X = 1301, y = 341 2 2, :, 324y, 85X - 324y 101, 85X, X ( A) ( B) ( D), ( C ) X, ( C) (4) : C1 : X 2 + y 2 = 4 C2 : X 2 + y 2 (A); (B); ( C) ; (D) + 6X + 4y = 0 : C1 (0, 0 ) C2, 177
C C2,, (C) : ABC, cos(a - B) + sin(a + B) = 2, ABC (A) ; ( B) ; ( C) ; (D) : cos(a - B) 1, sin(a + B)1, cos(a - B) + sin(a + B) = 2, cos(a - B) = 1, sin(a + B) = 1 -< (A - B) = 1 0 < A + B <. A - B = 0 A = B A + B = C = 2 2 ABC, ( C) (5) : z z + z = 2 + i z (A) - 3 4 + i ; (B) 3 4 - i; ( C) - 3 4 - i; (D) 3 4 + i : 2, z, z < 2 z < 2 178 z = a + bi,, a + bi + z = 2 + i; a + bi = (2 - z ) + i,
a z > 0, b = 1 (D) a = 3 4 > 0 b = 1, (D) : y = 2 - arccos 1 2X - 3 (A) [ - 2, ] ; 2 (B) [ - 2,0] ; ( C) [0, ] ; 2 (D) [ - )C0, 2 2 ] : (A) ( B) ( C) y = 0, y = 0, 1 arccos 2X - 3 = 2, 1 = 0, ( A) ( B) 2X - 3 ( C) (D) 2,, : arccos( - X) > arccosx ( ) (A)X[0, ) ; 2 (B)X(0, 2 ) ; ( C)X( 0,1) ; (D)X[0,1 ] : arccos( - X) = - arccosx, arccosx [0,]arccos( - X) > arccosx - arccosx > arccosx arccosx < 2 01(C) 3, 179
0 < a <, sina + cosa = 1, tga : 5 (A) 4 3 ; ( B) 3 4 ; ( C) - 4 3 ; (D) - 3 4 :, a : 0 < a, sina + cosa1, 2 2 < a <, (A) ( B) : sina + cosa = 1 5 > O, sina > cosa, ( C) 4,, : {an}, a5 a6 = 9, log3 a1 + + log3 a10 = (A)12 ( B)10 ( C)8 (D)2 + log35 : a5 = 3, a6 = 3, 1 3, 3log3 a1 + log3 a2 + + log3 a10 = log3 al a2 a10 = log3 10 = 10 ( B) 5, ;,,, : (X 1) (X - 2 ) 2 (X - 3) (X - 4 ) 2 (X - 5) > 0 ( ) 180 (A) {X 1 < X < 3 > 5} ;
B) {X 1 < X < 3 4 < X < 5}; ( C) {X 1 < X < 2 2 < X < 3 4 < X < 5}; (D) {X 1 < X < 2 2 < X < 3 X > 5} : X = 6, X = 6 ( B) ( C), ( B) (C), X = 2, X = 2 (A) (A)(D) 6,, : ABC, cosa + sina = 7, A 12 (A)0 < A < 45 ; ( B)45< A < 90 ; ( C)A = 45 ; (D)A = 90 ; ( E) 90 < A < 180 :0 < A < 45, sina > 0 cosa > 2 2, sina + cosa > 2 2 > 7 12 45 < A < 90, sina + cosa > 2 2 > 7 12 A = 45, SinA + cosa = 2 2 + 2 2 > 7 12 A = 90, sina + cosa = 1 > 7 12 (A) (B) ( C) (D), ( E) 7 : X 2 + y 2 + GX + Ey + F = 0 X, 181
(A) F = 0, G0, E0; ( B) E = 0, F = 0, G0; ( C) G = 0, F = 0, E0; (D) G = 0, E = 0, F0 : X 2 + y 2 + GX + Ey + F = 0 X,, ( - G 2, - E ) y, G = 0, E0, 2 (A) (B) (D), (C) 8 ;, : a b c d m n, P = ab + cd; Q = ma + nc b m + d n (A) PQ; ( B) PQ; ( C) P > Q; (D) : PQ( > < ) : 2 abcd n m bc + m n ad n m bc + m n ad n m bcm n ad = 2 abcd 182 PQ, (A) : sina + cosa = - 17, a 13 (A); ( B) ; ( C); (D) : a, 1sina + cosa1 < 1,
( B) (D)( C) 9,,, ) : z z - 2-2i 1 ( (A)2 1; ( B)3 2; ( C) 3 1; (D)4 2 : z - 2-2i11 Z P( 2, 2 ), 1 ( ), OP EOP F, E( 2 2, 2 2 ), F ( 3 2 2, 3 2 2 ), z = OF = 3, z = OE = 1, ( C),,, 10,, : a 6 sina1 2 (A); ( B); ( C) ; (D) : : sina = 1 a =? 2 6 ( C) 11 183
,,, : a = 6-1 5, b = 6 + 5, a (A)a > b; ( B)a = b; ( C) a < b; (D)a = 1 b ) 1 : b = 6 + 5 = 6-5 (6 + 5 ) ( 6-5 ) = 6-5 < 5, (A) 31 12,,, : BC = a, S A ( ) (A)BC ; ( B) BC BC S a ; ( C) BC BC 2S a ; (D)BC BC 2S a : : A, BC, BC, ( A) (D), (D) S = 1 2 ah, h = 2S a, (D) 13,,,, 184
( (A); ) ( B) ; ( C) ; (D) : (D) 1988 :,,,,,,,,,,,,,,,,,,,,,,,, 8 : 1, 185
, :, ( ) :, :,, ;,,,,,, 2, :,, =, =, >,,, ( ) ( ) ; 2(+ ) (- ) ( ) 2(+ ) ; ( ) ; + - ) ( )( + :,,,,,,,,,,,, =, =, =, = = 0, (- ) = 0; 2(+ ) 2( + ) = 2 ; ( - ) + = 0; + = 2 = =, ( ) ( ) (, ( ) 186
,,,,, 3,,, : 1 4, 1 3 ( ), :, 1, (1-1 4 ) 1 = 3 4 ;, 1, (1-3 4 )3 4 = 1, 3 4,, : 10% 900, 100, 90% ( ), 900 10%, 900 : 90010% = 90 () 900 100, : 900 + 100 = 1000 ( ), :901000 = 9%, 5,,, :18 24 236( ) 187
,, 18 24 : 18 1, 2,3,6, 9,18 24 1, 2,3,4, 6,8, 12,24,18 24 1 2 3 6 1236 1, 18 24, 6,, :,, 2 + 2, 2 + 2 + 2 + 2, 2 + 2 + 2 + 2( ) :,, [ () ], ( ), = 2 + 2 ; = 2 + (+ ) 2, = (+ ) 2 + 2 =, =, =, =,, =, =, = 188, 2 + 2,
+ 2 + 2 + 2, 2 + 2 + 2 + 2,, () 7,,,, : 1 4, 1 4, ( ),, 1, 4 1000, : 1000( 1-1 4 ) = 10003 4 = 750 (), 1 4 : 750(1 + 1 4 ) = 7501 1 4 = 937.5() 8,, : 1, 4 ( ),, (),,,2 ;, 4, 189
,,,,,,, 1,, :,,,, (, 1980),,,, ( ),,,, = 4 ( ) = 4, = (- ) ( - ) ( - )= 1 2 ( + + ) = ( - ) ( - ) ( - ) 2 190 :, 3 3,
+ 3? :, 3 + 3 3-3 :,, ( 3 + 3 ) ( 3 2-3 2 ) = - 3 2 + 3 2 -, ( 3 + 3 ) ( 3 2-3 + 2 ) = +, 3-3 3 2-3 + 2 3, :, 100,,,,,,, 4,,,, (1),, (2),, (3) 191
,, :,= 2 3, = 15, = 3 + 3, : = 1 2, = 2 = 6 + 2, 2 = 2 + 2-2= 3 = 2 2, = = 3 2 : = 120,, = 2 3 < 6 + 2 =, = 60 5,,,,,,,, : 15, (- 1 ) 2 + 5 -, 6-2, = 4, 4, 6-2- 2, :, 192
,,,, 1, 1 180, 200 7,,, 8,,,,,,,, :, 4(9-2)? :, > 1 9-2 < 1 > 1 3 :> 1 3 4(9-2), 0 < 4< 1 0 < 9-2 < 1 2 9 < < 1 4, 193
(9-2 ),, { > 1 3 } { 2 9 < < 1 4 }, 4(9-2 ) 9,,,,, 10,,,,,,,,,,,, 194 : 1 2-5 2-17+ 10 = 3 (1) : ( 1), 2-5 2-17+ 10 = 8 (2) (2), 5 2-17+ 10 = 2-64 (3) (3), 5 2-17+ 10 = (2-64 ) 2 (4) 2-239- 4086 = 0
- 23973465 2 :, 12,,, ( 1) (2 ) ( 2) (3 ) ( 0 )(3) 2-64 0, (3) (4 ) 1 2 2-64, 0, ; 0,,1 2, 11,,,,, 12 ;,,,, 13,,,,,, 195
? 0 1 2 3 4 5 : 6, 0 5 ; 1 3 5, 1 2 ( 6-5 ) = 300( ) : 6, 0 2 4 5, 0 1 3 5 4, 6-35 - 34 = 288 (),, 1 2 3 4 5 ( 5-4 ), 0 5, 4,, 1 2 4 = 12( ) 14, :355 2 = 12625, 300 2 = 90000 > 12625,,,,,,,,,,,, 196
,,, : 2 2 + 2 5-1, : : = 2 2 + 2 (1-2) - 1 = 2 2-2 2 + 1 = ( 2-1) 2 = 2-1,,,,,, 2 : 40+ 10 : = = = = = 1 40 + 80 1 40 + 80 80 240+ 80 80 40+ (40+ 80 ) 80 40+ 260 20 80 197
40+ 20 80 = 230 10 10 = 3 9 1 + 2 + 3 + + 100,,,, 3 2 ( - ) + 2 ( - ) 2 (- ) : =, = 0, -,,,, ( ), (- ), = (- )( - )(- ) = 1, = 0,= 2, 2 = - 2, = - 1 = - (- ) ( - ) (- ) 4 { }, 12 > 0, 13 < 0, 1, 2, 3,, 12 12 = 6(1 + 12 ) = 6 (6 7 ) > 0 : 13 = 13 ( 1 + 13 ) = 13(2 7 ) < 0 2 2 6 + 7 > 0, 7 < 0, 6 > 0, 7 < 0 198 6 (),,
() ), ( ), 5 2 + + = 0, 2, : 2 + + = 0, :, =,=, 0,0 2, 2 = 2 = 2 = 2 = 2 3 3 = 1 ( )3 = 1 = - 1 2 3 2 6 (- 2+ 3) 7 3 3 :, 3 3, : (- 2+ 3) ( - 2+ 3) ( - 2+ 3), 3 7 ;, - 2, 3 4, 3 1 1 3 3 : = 3 7[ 3 4 ( - 2 ) 3 ] ( 1 13 ) = - 3360 199
, 1 2 ( 1 > 2 ),, :, :,, = : 1 = + = + 2 : - 1 = 2-1 1 2 = - 1 2-1 = 4 1 4 1 + 4 2 8, ( 9, 20 ) ( 49, 55),, :,, 2 ( 49,55) 2 ( 49, - 55), 2 = 2, 1 2, : 2= 1 + 2 = 1 2 = 200
2 + ( - 55-20) 2 = 85,,?,,, 1 1 > > 0, = 1, : (2 + 2 ) (- ) 2 2 :, + 2 (0 )?, = 1? = 1,, 2 + 2 -, 2 + 2 = ( - ) 2 + 2, - = (> 0), ( 2 + 2) 2 2,,+ 2 2 2 + 2 2 = 2 2,, 2 (, )(0, )> > 0, (,0) :,, 201
( 0, ), 2 : = -, = -, - - + = = 1 + 1 + ( - ) ( - ) = - + - 2 = - 2 = =, =,,, (,,, :,,,,,,,, 0),,?,,,,, =, =,, 2 202
= 1, : (+ + 1) + (+ + 1) + ( + + 1) = 1 :, 1 : : [( + + 1) (+ + 1) + (+ + 1 ) (+ + 1 ) + (+ + 1) ( + + 1 ) ]/ [ (+ + 1) (+ + 1) ( + + 1) ] = 1 = (+ + ) (+ + 1) + (+ + ) (+ + 1) + (+ + )( + + 1) = (+ + 1) (+ + 1 ) + ( + + 1) (+ + 1) + (+ + 1 ) ( + + 1) = (+ + 1) (+ + 1 + + + ) + ( + + 1) ( + + 1) = (+ + 1) (+ + 1 ) ( + 1) + ( + + 1)(+ + 1) = (+ + 1) ( + + 1) (+ + 1) =, = 1,,, :, + + 1 = = + + 1, + + 1 = + + + + = 1, + + 1 203
50, 15 /,, 20 /,,,,? :, 20: 15 = 4: 3, 50 4 7,, 503 7, 50 1 7 4 7, 50 1 7 4 7 ;, 504 7 1 7 2,, = 200 1 7 + 2001 7 2 + 200 1 7 3 + + 200 1 7 1 7, = 1 1-200 7 = 1-1 = 100 3 7,,,, :,, 50/ ( 15 + 15) = 5/ 3 (), 5/ 3, = 205/ 3 = 100/ 3 ( ) 204
,, 3 5 = :,, : =, = 0, = 0 = 2 2 ( ),, = 0 1 2, 0 2 4 2, 4 2-2 2-0,, =, : 6,, + + - > + : -,,,,,, + + - > +, + + > + +,,, =, : + > 205
,,,, = + > + + +,,,,,,,,, 4 7, -, =, = =, = =, - -, - :,,, =, =,, - - - 206
,,, + + = =,, = = - = - + - ( + ), = 1 3 + 1 3 = 1 3 = 1 3 [ 1 2 ( + )2 ] 3 = 6 ( + ) 2 8 3 2-6 ( - 1) + 2 + 1 = 0 1 2 1 + 2 = 2, :, : ( 1) ; ( 2),,, : 1 2, 1 2 = ( 2 + ) 3 1 + 2 = 1 + 2 = 2 2 = 36( - 1) - 12( 2 + 1 ) 0 1 + 2 = 2( - 1 ) = 2 : = 0. > 0 12,, 1 2 1 = 2 = 1 = 36( - 1) 2-12( 2 + 1 ) < 0 12 = ( 2 + 1) = 1 3 207
= 2 0 2,,, 5 9 : 2 + 2 + ( 1 - ) 2 + 2 + 2 + (1 - ) 2 + (1 - ) 2 + ( 1 - ) 2 2 2 :, (, ), ( 0, 0 ) (, 0 ), ( 1, ), (0, 1), 2 2, 2 2,, +, +, 10, 2-6+ 5 = :,,,, 208
= 2-6+ 5 = 5, - 2 + 6-5 =, 2 - ( - 6) + 5 = 0, = 0, = 62 5= 6 + 2 5, = - 5 ( ) ; = 6-2 5, = - 2 + 6-5(15 ) < > 5, 2-6+ 5 =, 2 - ( + 6) + 5 = 0.= 0, = - 6 2 5= - 6 + 2 5,= - 5 () ; = - 6-2 5, = 2-6+ 5( < 1),, : - 6-2 5 < < 0 ; = - 6-2 5; < - 6-2 5= 0 > 6-2 5; = 6-2 5 ; 0 < < 6-2 5,,,,,,,,, 6 11 1 2 3 ( 0, 1 ) : 1 + 2 + 3-12 - 13-23 < 1 : 1 2 3, ( 0,),,,,,,,, 1 2, 3,,, : 3 =, ( ) = 1 + 2 + - 12-1- 2 = (1-1 - 2 ) + 1 + 2-12 - 1-2 = 0, () = 1-12 < 1; 209
1-20, ( ), (0) = 1 + 2-12 = 1 - (1-1 ) ( 1-2 ) < 1 ( 1) = 1-12 < 1 ( ) < 1, 12,,,, () ; (2) : :,, 1,,,, 2,, 1,, 3,, 210,,
),,, = = 60, = = 60, = 120,, 120 ( ),,,,,,, 7 13, : = 1 2 + 1 + + 1 22 2 2 2 :,,,,? 1 2 2 = -, 1 2 2 = 1 2 2 = 2 1 + (4 2-1 ) - 1 + + = (2+ 1) - (2-1) 1 + (2+ 1) ( 2-1) 1 2 2 = (2+ 1) - (2-1) = ( 3-1 ) + ( 5-3) + + [ (2+ 1) - (2-1) ] = (2+ 1) - 4 211
{ } 1 = 2 = 1, 3 = 2, + 1 + 2 1, + 1 + 2 + 3 = + + 1 + + 2 + + 3, 1 + 2 + + 4 :, { }, + 3 = + + 1 + + 2 + 1 + + 2-1, 4 = 4, 5 = 6 = 1, 7 = 2, 8 = 4, { } 4 : + 1 + 2 + 3 = + + 1 + + 2 + + 3, + 1 + 2 + 3 + 4 = + 1 + + 2 + + 3 + + 4, ( - + 4 ) ( + 1 + 2 + 3-1) = 0 + 1 + 2 + 3 1, + 4 =, { } 4, 4+ 1 + 4+ 2 + 4+ 3 + 4+ 4 = 1 + 2 + 3 + 4 4 = 4, 1 + 2 + 3 + 4 = 8 1 + 2 + + 4= ( 1 + 2 + 3 + 4 ) = 8,, ( ) :1.(1) ;2.( ) (+ 1) 8 15 ( 15 3- ) 15 :,,,,, : : ( 15 3- ) 15 = 3 15 ( - ) 15 = - 15 3 + ( - 1) = 2 ( 15 3- ) 15, = = 1, ( 15 3-1 ) 15 212
15 3-1 ) 15-2 16 1 : 4+ - 4 = 0, 2 : + = 0 3 : 2-3 - 4 = 0, :,,,,,, 4+ - 4 = 0 : + = 0 : ( 4 - ) = 4 4, = = 12 2 (4 - ) - 4 = 0 4, = ( 4 - ) - 4 ( 4 - ) 4 (4 - ), = - 4 8 2-3 - 4 = 0, (4 - ) (4 - ) + = - 1 = 2 3 = - 1 = 2 3,, ( -, - 1 ) ( - 1, 2 3 ) ( 2 3, - ),,,,, 213
,,,,,, 1 1,, : 2 = :,,,, :, =, =, = 2, = 2, =, = = = 2, = = 2 2 = = 2 2 (2 ) 2 = 2 2 2 2 2 = 2 1,, =,2=,3= :,, + - : 1, 1 + -, 2 = ( + - ) 214
1, ( 1), = 1 (1 - ) 1 - =, 2= 1 (1-2 ) 1 - =, 3= 1 ( 1-3 ) 1 - =, 2 = 2 1 (1-2 ) 2 (1 - ) 2 (+ - ) = 1 (1 - ) 1-1 (1-2 + 1-3 - 1 + ) 1 - = 1 2 (1 - ) ( 1 + - 2-3 ) (1 - ) 2 = 2 1 (1 - ) ( 1 + ) (1-2 ) ( 1 - ) 2 = 2 1 (1-2 ) 2 ( 1 - ) 2, 2 = (+ - ) 1, + -,,, + -,, ( - 1), 2? 3, = (1 + 2 ) 3 2? :,, 0 << / 2, 1 + 2=, = ( 0 << 215
), : = (0 << 2 ), = = (1 + 2 ) 3 2 (1 + 2 ) 3 2 2 = 2 4 = 1 2 ( 3 2 2 2 2 ) 3 = 3 = 2 1 2 ( 2 2 + 2 + 2 ) 3 3 = 4 27 2 2 = 2, = = 2 2, 2 4 27 = 2 3 9 4, 2 + 6, : 60, =, =, + + = + + = = = 120,, ( + + ) = = ( 2 2 + 6 2 ), = 2 + 6, = 2 216
3 5 = 5 2-6 2 + 20-30+ 7 := (9 2-12 + 4 2 ) + 20-30+ 3 = (2-3+ 5) 2-22 = [ 13 ( 2 13-3 13 ) + 5]2-22, = 2 13, = 3 13, = [ 13 (- ) + 5 ] 2-22 ( - ) = 1, = 16 + 10 13 ; ( - ) = - 1, = 16-10 13 6 : (= 1, 2,, ), (11 + 22 + + ) 2 ( 2 1 + 2 2 + 2 ) ( 1 2 + 2 2 + + 2 ), = (= 1, 2,, ), : = ( 2 1 + 2 2 + + 2 ), = 11 + 2 2 + +, = 1 2 + 2 2 + + 2, (2) 2-4 0, () = 2-2+, ( ) = (1-1 ) 2 + ( 2-2 ) 2 + + ( - ) 2 0 0, > 0, = (2) 2-4 0, = 0 () = 0 = 1 1 = 2 2 = = = 0, 217
,,,,, 4 7 :, = : = :, ;, : = = = = 8 + + 2 = 3 + + 2 = 3 + + 2 = 3 :,,,,, :,, 3-2 + + = + + = - = - - = 0, 218
= - ( + + ) = + +, 5 9 : :, : 2, 3,5,,, = 235 +, >, 2, 3, 5,,,,, 10 ( = 1,2,, ) 1 1 + 2 + + = 0 ( 2 ) :, : + :, + 1, + > 1, 1, 1,,, (,,= 1,2,, ), 2 3, 2, 3 1 + 2 + = 0 :, (,,= 1, 2, ) + 1, + > 1,, 1, - 219
2 -, + 2-2 > 0 + 2-2 - 2 > - = - (- ) = ( + 2-2 - 2 ) 2 > - 1 2 [0, 2 3 ] ( = 1, 2,, ) 2, 2/ 3 [0, 3 ], 1 ( 1 ) + 2 ( 2 ) + + ( ) > 0 1 + 2 + + = 0,, + 1, : (1) () ; (2), (,, ) ; (3) (2), ( 1); (4) 6 11 () = ( ) : ( 1) = 1, 1 ( ) = ( ) = = 2, 2 () = ( 2 ) = 2 2-1, 1 ()2 ( ) (2) = - 1, =, - 1 ( ), () - 1, 220
+ 1, + 1 () + - 1 () = [ ( + 1)] + [ ( - 1)] ( + 1 ) + ( - 1) = 2[ ] 2 ( + 1) - (- 1) [ ] 2 = 2( )() = 2 ( ), + 1 () = 2 ( )- - 1 () + 1 () + 1 (1) ( 2): ( ) = () 12 { }, 2 < - + 1, : < 1 : ( 1) = 1, 12 < 1-2, 1, 2 +, 12 < 1-2 < 1 12 < 1 1 < 1 = 1 1 = 1 (2) =,, < 1 = + 1, 2 < - + 1 + 1 < - 2 = (1 - ) + 1 < - 2 < 1 - + 1 > 1 - > 1 - > 0 + 1 < (1 - ) < (1 - + 1 ) < 1 (1 - + 1 ) 221
+ 1) + 1 < 1, + 1 < 1 + 1 = + 1, (1) ( 2), : (1) : 0 (0 ) (2) = ( 0, ), = + 1 (1) ( 2) 0,,,,,,,, :,,, : 1,,,,, :, 222
:,, 12345,,,,,, 1 2 3 4 5 1,,, 2, 32 5 3 4 (1),, 2,, 15, ; 2, (2),, 21 = 37;, 36 = 4 9, 12 = 34, 3 4 ; 3, 12, 4, 12 223
, : 1 ; :,, 1,, 4 9, (4),, = ;, ;, 5,,, 2,,,,,,,,,, : 3 5 367.5,8 10?, : 1 (1 )3 5 367.5,8 1? 224 (2) 3 1 367.5,
1? 2 (1) 3 5 367.5, 5, 12,? (2) 3 5 367.5,, 1, 3,? 3 (1) 3 5 367.5,8 1482,? (2 )1482, 3 5 367. 5, 5,?,,,,,,, 3,,,,,,, 6.28, :, 225
180-80 = 100 10080 = 1 1 4 6.281 1 4 = 7.85() : 1,, 6.2880 = 0.0785( ) 180-80 = 100 0.0785100 = 7.85( ) : / = 1 (),,, : 180-80 = 6.28 80 = 7.85,,,,, 4,,,,,,, : 226 (1) 500
; 200,? (2), 8, 6,,? (3), 8, 6,? (4), 8, 6,?,,,,,,,,,,,,,, :,,,, 1 227
: 2 + 3-2 - 3 2 + 3 = = 3 + 1 2 4 + 2 3 2 = = 6 + 2 2, 2-3 = 6-2 2 = 6 + 2-2 : ( 3 + 1) 2 6-2 2 ( 2 ) 2 = 2 = 2 + 3-2 - 3 ( > 0) 2 = 2 + 3-2 2 + 32-3 + 2-3 2 = 2 > 0, = 2, = 2,,,,,,,, 2, 228,, 0, > 1,
+ 2-1 ) 2 - ( 1 + 1 ) 2-4 2 1 + 1 : (+ 2-1 ) 2 - + 2-1 = - 2 (1) - 2 1-1 = + 2-1 = - (1) (2), = 1 2 ( - 2 + - 2 ), = ( 1 + 1 ) 2 2 (2) - ( - 2 + - 2 ) 2 + = ( 1 + 1 ) 2 - ( - 2 + 2 + - 2 + 2 ) 2 = ( 1 + 1 ) 2 - ( 1 + 1 ) 2 = 0,,,,,,,, + 2-1,, - 2-1 + 2-1,, 3, 53721 2-5372053722 229
= 53721 2 - (53721-1) ( 53721 + 1 ) = 53721 2 - (53721 2-1) = 1 53721 2 5372053722,,, 1 + 2 + 3 + + 99 + 100 (1 + 100) + ( 2 + 99 ) + + ( 50 + 51 ) = 5050, 4,,,, 2 =,, - 1 > 1-1 > + 1, + 1,,,,,,,, :,, 230
, 100,,, : ( ),,,,,,,,,,,,, :,,,, (),,,,, :,,,, :,,,,,, 231
,, :, :,, :,,,,,,,,,,,,,, :,,, : 232
) :,,, (1),, =,, =,,,, (2) 233
(3),, (4), ;,,, :,,, :,,,,,,,,,,,,,,,,,, 234
,,,, :,,, :, 46 : = ( ) = - 1 ( ) = : = 3 = 3 =,,,,,,,,,,,, :, 27 2, 2 : - 5> 6 : 2-5 > 6 2-5< - 6, < - 1, > 6,, 2 < < 3 235
< - 1}{2 < < 3}{ > 6} = { < - 1, 2 < < 3, > 3},,,,, 2-5> 6 ( 2-5) 2 > 36 ( 2-5- 6 ) ( 2-5- 6) > 0 ( + 1) (- 2) ( - 3) (- 6 ) > 0, { < - 1, 2 < < 3, > 6} -,,,, 70%,,,,,,, () 236,
?, :! 95,! :95? 6! ) ; :!! :?:? 181652 :181652?? () : 1( 8(; ; ; ; ; ; ; ) ; ) ; 1( 6() ; 5(: ; : ) ; 2() : ;, ( );,, :,, ;, 237
, 1996 ( ), 35, 45 98,116 95.6,,,,,,,,,,,,, : 10 2-4 - 3-2 - 4-3 - 4 5-3 - 3 10, 10 10, : 2: 2 ( ) 238 4:, 4 3:, 3 2:, 2 4:, 4 3:, 3 4:, 4 5:, 5 3:, 3 3:, 3,
,,,, :,,?,,, :,,,,,,?,,, = 1 3 ( 1 + 2 + 12 ) : - 1 1 1 2 3, 1 1 1, ()?, 239
1 3 1 + 1 3 2 + 1 3 12 = 1 + 2 + 1 2, 4 69 = 1 (1 - ) 1 -,,, = 1 1 - - 1 1 -, = +, + = 0,, { } = 2-1, 1 2 + 1 2 + + 1 2 =, { } 1 2, = 2, 1 = 1, { 1 2 }, 1 4, 1, = 4 3 ( 1-1 4 ),,,, = 1 +2 1 + = 1 +2 1 + = 1 1 + 1 + 1 + 2 = 1 1 + 1 + 1 + 2 240
2 1 2, = 1 + ( 1 - ) 2 = 1 + ( 1 - ) 2 (, ) (1, 1 ) ( 2, 2 ) = - 1 2 - = - 1 2 -, 2 2, 0, 0 = 1 + 2 2, 0 = ( 1 + 2 2 ) 2, ( 1 + 2 2 ) 2 1 + 2 2, 0 < 1 + 2 2, 0 = 1 + 2 2,,,, = + = (1-2 ) 2 + (1-2 ) 2 = 1 + 2 1-2 = 1 + 1 2 1-2, 2 + 2 + 2 = 2, 2 + 2 + 2 = 1 ( ),,, ( 1 + ) (1 + ) 1 +, 241
,, { }, : + = +, + = +,,,, : + = +, + = +, - 1 1 1 1 - = 1 3-1 1 1 (1 ),,, - 1 1 1, (2), - 1 1 1 + - = 1 3 1 1 1 -, - 1 1 1 ( 3),,,,, :,,,,,, ( 242
,,,,, 2-3,,,,,,,,, 126,,,,, ( ),,,, 1.6 = 0.4, 16 = 4 1 - + + = (1 ) (1 - ) 1 + + + = (1 + ) (1 + )1 - - + = ( 1 - ) ( 1 - ), - +, :,,,,,, : 243
,,,,,,,,,,,,,,,,,,, :,,, ; ;,,,, 1986 :,,,,,, ;,,,,,,,,, : 244
;,,,,,,,,,!,,,,,, : (1) ; (2) ; (3) ; (4),,,,,,!,,,,!,,, ( ),,,,, 245
,,,,,, :, ;, ;,,,,, : ( ) = ( > 0, 1, + ) [( 1 ) + (2 ) ] ( 1 + 2 ), (1994 ) 2 ( ) + ( ) : : 1 1 (1 + 1 ) () 2 2 : > 0 > 0 1 + 2 + 1 2 + 1 1 (1 + ) 2 2,,,,,,,,, 246
,, :, : : ( 2 + 1 2 ) (2 + 1 2 ) [( 0, 2 ) ] :,,,,,,,,, ;,,,, :,,, + + + = 1, : 1 + 1 + 1 9,,, + + 3= 1, : 1 + 1 4 3,,, ( + + ) ( 1 + 1 + 1 ) 9 (+ 3 ) ( 1 + 1 )4 3,, ;,, : 0, 2,0, 2;2,0, 2,0; 1,0, 1,0; 0,1,0, 1 9, 99,999,9999; 5,55,555, 5555;81,891, 8991,89991;,, 247
,,,,,,,,,,,,,, :,,, ( ),,,,,,,,,,,,,,,,,, 248,,?
,,,,,,,,,,,,,,,,, : (+ 3 ) 2 = 2,, 1: ( + 3) 2 = 2, 2 (+ 3 ) = 2 ( + 3) = 1, + 3 = 10, = 7,, 1, ( + 3) 2 = 2 2 (+ 3) = 2 (+ 3 ), - 3 ; 2 ( + 3 ), - 3, 2: ( + 3) 2 = 100, (+ 3) 2 = 100, (+ 3) = 10, 1 = 7, 2 = - 13,,,,, 249
,,,, :, = =, = 90, = 45, :,, = 90 = = 45 = = = = = = = 2, = 2 5 5, = 5 5,,, 250
=,, = = 2 135 45 = 2 = 2 5 = 2, = 5,,,= 5 5,,,,,,,,,,, ;,,,, :, : 2 2 + ( 2 + 2-2 ) + 2 = 0 < 0 : = ( 2 + 2-2 ) - 4 2 2 = ( 2 + 2-2 + 2) ( 2 + 2-2 - 2) = ( + ) 2-2 ( - ) 2-2 = ( + + ) ( + - ) ( - + ) ( - - ),, + + > 0, + - > 0, - + > 0, - - < 0, < 0,, : 251
2, = 2 2 + ( 2 + - 2 )+ 2 (2), = 2 2 + ( 2 + 2-2 )+ 2 (3) 2 2 + ( 2 + 2-2 ) + 2 > 0, < 0,,,,? 651, :, ( ),,,,,,,,,,, :,,, :,,,,,, 252
,,, :,,,,,, 655,,,,? :,,, ;,, ;,,, ;,,,,,,,, ;,,,,,,,,,,,,,,, 253
, ; ; ; ;,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, :, ( ),!,,,,, 254
,,,,,,,,,,,,,,, :?,,, ( ) :,,,,,,,,,,,, 70% 80%,, :,, 255
,,.. (1) (2) (3) (4) (5) (6) (7) (8) (9). (10) ;,, (2) ( 3) (6) ( 10 ), (2),,,,,,,,, (3),,, :, 256 (6)
.,.,,, 45 (10),,,.,.,,?,,,,, :,,,,,,, : (1), 257
(2),,,, (3),,, (4),,,,,,,, : (1),,,, (2),,,,, (3),,, (4),,,,, 85 95,, 657 258
,,,,,,,,,,? :,,,,,,,,,, /,,,, : = / ;,,,?!,,,, ( ),,, :,, : 22 + 2 23, 2, () 259
,,,,,?,,,,,,!, ( ),,,,,,,,,,,,,,,,,,, ( ), : : - 2 1 1 + 1-2 + 1 - + 1-2 2 = 3 : : 1-2 1 + =, = 2 2 1-1 - 2 =, = 0, ( )( ), 260,,,
,,,,,,,,,,,,,,!,,,,,, ( ),, 1988 :,,, 261
),,,,,,,, :,,,, ;,,,,,, 40%, 30%, 30%, 262
,,,,, 100,,,,,,,,,,,,,,,,,,,,, 7 4 :,,,,,,,,, 263
,,,? 1,,,, :,,,,,,,,, 264
,,,,,, 2,,,,, ;, ; ; ;,, ;,,, :,,,, ;,,,,,,,, 3,,, :(1 ), ( 2),,? ( 3) 265
?,,,, (4),,,,,,, 4, ; ; ;, : : 1,, :,,, 2,,,,,,,, 3, :, 266
,,,, :,,,, : 1,,, 2,, : = :, 3,, ( ),,, 4,, : 5 267
, :, (),, : 1,,,, =,,,,,, =, 2,, : =,, :,, 3,, 268
,,,,, 1,,,, 2,,,,,,,,,,,, 3, ;,,,,,,, 4,,, 269
,,,,,,,,, 5,,,,,,, :,,,, 6,,,, : ( 1) ; ( 2); (3),,, :, :,,,,, 270,,
,,, 1, (1),,,,,, (2),,,,,,,, 2, (1),,,, (2),,,,,, (3),, 271
,, (4),,,,,,, 1,,, 2,,,,, 3,,,,,, 272
, 1,,,,,,,,,,,,,, ;, : 1., ; 2.,,,,,,, :, 1., = 2/ 2 2., = +,,, 273
,,,,,,,,,,,,,,,, :, ;,,, 3,,,,,,,,, :,,,,,,,, 274
,,,,,,,, ;,, ;,, 1,,,,,,,,,,,, ;,,, 2,,,,, =, ;,, = ( ); 275
4,,,, :,,,,,,, :,, = + ;, : = - 5,,,,, : (1),,,, (2),, (3),,,,,, 276
,,,, ;,, ;,,,,,,,, :,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 277
,,,,,,,, :,, ;,, 1,,,,,, :,,,,, 278
,, =, ;,,, ;,, = /,,,,,,,,,,,,,,, ;,,,,, ;, 2,,,, 279
(1),,, (2),,, (3), (4),,,,,,,, :,,? (),,,,,,,, : 280
- = 0 - + = (2),, :, () ( 2) (3) () (3),,,,,, :, ( ), 17,,, :,,,,, (),,,, :, ;,,, 281
:,,,,,,,,,,,,,,,,,,,,,,,,,,,, 282
,,, 220, 220,, 220,,,,,,, ;,,,,,,,,,,, ;,, :,,, 283
;,,,,,,,,,,, =,,,,,,, ( ),,,,,,, :,,,,, 1 284
, ;,, :, 2,,, 3,,,,,,,,, :,, ;,,,,, 1 : 285
,,,,, 2,,, :,,, : (1),, (2),,,, (3),,, (4), 286
1 :,, ;,, : (1), ( 2),,,,,,,,, (3), 2,,, (1) ( ),,,,,, 287
,, (3),,,, (4),,, (),,,,,, : 1,,,,,,,, 288
,,,,,, ( ), 2,,, (1),,,,,,,, ;,,,, (2),,, :,,,,,,, 289
,,,,,,,,,,,,,,,, 1 2,, 290
,,,,, ;,,,,,, 1,,,,,,,,,,,, 2,, 291
,,, ;, 3,,,,,, 4,,,,,,, ;,,,, 5,,,,, ; ; 292
6,, : ; ; ; ; ;,,,, ;,,,,,,,, 3,,,,,,,,,,, :,,,, : 1,, 293
, : 12,,,,,,, 50 ( 1 ), 2,,,, 2, ( ) 1 47 48 ( ) ( ) 49 ( ) ( ) 2 3 294
2 1 3,,,?,,, 3,,, = / =, 4 54 1? 4,,,, ( 4 ),, 295
,,,,,, :,,,,,, :,,,, :,,,,,, :,,, ;,,,, 296
: ( 1), ; (2 ), : (1),, :, =,, =,, = ; ( 2), ; ( 3),,,, : ( 1),, ; (2),, : ( 1) = = = = = ; ( 2) ; ( 3),, ; (4) ; ( 5) 297
, ; ( 6), =,,,,, : (1 ),, ; ( 2) : 20% : (1 ),, ; ( 2),, ; ( 3) :,,,,,,?,,,,,, 298
, 8 : 1,,,,,,,,,,,,,,, 2,,,, ;,, ;,,,,,,, 3,,,,,,,, 299
,, 4,, : ( ) > 2 2> > < (),,,,, 5,,,,, ( ) 6,,,,, 300
,,,,, 7,,, : (1 )? ( 2)??? (3 )?? ( 4)??,,,, 8,,,,,, 1,,,,,, ;,,, 301
,,,,!,,,,,, 2,?,,, 80,,,,, 3 80,,,,,,,, 302
,,,, 1 () (2) (3),,,,, 2 : 303
;,,,, 3 (1),,,,,,, 30, :,,,,,,,, (2) 2; ; 304
; :, 2,,, ( ), ( ): 3 3, 2 5 : (3) : 305
:.... : :..,,, ;,,,, ; ; ;,,,, ;,,, 1.,, ;,,,,,,, 1,, 306 : 2 + 2 2 : 2 3 + 2 4 + 2
+ 2 2 2 3 + 24 + 2 : 2 + 2 2 2 3 + 2 4 + 2 2 4 : 2= 8+ 2() 2= 3+ 16+ (),,,, = 1,= 11, = 15 : 22 + 152 2 = 2 3 + 42 4 + 112 2., :,, () 1,,,,,,,,,,, 307
),,,,,,,,,,,,,,, ;,,,,,,,,,,, ;,,,,,,,,,,,, (), ;,,, ;,, 308
,,,, (2),,,?,,,, :,,,,, ;,, ;,,,,,,,,, 1, ;, 309
, ( ) ;,, ;,,, 2,,,,, (, )( ) ;,,,?,,,,,,,,,,,,, ;,,,,, 310
,,,,,, 1,,, ( ),,, 2,,,,,, 3,,,,,, 311
,,,,,, 1,,,,,, 2,,, 312 3,,
,,,,,,,,,, ;,,,,,,,,, 1, :,,,, 2,,, 313
,,,, 4,, 26,,,,,,,,,,,,,,,,,, 314
, 30,,,,,, 18, 7, : 1 :,,,,,, 2:,,,,,, ( ) 3:,,,,, (1 2.6 2,,, ) 4:, 5:, 6:, 7:,,, 8:,,, 315
:,,,,,,,, :, 26, 10: ; ; ;,, 8,,?,,, : 1,,,,,,,, 316
,, 4, : ; 100 ; ;, 2,,,,,,,,,,,,,,, 15, 5,,,, 3,,,,,, 30,,,,,,, 4 317
,,,,, 3 4, 2 3, 5,,,,,,,,,,,,,,,,,,,,,,, 6,,,,,, : 318
,,,, 8,,,,, ( ),,, 9 319
,,,,, ;, 10,,,,,,,,, :,,,,,,,,,,, : 1,,,, ( ),,, 320
,,, : ( 1),, ; (2); (3) ; (4 ) ;,, :,,,, 2,,,,,,,,, ;, ;,,,, : ( 1); (2) ; ( 3) ; ( 4), : 3 2,,,, ;,,,,,,, 321
,, ;,,,,,, :,,, :,,, : ( 1),, (2), ;,, :, ;,, 4,,, : ( 1),, ( 2), 2, ( 3),,, 322
(4),, 5,,, (),,??,?,,,,,,, ;,,, ;, ;, 323
,,,, :,,,, :,, :,,,, +, :,,,?,,,,,,,,,, 324
, :,,,,,,,,,,,,,?,,,,,,,,,,,,,, :,,,,,,,,,,,,,,,,, 325
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 326,
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ;,,,,,,,,,,, 327
,,,, : 1; 2 ; 3 ; 4 ; 5,,,,,,,,,,,,,,, :,,, 328