γλώσσες
Σελίδες
Νομικός
- - 2013 3,5
2013
"" 2013:
...
4 SEEMOUS 2013
.
1 . 1099/96 .
IMABHMATIIJKH ETAIPEIA
ETAIPEIA 88 2013 : 3,50
e-mail: [email protected] www.hms.gr
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. 2.
. 3. .
1202 (. 1 1 70-. 1 240) (F ibonacci), Lber avac I 1 ,2,3, . . . ,9 () .
Guilielmo Bonacci. "filius Bonacci" Bonacci. - Fibonacci "filius Bonacci". , Bigollo "".
( ) . .
529 . ,
. . ""
900 " " "" . .
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XVI , , .
529 . 1 500 . . a - (Buggia). , . -
' 88 .4/1
--------------- . . . ---------------. , , , , .
, . Liber Abbaci. . / . . ,
' 1 4 ' ' ' 2 ,
4.!_. F ibonacci 2
/ , . .
F ibonacci . . fn - : h h h h h h h h 2 3 5 8 1 3 2 1 34 55 . . .
: fn+2 = fn+l + fn n ' fo = 1 'f = 2
F ibonacci , 2 . - !12 = 377
:
Fn+2 = Fn+l + Fn F0 = 1 F1 = 1 F ibonacci , , , 2, 3, 5, 8, 3, 2, 34, 55, 89, 44, 233, 377, 6, 987, ...
- .
---------------------------------------- 1
2
' 88 .4/2
--------------- . . . .
+ b -- =- b
.
_____ ____ -----
b----- -v--+b
+b . -- =-= b
: b 1 1 +-=-=>1 +-= b
2 --1 =0 :
1 +./5 , 1 -./5 =-- =--2 2 ' = -1
= 1 ,61 8033988749894842 . . . . ' =-0,6 1 80339887 49894842 . . .
Fibonacci . ( ) . .
n n+1 (n+ 1)/n 2 3 1,5 3 5 1 ,666666666 ... 5 8 1,6 8 13 1,625 .. . . .. . . . .. . . . . . . . 144 233 1,618055556 . .. 233 377 1,61802575 1 . . . . .. . .. . . .
.
.
Fibonacci ;
..,.,---------
1
1/2
2
.. .. ..
. .
. . , .
= = =
72 36 .
2. yv p
b ( ). -
' 88 .4/3
---------------- . . . ( ) . + b .
h
+h . , , -
. "" . . .
, ,
3.
. -
. .
p p Fbonacc.
. . . ( ) ( , "'" .
Fibonacci. .
Fibonacci.
. 88 .4/4
--------------- . . .
4. 6 libonacci 2 - - 1 = - = (1 + .J5)/2 = (1 - .J5)/2 . +' = 1 . ' = -1 .
2 = + 1 3 = 2 + = ( + 1) + = 2 + 1 4 = 3 + 2 = (2 + 1) + ( + 1) = 3 + 2 5 = 4 + 3 = (3 + 2) + (2 + 1) = s + 3
;
= J5+l =lim Fn =1+----2 n->oo F n-1 }+ --- 1+--1 1+-
, .
=l+Jl+l+ , .
5. t) :: .,/; Pascal
. - . :
I ; I / I
I I I I I 1 I I I I
I
to 6 1 I
I I )5 s 21 7 1
I I
1 I 70 56 28 8 1 I I I
I I I
. .
2.
3. l . 2 .
4. 1 2 3.
5. , 3 , 1 5.
6. 1 , 4, 3 8.
7. , 5, 6, 1 3.
8. 1 , 6, , 4, 2 .
9. 1 , 7, 15 , 10, 1 34. Pascal.
' 88 .4/5
--------------- . . . --------------
1 2 3 4 5 6 7 8 9 10 1
1 1 1 2 1
1 3 3 1 1 4 6 4 1
1 5 10 10 5 1 6 15 20
1 7 21 1 8
1 2 3 5 8 13 2 34 55
F ibonacci . ( F ibonacci. Pascal. . 1 .
F ibonacci. ;
1 1. 1. 1, 2, 3, 4, S, . . . H , 3, 6, 10, 15,. 5. .
F ibonacci . . . F ibonacci: , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 8 9, 144,233, 377, 61 , 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657,75925, 121393, 196418, 317811, . . .
( ) . , 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, , 7, 7, 4, 1, 5, 6, , 7 , 5, 3, 8, 1, ... 150 F ibonacci. 150
60 F ibonacci . 60.
. , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 75925, 121393, 1964 18, 3178 11, . . . 600 F ibonacci 2 300. , , .
. 3 ; 3, 4, 5 . 3 . 4 F ibonacci. , , 1, 2, 3, 4 . b .
" :
: 1.
. 23 = 6 .
2. : 26 = 1 2 .
3. . 5 = 5. .
4. . 22 = 4 , 3 2 = 9 . 13. 13.
5, 12, 13
' 88 .4/6
---------------------------------- . . . "" F ibonacci 2, 3, 5, 8. . , , 30, 1 6, 34. , -
6. Fibonacci l 000 1 999 F ibonacci - .
. 1299 - - . / - , - - - . " - - ;" - . - B iggolo ( - ) - . . 300 - - . - . -
F ibonacci - . Liber Abaci / , . 1 , 2, 3, 4, 5, 6, 7, 8, 9, 0.
.
1, s, 10, L 50, c 100, D 500, = 1000
, . . . . , . . - .
Fibonacci : 1. Lber Abbac ( ), 1 202 2. Practca Geometrae ( ), 1220 3. Lber Quadratorum ( ),
1225. 4. Flos ( ), 1 225.
5. D mnor gusa ( ).
. . 6. Commentary on Book of Eucld's elements. -
. Boyer ' history of Mathematics : " 900 .
Fibonacci 1240 20 " , "
. Fibonacci; ; ; ; (
:
Fibonacci JA VA.
Takis. [email protected]
' 88 .417
;
;
. :; ; . ' .
, Cambridge The Hub Eents 1 : ; , . :. : : ; ; ;
z 2
. , . , :. , .
. ,
, , , , . , , . , , , ; . , .
. a .
:. . , : (MEG scanners) . , . .
' 88 .4/8
a yv tJa
...
30 ' "
23 2013 1 ( ) , = 1, 2, 3, ...
1 = 2 . = ( + 1) ( 1 + 2 + + ._1 ) , 2. -1 2013 (1 ) : 1 = 2, 2 = 1 = 3 2, 3 =
4 ( 1 + 2 ) = 4 4 2 = 4 22 ,
1 2 2 5( ) 5 3 6 ( ) 6 4 4 =- 1 + 2 + 3 = - 24 = 5 2 5 = - 1 + 2 + 3 + 4 =- 64 = 6 2 . 3 3 4 4
n = ( n + 1 ) 2n- , n = 1, 2, 3, . . . , k .
n = k + 1 , : k+ = ( k + 2) 2k . k + 2 ( ) k + 2 ( 2 ( ) k-1 ) , k+ = -k- ,+ +G:J ++k = -k- 2+32 +42 ++ k+1 2 k + 2 ( 2 ( ) k-1 ) : k+ =-k- 22 + 32 + 42 ++ k + 1 2
( 1 ) 2,
2 = k+ 2 (21 + 322 +423 + + (k+ 1)2k ) k+l k ' ( 1 ) (2) , :
k + 2 ( 2 3 k- 1 ( ) k ) k+ =-k- -2 - 2 - 2 - 2 +- 2 + k+ 1 2
( 1 )
(2)
a,,, k;2 { -1-1;::_; + (k+ 1)2'} k;2 ( -2' + (k+ 1)2' ) (k + 2)2'. : 20 1 3 = 2014 22
0 12 2
( n + 1 ) n = - ( + 2 ++ n- ) , n 2, n-1
( n + 2) n+l = -n-( + 2 + . . . + n ) ' n 1,
' 88 . 4/9
(3)
(4)
(5)
-------- - --------1 + 2 + + n = (-n-)n+I, n;:: . (6)
n+2
( 5) ( 6) n = (-n-)n+
I - ( n - 1 )n => n+ = ( 2(n + 2))n, n;:: 1 (7) n+ 2 n + 1 n + 1
n = ( 2(nn
+ 1) )n
-
] =( 2(nn
+ 1) )(
n
21)n_2 = . . .
= ( 2(nn
+ 1 ) )(n
21) . . (
2 4) ( 23 ) = ( n + 1) . 2n-2. = ( n + 1) 2n-I' ' ' 2 ' 2014 220 12 1 = . 2013 = . 2 : y = 22 + 5xy + 3y2 (1 ) y = 22 + 2xy+ 3xy+3y2 y = ( + y ) ( 2+ 3y) ( 1 ) + y = z , z ( 1 )
y = z ( 2z + y) y = 2z2 + yz ( z - 1) y = - 2z2 . ( 2) z = 1 ( 2) y = -2 ().
2z2 2 ( z2 - 1) + 2 2 z-::f:.1 y = --= - = -2 ( z+ 1) -- . (3) z- 1 z- 1 z- 1 y z - 1 2,
z- 1 {-1, 1, - 2, 2} {0, 2, - 1,3 } . z = , ( , y) = (, ). z = 2, (x,y) = ( 10, -8) . z = -1 , ( , y) = ( -2, 1) . z= 3 , (x,y) = (12 ,-9) .
20, ' y = 22 + 2xy+3 xy+3 / ( + y-1) ( 2x+ 3y+ 2) = -2 (4)
, , ( 4) :
:::::
12} ::
: } ( , ) (, ) . ::;: :1- 2} :::} (x, y) (10, -8) :++12
2-1} :: :3-3} (x,y) (1 2, -9)
. {;+;;; :21} ::y - 1} (x, y) ( -2, 1) 3 " 2x2 +5)-X+ 3 /-y =O, (4)
l't> . t , = / + 8 y = y ( y
+ 8) = p2, .
' 88 . 4/10
--------- - -------- p = , = y = y = -8 . y = , (4)
= , (, y) = (, ) . y = -8 , ( 4) = 10, (, y) = ( 1 , -8) .
p -:f:. , / + 8 y -p2 = , ( 6) y p . , ( 5) . ' = 64 + 4 p2 = 82 + ( 2 p )2 = w2 , ( 8, 2p, w) .
(k (m2 - n2 ) ,k 2mn,k(m2 + n2 )) , k,m, n , m>n . : k ( m2 - n2 ) = 8, k 2mn = 2p (7)
k 2mn = 8, k ( m2 - n2 ) = 2p. (8) . k = 1 ( 6) ( m, n) = ( 3, 1 ) , p = 3 . ( 5) / + 8 y -9 = y = 1 y = -9 , ( 4) = -2 , y = 1 = 1 2, y = -9 , : ( x,y) = ( -2, 1) ( x,y) = ( 12, -9) . k?::. 2 , (6) p . , (7) p .
, ' = 64 + 4 p2 = w2 : '= 64 + 4p2 = w2 w2 -4p2 = 64 ( w- 2p ) ( w+ 2p) = 64 .
, , : ( w+ 2p ) + ( w- 2p) = 2w= 2
( w+ 2p )- ( w- 2p) = 4p = 4. w p :
{w+ 2p= 1 6} ( w, p) = ( 1 0, 3) . ( 6) :
w- 2p= 4 y2 + 8 y -9 = y = -9 y = 1 ,
(x,y) = (12, -9) (x,y) = (-2, 1) .
{w+ 2p = 8}
( w,p) = (8, 0) . (6) : w- 2p= 8
/ + 8 y = y = y = -8 , ( x, y) = (, ) (x, y) = ( 10, -8) .
: y 'y,
. , = 2 -10+24 = (-5)2 -1, (-5)2 -1 = 2 (-5 -2 = 1 (-5-)(-5+) = 1 , .
3 , 2, , 60 I I = i, i = 1, 2, .. , 160 .
,2, ,0 : , 2, , 60
' 88 . 4/1 1
--------- - ------- . 1 2 1,2, ,160 3, ,0
t k1 ,
k2 n - kn . 1 , 2, , An , = IA;i - 1,2, . . . , 1 60 , k1 ,k2, ,kn. { k1 , k2, , kn} 2
n ,
, . 2n :?: 1 60 , n:?: 8 n 8. n = 8 , n 8. 81 , .60 80 . 80 80 = 6400 . 80 , . . . ,:60 ; AO+i' = 1,2, . . . , 80 . 4, . . . ,80 , :2, . . . ,:60 40 . 2 80 40 = 3200 .
40 , . . . ,0 2, . . . ,60 ;, +'+;2+;, = 1,2, . . . ,40 . 2, . . . ,40 , Ao+; A:oo+i , A40+i, = 1, 2, . . . , 20 20 . 3 80 20 = 1 600 . , +' AO+i'A;o+i' A;O+i' AO+i' +' A30+i' Aso+i' = 1,2, . . . , 1 0 , 1 . 4 80 1 = 800 . 1 . , A+i(modo) = 1,2,3,4,5 f = 1,2,3,4 5 , 5 80 5 = 400 . 32 . A+(modS)' = 1,2,3 , 96 , f = 1,2, . . . ,5 3 , 6 96 3 = 288 . 32 A1
--------- - -------
. 7 128 , 8 64 . :
8, . 81 80 , 81 , 160 80 . a 81 80 = 6480 .
4 c(O,R) (
R ) ( ). ( c1 ) c(O,R) . , c2, c(O,R) . , c3 , c(O, R) . .
c3
. c2 : 01 = . c1 : 02 = . : 01 +02 =+ => .z=+f'=18 -, ( 1 80 ). c1 , c2 , c3 . c1 ; 2 = 2 c 2 : : 2 = 2 = 1 , c1 , c2 c3 . c1 , c2 , c3 ,
c,
1
. c1 c2 , 1 1 ( = = R ), : zl = I . , , . , ,
, , . = = ( ) = = ( c(O, R) . = = = . .
. , Miquel ,, . .
' 88 . 4/13
-------- - -------- 86 87
t I . ABCD . 1 , CD . 2 , C DA F . 3 C , D G . , 4 D , BC . EG l_ FH . [, 2012]
ABCD , . EG l_ FH . ABCD BC AD . 2 AD F , : XF2 = XC . ( 1 ) 2 = XD . (2) ABCD , :
XB XC = XA XD (3) ( 1 ), (2) (3) XF = , XHF . BHF = ( 1 80 -CXD) = ( C + b) , ( 4) C,D ADCD .
1 ( ) BGE =l
A +D (5) GE EF , BGYH :
GYH = 3600 -(+( C+D)+( A+n))= 360o -( B+D+( A+c)) = 360 -(s + 1 8 ) = 90. 12. ABC > C D, , F BC,CA, AB, . BC l_ BC . BAC , ABC , DE DF L, . M, L, H . [, Western M0, 2011]
' 88 . 4/14
-------- - --------
c
2 CL,BI,DI,BK . CL L .
BiK = ! + =( BAC+ ABC) = (1 80 - ACB) (1 ) CD, CE , C ABC , CD =CE . , DEC
EDC = DEC = ( 1 80 - ACB) . (2) ( 1 ) (2) BiK = EDC EDC = BDK ( ), BiK = BDK , B,K,D . Bh = BDI = 90 , (3) D BC ABC . (3) _l . , ic = ( 1 80 -) = BDL , L, D, C IiC = IDC = 90 , CL _l AL . AL , L CN . BC , 1L 11 . BiA = = 90 , ,, , Mif = = lv.fi ( 1L 1 1 ), M,L,H . 6. ( 1), . , . , L-, L , . , 1 1 L- . . , l O x l O
' 88 . 4/15
--------- - -------
1 00 , . 3, 1 , 3, 5, 7, 9 , 2, 4, 6, 8 1 0 . L- . , L L .
3 L- , : 3 3 .
L- , L- 2 . L. L - 4 , 1 00, 25 L- . L- 2 . 25 L- , . 25 . . . 7. n n n . (, b ) , , b b n n n ( n2 - 1) . [ , 2012] . '
n ( n2 - 1) , n . , n , , , n n . ,
' 88 . 4/16
-------- - ------- , 2 [( n - 1) + 2( n - 2) + 3 ( n -3 )+ . . . + ( n - 2) ( n - n + 2) + ( n - 1) ( n - n + 1)]
n-1 n-1 n-1 n-1 n-1 (n - l )n (n - 1) n (2n - 1) = 2 (n - ) = 2 n - 22 = 2n - 22 = 2n - 2 ....:...,__---'----....:...,.._
i=1 i=1 i=1 i=1 i=1 2 6 = n2 (n - l) -
( n - 1) n (2n - 1) = (n - 1) n (3n - 2n + ) = n ( n2 - 1) .
3 3 3
.
: ( ) ( ) () . . . ( . . ); . }
( 1 ) ( ) ( )() . . . ( . . )
12 t (1 ) , , -
( ) ( )() . . . ( . . );. } (2) ()()() . . . ( . . )
1-1 .
n ( n2 - 1) . 8. { 1, 2, 3, . . . , 201 1 } : , , , b lb b . . [, Western , 2011] ' = {, 2, 22, 23 , . , 21 0 , 3, 3 2, 3 22 , . , 3 . 29 } , lM I = 2 1 . IMI 2: 22 1 < 2 < . . . < , lM I = 2: 22. : an+2 2: 2an , n . ( 1 ) , n :-::; k- 2 an < an+l < an+2 < 2an , an , an+I > an+2 . ( 1 ) :
4 2: 21 2: 4, 6 2: 24 2: 8 = 23 , , 22 2: 220 2: 21 1 > 20 1 1 , . IMI = 2 1 . 1 6. (xn ) "''. 1 = .!.. n+ = n + 1 (xn + .!..) , n 2: 1 . -" 6 n + 3 2 201 3
' 2 =(.!..+.!..) = 3 = (+.!..) = . . . . xn =!!:_ r 4 6 2 6 ' 5 6 2 6 ' 6 ' 88 . 4/17
------- - ------- n = () (n +_!_) = . n+I n + 3 6 2 6 , xn = n , n * 6
, , (-b)2 (b-c (c-)2 (-b)2 17. , b, c > O, : ( ) ( )+ ( ) ( )
+ ( ) ( ) ;;::: 2 2 2 c+ c+b +b +c b+c b+ +b +c [, Western 201 1]
(1 ) Cauchy - Schwarz, : [ ( -b)2 (b-c)2 ( c-)2 ][ J ( ) ( ) + ( ) ( ) + ( ) ( ) (c+) (c+b) +(+b)(+c) +(b+c) (b+) c+ c+b +b +c b+c b+ ;::: ( l-+lb- +lc-)2 ;::: (l-+lb-c+c-)2 =4(-b)2
( -b)2 ( b-c)2 ( c-)2 4( -b)2 + + 2:: ( 1 ) ( c+)( c+b) ( +b)( +c) ( b+c)( b+) ( c+)( c+b)+( +b)( +c)+( b+c)( b+) ( c+)( c+b)+( +b)( +c) +(b+c)(b+) =( d +b2 +c2) +3( ab+oc+ca) 4( d +b2 +c2) (2) (2) ( 1 )
( -b)2 ( b-c)2 ( c-)2 4( -b)2 4( -b)2 ---'----':..__...,.. +--'---.,..___:__ +---"----'- > >---,---'---:-(c+)(c+b) (+b)(+c) (b+c)(b+) (c+)(c+b)+(+b)(+c)+(b+c)(b+) d+Zl+c) (2 ) _!_( -2b )2 +_!_( - 2c )2 + ( b -c )2 2:: , 2 2 3 ( 2 + b2 + c2 ) 2:: 22 + 2b + 2bc + 2c = 2 ( + b ) ( + c ) ( + b ) ( + c ) %( 2 + b2 + c2 ) . (3) ( b + ) ( b + c) ( 2 + b2 + c2 ) (4)
( c + ) ( c + b) %( 2 + b2 + c2 ) (5) (3) - (5)
(-b)2 (b-c)2 (c-)2 2 (-b)2 +(b-c)2 +(c-)2 2 (-b)2 +(b-c+c-)2 --'---,--'-----.,-+ + >-. >- ---,--'=----,---(c+)(c+b) (+b)(+c) (b+c)(b+) -3 d +b2+C - 3 (d+b2 +c2) 18. f () = ( + ) ( + b ) , , b n 2:: 2 . 1 , , n 1 + 2 + . . . + xn = 1, F = min {f(x; ) ,f (xj )}.
Ii
-------- - --------
J(xi +) (i +b) ( 1 +)( 1 +b) ( (xi +) (i +b) + (x1 +) ( 1 +b )) =xixJ +( +1 ) ( +b)+b.
F '""' min (f(x, ) ,J(x1 )j ,; "'"' ,1 + a;b 2;;. (, +1 )+()ab
1 [( n )
2 n ] + b n (
n)
1 [ n ] n 1 (
n) =- i -; +-(n-1)xi + b =- 1 -; +--=-(+b) + b 2 i=l i=l 2 i=l 2 2 i=l 2 2
1 [ 1 (
n )
2] n -1 (
nJ
1 [ 1 ] n -1 n ( n - 1) n- 1 (
1 ) - 1-- i +-(+b)+ b =- 1 -- +-(+b) + b=- -++b+nab . 2 n i=t 2 2 2 n 2 2 2 n
, , ' 1 ' ' ' F ' , = 2 = . . . = xn = - . n Fmax = n 1
( + + b + nb) .
: : x1 , x2 , . . . , xn , + 2 + . . . + xn = s , s , F = min {f(i ) ,f (1 )} , , = 2 = . . . = xn =
!.... . ISi z + y + z . , , y, z < y < z, + y > z + y + z = x +y- z b = x+ z-y c = y+ z -x 2 ' 2 ' 2 ' , b, c S = + b, y = b + c, z = + c . :
S ,
IA I = k,
x,y, z E A ,
x < y < z, x +y > z x + y + z
. (*) , = {1, 2, 3, 5, 7,9, . . . , 20 1 1} , IAI = 1007 (*), k ;::: 1 008 . S 1 008 (*). :
n ;?: 4,
,
IA I = n + 2 ,
' 88 . 4/19
-------- - -------T2n = { 1, 2, 3, . . . , 2n}
(*).
. n = 4, 1'g := {1, 2, .3, . . . , 8} 6 , := n { 3, 4, 5, 6, 7, 8} 4 . , 4, 6 8 (*). , . x,y {3, 5, 7} , ( 4, , y) , ( 6, , y), ( 8, , y) (*), . ,
3,5 7, ( , 5, 7) (*). n = 4 . n 2:: 4 n + 1 . , iAi = n + 3, J;n+2 := {1, 2, 3, . . . , 2n + 2} . C := A n {1, 2, . . . , 2n} . !Ci 2:: n + 2 , , , . c = n + 1 2n + 1, 2n + 2 , J;n , (, 2n + 1, 2n + 2) (*), , J;n , 1 , = { 1, 2, 4, 6, . . . , 2n, 2n + 1, 2n + 2} , 4, 6, 8 (*). k 1 008. 1 3. C,D . C,D . AD BC F , EF . E,C, M D . 1 9. 1 , 2 , . . . , n , b1 , b2 , . . . , bn n n n : ( + bi ) = 1, (2) ( -bi ) = , (3)
2 ( + bi ) = 1 0.
i=l i=l i=J
: max {k , bk } :::; 1 0 2 , k = 1, 2, . . . , n . l O +k 1 3. k b b + 1 + + 1 = k b . n , n 2:: 3 . p = (x1 , x2 , . . . , xn ) {1, 2, . . . , n} 1 xk , < j < k . , ( 1, 3, 2, 4) 3 1 4, 4 1 2. S = {p1 , P , Pm } {1, 2, . . . , n} . {1, 2, . . . , n} , pi S. m .
' 88 . 4/20
MATHEMATICUS Homo Mathematicus , : ) , 2) , 3) , 4) , 5) .
: ! , , .
. " ; " . , Maurits Cornelis (M.C.) Escher Victor Vasarely, , , ' ,
:
.
[ : " " "+"]
Maurits Cornelis (M.C.) Escher [17/6/1898 - 27/3/1972] . . . ' . . ,
"
".
._,... ,
.
,
.
.
,
,
, ,
.
.
, ,
'
, . . . Victor Vasarely [9/4/1906 - 15/3/1997] . . . Belle - Isle (
) ,
,
.
,
. ,
,
,
,
. . . . .
' 88 .4/21
------------- MATHEMAiCUS -------------1 11 " , " Nagel ( 1 )
, . " Nagel"
Cvienne (. Poulain) ( 2) . , " " (J, Neuberg)
agel Cvienne
Ceienne
( 3)
t:'j. r IOll
------------- MATHEMAiCUS -------------
y z 3 -- + -- + -- -c - x c - y c - z 2
Oz,Oy,Ox O.xyz ,, ===c. x,y,z (Oy,Oz),(Ox,Oz),(Ox,Oy) ,
, 1 y 1 z 1 , y z 1 1 1 : - = -- , - = -- , - = -- , - + - + - = -- + -- + -- , x+y+z=c. c 1 c 1 c 1 c c c 1 1 1
__ = 1 __ =
1 _z_ = 1 , y z 1 1 1 -- + -- + -- = -- + -- + --c - x MA ' c - y MB ' c - z c - x c - y c - z
3 y z 3 , , --1 + --1 + --1 - , -- + -- + -- - , 2 c - x c - y c - z 2
2" : MathsJams ! ! ! ; () [Thales and friends]
MathsJams,
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, , , , , , , , .
, , , post-it, , , , , IV. - 1 . . , , . - . - , ' , ' - -
.
MathsJam . stand-up , Matt Parker. , , , , , .
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, . . . - , 1 0 1 - ,
' 88 .4/23
------------- MATHEMAiCUS -------------- , 10 , . . , . - , ' , , . ' '", 2014 ! ! ! ! ! ! ! ! ! ! ! 2 . 5/12/2012 "" 104 . , , , . 1 988 . . , . ,
Metropo!itana Nossa Senhora Aparecida , . . .
1 2 3 3 . 125 ( ) R [ 1 887].
, . . . 1 904
, . 1 9 1 2
4 . ,
() , Scence, New Scentst Nature 0,84087 femtometres
IIIJo ;;;
. Sir Francis Spring S.N. Aiyar , . . F. Baker, . W. Hobson G. . Hardy. ' , John Lttlewood ( 1 885- 1977), , .
( ).
, , 4% [http://www.ethnos.gr/article.asp?catid=22769&subid=2
&pubid=63772395]
111 1 1 6 , , , 641 2". : JF: = 65536 , , 65536, 6552 1 , 6542 , . Euler 6542
' 88 .4/24
I
. #-1 1 :
Ss = 8284 4 = 6S2 + 2 ) 1 ; ) . ) =, =1
=2, (, ; ) , . ) = 12 ) , iii) 3 , . : ) = S8 = 82S4
8=824 => = , . ) ')..;f:- 1 , ')..;f: 1 , :
Ss = 82S4 => (8 - ) = 82 (4 - 1) =>
- 1 - 1 8- 1 = 82(4- 1 ) => (4- 1 )( 4+1 ) = 82(4- 1 ) => 4+ 1 = 82 => 4 = 8 1 => =-3 =3.
) i) = + = + 2 = 4 =12
( = 3 = 3 > ) r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1!
4
-
2 4
8
ii) .
= =2 . = - = = 3 .
. = 90 , =60
r=30, 2 = + . iii)A2 =2-2 = (2 )2 - ()2 = 3()2
= 3.32 = 33 = 3. = z = 3 .32 = 33. = = 33.
2. (2, ) = 5 , ' - , . ) (, ) ' (, ),
r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - , 8 I I I I I I I I I I I I I
I
I I
/
2
- - - ..... '
(2, 0}
' '
' \
\ I
"" ' \
-2 4 I
)
\ '
. -2
' -4 ' _ _ _ _ .,..,
. d(-3,0) = l -31 = 3 2. d(2,7) = 1 7-21 = 5 3. 7-(-3)= 10 4 . d(x,O) = I xl 5. d(x,7) = I -71 = 1 7-xl 6. d(x,2) = I -2 1 > 5
) i ) : 2 -4:;t:O 5-1 2-xl
2 -4:;t:O x:;t:-2 x:;t:2.
I I
5-1 2-xl 1 2-xl 5 -3 7 - 3 + 7 ( 2=
2 [-3,7] , ,
8
5 7 - (- 3) , , 2 , = 2
.
5, xe [-3, 7]). Dr = [-3 ,-2)u(-2,2)u(2,7].
f(x)=O 5 - 12 - xl = l2 - xl = 5 -2=-5 -2=5 =-3 = 7.
(-3,0), (7,0).
i i ) I I 2 -2 2 ( 2, [ -2, 2]). = [ -2, 2]. , Dr , nDr = (-2, 2) {- , , } . i i i ) Dr = {-3,- ,, ,3,4,5,6,7 } .
f(x)= O 5 - 12 - xl = - 21 =5 -2= -5 -2=5 = -3 = 7
f(x)= , ={-3,7} . , , f(x)= ,
( )= () = . () 9
3. f(x) = x2 - ( +
) + , e IR*
i) :
I + -
+ 11
eR.
) : ll + 2 , . ;
i i i ) f(x) = , , x2e IR < 2 + 2
' 1 : = 2 = - .
(,
2( + ) ). ) . e IR* , . ) I I =, f(x) , 2, 3, 4 [, 2] , (3, 4) 2
t ( +
) 2 4 ,
( - ) 2
, .
1 , -- = ,
2 = 1 , {- 1 , 1 } .
iii)
, 2 . 1 + 2 = +- .
1 ' 1 1 3 = 2 = , 2 =
= =F- 4 ,
.
i) (0, ), f(xo) = R* ( 1 ).
( 1 )-4 2+(4 2+3-4)-4 =, R* ( 1 ).
( 1 ) = - 1 , : ( 1 ) => -4 2-8 -40+3=0 (i) = 1 , : ( 1 ) => -4 2-8 +40-3=0 (ii)
(i), (ii), - 16 0 =0,
= . : (i) => -4+3=0 => = 4
:
= , = , ( 1 ). 4
3
(, - ). 4
) l l =l = - l =l . = - ! ,: f(x) = 2 +2 + .
4
f(x) - - .!. . 2 2
f(x) 3 1 [-2 ,- 2 ] .
= 1 , : f(x) = 2 -2 + . 4
f(x) .!. . 2 2
f(x)
:
- 312 - 1/2 1/2 3/2
3 1 1 3 ' ' -2 , - 2 , 2 , 2 ,
:
1 ' ' ' 3 = , - 2
1 ' ' ' 3 =- , 2 . )
( , ) , '(-.!_ ,0), (.!_ ,0) , 4 2 2
(, ) , '(- ,0), ( ,0) 4 2 2
r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 1
'(-3/2,6)' -;(-1/2, ) (1/2,.' ) ,'8(3/2, )
I I - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
, , ', , , '.
86 37 2+=(2+ )=
' 88 .4/27
'
. .
f()=2++y ,, R 1#0, , =2 - 4. , f()=(- )(- 2), 2:0 1 ,2 .
' :
{ = -72 ) {, } . = -72 + = - 1
, : { 1 ,-72} , {2,-36} , {3, -24} , {4, -1 8} , {6,-12} , {8,-9 } , {-1 ,72 } , {-2, 36 } , {-3,24} , {-4, 1 8 } , {-6,12} , {-8,9} . ' {8, -9} +=-1 . . : f(x)=x2 +8 -9 - 72= (+8) - 9(+8) = (+8)(-9). =-1 =8-9. ) {, } = 7{-4)=-28 + =-12. {-14,2 } , : f(x) = 72 - 14 + 2-4 = 7(-2)+2(-2) = (-2)(7 +2). =-12 =-14+2. ) f(x)= 1 52 + - 6, : =1 5(-6) = -90 += l . , =l, =-9, : f(x)= 1 52 + 1 -9 - 6 = 5(3+2)- 3(3+2) = (3+2)(5 - 3).
f(x) = 92 - 12 +2 . : =72>0 1 12 + ..fii 12 + 6Ji 2 + Ji 2 - Ji '
1 8 =
1 8 3 , 2 = 3
, :
2 + Ji 2 -Ji [;; [;; f(x) = 9(- )( ) = (3-2-2 )(3 - 2 + 2 ). 3 3
: =3( Ji -2), =-3( Ji +2) =92=1 8 +=-12, : f +12t + 18=0 (1 ). , f(x) ( 1 ). , , f(x) , () =3( Ji -2), =-3( Ji +2), ,
: -12= + 2= . 9
, : f(x) = (3)2 - 232 + 22 - 22 + 2 = (3-2)2 - ( Ji )2 = (3-2- Ji )(3 - 2+Ji ). . : :t{)=4 +2 + ,
' , :).
: ) : 1 52+ - 6 = (3+2)(5-3),
=R -{- , } . {:, } := 3(-4) = -12 3 5
:+=-4 {-6, 2} . : 32-4 -4 = 32-6+2- 4 = 3(-2)+2(-2) = (-2)(3+2). ( - 2)(3 + 2) - 2 : = = -- . (3 + 2)(5 - 3) 5 - 3
2 ) : (i) - 2 = 19 -38 = 40 - 24 2 1=-14 = - .!i 5 - 3 19 21 2 ' ' ' 2 ' ' ( ") ' ' = - - . J. - - , . 3 3
, , , ' - 2 8 (" .) , R { 3 } , , 2 , , : -- = - = - - , --
5 - 3 19 5 3 () , . (), () r=. (), ()
' (") 2 ' ' ' 2 t - 3 , - 3 . - 2 3) : ( 1 ) -- = (2) : (2) (5-1 )= 3-2. 5 -3
) 5-1=0, =..!. , (2) , : 3-2 = -2 =- 2. 5 5 5
' 1 3 - 2 ) 5-1:f;, :f;- , : (2) = -- . 5 5 - 1
, 5;."""" , . (2) , 3 - 2 , 3 - 2 : vc.t. :
5 _ 1
5 _ 1 ,
nJ ' 3 - 2 3 ' ' ' ' 3 - 2 3 1 5 10=15 30 7 ' 5 _ 1 -5 , , 5 _ 1 =5 - - = : .
. (2). ' ' ' ' ' ( 1 ) ' 3 - 2 ' ' :
5 _ 1 ,
' , 3 - 2 3 , 1 , , , , , , , 5 _ 1
5 : 5 , . : , (2) ' ( 1 ).
' 3 - 2 3 3 - 2 2 ' 3 - 2 2 ' ' ' ' : 5 _ 1 -5 5 _ 1 -r-3 , 5 _ 1 -r-3 , , u.
: 3 - 2 = - 9-6= -10 + 2 1 9=8 = . = 5 - 1 3 19 19 '
, ( 1 ) . ( 1)
1 ' ' ' ' ' 8 ' ' ' = 5 , = 1 9
. :
' (")
3 -2 ' ( 1 ) , . : = 5 _ 1
, :
R - { ..!. , } , ( 1 ) : { ..!. , } . 5 1 9 5 1 9
' 88 .4/30
,
1 . . . =+ .
=. =. . LEAH=LHIA= L. = L. == .
, =. =, . . = . L. , j_ . : + L. = + = 90 L. + L. = + = 90 , = .
= = .
2. C1 C2 . C2 C1 . C1 , =. .
. =
AR = ABC . ) : 1 = 1 :=::> C1 -Q1 = 1 :=::> -\ = :=::> 1\ = +\ :=::> ABR .
4.
=2. . :
) . )
, . ) , ,
, , .
)
. ///, LAE = LAEZ = x LZEB = L = y , , ==,
= = = 2
.
==, + y = 90 , ( : + y = 90 ).
) ( .),
) , LAEB = + y = 90 , LEMZ=LENZ=90 ( ).
5.
. .
L = L
. L = L L ( LBA=LBMA=90). LMZN = L ( ).
LMZN = L = LBAM = L .
6. ,,
,, . : ) ,, ) L = LA + L .
.
'
' '
\ \
' '
., , - - - - - ... .
. ' \ \
\ \ .
---------'
\
' ' ..
... ... .. _ .. ,
(),(). () . , :
' 88 .4/32
' L + LA = 1 80 L + LB = 1 80 ,
L + L + LA + LB = 360 360 - L + 1 80 - L = 360 L + L = 1 80 . () .
) L = L + L L = L + L = = (180 - LB - LEI ) + (1 80 - L - LE2 ) = = (1 80 - LB - L) + (1 80 - LEI - LE2 ) L = LA + L . 7. .l , .l . + = , . .
I! .l ,
, =. + = + = =
, (, , =). LMB=LB
L = L ( 11 ) LB = L
. (* ).
8. , , , =4. . .
B --------------,r =4
=2, . .
I I = I I = 2
, . 9. (=) . . , , . =2. .
- - - - - - - c .- _ _ _ _ _ _ _ _ _ : : : ,Z 1.1
. , ==. L = 2LZ = 2(90 - L) =
1 80 - 2(- +-) = 1 80 - - = 2 2
L - LB = + L - LB = 2 2
(=). =2. 1 0. , , . .l , .l .l , =+. .
, , . =.
EI,EM , . =.
, .
= + = + ( 1 ) 2 2
=2. ( ) =2. ( ). ( 1 ) : =+.
' 88 .4/33
,
I 73+332- 3+35=0 ().
( +
) 3 =y , , . +
(). :
, ' ( + ) 3 * - -- =y + (-1)3+ 3( - )2+ 3(2 -2)+(3-3)=0 (2). (2) ( 1 )
- 1 = 7 = 8 - = 1 1 = 8 - 1 1 ( 1 ) 2 2 , 2 2 ( ) - = -1 8 - + 1 = 2
3 - 3 = 35 83 - 3 = 35 (3)
( 1 ) , (2) (, ) = ( -3, 1 ) (,) = (, ; ). 83 - 3 =35. ( 1 ) =-1
: ( 1 ) ( - 3) 3 =8 - 3 =2 = -5. x + l + 1
2. ( ) = 3++ R (). ) > ()>
' J ) . 15 +1+1>/+2+ 1 , f(x1)>f(x2). f 2) R.
. g(O)=g( 1 ), g 3 ) f(x)+g(x)=O. R. i11 : 3) g(x) g()= (2) 1- f() = (2++1)g()= 5++1=0.
::=: 5++ 12:1 >0, . xz+
J 5. 22: { + =
2 {2 + y2 = 3
: z: , eR, + y = 1 2 + y 2 = 1
. ) 2 , . ) =l =-1. #l :
>, >, Xz>O, Yz>O. :
1 1 2 ) D=D2 = 1 = -1 = (-1)(+1).
: 1 D1:f0 (-1)(+ 1 ):f0 :f-1 :f1 Dz:fO 2 .
' 2 1 3 2 Dx1 = 1 = -1 = (-1 )( ++ 1 )
2 D I = =- 2 = -(-1 )
1 1
"! Dx2 = 1
"! Dyz =
1 1 =- 3 = -(-1)(+1 ).
D 1 = _, = .. ,
D,
Dy, , , = -- = --- , 2 D1 + 1
Xz = Dx2 = 2+ 1 , Yz = DY2 = -. D1 D1 ) =1 D1=D2=0
D = Dy = Dx2 = Dy2 =0. =1 , z : { + = 1 {2 + Yz = 1 , , + = 1 Xz + Yz = 1
' 88 .4/35
' + = Xz + yz=O.
: { : t {2 : t , tR. - 1 - t y2 - 1 - t
=-1 D=D2=0 Dy = - 2 i= . , Dx2 = Dyz = =-1 2 {- Xz + Yz = - 1 , ,
-Xz + Yz= l . , z Xz - Yz = l
: {Xz = t
' tR. Yz = 1 + t
) -1->0, + 1 _ __i__>O, -> 2++ 1 , 2 +1 + 1 . , + 1>0, (+ 1 )
'
'
1 . : ()= 4+(+)3+(++ )2+(2+2)+, ,IR : [(1)]2+[(-1)]2=0 (1), : = zoos +zoos. : {) (1) = (- 1)=0 :::::.
1 +( +)+( ++)+( 2+2)+= 1 -( +)+( ++)-( 2+2)+=
:::::>
{ 2+2+2+2(+)+1=0
:::::. { :::::. {
2+2-2 = 1 ( +)2+ 2( +)+ 1 = (-)2= 1 (++1 )2=0 - = 1
i
+ = 1 { +=- 1 -=1 -= - 1
:::::. ,) = (0,-1) (,) = (- 1 , ) . = 1 . 2. ()= 2+-2, I*. ) =-1, . ) (2)=70, . ) : (- 1 )=0 :::::. (- 1 )2+(- 1-2 =0:::::. (- 1 )2+(- 1 )=2 , , ( - 1 )2= 1 ( - 1 )=- 1 , 1 +( - 1 )=2, ' , . ) (2)=70 :::::> 22+2-2=70 :::::>(2)2+2-72=0. 2=>, 2+-72=0, =8. 2=8 v=3. ( ) =6+3 -2 6. () = 6+3-2 = 6- +3- 1
= ( 3)2- 1 + 3- 1= (3- 1 )( 2+2). 3. () -, -, -, , , , ( * * * ), (): (-)(-)(-) , ( ) = . ( ) , ( ) : () = 2+ +. : ()=(-)(-)(-)()+2++ R. = , = , = ,
2+ + = } 2+ + = ( 1 ) 2++ = , (2- 2) + (-) = -} (2- 2) + (-) = - :::::> ( +)( -) + ( -)=- } (+)(-) + (-)=- :i' * : (+)+=1 (+)+=1 . (-)=. *, =. (+)+=1 =1 . = =1 , ( 1 ) : =, ()=2+ 1 +. ():(-)(-)(-), ()= . 4. : 912 - 25\ I, 7. ()=9 - 2 5v. (5)= 75. (): (-5), () (5), ()=(-5) ()+75. ( 12)= 9 12- 25 =(1 2-5)( 12)+ 75
= 7[(12)+5]=/ 7.
q(x) = 9 1 2-2xv. , ... ' . 6(ii), (), ( ! 50) 6, ' ( 1 80). .. 14 I 34+ +52 + 8 I 52 + 7 1 4 I 9 8 1 + 5 25 8 I 25 + 7 . . . 5. : (22+-1 )( 22+-2)=72 (1).
22+- 1 = 22+- 1 = ( 1 )
}
} (- 1 ) = 72 {-8, 9}
22+- 1 = -8 22+- 1 = 9 22++7 = 22+- 10 =
22+-1 0 = 0 {-% ,2} . 6. :
.Jx + 2 = .J4x + 1 - rx=l (1) ' 88 .4137
' ) ( ), : +2 , - , 4+ , . : (1) rx+2, + = .J4x + +2+- +2 ( + 2 - )= 4+
x:O .Jx z + - 2 = z+-2 = xz = 2, .
4 7. a: r -+
r , R. ( )
IR. ( )2 -++ 4
'
)
, . () , ( ) = ( + ) ( ) + .
= _
[- + ] (-) + =+ = , = (-), (): ( +} .
) , -,
= (-} Q(x) ()
+ ,
() = ( + }Q() + () = ( :}Q() +
( + ) . Q( ) () = +
() = ( + ) Q(x) + .
() +
( ) = Q(x) .
) Homer ()
+ , , (
' 1 37),
( ) +
Q(x). , () +
( ) = Q(x) ,
() Q(x) . a .
)) (83 - 62 + 3 - 5) : (2 - 1) Homer.
2-1 = ..!.. , 2
-4 Q(x) Q(x) = 82 - 2 + 2 .
() = Q() = 82 - 2 + 2 4xz - x + l .
2 i i)
( -64 - 3 + 1 62 - 2 - 8) : (2 + 3) : 2+ 3 = - ,
2 Homer( . . . )
4 Q( )
Q(x) = -6x3 + 82 + 4 - 8 .
( ) = Q(x) = -63 + 82 + 4 - 8
2 =-33 + 42 + 2 - 4 .
i i i) (325 - 764 -93 + 1052 - 53 + 7) : (-4 + 7) :
-4+7 = 2 , 4
Homer( . . . )
Q(x) Q(x) = 324 - 203 - 442 + 28 - 4 .
() = Q(x) = 32
4 - 203 - 442 + 28 - 4 -4
= -84 + 53 + l lx2 - 7x + l .
' 88 .4/39
,
1
, .J3 = = -- .
4 ) , ,
> . ) , .
, 1 1 J2 ) : - + - = - . y :
) =, =y,
: > => > =>y > . : +y= 32 =2 = - . 1 6
, y : 32 , , 3 -t+- = y>x, y=- ,
1 6 4 , 3 =- =- =- . 4 ' 4 4
2 = - => = - = 2 => =30 => =60 4 2 I => =30. :
.J3 -4- , =- = -- = 3 '
4 0 J3 =30 =2=-- .
2
- r 2 ) : = -- = = = + r a.J3 .J3 + 1 --+-2 2
(.J3- 1) , 2 2 2 _....:....:..._ _ __:... . : = + -2 260 = 2 2 ( .J3 - 1)2 ( .J3 - 1) 1 - + -2 - -- 4 4 2 2 2 2 2 - [1+( .J3 -1)2 - ( .J3 -1 )]= - (6-3 3 ) = 4 4 32(2 -{3) . = .J3 J2 -.J3 . 4 2 ) :
2 2 J2 ' --
+ - = .J3 .J3 2 _ .J3 ' 2 .J3+ 1 J2 ' = .J3 J32 -.J3 ' 4 + 2.J3 = 2 ' 2+ .J3 3 3(2 -.J3) ' 22 - ( J3 )2 = 1 , .
2 -J3 : z -JJ (JH +()' - 2K
= (-)' = -= z -JJ =J4 - zJJ =J
, , = = .fi , :
1 1 1 , 1 1 1 ,
+ y = '
+
=
'
+ = 1 . : // =:> = - -= -= - . :
+ + = = 1 .
: = .fi = .fi . 2
2 . ()=5, ()=( ) = () = 4 ()=6. ) . ) . ) ', , . :
4
4 4
) \ = , , 1 = , , ( ) ' 1 , , , 1 80 . :
( + ()2 - ()2 = 1 2() ()
25 + 36 - 16 3 2 5 6 4
( + ()2 - ()2 = 2() ()
16 + 36 - 16 3 ----=- . 1 = 1 . 2 4 6 4
v1 =
5 - 4 1 ()=()= -2-= 2 .
2 1 63 3..fi () =16 -4
= 4 () = -2- . 5 + 4 3..fi 27..fi () = -2- -2-= -4- .
) : ()2(') = . : 4
= =:>()= () =3. 2
: 1 5
( ') 4 3 (') 3 -'---- = -= - -- =- () 5 4 ' () 4 ' = f" =90 , '
' . ( ) : 1 = , 2 = , 1 = 2 , .
3. =
()2 = =90 . ( )= -- . 2
:
(. )
(O,R) AO_i . a (. ), :
( )= () + () =
__!_ ()() + _!_ ()()= __!_ ()[()+()] ) = . : 2 2 2 ( ' 88 .4/41
_!_ ()[()+()] = _!_ ()() = _!_ ()2, 2 2 2
: = 90 .
()2 ' : 2
(). 2 = ( . ) . , : =, 2 = = , . : ()=() +()= () + ()= ()
()2 2
4. (0, R) =3, =. , : ) . ) . :
) ': = 3 = R.J3, = 4 = RJ2 , 6 = =120 6 = =90 3 ' 4 ' .
6 =360 - 120 -90= 1 50. : 2 = 2+02 - 20150= 2R2+R2.J3 = R2(2+.J3 ), AB=R2 +.J3 = R .J3 + 1 =R J6 +J2 J2 2 ( 1 : r-2_+_.J3=3 = .J3+ 1 ' J2 ).
++= R(J6 +J2)+R.J3 +RJ2 = 2
R(J6 + 2/3+3J2) 2
=60::::> ()= _!_ ()()60 2
= _!_R J6 +J2 RJ2 .J3 = R2(6 + 2.J3) 2 2 2 8 R2(3 + .J3)
4
' : ()=()+()+() = 1 2 R2 1 .J3 =- R (120 +1+150 )= - (- +1+- )= 2 2 2 2
= R2(3 +.J3) 4
: 6 =90 = = 1 800 - 1200 30, I I I 2 =, =2 2= 2+2
R2 R.J3 ::::>4x2=x2+R2 => 2= - => = -- => 3 3 =
R.J3 =
2R.J3 . 3 ' 3
, , , .J3 : 1= -::::>-=- ::::> R.J3
3 R = -- .
3
, R.J3 R(3 -.J3) =R-= R- --= . 3 3
', ++=R.J3 +R=R( .J3 + 1 ) ()= _!_ ()()= _!_ R R .J3 = R 2 .J3 . 2 2 3 6 ) 6 =120- 6 =30 = ::::> == 2 I I
' 88 .4/42
RJ3 , +=+=R. 3
: 30 R , , l= --2R = - .
360 6 :
R = + + l =R + - . 6
: = (0. ) - ()=
30 2 1 360
R - 2 ()()30 = nR2 _ _!_ R RJ3 _!_ = R\ -J3) . 12 2 3 2 1 2 5. : +=2 = =, . :
) a=, =, +=2 => 2=2 =>=. = == , =, Ay
600 1 = 2 =30 = -- () 2 ,.
)
'
,
1 2 + y2 - 2( - 1 ) + 2 y- 2 + 1 = : ) R ) ) , . ) y'y ) 2 + 2 - 4 = 4(- 1 )2+42-4(-2+ 1 )=4[(- 1 )2+2-(-2+ 1 )]= 4[2-2+ 1 +2 +2- 1 )]=82>0 R* ( = ) ) ( - 1, -) =- 1 =+ 1 y=- y=-(+ 1) y = - - ) 2 + y2 - 2( - ) + 2y - 2 + = (-2 + 2y - 2) + (2 + y2 + 2 + ) = R* -2 + 2y- 2 = 2 + y2 + 2 + = . -2+ 2y - 2 = y = + 1 2 + y2 + 2 + = 2 + ( + )2 + 2 + = 22 + 4 + 2 = 2( + )2 = = - y = ( - 1 ,0) ) y 'y lxo l = - 11 = ll h - 1 = '2 - 1 = - '2 = - '2 = - - J2 ( ) 2 1 : 3 - y - 6 = 2 : + y - 6 = , 1 , 2 ) ) ) ' '. ) 1 , 2 .
{3- y - 6 = {4 = 2 { = 3 (3,3)
x + y - 6 = 0 x + y - 6 = 0 y = 3 1 y= 3=6=2 (2,0) 2 (6,0) =4.
( 9 3) 0 - 3 -3 1 ' ' ' ' 2, 2 = 6 _ 3 = 3 = -
= y - = 1( - !) y = - 3 2 2
,
{ = 4
{ = 4
(4,) y = x - 3 y =
' 88 .4/44
' ) , '. =l y-O=l(x-2) y = -2 { = - 2 . . . { = 4 (4,2) x+ y - 6 = 0 y = 2 ' (1 , y ) , ' 2 + = 4 0 + = 2 1 = 6 = 4 ' ( 6,4 ). 2 2
) =(- 1 ,2) , ' =(2,3) = -i ldet(A,B ')I 1 -1 2 7 = 2 1 2 3 I = 2 1-l 3 - 2 21 = 2
' = 2 2
- t
-t
-2
l
10 11
c 0(0,0) u c2 0(0,0) ) . ) . ) . 2 x(x- 5) + (y - 7)(y - 2) + (x + y - 7) = 0 ) e IR )
' 88 .4/45
'
8 ' .
0 1 ; = lfi l + l lfi, = , = = IPI . = ( => l = l = fil) ' , .
1 1
. ------ --- -- -- = + . ,
1 . 1 .
: - -- -- -' - 1 2 n , . = +2 + . . . +n = 0, , 2.
=+1
: n-
, .
t?------.,_ __ _..;.; ,
. 2
(OA2 ,0An ) , (OA3 ,0An_1 ) , , (OAv ,OAv+ ) , - - - - - -OA i i Ox, OA2+0An I I Ox, . . . , OAv+OAv+ i i Ox => x i Ox.
11 Oy . = ( . 2) : ( ). 02. , , -- -- , , (), t , = t . : . , , =4 =S. , , . : . ,, , . ' = 4. : 20 = + = +++ => 20=4+4 (1) . ,
20=+=+++ => 20 = + (2) .
' 8 8 .4/46
'
( 1 ), (2) = - 40, ! . ' :
=4::::>-=4(0-)=>= 4; - - - - (- -) - 40+ , =5::::>-0=5 - =:>=
5 :
r =- => 40 + = -40- => + = -4( + ) => => 2 0 =-4 20 => =-4 0 .
03. , . . : ' = 2 ( -y) , JR., , Oy , 16. ( -2, 2) 2 -y = 4 . 2 -y = 4 = , . : a . = 2 ( -y) , JR., ( 2, ) Oy (, ) , < . 2 = 16 = -4 (-8, 0) , (, -4) . + 2y + 8 = 0, 2 -y = 4 . 2 -y = 4
y = ..!_( + 8) , - 3y + 8 = . 3
x-3y + 8 = 2-y = 4 , ( 4, 4) , = .
er = _ _!_ _ 2 = -1 L = 90. 2
.
2 6 = - - = -l .l . "' 6 _2
8 (0 ,-4)
: 2 = (0+ 8)2 + ( -4 + 0)2 = 80.
' 88 .4/47
1 [ ] (R) 200 55cm. ) 5 5 887 ,5cm. ) ) : = 20 2 = 50 , 3 = 60 , 4 = 50 5 = 20 . . 50%. ) c :
c = R . : c = 55 = 1 1 ( cm) . 5
x i , i = 1 ,2,3, 4, 5 - 0 = c = 1 1 = 5 , -
: Sv = [2 + ( - ) ] , : 2 5 887,5=-[ 2 + (5-l)c }=> 1775 = 10 +20 1 1 2
= 1 55,5 . :
[ 155,5- ,155,5+ )= 150, 161) , [1 55 + 1 1, 1 6 1 + 1 1) = [1 6 1, 1 72) [ 1 6 1 + 1 1, 1 72 + 1 1) = [ 1 72, 1 83) [ 1 72 + 1 1, 1 83 + 1 1) = [ 1 83, 194) [1 83 + 1 1, 1 94 + 1 1) = [1 94,205) ) ( ) x i vi fi % F;% [1 50, 1 6 1) 1 55 ,5 20 1 0 1 0
[ 161, 1 72) 1 66,5 50 25 35
[1 72, 1 83) 1 77,5 60 30 65 [ 183, 1 94) 1 88,5 50 25 90
[1 94, 205) 1 99,5 20 10 1 00 200 1 00
. 50.
Fi%
1 00
.... ................... ! ...
"
150 1 6 1 17 1 183 194 205 ( cm)
1 83 - 1 72 = 172+ = 1 72 + 5, 5 = 1 77,5 (cm) 2
: 1 77,5 ( cm)
2 [ -]
f(x) = 3 -2 + 3 + 201 3 , JR . 2
, . ) . ) : (1, f(l)) f ' . ) , :
'
"" 1 2 3 4 5 I ( 1,1) (1,2 ) (1,3 ) (1, 4) (1, 5 ) 2 (2, 1) (2,1) (2,3) (2, 4) (2, 5) 3 (3, 1) (3, 2) (3, 3) (3,4) (3, 5) 4 ( 4,1) ( 4, 2) ( 4,3) (4, 4) ( 4,5) 5 ( 5,1) (5, 2) (5,3 ) (5, 4) (5, 5) 6 ( 6,1) ( 6, 2) ( 6, 3) (6, 4) ( 6, 5) : = {(1, 1) , (1 ,2) , (1,3) , . . . , ( 6, 5) , ( 6, 6)}
{) = 36
6
(1, 6) (2, 6) (3, 6) (4, 6) (5,6) ( 6,6)
) .
' : ={(1,1) ,(1,2) ,(1,3) , . . . ,(4,3),( 4,4)}
() = 16
) f IR . f'(x) = 62 - 6 + 1 , IR . :
) f'() = 32 - 3 + 3 , x E IR 2 = ( -6) - 4 6 62 - 4 32 2 . : f'(l) < 3 - 3 + 3 < - < -1 : : = {(1,3) , (1 ,4) , ( 1,5) , ( 1, 6) ,(2,4) ,(2,5) ,(2,6) , 1,....;..::..:..:...:,_:_:_1 2 "--t'----3-----4--,
(3,5) , (3,6) ,( 4,6)} () = 10
: {) = {) )tOl () [.lli
) = {(1,1 ) , ( 2,2) ,(3,3) , ( 4,4 ) , ( 5,5) , ( 6,6)}
() = 6 . : {) = () )] () [.lli
, AnB =0 . : P(AnB) = O . . = {-) ( -)
{-){-)=J {) = [{ -) { -)] =
(-)+(-) =() +() - = 10
3:6 = : =
3 [ -] : f(x) = 23 - 32 + + 201 3 , IR , : {1, 2,3 ,4} , : ) : f IR . ) f f" . ) : f : y = - + 1 .
1 2 3
2 2 4 6 8 32 3 12 27
: = {(2, 1 ) , (3, 1 ) , (4, 1 )} {) = 3
: () = () =\31 ()
[lli
) IR , f"(x) = 12 - 6
f"(x) = 0 12 - 6 = 0 =1._ 2
4
48
) : f'()=-16(:)2 -6(:)+1 =-1 62 62 32 ---- + 1 = -1 .:..____ = -2 4 = 32 42 2 2
: 2 3 4 12 27 48
{( )} () -() _il : = 3, 2 - () - [lli
) n = 0 , . ,
() = () + () = 1 + l =[i] : ( )'
': ()' = 1 -P(AuB4) = 1 -.!_)ll 4 ]
' 88 .4/49
4 [-]
! 3- , 2: 0 ;t: f(x) = - 1 - , = 3 . ) f 0 = () . ) g( ) = 2 ln ( 2 - ln ) , > , . ; ) ( 5,1 1) . ) 15 . ) f 1 :
limf(x) = f(l) lim ,- 1 =.!.
->\ ->\ v - 1 3 lim(ix)3 -1 !
[()2 ++12] 1
->1 -1 3 -->1 3 lim[{)2 + + ] = .!. 3 = .!. = 9 -+\ 3 3 = = 9 . ) g (, +) g'(x) = (2 - ln x) + 2 ln x (-!) = !( 1 - ln )
g'(x) >
'
6 []
fJ..Erl.o , , . s m R = 1 .
i) . i i) : 82 } . 2
1 i ) ' . R = 1 , = ( ), , :
R = 2 - , 1 = 2 - 2 = 1 = 1 , , = 2 = 1 . i ) . ' _ , , + 2 2 + 1 : = = =--
, + 2 + +
(0 -- )2 + (1 -- )2 , 2 + + 8 = =
+ 2 + 2 _ ( + )
_ = ( + )3
-( + )3
-( + )
2
< =.!_ 2 + 2 + 2 - 2 2
s ,
+ ! 7 []
; , i = 1, 2, 3, 4, 5 - ; , f; , ; F; , i =l,2,3,4,5. i ) O, l, f2 , F2 , ,2 , N5 , F5 i i) :
; 1 2 3 4 5 f; 0,1 0,2 0,2 0,4 0,1
. i i i) . ) , . i) : f2 , F2 = f, + f2 f2 F2 F5 , , 1 1 = 1 ::::> 1 2 5 = , : O, f2 , F2 0 F5 , l, , , N2 ,N5 , . , = 1 . ii) :
f + f2 + f3 = 0,50 : , +2 +3 = . 2 , . . i i i) f, + f2 + f3 = 0,50 ='3,5 i ) '
. ; 1 2 3 4 5
f; 0, 1 0,2 0,2 0,4 0, 1 1
; f; ;2 . f; 0, 1 0, 1 0,4 0,8 0,6 1 ,8 1 ,6 6,4 0,5 2,5 3,2 1 1 ,6
1
, : 5 = ; f; = 3, 2 :
i=l 5 82 = ;2 f; -2 = 1 1, 6- (3,2/ = 1,36
i=l ,
CV = 136 = 0,3644 = 36, 44% 3,2
. ; = ; +
.
: 8 = 8 = 1,36 y = + = 3, 2 +
: CVY 0,111 17 ... 0, 113,2+ 1 1,7
3,2+\j 8, 5 -14,9 . 9.
' 88 .4/5l
'
. -
() ()
:
1) /
f '( ) > f .
2) :
lim (- J = lim (- J2 = (----=!_J2 = .!. --+(-1)2 - 2 x+l - 2 -1 - 2 9 3)
0 : , (!) ( ) = f' (x0 ) g (x0 ) + f(:0 ) g'(x0 ) g (g(x0 ))
4) f(x)
(xo ,f(xo )) cf ' f'(x0 ) .
5) f(x) = x2 ln x , Cr , (x0 , f(x0)) , y - / ln 0 = 2( - 0) .
6) f , f'(x) > , f .
7) g(x) , , g'(x) .
8)
. 9)
.
1 ) .
1 1) 1 , 2 , . . . , sx ; = ; - c z; = ; c , = 1, . . . , c (0, +) :
) y = ) sY = sx ) = ) sz = sx c
12) , () :# ( ) ' : ( ) - () = 1
( ) - ( U ) 1 3) ,
: (A B)' U (AU ) =
14) : = {, ,2 , . . . ,v }
( ; ) = .!_ = 1, 2, . . . , .
15) .
1 2 16) , () = 3 () = 3 , .
17) .
1 8) -' .
19) . . : P(AuB) =P(A) +P(B)
20) = {0,3, 4} , = {4,3} : - = (/) .
21 ) , : ( ) + () > 1
22) , : P(AUB)-()
'
. 1 (] 200 : 28% (120km/h), 160 km/h 0,02. , 40km/h 80km/h 160km/h. 200km/h. 40km/h f(x) = 3 - 42 + 4 + 34, IR , : a. , , (0,40), (40,80), [80,120), [120,160) (160,200).
82 - b. lim
2 , 1 -1
1 2 . c. , 40km/h 160km/h; 40kmlh 160km/h, .
a. 5 40 . : f4 %+f5 % = 28 => f4 + f5 = 0.28 , f5 = , 02 . : f4 = , 26 : 2 = 8v 5 => f2 = 8f5 => f2 = , 16 . f R : f'(x) = 3x2 -8 +4 , :
f'() = =% = 2 , f'() > 2
3
f'() > 0 3. < < 2 . : 1 = 2 => f1 = 0.01 3
: f3 = 1 - (f1 + f2 + f4 + f5 ) = 0, 55 . :
[ ... , ... ) f Nd F f% F% 0-40 2 0,0 1 2 0,0 1 1 1 40-80 32 0, 1 6 34 0, 1 7 1 6 1 7 80- 1 20 1 1 0 0,55 1 44 0,72 5 5 72 1 20- 1 60 52 0,26 1 96 0,98 26 98 1 60-200 4 0,02 200 1 2 1 00 200 1 1 00
b. : . 82 -2 82 - 32 . 8(
2 - 4) lm = lm = lm = xv1 -1 2 -2 2 -2
= tim 8( + 2)( - 2) = Iim8(x + 2) = 8 4 = 32 2 -2 2
c. : ( ) = f2 + f3 + f4 = , 97
: f +f5 =0,01 +0,02=0,03 2 [] =
s=2, :
0.2 . . 0.14 0. 13 .
...
....
.... .::
, ' , : a. () ={(,y) (-,)} b. () ={(,y) (-2,2)} c. () ={(,y) (-2,4)} d. ( - )
a. () = 0,5 50% 100% . b. () = 0,68 68% ( -s, + s)
c. () = 0,68 + 095 - 068 = 0, 8 1 5
2 (- s, + 2s) d. P(A-B) =P(A) -P(AnB) =0, 5-0,34=0,16 .
' 88 .4/53
' 3 [-] 20 :
[ .. -.. ) ; [0,5) [5, 10) 2 [ 10, 1 5) 3 [1 5 ,20] 4
a. 1, z, 3, 4 :
[0,5) : . 2 - 5 + 4
1 = m -+4 - 4 [5, 10)
: f(x) = 3 - 62 + 9 + 1 .
( 1 15) ( 15 20). ( ) = 2() .
b. .
c. , . ;
d. . .
a. : 1 + 2 + 3 + 4 = 20 (1) [0,5) :
2 - 5 + 4 ( - 4) ( - 1) = lim = lim = -+4 - 4 -+4 - 4 = Iim ( - 1) = 4 - 1 = 3 1 = 3 ( 2) -+4
[5, 10) f(x) = 3 - 62 + 9 + 1 ,
f'(x) = 3x2 - 12 + 9 = 3 (2 - 4 + 3)
f'() = ( -1) ( -3) =0= 1 = 3
3 + +
'--> I
f(x) f(x)
, f m = 1 , : 2 = f(l) 2 = 13 - 6 1
2 + 9 1 + 1 2 = 5 (3) :
2 () = 2() -3 = --4 3 = 2 4 (4) 20 20 ( 1 ), (2), (3) (4) :
4 = 4 3 = 8
c. : 4 - .
= i=l I I
= 2 1 5 = 10 75 20 '
: 4
v, (, - }' J463,75 s = i=l = = 4 8 1 5 20 ' :
cv =!.. = 48 1 5 = 44 79% > 1 0% 1 0,45 ' . d. .. :
I I
9 8 7 6
- 5 I > 4 i 3
2 1
. I I I ------------I 11 [0,5) [5,10) [10,15115,20] i
I - I ____ _ .. __ __________ _ ___ __ _____ _______ _ _ __ ______ __ __________ ... 1 1 1 ,25 ( ). // ////
, ' , .. -=- - = - ' 88 .4/54
100 90 j l 80 (-----J--->l z 70 60 ::5 - - - - 40 --.,......- 30 20 10 --.-
[0,5) r5,10) [10,15) [15,20] ( c; i i ' -------- ----- - _ !!-!-!:1---- - - -- ---- - --'
- 10 50-40 - =---=-- . . . = 1 1,25 1 1 5- 10 80-40
4 [- ] : f() = 3- , x E R a.
. ; b: , .. .
. , : 3( )+ () S 3() + ()
. () =
, 6
2f(()) s --J3 . . f ( ( 11 )) -1 .
c. :
r(o), r(:} r(;} r(;} r(;}
r( 23
} r(3
4} r(s;} r () 1J
. :
f'() = 3 + , x E IR , f' () > JR (;)
IR , .
b. i) f/' ( ) P(B)f(P(A)) f(P(B))
3() - () 3 () - () 3 () + () 3() + ( )
ii) : () () =
f(()) f ( ) = - J3 6 2 2
2 f(( )) - J3 iii) ( )
r/' f(P(AnB)) f(O) f(P(AnB)) -1
c. :
_ f(O) + f( ) + r( ) + . . . + r( ) + r()
= 9
() (
5) 5 f - + f - =---+-+-=3 6 6 2 6 2 6
f - +f - =3---+3-+-=3 ' () (3) 3 4 4 4 4 4 4
r()+ r(2)= 3- +3 2 +=3 3 3 3 3 3 3 f () + f ( ) + f () = -1 + 3 . + 3 + 1 = 9
2
9 3 + 3+ 3 + - 3 = 2 = 9 2
5 [ - ] : (1) (),()
' 18- 102 : ( )
(9 + 7) =
10
(2) [()]2 2 + ()
+ ()
= 0 4 ()
, (),()> . : .
(),() . 1). , ,
(),() ;
. 2 c. f(t) = t - -t + c ,
e c 1R c t = t0 , f f(t0 ) () a b.
' 88 .4/55
a. () = 1 1 :;e :;e :
1 8 - 102 1 (9 + 7) = 1 002 + 70 - 1 8 = 0 l Ox
= =-.2_ -- > e - 2t > t < -et 2
t e / 2 + f1(t) + -f(t) I
f t0 = . 2
a b : 1 e 2 e f(t0) =5 m2 -;
. 2 + c = s
ln e - ln 2 - 1 + c = .!.. c = .!.. + ln 2 . 5 5
6 [ - ] = {1,2,3, . , 200} . e ( ) : {..7.!__, 200 ( ) = , R
2 -, 50 a. . b.
.
c. 7.
d. 7.
e. f(x)= 4r + 7x- 2 , x eJR = {1 ,2, , 2} ;
2 - f'(x) () = lim
-1 3 -
a. : = { } = { } , : = . 1 00. () + () = () = 1 . ,
2 7 7 1 00 -+ 100 - = 1 22 +- - 1 = 50 200 2
42 + 7 - 2 = = -2 < (.) =.!_ 4
{ 7 ' - ,m 111:; b. : ()= 8008001 ,mn; :
1 1 () = 100 ( / ) = 100 -=-800 8
c. : = { / 7} = { = 7, }
' 88 .4/56
' ::; 200 7 ::; 200 ::; 200 = 28,57 7
max = 28 () = 28 = {7, 14, 2 1, . . . , 1 89, 1 96} , 14 , , 14 , . :
7 1 8 14 () = 14 . 800 + 14 .
800 = 14 .
800 =
100 = , 14 .
d. :
( ) = ( ) + () - ( n ) n={ / 7} 14 . :
( n ) = 14 -1- = _2_ : 800 400
( ) = .!_+-_2_ = 50+ 56 - 7 = 99 . 8 1 00 400 400 400
e. , : 1 ::; 2 ::; 200 0,5 ::; ::; 1 00 { 1, 2, . . . , 1 00} .
f() = (42 + 7 - 2)' = 8 + 7 , x ffi. , 2 - f() 2 - 8 - 7 -8 - 5 : = = ---
3 - 3 - 3 - , , 1. 2 - f() 1. -8 - 5 : m = m =
-->-1 3 - -->-1 3 -
= -8(-1) - 5 = 8 - S = () = . 3 - (-1) 3 + 1 4 4
2 , () = _% () = ( {1 ,2,3, . . . , 2} (2) ) =
= ( {1 ,3, 5, . . . , 2 - 1} ) + ( {2,4, 6, . . . , 2} ) =
7 8 3 = .-+ .- =-=-=- = 75 . 800 800 800 100 4
7 : [ - ]
f(x) = 100 (: } ( 0,12 ) . a. 0 ,
f . b. 0 a,
, , li 1 c, c = m --.. f(x)-6 f ()-8 c. ,
, = 0 1,2, . . . , , ,
(BullsEye ). , m ) - ;
i = 1, ... , , . : (; ) = ; ( - i + 1) c i = 1, ... , , ; , c b. : i) i = 1, ... , ii) ( ). : a . f f' . , , 0 f . f (0, 12),
f() = (100 ( )) = 100 ()
I
f"(x) = ( f() ) = ( 1 00 ( )) = = too[ -( )}-: ( ) : f"(x) = ( ) = = 6, ', (0, 12) = 1 = 6 f"(x) > . . . 6 < < 12 :
6 12 f"(x) - + f() I ,-/7
0 0 = 6
b. : c = lim 1 = --. f(x) - 6 f() - 8
' 88 .4/57
'
6 = lim ( ) ( ) = ( . . . )
6 - 6 1006 6 -8
= = 1 c = __ 1 00 0 - ( -1) - 8 92 92
c. i) = 6 . : i = 6 ; = =62 - 52 = l l
1 1 1 (6 ) = (n -6+ 1) c = l l(6- 6+ 1)- =- 92 92 i = 5 ; = , = 5
2 - 42 = 9
1 1 8 (5 ) = (n - 5 + l) c = 9(6 - 5 + 1)- = -s 92 92 :
2 1 20 1 5 6 (4 ) = 92 , (3 ) = 92
, (2 ) = 92 , ( ) =
92.
i i ) . . , 7 : 6 (). 7 . : 1 = () + (1 ) + (2 ) + . . . + (5 ) + (6 ) 6 ( ) () = 1 - (; ) () =
i=l
=l-(i+ 15 + 20 + 21 ++.!!)=-1 . =_.!._. 92 92 92 92 92 92 92 92
8 : f{x) =ar +3x, g(x)=ar -2x h{x) =(-l)x, , JR 20 , ' : :
[ . . . -. . . ) ; ;
[0,2) = f' { ) [2,4) 2 2 = g' ( 2 ) [4,6) 3 3 = h' ( 3 )
20 , f, g, h , . a. , 2 , 3
f, g, h , 2 , 3 .
b. . ;
c. , : , : , () = .!. .
4
a. 1 = 1 , ;=3, 3 =5 . : f{x)=3i +3 , f{x,) =f{1) =3:1+3, g'(x) =2-2, g'( 2 ) =g'(3) =6-2 . h'(x) = - 1 . , h'( 3 ) = h'( 5) = - 1 . b. : 1 + 2 + 3 = 20 10 = 20 = 2 . : 1 = 9 , 2 = 1 3 = 1 . :
3 .
' ' ' - "-1 44 2 2 : = - = - = , .
: s =
20 3 2 ; (; -)
....ci="'-1 ----- = 1, 1 7
CV == 53%> 10% .
c. , =
[0,2). , () = _2._ . 20
= , : () = ( u ) = () + () () =() -() =1__2_ =! P(A)=Ys 20 20 5
' 88 .4/58
l f()= , (0, ) . . (0, ) f 1-1 1 a . . f1 (f1(x)) ' 1 . . C, M(l,f -1(1)). . f1 . . - C ' = , =1 C, M(l,f -1(1)). : . (,) f()=()'
1 2 = -- =-1- < 0. 2
f ; (,), n m 1-1. ': lim f( ) = lim =
---*>+ ---*>+
lim(-1 uvx) =(-too) 1 =-too
-*>+
limf(x) = limox = lim(-1 uvx) =(-too) (-1) =- f(' (y)) (f-' )'(y) = 1 => ( -1 - 21 (y)) (' )'(y) = 1 => ( - 1 - f2 (f-' (y)) (' )'(y) = 1 => (-1 - y ) (f-' )'(y) = 1 => (' )'(y) =
- . (' )'() = --1-2 , JR 1 + y 1 +
. ': (f-1)'(1) = _ _!_ (f-1 )(1) = ::::> f() = 2
= 1 =!:, (,). 4
C r- (1 ,' ( 1 )) y-' ( 1) = (' ) '() (-1)
1 1 +2 y-- = - - (x - 1) y = - - x +--4 2 2 4 . IR (' )'() =
. __ 1_ => (' )
"()
= (l + x2 )' = 2 1 + 2 (1 + 2 )2 (l + x2 )2
: (' )"() = > => (f-1 )"()>, (' )''() < . ' (-,] [,+) (o,r- ' ()). : ' () =
=> f() = 0 => = => = , (, ). 2
.. (0, ). 2
. r- [0, 1 ].
1 + 2 ' y= - 2
+-4- C,
( l ,1 ( 1 )) C, . I
: ()= j(1 (x)-y)dx=
I I 1 + 2 I J1 (x)dx- Jc--x+ --)dx = J(x) '1 (x)dx 2 4 [2 + 2 ]I J
l -1 . 1 + 2 + - - - = x (f ()) dx +--- = 4 4 4 4 I 1 - 1 - [ ()J - Jx (- 1 + x 2 )dx +-4- =
1 1 ( 1 )- ' () +_!_ J' (2x)'dx + - 1 - = 2 0 1 + 2 4
1 [ 2 ]' 1 + 1 1 - +- ln(1 + x ) -- = -ln 2 --. 4 2 4 2 4 2
.. : F(x) = J t2 (e1 - l)dt
' 88 .4/59
: . AF = [, + ). . F . . F(x) AF .
. 0 (0, 1) : F'(x0)= 3e- 7 . 3
. F(x) xFx(eFx - 1) AF . : . f(t)=t2(e1- 1 ) . R , fx 1R. , . F AF = [,+) f(t) , . . f(t)=t2 (e1 - 1) [O,.fx:"] (,+ ). F (,+) F '(x)=( )2 (eFx - 1)( )' = =( .fx:")2 (eFx _ 1)_1_=-Fx:" (eFx - 1) .
2-Fx:" 2 0 = . > : F(x)-F(x0) = F(x)-F(O) = F(x) lim F(x)
- 0 -0 , , , -
(eFx - 1) F'(x) ., F'( )
: -- = 2 lim __ = () ' 1 --+0. ()' '
lim F(x) =0, F '(O)=O. { (eFx - 1)
' F'( ) , > = 2 , 0 = 0
e'.. . ': > > ex >eo e >1 (eFx - 1) => 2 > => F '(x)>O. F
[0, + ) . z => F(x) z F(O) => F(x) z . ' : > F [,]
F (,).
(,) : F '() = F(x) - F(O) F(x) - 0
F '() = (eFx - 1) > 2
> F '() > F(x) > F () > . 0 = F () = , F () ;::: . , : > F [,] F (,) .....
(,) : F '() = F () - F () = - 0
F () , , , = --. F () >
x>O F '() > F () > O F () > . 0 = F () = . F(x) z . . F [0, 1 ] F (0, 1 ) .....
0 (0, 1 ) : F '(0) = F (1) - F () = F (1) =
1 - 0
Jt2 (e1 - l)dt =Jt2e1dt - Jt2dt =Jt2 (e1 )'dt -[f] = 3
= [ t2e1 ] - f2e1tdt _ ..!_ = e-.!.-2 J(e1 )'tdt = 3e- .
3 3 3
. t>O f(t) = ( t2 (e1 - )) ' = =t(2e1 -2+te1). ' g(t) =2e1 -2+te1 , t>O. g'(t) =2e1 +te1 +e1 =te1 +3e1 >=>g :; (0,-t). : t>O=>g(t)>g(O)=>g (t)> O => t g ( t )>=> f( t )> t>O f [,+) f:; [0,-+). : t.. =>t .Jx =>f(t) f(fx) =>
Jf(tXit Jfcfxxt =>F(x) f(fx) fdt
=>F(x) (fx)2 (e -1) [t]; =:>F(x)x.Jx. (e -1).
' 88 .4/60
\
' f , . :
f , F(x)=J: f(t)dt , F'()=f(), .
. :
1 !! : f: , : 01 ex-1 f(x)dx = f(x) + ex ( 1 ). ( 1996) 2!! : f: :
f(x) = (103 + 3) i2f(t)dt - 45 : f(x) = 203 + 6 - 45 (2008) 3!! : f: [0,1] v :
f.1 1 0 f(x)(x - f(x))dx = 12
. 4!! : f: [0,1] :
01 f(x)dx = + 01 [2 (x2)dx () -
; : , G. Polya How to Sole it ( : , ! 1 998 , )
, , . - : f(x)dx ; . -
;
Im! . 01 ex-1 f(x)dx = c, : ( 1)=> c = f(x)+ex => f(x) =c-e' . : ( 1 ) =>
1 ex-1 (c - ex)dx = c 1 cex-1 dx 1 e2x-1 dx = c
[e2-]1 e [cex-1]5 - -- = c - ce-1 - -2 2 - 1 1 +=-=c=>c = - (1 - e2) . : f(x) = -ex + 2 2 (1 - e2) .
2 , .
2!! . . : f(x)dx = c ( 1 ) , :f(x) = (103 + 3x)c - 45 (2) : ( 1 )=> 02{(103 + 3x)c - 45}dx = c [ 4 + 2) c - 45] =c 46c - 90 = c5c = 90 c = 2 . c=2 (2) f(x) = 403 + 6 - 45 .
. 3g . . :
01 f(x)(x - f(x))dx = c , : 01 f(x)(x - f(x))dx =
112 , c = 1
12
- ; . -
;
. -
;
-
-
;
. f,g ,, , : : (f() + g(x))dx = : f(x)dx + : g(x)dx
' 88 .4/61
: f(x)dx+ : f(x)dx=I: f(x)dx
2. f [,] f(x)2:0 [,] : f(x)dx 1
[, ] f() * , : f(x)dx > .
. 2!:1 :
f: [, ] 4 Iffi. : * * f(x)2:0 , [,] *: f(x)dx = f(x)=; : f(x)=O, [,], [,] f(x):f , : f(x)dx > . ' : 3!!!1 . J,l . 0 f(x)(x - f(x))dx =
12 0 (xf(x) -
f2(x))dx = _!_ - J,01(f2 (x) - xf(x))dx = _!_ 12 2 f 1 1 1 1 - (f2 (x) - 2 -xf(x) + -2 - -x2)dx = - 2 4 4 12
- J1 (f(x) - x)2dx + J1 x2dx = 2 4 12 f 1 1 1
- (f(x) - -x)2dx + [-3] = - 2 12 12 f 1 1 1
- (f(x) - -x)2dx + - = - 2 12 12
- J (f(x) - x)2dx = 2 f(x) - = f(x) = [0, 1 ],
1 [,1] {(
) *
(f() - )2 > ,
I01(f(x) - x)2dx > , . 4!!!!
0 {2 (x2)dx 3!!!! , : f 2
-
;
u= 2 u2=x . x=u2 .
4!!!1 = u2 = g(u), O=g(O), l=g( 1 ) g'(u)=2u [0, 1 ] .
( 2 1 ) : f(x)dx = I:ci f(x)dx = f(g (u))g'(u)du = f(u2)2udu. ': ( 1 ) 2xf(x2)dx = + 01 f2 (x2 )dx
f (f2 {x2) - 2xf(x2) + 2 - x2)dx = - 3 fl 2 2 3 1
1 0 (f(x ) - ) dx - [3]0 = - 3 I01(f(x2) - x)2dx = f(x2) = , [, 1 ] . ...[, ( ) :f() = ..JX, [, 1 ] . 1. Andreescu , G Dospinescu, Problems From the Book, Press, USA, 2008. 2. R.L. Finney, M.D. Weir , F.R. Giordano, Tomas' Calculus, Tenth edition, Addisson Wesley Longman, USA, 2001 (- : THOMAS , . , , 2005). 3. G.Polya, Mathematical Discovery, John Wiley & Sons, lnc., USA, \96\ ( : , . , . , , , 200\ ). 4. J. Stewart, Calculus, Seenth Edition, Brooks/Cole, USA, 200 1 . 5. . , . , 200 - . , , , 2006.
. . .
, a , , , , .
. . . , , . . .
.. '.
. . . [ 1 935 , 1 9 1 3] .
, .
' 88 .4/62
. . .
. . []
f [, ] (,) , , i =1 ,2, . . . , (,) f'(), f'() , ...
:
) t f 1(t) + 2 f 1(2) +. + v f1( v)=(t+2+ . . . +v)f'() (I) )
1 + 2 + + v _ 1+2+ + , f' . _1 2 ( ) (z ) (v ) - f'() (11), () :f (, ) ;>, - , , . . . , .
, . . f [ 1 , 6] ( 1 ,6)
) , 1 ,2 , (1 , 6) : 2 f I ( ) + 3 f I ( 2) + 5 f I ( 3) = 10 f' () () ) f 1 () > , (1 , 6) ,
, ' ' 2 + 3 + 5 - 10 1 , 2 , 3 (1 , 6) : f' ( 1) f' ( 2) f' ( 3) - f'() (i)
(11)
, xz ( 1 ,6) < 2 , z ,3 (1 ), C 2), (2 6) ..., : f'() = f( ) -f() f'(z) = f(xz ) -f(X), f' (3) = f(6 ) -f(Xz) - Xz - 6-2 2f' ( ) = 2 f( ) -f() = f( ) -f() 3f'( ) =J f(Xz ) -f() = f (Xz) -f() 2 z - 2 Xz - -3 -S f' ( ) = 5 f (6) -f(X2) = f (6) -f(X2) 3 6_2 6 2 5 : - 2 - 6- 2 1 -+2- +6- 2 6- , , , -- - -- = -- = = - = - . 2 3 5 ' 2+3+5 2 1 = 2 + 1 = 2 2 = 3 + 1 = 5 + 1 = 2. . 2 f1 ( ) + 3 f1 ( z) + 5 f1 ( 3) 2 = f(x ) -f()+f (xz);f()+f (6) -f(Xz) = f(6)f() 2[f(6) - f(1)]=2(6-1)f()=10f(), (1 ,6).
2 (ii) f [ 1 ,6] ( 1 ,6)
[ 1 ,6], f( l )< f( ) < f(z) < f(6) ,z( 1 ,6) 0, f(6)=1 0+f(l ), f()=2 +f(l ), f(z) = 3 +f( )= 5+f(l) . , 2 ( 1 ,6) f( ) =2 +f( l ), f(z) = 5+f(l ) =2 +f( l ) 2= 5+f(l ) ( f( l ), f(6)), (;). , .. ( f( l ),f(6)) 0 (16) f() = , , z ( 1 ,6) f( ) = f(z) =2
' 88 .4/63
' , 2 + __ 3_-t- __ 5_ = 1 -l+x2-x1 +6-2 f' ( 1) f' ( 2) f' ( 3) ( 1 ,6).
6-1 5 f(6)- f(l) 10
50 50 10 --- - -f(6)- f(l) (6-l)f'() f'()
2,3,5 [ 1 ,6] 1 , 2 () [f(1 ), f(6)] 2,3,5 . 2, 3 (ii)
: (ii), ( ) 7= , f(; ) 7= , = 1, 2, 3 => f(x1 ) 7=f(1), f(x2 ) 7=f(x1), f (6) 7=f(x2) 40, f(1 )=f(6), (. R.) 0 (1, 6) f(0 ) = , . : [ -f(1)][ -f( 6)] = 2( -8) = -16 < 0 [ 2 - f(1) ][ 2 - f ( 6) J = 5( -5) = -252 < 1 , 2 f( 1 ), f( 6)
. () [, ] d=-
d; ; , = 1, 2, ... , , d; = ; d = ; , , + 2 + .. . + "
, = +d1 = + , , 2 = , +d2 = +( +2 ), . . . , - = +(, + + ... +v- ) - (, , ) , ( , 2 ) , . . . , ( _ , ) .. '
f()+z f 1(z) + . . . +vf1( )= {, + + ... +v ) f{)-f{ ) =(, + + ... +) (-)f{) ={, + + ... +\)f{) - -
(11), f () < f () [ f () , f () J d; ; , i = 1, 2, . . . , , = f( ) = f(1 ) + , , 2 = f( 2 ) = f( ) + 2 = f(1 ) + ( + 2 ) , . . . , v- = f (1) +( + 2 + ... + - ) - 1 , 2 , . .. , - (,) ; = f (x; ) . i = 1, 2, ... , ( - 1) (, 1 ) , ( , 2 ) , ... , (v- ) ... ' f/:)
z v _ ( ) - ( ) - , + 2 + ... + + f/(z ) + . . . + f/(v ) - , + 2 + ... + f () -f () = + 2 + ... + ( - )f() = f()
. 1. f [, ], f [, ] m. 2. f [, ] f, [, ]. 3. f [, ], (, ) (, ) , : i) =0e (, ) , f, [, ]. ii) =e(, ) , f, [, ] . . 1
, , eR :::;, , ::;. : + + - - - ::: (). : f(x) = + + - - - xe [0, 1 ] . f: f(x) =( 1- -- )+ + - . : +=1, f{)=1-:::1 , iJy . f{x):::1 xe[0,1 ], f{):::1 (1). +:;f 1 , f . [0, 1 ] . f()=+-= 1 ++--1 = 1 +( 1-)( -1 ):::;1 . f( 1 ) = 1- - + + - = l - :::;1 .
f(x):::;1 xe [O, 1 ], ++----::::l :::;, , :::; . 2. , , eR ::S, , ::;, : 2 + 2 + 2 ::::; 2 + 2 + 2+1.
' 88 .4/64
: f()=2( 1-)-2+2-2 - 1+2 [, 1 ] . : f '() = 2( 1-) - 2. z z z ( 1 ,2]. h [1 ,2].
1 : h(z)h(2) = 4(2+"2 ) = 10 z [ 1 ,2].
: f(x) f( l ) = g() g( l )=h(z)h(2)=10. .
' 88 .4/65
' rr :;:
. . .
: = = = - .
.
: , ; : . 7 1 1 . ; : 6. : 6 . : , -
: 8 ; : . 2 . : . . . . ' , 5 8 . : . 8+5=1 3, . . . , . : 7 5 ; : . : , 3 4; : , . : 2, - 12 , - 12= {0, , 2, . . . } . - . , , 365 -
365: 12 5, 365=30xl 2+5. 5 12+5, . 12 . , - 35+14, 1 1+2, zl2 . , , , . -, f, g: IR IR
f(x) g(x)=O R, ; : - ' : ; : f(x)=O g(x)=O. : . ; : f(x)=O R,
g( )= IR. : . , , . : ! : . . . -
, f1 ) {0, 3 ( ) {-2, 3 , _ = 3 g = 0 3 . -5, < ' < f(x) g(x); : R . : ,
' 88 .4/66
------------------------------------------------------ -----------------------------------------------------
f(x)=O g(x)=O, xEIR. : -
: , , . . : , , . . . . . , . ; : , , . , , a. , ,
fil ) {0, Q ( ) {1 , Q
\ = g = .
1 , x E JR -Q O, x E JR -Q : . . JR; : -
f(x)= g(x)= . , {, 2: {0, >
< - x S O : . JR, . : ,
f(x) = ' - , g(x) =
' - . {) < { <
, > 4 , > f xEJR f2 (x) = ; : ; : . , , , , . , , ; : . . , f():;:. f2 () = , -, f2 (x) = . , . , ' , , .
; : . f -
> (f(x)-lnx)(f'(x)-..!_ )= f( l )=O, f(x)-lnx=O
>, f'(x)= ..!._ , >, . , ! : .
(f(x)-lnx)(f'(x)- ..!._ )= 2(f(x)-lnx)(f'(x)- ..!._ )= [(f(x) - lnx)2 )' = (f(x) - lnx)2 = c
f( l )= c=O, (f(x)-lnx)2=0 >, . f, g, xEJR, (f(x)- 1 )(g(x)-3)=0. , f(x)=1 xEJR
g(x)=3 xEJR; : ,
fi"{ ) {1 ' 2: 3 ( ) {-2, 2:
3 \ = g = . 5, < 3 3, < 3
: , ; :
' ' fil ) {1 ' 2: \ = - + 1 , <
g(x)= {2 +3, 2: .
3, <
: , xEJR, (f(x)- 1 )(f(x)-3)=0; : ,
fi"{ )= {1 ' 2:
3 \ 3.
3 , < : . : ; : f(x)=1 , xEJR
f(x)=3, xEJR
(f(x)-1 )(f(x)-3)=0, xEJR;
' 88 .4/67
------------------------------------------------------ -----------------------------------------------------: - . , - : g(x)=h(x) , - , , g(x)-h(x) . f : - , f() . ( ). (x)=g(x)-h(x) , , : ; -: ()> !R. f(x)=g(x), !R
. f(x)=l , !R f(x)=3, g(x)=h(x), !R
!R. : , . ;
: . , , !R f()=l f()=3. f 1 3, 2, - !R f()=2. = (f(x)- 1 )(f(x)-3)=0 (2- 1 )(2-3)=0, . , f(x)=l !R,
f(x)=3 !R. : ; : .
( f( )-g( ))( f( )-h( ) )= ,
f(x) = {g(x), h(x), x !R- A
. . : g(x), h(x) ; . ; : . g(x), h(x) , . ;
. . : . , f(x) , . , , . : . ; : , f()=h() f()=g(). : , ()=2f()-g()-h()=h()-g() [h(x0 ) - g(x0 )] [g(x0) - h(x0)] =
=> [h(x0 ) - g(x0 )]2 = => h(x0) = g(x0 ) !R, g(x):;t:h(x).
,
: : ' 88 .4/68
------------------------------------------------------ -----------------------------------------------------
, f(x)=g(x) !R,
f(x)=h(x) !R. : , , , ; : ; : . ex ;::: + 1 . : f(x)= ex - - 1 . IR f(x)=ex- 1 f(x)>O > f(x) 2 g(x), > . ; : ; : , , g( )=h() .
: f3 (x) = {g(x), < h(x),
f4 (x) = {h(x), < ; g(x), : ; : ; :
f () = {g(x), ::; f () = {g(x), < . 1 h(x), > 3 h(x), :
f1 (x)=h(x)=f3(x) = f(x)=g()=h()=f3(x).
: , . . . , . , . = . . : . :
f(x) = {g(x), * . h(x), =
: ! h( )=g( ). ; : , . . . , , f(x)=g(x). : g(x), h(x) . ; : , , , . , , , -
' 88 .4/69
------------------------------------------------------ ----------------------------------------------------- 4. g(x), h(x) ; : . . . . : ! . g(x)=x2+4 h(x)=Sx-2. ; : g(x)=h(x) 2-5+6=0 , 2 3 . : . , , -, xJR
(f(x)-x2 -4)(f(x)-5x+2)=0. : ; : . .
: f1 (x) = {2 +4, 2 , 5 - 2, > 2 f2 (x) =
{2 +4, 3 ' f3 (x) = {xz + 4, 2 2 5 - 2, > 3 5 - 2, < 2 f ( ) {2 +4, 2 3 ' ' , 4 = . = -5 - 2, < 3
. : . . ; : . . : ; :
t;(x)={i-+4, 223 f ()={2 +4, 23 5-2, 2
------------------------------------------------------ -----------------------------------------------------: , . , 910- 10- 10=9.() . : , 9.999 a , 999, 9.000. : . . n ; : ; : , . : ( ). . : ! , . : , . , ( ), , , . 9 9 8 7=4536. : , , ; : . : , , , 4536! : . , , , , , . , 9 8 7 6 3 9 8 7.
9 - 8 - 7 - 6+3 . 9 - 8 - 7=9 - 8 - 7 . (6+3)= 4536 . : . ; : . n g(x) h(x) . , , m . , m , .
IR - . , , 1, 2, 3, . . . ,xv_ 1 , .
- ro :( .., 1 .- \,. 1 '!\.- + ro
IR
; : . . 2; : . ; : , . : + . f(x); : g(x) h(x). . : , ; : + 1 2 2+ . . : ! : ; : ; : , , g(x) !R h(x) !R. : ! : ; ; : . , . g(x)=h(x) -, . . : 2 . 2. : . : . .
' 88 .4171
-------------- . . . -------------
.
P. R. HALMOS
201 ( 82 )
: . . - . - . . I
f() ?: > , ,
:
2 2 ( 1 ) + ( + 1 )y = + .
2 2 2 (2) 10 + ( 10 + + 1 ) = 1 0 +2 + 1 0. ( - )
( -)
(1) + ( + l)y = 1
0 = -1 , y0 = 1 , (-1) + ( + 1) 1 = .,
[-(2 + 2 )] + ( + )(2 + 2 ) = (2 + 2 )
0 = -(2 + 2 ) , y0 = 2 + 2
2 2 : = -( + ) - ( + I)t
y = (2 + 2 ) + t , t E Z .
2 (2) 1 = k kx + (1 Ok + I)y = 100k + 1 0 . (k , 1 0k + ) =
. kx + (10k + I)y = 1 0 = - 10 , y 0 = 1 ,
k( -10) + (1 0k + 1) 1 = 1 ,
[,] .
[0,1] ,
4jf(x)j ?: , .
( - ) .
( - )
g(x) = f) ,
: g
[,] g() ?: 1 ,
[0,1] , [,]
, 1 , , , I c )j 1 - , u: g ?: - , 4 4 . g() = f
() > ,
[0,1] , g
[0,1] [g(O),g(l)] .
.!. , 4
1 : jg( 1 )j < .!. . 4 k( -10)(100k + 1 0) + (10k + 1)(100k + 10) = 100k + 1 0 3 [0,-] , [- ,] . . (2)
0 = -10(100k + 10) , y 0 = l OOk + 10 .
(2) :
2 2 2 = -10(10 + + 10) - ( +I + I)t ,
2+2 2 = 10 + 10 + 10 t ' t z .
- .
202 ( 82 )
f [0, 1 ]
4 4
. t 1 E [O,.!.] , t 2 [ ,] , 4 4
: jg( t 1 )j < .!. , jg( t 2 )j < .!. . 4 4 [t 1 , t2 ]
g ,
' 88 .4!72
-------------- . . . --------------g(t2 ) - g(t 1 ) :?: t2 - t 1 (1) . , ,
( ) ' ' t t ' ' ' , : 2 - 1 < - , , 2
, , , ,
, 2
. :
, [,] ..!._ 4
, : Jg( )j :?: ..!._ , . 4
: - , -
, - ,
- .
203 ( 82 ) .
, ,
1 , 2 , 3 . :
1 + 2 + 3 = - .
( - ) ( )
,
. = = , = = = = .
. .
. R
, :
= = = = R . = =
, : 2
2 = 02 -2 = R2 - (1) 4
2 2 = 02 -2 = R2 _ l_
4 (2) .
2 + 2 = 2 ( ) , (2) 2 2 2 2 2 2 R2 -+ R2 _r_ = - ::::::> R2 = + +
4 4 4 8 ::::::> 8R2 = 2 + 2 + 2 (3)
:
::::> 2 + 2 - 20() = 2 ::::> R2 +R2 - 2R21 = 2 ::::> 2 = 2R2(1 - 1 ) (4)
, , :
2 = 2R2 ( - 2 ) , 2 = 2R2 (1 -3) (5)
(4) (5) :
2 + 2 + 2 = 2R 2 (3 -1 - 2 - 3)
(3) ::::::> 8R 2 = 2R 2 (3 -1 - 2 -3 )
= = = = . ::::::> 3 -1 - 2 -3 = 4 ,,, => + 2 + 3 = -
' 88 .4/73
-------------- . . . -------------- : 6 6 - , ' ] - .
204 ( 83 ) a R ,
( 1 ) : 6 + 4 + 2 + ax - 1 = , , .
( 2 - )
( ) ) - )
( 1 ) :
5 + 3 + _ _!_ = -a f(x) = -a ,
1 f(x) = 5 + 3 + - -
R , ,
(-,) , (,+)
" ' 1 1 1 2 + 2 + 2 ( + 1) ( + 1) ( + 1)
2( + + ) + 6 , , , 1 1 1 . ; + + - (-+-+-)
( - )
220. ( . ,0) () 2
= -. . 2
iryo
> 0 . (C - . )
2 2 1 . == =2.
==. f'(x) = 54 + 32 + 1 + > , R . lim f(x) = - , lim f(x) = +
x-roo
f((O,+oo)) = R , y = -a
cf (,+) ,
f(x) = -a . Cr
f(x) = -a ( -,) ,
. v - , ] - , ; : - ,
] - .
> 2 1 8. = = ,
= = 40 ,
= . : = 1 0 .
( - .)
2 1 9. , , R - {- 1} = - 1 ,
.
. ( - )
222. C1 C2
C1 .
C1 C2 .
C1 . : = , = , =
+ + = k (cm) , (AB) = k(cm2 )
= k2 - 4k (cm3 ) , :
1 ) ,
,
k.
2) k C1 , C2
( - )
' 88 .4174
. :
,
,
: : G. . Hardy ( . )
: 0 , , . , , . : .
: / ;
, . .
( . ) . . . , . = combasso= t , computer = . : ;
20 . 1 873 William Watts( ) . . . . 1 968.
' 88 .4!75
------------ -----------: ;
3 .. , =4/3(3) =42 . (
). V=23, =4/3(3) V- V=23-4/3(3)=2/3(3)=1/2 , , , , . , . 5 1 3 . .. . 1 930 12 , 2006 3, 4, 5 .
: ;
, , . , . . , . Pascal Fermat Mere 1 865 . Mere: 6 4 6 24 ; Mere: , 3 . 2 1 ;. : ;
2012 5 , 5 5 ; : 23
50%, 367 100%; . 23, 367
. : ,
;
, 521 (52 1 ). 37 DNA 1 3 v.
( ) .
' 88 .4/76
------------ -----------
23 . ,
. .
-1
?-.
' .:wY
4' : 17 ;
1 7 , . 57 1 3 . 5,7, 1 3, 1 7 . . . 3 7 1 3 1 7 29 .
- 76 76 76,
7 6 , 7676=5776, 7625=7632=2432. 76228;
; 6 1 1 1 5 ; Ferma ,
. ;
' . . . ; ' ;
' 88 .4177
-------------
1 38 37 73
. 5 (13837) .
. ;
: '
, . ; ,
[ L ___ -- ---- - -
; ;
- 76 76x228=76x210x2 10x28=76x l 024x 1 024256=76 1 048576256= . . . 76256= . . . 56.
, v = 42=32 >
, 6 1 5 . 1 1 5 6 . 6 5 1 1 5 . 26. 26 25=52 27=33 : :
. ,
, .
, .
.. 1 8 1 3837x(l 8)x73= 1 3 837x73x 1 8=1 0 1 0 1 0 1 x 1 8= 18181818.
. . 1 884 Herman Schwartz , . > i , . .
. , . . , , .
' 88 .4/78
) y m . , . 15 ) (
). . , . , . , ) , , . . : , )), ), 9) .. . t : , ( 4 , - qq\.) ), ), .. q ;: 'f\. ) , J
24-2-2013 a . . . . . . . . .
84 a . .
, , , . .
3
IAPAPTHMA .\l )' Uf!t.; 1).11- lJt !1rJlrr-prrl:#t-f1'.S1 W> -.. -.t>. f.o.. t. ..._ .. ._..
-6.. !1. --" 11Jt- IZAI H_,.o.rDJrdDfulw llvp/tAm;-F
n;A w'l! *"""'"- -...,. 11."-J' .. L ._.......,.,..,_ ,..
1.2.41 - U.H Qrrull>u vJ - {..- Aioe ""'') -.'"-- .. _,_,.....,._
JUt- 1),15 :nfi r f/dllmvr;d> !16' n
. -
6