eykleidhs_b_t92 2013-2014
DESCRIPTION
ΕΥΚΛΕΙΔΗΣ Β ΤΕΥΧΟΣ 92TRANSCRIPT
-
92
- - 1420 3,5 - - 1420 3,5
BB
h h i
... ... ... ...
2014
96 ...
.
.
.
.
.
.
4156
4156
Hellenic
Post
Hellenic
Post
.
1099/9
6
.
.
.
1099/9
6
.
.
5
5
5
00
5
5
5
00
0
5
25
75
95
100
0
5
25
75
95
100
-
92 - : 3,50- - 2014 -e-mail: [email protected], www.hms.gr
34106 79 .: 210 3617784 - 3616532Fax: 210 3641025
210
:
:
/
o / .
:
Rosetta, 1................................................................................................................................................................................
.....................................................................................................................................................................
..............................................
.......................................................................................................................................
........................................................................................
................................................................................
, 2, 4 , 11
20
: , 24: , 29
: , 31: , 36: , 39
: , 45: , 49 , 66
Homo Mathematicus, .............................................................................
..............................................
, 70 , ... 74 , 77, 79
......................................................................................................................................................
...............................................................................................................................................
A ,
. .
,
4 .
4
.
,
2014.
, .
.
31
: :
.: 2054ISSN: 1105 - 7998
: 3,50
,
(12,00 + 2,00 = 14,00). 12,00
(1).
(2). (3).
(4).
...
[ , , .] ,
... B. .
: ... . 54 ..
30044 ..., EUROBANK
... , , .
,
.
: (A. & ). .: 210 6623778 - 358 . ROTOPRINT :
:
:
:
:
::
:
: 11 . 2014.
:
...
, ...
:
12 2014
: e-mail: [email protected]
13 2014
0
5
25
75
95
100
0
5
25
75
95
100
0
5
25
75
95
100
0
5
25
75
95
100
-
92 .4/1
2014 -. Rosetta 2004. 800 -
- 950 - . Rosetta 20 - - -. 4,6 - , .
(ESA) - - - . 218 - - .
project Rosetta .
Rosetta 20 - Philae , . 30 2014 - . - - 2014.
-
92 .4/2
T 1820 F. Belinghouzen M. Lazaref Vostok Mirny - . 98% 1,5Km -93 C. , . , BICEP2 - , CMB .
BICEP2 , - 1981 - Alan Guth Andrei Linde. Alan Guth - , - . , - . , :
1. .
2. - , - .
3. ; ;
4. ; - ;
Alan Guth ( ) - - .
Andrei Linde - - . - - . - - -. - -. , . Linde. , - ( - bel 1993), BICEP2 - - 10-35 -. Chao-Lin Kuo BICEP2 - Andrei Linde ( ) , , - BICEP2. , - - , (-, , - ) -. - - , - , , - - ..
. 1916
-
-------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------
92 .4/3
. . . - , -. - . , - 10-34 .
BICEP2 BICEP2, , - (CBM). - - - 380.000 ( 13,8 . ). , - (- ) - CBM. - () , CBM - 13,8 . . - BICEP2 - , - . , - , - . , , , , - . 15 30 , -
. , ( ), - .
- , . 3.000 , -, . , . , ` , - . , "", "". o - . , - , . - - .
-
- - . , - , - . - - .
-
92 .4/4
.. ; :
- Jacobi , - Hardy [Nobel (Shaw)]: ;
. : , -. - , . . -
, . -
. - ; : , -, , -. , - . , - 0, , , - . - - . . , - ( ), - ( ), (.., ISBN). , , -.
6 .. .. - , - .
. - , 1,2,3,4. - ' - . - , -, -. , , . - . , , . , , ' - . [1] , - . [2] - . [3] - -, , ... - -, . , -. . . . , - , -. 10=(1+2+3+4). -
-
----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/5
*. , , - . - . - . . (2+2=2 ), , , * . . . 0,1,2, ,9 : (- )3,14, ( , )1,618, e ( Euler , , 2,718. e , . -, - , . , , , , , , , , , , , , .. -. , . .. - 300 . ; ; 150 ; ; 3 .. 1064 . 20 Kanser Newman Mathematics and the Imagination googol . googol , 100 . - (Milton Sirotta) 1938 10100 . Google - googol - Inter-net 10100 -, , -. *, , . * !
: - 3 - . () - - .. - - - , - -. Srinivasa Ramanujan 20 -. Evariste Galois 19 -.
[]
, . . - . -: , , , , , , . -, 1. : 19 - , , 4 1. 12 + 92 = 82, 82 + 22 = 68, 62 + 82 = 100, 12 + 02 = 1. 82, 68 100 - 19 - , 28 86. - , 100 20. 1, . - 4 , 4 - 4. - 131 3 : 1 + 3 + 1= 1 + 9 + 1 = 11, 1 + 1 = 1 + 1 = 2, 2=4 24 -, 42. -
-
----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/6
( ) . : 220 284 : 220 : 1,2,4,5,10,11,20,22,44,55,110, 1+2+4+5+10+11+20+22+44+55+110 =284 . 284 : 1,2,4,71,142 1+2+4+71+142=220 220 284 - - . - - -. 220 . , , , . 1636 Fermat - 17.296 18.416. 9.363.584 943.705. Euler - - 62 . 1866 - . ( ) 1.184 1.210 . - . : . : , . . 162 = 256 2+5+6=13 132 = 169 1+6+9=16. , , ( -), . Poulet 1918. : 12.496 (12.496,14.288 , 15.472 , 14.536, 14.264). 12.496 1,2,4,8,11,16,22,44,71,88,142,176,284,568,781,1.136,1.562,3.124,6.248 : 1+2+4+8+11+16+22+44+ +71+88+142+176+284+568+781+1.136+1.562+3.124+6.248=14.288 . : 14.288 15.472 14.536 14.264 12.496. 3 . - , , . , -, . 6, : , 1, 2, 3 6 (1+2+3=6) - 28 (1 + 2 + 4 + 7 + 14=28), 496 (1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248=496 ), 8.128. . : 6=1+2+3, 28=1+2+3+4+5+6+7,496=1+2+3++30+31, 8.128(1+2+3+.+126+127).
Abel 2014
Yakov G. Sinai
Abel - - Abel . - - Abel 2014
Yakov G. Sinai -. Sinai Princeton - Landau . 1935 -, Benjamin Fedorovich Kagan, . 20 , 6 ( 750.000 ). Yakov G. Sinai - - . - , - : ) Kolmogorov Sinai, . Kolmogorov, , - . ) Sinai, ) Sinai .. Sinai . , . - () - () . - , , , . , - . - - , -. - , - , - . -, , - . 2013 Pierre Deligne, - Institute for Advanced Study Princeton Deligne - .
-
----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/7
- - . 28 - 28 . 6 - 6 - , -
- . - 9 - 2n1(2n 1) 2n 1 - . - ,
48 . ( 48) - 17.425.170 . - - . - () . - . . 8 - 1+2+4=712. : n , f(n) : f(n)=10 * f(n-1)+n f(0)=0. , - . n=1, n=2, n=3, n=4, n=5 : f(1)=10 * f(0)+1=10 * 0+1=1 f(2)=10 * f(1)+2=10 * 1+2=12 f(3)=10 * f(2)+3=10 * 12+3=123 f(4)=10 * f(3)+4=10 * 123+4=1.234 f(5)=10 * f(4)+5=10 * 1.234+5=12.345 1997 . . n . - f(49) 500 .
.. : 153 = 13 + 53 + 33 , 370 = 33 + 73 + 03 , 371 = 33 + 73 + 13 407 = 43 + 03 + 73 . - . ( ) + , , * , , ^ , , ! . : 24 = (24)!=(2+4)!, 36 = 3! 6, 71 =(7!+1), 119 =-1+(-1+9!)!, 120 = ( (1 + 2)! - 0! )! = (-1 + (2 + 0!)!)!, 143 = -1 + 4! *3! - , . . . , , . . , . , () : .. - , - 0 1 -, () (). , , , . : 1, 1 + 2 = 3 , 1 + 2 + 3 = 6 , 1 + 2 + 3 +4 = 10 , . . 1=12, 1 + 3 = 4=22 , 1 + 3 + 5 = 9=32 , , 1 + 3 + 5 + .... + (2 3 ) + (2 1 ) = 1 3. 1, 1 + 4 = 5, 1 + 4 + 7 = 12, 1 + 4 + 7 + 10=22, - 1 4. 1, 1 + 5 = 6, 1 + 5 + 9 = 15, 1+5+9+13=28, - , 1 .
-
----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/8
: 13+23+33++3= . .. 2+4=2x3, 2+4+6=3x4, 2+4+6+8=4x5,
- , . ( ) . . - 2 () . - (). 100 25 : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. . , Riemann ( - )
Goldbach ( - - 2, ), -. - . -
- . - 2, ... - . - . : 1) . 2) - , . 3) , 2 5, 1, 3, 7 9. 4) p . , p (). 5) p , p p ( Fermat). 6) p>1 - (p-1)!+1 p ( Wilson).
Fermat [] Fermat Fn=2 1, n () n=0,1,2,3,4. Euler F5=6416700417 116 6542 .
[]
- 2, .. 11 13, 17 19. 1,3 7,9 9,1. () - . - , 6 5 ( 3,5). 12=6+5 1 (1+2)=6+5 12+21=6+5 .. 11.13=6.23+5. . - 1, 2 , . . -(1+2)=1. 12-(4+12)1-12-5=0 1=2+6+342 4 1 1=12+5 2=12+7. 1 2 1 2
=
12
21
1 22 12
60 5 7 - - - 1,4 6 0,5 11 13 48 4 286 1,1818 12 1 17 19 36 2 323 1,1176 18 2 29 31 30 1 449,5 1,0689 30 3 41 43 28 2
3 587,666 1,0487 42
- - .
[]
- - . - - . (-). , , -
-
----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/9
!! ( , .) - 2 . . 1 - .. - ( ) . . ! - Enigma. Alan Turing - . - . 2400 - . - ( - ) . - RSA 1977, ( -) . - ( 2) 4 - 3 1. (public key cryptography), - -, (key distribution problem). , - , : Bob Alice, . Alice , Bob. Bob , . , , Bob . - , -. -, - . -
(11 x 13 = 143), 2 - (? x ? = 143). - ( ). RSA Ronald L.Rivest, Adi Shamir, Leonard Adleman ( - RSA) 1977 Massachusetts Institute of Technology. - RSA - - 65 ( 129 ) . 1994 (17 ) 1600 . RSA ( !) : - - /; ( 2) 4 3 1. 3 - 1 , Dirichlet, - . , - , 4 - 3 50% 1. 1 4 (.. 17 41) - 1 4. 3 - 4 - 1 4. 4 - . - - 1= 3= - . - . . - , /. - ( - 300 ) .
-
----------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/10
- RSA : p, q d.e1 mod[(p-1)(q-1)]* (me)dm mod(pq). ( ) 1 a. - ( ) . (block) . b. 2 p,q - n=pq . c. e & d 1 (p-1)(q-1). .. RSA. m=65 . (n,e)=(427187,11) (.. - ). p=677,q=631. (e,(p-1)(q-1))=1. . : m^{e} n - c=178975 (m^{e} c mod n) c . O - d d.e (p-1)(q-1) 1. d=309731 ( e (p-1)(q-1)** . : : c^{d} n 65 (c^{d} 65 mod n). 2 () 1.
p, q. p=47, q=59. =p.q=2773. 2. ()=46x58=2668
e 2668 . e=17
3. *** * : a b modn a - n b ** ...(r,n)=1 k Z: r.k 1 mod n a r (r,n)=1. k : (a.r) ka(r.k)a1a mod n *** : 1 p=3, q=5, pq=15, (p-1)(q-1)=8 c=7 d : (p-1)(q-1)+ cd=1, 8+7d=1, 8=1x7+1, 7=1x7+0 =1 : 1=8-1x7 . =1 d= -1. d 1 (p-1)(q-1)
x & y 2668x+17y=1. (- (2668,17)=1 ). - y=157. d=157.
4. N e, d. e d p,q - .
5. m=31, :
: cm^e (modN) . 58731^17 (mod2773) 587 - d :
: mc^d (modN) 31587^157 (mod2773) , n e ( ) d. e n d - n . p,q - n d - . - . , - . - , - , . Henri Poincare : - , - - . - . -17mod8 d=7. 2 p=7, q=17 pq=119, (p-1)(q-1)=96 c=13. : 96+13d=1, 96=7x13+5, 13=2x5+3, 5=1x3+2, 3=1x2+1, 2=1x2 (96,13)=1 : 1=3-1x2=3-(5-1x3)=2x3-5=2x(13-2x5)-5=2x13-5x5=2x13-5x(96-7x13)= 37x13-5x96 = -5 d=37. 37x131mod96
-
92 .4/11
E.M.E.
31""222014
1: x 2 2x 6x 8 x x 2x x 2 , x . : x 2 x 4 x x x 2 x 2 , x , (1) , x 0, 2 4 : 0 2 2 0 . x x,x 2 x 2 , : x x x 2 x 2 Q x , (2)
Q x . (2) (1) : 2x 2 x 4 x x 2 Q x x x 2 x 2 x x 4 Q x 2 , (3) x . , x x 2 x 2 x 4 x 2 Q x xQ x 2 0, x . (4). (4),
x x 2 x 2 x 4 , : x 2 Q x xQ x 2 0, x . (5) x 0 Q 0 0, Q x xR x , R x . (5)
(4) : x 2 xR x x x 2 R x 2 0 x x 2 R x R x 2 0, , x x 2 0 x , : R x R x 2 , x . R x R x 2k , k , x , R x x , c R 0 R 2k ,k . R x c, x , 2 2x x x 2 x 2 Q x x x 2 x 2 xR x cx x 4 .
2: n 8n 25n 5
. (1 ): p,q , q 0, p,q 1
38n 25 p
n 5 q . (1)
3 3p ,q 1 , (1) : 3 3q 8n 25 p n 5 , (2) 3 3 3 3p 8n 25 q n 5 k : 8n 25 kp n 5 kq . (3) , 38n 25 kp , k , (2) (3). (3) : 3 3 2 28 n 5 8n 25 k 8q p k 2q p 4q 2qp p 65. 2 2k, 2q p 4q 2qp p 65.
22 2 24q 2pq p 3q p q 1 mod3 22 2 24q 2pq p 3q p q 3 , , 2 24q 2pq p 65
22 2 24q 2pq p 13 3q p q 13, : 2 24q 2qp p 13, k 1, 2q p 5. : p 2q 5 224q 2q 2q 5 2q 5 13
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/12
2p 2q 5, 2q 5q 2 0 p 2q 5, q 2 p 1, q 2 .
3p 1q 8
, : 8n 25 1
n 5 8 8 8n 25 n 5 n 3 .
2 : (2) n , : 3 33 3
5 5q pn
8q p . (4) 3 3 3 3d 5q p ,8q p . 3 3 3 3 3d 5q p 8q p 13q p,q 1
d 13 . 3 3 3 3 2 28q p 5 8q p 5 13 2q p 4q 2pq p 65d . 22 2 24q 2pq p 3q p q 1 mod3 22 2 24q 2pq p 3q p q 3 , ,
2 24q 2pq p 65 22 2 24q 2pq p 13 3q p q 13. (5) (5) q 2 , : q 0 q 1 , 2 2p q 13 p q 10 , p,q . q 2 , 2p q 1, :
p,q 1,2 p,q 1, 2 p,q 3,2 p,q 3, 2 . (2q p) 65 , p,q 1,2 p,q 1, 2 , n 3 .
3: n n , , , , , , , . 1S 2S . n , , : 1
2
S 39S 64
: 1 1 239S S S103 . : 2 221 2
n n 1S S 1 2 n
2 1S ,
2 2n n 1103
2
. 103 4+3, 103 2n 1 .
2n , 103 n . n , 206, : *n 206k,k . 206 .
1S :
2 2
2
n n 12 2n1 2 A
2 2
, : 2 2
2n n1 1 n B2 2
.
, 1 1 239A S S S B103 . 1S A,B , A 1
2 2n n1,2,..., 1, 12 2 . O
A 2 2 2n n1,2,..., 1, 2
2 2 ,
. 2
2n1,2,..., 1,n ,2
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/13
, 2 2
2n n1,2,..., 2, ,n ,2 2
2 22n n1,2,..., 2, 1,n ,
2 2 .
1, , . 4: c(O,R) ( O R)
, R AB 2R . c1(A,r) ( A r, 0 r R ), c(O,R), C D ( C AB). B, BE BF c1(A,r), E, F, E c(O,R). EC DF M. BCFM . (1 ): AEBF ( BE BF , : AEB AFB 90 ) . CD c . AB c . EF . , L , . AB EBF ( BE BF ). AB CBD , ABC ABD c O,R AC AD . EBC FBD . EBC FBD , BCE BFD . : 2BC BF a y xy a
BE BD x a ,
( ): BE BF a , BC x BD y . LCB LAD : LC CB LC x (1)
LA AD LA r .
LCA LBD , : LB BD LB y (2)LC CA LC r
.
1 2 : 2LB xy (A)LA r. ABE , AL , :
2 2
2 2
LB EB a (B)LA EA r
. A B : . BCE BFD , ( ). BCE BFD , : C F . BCFM .
2 : Q EF,CD Pascal EEDFFC , T ED,CF , T,B,M . , ECFD , T,M Q . TM Q , AQ ( ), ABT 90 . , EF TM AB EF ,
BMC MEF CFB , . : BCE BFD , BEDM .
B,A
2c)c( 1 )c( 2
)c( 1 )c( 2
)r,A(c1
2axy
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/14
E.M.E.
9127. : 2 nn! n , n 2. : : 2n! 1 n 2 n 1 k n k 1 n 1 [n ] nn n n n, [n ]. k n k 1 n, k 1,2,...,n . (1) , 2k n k 1 n nk k k n n k k 1 0 , (2) 1 k n , (1) k 1,2,...,n, 2 nn! n , n 2. (3) (2), k 1,2,...,n 1 k n , k n k 1 n , (3) . n 2 k , : 2 nn! n , n 2. 28. : 1 4a 1a a a ,
2 a 0 .
: 1x a, nx a a a ( n ). n n 1x a x , n 2,3,... , 2n n 1x a x , n 2,3,... . nx n 1x a
a a , : n n 1x x , n 2,3,... .. , nx . 2n n 1x a x , n 2,3,... 2n nx a x , n 2,3,... , nx , n 2,3,... 2x x a 0 . (1) 1 21 1 4a 1 1 4a,2 2
, 1 n 2x ,
n 2,3,... 1 1 4a 1x a 2 , .
29. : 2 2 2 2 2 1 ,42 2 2 2
n
n 1 . : kx 2 2 2 2 , k . 2 2, :
kx 2 2 2 2 2 2 2 2 2, k 2,3,.... 0. n n 1x 2 x :
n 1 n 1
n 1 n 1 nn 1
2 2 2 2 2 2 2 x 2 x 2 1 1A2 x (x 2) 4 x 2x 2 22 2 2 2
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/15
n nn
1 1x 2 x 2 4 .x 2 4
30. nnx n, n 1,2,3, ... . : 31 2 3x 1, x 2, x 3 31 2 3 . : n n 1n n 1, n 3 . , : n nn n 1 n n 1 nn 1n n 1 n 1 1n n 1 n n 1 n 1 n,n n n 3.
n n
kk 0
n1 11kn n .
k k
n n n 1 n 2 n k 11 1 1 2 k 1 11 1 1k n k!n k! n n n k!
, n 3 ,
nn
n 1 n n 1
111 1 1 1 1 1 1 1 121 1 ... 1 1 ... 1 1 2 1 3 3.1n 1! 2! 3! n! 2 2 2 212
, n 3 , : n11 3 n,
n
n 3nx n 3, n 4. ,
nx 33x 3. 19. p
p 12 1p
.
: p 1
22 1 x ,p
x . p 2 p 12 1 1 ,
p 2
p . : p 1 /2 p 1 /2p 1 2 22 1 px 2 1 2 1 px . (1) p 1 /22 1 , p 1 /22 1 2 , , p 1 /2 p 1 /22 1, 2 1 1 . , (1) : p 1 /2 p 1 /22 2 *2 1 y 2 1 pz , y,z p 1 /2 p 1 /22 2 *2 1 py 2 1 z , y,z . . p 1k
2 . , k 1 p 3 , : y z 1,
p 3 . k 1 , k 2 k 1 k 2 22 1 y 2 2 ... 2 1 y . y , y 2 1, , :
2k 1 k 2 k 1 k 22 2 ... 2 1 2 1 2 2 ... 2 4 1 , . , k 22 1 y , k 1, mod4. . p 1k 1
2 . p 1 /2 2 k 22 1 z 2 z 1 z 1 z 1
mz 1 2 z 1 2 , m k,1 m. z m m2 2 2 2 2 1 2 1 m 1 m2 2 1 1 2 1 2 1 1 1 0 m 1 1, m 2,
p 7 , p 7 p 1 /2 22 1 py , y 1, . , 3 7. 20. n . p
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/16
1 2 nx x ,... x :
1 2 n
1 2 n... px x x
. [ , 2013] : p
1 2 nx x ,... x 1 2 n
1 2 n... p.x x x
1 2 nx ,x ,..., x
1 2 nx x ,... x , : kx k k
k 1,x
k 1,2,...,n.
p , , : 1 2 n
1 2 n1 p ... n.x x x
p 1,2,...,n .
1 n , : 21 2 n1 ...n 2n n 1 2 nn ...1 2 n
.
2 p n 1 : 1 2 n 1 2 p 1 p n
1 2 n 1 2 p 1 p n... ... ...x x x x x x x x
,
1 2 nx ,x ,..., x 1 2 p 1
1 2 p 1... p 1x x x
p n
p n... 1x x
. , kx k, k 1,2,...,p 1 kx k n p 1 , k p,p 1,...n . 1 2 nx x ,... x 1,2,...,n . 18. ABCD 1. : AB BC CD DA 4 , ABCD . : AB a,BC b,CD c,DA d , : abcd 4 . (1) 1, : ac bd AC BD 4 , (2) . ac bd 2 acbd 2 4 4 . (3) (2) (3) : ac bd 4 , (4) ac bd AC BD = 4 AC 2, BD 2 , : AC BD 2 , . , a c b d . (5) ,
2 2 2ac bd ac bd 4abcd 4 4 4 0 ac bd , (4) : ac bd 2 . (6) (5) (6) : a c ac 2 b d bd 2 , ABCD . 19. ABC 90 c . 1c AB, AC c . 2c , CA c . 1 2r , r 1 2c ,c , , : 1 2r r 4 ABC . : BC a, CA b AB c , 90 , : 2 2 2a b c . 1c c K AB,AC M L , . : 1PL PM PK r ABC 1R OK OP r (1) c 1c . BC , aR
2 ,
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/17
AMPL 1r . : 1BM AB AM c r 1LC AC AL b r . BMP CLP
22 2 2 21 1PB PM MB r c r (2) 22 2 2 21 1PC PL LC r b r (3)
PBC 2 2 2 2PB PC 2 PO CO .(4) (2) (3), (4)
2 22 2 2 21 1 1 1
2 2 22 2 21 1 1 1 1
r c r r b r 2 PO CO
r c r r b r 2 R r R
: 1r b c a . (5) 2c : 2r b c a .(6)
2 2 2 2 21 2rr b c a b c a b c a b c 2bc a 2bc 4 ABC . 20. ABC P c ABC . Simson P . [ , 2013] : AC , ( ). BC CA, , ABC . c S . . S AC : NH NS . , : BT MN . , CPMN CPBT : BMN CPN CBT , . , BTNZ , ZNPS ( ). , ZP NS NH
ZSN (1). , S TP . (2) (1) (2), . , HZPN , . Simson . . Euler ABC. Euler . 11. n M 1,2,3, ...,2n 1 . M A,B, C , : () a A b B , a b C. () c C a A b B , c a b . : a b, a A,b B ., a b, a , a A C, . A M 2n 1 A . c C. ,
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/18
a A b B , a bq c, q . , a A : a c 1 1 c 2c 1 . , : 2n 1 2c 1 n c . n 1 B. n 1 A, n 1,n 2,...,2n 1 A. , b B, b n A n 1 , b . , A B 0 M . n 1 B ,
k n 1,n 2,...,2n : A 2n 1,2n,...,k 1 . c C c n, : n 1,n 2,...,k B . A n 1,n 2,...,k B 1,2,...,n .
B n 1,n 2,...,k C 1,2,...,n . k n 1,n 2,...,2n , n M . : , , , . , , , . . . , . , , 12, 3 10. , . . . , : 2, 3, 17, 4, 5 3 2 ,
. 16
.
. :
91
16 BC .
,
C .
31. a,b,c a b c 1 , :
2 2 2 2 2 2
3 2 2 3 2 2 3 2 2
a b b c c a 3ab bc cac a ab b a b bc c b c ca a
.
32. f : f x y y f f f x , x, y .
33. 1 2 4... , , 1 8 2 4 4 2, 6 10 , . , 1 1 1, , , (1) , :
-
-------------------------------------------------------------------------------- -----------------------------------------------------------------------------
92 .4/19
3 28x 4x 4x 1 0. (2) [ ] 21. () p,
p 17 1p
.
() p, p 111 1p
.
22. p,q : 3 2q p p 1 . 23. x, y,z : x z y7 3 2 . 24. 2 , . , 4 216 2 ,49 7 , 2 2 4 4 41225 5 7 ,810000 2 3 5 , , , , 1021 3 7, 1024 2 , , ., . 2 . [ ] 25. , , . , 249 7 67() 2 ,
101024 2 4 , 30 2 3 5 143 11 13 . , . [ ] 21. , AB,BC , , ABC. K,L ABC AB,AC, . KL , ACB . 22. ABC BAC = 60 . Euler BAC. [ ] 23. ABC BAC = 120 . Euler BAC. [ ] 24. ABC A = 45 A = 135 , AC, E Euler C C . c , C , . [ ] 25. ABC A = 30 A = 150 E Euler . : C . [ ] 26. 1 2 3 4 5, , , ,
1 2 4 5 2 3 4 5, . p 2 3 , 1 P 1 2 . Euler , [ ] 12. 2009 , . , . 13. n 2. 1 2 n, ,..., n : i j i j, i, j 1,2,...,n . n 1 2 nS ... . [ X X X Y: a :a X,a Y a :a X,a Y , X,Y , X,Y. ] 14. n 4 . n 1 , n . n.
-
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- HEMATICUS -------------------------------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/20
HEMATICUS
Homo Mathematicus , : 1) , 2) , 3) , 4) , 5) - .
: I. " ;" Albert Einstein. - - , . - . - , - . - , . , - - ,
. : . , - , - . . -, ( - ). . . .
. " , "
Mention Kariya
Mention
.
Kariya . ===2 ( -
) . ,, (Kariya).
-
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- HEMATICUS -------------------------------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/21
- . C1,C3,C5 ,, ,, ,,. C2,C4,C6 C1,C3,C5
,,, . , .
. ; ; [ ] V. " " . - . 1 . ErdsStraus ErdsStraus : n 2, n/2 -
, :
n 2 , x,y,z 4 1 1 1= + +n x y z
Paul Erds Ernst G. Straus 1948. Erds
2 . ndrew Beal x y z B C+ = ,,C,x,y,z x,y,z>2, A, B, C -
1993, Fermat
V. "O -" 1 () . - , . : 5 ,,,, - . , , . ,,, , ( ). , - (k1), : (k1.) (k1.). ,,. - : ) , , . 1.
-, , . - . ) . . , (k1.) 2. 2. ) . , , -
-
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- HEMATICUS -------------------------------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/22
) - (k1.)
- (k2) (k3)
2 , (), - . -. .
:
, - , -. F - F, . - F - F, - k. O . , .. - F - . F, -, F. - - k, ' ()/()=k.
F, F, - F. " .. 2 ABDC 'DC', 4 DC. - F F. - F - F.
F - F [ ()/(AD)=(AB)/(CD)] .
-: -, , 40 - 40.
- - . .. - : , -, , , G. - ' G, G - . , 2 - ( G' 2 - - G.
3 () , "" , : Homo mathematicus () (1),(2). (1) ,. (), ,, (2).
, , - , .
VI. 1. + 24/9/2013. : , ; - -
; - , Connecticut, Exeter ational Institute for Mathematical and Biological Synthesis (NIMBios)
-
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- HEMATICUS -------------------------------------------------------------------------------------------------------------------------------------------------------------------------
92 .4/23
2. Nobel Nobel2013,AliceMarlow . Toomuch happiness,
,
3. , - , . , , - . . - , , . ,
- ( ) - . , . http://thalesandfriends.org/el/2013/09/25/diastimiko-parkarisma
4 . ( , 29x36, 1889, van Gogh. , ..) - magnus opus, van Gogh - . - - - -.
, -, . - van Gogh - . 1941 ( ), - - - , , .
VI. " ;" - , (1869-1930) (1883-1950), , ,
- . 1920 . [: Loren Graham Jean-Michel Kantor, . , 2013]
-
92 .4/24
. . : .. , .. , , , ..
.. 1.
f(x) = 2 2x -10x + 25 4x - 4x + 1+
x - 5 2x - 1
. . f . . f. . Cf : .
2
2
x 10x 25 04x 4x 1 0
x 5 02x 1 0
.2
2
(x 5) 0(2x 1) 0
x 5 02x 1 0
x R
1 1A ( , ) ( ,5) (5, )2 2
. . : f(x)=
2 2(x - 5) (2x -1)+
x - 5 2x -1=
| x 5 | | 2x 1|+x - 5 2x -1
: 1
2
12
2 xf ( x ) 0 x 5
2 x 5
. :
. fC yy. : f(A) { 2,0,2} 2. ( 2, 6) (1, 2+), R ) .) .)
, .: . xOy, R :
x 1y 2
x 1 y 2 x 1
y 2
y x 3 (): y = x + 3. . : Mx 3 2 3 5 6 y ,
M . . . () . 1 1 (): y 6 = 1(x 2), (): y=x + 8. 3. f :
. . . . Cf : . fC xx. A R . . x 1 y = x = 135 = (180 45) = 45 = 1. x 1 y = x . 1< x < 2 y = 1. x 2 y = x + k = 45 = 1 (2,1) , : 1 = 2 + k, k = 3. x 2 y = x + 3
: x x 1
f (x) 1 1 x 2x 3 x 2
. fC yy
-
---------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------
92 .4/25
f(A)=[1, +) . xx 0 3 3, 1 1, . 3 1E 1 2
2
4. f(x) = x2 + x + . , , 3, 3 ( ||, 3) y = x2 1 . f xR :
20, 3, 3, 3 1,2 4
2, 12, 15 , : f(x) = 2x2 12x + 15. f(x), x2 1. : f(x) x2 1 2x2 12x + 15 x2 1 x2 12x + 16 0 6 2 5 x 6 2 5 . fC y=x21 x 6 2 5,6 2 5 , x ( ,6 2 5) (6 2 5, ) x 6 2 5 . 5. f(x)= x2 4x + 3. f(x) = 0 : . ( ).. .. .. .. x1+x2>1.
2 21 2 1 2x x + x x +16 < 0
: . : 00
2 0 03 4 0 1 4
[-1, 0) (0, 4]=. . : 0 0 3 0 (0, 3).
. : 0
0P 0
[ 1,0) (3,4]( ,0) (3, )
0 0 0 . . :
P 0 S 0
, ( ,0) (3, )
4 0
( ,0) (3, )(3,4]
0
.
. : S 1 1 41 1
4 0 ( + 4)>0 (, 4) (0,+ )(0, 4] . :
2 21 2 1 2x x + x x +16 < 0 2 (2) x1x2(x1 + x2) + 16 < 0 3 4 16 0
2
2
4( 3) 16 0 42 + 3 > 0
3( , 1) ( , )4
3( , 4]4
.
6. f(x)=29x + 18x + 9
3 | x + 1 |+2x
. .. .. g(x)=-x2+ 6x + 6 f g.. x f g.. : f(x)2015(x2+x+1) 0 : . 29x 18x 9 0 ,
| x 1 | 0
2(3x 3) 0x 1
x 1 .
: = ( , 1) ( 1, +)
. : f(x) =29x +18x + 9 2x
3 | x +1| =
2(3x + 3)
2x| 3x + 3 |
| 3x + 3 | 2x| 3x + 3 |
= 1 + 2x. . : f(x)=g(x) (1) : (1) 1 +2x = x2 + 6x + 6x2 4x 5 = 0 ( x = 1 x = 5 ) x = 5 1.
-
---------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------
92 .4/26
(5,11). . fC gC , f(x) 0 x R ( > 0 < 0) 7. f(x) = x 1 . f(2 + 1) > 0 . : 2 2f () + f( ) =f(f(2))(1) . : f()x2 + f(+1)x + f(2) = 0 (2) . . (2) x1, x2 : 2 2
1 2 1 2
1 1 2+ + = 4( -1)x x x x
(3)
: . f(2+1)>02+11>0 2 >00 R*. . f(2) = 1 f(f(2)) = f(1) =1 1 = 0, :
2 2 2 2f () + f( ) = f(f(2)) ( -1) + 1 = 0 (1) (1) (1)20 210, (1)(+1)0 (1 1). (1) =(,1][ 1, +) : 22( 1) 01 11 0
.
. f() = 1, f( +1) = , f(2) = 1 (2) (1)x2 + x +1 = 0. (2)
1 00
1 00
2 4 1 01
1
221
1 1 10
. . 1, (;), : x1+x2=
1 , x1x2=
11
21 2 1 2
2 21 2 1 2
3 (x +x ) 2x x 2+ = 4( -1)x x x x
24+4=0=2 1. 1 2x x 1.
-
1. 500 250 350 ,
25%. : : . : . i. P(A),P(B) . ii. P(A B) . iii. : . P(A B) . . P( ) . iv. : . . . . i. N A 250, N 350 500 ,
N A 250P A 50%N 500
350 7P B
500 10 .
7 3P B 1 P B 1 30%10 10
.
ii. 1P A B 25%4
. P A B P A P A B
1 5 5P A B P A B 25%4 10 20
.
N A B N A B5P A B
N 20 500
N A B 125 , 125 . iii. . P A B P A P B P A B
50% 30% 25% 55% . P( ) P B A P B P A B 30% 25% 5% . iv. . P A B 1 P A B 1 55% 45% . . P A B P A P B P A B
P A P B P A P A B
P B P B A 1 P B P B P A B
1 P A B 1 25% 75%.
2. : 2f(x) x x 1 , . i. fC . 2 x x ,, . , fC y y .
-
---------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------
92 .4/27
ii. 2 . iii. f . iv. . 1 . . . . y y 0, 5 . : i. 0 0M x , y 0 0f x y (). 20 0 0I x x 1 y ,
20 0 01 x x y 1 0 0 0
200 0
1 x 0 x 0y 0x y 1 0
. fC (1,0). fC (1,0) f x 0
1x 1 2x 1 2 2x x 1 x 1 1 , 2 . A M
(1,0) (-1,0). 2f 0 0 0 1 1 , fC
y y 0, 1 . ii. 1, 1 1 , 22 2 2 221 1 1 1
2 22 2. 2 2 2 22 2 2 21 1 2 1
2 22 4 2, 2 4 2 2 22 2 2 4 2
2 2 22 2 2 4 2 2 0
2 0 0 2. iii. 2f (x) x x 1
221, 2 2 4 4
,
f ,2 ,
,2
x2
22f
2 4 .
iv. . 2
22f 1 1 2 42 4
2 2 2 2 4 0
. 4 2, 1 . . fC yy 0, 1 . y 5 1 5 4 2f x x 4x 5 , 2, 1 .
- .. 1. , . : : . P(A),P(B) . P(A), P(B). . = . . . =- =-6 i) B ii) N iii) 4019. .
> P(A)>P(B)
P(A)= P(B)=.
. P(A)= N()=3 P(B)=()=43=4. . . P( )=P(A)+P(B)= . i) : 8-8+8=-6 P(A)=
-
---------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------
92 .4/28
2--=0. 3=4 =10, =8,=6 24. ii) , , , 4 2 4 50
S= 50 4700. iii) , 2 0 4019 4019 2 1 4025 4025 2 2 2014. 2. , ,
-
92 .4/29
. - . : .. , - ..
. . .
1. ( = ) o 108 . -
- . : . . . . =
. -, .. .
o 108 , o 36 - o 72 . 36 . .
o o o o 180 180 108 72 o 72 . =. = ( ) = ( ) =. 2. // . - - . - . . . . . . . - , 1 2 , 1 2 .
// o1 2 1 2 180
2 2 2 2 2A 2 180 A 90 90 .
. 90 - -. - , . . AE
2 ,
- . BZT
2
, ABT
2 EZ E T TZ
A AB B2 2 2 2 -
.
.
1. . =// =//. , , , -, : i. , , . ii. , iii. , , . : i. =//=//.(1) =// =//. (2). (1) (2) // , . , , .
-
-------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------
92 .4/30
ii. //, // //, //. , . = (3) = (4). , = (2), = (3) =-=-=. .
iii. , , . - , , . 2. , . , , : i. . ii. . iii. .
: i. To AB . . =. -. , = (1) ii. - . , . iii. , (1), , 3. . , -. : i.
ii. =
iii. =
: i.
90 90 180 ii. 90 , 90 . - , = iii. =-=-= //= . ( =90) - . =. 4. =120. , == , . :
i. . ii. . iii. .
:
i. - (=, =) ( ). . ii. , - , . =2. - . iii. =30. , 2 =60. - .
-
_______________________________ ______________________________
92 .4/31
-. -. : .. - . , ,
. 1. f(x)=(
2x)+1, < 0 -
-1 =3. . . . =2/3 = 2, i) f
fC xx [0, 3]. ii) fC (, 1+ 3 ). iii) f - x . iv) x Cf - y=3. . v) g - - f 2 . ; : . : = 2 3=
2 =
23
f(x)=(2 2
3x) +1= 2
3 x+1
: 23
x 10
23
x 23
x +1+1 f x 1 f(x) = +1 2
3 x = 1 2
3x = 2 +
2 x=3+3 ,
4 . +1=minf(x).
: +1=1 =2, o: f(x)=2 2 13x .
. i) : f(x) = 2 23x +1. -
Cf xx -: f(x)=0 (1). (1) 2
3x =
6
23x = 2 +
6 2
3x = 2 +
6
x = 3 +4 x = 3 + 5
4
x[0,3] : x =4 x = 5
4
ii) f() = 1+ 3 2 23a +1=1+ 3
23a = 3
2 2
3a =2
3 2
3a=2++
3
= 3 2 = 3 + 2
iii) =-2 f(x) = 2 23x +1.
2x 2x1 1 2 2 23 3
2x3 2 1 1 3 f x 13
f(x) = 3 2 23x +1 = 3
23x =1 2
3x =2
2 x=3 3
4
f(x)=12 23x+1=1 2
3x=1
23x=2+
2 x=3+ 3
4 .
f 3 x=3 34
-1 x=3+ 3
4 .
iv) : f(x)3 x x Cf y = 3
-
_______________________________ ______________________________
92 .4/32
v) g : g(x)= f(x ) 2 = 2( 2
3x )+12=
= 2 23x 1 = f(x). Cf Cg -
xx
2. (x) = (2 2)x4 +x3 +(21)x2 + 3x +2+1, R , p(x)
-
_______________________________ ______________________________
92 .4/33
(log50) > 0. : P(e) < P(log50) . Q(2) = 0
322
222+ 2
2
+3=0
3 22 x 2 xx x 2x 3 0 x 1, 32
2,2
2 =2 =1, . Q(x) 3 . 4. f(x) = 1xlnx, x>0. . f . x Cf xx . x f(ex) > 1 e2x-5 5
. ( 4
, 2
), >ln(e) . 0f(x2) f . 2 : 00 lnx2lnx1>0, lnx . f(x1)>f(x2). f . 3 : h x x ln x 0, x, lnx . A: h(x) f(x)=1h(x) 0, . . . Cf xx x f(x)>0. f(1)=0, 0, : f(x)>0f(x)>f(1) 0 < x < 1 . Cf xx x(0,1) . f(ex) > 1e2x-5x 1exlnex >1e2x-5x exx>e2x-5x ex ln(e), >1+ln(), > 1+ln( 1 ), > 1 ln(), 0 > 1 ln(), f() < 0, f() < f(1), > 1, f 0, .
(4 ,
2 ).
5. f(x) = 2 x
2e 2 x
2e+ 5,
g(x) = ln2x + lnx2 + 3, x > 0. . f(x) 4 x x . . g 4 x g . x Cg xx. . = ln2( 3
10) + 2ln( 3
10) + 3
. x f(x)=g(x); . . f(x) = 2
2xe
22xe
+ 5 = =2
2xe
22xe
+1+4=(2xe
1)2 +4 4 , (
2xe
1)2 0 x . : f(x)=4 (
2xe
1)2 =0 2xe
=1 x=4e+e, . . g(x) = ln2x + lnx2 + 3 = ln2x + 2lnx+3 = = ln2x + 2lnx 1 + 4 = (lnx1)2 + 4 4, : (lnx-1)2 0 x > 0. : g(x)=4 lnx=1 x=e. g(x)4 f(x) x > 0. g x0 = e 4. . Cg xx x : g(x) < 0, : g(x) < 0 ln2x + 2lnx + 3 < 0
2
lnx lnx
1 32 3 0
lnx < 1 lnx > 3 0 < x < 1e
x > e3 2 : g(x) < 0 (lnx-1)2 + 4 < 0 (lnx-1)2 > 4 ln 1x > 2 lnx1 < 2 lnx 1>2 lnx3 0e3
-
_______________________________ ______________________________
92 .4/34
. = ln2( 310
)+2ln( 3
10
)+3 = g( 3
10
)
: 0< 310
3 3
10 > 1
2> 1e
310
-
_______________________________ ______________________________
92 .4/35
. 1f(1) 1 ln e2
2 1f(2) 2 ln e
2 .
2 2 21 1 1 1e e e e ln e ln e 12 2 2 2
, ln x , 1 2 2 , (1) (2) f (1) f (2) . . 92
4 32
2 .
3 1 ln 2,2 1 2 ln 2 , lne ln4
2 e 3 . : 3 2 x 1f (x) x 2x ln(e )
2
x 3 2 x
3 2 2
1 1x ln(e ) x 2x ln(e )2 2
x 2x x 0 x x 2x 1 0
x ln 2,1 2 0,1 2 2 1 ln 2 ln 2 1 2 .
. -
1. f x xf x
1 x 1 x
K , 24 .
) . ) . ) f(x)=2. ) f ) x 0,1, 1 x
2 x
A ,2
) f 24
4 4 21 1
4 4
22 21 1
2 2
()
2 2 2 2 11 1 2 22 2 2 2 2
2 2 2 ) x xf x
1 x 1 x
1 1x1 x 1 x 2
2 xx1 x
2
x 2 xx x
2
x
f x 2 x 1 x 22
2 2 2 1
2 2 2 2
) x A x A
2 2f xx x x
2 2 f xx x
. 2. ) ln y ln xx y x,y 0 (1) ) : ln x ln 24 4x 32 0 (2) ) :
ln y ln x 4x y 2e
ln xy 2
(3)
) : lny lnxln x ln y , lny lnx lnx lny ) x 0 ()
ln x ln 22 x . : lnx2 lnx2 2 4 2 32 0 2ln x ln x2 4 2 32 0
ln xln x
ln x2
ln x 3 3
222 8
1,44 32 0
2 2 ln x 3 x e
) x, y 0 () :
ln y ln y 4
12
x x 2e3
ln xy 2
ln y 42x 2e
1 ln xy 22
ln y 4x eln xy 4
lny 4lnx lne
lnx lny 4
2ln y ln x 4 ln x ln y 2 x y eln x ln y 4
ln x, ln y 22t 4t 4 t 2 .
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/36
. - . : - ,
1. =5, =6 =600. ) 31 . ) =. ) : 2=2+22600= =52+62256 1
2=25+3630=31 = 31 .
) 2=2 2 22 2
4 == 91
4, = 91
2.
2. , 0 135 . ) / 2 2 . ) - . ) AB 2 .
) , : 2 2 2 2 , :
2 2 2 0 2 2 02 135 2 2 45 2 2 2 2 222 2 2 2 2 22 .
: 2 2 2 2 , 2 2 . ) . .: :
2 2 22 2 2 2
4 , :
2 2 22 2 22 4 2 2 2 22 24 4 4
: 2 22
.
) : 1 1 1 2 22 22 2 2 2
22 2 2 22 2 4 2 24 4 4
,
: 2 22 2 2 2
4 4 .
3. =4, 17 =5. ) . ) . ) - . ) : > , : 2=25 2+2 = 16+17=33 2 < 2+2 < 900 . )
2 2 22 2 2
4
2 17 2 25 16 ...4
68 174
17 ) () : = . =
2 12 2 . =+=2+1=3. - : 2=2 2=171=16 =4. : 20 . 4. , ().
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/37
: ) = ) A =
2
2
(1) :
i) 2 + 2 =22 ii) = 32
: ) ( 0180 ) , : = ) i) :
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/38
ii. . - - -. 1
5 .
2 152 2 10 10
2. =6
=8. . i. ii. . : i. .
2=2+2=10. .=2.10=36=3,6 ii. =3,610=8=4,5
3. A =60
. , . : i. == B
2
ii. iii. ()() : i. ,
, == B
2
ii. ====2 -
R=2 ,
, , . =30 - . - =2.30=60. - . iii.
60 ..
,
==30, 2
. .2 2
.2 2
.
()(), 2. 2=2+2260=2+2. 2+2, ()20 . 4. =90. ,
( ) .
: 2
2
, 2
2 .
2 2 2
1 1 1
,
2 22
2 2 ,
2 2
2 2 ,2
2 2 2 , 2 0,
5. O =R 3 =R
30. =2 7 , .
: o60 =R 3 =l=3. 30 =R=6, B =x x
2 .
180 x x30 30 60 1802 2
ox 60 =6=R, 90 =2R.
: =2R22R3R= =(2 7 )2R27R2=28R=2.
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/39
. - . : , 4 -, , 65 .
1. , ( , ) v ( , )
, w ( , ) . : ) v w ) v w
) 21
v // w
) v 2w 4 ) v w v w 0 ( , ) ( , ) 0
( ) ( ) 0 ( )( ) 0 0
) 2 2v w v w 2 2( ) 2 2 22( )
) v // w 0
2( ) 0 2 2 2 0 2(1 ) 0 21 0
2 21
1 (1 ) 0
) v 2w ( , ) 2( , ) 2
42
2. (1,3),(2,1),(4,1) v (k 2002, 2012) .
) . ) A BM . ) N : 4(A ,BM)
5 .
) v B 2AB , : i) v 2AB B ii) k = = 2014 ) ()2 = (41)2+(13)2 = 25, ()=5 , ,
M M4 1 5 3 1x , y 1
2 2 2
()2=( 52
+2)2+(11)2=(92
)2, ()= 92
) 9A (3, 4),BM ( ,0)2
27A BM2
) : 27
A BM 329 5A BM 52
,
2 = 12 = 19/25 = 16/25, - [0,], = 4/5. ) i) v B 2AB v 2AB B (v 2AB) B (v 2AB) B ii) v B 2AB v = (6,2)2(3,2) v = (12,2) k 2002 12 k 2014
2012 2 2014
3. :x+y+=0, C1: x2+y2+x+y+=0
C2:y2= x, R*, ,R. C1 -, : ) = 0. ) C1. ) C2 . ) C2 (4,2), .
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/40
) C1. , - :
2 2 x y 0x y 0( )
Ax By 0Ax By 0
(), , = 0. ) C1, (0,0). , C1 . ) :
2
2 2
Ax By 0 y By 0y Ax y Ax
A 02
2
y By(y B) 0 y 0By Ax x 0 xA
C2 (0,0) (2/,). ) , :
2B A 14A B 2B 2
: (/2,/2) (,1) C2:y2 = x, ,
= , E( ,0)2
, 1E( ,0)4
. 4. (,), B(+,4), ,R . ) OA OB , (,) C1. ) ()2 = 4, (,) C2. ) C1 C2. ) ,R, (,), : OA OB 0 2 2(OA) (OB) 4 (I). ) OA ( , ),OB ( ,4 ) OA OB OA OB 0 ( )( ) (4 ) 0 2 4 , C1:y2 = 4x. ) ()2=4(++)2+(4)2=4 (2)2+(42)2=4 42+1616+4x2 = 4 2+44+2 = 1 (2)2+2 = 1. , C2:(x2)2+y2=1. ) C1, : 2=4 =2,,
E Ex 1, y 02
(x2)2+y2=1. C2. ) ,R, -
(). : ( ) 1OA OB 0 OA OB M C
OA OB 02 2 2 2OA OB 4 OA OB 2OA OB 4
2 22
2(OA OB) 4 BA 4 AB 4 M C
C1,C2, - . C1,C2 , (
2
2
y 4x 3
).
5. (1)x2+(3+1)y2+(+1)x = (3)(1),R (1) ) = 0, (1) C1, . ) =1, (1) C2, . ) =1, (1) C3, . ) C4, , - C1 6
7.
) = 0 : (1) x2+y2=3 x2y2 =3
2 2x y 13 3
,
, 2 2
1x yC : 13 3 .
2=2+22=6= 6 , 6 23
. ( 6 ,0), ( 6 ,0). ) = 1 : (1) 4y2 +2x = 0 y2 =
12
x, , C2:
y2 = 12
x 2 = 12
, = 14
,
( 18
,0) , : x = 18
.
) = 1 : (1) 2x22y2= 8 x2+y2= 4, C3: x2+y2=4 (0,0) R=2. ) C1,C4 , = 6 2 4C :
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/41
6 6 6 77 7
.
2 = 22 = 76 = 1, , C4: 2 2x y 1
7 1 .
6. C:x2+y2+x+y6=0, A,BR, (1,1). ) (3,3) C. ) x+y2 = 0. , , : ) . ) C, : 2K (KM KN)K KMKN 0 . : C1+16=0+ = 4 ) : 9+9+3+36=0, + = 4, . ) ,. - - . , ,
K K3 1 3 1x 1, y 1
2 2 .
= 3 1 13 1 , ,
1, y1 = (x1), x+y2 = 0. ) R , : R2=(K)2=(1+1)2+(1+1)2 =8, R= 2 2 ) -. ( -), : M N M N 0 (KM K )(KN K ) 0
2KMKN KMK K KN K 0
2K (KM KN)K KMKN 0
7. x2+y22x2y+ = 0 (1) (,) R. ) = 2, (1); ) (1) C, , < 2. ) 0 4+44 > 0 < 2. ) i) : (1,1) R 2 . C, -: D=()2R2=(1)2+(1)2(2)=(23) D > 0, < 0 3/2
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/42
2 22 2
M2p 2tx p t
2 , 4p 4t 2(p t)
2
2
2 2 2 2(p t) p t 2pt ( ) x 2pt2
2
21 1x t p 5 x 54 4 2
.
: (c): 21 1x 54 2
2. ( 1c ):2 2x 1
16 9
(4 ,3 ) , (0,2 ) i) . ii) - . iii) 3, ,
2 2
() -: () 4 . 3 7 iv) () ( 7 3
8, 0)
: 2 22164 , - . v) () . 2(c ) vi) 1 2(c ),(c ) -. :
2 2Mx
16 9
2 2(4 ) (3 )16 9
2 216 916 9 2 2 1 . -
( 1c ). i) M M9xx 16 135 9x 4 16 3 144
3 x 4 12 0 ii) ii) : 3
4 .
() - 4
3 , -
:
43 (x 4 )3
4 x 3 7
iv) () ( 7 38
,0)
: 7 3 34 78 2
.
: 2 2 2 2
22 2 2
9 71 116 16
.
2221 1 21 16 3. .64 64 7 4 .
iv) x=0 : 73 . 73
.
7(0, )3
. =0 :
7x 7 x4
. 7( ,0)4
.
: 7x8
8x (1)7 67 (2)
6 7
.
2 2
2 2 64x 361 149 49
2 2
2 2
x 17 7( ) ( )8 6
. . . 2(c ) :
2 2
2 2
x 17 7( ) ( )8 6
1 2c ,c -
: 2 2
1 2
7 7( ) ( )7 76 8, 74 46
3. 2 2 2x 2 x 6 8 (1) . ) (1) - () (). ) . ) 0
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/43
:x 4 0 : x 2 0 . ) : 1, 1 . 1 , 90 . ) : x+-4=0 x-+2=0 x= =3, M M
x3x .
3
=3x 0
-
------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------
92 .4/44
9A ,62 . : 2 2B x ,y
2C , 2 22 25x 9y 45 - 2 25x x 9y y 45 2
2 2
5x 2 45, 39y 3 9y
.
5B 2,3
( ). 3. - 1C 2C - 9 ,6
2
52,3
. 090 , , 0 .
9 52,6 ,62 2
5 52 2, 4,
3 3
,
5 54 6 10 10 0.2 3
3. 22C : x y 2 4 , 4 . - , 0,0 . 1. ,. 2. . 3. C . 4. 1 3 , . 1. C 0,2 - 2 . 4 2
xK 02
y22
Ax y 4 .
2. , . 2 2 24 0 4 4 , 4 , .
3. : x y x y . ,
C : 22 2 4 2 2x y 2 4 22x y 2 4 .
22C : x y 2 4 , - (C) . 4. (1),(2): ,4 , , . 2 ,2 4 , 0, 4 . :
1 det ,2 2 2 412 0 1 .
3 3 . : 22 2 4 3+ 22 4 3 1 . - : 3,1 , 3,1 , 3,3 , 3,3 .
65 2y 2px , - (). : ) y 'y ) : ) p , 0
2
1p , y2
.
: p p2 2x 0
2
. y'y ) 1y
p ,
1 11
py
. - () , 1 1 1x , y . : 1M c . - ( c) 1 1y y p x x 2 1
1
py
. 1
-
92 .4/45
. - . : 3 , - 5 ,
3 1. , ,
x 1
( ) limf x , 2
2
1 x x 1f xx x
P(B) CV , CV - 5 . i) . ii) , . iii) 5 4 2 P(A B) 0,5
10 .
i) :
2
2
1 x x 1f(x)x x
2
2 2
1 (x x 1)(x x )(1 x x 1)
=
2
2 2
(x x )(x x )(1 x x 1)
21
1 x x 1 .
2x 1
1 1P A lim21 x x 1
.
P(B) : 1 3 5 7 9x 55
, 2 2 2 2 2
2 21 3 5 7 9s 55
165 25 33 25 85
.
8 2 2CV5 5
2 2P(B)5
ii) , : P(A B) P(B)
1 2 2 5 4 2 12 5 10
, ( 4 2 5 ), iii) A B A , : P(A B) P(A)
1P(A B)2
. P(A B) 1 P(A) P(B) P(A B) 1 P(A) P(B) 1 P(A B) 5 4 2 P(A B)
10 .
5 4 2 P(A B) 0,510
2. xf (x) e x 1 f y. -
f y=x - 2. i) , ii) =1 =2, , f. iii)
f(2), f(1), f(0), f(1), f(1/2) xf (x) e x 1 x=0 -
=+1. (0,+1) Cf yy yA>0 +1>0>1 (1).
=f(0) xf (x) e =f(0)=+. A
y=x =1, - += 1 (2) : y = x+(+1) (0,+1) (+1,0). : ()=2 21 ( 1) 2
2
21 4 1 2 +1 = 2 =1 =-3 =1 (1) : (2)=2.
ii) xf (x) e 2x 1 xf (x) e 2 , f (x) >0 xe 2 0 x>ln2 f (x)
-
-------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------
92 .4/46
A B = 3 - () (U)
i) 1(1) (1 1) (0)
1
1 1 1(2) (2 1) (1) (0)2 2 2
1 1 1(3) (3 1) (2) (0)3 3 6
1 1 1(4) (4 1) (3) (0)4 4 24
P(0) P(1) P(2) P(3) (4) 1
1 1 1P(0) P(0) P(0) P(0) (0) 12 6 24
65 P(0) 124
24P(0) 65 . 24P(1)65
, 12P(2)65
, 4P(3)65
, 1P(4)65
ii) =3 21=321=8 . 2
1=3=2, . 2,3 3,0 =={0}
U={0,2,3}, ()= 24P(0)65
(U)=40 865 13
4. (//) =2 30. - .
: 1+2+=302++=303+=30.
=x , 3=30x. 1 2( )
2 (2 )
2
(30 x)x2 2115x x (x)
2 , (x) 15 x .
min=E(15)= 225 112,52 . 5. . - (..>90/100) 75 s=5, 72 s=9 : 90 (90= x +3s) 0,15% . 90= x +2s 90 2,5% 6. , .. A B - (), () :
2 3x ( 1)x 0 (1)
i) (),() . ii) ()=25% (),()
i) (1) = 2( 1)2 0 - x1= x2=2. - [0,1] 2. A B ()(), ()=2, ()=. ii) A B , A B A . ()=0,25()()=0,252=0,25 2+ 1
4=0(21)2=0= 1
2 ()= 1
4, ()= 1
2.
7. x1,x2,,x20 - =25 30% - AX =15 CVA=20% 40% BX =30 CVB=10% -.
30%
(6,) , 40% - (8,), 30% (10070=30) .
ti=615+625+830=90+150+240=480 480x 2420
6 -
: AAA
SCV ,X
A A AS CV X = SA=0, 2015=3 SB=0,1030=3. :
2 2 22 21 2 6A
x x ... xs 156
2 2 2 2 21 2 6x x ... x 6 (3 15 ) =1404
6 , 25, 2 2 2 27 8 12x x ... x 6 (25 ) 3750 . 8 - :
2 2 22 213 14 20x x ... xs 30
8
2 2 2 213 14 20x x ... x 8 (9 30 ) 7272 2 2 21 2 20x x ... x 1404 3750 7272 =12426.
: 2 2 2
2 21 2 20x x ... xs 2420
= 212426 24 621,3 576 45,3
20 ,
s 6,8 S 6,8CV 0,1X 24
.
-
-------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------
92 .4/47
EM 5 -
1. 1,2,3, ...,9
1 log , 1,2,3,...,9 .
) - - . ) p p p . )
pCV 0,09 , - ps p , 1,2,3,...,9 . ) f 1f (x) 1
x
x 0 A / fC
,f () 1
64 , p
6 / log :
i) , -. ii) T , , . iii) T , . : ) : 0 p 1 , 1,2,3,...,9 1 2 3 9p p p ... p 1 . ,
10 p 1 0 log 1 1log1 log log10
11 10 1 10 0 1 9 ,
19
. : ii 1
2 3 1p log log ... log1 2
2 3 1log ... log 1
1 2 ,
. 9 :
1 2 3 9p p p ... p log10 1 ) 9 i
i 1
1 1 1p p 19 9 9
9 - . :
1 2 1p p log log 1
2
1 2 2 2 2log log 0 p p 2 1 1
1,2,3,...,9 . : 9 8 1p p ... p p 5 6 p log 5
) p 2p ps
CV 0,09 0,09 s 1019
) fC -: 21f . :
22
1 1 64 1 8 64 . :
7 ii 1
1,2,3,4,5,6,7 P A p log8
6 6 6 6log log 5 5 5 : 6,7,8,9
6 7 8 9P B p p p p 10log 1 log66 . i) 10log8 log
6
. ii) A B 1,2,3,4,5 P A B log6 iii) A B 6,7 P A B
7 8 8 4log log log log6 7 6 3 . :
4P A B 1 P A B 1 log3
2. f
2 2x lnt t lnt t x tf (x)
x t , t 0
) : x tlimf(x) t lnt t
) g x t
1g(t) limf (x)t
, t 0 (i) o - g . (ii) () g 1,g(1) . (iii) i ix ,y , i 1,2,3,..., (). 1) x 9 xs 1 iy i 1,2,3,..., 2) -
-
-------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------
92 .4/48
iy , i 1,2,3,..., . 3) c iy, -. : ) :
2 2 2 2x ln t t ln t t x t x ln t t ln t x t x t x ln t x t t x t x t x ln t t .
: x t x t
x t x ln t tlimf (x) lim
x t
x tlim x ln t t t ln t t , t 0 ) (i) () 1g(t) t ln t t
t , t 0
t 0 , 2 21 1g (t) ln t 1 1 ln tt t
2
3 3
1 2 2 tg (t)t t t
2g (t) 0 2 t 0 t 2 2g (t) 0 2 t 0 0 t 2 2g (t) 0 2 t 0 t 2 t 2 fC 22, 3 ln 2
2
(ii). g (1) ln1 1 1 -: y x . 1,g(1) 2 : 2 1 3 . - : y x 3 . (iii) (1) : i iy x 3 , i 1,2,3,..., : y x 3 9 3 6 y xs 1 s 1 (2) :
yy
s 1 1CVy 6 10
. o. (3) z . i iz y c , i 1,2,3,..., c 0 . z y c z ys s 1 . :
z1 1 1CV c 6 10
10 c 6 10 c 4
c 16 . c 16 c 16 .
1. 2xf x ln x 1
2 , x 0 .
i. f. ii.
x 1
x f xlim
1 x
. iii. x 1g x x e , x , g x 0 . iv. f x g x x 0 . : i. 0, : 21 x 1f x x
x x ,
f x 0 x 1 f x 0 0 x 1 . f
1, f 0,1 , 3minf x f 1
2
3f x2
x 0, (1). ii. x f x
1 x
=
2x 11 x
= x 1 1 x ,
x 1
x f xlim
1 x
= 1 1 1 1 4 .
iii. x : x 1g x (x e ) = x 1 x 11 e x 1 1 e , g x 0
x 11 e 0 x 11 e 0 x 1 x 1 0 x 1 . g x 0 x 1 . g ,1 g
1, . maxg x g 1 0 ,
g x 0 x (2). iv. (1), (2): 3g x 0 f x
2 x 0 .
2. 2f x x : y 0 . i. , - 1 2, fC , . [A.: 1
4 ]
ii. fC , . [.: 2 12 , 1
4 ]
iii. i ix ,y , i 1,2, ...,10 fC x 2 s 14 . .
[.: 22s y x , y 18 ]
-
____________________________________ ___________________________________
92 .4/49
. - . : - , . , , , . , . N. , .... - 1
. : . 2 2f (x) x 6x 9, x . . 2 (x) 5(x) 6 0, x . 1 . g
2g (x) 2g(x)x+1 ,x g( )=1+ 22
. 2 g(x) x 1 x , x g(x)= -x ( ,0) .
2
. , 3 3
0, f
x x lim f x lim f x 3 3 3 f 0x 3 - ,0 0, . A [3, ) ( ,3) f (x) x 3 x , f (x) x 3 x , 1 : - 88 .
x 3 , x 3f (x)
x 3 , x
-
____________________________________ ___________________________________
92 .4/50
2 t( ) g( ) ( ) 1 ( ) =2 2 2 2 2 2
2= 1 2 0 t(0) g(0) 0 1 0 =1>0.
2
0 .olzano x ( ,0)2
0 t(x ) 0, g(x)= x
( ,0).2
. :
3 3 3
x(1) g(x)dx g(x)dx3 3
2 2
g(x)dx x dx (g(x) x )dx 0 (2).
x2
F(x)= (g(t) t )dt , x [,] , F [,]
2
2
(2)2
2 2
(,) F (x)= g(x) x .
F() F() (,) F ()= 3 . (g(x) x )dx
3 g() 0 g() 0 g() .
2 . C () |Re(z)+1| = |z 1|. . M C, w = 2 + 6i . f Cf C 0. - Cf - Cf, - , . . f - : 1f f x x 2 f x x , [0,+) f(x), f1(x), 1
3 =4.
. : 3 6
f x f x dx = 0 : . z=x0+y0i . |Re(z)+1| = |z 1| |x0 + 1| = | x0+y0i-1||x0+1|= 2 20 0x 1 02 = 4x0. C, (x0, 0) z, 2 = 4x (1,0) x=-1
. (x,) C. w (2,6) : (AE)= 2 22 x 6 =
222 2 6 d
4
, (AE) -
d() . d()= 3 124
d()=03=86=2 3 6 d()>0 >2 3 6 d()
-
____________________________________ ___________________________________
92 .4/51
() :
3 363
30
1E x 6 2 x dx6
3 362
333
0
1 x 46 x x2 36
3 43 23
3
1 6 46 6 36 3 6 8 12 36
. f 1-1 -: 21 xf x
4 , y 0
2 x y x 0 2yx
4 ,
: f f x x =2 1f x x f f x x = 1f f x x f 1-1 f(x)= 1f x
2 x =2x
4 (x=0 x=4).
. : ' 3 3
6 6
I f x dx f x x dx2 2
3
6
h g x g' x , h t f t g t t.
2
g 3 6 3 g3 66
I h x dx f x dx f x dx
. 3 f:IRIR z = x
0f t dt + f x 1 i , -
: z+3=z+3i. . f, - :
1 0
f x dx + e -1 1 f x dx =e. f -1(x) < x, x>0.
. ) Cf - >xM. ) - Cf - 1fC . . h:(0,+)R h(x)=
x2 -x
t tf tdt
t 1 . N :
) h(x) = f -1(x2+1) >0 - h , E < eln
2.
) [0,1] : 2h( 2
) h(0)+h() . : =
2 2
2 0
x x dxxf ln x 1
, < ln 2
. x
0f t dt = f x 1 ,
z i z 3 z 3i i +3= i +3i 2 22 2 3 3
x 0
f t dt = f x 1 f(x) = f(x) f(x) = cex ( 252). f(0)=1 c=1. f(x) = ex, xIR f -1(x) = lnx, x>0.( 154). 1
0f x dx + e -1 1 f x dx =
1 x
0 e dx + e 1 ln xdx = 1x 0e + e1x ln x x = = e.
x>0 : lnxx-1lnx-x -1
-
____________________________________ ___________________________________
92 .4/52
. 11 Cf 1fC . ! .) 1 : h(x) = x 2 -x t tf t dtt 1
=
x
2 -x
tdt
t 1 + x
2 -x
tf tdt
t 1 = x
2 -x
tdt
t 1 = x
2 0
2t dtt 1 = 2 x x 22 0 0t 1 'dt ln t 1t 1
ln(x2+1) = =f-1(x2+1), 2tf tt 1 -, 2
tt 1 .
2 : h, h(x)= 22xx 1 =(f
-1(x2+1)), h(x)= f -1(x2+1)+c. h(0)= f -1 (02+1) = 0= c
. ) x>0, h(x)= 222 1 x 1 x
x 1
h(x)>00
-
____________________________________ ___________________________________
92 .4/53
. x 0 >0 2g() ln( x 1 x) x . .
x 0limh(x)
xt
00
x2
0
1( du)dtg(u)
h(x) . ln( t 1 t)dt
E..1
0
f (t) dt .g(t)
. 1
102
0
f (t) dtg(t) : (x) 2 x 2 (x)dx (4).
ln ( 2 1)
: . :(1) f (x)=
(2)
2 2
1 1 g (x) 1(g (x) 1)g(x) x g(x) g(x) xg(x)(g (x) 1) g(x) x g(x)g (x) g(x)
g(x) x 2g(x)g (x) 2x (g (x)) (x )
2 2g (x) x c x=0 c=1.
2
2
g(x) 0 g(x) x 1 f(x)=ln( x 1 x) (1),(2).
. (0,+) : 2 2(3) f (x 2) f (x ) f (2x 2) f (2x)
2h x h 2x 5 , : h : h(t)=f(t+2) f(t) . h h (t)=f (t+2) f (t) . (5) h -. h, h(t) - f(t+2),f(t), - f (0, +). f (x)
2
2 22
1 ( x 1) 2x( )x 1 x 1x 1
f(x)0 ( ,0). f 0, f ,0 , f 0. (0, ) : t
-
____________________________________ ___________________________________
92 .4/54
2 [ 1
91]
, . 1 ,
, ,
1. , , 2. 3. f (x) 0
x 0,2
. 1. ) :
0,2
0 0
.Rolle, 0 x 0, 2 ,
0. : 0, 0, 0,
2 , 1-1.
.
) : f 0,
2
2f (x) x . : 2f (0) 1 0
2f ( ) 02
. Bolzano 0 x 0, 2
0f x 0 . ... 2. f 0,
2 , :
0 0 0
0
00,x 0 x , 2
, 0x 0f x . f 0 f (0) 0 f 0
2 .
3. f (x) 0 x 0,
2
x 0, .2
f x 0 x 0,2
. 2 1. : , , 0 : ) 0, :
) C (1,1), C Q(0,1) . 2. , ,, : i) 0 0 ii) , , , , :
, , , ,, , , 1.) (0,+)
22 (x) x (x) 2x (x) x (x)
-
_______________________________ _____________________________
92 .4/55
'2 2
4 2
22
' x x x ' x x0 0
x x
xc x c x
x
, 0,, : 0 lim lim 0 , 0 (x)>0 (0,+)
(x) 22 (x) x (x) ln (x) 2ln x
(x) x
12ln x c1ln (x) 2ln x c (x) e
2
1c ln x 2(x) e e cx . ) (1) 1 c 1 2(x) x . , Q(0,1)
22 2 4 20 0 0 0(QM) x 0 x 1 x x 1 1, x 3 2k ' x 4x 2x 2x 2x 1 .
: (QM) k(x) 0, : 2 1 2k ' x 0 x x
2 2
2k ' x 0 0 x .2
0, , , . . ,
Q. 2. 0, , 0 :
t 0, '1E t f t . / O 0,0 , A t,0 ,
M t,f (t) , B 0,f (t) 2E (t) t f (t) , t 0
t1 2 01 1E E f (x)dx tf (t)3 3
1f (t) f (t) tf (t) 2f (t) tf (t)
3 .
: 2f (t) t , t 0 ( 2f (x) x , x 0 ).
: t3 3t 2 2
00
x t 1x dx t t3 3 3
3 : , , : (1) (1, 1) 1.
2. 3. :
2
2
x 22ln 2x 1x 2x 3
4. , 5. , , 1. 0, : 21 f ' 2 'f x f x-x -xx e xe x e e 2f x2 2 2 xx' ' e ce 'f x-x -x -xx e e x e x e
(1, 1), 0. : (1)
ln 2 ln
-
____________________________________ ___________________________________
92 .4/56
2. 0, , . : 0 0 2 0 2 f 2 0,2, 2, . 2. ,
x 0 x 0lim f (x) lim 2ln x x
2ln xf (x) 2ln x x x 1x
.
x
2ln xlimx
.
2ln x 2xx
x
2lim 0x .
x
2ln xlimx
0, : xlim f (x) 1 .
1A 0,2 f1 x 0f A lim f (x),f (2) ,2ln 2 2 . 2A 2, f2 xf A lim f (x),f (2) ,2ln 2 2 .
: 1 2f A f A f A ,2ln 2 2
x 0lim f (x)
f x 0 yy.
xlim f (x)
. : f (x) 2ln x x 2ln x 1,
x x x
x
f (x)lim 1x
xlim 2ln x x x .
. 3.
2
2
x 22ln 2x 1x 2x 3
2 22 ln x 2 2ln x 2x 3 2x 1 2 22 ln x 2 x 2 2 22 ln x 2x 3 x 2x 3
2 2f x 2 f x 2x 3 (1) 2x 2 2 22x 2x 3 x 1 2 2 2, f 1-1, :
2 2(1) f x 2 f x 2x 3 2 2 1x 2 x 2x 3 x
2
4. :
, i) 2 ln 2 2: , 0,2 2, , 1 f A 2 f A x 2 . ii) 2 ln 2 2: , 2 iii) 2 ln 2 2, f A . R :
5. 0 0 0( , 2 ln )M x x x . () : 0 0 0 0 0x 2ln x xd M, 2 x ln x2
= 0 02 x lnx , x x 1 lnx x>0.
: d(x) 2 x ln x , x 0 . d 0, 1d (x) 2 1
x .
d (x) 0 0 x 1 . d
0,1 . d (x) 0 x 1 . d
1, . d 1, 1 2. fC 1,1
-
_______________________________ _____________________________
92 .4/57
1. f
2x 1
2
e e x kf(x)
ln(x 1) x k
f
. k=0 . f . f 1 fC 0x 1 . x 1 f(x)+1x+ln2 . fC 1fC . : f R
, 1f x f x f x x
( 1f f R ). . f 0x k .
2
2 2
x 1
x k x k x k
2 k 1 2 k 1
x k
lim lim lim e
lim ln(x 1) e e ln(k 1) e e
f (x) f (x) f (k) (e )
g(k)= 22 k 1ln(k 1) e e . g(0)=0 g(k)= 22kk 1 2k
2k 1e
=2k( 21
k 1
2k 1e ) . k>0 g(k) >0 k0 (0, ) 0 f . f 11 . x
-
_______________________________ _____________________________
92 .4/58
f (x) 1 x ln 2 x1. . f
f1(x)=f(x) f(x)= x x0. A h(x)
. h (,0) h((,0))=(
x x 0lim h (x), lim h (x) )=
=(, 1). 0 ( ,1) ,
-
_______________________________ _____________________________
92 .4/59
, x 0 0, x
, . , . (1)
, (1),(2), , , C, C , C, C.
.
1. : 3 2
2
x 6x 9x 2, x 3f (x)
x 4x 5, x 3
. f. . f(x) = 0. : . f ( x = 3 ). f . (-,3) : 2f (x) 3x 12x 9
f (x) 0 x ,1 f (x) 0 x 1,3 (3,+) f (x) 2x 4 2 x 2 0, . f (-,1], [1,3] [3,+), f :
x xf (IR) ( lim f (x),f (1)] [f (3),f (1)] [f (3), lim f (x))
( ,6] [2, 6) [2, ) ( , ) IR
. 0 f (-, 1] f . f(x) = 0 . :
23x 12x 9, x 3f (x)
2x 4, x 3
f x0=3. f x0=3. ( f x=3 ). 2. f IR Cf (0, 0) y = 2x 5, :
x
0
x 0
f (t)dtlim
1 x
: 00
de L Hospital, , :
x
0
x 0 x 0
f (t)dtf (x) 0lim lim
1 x x 0
de L Hospital, x 0 , f 0.
x 0 x 0 x 0
f (x) f (0)f (x)f (x) f (0)x 0xlim lim lim 2x xx 1
x x
x
0
x 0
f (t)dtlim 2
1 x
.
3. f:(0, +)IR f (1)=1 x>0 :
x1
f (t)f (x) x dt 1t
. f. : f , , : f (x)1 f (x) 1
x 2xf (x) f (x) 1x x
'
'f (x) (ln x)x
f (x) ln x c
x
x>0 (2). f(1)=1, : f 12 ln1 c c 1.
1
f x lnx 1 f x x lnx 1x
x>0. : f (1). :
0 0
g x h x 1g x h x
2g x h x
:
1 g x h x c (3) 2 30 0g x h x c c 0 g x h x . . ( .. , ).
-
_______________________________ _____________________________
92 .4/60
. N.
1. f : , f(1)=1
x1
g(x) f (t) t dt . A. 1) g Rolle [1 1]. 2) g (x 1 lnx) 0 (1). 3) 2g (2x 2 lnx ) 0 . 4)
1
1
5g(t)dt3
. B. G :[ 1,1] G(x)=g (x) . 1) G, 1G . 2) 2x 1
x 0lim e 1 ln|x| G (x) .
: 1. f . : i) f , f
. ii) f , f
( ). 2. f . : i) f f (x)dx 0 . ii) f 0
0f (x)dx 2 f (x)dx 2 f (x)dx
,
. ( t
tg(t) f (x)dx
t t -t 0h(t) f(x)dx-2 f(x)dx , ,
g (t) h (t) 0 g(0) h(0) 0 . g(t) h(t) 0 t . g( ) h( ) 0 . t . 1) Dg=. g [1,1], (1,1),
11
g( 1) f (t) t dt 0
.
g(1)=g(1). 1 g(1)=g(1), g .
: g (x) f (x) x g ( x) f ( x) x f (x) x f (x) x g (x) .
g . g . 2 : 1
1g( 1) f (t) t dt 0
.
11
g(1) f (t) t dt 0 h t f (t) t . : t, : t h(t)=f(t)+t=f(t)+t=(f(t)t)=h(t). 2) : g (-x)=-g (x) .
x=0 g (0)=-g (0) g (0)=0 , : g (x 1 ln x) 0 g (x 1 ln x) g (0) (2). (2) g . :
H f f(1)=1, f(1)= 1 , , f .
g (x) f (x) x , : x1f(x2) x1>x2 1 1 2 2 1 2f (x ) x f (x ) x g (x ) g (x ) . g . : 2 x 1 ln x 0 ln x x 1 , x=1. ( ln x x 1 x>0. x=1).
3) : 2g (2x 2 ln x ) 0 2 2g (2x 2 ln x ) g (0) 2x 2 ln x 0
2ln x 2x 2 2ln | x | 2x 2 ln | x | x 1 .
x>0, ln | x | x 1 lnx x 1 x 1 . x0. 2 2 2 2 2
1 1 1 1 1ln 1 2 1 1 0e e e e e
211, e ,
Bolzano o 2
1x 1,e
.
, xo .
-
_______________________________ _____________________________
92 .4/61
2g (2x 2 ln x ) 0 . 4) (1) g ,
1 1
1 0g(t)dt 2 g(t)dt
.
:1 1
0 0g(t)dt (t) g(t)dt
1 1100 0
tg(t) tg (t)dt 0 t f (t) t dt 1 2
0t tf (t) dt 1 120 0t dt tf (t)dt
13 1 1
0 00
t 1tf (t)dt tf (t)dt3 3
, : 1 1
1 0
2g(t)dt 2 tf (t)dt3
. 1
0
2 52 tf (t)dt ,3 3 10 1tf (t)dt 2
, 0 t1 f [0, 1], : f(t)f(1) f(t)1 tf(t)t. t= 0 t=1,
1 1
0 0tf (t) t tf (t)dt tdt
121 1
0 00
t 1tf (t )dt tf ( t)dt .2 2
1) (1) g R, G =[1, 1]. G. 1G G. G( ) g ( ) . g =[1, 1], g ( ) [g (1),g ( 1)] = [f(1) 1,f( 1) 1] 2,2 . : 1GD 2,2 . T 1G G, =[1, 1]. 2) 2x
x 0lim e 1 0 x 0lim ln | x | . 2 x2 x
x 0 x 0
e 1lim e 1 ln | x | lim x ln | x | .x
2x2x
2x
x 0 x 0 x 0
e 1e 1: lim lim lim 2e 2x x
.
x 0 x 0 x 0
ln | x |ln | x | lim x ln | x | lim lim1 1x
x
x 0 x 0
2
1xlim lim( x) 01x
. : 2x
x 0
e 1lim xln|x|x
2 xx 0 x 0
e 1lim lim x ln | x | 2 0 0x .
: 2xx 0lim e 1 ln|x| 0 . 1GD 2,2
G1 =[1,1], 1Gx D = -2, 2
11 G (x) 1 1G (x) 1 : 2x 1 2x 1 2xe 1 ln|x| G (x) e 1 ln|x| G (x) e 1 ln|x| 2x 1 2xe 1 ln | x | G (x) e 1 ln | x | . 2x 2x 1 2xe 1 ln|x| e 1 ln|x| G (x) e 1 ln|x|
: 2x 2xx 0 x 0lim e 1 ln| x| 0 lim e 1 ln|