1, : ( ),?, :,,,, ( ), 1 180,, ( ) 1 1,, 2 180 ;,, 3 180 ;, n ( n - 2 ),, ( n - 2) 180 1 1, : ( ),.,, 2, (, ) 1
, 3 x + y = 14, 2 x - y = 6 : 1 ( ) : + 5 x = 20, x = 4 x = 4 y = 2, x = 4, y = 2 2 ( ) : y = 14-3 x, 2 x - (14-3 x) = 6, x = 4 x = 4 y = 2 x = 4, y = 2,, :, ( ),, 3,, : :,,?, :,,, :,,? :,,,, : ;, :, : 2
, :, *,, :,, : 4 : : 1: x 2 + y 2 + 7 x y + 4 x + 4 y = 31, x y = 2-5 x 2 + y 2 + 2 x y + 4 x + 4 y = 21, ( x + y) 2 x + y = u, u 2 + 4 ( x + y) - 21 = 0 + 4 u - 21 = 0 u1 = 3, u2 = - 7; x + y = 3 x + y = - 7, : x + y = 3, x y = 2 ; 2: x + y = - 7, x y = 2 y = 2 x, : x 2 + 2 x 2 + 7 x 2 x + 4 x + 4 2 x = 31. x 4 + 4 x 3-17 x 2 + 8 x + 4 = 0,,,, 2, 1,, 1,,,,, 3
,, (, :,, ) ;,,, : S1 = 1 + 2 + 3 + + 99 =? 1 2, *,, : 2 S1 = 1 + 2 + 3 + + 99 + 99 + 98 + 97 + + 1 = 99 100 = 9 900 1 3 S1 = 1 2 9 900 = 4 950, S2 = 1 2 ab, S3 = 30 2 = 15, 5 ( ), :?,, :,,, 1 4 4
(, ),, (, S = 1 + 2 + 3 + + 99, : 99 + + 3 + 2 + 1 ),,,,,, 6 x 2,, a x 2 = m ( m 0 ) + bx + c = 0 ( a 0 ) ( ), : 7 ( 1),,,, ;,, ( 1 5 ),, 1 5,, : ;, ;,, : ( 1 6), : ; ; 5
1 6 1 7,, ( 1 7 ), : ; 90 ;,,, : ; 90 ;, ( 1 8 ) 1 8 8,, n ( y ( n) + p1 y ( n - 1 ) + + pn y = 0) ( n + p1 n - 1 + + pn = 0),, 6
,, ( ) ( ),,,, S, 2, 2 S;, G., 9 :, 50 140,?, ( : ) :,,,,, 70 ( ) ;, 1, 1, ( 70 ) (50), 70-50 = 20 ;, 50-20 = 30 10 1, 9, A BC, : D L = 1 3 m, B L = 2 3 n, CL = 2 3 p(, m, n, p BC, AC, AB ), L D G, DG = L D, LG = 2 3 m, B L = 2 3 n, BG = 2 3 p ( BDG CD L ), 1 9 ( B LG),,,,, 7
, ( )?, :,,,, : (, ) 1 1 10 : 1 10 8
S = S - S, 1 11 : S = S + S 1 11, ( ) :,, 2 ( 1 12 ),, A CB = OBC, AOB = A CB + OBC = 2 ACB,,, ( 1 12 ), ACB = DCB + A CD( ACB = DCB - A CD), 1 12 9
, :,,,, ( ) 3 : n x1, x2,, xn, n y1, y2,, yn, f ( x), : f ( x i ) = yi ( i = 1, 2,, n),, n : n x1, x2,, xn, yk ( k = 1, 2,, n), f k ( x), n : f k ( xi ) = 0 ( i = 1, 2,, k - 1, k + 1,, n), f k ( xk ) = yk, : x1,, xk - 1, xk + 1,, xn f k ( x), f k ( x) ( n - 1) ; ( x - x1 ), ( x - xk - 1 ) ( x - xk + 1 ), ( x - xn );, fk ( x) ( n - 1) ;, ( n - 1), : c( x - x1 ) ( x - x2 ) ( x - x k - 1 ) ( x - xk + 1 ) ( x - xn ) ;, f k ( xk ) = yk c =, yk ( xk - x1 ) ( xk - xk - 1 ) ( xk - xk + 1 ) ( xk - xn ) f k ( x) = yk ( x - x1 ) ( x - xk - 1 ) ( x - xk + 1 ) ( x - xn ) ( xk - x1 ) ( xk - xk - 1 ) ( xk - xk + 1 ) ( xk - xn ),, : f ( x) = y1 ( x - x2 ) ( x - x3 ) ( x - xn ) ( x1 - x2 ) ( x1 - x3 ) ( x1 - xn ) yn ( x - x1 ) ( x - x3 ) ( x - xn - 1 ) ( xn - x1 ) ( xn - x2 ) ( xn - xn - 1 ) + y2 ( x - x1 ) ( x - x3 ) ( x - xn ) ( x2 - x1 ) ( x2 - x3 ) ( x2 - xn ) + +,,, ( ), ;, ( ), ( ) 4 A BC, 1 13 10
OA = OB = OC O( 1 13 ) OA = OB( ) OB = OC( ),, AB BC,, :,,,, : ( ),, ( ),,, ( x, y), ( x, y, z), ;,,,, ;,, (,, ),,,,, :,,,, ;,, 11
5 1 14, a, b, X, Y a b, d = X Y, a, b, Y, X ( Y ) ( a), a X X Y a, X Y Y a, Y a, X Y, X Y 1 14 : b Y Y X b, Y X b X, X, Y,, X, Y X Y a b, d = X Y, X Y a, X ( Y ) X Y, X Y ;, X Y b, Y ( X) X Y, X Y a b, d = X Y,, a, b 6, 1 15,, ( X), V, W,, Z,,, X 1 15, : ( n - 1) V W Z, VX Z X, : ( V Z), X ( ), V X Z,, VX Z, X, VZ X H, X X V = X Z, V X Z, :,, 12
: :,,, : y = f ( x1,, xn ) ( x1 0, x2 0,, xn0 ), f ( x1 0, x20,, xn0 ) y, : f x 1 ( x1 0, x20,, xn0 ) = 0, f x 2 ( x1 0, x20,, xn0 ) = 0, f x n ( x1 0, x20,, xn0 ) = 0,,,,,,,,,,,,,,,, ( x2 a 2 + y2 b 2 = 1, x2 a 2 - = 1, y 2 = 2 p x), ( a, b, c, p, ),, ( x1, x2,, xn ) ;, xi ( i = 1, 2,, n), x2 y 2 b 2, x1 ; x3, x1 x2 ; 1 16 13
: 1 16,,,?, : * ;, * *, ( ) : 7, :,,,,,, :, ( ) 8 y = x 2, : 14
, y = x + 0) : b ( 3 a 9 ( ) ( ), ( x lg x), : 10,,, ( ) ( ),,, :,, : : 1 17,, BC x, BC AD y A, B, C A ( 0, a), B( b, 0 ), C( c, 0 ), : kbc = 0, kca = - a c, kba = - a b, : AD: x = 0, B E: c x - ay - bc = 0, CF: bx - ay - bc = 0 1 17 15
x = 0,, y = - bc a,, :, ;, ;,,, ;,,,, 11, ( ) 1 18, : x, x 2 = (2 a) 2 + a 2 = 5 a 2, x = 5 a,, 1 1 18 19 1 19, :, ( ),,, (, ), ( ) 16
12,,,,, ( ) :, ( 5 ),,,, ;,,,,,,,, ( ),,,, 1868 :, ( ),, ( ) ( ) ( ), :,,,,,,,,, ( : ) ; : 13, :,, ; 17
,, 3 2 ;,,,,,,,,, ; (, 3 2 ),,, ( ),,,,, 3 2,,, 60 : 60, cos20 sin20,,,, 60 : co s3 x = 4 cos 3 x - 3 cos x, x = 20, : cos60 = 4 cos 3 20-3cos20, 4co s 3 20-3 cos20-1 2 = 0, cos20 : 4 x3-3 x - 1 2 = 0, :,, co s20 sin20 18
,, 2 000,,,,, :,, ( ) ( ),,,,,, ( ),,, 14,, : x 3-5 x 2 + 6 x = 0 x 3-5 x 2 + 6 x = x( x - 2) ( x - 3 ), x( x - 2 ) ( x - 3) = 0, x1 = 0, x2 = 2, x3 = 3 15, : sin x - 3cos x = 2 sin x - 3cos x = 2 1 2 sin x - 3 2 cos x = 2 cos sin x - sin 3 3 cos x 19
, 2 sin sin x1 x - = 2 3 2, = 2sin x -, 3 x - = 2, 3 = 2 k + 4 + 3, x2 = 2 k + 3 4 + 3 ( k ) 16 : a0 x n + a1 x n - 1 + + an - 1 x + an ; bx + c x 2 + p x + q ; bx + c ( n > 1) ( x 2 + p x + q) n ( n, a, b, c, ) b x - a ; b ( n> 1) ; ( x - a) n, (, ), : : 2 x + 3 x 3 + x 2-2 x d x : 2 x + 3 x 3 + x 2-2 x = - 3 2 x + 5 3 ( x - 1 ) - 1 6 ( x + 2 ), 2 x + 3 x 3 + x 2-2 x d x = - 3 2 1 x d x + 5 3 1 x - 1 dx - 1 6 1 x + 2 dx = - 3 2 ln x + 5 3 ln x - 1-1 6 ln x + 2 + C 17, sin x = 2 u 1 + u 2, cos x = 1 - u2 1 + u 2, tan x = 2 u 1 - u 2,, 20
, u = tan x 2 tan x 2 ;, ( tan x 2 = u), tan x 2 u ;, 1 + sin x dx sin x( 1 + cos x), u = tan x 2 : 2 u 1 + 1 + sin x sin x( 1 + cos x) = 1 + u 2 2 u 1 + 1 - = u2 1 + u 2 1 + u 2 ( 1 + u) 2 ( 1 + u 2 ) 4 u tan x 2 = u, : 1 + sin x sin( 1 + cos x) dx = (1 + u2 ) (1 + u) 2 4 u = 1 2 ( u + 2 + 1 u- ) du = 1 2 2 1 + u 2 du u 2 2 + 2 u + ln u + C = 1 4 tan2 x 2 + tan x 2 + 1 2 ln tan x 2 + C,,, ;, : ( u( x) v( x) ) = u ( x) v ( ; ( u( x) v( x) ) = u ( x) v( x) + u( x) v ( ; u( x) v( x) = u ( x) v( x) - u( x) v ( x) ; ( v( x) 0) v 2 ( x) ( u( x) v( x) ) dx = c u( x) dx = c u( x) dx ; u( x) dx v( x) dx ; u( x) v ( x) dx = u( x) v( x) - u ( x) v( x) dx, 18, : ln x dx = x ln x - x (ln x) d x = x ln x - x + C 21
, 19 a x 2 + bx + c = 0 ( a 0),,,,?,, 16,, x 3 + p x + q= 0 ( 8 : a x 3 + bx 2 + c x + d = 0, a 0, ), u v, x = u + v : ( u+ v) 3 + p( u + v) + q= 0, u 3 + 3 u 2 v + 3 uv 2 + v 3 + p u + pv + q = 0, u 3 + ( 3 uv + p) u + (3 uv + p) v + v 3 + q= 0, 3 uv = - p, u v : u v = - u 3 + v 3 + q = 0, 3 uv + p = 0 p 3, u3 v 3 = - p3, 27, u 3 v 3 y 2 + qy - p3 27 = 0 : u 3 = - q 2 + q 2 4 + p3 27, v3 = -, : x = 3 - q 2 + q 2 4 + p3 27 + 3 - q 2 - q 2 4 + p3 27 q 2 - q 2 4 + p3 27,, u v : u, v (, x),, ;, u v ( 3 uv = - p), :, ;,,,, 22
, ( ), (, ),,,, :,, 20 : : g( x) [ a, b], ( a, b),, g( a) = g( b), c( a < c < b), 0, g ( c) = 0. f ( x) [ a, b], ( a, b), c( a < c < b), f ( b) - f ( a) = f ( c) ( b - a),, ( x) = f( x) - f ( a) - f ( b) - f( a) b - a ( x - a) ( x) ( ( a) = ( b) = 0 ),, : ( a, b) c, ( c) = 0; ( x) = f ( x) - f ( b) - f( a) b - a, f ( c) - f ( b) - f( a) b - a f ( b) - f ( a) = f ( c) ( b - a) = 0,,,, : ( x)?!,,, ;, 23
g( a) = g( b), ( x), : f ( x),, ( a) = ( b) (, ( a) = ( b) = 0 ), ( x) = f( x) - f( a) + k( x - a), ( a) = 0 ;, k,,, k, ( b) = 0, ( b) = 0, : f ( b) - f ( a) + k( b - a) = 0, k = - f( b) - f ( a) b - a, ( x) = f ( x) - f ( a) - ; f ( b) - f ( a) b - a ( x - a),, :,, :,,,,,,, : 1 ( 17 ), : 24
2,, 1 20 : ( 1) V =? (2 ) V1 =? ( 3) x =? ( 4) x =? V2 =? H =? 1 20, : 1 21,,, 25
,, x ( R, r, h), ( ) ( ),, :, ( ),,,,,,,, ( ),,,,, :,,,,, :?,?,?,,,,, :,,,,,, ;,,, (,, :, 26
),,,,, : ;,,,,,,, (, ),,,,,, 2 27
2 :,,,?,, ( ),,, :?,,,,, ( 3 ),,,,,, : ( ) ( ) (,,, ) : ;,,, ;,,, ;,,, :,,,, :,,,, :,, 28
, : 1 ( ), :, x 2 + y 2 2 x y, : x 3 + y 3 + z 3 3 x y z x 0, y 0, z 0,, : x 3 + y 3 + z 3-3 x y z = ( x + y + z) ( x 2 + y 2 + z 2 - x y - y z - x z) = 1 2 ( x + y + z) [ ( x - y) 2 + ( y - z) 2 + ( z - x) 2 ] 0 2 2 1, ( ) ( ),, : 2 1 3 2 2,, :, 2 2 4,, 29
:,,,?, :,,!,,,, ;,,, 5 ( ), :,,, :, ( 2 3) 2 3 6, ( 2 4 ) : ( ) ; ( ) ( ),, : 2 4, :,,, ;,, ( ), : 30
,,,,,,,,,,,,,,,, 7,,,, ( ), ( ), :,, ( ) ;,, : ( ), ( ),,, : ( x - 4) - 1 > 5 x + 6 3 2 : ( ) ( x - 4) 3-1 = 5 x + 6 2 : 2 ( x - 4 ) - 6 > 3( 5 x + 6 ) 2 ( x - 4 ) - 6 = 3( 5 x + 6 ), : 2 x - 8-6 > 15 x + 18, 2 x - 8-6 = 15 x + 18, : 2 x - 15 x > 18 + 8 + 6, 2 x - 15 x = 18 + 8 + 6, : - 13 x > 32, - 13 x = 32, - 13 : x < - 32 32 x = - 13 13,,,, :,,, 31
, :,,, ( ),,,,, 8, :, :, : 9 1, ( 20 ), 32
: f( x) F( x) [ a, b], f ( x) F ( x) ( a, b), F ( x) ( a, b), a, b c, f ( b) - f ( a) F( b) - F( a) = f ( c) F ( c), ( x), f ( x),, ( a) = ( b) : :,, ( x), f ( x) F( x),, ( a) = ( b), ( x),, ( x) = f( x) - f( a) + k[ F( x) - F( a) ], :,, k = - ( x) = f ( x) - f( a) - f( b) - f ( a) : F( b) - F( a) f ( b) - f ( a) [ F( x) - F( a) ] F( b) - F( a),, 33
, 10, ( ) 5, :, ( ),,,, 1 3 ; :, ( 2 5 ) 2 5 2 6,, 1, A H = 6 x, O H = OP 3, x (, H, P BCD A CD ) : O H = OP = x A H = 6 3 x, B P = A H = 6 3, BO = B P - OP = 6 3-6 3 x = 6 3 ( 1 - x) OB 2 = O H 2 + B H 2, : 6 3 2 (1 - x) 2 = 2 6 3 x 2( 1 - x) 2 = 2 x 2 + 1 + 3 3 2 B H 3 3, : x = 1 4 1 4., 34
a, : 1 4 6 3 a= 6 12 a,, 11 ( ) ( ), : A x + By + C = 0,, : A x + By + Cz + D = 0, l1 ( A1 x + B1 y + C1 = 0) l2 ( A2 x + B2 y + C2 = 0 ) A1 A2 = B1 B2, : 1 ( A1 x + B1 y + C1 z + D1 = 0 ) 2 ( A2 x + B2 y + C2 z + D2 = 0), A1 A2 = B1 B2 = C1 C2, n, :, ( x1, x2, x3, x4 ) ;,,,,,, ( 2 12 7 ), : 2 7, ( 2 8 ), : ( ) :, ( 2 8 ) 35
2 8,, :,,, ( n ), ( ) 13 8, ax 2 + bx + c > 0 ( < 0 ) ax 2 + bx + c, ( a x 2 + bx + c= 0 x1 x2, x1 < x2 ) : x x 1 x 2 ax 2 + bx + c a > 0 ax 2 + bx + c a < 0 + 0-0 + - 0 + 0 - : ax 3 + bx 2 + c x + d > 0 ( < 0 ) ax 3 + bx 2 + c x + d, ( a x 3 + bx 2 + cx + d = 0 x1, x2 x3, x1 < x2 < x3 ) : x x 1 x 2 x 3 ax 3 + bx 2 + c x + d ( a > 0) ax 3 + bx 2 + c x + d ( a < 0) - 0 + 0-0 + + 0-0 + 0 -, 36
,,,,, 14, : ( x1, x2,, xn ), ( y1, y2,, yn ) n, ( x 2 1 + x 2 2 + + x 2 n ) 1 2 ( y 2 1 + y 2 2 + + y 2 n ) 1 2., n : x1 y1 + x2 y2 + + xn y n ( x 2 1 + x 2 2 + + x 2 n ) 1 2 ( y 2 1 + y 2 2 + + y 2 n ) 1 2 ( x1 y1 + x2 y2 + + xn y n ), 15 ( ) :,,, 16,, ( ),, :, :, S = 1 2 C R = 1 2 ( 2 R) R = R2 ( 2 9) 2 9 17 n a1, a2,, an, a1 + a2 + + an.,, a1 + a2 + + an + 37
,,,, a1 + a2 + + an a1 + a2 + + an : : ai ai i = 1 i = 1,, b f ( x) dx a b f ( x) dx ( a < b) a,,,,,,,, ;,,, 0 = 1 : [ 1 + ( - 1) ] + [ 1 + ( - 1) ] + [ 1 + ( - 1) ] + = 0 + 0 + 0 + = 0, 1 + [ ( - 1) + 1] + [ ( - 1) + 1] + [ ( - 1) + 1] + [ ( - 1) + 1] + = 1 + 0 + 0 + 0 + = 1,, 1 + ( - 1 ) + 1 + ( - 1 ) + 1 + ( - 1 ) + 1 + = [ 1 + ( - 1) ] + [ 1 + ( - 1) ] + [ 1 + ( - 1) ] + = 1 + [ ( - 1) + 1] + [ ( - 1) + 1] + [ ( - 1) + 1] + 0 = 1 + 1 18 n = 1 n 2, ; + n = 1 1 = 1 + 1 + 1 + + 1 + n 2 2 2 3 2 n 2, 18, : 1 + 1 + 1 + + 1 + = 2 2 2 3 2 n 2 6 : 38
, a1, a2,, an n a0 + a1 x + + an x n = 0 ( an 0 ) n 0, : - n, a0 1 - x a1 1 - x a2 1 - x an,, : 1, - 1 ; 2, - 2 ; ; n, 2 n a0 1 - x2 2 1 1 - x2 2 2 x2 2 n,, x 2 a2 = - a0 1 + 1 + + 1 2 1 2 2 2 n sin x = 0 : 0,, -,2, - 2,, n, - n,,,, x, sin x x 2, - 2 ; ; n, - n ;, : sin x x = a0 1 - x 2 1-2 x 2 1-4 2 x 2, 9 2, x 2 - a0 : 1 + 1 + 1 +, - a 0 1 + 1 + 1 + ; 2 4 2 9 2 2 2 2 3 2, sin x : sin x x = 1 - x2 3! + x4 5! - x6 7! +,, x2-1 3!, - 1 6, - a0 1 + 1 + 1 + = - 1 2 2 2 3 2 6 ;, a0, : 1 1 + 1 + 1 + = 1 2 2 2 3 2 6, = 1 1 + 1 + 1 + + 1 + = 2 2 2 3 2 n 2 6 = 0, - ;,,, 39
( 1.644 934 ),,,,, I. :,,,,,,,, : 1 1, x 2 + y 2 2 x y, : x 3 + y 3 + z 3 3 x y z( x 0, y 0, z 0 ), x 4 + y 4 + z 4 + w 4 4 x y zw ( x 0, y 0, z 0, w 0 ),, : x n 1 + x n 2 + + x n n n x1 x2 xn ( x1, x2,, xn ), x n 1 = A1, x n 2 = A2,, x n n = An, : A1 + A2 + + An n n A 1 A2 An : n 2 0 < a < 1,0 < b< 1, : ( 1 - a) ( 1 - b) = 1 - a - b+ ab> 1 - a - b, : 0 < a < 1, 0 < b< 1,0 < c < 1, (1 - a) ( 1 - b) ( 1 - c) > 1 - a - b - c, : 0 < a1 < 1, 0 < a2 < 1,, 0 < an < 1, (1 - a1 ) ( 1 - a2 ) ( 1 - an ) > 1 - a1 - a2 - - an 40
3, : 2cos 2 = 2 1 + cos = 2 + 2cos ( 0 < < 90 ) ; 2, : 2cos = 2 + 2cos = 2 + 2 + 2cos 2 2 2, : 2cos 2 n = 2 + 2 + + 2 + 2cos n 2, ( ),,, ;,,,,,, 3 + 3 = 6 6 3,,,,,,, :,,, C. :,, 4,,,,,,,, 6 = 3 + 3, 8 = 5 + 3, 10 = 7 + 3, 12 = 7 + 5, 14 = 11 + 3, 16 = 13 + 3, 18 = 11 + 7, 20 = 13 + 7,, : 4, 41
, 2 2-1 = 3,2 3-1 = 7,2 5-1 = 31, 2 7-1 = 127, 1600 : 2 P - 1 (, P ) (, M P ),, P = 67, M67 = 2 67-1, 2 67 2 6 7-1, : 2 2 1 + 1 = 5, 2 2 2 + 1 = 17, 2 23 + 1 = 257, 2 24 + 1 = 65 537, : 2 2 n + 1( n ) (, Fn ), 16, : F5 ( = 2 2 5 + 1 = 4 294 967 297), 641 6 700 417 5, ( V ) ( E) ( F) ( 2 10), : 2 10 V E F V E F 4 6 4 7 12 7 8 12 6 10 15 7 6 12 8 8 12 6,, : V + F = E + 2, ( ) 42
,,,,,,,,,, :, 6,? (, ),? : 100 26, 1 000 168, 1 000 000 78 498, 1 000 000 000 50 847 478,, : (1 N) 1 100 26 0.260 1 1 000 168 0.168 1 10 6 78 498 0.0784 98 1 10 9 50 847 478 0.050 847 478, ;, ln N ( ), : 1 N, N ( 1 N) 1 ln N ( % ) 1 100 0.260 0.217 20 1 1 000 0.168 0.145 16 1 10 6 0.078 498 0.072 382 8 1 10 9 0.050 847 478 0.048 254 942 5 19 7 : 43
S1 ( n) = 1 + 2 + + n= n( n + 1 ) 2 ( 7 ) ;, ( S2 ( n) = 1 2 + 2 2 + + n 2 )? ( S3 ( n) = 1 3 + 2 3 + + n 3 )? S1 ( n) S2 ( n) : N 1 2 3 4 5 6 7 S1 ( n) 1 3 6 10 15 21 28 S 2 ( n) 1 5 14 30 55 91 140 S2 ( n) S1 ( n) : N 1 2 3 4 5 6 7 S 2 ( n) - S 1 ( n) 0 2 8 20 40 70 112, :, S2 ( n) S1 ( n) N 1 2 3 4 5 6 7 S 2 ( n) S1 ( n) 1 = 3 3 5 3 = 5 3 14 6 = 7 3 3 = 9 3 55 15 = 11 3 91 21 = 13 3 5 = 15 3 : S2 ( n) S1 ( n) = 2 n + 1 3, S1 ( n) = S2 ( n) = n( n + 1) 2 n( n+ 1) (2 n + 1 ) 6, :, S2 ( n), : S3 ( n) S2 ( n), S3 ( n),, : N 1 2 3 4 5 6 7 8 S2 ( n) 1 5 14 30 55 91 140 204 S 3 ( n) 1 9 36 100 225 441 784 1296 S 3 ( n) S2 ( n) 1 = 3 3 9 5 = 9 5 18 7 = 2 9 7 10 3 = 3 10 9 45 11 = 5 9 11 63 13 = 7 9 13 28 5 = 7 12 15 103 17 = 12 9 17, S3 ( n) S1 ( n), : 44
N 1 2 3 4 5 6 7 S 1 ( n) 1 3 6 10 15 21 28 S 3 ( n) 1 = 1 2 9 = 3 2 36 = 6 2 100 = 10 2 225 = 15 2 441 = 21 2 784 = 28 2 S3 ( n) = ( S1 ( n) ) 2 S3 ( n) = n( n+ 1) 2 2 = n2 ( n+ 1) 2 4, :,,,,,,,,,,,, 8 5,,,,,,, ( 2 11) 2 11, V E F 2 11, V, E, F 1, 4, ( 4-1) ; 1 + (4-1 ) = 4, k,, V, E, F 1, k, ( k - 1), 1 + ( k - 1) = k. :, ( ), : ( ), ( V + F - E),,,, V = 4, F = 4, E = 6, ( V + F) - E = 2 9 n (,, )?,,,, n (, )? 45
,, n, Sn 2 12, n 3 4, Sn 4, l4,, 2 12,, : ( n - 1) ln, ln ( n - 1 ) n ( ( n - 1 ), 2 13 ),, n, 2 13, n 1 + 1 + 2 + + n = n( n + 1 ) 2 + 1,,,, : ( ), ( ), : ( n - 1 ) n, V n n ( n - 1),,, ( n - 1) n ( n - 1 ) ;, ( n - 1 ) n ( n - 1 ) [ ( n - 1) + 1] n 2 + 1 = n( n - 1 ) 2 + 1, n( n - 1) n, 2 + 1 46
,,, n 1 + 1 + 2( 2-1 ) 2 + 1 + + = 1 + 12 2-1 2 + 1 + 22 2-2 2 n( n - 1) 2 + 1 + 1 + + n2 2 - n 2 + 1 = 1 + 1 2 (12 + 2 2 + + n 2 ) - 1 2 ( 1 + 2 + + n) + ( 1 + 1 + + 1) = 1 + 1 2 n( n + 1 ) ( 2 n + 1 ) 6 = 1 + n( n2 + 5 ) 6-1 2 n( n + 1 ) 2,, + n,,,,, ( ),,,,,,,,,,,, 1, : : 0 f ( x) c, x ( a, b), f ( x) 0, x ( a, b) :, f( x) = kx + h, x ( a, b), f ( x) k, x ( a, b) 47
, : ( * ). f ( x) 0, x ( a, b), f( x) c, x ( a, b), : ( *, f ( x) k, x ( a, b), f( x) k x + h, x ( a, b), : ) f ( x) k, x ( a, b) g( x) = f ( x) - k x, : g ( x) = f ( x) - k 0, x ( a, b), *, : g( x) h, x ( a, b) f ( x) = k x + h, x ( a, b) *, :, :,,,,, :,,,,,,,,,, ( ), ( ),,,,,,,,, : : ( )? : ( )? :??? 48
, : (, )??,,,,,, :????,,,,,,,, :????, :,,,,,?, :,,,,, :,,, : 49
,, ( n ( n + 1 ) ),,,,,,,,,,, 2 2 : ( 1 - a1 ) ( 1 - a2 ) (1 - an ) > 1 - a1 - a2 - - an ( n 2, 0 < a1 < 1,, 0 < an < 1 ) : n= 2,, ( 1 - a1 ) ( 1 - a2 ) (1 - ak ) > 1 - a1 - a2 - - ak : ( 1 - a1 ) ( 1 - a2 ) ( 1 - ak + 1 ) > 1 - a1 - a2 - - ak - ak + 1 : ( 1 - a1 ) ( 1 - a2 ) (1 - ak ) (1 - ak + 1 ) > (1 - a1 - a2 - - ak ) ( 1 - ak + 1 ) ( ) = 1 - a1 - a2 - - ak - ak + 1 + ( a1 ak + 1 + a2 ak + 1 + + ak a k + 1 ) > 1 - a1 - a2 - - ak - ak + 1,,,,,,, :,,,,, ;,,,, S3 ( n) S1 ( n), S2 ( n),, ( 5 8 ),,, 50
:,, ( ), ( ),,,,,,, : ( ) ( ),,,,,, : ( ),, :,,, ;,,,,,,,,,, 51
:, ;,,,,,,,,,,, + 1 n = 1 n 2 = 2 6 4 P ( P + 2) ( 5 7, 11 13 ),,,,, 17 19, 29 31,41 43, ;,, ( ) : P ( P + 2 ) ; n 1 1,, n. ln n ( ln n) 2 5, V a : 2 V - a = 4., 2 14 : 2 14 52
V a 2 V 2 V - a 8 6 2 = 12 16 4 4 4 = 4 8 4 6 8 = 8 12 4 10 5 2 + 2 3 = 16 20 4 6 3 2 + 2 = 8 12 4,, : S1, S2,, SF F, n ( n - 2 ). a = ( S1-2 ) + ( S2-2 ) + + ( SF - 2 ) = ( S1 + + SF - 2 F) ( S1 + S2 + + SF ) F, S1 + S2 + + SF = 2 E : a = 2 ( E - F). 2 V - a = 4, 2 V - 2 ( E - F) = 4, F + V = E + 2.,,, ( ) : A, A B C D, A ; B C D, A, :, ;,,,,, ( ),, I. : 53
,,,,,,,,,,,,,,,,,,,,,,,,, ;,,,, 54
3 : 1900, :, ;,?,?, :,,,,,,,,,,,, :,,,,,,, :,,,,,,,,,, ( ) ;,, 55
?,, :,,,,,, :,,,, ( ), :,,,,,,,,,,,,,,,,,,,,?,?,, :,, :,, ( ) (, :,,, ), :,,, 56
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1, ( ) (, ), ( ) 2,,,,, ( ), :, :,,,,, :,,, ;, ;, :,,,,,,,, :,, (, 4 ),, ;, 3,,, e(, 2. 718 2 ), 10 ( )?,,,,,, ( loga N = b) ( N = a b ), 10, (, 34 617 0. 001 27) (4-3),,,,, N lg N, N 1 10 000 58
, lg N ( 4-0 = ) 4,, N N( ) b = lg N( ) b( ) 10 1 0. 301 0 20 30 1. 301 0 1. 477 1 0. 176 1 40 = 10 1. 602 0 0. 124 9 90 1. 954 2 100 2 0. 045 8, (,, ), 1. 1, 1. 1 0 = 1, 1. 1 1 = 1. 1, 1. 1 2 = 1. 21, 1. 1 3 = 1. 331, 1. 1 4 = 1. 464 1, : b( ) b N = 1. 1 b ( ) N 0 1 0. 1 1 2 3 = 1 1. 1 1. 21 1. 331 0. 11 0. 121 4 1. 464 1 0. 133 1, 10 : b b N = 10 b N 0 1 9 1 2 3 = 1 10 100 1 000 90 900 4 10 000 9 000 ( ) 1. 001, : : 59
b b N = 1. 001 b N 0 0 0. 001 1 2 3 = 1 1. 001 1. 002 001 1. 003 003 001 0. 001 001 0. 001 002 001 4 1. 004 006 004 001 0. 001 003 003 001,? 1. 001, 1, 0. 001,,, ( 1. 001) 1 0 00,, : ( 1. 001 1 0 00 ) 0 = 1. 001 0 = 1; ( 1. 001 1 00 0 ) 0. 001 = 1. 001 1 = 1. 001; ( 1. 001 1 000 ) 0. 0 02 = 1. 001 2 = 1. 002 001; : B b N = ( 1. 001 1 000 ) b N 0 1 0. 001 1. 001 0. 002 0. 003 = 0. 001 1. 002 001 1. 003 003 001 0. 001 0. 004 1. 004 006 004 001, : 1. 1 10,1. 01 10 0,1. 001 1 000, 1. 000 1 10 0 00,,, 1 + 1 n n,, lim n 1 + 1 n n e, (, e ),,, (, ),, ( ),,, 60
,, ;,,, 5-1 0. 618, 2, ( ),,,,,,,, : :,, :,,,,,,,,,,,, :, ;,,, :,,, :,,,,,,, 4 + 4 + + 4 = 7 4, 4 4 4 = 4 7 7 4,, 7 4 61
, ( ),,,,,,,,,,,, ( ), 1,,,,,,, : ( ) ( ), ( ), ( ) ( ),, ( ) : :, 1, ( ) ; : 1 : : : 1 : S 1, S,, S,,, :, 62
, :, :, :,, ( ),,,,, :,, 4,, ;, ( ) :, 0 1,, 1, 2, 3,, 10, 11, 12, 13 : 1 1 2 10 3 11 4 100 5 101 11 1 011 12 1 100 13 1 101, ( : 1,, 0,,, 0 1,, 63
, 1 011 011 001 729 ) ;,,, ( ) 1 0 ;, ( ),, 1 023 : ; 729 : ;, ( ),,,, ;,,,,, ( ) ( ) ( ) : V = 1 3 h( S + S + S S ) ( h, S S ), : sin x = t an x = 2 u 1 + u 2 ; cos x = 1 - u2 1 + u 2 ; 2 u 1 - u 2 ; u = tan x 2, R( sin x, cos x) tan x 2,, : ( x0 ( a, b) n+ 1 ) : f ( x) = f ( x0 ) + f ( x0 ) ( x - x0 ) + f ( x0 ) 2! ( x - x0 ) 2 + + f ( n) ( x0 ) n! + f ( n + 1 ) ( c) ( n + 1 )! ( x - x0 ) n + 1 ( c x x0 ) ( x - x0 ) n,, :,, 64
( ) : : 5,, a 0 = 1 ( a 0 ) ; a - n = 1 a n ( a 0, n ),, a 3 a 3 = a 3-3 = a 0 = 1; a3 a 6 = a 3-6 = a - 3 = 1 a 3, ( ),, :, ( ) :,,,,, 6 F. F., F. 1872, F. :,,,,,,,, F. : ( ),, ; ;, F. M. : F.,, 50,,,, 65
,,,, :,, 20 30 (, ), ( ) :,,,,,,, :,,,, ( ), ( ),, ( ) ( ) ( ),,,,, : :,! :,,,?,,,, :,,,,,,,,,,,,, : 20 60,, 66
, :,,,,,,,, :!, ;,,,, ( ),,,,,,,, :,,, 2 (, ), ( 2 : 2 = p q ( p, q ), 2 = p2 q 2, p 2 = 2 q 2, p p = 2 m, : ( 2 m) 2 = 2 q 2, 2 m 2 = q 2, q p, q 2,,,, M. :, :, ;,,,, ;, 7, 67
,,,, ;,, ;,,,,,,, :,,,,, ( ),,,, 8, ( ),,,, ( ),,,, 9,,,,,, (, )?,,, : 1843 10 16,,,,,, i, j, k ;,, a + bi + cj + dk, a, b, c, d, i, j, k i ; ( i 2 68
= j 2 = k 2 = - 1; i j = k, k i = j, j k = i; j i = - k, i k = - i, k j = - i), p = 3 + 2 i + 6 j + 7 k, q= 4 + 6 i + 8 j + 9 k, : p q= ( 3 + 2 i + 6 j + 7 k) (4 + 6i + 8 j + 9 k) = - 111 + 24 i + 72 j + 35 k; q p = ( 4 + 6 i + 8 j + 9 k) (3 + 2i + 6 j + 7 k) = - 111 + 28 i + 24 j + 75 k,,,,,,,, M. :,, ; 18 70 80,, :,?,,,,,,, ;,,,,, ;,,,,,, :,,,,,, :,,, ;,, 69
,, (, ),,,, ( ) ;,, :,,,,,,,,,,,,,,,,,, A. :,,,, ( ),,, ( ), ( ),,,, ;,, ( ) (, ),,,,,,,,,, 70
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,,,, (, :! ),, ( ), :,, ( ),,,, ( ),,,,, : ( ),,!!!, ( ),,,,,,, :,,,,,,,, :, ( 11 ) : 2 11, 2, 3, 5, 7, 11 2 3 5 7 11 = N N 1,, N, 72
: ( ),,,,,,,, (, ),,, :,,,,,,,,,,,,, : (, ), ;,,,, :,,,,,, ( ),,,, ;,,, :?!,, :, ;,,,, 73
,, :,,,,,,,, :,,,,,,,,,,,,, ;,,, 1, k, l, m r, O;, l m A, m k B, k l C ( 3 e r 1 ) : A, B, C 3 1 3 2 3 3 e P, : P A = PB = PC = r ( 3 2), k, l, m ( K, L, M ) ( O, A, B, C ) 3 3, 3 4 :, (?),,,,, 74
, :? : 3 3 4 3 5 3 6, 3 4 3 5, 5 ;,, 3 4 ( 3 6 ),,, r( 3 6 ),, ( P) r, P A = P B = P C= r : : 3 4,,,,,,,,,,,,, ( ),, ( ), :, 75
, ( )? :,, :,,,,,,,,,,,,, :,,,,?,, ;,,, :,,, :,, :,,,, ;,,,,, ( ) 76
,,,,,,, :, ( ), 2,,,,,, :,,,,,,, x 2 + y 2 2 x y x 3 + y 3 + z 3 3 x y z, 3 x y z 3,,,,,,,,, ( ),, ( ) : ( ), ( :, ) ;,, :,,,,,,,,,,, 77
,,,,, :,,,, :,,,,,,,, ;,,,,,,,,, :,,,,,,,,,,,,,,,,,,,,,,,,,, 78
,,,,,,,,, (,, ),, :,,,, :,,,,,,, :,,, 1 3 6 3 7 3 6 79
(1 ) S =? (2 ) S = 2 ( S 1 - S 2 ) ( 3) a =? h =? 3 7 (4 ) sin 2 = L 2 R, h = R 2 -,,,,,, :, ; L 2 2 3 8 80
( 3 8 ),,,,,,,,,,,,,,,,,,, 81
4 1, ( ),,,?, A.. : :,,,,,, 1? ( ), 1 (, 1 ),, ( ),,,,,,,, :,,,,,, ;,, ( ),, 82
, ;,,?,,,,,,,,,,,,,,, 1, ( ),, (, ),, ( ), ( ),,,,,, ( ), ( ),,,, (, ),, :,,,,, ;,, a, b x, y 83
,,,,,,,, F.,,,,, ( ),,, :,,,,,,,,,, :,,,,,,, 1, 2, 3, 4,,,,, ( ),, : 1, 2, 3,, 101, 102,,,, 84
(,,,, ), ( ),,, (, :,, ), i 1, 1545, x ( 10 - x) = 40, ( 5 + - 15 ) (5 - - 15 ) ( 5 15i), ( 5 + - 15 ) ( 5 - - 15 ) 40, 10,,, ( 5-15 ),,?, ( ), : 0, 0, 0,,,,,,,, (, ), ( 1799, [ ] n ),,, 1 : 1 - x 2 + x 4 - x 6 + x 8 - = 1 1 + x 2,, x = 1 x = - 1,,, : x = i, 1 1 + i 2 = 1 1 + ( - 1 ) = 1 0, 1 1 + x 2 ( 0, 1 ),, 85
1 ( 4 1, 1 - x + x 2 - x 3 + x 4 - = 1 1 1 + x : 1 x = - 1 ) 1 + x, : i, e i + 1 = 0 i 4 1 2 i, 3,,,,,??,, :, (, ),,, ;,,,, ( 4 2 ) 4 2 : ( ),,,,!, ( ) ( ),, 3, ( ),, : 86
; ( ), ( )?,, (, ), :, ( ) ;, ;,,,, M. : 1700,,,,,,,,,,,!, ( ), 1, 2,, : ( ),, x, y, z,, R : R x y x y R,, R : : x y, Rx y R y x; : x y z, : R x y R y z R x z; : x, Rx x, R : R, R,, 87
,, : 1 ; 2 ; 3 ;,,,, = c = 1( c ) ( x) ( x c ( " y) ( y c y = x) ), = c = 2 ( x) ( y) ( x y x c y c) ( " z) ( z c z = x y = z) ), = 1 = { c c = 2 = {c c = 1}, = 2},, 3 (, ) (, 3 ),,, : [ 0] = { x x 3 0} ; [ 1] = { x x 3 1} ; [ 2] = { x x 3 2}, : [ 0] [1 ] [2] [ 0] [ 0] [1 ] [2] [ 1] [ 1] [2 ] [0] [ 2] [ 2] [0 ] [1] [0] [ 1] [2 ] [ 0] [0] [ 0] [0 ] [ 1] [0] [ 1] [2 ] [ 2] [0] [ 2] [1 ],, :,,,, ( ) ;, 88
,,,,,, 4 :,,,, 4 3 : 4 3, :,,,,,,,, ( ),,,, : 4 4, :,, ; 4 4,, :,,,, ( ),,, ( ),, ( 89
),, 4, 4, (,, ),,, ( ), ( ), :,, ( ), ( ) ;,, ;,,,,,, (, ),, ( ),,,,,, ;, ( ),,, ( ) ( ),,,, :,,,,, 4 5,, 90
4 5 ( ),, :, ( ) (,, ),,,,, i, ( ),,,, ( ),,, 5, (, ),,,, 19,, - 1,,,, ( ), 91
,, : ( a, b) a + bi,,,, ( 4 6) : 4 6, : ( ), x ( ) :,,,,, (, ), ;,,,, (, ) 92
, : ;, ( ), ( ), ( ), ( ),, :,,,,,, (,, 180 ) ;,,,,,,, 18, ( ) ;,,, ; 93
:, ( ), : 11,,,,, :,, :, ( ),, (,,,,,,,,, ), :, ( ) ;, ( ) ;, ( ) ( ) ;, ; ( ),,,, (, :, ) :,, ;,, ;,, ;, ;,, ( ),, 467 ( ) (, ), ( ),,, 94
(, ),, ( ),,,, ;,,,,, :,,,,,, (, ), :,,,,, 3,,, ( ; ),,,, :,,, a : a b - b a + c d - d c a b - b a c d - d c b - b a = a+ c b + d - ( b+ d) a + c ; = ac - bd ad + bc - ( ad + bc) ac - bd,,, 95
,, ( a b - b a a + bi ), : 19 ;, :,, ( ),,,,, ( 1899 ), :, ;, (, ),,,,, ( ) (, : ),,, :, ( : ),,,, : ( 1 8 ), 8 ( 1 4 ), 1, B C, A : A, B, a, A, B : B A, C, A, B, C ( 1 5 ), 2 :, ( ) :,, ( 1 2 ), 1 ( ) : A B CD, AB A1, A2,, An, A A1, A1 A2, A2 A3,, 96
An - 1 A n CD, B A A n,,,,,,, ( ) (,,, ;,,, ),,,,,,, ;,, 20 20, :,, ( ),,,,,,,, ;,,,, ( ) :,, :, ; (, ),,, 97
:, ( ) ;,,,, ( ) ( ), : ( 1) : ( ) ; ( ) ( 2), ; ( ),,,,, :, : ; ; ;,,,,,,,, ( ),,, ( ), ( 5 ), : :,,,,,,,,,,, :,,,,,, ( ) 98
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5 1,,, ( 5 2 ), :, 5 2,,,,, C. : 5,,,,,,,, 105
:, ;,,,,,, ;,,,,,, 16,, ( ),,, : 1, 15 ( ) : ( 5, OA B 3 A B ),, A B, OA B, OA B, 1 2 l r(, l A B, r ) ;,, 1 l 2 lr = r 2 l = r l 2 2 r = r2., 5 3 2,,,, t, s;, s,, ;, t t 0, ( dt), ds dt,, (, ;, ),, ( ) (, 106
) ;,, ( ),,, ( ) ;,,,,, 19,, :,,,,,,,,, ; 20 60, ( )?, G. :,,,, 1874,, :, ( ),,, ( ), ( ), 0 ( ), : 1, 2, 3,, n, 2, 4, 6,,2 n,, 0 107
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( 0, 1),, (0, 1 ) n : an = 0.an1 an2 ann, ani ( i = 1,2,3, ) 0, 1, 2, 9 ( 0, 1 ) : a1 a2 a3 an a1 1 a1 2 a1 3 a1 n a2 1 a2 2 a2 3 a2 n a3 1 a3 2 a3 3 a3 n an1 an2 an3 ann, a11, a2 2, a33,, ann,, b= 0.b1 b2 bn,, bn = ann + 1, ann 8 ann 9 1, ann = 8 ann = 9 b, an ; b ( 0, 1),,, an,,, ( 0, 1) ( ( 0, 1 ) : {0.an1 an2 ann }),,,, : M N, N M, M N, M N ( ),, : 0, c, c > 0,, : S P S ( S ) 1 1, S = {1,2}, P S : ( ), {1}, {2}, {1, 2} 109
S P S ( S ),,, : : S P S ( f ), S ( ) S ( P S ) S x S M f x f x x f x, : x f x, x f x, : S = { x x S x f x} S S,,, z S, S = f z z S, z S z S, S z f z, z S, ; z S, z f z, S z S,,, S P S 2 a 2 a > a; a, :,, : 0, 2 0,2 2 0,,, ; ( ),,, : 12,,,, : ( 1),,, + 1, + 2, 12, ;,,,,, 110
( 2),,, 1,2,3,, ; + 1, + 2,, +, 2; :,,, + 1, + 2, ( ) ; +, 2 ( ) ; 2 + 1, 2 + 2, ( ) ; ( 2) +, 3 ( ) ;,, 0 (, ),,, 1,, 1,, : 1 + 1, 1 + 2, (, 1 ),,, : ( 3),,,, :, 1, 2, ; : 0, 1, 2,,, ;,, 1897, A. ;,,,,,,, :,,,,,,, 111
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19,,,, :, :,,,,,, : y = f( x) x x = i; y f( x + i) - f ( x) =, x i i 0 ( ) y x,, 1872,,,, : x = x0 f ( x),, x0 x, f( x) L, L f ( x) x0,,, :,,,,,,,, ( ),, (,,,,, ),,, 19,, 115
,,,,,,,,,, ;,,,,, (, an = 0.an1 an2 ann ), ( b= 0.b1 b2 ),,,,,,,, ;,,,, ;,,,,,,,,,, 1902 :, S0, x S0 x x S0, S0 S0, S0 S0, S0 S0, S0 ( ), S0, S0 S0, ; S0 S0, S0 ), S0 S0, S0 S0, (,, : 116
, M PM M M M, P M > M S( ),, PS > S; S, PS, PS S, :,, ( + 1 ) W, W, Q : Q ; Q, Q > Q,,, ( ),,, :,,,,,, :,,,,,?,, :?, :,,,,,,,,,,,, 117
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,,, 1 : 1 :, : 1 + 2 + 3 + + 99 =?,, : 4 950,,,,,,,,, ( 7 1) : 7 1, S = 1 + 2 + 3 + + 99 =? 2 S = 1 + 2 + 3 + + 99 + 99 + 98 + 97 + + 1 = 100 99 = 9 900 : S = 4 950 2 :, :, 50 140,?, :,, ( 9),,,,, : 154
, 50 x + y = 50 140, 2 x + 4 y = 140, : x + y = 50, 2 x + 4 y = 140, : x = 30, y = 20, h 50, f 140, h f, : x + y = h, (1 ) 2 x + 4 y = f ( 2) : ( 2) 1 2 x + 2 y = f ( 3) 2 ( 3) - (1 ) y = f 2 - h,,, ( ) 3 : 7 x y 2 1, 2, 3, 4, 5, 6, 7, 8, 9,?, :?,, :,,?, : S,, 3 S, 1, 2, 3, 9, 45, S = 15 155
:, :? :, 7 2 7 3 7 4 9? ( 7 3 ), 8 : 8, 9 8, 15 (, 9?), 8 7 6 1? ( 7 4 ), 2, 15, 12, (, 1?), 2, 3, 4, 5 : : 5, 1 7,, : 5 7 5 7 6, 1 ( 7 6), 9, 156
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,,,, :,,, 1 2 : ( 1, 5, 9) ; ( 1, 6, 8) ; ( 2, 4, 9) ; ( 2, 5, 8) ; ( 2, 6, 7) ; ( 3, 4, 8) ; ( 3, 5, 7) ; ( 4, 5, 6) : : (,, ),, 5,, 5,?,,, 2, 4, 6, 8, 1,3,7,9,, 2, 4, 6, 8, 1, 3, 7 9,, :,?!, : ( 1) 1 9, 2 10? 87 95? ( 2) 3, 6, 9, 12, 18,, 27 1 9? ( 3) 5,8,11, 14,17, 20,, 29?, ( ), 158
, 4, (, ) ( 7 14),, :??,, CE, 7 14 (,,, 1992 ),, :,?,,, (, A B AC = 1 1, BD DC = 1 1, AB AC = BD DC, 7 15), :,?, :,,, A B A C, A BD DC??,,? 7 15,,,,,, ( A B A C= 2 1 AB A C= 3 1 ), : ;,,,, 1 1, A B AC = 1 1 BD DC = 1 1,, :,, : 159
? : : ( 7 ( DE DF ) ; 16), 7 16 7 17 : ( 7 17, S AB M = S ACM ),, ( ) : A A K A BC (, 3 2 )?,, ( 7 18),, : A B AC,? 7 18 7 19, ( 7 19 ), AB A C,, :,?, ( 7 20, A H = AC, H C AD) 7 20 160
,, AD A, 7 20 : AD H C,,?, AD A, :?,, A AD, A B A C, ( ),,, A H C AD, C AD CH,,, CE,,,, (,, ), AB A C = BD DC( AD A ), BD DC, BD DC A BD ADC,, A B A C A BD ADC,, AB AC ABD ADC, ( ) ( 7 16 DE DF),, 5 4, (,, 1992 ),,,,, : P l, Q, PQ P l ( 7 21 ),, : P l, P? P,, 161
?, : :?? : P l l, l l P l ( 7 22 ),,,, P l l l 7 21 7 22,? :,, ( M) ( l ; l ; l l ; l l ), :, l l ( M l l d d,, l l d + 0; d + 0; d + d ; d - d d - d),?, l,, ;, P( x0, y0 ) l: A x + B y + C = 0,,, ;, :?,?,,,,, : M l l?,, l,, P l l l,, l l, M( P) l l, P l, 162
?!, : l, (, ),,, :,?,, : P l l, l l, P l,,,,,,, 7 21 ( P l, Q, P Q ),,,, ( ),,,,,,,,,,,, :, ;, :!,,, :,,,, (, ), :,,, 163
,,, (, ) (,, ),,,,,,,,,,,,,, 6,, y = a x ( a> 0) : y = a x + x - a x = a x ( a x - 1 ), y x = ax - 1, x a x y y = lim x 0 x = ax lim x 0 ( a x lim x 0 a x - 1 x a x - 1 x - 1 ) = t, x = loga ( 1 + t) ;, x 0, t 0, : = lim t 0 y = a x ln a t log a( 1 + t) = 1 lim t 0 log a(1 + t) 1 t = 1 = ln a loga e,, t = a x - 1??, : y y = lim x 0 x = ax lim x 0 a x - 1, t x 164
,,!,, :,, :,,, : ( log a x ) = 1 x log ae,,, :, :,?,,, :,, : ( a x ) = 1 loga y = 1 loga e y = a x ln a,,,,,,,,,, 7,,,, : sin ( + ) = sin cos + cos sin 7 A D BC. S A DC 23, B AD, DAC, AD = m, : S AB D = bmsin, S ABC = bcsin( + ) S AB C = S AB D + S AD C, bcsin( + ) = cmsin + bmsin = cmsin, bc : sin( + ) = m b sin + m c sin, 7 23 165
sin( + ) = sin cos + cos sin, ( ) ( 4,,, 1994 ),, 8,,,,,, ( ),,,,, :,?,,, ;,,,,,, : ( ),,, : (, 7 ) ;,,,,,,,,,,,,,,, 166
, ( ICMI),,,,, :,,,,,,,,,,,, 6,,, ;, ;,, 18 7 24,,,??? 18, 167
, :, : (, 7 24 ) 7 24, ;, ;,,,,,,?, :,,, 6,,,, 20 80,,, ( ),?,?,, :,,,,, ( ),,,,,,,, 168
,,,,,,,,, (, ),,,,,,,,,, S. M. :,,,, ;,, ;, :,?, ;,, :,?, ( What if not ) 1 ( ), : : :,?, : 169
( 1.1) ; ( 1.2) ; ( 1.3), a 2 + b 2 c 2 : :,?, : ( 2.1) ; ( 2.2) ; ( 2.3) : :,? : ( 3.1) ; ( 3.2) ; ( 3.3), : :,?, : ( 4.1) ; ( 4.2) ( ) : :,?, : ( 5.1) ; ( 5.2) ; ( 5.3) ; ( 5.4) :,? : ( 6.1), a 2 + b 2 = c 2 + d 2 ; ( 6.2), a 2 = b 2 ; ( 6.3), a 2 + b 2 = c 2 + n; ( 6.4), a 2 + b 2 = n :,? 170
: ( 7.1) >, a 2 + b 2 > c 2 ; ( 7.2) <, a 2 + b 2 < c 2 ; ( 7.3), a 2 + b 2 c 2 ; ( 7.4), ( a 2 + b 2 ) c 2 ; ( 7.5) a 2 + b 2 c 2 ; ( 7.6) a 2 + b 2 c 2 : :? : ( 8.1) a 2 - b 2 = c 2 ; ( 8.2) a 2 b 2 = c 2 ; ( 8.3) ( a 2 ) b 2 ( 8.4) a 2 b 2 = c 2 ; = c 2 : : ( 9.1) a 3 + b 3 = c 3 ; ( 9.2) a 4 + b 4 = c 4 ; ( 9.3) a 5 + b 5 = c 5 : : ( 10.1 ) a 1 2 + b 1 2 = c 1 2 ; ( 10.2 ) a - 1 + b - 1 = c - 1 ; ( 10.3 ) a 2 + b 2 = c 2 : : ( 11.1 ) a + b 2 = c 2 ; ( 11.2 ) a 2 + b 2 = c 3 ; ( 11.3 ) a 2 + b 3 = c 5 ; ( 11.4 ) a 2 + b 2 = c,,, : ( 7.1) a 2 + b 2 > c 2, : ( 1)? ( 2)? ( 3) a 2 + b 2 c 2, 1? 171
( 4) a 2 + b 2 > c 2?,, :?????????????????,??????????????,?? 172
?? ( )?,,,,,,,,,,, : (, ),,?, :, :,,,,,,,, ( ),,,,,,,,, 6,,,,,,,,, 173
,,,,,,,,,,,,,,,,,,,,,,,,,,, (, 3 ), 2, ( ), ;, ;, : ( 7 25)?, : ( 1), (2 ), ( 3), ( 4) 174
7 25 7 26 7 27,, 3 10 ( 7 26), 3, ( 7 2 ( 10-2 3 ) : = 4 10 2 25 27) ;,,,, : R, S,, ( S - 2 R) 2 S 2,, : ( 1 2 ),? : ( S - 2 R) 2 S 2 = 1 2, : 1.46,,, :,,?, 20, 7,? 10,?, 1 2,?,,, 175
3 2000,, ( ) : ABC, D, E A C, AB ( 7 28), : BD = CE,,, 7 28, : : ABC, B, C AC D, A B E : BD = CE : A BC, D, E A C, AB : DBC = ECB ABCD O : AC = DB 7 29 7 30 7 31,, 40 84, ( ) 28. :,,,,,,,, :, ;, ;, ;, ; 176
,,, :,, :,,,, ( 7 28 ), : :,,?,, ( 7 32 7 33) ( A ;, ) : A : ABC, B, C AC D, A B E : BD = CE B : A BC, BD, CE A C, AB : BD = CE 7 32 7 33,, ( 7 34 7 35 7 36) : C : A BC, A D = A E, BD, CE D : A BC, BD, CE C A, B A E : A BC, BD, CE AC, A B, B, C AD AE B F, CG : BD = CE : BD = CE : BF = CG 177
7 34 7 35 7 36 :,,?, ( 7 37 7 38 ) : F ( ) : A BC, G : A BC, BD, CE A C, BD, CE A C, AB F : DBC = ECB AB F : BCF 7 37 7 38 :,,?,, ( 7 39 7 40) : H ( ) A BCD, O : AC = BD I A BCD, D A B, BC DE, DF : DE = DF 178
7 39 7 40,,,,, 4 ( 4 5 ),,,,,,,,, 2000 : ( ),,,,,, :,,?,, :,?,,, (, a 2 = c 2 - b 2 ), ( 7 41) :, b, c A, a? ( * ) :, 7 41 179
a 2 = c 2 - b 2 = c 2 + b 2-2 b 2 = c 2 + b 2-2 bccos A ;,,, : : A,,!,,, ( * ),,, b, c a, b, c A a?,?,!, :,?, : ( )?,, :!,,!,,, ;,,, :,, ( a 2 = c 2 - b 2 c 2 + b 2-2 bccos A), (,,? ),, (, ),,!,,,,, :,,,,,, :,,, 180
,,,, :,?,,,,,, (,,,,,,,,,, ), :,,,,,,, 2,,,,,,,,,,,,,! 5 4, : 5, :? (, 1999 ), (, ),, ( ),,,,,,, ( ),, 181
,,,,,, (,, ),,,,,,, :,, (, ) (,,, ),,,,,,, ( ) 3, 1, 2,, 9,,,, 5,, :? 5? 6 4?, 3,,,, ( 7 42 ) : 7 42 182
,, : 2 3 R3 (, 4 3 R3 ),, :, 2 3 R3,,, :,, :,, 7 42,,, :,,,?, 2 6,,,,,,,,,,,,,,, 6,,,,,,,,,,,,, 183