icosidodecahedron9

To {Eikosidwdekedron} parousizei jmata pou qoun suzhthje ston isttopo http://www.mathematica.gr.H epilog kai h frontda tou perieqomnou gnetai ap tou Epimelht tou http://www.mathematica.gr.Metatrop se LATEX: Fwtein Kald, Anastsh Kotrnh, Leutrh Prwtopap, Aqilla Sunefakpouo , Sqmata: Miqlh Nnno, Qrsto Tsifkh Seli-dopohsh: Anastsh Kotrnh, Nko Mauroginnh, Exfullo: Grhgrh Kwstko. Stoiqeiojetetai me to LATEX.Mpore na anaparaqje kai na dianemhje elejera.

Eikosidwekedro filoteqnhmno ap ton Leonardo da Vinci

To eikosidwdekedro enai na poledro (32-edro) me ekosi trigwnik dre kai ddeka pentagwnik. 'Eqei 30 panomoitupe koruf st opoe sunantntai do

trgwna kai do pentgwna kai exnta se akm pou h kje ma tou qwrzei na trgwno ap na pentgwno. Enai arqimdeio stere - dhlad na hmikanonik

kurt poledro pou do perissteroi tpoi polugnwn sunantntai me ton dio trpo sti koruf tou - kai eidiktera enai to na ap ta do oiwne kanonik

- quasiregular poledra pou uprqoun, dhlad stere pou mpore na qei do tpou edrn oi opoe enallssontai sthn koin koruf (To llo enai to kubo -

oktedro). To eikosidwdekedro qei eikosiedrik summetra kai oi suntetagmne twn korufn en eikosadrou me monadiae akm enai oi kuklik metajsei

twn (0, 0,),(

12,

2,

1+2

), pou o qrus lgo

1+

52

en to duadik tou poledro enai to rombik triakontedro.

Phg:http://en.wikipedia.org/wiki/Icosidodecahedron

Apdosh: Pno Giannpoulo

O diktuak tpo mathematica.gr ankei kai dieujnetai smfwna me ton kanonism tou pou uprqei sthn arqiktou selda (http://www.mathematica.gr) ap omda Dieujunntwn Meln.

Dieujnonta Mlh tou mathematica.gr

Suntoniste

Airet Mlh

1. Fwtein Kald (Fwtein) Genik Suntonstria

2. Miqlh Lmprou (Mihalis Lambrou) Genik Sun-

tonist

3. Nko Mauroginnh (nsmavrogiannis) Genik Sun-

tonist

4. Spro Kardamtsh (Kardamtsh Spro)

Upejuno Enhmrwsh

5. Qrsto Kuriaz (chris gatos)

Upejuno Programmatismo

6. Mlto Papagrhgorkh (m.papagrigorakis)

Upejuno Oikonomikn

7. Girgo Rzo (Girgo Rzo)

Upejuno Ekdsewn

Mnima Mlh

1. Grhgrh Kwstko (grigkost) Diaqeirist

2. Alxandro Sugkelkh (cretanman) Diaqeirist

Epimelhte

1. Strth Antwna (stranton)

2. Andra Barberkh (ANDREAS BARBERAKHS)

3. Kwnstantno Btta (vittasko)

4. Nko Katsph (nkatsipis)

5. Anastsio Kotrnh (Kotrnh Anastsio)

6. Jno Mgko (matha)

7. Girgo Mpalglou (gbaloglou)

8. Rodlfo Mprh (R BORIS)

9. Miqlh Nnno (Miqlh Nnno)

10. Leutrh Prwtotopap (Prwtopap Leutrh)

11. Dhmtrh Skoutrh (dement)

12. Mpmph Stergou (Mpmph Stergou)

13. Swtrh Stgia (swsto)

14. Aqilla Sunefakpoulo (achilleas)

15. Kwnstantno Thlgrafo (Thlgrafo Ksta)

16. Serafem Tsiplh (Serafem)

17. Qrsto Tsifkh (xr.tsif)

18. Dhmtrh Qristofdh (Demetres)

Melh

1. Spro Basilpoulo (spyros)

2. Ksta Zugorh (kostas.zig)

3. Girgh Kalajkh (exdx)

4. Qrsto Kardsh (QRHSTOS KARDASHS)

5. Jansh Mpelhginnh (mathfinder)

6. Jwm Rakftsalh (Jwm Rakftsalh)

7. Kwnstantno Rekomh (rek2)

8. Girgo Rodpoulo (hsiodos)

9. Staro Staurpoulo (Staro Staurpoulo)

10. Baslh Stefandh (bilstef)

1 ( ) 100 , 99% .

98% .

2 ( )

() -

( )

,

...

6210001000.

.

3 ( )

B, B, A.

;

4 ( )

A = 62006 + 32003 + 182001 + 92005

30.

5 ( ) .

AB 6m 4m. 9m .

()

.

() .

6 ( KARKAR) A , B , ,

.

A : ! B : ! . :

!

( -

) B

7 ( )

:

P(x) = x15 2012x14 + 2012x13 ... 2012x2 + 2012x. P(2011).

8 ( ) ABC A = 900

ab, bc, (a + c)(a c) .

,

9 ( )

m (m 1)x4 5x2 + 3m 2 = 0 .

1

10 ( ) :

x = 111...12

y = 111...1+1

z = 666...6

,

: x + y + z + 8 .

,

11 ( )

AB O . A O E, Z EO 24, Z.

12 ( KARKAR) S CD ABCD. BS AD T . AM,T M,CN, S N . MDN = A = .

,

13 ( )

() 1 = 4

+1 =8 9 + 2

N.

()

=1

3 N .

(). - ().

14 ( KARKAR) : 2x = 3a = 6b ,

: x =ab

a b .

,

15 ( )

(O,R) , (K, r) E. AB, , OK AB .

.

16 ( )

ABCD AC, BD CAB,BCA,CDB,BDA 70, 30, (50 a), a , , . BD CBA a.

,

17 ( KARKAR) ,

A(1, 3) B(4, 2), .

18 ( ) n,

n 2, :

1 +12+

13+ + 1

n>

n.

,

19 ( )

f (x) = x x + 1 x 0 = {1, 2, . . . , n}, P(k) = 12

5f (k)

k .

2

() f .() n.

()

x = n. 16% 33, .

20 ( Parmenides51)

lit

< , R. 1, 6 lit 20% 1, 4 lit 90% .

[, ] :() , .

() .

()

,

1000 2 lit.

, ,

21 ( )

f : R R z z , 1/2

f 2(x) + sin2(x) = 2x f (x) x R

limx0

f (x)x

= m,

m = |z2||2z1| .

() |z 2| = |2z 1|.() z

() limx0

f (sin x)x2 x .

() g(x) = f (x) x (,0) (0,+).

() f .() (|z + 3 4i| + 5)x = x3 + 10

[1, 2].

22

z = (k t) + (k t)i t R k > 1.

(1) z.

(2) w

y = x (k 1) , k |z w|min = 52

2 1.

(3) k (2) z |z z|.

(4) k (2) |w 3 + 4i|.

(5) u

u = (1 + mt) + (1 + mt)i, m

u .

(6) k,m (2) (5) , |z u|.

, , ,

23 ( )

f (x) = ex ln x + 2 1 1, ex

21 ln(x2 + 1) = 1e,

e1x + 2 ln x > ex.

24 ( ) f

h(x) = e f (x) f 3(x) + 2 .

f (x) :(12

) f (x2x) ( 12

) f (4x)> 0

, ,

25 ( )

f [a, b], (a, b) f (a) , f (b). 1, 2 (a, b) 1 , 2 : f (1) f (2) =

( f (b) f (a)b a

)2.

26 ( )

f : [1, e] R f (1) = 0, f (e) = 1 f (x) + e f (x) = x + 1

x, x [1, e] . :

f (x) = ln x x [1, e] .

, ,

27 ( ) :2

0

x10 + 210

x15 + 215dx < 181

2816.

28 ( )

f : R R :( f (x))2 + ( f (x))2 x2 + 1, x R.

:

1. ( f (1))2 ( f (0))2 43,

2. | f (1)| 43,

3. |F (1) F (0)| 109, F f .

3

, ,

29 ( )

f (x), g(x) : [a, b] R g(x) > 0,x [a, b].

(x a) f (x) = (x b)g(x),x [a, b] 1)

g(x) > 0 f (x) < 0, x (a, b)

2) b

af (x)dx = b

ag(x)dx

) f (b) = g(a)) (a, b) : f () = g()

30 ( parmenides51)

z1 = a + eai, z2 = b + bi, z3 = c + i ln c a, b R, c > 0.

1.

z1, z2, z3,

2. |z1 z2 | - z1, z2 ,

3. |z1 z3| - z1, z3 .

, ,

31 ( )

f : [0,+) R

f (x) + x f (x) = 12

f ( x2

) x [0,+) .

32 ( )

f (x) = ln(x2)

x2+1, N , N ,

.

33 ( )

f : [0, 1] R

f (0) = 0 10

e f(x) f (x) dx = f (1).

f (x) = 0,x [0, 1]

34 ( )

f : (0,+) R f

(x

y

)=

f (x)y

x f (y)y2

x, y (0,+) 1, f (1) = 1

Juniors,- -

35 ( )

n,

sin2n x + cos2n x + n sin2 x cos2 x = 1, x R.

36 ( ) a, b, c a + b + c = 3,

a2

(b + c)3 +b2

(c + a)3 +c2

(a + b)3 3

8.

Juniors,

37 ( )

ABC BC, I . BI AC D E, D CI. EI AB Z. DZ CI.

38 ( )

ABC E M, .

M, ABC ,

E1, E2 , E3. 1

E1+

1

E2+

1

E3 18

E.

Seniors,- -

39 ( )

f :(x y) f (x + y) (x + y) f (x y) = 4xy(x2 y2)

4

40 ( ) -

f : {1, 2, . . . , 10} {1, 2, . . . , 100} x + y|x f (x) + y f (y), x, y {1, 2, . . . , 10}.

Seniors,

41 ( )

ABCD (O) P AC BD.

ABCD Q,

QAB + QCB = QBC + QDC = 90o.

P, Q, O , O (O).

42 ( )

P ABCD, AB, BC, CD, DA, . ABCD -.

43 ( - 1

1995) 6-

5. ,

6 -

44 ( )

ABC AB AC D E , DE

. :ADDB

+AEEC

= 1

45 ( IMC 1996) n

.

sin nx(1 + 2x) sin x dx.

46 ( ) (an)nN an >

1n

n.

+n=1

an .

47 ( ) M = R \ {3} x y = 3(xy3x3y)+m, m R. m (M, ) .

48 (vzf) m n n .

, n , -

.

49 ( )

ln(1 aix)x2 + m

dx , a > 0, m > 0

50 ( ) an = 1 +n

j=21

ln j lim( nn)an

51 ( ) An = {n, 2n, 3n, ...} n N

J N, iJ

Ai

52 ( )

A, B AB = (A \ B) (B \ A), - .

A1A2...An Ai

53 ( ) p Pp > 5 (p 1)! + 1 p( pk k N).

5

54 ( dimtsig) -

.

55 ( ) S ,

s1, s2, ..., sn s1+s2+...+sn b ab > b2, a > c ab > bc, ab.

a2b2, b2c2, b4

a2b2 = (b2 + c2)b2 = b4 + c2b2, , .

10

:

9 ( ) -

m

(m 1)x4 5x2 + 3m 2 = 0 - .

http://www.mathematica.gr/forum/viewtopic.php?f=19&t=24722

( ) m = 1

: 5x2 + 1 = 0 x = 15.

.

m , 1 x2 = z

(m 1)z2 5z + 3m 2 = 0

(z) = (5)2 4(m 1)(3m 2)= 25 4(3m2 5m + 2)= 25 12m2 + 20m 8= 12m2 + 20m + 17 0 12m2 20m 17 0 (1)

(m) = (20)2 4(17)12 = 400 + 816 = 1216

m1 =20 +

1216

24=

5 + 219

6,

m2 =20

1216

24=

5 219

6.

(1) 5 219

6 m 5 + 2

19

6.

, ,

. Vieta :

z1 + z2 = ba=

5

m 1 > 0 m > 1

z1z2 =

a=

3m 2m 1 0 3m 2 0

3m 2

m 23

m > 1

1 < m 5 +19

6.

10 ( ) :

x = 111...12

y = 111...1+1

z = 666...6

,

: x + y + z + 8 .


( ) :

x = 111 1 2

= 1 100 + 1 101 + 1 102 + + 1 1021

=102 1

9

2

a1 = 1 = 10.

:

y = 111 1 +1

= 1 100 + 1 101 + 1 102 + + 1 10

=10+1 1

9

=10 10 1

9

11

z = 666 6

= 6 100 + 6 101 + 6 102 + + 6 101

= 6 10 19

=6 10 6

9.

x + y + z + 8 =102 1

9+10 10 1

9+6 10 6

9+ 8

x + y + z + 8 =102 + 16 10 + 64

9

x + y + z + 8 =(10 + 8

3

)2.

12

:

11 ( )

AB O -

. A O

E, Z EO 24,

Z.


( ) ON//Z.

AZ A = 45,

ZA = = 22, 5.

AEO = 67, 5 (1) ,

AOE.

ENO = EAB = 22, 5 + 45 = 67, 5 (2) ,

- . (1) , (2) ONE (OE = ON = 24). O A ZA

Z .

Z = x = 2ON = 48.

2 ( )

.

A x

a x =A

A=

2, x =

2

1 +2

a.

,

AO OE =1

2

2

1 +2

a

, x = 2OE = 48. 12 ( KARKAR) S CD ABCD. BS AD T . AM, T M,CN, S N . MDN = A = .

13


(Antonis-Z) BM I BM, DC. BN AD E. N, M , BN

S BC = S BN + NBC = 2x BM ABS = ABM +MBT = 2y.

DCN = MAD = x + y. MABE , ABM =AEM = y.

BEA = EBC = x

. IBCN , CBN = CIN = x, DIN = NED DIEN . DIB = ABI = y, DIB = DEM, MIDE .

DIEN, MIDE MDNE ,

MDN = = 180 x y = 180 B2= 90 +

2.

14

:

13 ( )

() 1 = 4

+1 =8 9 + 2

N. ()

=1

3 N .

(). - ().


( )

:

v+1 =1

av+1 3=

18av9av+2

3

=1

8av93(av+2)av+2

=av + 2

5av 15

=av + 2

5(av 3), v N, :

v+1 v =av + 2

5(av 3) 1

av 3=

av + 2 55(av 3)

v+1 v =av 3

5(av 3) =1

5:

. ,

1 =1

a1 3= 1 v = 1 + (v 1) 1

5=

v + 4

5, v N.

:

S v =v

2(1 + v) = v

2

(1 +

v + 4

5

)=

v

2

v + 9

5=

v2 + 9v

10.

:

v =1

av 3 v + 4

5=

1

av 3

av 3 =5

v + 4 av = 3 +

5

v + 4 av =

3v + 17

v + 4, v N.

14 ( KARKAR) : 2x =3a = 6b , : x =

aba b .


1 ( f reyia)x ln 2 = a ln 3 = b ln 6 = b ln(2.3) = b ln 2 + b ln 3

x ln 2 = a ln 3

(a b) ln 3 = b ln 2 ln 2 = a bb ln 3. :

x ln 2 = a ln 3 xa bb ln 3 = a ln 3 x =ab

a b

2 ( )

2x = 3a = 6b = m

x =m

ln 2 , a =m

ln 3 , b =m

ln 6 , a , b

ab = m2

ln 3 ln 6, a b = m ln 2

ln 3 ln 6

aba b =

m

ln 2 = x.

3 ( )

2x = 3a = 6b x ln 2 = a ln 3 = b ln 6ab

a b =x ln 2ln 3 x ln 2ln 6

x ln 2ln 3 x ln 2ln 6

=x2 ln2 2

x ln 2 ln 6 x ln 2 ln 3 =

15

x ln2 2

ln 2 (ln 6 ln 2) = x ln2 2

ln 2 ln 2 = x

4 ( )

3 = 2xa , 6 = 2

xb 3 = 2 xbb x

a=

x bb x =

aba b .

5 ( ) x = a = b = 0 - .

a, b, x , 0 a , b, {a = b3a = 6b

3a = 6a .

16

:

15 ( )

(O,R) , (K, r) E. AB, , OK - AB .

.

http://www.mathematica.gr/forum/viewtopic.php?f=22&p=127551#p127551

1 (KARKAR) R2 + r2 AL = CL . E = E

1

2E

1

2E E E

=1

2(R2 + r2) E E .

E !

2 ( ) LA = LC =R2 + r2 , (1) ,

L LA = LB. , (1)

(

R2 + (LO)2 = (LA)2 = (LC)2 = r2 + (LK)2 R + r = OK = LO + LK ).

16 ( )

ABCD AC, BD CAB, BCA,CDB, BDA 70, 30, (50a), a , , . BD CBA a.


1 ( ) :

M AC BD,C BD = A BD = x,C MB = 70 + x

x = 40o

-

ABC, BCD, CDA, DAB

sin 70sin 30

=sin(50 a) sin(70 a)

sin a sin(60 + a) sin 70sin 30 =

cos 20 cos(120 2a)cos 60 cos(60 + 2a)

2 cos 20 =cos 20 + cos(60 + 2a)

12 cos(60 + 2a)

cos(60 + 2a) = 0 a = 15o

2 ( ) ABC (70, 80, 30).

17

C BD, BA E. BCE (80, 50, 50) ( - ),

BCDE AEDC (CEA = CDA = 50). ADE = ACE = 20, DCE (70, 55, 55) BCD : a = 15.

3 ( ) A BD CB E ADCE( 50 ),

DEB, DAB a = 15.

18

:

17 ( KARKAR) ,

A(1, 3) B(4, 2), .


1 ( ) > 0

C(, ), D(,). AC, BD :x + y = 2 x y = 2, . A B : + 3 = 2 4 2 = 2 . =

2

2 =

2

2,

:

2 + 2 = 2 . . . 2 = 2, c x2 + y2 = 2.

2 ( )

a ()

r > 0 .

(AC) : y 3 = a(x 1) ax y a + 3 = 0 (BD) : y 2 = a(x 4) ax y 4a + 2 = 0.

, :

d(O, AC) = r |3 a|

a2 + 1= r

|3 a| = r

a2 + 1

d(O, BD) = r |2 4a|

a2 + 1= r

|2 4a| = r

a2 + 1.

|3 a| = |2 4a|, a = 1 a = 1

3.

a = 1 r =2,

C : x2 + y2 = 2 ( ) a = 13

r =10 C : x2 + y2 = 10 ( ).

,

.

3 ( ) y = x + , , R 1 A xx. 3 = 1 + = 3 .

yy K (0, 3 ). y = x + , , R 2

B 1. 2 = 4 + = 2 4.

yy L (0, 2 4).

O R C, D , - KON, LOD , CO = DO = R, CON = DOL ( ), OK = OL |3 | = |2 4| . . .

( = 1 = 1

3

).

R = d (O, 1) = ...2 R =

10.

4 (parmenides51)

AC, BD - .

M AB.

19

M(1 + 4

2,3 + 2

2

)=

(5

2,5

2

).

OM AC, BD, AC, BD, AC BD.

xM =5

2, 0 = xO

OM =yM yOxM xO

=

52 052 0

= 1 ,

OM y 0 = 1(x 0) y = x. -

AC,OM, = d(AC,OM) = d(A,OM) = |yAxA |

(1)2+12=

|31|2=

22=2.

x2 + y2 = 2 =22= 2.

AC, BD

, AB ( C D). xA = 1 , 4 = xB

AB =yB yAxB xA

=2 34 1 =

1

3,

AB y 3 = 1

3(x 1) 3y 9 = x + 1 x + 3y = 10.

AB, = d(O, AB) = |0 + 0 10|

12 + 32=

1010=

10.

x2 + y2 = 2 = 10.

18 ( )

n, n 2, :

1 +12+

13+ + 1

n>

n.


1 ( ) k +

k 1 >

k k > 1 k = 1

.

1

k

k 1>

k,

1k>

k

k 1.,

1 +12+

13+ + 1

n>

1 +2

1 +

3

2 + +

n 1

n 2 + n

n 1 = n. 2 ( )

nk=1

1k> n n

n

k=1

1k=

n2n

n!.

n

2nn!

>

n,

n >nn!.

-

n2 >n(n + 1)

2= 1 + 2 + 3 + + n >

nn1 2 3 n = n n

n!,

.

3 ( )

1 +12+

13+ + 1

n+

1n + 1

>

n +1

n + 1,

>

n + 1.

(

) n + 1 n = 1

n + 1 +

n 0

.

()

P(1) + P(2) + + P(n) = 1

f (1) + f (2) + + f (n) = 512

f (x) x = 1, . . . , x =n

5

12=

(1

2 122

)+

(1

22 123

)+

+

(1

2

n 12

n + 1

)

=1

2 12

n + 1

n = 35.

() 16% 33 x s = 33 s = 2.

CV = sx 100% = 5, 71%

CV < 10% -.

20 ( Parmenides51)

lit < , R. 1, 6 lit 20% 1, 4 lit

90% . [, ] :

() , .

() .

()

,

1000

2 lit.


1 ( )

() X . A = {X 1, 6} B ={X 1, 4}. X , - -

[, ]. -:

P(A) = 1, 6 a a = 0, 2

P(B) = 1, 4 a = 0, 9.

= 1, 2 = 3, 2

21

()

a +

2= 2, 2.

()

= {X 2}

N()N() = P() =

2 1, 23, 2 1, 2 = 0, 4

N() = 1000 N() = 2500. 2 (parmenides51)

[, ], ,

.

() :

100% . 1, 6 20% .

1, 6 =

100

20. (1)

100% . 1, 4 90% .

1, 4 =

100

90. (2)

(1),(2)

= 1, 2 = 2, 2.

:

[, ] 100% [, 1.6] 20%

, [1.6, ] 80% . [1.4, ] 90% [1.4, 1.6] 10% . 1, 6 1, 4 = 0, 2 10% . 2 0, 2 = 0, 4 2 10% = 20% 1, 6 20%

1, 6 = 0, 4 = 1, 6 0, 4 = 1, 2.

10 0, 2 = 2 10 10 = 100% [, ] 100%

= 2 = + 2 = 1, 2 + 2 = 3, 2.

()

,

x = +

2=

1, 2 + 3, 2

2=

4, 4

2= 2, 2.

() 3, 2 1, 2 = 2 ( ).

2 1, 2 = 0, 8 1000 .

2

0, 8=

1000

=2000

0, 8= 2500

.

22

:

21 ( )

f : R R z z , 1/2

f 2(x) + sin2(x) = 2x f (x)

x R

limx0

f (x)x

= m,

m =|z2||2z1| .

() |z 2| = |2z 1|.

() z

() limx0

f (sin x)x2 x .

() g(x) = f (x)x

(, 0) (0,+).

() f .

() (|z+34i|+5)x = x3+ 10 [1, 2].

http://www.mathematica.gr/forum/viewtopic.php?p=111574#p111574

( )

() x , 0

f 2(x)x2

+sin2(x)

x2=

2 f (x)x

,

limx0

( f (x)

x

)2+

(sin x

x

)2 = limx0

2 f (x)x

,

m2 + 12 = 2m m = 1.

m = 1 |z2||2z1| = 1 |2z 1| = |z 2|.

()

|2z 1| = |z 2||2z 1|2 = |z 2|2(2z 1)(2z 1) = (z 2)(z 2)4zz 2z 2z + 1 = zz 2z 2z + 43zz = 3|z| = 1.

()

limx0

f (sin x)x2 x = limx0

f (sin x)sin x

sin xx

1

x 1= 1 1 (1) = 1.

limx0

f (sin x)sin x

sin x=ux0u0= lim

u0f (u)u

= 1.

()

f 2(x) + sin2 x = 2x f (x) f 2(x) 2x f (x) + x2 = x2 sin2 x( f (x) x)2 = x2 sin2 xg2(x) = x2 sin2 x.

| sin x| 6 |x|, x R x = 0 x , 0

x2 sin2 x , 0, g2(x) , 0. g (, 0) (0,+),

.

() 4 f .

(1) f (x) = x +

x2 sin2 x.(2) f (x) = x

x2 sin2 x.

(3) f (x) =x +

x2 sin2 x x > 0,

x

x2 sin2 x x < 0.

(4) f (x) =x +

x2 sin2 x x 6 0,

x

x2 sin2 x x > 0.

23

()

h(x) = (|z + 3 4i| + 5)x x3 10 x [1, 2]. h .

h(1) = |z + 3 4i| + 5 11= |z + 3 4i| 6 6 0

|z + 3 4i| 6 |z| + |3 4i| = 6.

h(2) = 2 |z + 3 4i| + 10 8 10= 2(|z + 3 4i| 4) > 0

|z + 3 4i| > ||z| |3 4i|| = 4. h(1)h(2) 6 0. h(1)h(2) < 0 Bolzano x0 (1, 2) h(x0) = 0.

(|z + 3 4i| + 5)x = x3 + 10 (1, 2). h(1)h(2) =0, x = 1 x = 2 .

(|z + 3 4i| + 5)x = x3 + 10 [1, 2].

22

z = (k t) + (k t)i t R k > 1.

(1) -

z.

(2) w

y = x (k 1) , k |z w|min = 5

2

2 1. (3) k (2) z |z z|.

(4) k (2) |w 3 + 4i|.

(5) u

u = (1 + mt) + (1 + mt)i, m

u

.

(6) k,m (2) (5) , |z u|.

http://www.mathematica.gr/forum/viewtopic.php?p=110457#p110457

( )

(1) z = x + yi, x, y R

x = k t, y = k t.

,

2(t) + 2(t) = 1 (x k)2 + (y k)2 = 1.

z A(k, k) 1 = 1 k > 1.

(2) d(A, ) = |3k1|2

> 1 k > 1, .

|z w|min = d(A, ) 1 = |3k 1|2

1

|z w|min =52

2 1 |3k 1|

2 1 = 5

2

2 1

|3k 1| = 5 k = 2,

k > 1.

(3) k = 2

z = (2 t) + (t 2)i.

z = + i, , R = 2 (t) = (t) 2

2(t) + 2(t) = 1 ( 2)2 + ( + 2)2 = 1.

24

z B(2,2) 2 = 1 = 1. : (1) k = 2

|z (2 + 2i)| = 1 |z (2 2i)| = 1 z B(2,2) 2 = 1 = 1. |z z| == |(2 t) + (2 + t)i (2 t) + (2 t)i|= |2(2 t)i| |z z|min = |2 1| = 2 |z z|max = |2 3| =6.

(4) |w 3 + 4i| = |w (3 4i)| w K(3,4). |w3+4i|min = d(K, ) = |34+1|

2= 0 (

K ).

(5) u = + i, , R.

(t) = 1 + m

, (t) = 1 + m

( m , 0 m = 0, u

(1,1)). ( + 1)2

m2+

( + 1)2m2

= 1 ( + 1)2 + ( + 1)2 = m2

u

(1,1) 3 = |m|. O(0, 0) (0 + 1)2 + (0 + 1)2 =m2 m2 = 2 m =

2.

:

w

m R

u = 0 + 0i { 1 + mt = 01 + mt = 0

t =1m

t = 1m

2t + 2t = 1 (1m

)2+

(1m

)2= 1

2m2= 1 m2 = 2 m =

2.

(6)

c =z u=(k t) + (k t)i (1 + mt) (1 + mt)i

=(k t) + (k t)i + (1 mt) + (1 mt)i=(k t + 1 mt) + (k t + 1 mt)i=[k + 1 (1 + m)t] + [(k + 1 (1 + m)t)] i

c = x + yi, x, y R

x = k + 1 (1 + m)t x (k + 1) = (1 + m)t,

y = k+ 1 (1+m)t y (k + 1) = (1+m)t.

(x (k + 1))2 + (y (k + 1))2=(1 + m)22t + (1 + m)22t

(x (k + 1))2 + (y (k + 1))2 = (1 + m)2k=2

m=2

(x 3)2 + (y 3)2 = (1 2)2

M(x, y)

(x 3)2 + (y 3)2 =(1 +

2)2.

25

OK, A, B. (OK) = d(K,O) =

32 + 32 =

32. OB

(OM) (OA). |C|max = (OA) = (OK) + = =

= 32 + 1 +

2 = 4

2 + 1

|C|min = (OB) = (OK) + = == 3

2 1

2 = 2

2 1.

M(x, y) (x 3)2 + (y 3)2 =

(2 1

)2.

OK, A, B. (OK) = d(K,O) =

32 + 32 =

32. OB

(OM) (OA).

|C|max = (OA) = (OK) + = == 3

2 +

2 1 = 4

2 1

|C|min = (OB) = (OK) + = == 3

2

2 + 1 = 2

2 + 1.

26

:

23 ( )

f (x) = ex ln x + 2 1 1,

ex21 ln(x2 + 1) = 1

e,

e1x + 2 ln x > ex.


( ).

) x > 0.

0 < x1 < x2 x1 > x2 ex1 > ex2 (1)

(y = ex . )

0 < x1 < x2

ln x1 < ln x2 ln x1 + 2 > ln x2 + 2(2)

(y = ln x . )

(1) , (2) f (x1) > f (x2) f 1 1

) ex21 ln

(x2 + 1

)=

1e

e(x2+1) ln

(x2 + 1

)+ 2 = 1

e+ 2

f(x2 + 1

)= f (1) f :11

x2 + 1 = 1 x = 0 x > 0, .

) e1x + 2 ln x > ex

e1x + ln x + 2 > ex ln x + 2

f(1x

)> f (x) 1

x< x

x>0 x2 > 1 x > 1

24 ( )

f

h(x) = e f (x) f 3(x) + 2 .

f (x) :

(12

) f (x2x) ( 12) f (4x) > 0http://www.mathematica.gr/forum/viewtopic.php?f=52&t=12656

( ) h R, x1, x2 R x1 > x2

:h(x1) > h(x2)

e f (x1) f 3(x1) + 2 > e f (x2) f 3(x2) + 2

( f (x1) f (x2))( f 2(x1) + f (x1) f (x2) + f 2(x2))

f (x2) () ()

, .

: f (x1) < f (x2), f R.

27

g(x) = x, 0 < < 1 R,

:

(12

) f (x2x)>(12

) f (4x) f (x2 x) < f (4 x) x2 x > 4 x x2 > 4 x < 2 x > 2.

:

h R : x < y h(x) > h(y), x, y R.

* .

* .

- x = y, h(x) = h(y), .

- x > y h , h(x) < h(y), .

x < y.

2 ( ) ( )

f

. .

f .. x1, x2 R x1 < x2 f (x1) 6 f (x2).

e f (x1) > e f (x2) f 3(x1) > f 3(x2) h(x1) 2 > h(x2) 2 h . .

x1, x2 R x1 < x2

f (x1) > f (x2), f . .

28

:

25 ( )

f [a, b], (a, b) f (a) , f (b). 1, 2 (a, b) 1 , 2 : f (1) f (2) =

( f (b) f (a)b a

)2.


1 ( )

a < x < b f (x) f (a)

x a =f (b) f (a)

b a := ,

f (b) f (x)b x =

-

[a, x] [x, b].,

> 0

f (x) f (a)x a >

f (b) f (a)b a >

f (b) f (x)b x ,

a < x < b.

g(x) := f (x) f (a)x a) (a, b]

[, c) ( ) ,

h(x) := f (x) f (b)x b [a, b)

(d, ] ( ).

k > 1 g() = k h() = /k , (a, b). [a, ] 1 (a, ) f (1) = g(). [, b] 2 (, b) f (2) = h().

k .

:

f [a, b], (a, b) f (a) , f (b). 1, 2, . . . , n (a, b)(n N)

f (1) f (2) f (n) =( f (b) f (a)

b a

)n.

2 ( )

f

f (b) f (a)b a = f

() > 0. U y : f (x) > 0, f (x) U . ln f ()

1, 2 U : ln f () = ln f(1) + ln f (2)

2, f (1) > 0, f (2) > 0

(1, 2) = V . ,

1 < < 2.

3 ( )

1 (a, k), 2 (k, b), :

f (1) = f (k) f (a)k a (1)

f (2) = f (b) f (k)b k (2)

f (1) f (2) =( f (b) f (a)

b a

)2,

(1) (2),

:

f (k) f (a)k a

f (b) f (k)b k =

( f (b) f (a)b a

)2(3)

- =f (b) f (a)

b a , (3) k,

29

:

f (k) f (a)k a

f (b) f (k)b k =

2

( f (k) f (a)) ( f (b) f (k)) = (k a) (b k) ()

k (a, b), :

f (k) f (a) = (b k) (4)

f (b) f (k) = (k a) (5).

k (4). (5) ...

(4):

f (b) f (k) = f (b) [ f (a) + (b k)]= f (b) f (a) (b k)= [b a b + k] == (k a) (2)

(5)

k. (4), ( )

h(x) = f(x) f(a) (b x), x [a, b]

Bolzano

h(a) = (b a) = f (a) f (b)h(b) = f (b) f (a) 0

= f (b) f (a).

:

h(a) h(b) = ( f (a) f (b))2 < 0, f (a) , f (b). .Bolzano h [a, b]

k (a, b) : h(k) = 0,

(4)

(5).

,

[a, k], [k, b], ,

.

(atemlos)

..

26 ( )

f : [1, e] R f (1) = 0,f (e) = 1 f (x) + e f (x) = x + 1

x, x [1, e] .

: f (x) = ln x x [1, e] .


1 ( )

h(x) = f (x) lnx, x [1, e],

( ) -

.

x0 (1, e) h(x0) , 0, , , f (x0) > 0, (

) x1

[1, e] ( ), Fermat

h(x1) = 0 f (x1) = 1x1,

x = x1

e f (x1) = x1 f (x1) = lnx1 h(x1) = 0,

.

2 ( )

,

g(x) = f (x) lnx [1, e].

g [1, e] .

: g(1) = g(e) = 0 :

g(x) = x(eg(x) 1), x [1, e] (1)

g .

.

30

, x1 ,

g(x1) > 0 .Fermat

g(x1) = 0.

, (1) . . ; ...

x2, (1) Fermat g(x2) = 0. .

g . g [1, e]...

g(x) = 0, x [1, e] f (x) = lnxx [1, e].

3 ( )

. -

:

g(x) = f (x) lnx g(x) = x xe f (x) xeg(x) + (eg(x)) = x

h(x) = eg(x), xh(x) + h(x) = x,

ex2

2

(e x2

2 h(x)) = xe x2

2

e x2

2 h(x) = e x2

2 + c elnx f (x) = 1 + ce x2

2

x = 1, c = 0.

elnx f (x) = 1 f (x) = lnx.

.

4 ( ) x (0,+): f (x) + e f (x) = x + 1

x(1).

x+1

x g(x) = x

2

2+lnx, x > 0.

(1) :

e f (x) = (g(x) f (x)) e f (x) (g(x) f (x)) = 1. eg(x) = x e x

2

2 : eg(x) f (x) (g(x) f (x)) = xe x

2

2 , (eg(x) f (x)

)=

(e

x2

2

),

eg(x) f (x) = ex2

2 + c.

x = 1 c = 0

f (x) = lnx, x > 0.

5 ( ) h : (0,+) , h (x) = ef (x)

x.

h (1) = h (e) = 1 f (x) = h (x) + xh

(x)xh (x) =

1

x+

h (x)h (x) ,

1

x+

h (x)h (x) + xh (x) = x +

1

x

h (x) = xh (x) (1 h (x)) .

,

x = x0,

h (x0) = 1 h (x0) = 0 (.Fermat) .

h h (1) = h (e) = 1, . h (x0) = h (1) = h (e) = 1.

h (x) = 1, x [1, e] e f (x) = x f (x) = ln x,x [1, e].

6 ( )

x : lnx , f (x), lnx > f (x). f (x) 1

x= x e f (x) = elnx e f (x) = (lnx f (x))e

lnx, f (x) ex. lnx, f (x) [1, e] {1, M} e max{e, eM} = k ( ). ( f (x) lnx) = (lnx f (x))e (lnx f (x))k

(( f (x) lnx)ekx)) < 0 1 x e, 0 = (( f (e) lne)eke)) (( f (x) lnx)ekx)) (( f (1) lnx)ek)) = 0, f (x) = lnx . lnx < f (x), lnx = f (x),x [1, e].

7 ( ) f f + f e f = 1 1

x2

f = f (x +

1

xx f

)+ 1 1

x2=

31

= ( f )2 (x + 1/x) f + 1 1/x2 (

e f ) y = y2 (x +

1

x

)y + 1 1

x2, y = f (

Ricatti).

y =1

x+

1

u, ... u + u

(1

x x

)= 1,

(xe x2

2 y) = (ex2/2) y = 1

x+

a

xe

x2

2 .

f (1) = 1 y(1) = 1

a = 0 f (x) = lnx + b f (1) = 0 b = 0 f (x) = lnx, x [1, e] ( f (e) = 1)?

32

:

27 ( )

:

20

x10 + 210

x15 + 215dx < 181

2816.


( )

20

x10 + 210

x15 + 215dx = 1

32

20

1 + (x/2)101 + (x/2)15 dx

=1

16

10

1 + y10

1 + y15dy

=1

16

(1 +

10

y10 y151 + y15

dy)

(1 + x2

) (( f (x))2 + ( f (x))2)

>( f (x) + x f (x))2 (

1 + x2)2>( f (x) + x f (x))2 f (x) + x f (x) 6 1 + x2

(1 + x2

)6 (x f (x)) 6 1 + x2

10

(1 + x2

)dx 6

10

(x f (x))dx 61

0

(1 + x2

)dx

| f (1)| 6 43

3. -

t > 0

t0

(1 + x2

)dx 6

t0

(x f (x))dx 6t

0

(1 + x2

)dx

t t3

3 t f (t) t + t

3

3 1 t

2

3 f (t) 1 + t

2

3.

f t = 0 10

(1 + t

2

3

)dt 6

10

f (t)dt 610

(1 + t

2

3

)dx.

.

33

:

29 ( )

f (x), g(x) : [a, b] R g(x) > 0,x [a, b].

(x a) f (x) = (x b)g(x),x [a, b]

1)

g(x) > 0 f (x) < 0,x (a, b)

2) b

af (x)dx = b

ag(x)dx

) f (b) = g(a)) (a, b) : f () = g()


( )

1) g(x) > 0, x [a, b] g [a, b] x a

(x a) f (x) = (x b)g(x) 0 = (a b)g(a) g(a) = 0 x > a g(x) > g(a) = 0

(xa) f (x) = (xb)g(x) a < x < b g(x) > 0 x a > 0, x b < 0 f (x) < 0

:

g(a) = 0, f (b) = 0 [a, b] g(a) 0, f (b) 0

2) )

(x a) f (x) = (x b)g(x) b

a

(x a) f (x)dx =b

a

(x b)g(x)dx

[(x a) f (x)]ba b

a

f (x)dx =

[(x b)g(x)]ba b

a

g(x)dx

b

a

f (x)dx =b

a

g(x)dx

[(x a) f (x)]ba = [(x b)g(x)]ba

(b a) f (a) = (b a)g(b) f (a) = g(b)

2.)

h(x) = f (x)g(x), x [a, b] [a, b]

h(a) = f (a) g(a) = f (a) f (b) > 0

f ( (1)) [a, b] f (a) > f (b) h(b) = f (b) g(b) = g(a) g(b) < 0 (1) g [a, b] g(a) < g(b) h(a)h(b) < 0 Bolzano x0 (a, b)

h(x0) = 0 f (x0) = g(x0)

h(x) = f (x) g(x) < 0, x (a, b) h [a, b] x0 .

30 ( parmenides51)

z1 = a + eai, z2 = b + bi, z3 = c + i ln c

a, b R, c > 0.

1.

z1, z2, z3,

2. |z1 z2| z1, z2 ,

3. |z1 z3| z1, z3 .

34


( ) ) z1 z = x + yi, x, y R, x = a, y = ea, a R y = ex. z1 y = ex.

z2 z = x+yi, x, y R, x = b, y = b, b R y = x. z2 y = x.

z3 z = x+yi, x, y R, x = c, y = lnc, c > 0 y = lnx. z3 y = lnx, x > 0.

) z1 = k + eki, k R |z1 z2| ,

L(k, ek) y x = 0, : d(L, ) = |e

a a|12 + (1)2

.

f (x) = ex x R

ex () x () f (x) = ex 1. :* f (x) = 0 x = 0* f (x) > 0 x > 0, * f [0,+)

* f (, 0]* f x = 0 f (0) = 1. d(L, ) a =0 ( L(0, e0) L(0, 1)) 1

2=

2

2.

1. L : y 1 = 1(x 0) y = x + 1. y = x, y = x + 1 (x, y) =

(1

2,1

2

),

z2

2

2.

|z1 z2| z1 = i z2 =

1

2+

1

2i.

)

y = ex, y = lnx - y = x,

|z1 z2| ,|z2 z3| 2

2

z3 L(0, 1) y = x, N(1, 0), z2 M

(1

2,1

2

)( ()

).

|z1 z3| :z1 = i z3 = 1 |z1 z3| = 2

2

2=

2.

35

:

31 ( )

f :[0,+) R

f (x) + x f (x) = 12

f ( x2 ) x [0,+) .


( ) g(x) = x f (x) ,x [0,+) [0,+)

g(x) =1

2

g(x/2)x/2

=g(x/2)

x, x > 0

0 , x = 0(1) .

..... g [0, x/2] 0 < 1 < x/2 ,

1

2g(1) = g(x/2) g(0)

x=

g(x/2)x

(1)= g(x) .

..... g [0, 1/2] 0 < 2 < 1/2 ,

1

2g(2) = g(1/2) g(0)

1=

g(1/2)1

(1)= g(1) .

.....

n , n N , 0 < n+1 < n/2 .

g(x) = 12

g(1) = 122

g(2) = . . . = 12n

g(n) (2) .

limx0

g(x) = limx0

g(x/2)x

00= lim

x01

2

g(x/2)x

=

limx0

1

2g(x/2) = 1

2limx0

g(x) limx0

g(x) = 0 = g(0) . g [0,+) . n + n 0 , 0 < n+1 < n/2 0 < n < 1/2n1 . lim

n+g(n) = 0 (3) .

, x [0,+),

limn+

g(x) (2)= limn+

1

2ng(n) = lim

n+1

2nlim

n+g(n)

(3)= 0 0 = 0 g(x) = 0 g(x) = c x f (x) = c 0 f (0) = c , f (x) = 0 , x [0,+) .

2 ( ) f x1, x2 [0, ] , > 0 m M, , [0, ] .

m f (x) M , x [0, ] .

m f ( x2 ) M , x [0, ] .

(x f (x)) = 12

f ( x2 ) M2 (x f (x) Mx

2

) 0 . g(t) = t f (t) Mt

2, t [0, ] .

x > 0 [0, x] (0, x)

g() = g(x) g(0)x 0 = f (x)

M2.

g() 0 f (x) M2.

x = x1, x1 ( x1 = 0 ,

M = 0)

f (x1) M2

M M2

M 0 .

, (x f (x)) = 12

f ( x2 ) m2 (x f (x) mx

2

) 0 h(t) = t f (t) mt

2, t [0, ]

, f (x) m2.

x = x2, x2 ( x2 = 0 ,

m = 0)

f (x2) m2

m m2

m 0 . 0 m f (x) M 0 f (x) = 0 , x [0, ] . , [0,+). , .

36

32 ( )

f (x) = ln(x2)

x2+1, N , N ,

.


( ) f (x) = ln(x2)

x2+1,

f (x) = 2 (2 + 1) ln(x2)

x2+2

f (x) = 2(4 + 3) + (2 + 1)(2 + 2) ln(x2)

x2k+3,

x R , N , N . f f .

1.

f (

) .

f A (x1, f (x1)) , B (x2, f (x2)) , 0 < x1 < x2. f (x1) = f (x2) = . f [x1, x2] (x1, x2) , f () = f (x2) f (x1)

x2 x1= .

0 < x1 < < x2

f (x1) = f () = f (x2) . . Rolle f [x1, ], [, x2] f

. ,

f .

2. -

f () A(x1, f (x1)) , B(x2, f (x2))

y f (x1) = f (x1)(x x1) (1) . :

) f (x1) = f (x2) 1. f x2 = x1 . B (x1, f (x1)) = (x1, f (x1)) .( - - x1 > 0)

) B (1) f (x1) f (x1) = f (x1)(x1 x1) f (x1) = x1 f (x1) (2) ln(x21 )x2+11

= x12 (2 + 1) ln(x21 )

x2+21

ln(x21 ) = 2 (2 + 1) ln(x21 )

(2 + 2) ln(x21 ) = 2 ln(x21 ) =

+ 1

x21 = e

+1 x1 = e1

2+2 .

(1) :

(1) (2) y = f (x1) x f (x1) = e( + 1)

: y =

e( + 1) x .

.

37

:

33 ( )

f : [0, 1] R

f (0) = 0 10

e f(x) f (x) dx = f (1).

f (x) = 0,x [0, 1]


( ) 10

e f(x) f (x)dx = f (1) = f (1) f (0) =

10

f (x)dx

10

(e f (x) f (x) f (x))dx = 0 10

f (x)(e f (x) 1)dx = 0

, f (x), e f (x)1 .

x (0, 1), f (x)(e f (x) 1) 0

x0 [0, 1] f (x0) , 0

(. f (x0) < 0),

10

f (x)(e f (x) 1)dx > 0, .

, f (x) = 0,x [0, 1]

f (x) = c,x [0, 1].

f (0) = 0, f (x) = 0, x [0, 1] 34 ( )

f : (0,+) R f

(x

y

)=

f (x)y

x f (y)y2

x, y (0,+) 1, f (1) = 1


( ) y x f (1) = 0.

:

f (1) = limx1

f (x) f (1)x 1 = 1

limx1

f (x)x 1 = 1 (3)

x0 > 0.

x x0, h =x0

x 1.

:

L = limxx0

f (x) f (x0)x x0

= limh1

f(

x0

h

) f (x0)

x0

h x0.

( x = x0 y = h):

L = limh1

f (x0)h

x0 f (h)h2 f (x0) x0(1 h)

h ,

,

(1) :

L =f (x0)

x0+ 1.

x0 , f / .. :(

f (x) = f (x)x

+ 1,x > 0) f (x)x (x) f (x)x2 =

( f (x)x

)=

1

x= (lnx) ,x > 0

( f (x)x

= lnx + c,x > 0).

38

f (x) = xlnx + xc,x > 0 .

x 1 : c = 0.

f (x) = xlnx,x > 0

.

39

, , :

35 ( )

n,

sin2n x + cos2n x + n sin2 x cos2 x = 1, x R.


( ) x =

41

2n+

1

2n+

n

4= 1

2n(n 4) + 8 = 0. (*) n 4 2n(n 4) 0 2n(n 4) + 8 > 0 0 n < 4, n = 2 n = 3 (*).

sin4x + cos4x + 2sin2xcos2x = (sin2x + cos2x)2 = 1 x R Euler sin2 x + cos2 x + (1) = 0(sin2 x)3 + (cos2 x)3 + (1)3 = 3 sin2 x cos2 x(1), sin6 x + cos6 x + 3 sin2 x cos2 x = 1, x R., n 2 3.

36 ( )

a, b, c a + b + c = 3,

a2

(b + c)3 +b2

(c + a)3 +c2

(a + b)3 3

8.


1 (BillK) f (x) = x2

(3 x)3 , x (0, 3) f (x) > 0 . Jensen f (a) + f (b) + f (c) 3 f (a+b+c3 ) f (a) + f (b) + f (c) 38

a2

(b + c)3 +b2

(c + a)3 +c2

(a + b)3 3

8

2 ( )

CS Nesbitt :(a2

(b+c)3 +b2

(c+a)3 +c2

(a+b)3)

[(b + c) + (c + a) + (a + b)] (a

(b+c) +b

(c+a) +c

(a+b))2 94 .

3 ( )

a2 (a + b + c)(b + c)3 +

b2 (a + b + c)(c + a)3 +

c2 (a + b + c)(a + b)3

9

8

a3

(b + c)3 + a2

(b + c)2 9

8.

:

Chebyshev 9(x3 + y3 + z3) (x + y + z)3. Nesbitt

9 a3

(b + c)3 ( a

b + c

)3 27

8

3 a2

(b + c)2 ( a

b + c

)2 9

4

40

:

37 ( )

ABC BC, I . BI AC D E, D CI. EI AB Z. DZ CI.


(KARKAR) C, E BC CEI = CDI = 90o+ B

2= DIE = 90o , (1) (1)

A = 90o DIZA IDZ = IAZ = 45o , (2)

(2) ADB = 90o B2, ADZ =

C2= ACI , (3) (3) DZ CI

.

2 ( ) E, - D CI BC, FD = FE , (1) F CI DE. DIC = EIC , (2)

(2) DIC = 180o IDC C2

= 180o

90o B2

C2

= 45o, DIE = 90o =BD EZ , (3)

(3) = IE = IZ , (4) BD B.

(1), (4) DEZ, DZ FI CI .

38 ( )

ABC E M, .

M, ABC , E1, E2, E3. 1

E1+

1

E2+

1

E3 18

E.

41


( )

MA1A2, MB2C2 A1MA2 = B2MC2

E1(MB2C2) =

(MA1)(MA2)(MB2)(MC2) , (1)

, E2

(MA1B1) =(MC1)(MC2)(MA1)(MB1) , (2)

E3(MA2C1) =

(MB1)(MB2)(MA2)(MC1) , (3)

(1), (2), (3) = E1 E2 E3 =(MA1A2)(MC1C2)(MA2C1) , (4) (4) , .

- :

1

E1+

1

E2+

1

E3 3

3

E1 E2 E3=

36

E1 E2 E3 (MB2C2)(MA1B1)(MA2C1)

18

E1 + E2 + E3 + (MB2C2) + (MA1B1) + (MA2C1) 18

E

42

, , :

39 ( )

f :

(x y) f (x + y) (x + y) f (x y) = 4xy(x2 y2)


1 ( ) x + y = a

x y = b x = a + b2

y =a b2

.

b f (a) a f (b) = ab(a2 b2)

a, b R, a, b R

f (a)a

f (b)b = a2 b2,

f (a)a

a2 = f (b)b b2.

h(x) = f (x)x

x2 x R , f (x) = x3 + cx, x R. f (0) = 0, f (x) = x3 + cx, x R, . 2 ( ) x = y + 1

f (2y + 1) = (2y + 1) f (1) + 4y(y + 1)(2y + 1).

f (1) = c + 1

f (2y + 1) = c(2y + 1) + (2y + 1)(4y2 + 4y + 1)= c(2y + 1) + (2y + 1)3,

f (x) = x3 + cx x R. 40 ( )

f : {1, 2, . . . , 10} {1, 2, . . . , 100}

x + y|x f (x) + y f (y), x, y {1, 2, . . . , 10}.


( ) , x 1, 2, . . . , 9, 2x + 1 x f (x) + f (x + 1)(x + 1) 2x f (x) + 2(x + 1) f (x + 1). , 2x + 1 f (x + 1) f (x). ,

99 f (10) f (1) =9

x=1

f (x + 1) f (x)

9

x=1

(2x + 1)

= 99.

, f (x+1) f (x) = 2x+1 f (1) = 1. f (x) = x2 x 1, 2, . . . , 10.

43

:

41 ( )

ABCD (O) P AC BD.

ABCD Q,

QAB + QCB = QBC + QDC = 90o.

P, Q, O , O (O).


( .) QBC + QDC = 90o= QBD + DBC + QDB + DBC = 90o , (1)

(1) QBD + QDB = 180o BQD DBC +BDC = A, BQD = 90o + A (2)

, AQC = 90o + D , (3) (2), (3), Q

(K), (M), BD, AC -

BQD, AQC R, Q, (K), (M.)

(K) , BQD + BKD

2= 180o , (4)

K (K). (2), (4) BOD = 2A,

BOD + BKD = 180o , KBOD

OBK + ODK = 180o = OBK = ODK = 90o, KB = KD OB = OD.

, (O) ABCD, (K) , (M)(

MAOC , M (M) ).

, O (O) (K), (M) , QR, ,

P AC BD,

44

(O), (K), (M) . 42 ( )

P ABCD, AB, BC, CD, DA, -. ABCD .


( .)

, .

. - P BC ABC, . P, ABCD a, b, c, d, AB, BC, CD, DA,.

P A, (a + d) ,

, (b + c) , P C.

, -

a + b + c + d , A, C .

,

B, D .

ABCD ,

.

,

,

, .

45

:

43 ( -

1 1995)

6- 5. ,

6

http://www.mathematica.gr/forum/viewtopic.php?p=120445

1 ( )

. .

b 599.997 106 b2 599.997 103 > 774 103

2b > 1500 103.

(b + 2)2 = b2 + 4b + 4 > 599.997 106 + 3 106 =600.000 106.

b2 (b + 2)2 .

b2 < 599.998 106 < 600.000 106 < (b + 2)2.

(599.998 106, 600.000 106) ,

599.998 599.999. .

2 ( ) 100000

5 .

-

. ,

(700000, 790000) (70000)2 < 5 1011 (79000)2 > 6 1011. 89999

.

44 ( )

ABC AB AC D E , DE

. :ADDB

+AEEC

= 1

http://www.mathematica.gr/forum/viewtopic.php?p=111876

1 ( ) H BO W CO AH OD AW OE. WAH = DOE = DAE = 60, WOH = 120.

A,W,O, H

AWH = AOH = 60 = AOH = AHW.

AWH . (

) : OW + OH = OA(= OB = OC).

ADDB

=OHOB =

OHOA

AEEC =

OWOC =

OWOA

ADDB

+AEEC

=OH + OW

OA=

OAOA

= 1.

: 1) D S E A ADDB

+AEEC

=AB/2AB/2

+0

EC= 1.

2) 1) E AC , D A. 2 ( )

: BD = x CE = y, : AD = x AE = y. , , : ED + BC = BD + CE,: ED + = x + y : ED = x + y .

46

ADE : ED2 = AE2 + AD2 2AD AF (1) F E AD. FEA = 30, AFE: AF =

1

2AE =

1

2( y). , (1),

: (x + y )2 = ( y)2 + ( x)2 ( x) ( y) =

3xyx + y

. -

: AD = x = x (2y x)x + y

AE = y = y (2x y)x + y

. :ADDB

+AEEC =

x (2y x)x (x + y) +

y (2x y)y (x + y) = 1.

3 (Math Rider ) c

-

. H DE F,G AB, AC :

:

AF = BF = AG = CG = c2

(1)

FD = DH, GE = HE (2) ( ) c = 2

3 (3) ( . . -

OGC)

+ = 60, = DOH = HOE( FOG = 120 FOD = DOH, HOE =EOG)

+ = 60 ( + ) =3 +

1 =3

3( + ) = 3(1 ) (4)

ADBD

+AEEC =

AF FDBF + FD

+AG GECG +GE

(1)=

c2 FDc2 + FD

+

c2 GEc2 +GE

=c 2FDc + 2FD

+c 2GEc + 2GE

(2)=

c 2DHc + 2DH

+c 2EHc + 2EH

(3)=

23 2DH

23 + 2DH

+23 2EH

23 + 2EH

=

3 DH

3 + DH

+

3 EH

3 + EH

=

3 3 +

+

3 3 +

=6 2

3 +3( + ) +

(4)=

6 23 + 3(1 ) +

=6 26 2 = 1

47

:

45 ( IMC 1996)

n .

sin nx(1 + 2x) sin x dx.


( )

In =

sin nx(1 + 2x) sin x dx

=

0

sin nx(1 + 2x) sin x dx +

0

sin nx(1 + 2x) sin x dx

=

0


0

sin nu(1 + 2u) sin u du

=

0


0

2u sin nu(1 + 2u) sin u du

=

0

sin nxsin x dx.

sin a sin b = 2 sin(a

2 b2

)cos

(a

2+

b2

)

In In2 = 0

sin nx sin (n 2)xsin x dx

=

0

2 sin x cos (n 1)xsin x

dx

= 2

0

cos (n 1)x dx = 0.

I0 = 0 I1 =

In =

0 n = 2k, n = 2k + 1.

46 ( )

(an)nN an >

1n n.

+n=1

an .


1 ( ) kn .

+n=1

an +n=1

(kn kn1)akn >+n=1

(1 kn1kn

).

1 kn1kn 0 ln xx1 1 x 1,

ln(kn) = ln(k1) +n

i=1ln(

kiki1

),

.

2 ( )

, kn kn > 2kn1 n. 1 kn1kn 0 .

3 ( ) n

k=1ak

. n

k=1(akan) (

),

nan. , an >cn

c > 0 n,

. limn+ nan = 0,

.

48

:

47 ( )

M = R \ {3} x y = 3(xy 3x 3y)+m, m R. m (M, ) .


( )

,

e. :

1 e = 1 3e 9 9e + m = 1 m = 6e + 10

2 e = 2 6e 18 9e + m = 2 m = 3e + 20.

e = 103

m = 30. m = 30

.

a, b M

a b = 3ab 9a 9b + 30= 3(ab 3a 3b + 27) + 3= 3(a 3)(b 3) + 3 , 3

a , 3 b , 3. . a M

a 103=

10

3 a

= 3a10

3 9a 9

= 10a 9a 30 + 30= a

103 .

a M

a x = x a = 103

3ax 9a 9x + 30 = 103

x = 27a 809(a 3) , 3

.

a, b, c M a (b c) = 9abc 27(ab + bc + ca) + 81(a + b + c) 240

= (a b) c

(M, ) . 48 (vzf) m -

nn . , n

, .


( )

k > n A1, A2, . . . , Ak n n . :

,I{1,2,...,k}

(1)|I| det

jIA j

= 0 (1) , k = 3, (1) 2 2 A1, A2, A3

det (A1 + A2 + A3) det (A1 + A2) det (A2 + A3) det (A3 + A1) + det (A1) + det (A2) + det (A3) = 0.

(1) : m > n A1, A2, . . . , Am nn .

r

, r 6 n, .

, (1) k = n + 1 r , r 6 n + 1, .

, (1) k = n +2, . . . ,m A1 + A2 + + Am .

(1). [k] := {1, 2, . . . , k} :

,I{1,2,...,k}

(1)|I| det

jIA j

=

,I[k]

(1)|I|S n

sgn ()n

i=1

jI

A j (i, (i)) .

49

, :

,I{1,2,...,k}

(1)|I| det

jIA j

=

S n

sgn () ,I[k]

(1)|I|n

i=1

jI

A j (i, (i)) .

, , S n. ai j = A j (i, (i)) . :

,I{1,2,...,k}

(1)|I|n

i=1

jI

ai j

= 0. (2) (2) aJ := a1, j1a2, j2 an, jn , J :=[ j1, j2, . . . , jn] ( j1, j2, . . . , jn ,

, J

j1, j2, . . . , jn.) , I {1, 2, . . . , k} J I,

aJ (2)

, (1)|I|. |J| 6 n < k, r = |I| |J|, 0 6 r 6 k |J|, aJ (2)

k|J|r=0

(1)|J|+r(k |J|

r

)

=(1)|J|k|J|r=0

(1)r(k |J|

r

)

=(1)|J|(1 + 1)k|J| = 0.

(2), (1), .

50

:

49 ( )

ln(1 aix)x2 + m

dx , a > 0, m > 0

http://www.mathematica.gr/forum/viewtopic.php?f=9&t=7842&start=160

1 ( )

I1 = 0

x2(x2 + b2) (x2 + c2) dx = 2 (b + c) , b, c > 0 .

I2 = 0

ln(1 + x2)x2 + m2

dx = m

ln(m + 1) ,

I2 = 0

ln(1 + x2)x2 + m2

dx = 0

1

x2 + m2

( x0

2y1 + y2

dy)

dx y= xt===

0

1

x2 + m2

( 10

2x2t1 + x2t2

dt)

dx =

2

10

t

( 0

x2(x2 + m2

) (1 + x2t2

) dx) dt =2

10

1

t

( 0

x2(x2 + m2

) ( 1t2+ x2

) dx) dt I1==1

0

1

t

m + 1t

dt = m

ln(m + 1) .

ln(1 axi)x2 + m

dx ax=y====

a

ln(1 + yi)y2 + a2m

dy = a2

ln(1 + y2) + i arctan yy2 + a2m

dy

= a

0

ln(1 + y2)y2 + a2m

dy + 0 = a 0

ln(1 + y2)y2 +

(a

m)2 dy I2==

m

ln(1 + a

m).

2 ( ) -

f (z) = ln(1 aiz)z2 + m

, z C, a > 0,m > 0

, branch cut

z1 = ia , .

ln(1 aiz) = ln(az + i)ln i = ln |az + i|+i arg(az + i)i2

ln i = ln |i| + i arg(i) = i2.

C : [0, ] C A (R, 0) , B (R, 0). branch cut,

f ( ),

Cauchy:

C

f (z) dz = 2i Res( f , im ) , i

m.

C

f (z) dz = 0

f (Reit) iR eit dt + RR

f (z) dz .

0

f (R eit) iR eit dt = 0

ln(1 aiR eit)R2e2it + m

iR eit dt .

R

0

iR eit f (R eit) dt 6

0

R

ln(1 ai R eit)R2 e2it + m dt . R2 e2it + m > R2 |e2it | m = R2 m , R > m,

ln(1 aiR eit) = |ln(1 + aR sin t aiR cos t)| =ln 2 + 2aR sin t + i arg(1 + aR sin t aiR cos t) 63

2+

1

2ln(1 + 2aR sin t) 6 3 + ln(1 + 2aR)

2.

0

R

ln(1 aiR eit)|R2 e2it + m| dt 6 R

0

3 + ln(1 + 2aR)2R2 2m dt =

R 3 + ln(1 + 2aR)

2R2 2m 0 , R .

limR

0

f (Reit) iR eit dt = 0

limR+

RR

f (z) dz =

f (z) dz .

f (z) dz = 2i Res( f , im ).

51

Res( f , im ) =ln(ai

m + i)

2i

m=

lni (am + 1) + i arg (i (am + 1)) i/2

2i

m=

ln(1 + a

m)

2i

m.

ln(1 aix)x2 + m

dx = m

ln(1 + am ) . 50 ( ) an =

1 +n

j=21

ln j lim( n

n)an


1 ( ) 2 j n 1

ln j =1

ln n1

1 + ln( j/n)ln n=

1

ln n(1 + O

( ln( j/n)ln n

))=

1

ln n + O( ln( j/n)

ln2 n

), 1 < ln( j/n)ln n 0.

n

j=2

1

ln j =n 1ln n + O

(1

ln2 nn

j=2 ln( j/n)).

ln(x/n) [2, n].

n2

ln(x/n) dx + ln(2/n) n

j=2ln( j/n)

n+12

ln(x/n) dx ,

nj=2

ln( j/n) = n2

ln(x/n) dx + O(ln n) = n + O(ln n) .

n

j=2

1

ln j =n 1ln n + O

( nln2 n

)

( nn)an

= exp(ln nn

(1 +

n 1ln n

+ O(n ln2 n))) =

exp(1 + O(ln1 n)

) e .

2 ( ) (

n

n)an

= eann

ln n .

an

nln n = ann

ln n.

an

, 1 +n

j=2

1

ln j > 1 +n

j=2

1

j 1 .

n

ln n n

. :

an+1 ann+1

ln(n+1) nln n=

1ln(n+1)

(n+1) ln nn ln(n+1)ln(n+1) ln n

=ln n

ln[( nn+1

)nn] =

ln nln 1(

1+ 1n

)n + ln n = 1 ln(1+ 1n )nln n + 1 1 Cesaro-Stolz : lim

nann

ln n= 1 .

e .

52

:

51 ( ) An = {n, 2n, 3n, ...} n N

J N,

iJ

Ai


( ) : .

A j , N. , m N. J , j J j > m( J {1, 2, ...,m} = ). m < A j, . .

52 ( )

A, B AB = (A \ B) (B \ A), .

A1A2...An

Ai


( ) A x

A(x) = 0 x < A A(x) = 1 x A.

(AB)(x) A(x) + B(x) mod 2.

x A1 An (A1 An)(x) = 1 A1(x) + + An(x) 1 mod 2 x Ai.

modulo2.

53

:

53 (

) p P p > 5 (p 1)! + 1 p ( pk k N).


( )

p > 5 , (p 1)! + 1 = pk k. ,

(p 1)! = pk 1.: (p 1)! 0

(mod(p 1)2

).

,

(p 1)2 = 2 p 12

(p 1) ,, p > 5,

2

(p 1)p1. ,

pk (p 1)p1 + 1 > (p 1)! + 1 = pk, . .

54 ( dimtsig)

.


( ) -

1 .

m

111...1n

= m2, n > 1.

, m

1 9.

m m = 10a 1, a N.

,

111...1n

= 100a2 20a + 1,

111...10n1

= 100a2 20a,

111...1n1

= 10a2 2a.

, ,

.

54

:

55 ( )

S ,

s1, s2, ..., sn s1 + s2 + ... + sn < 1. S - ;


( ) n N An = {sk : sk 1n }. ( n

). S = An, S -.

56 ( -

) (entire) f , g

f (z)2 + g(z)2 = 1 z C.


( ) f 2 (z) + g2 (z) = 1

( f (z) + i g (z)) ( f (z) i g (z)) = 1

f (z) + i g (z) , 0 z C

(z)

( C) f (z) + i g (z) = e(z) . 1

f (z) + i g (z) = f (z) i g (z) = e(z) .

{ f (z) + i g (z) = e(z)f (z) i g (z) = e(z)

}

f (z) = e(z)

+ e(z)

2g (z) = e

(z) e(z)2i

(z) = i (z) ()

f (z) = cos ( (z)) g (z) = sin ( (z))

(z) .

55

:

57 ( )

BE B O

BOOE

=

32.

.

http://www.mathematica.gr/forum/posting.php?mode=edit&f=27&p=119266

( )

ABE, BEC :32=

BOOE

=ABAE

=BCCE

=AB + BC

AC.

:

sin C + cos C =32

1 + sin 2 C = 32

C = 150 A = 750

2 ( ) (ABC) = (ABE) + (BEC).

1

2AB BC = 1

2AB BE

2

2+

1

2BC BE

2

2.

BE2=

AB BCAB + BC (1).

r

ABC BO =2r

EO = 23

3r.

1

2AB BC = 1

2(AB + BC +CA)r (1)

2 +

23

32

=AB + BC +CA

AB + BC .

1 +

23= 1 +

ACAB + BC

. -

AB2 4AB BC + BC2 = 0 ABBC = 2

3.

ABBC = 2

3 = tan 15o

sin Csin A = tan 15

o .

C = 15o A = 75o.

ABBC

= 2 +3 = tan 75o

.

58 ( )

sin(

arctan(1

3

)+ arctan

(1

5

)+ arctan

(1

7

)

+ arctan(1

11

)+ arctan

(1

13

)+ arctan

(111

121

) ).


( )

arctan (x) + arctan (y) = z tan z =

x + y1 xy

z = arctan(

x + y1 xy

).

56

tan (x) = z 1

z= tan

(

2 x

)

arctan(1

z

)=

2 x =

2 arctan (z)

arctan(1

z

)+ arctan (z) =

2.

arctan(1

3

)+ arctan

(1

5

)+ arctan

(1

7

)+ arctan

(1

11

)+

arctan(1

13

)+ arctan

(111

121

)=

arctan(4

7

)+ arctan

(1

7

)+ arctan

(1

11

)+ arctan

(1

13

)+

arctan(111

121

)=

arctan(7

9

)+ arctan

(1

11

)+ arctan

(1

13

)+ arctan

(111

121

)=

arctan(43

46

)+ arctan

(1

13

)+ arctan

(111

121

)=

arctan(121

111

)+ arctan

(111

121

)=

2.

sin(

arctan(1

3

)+ arctan

(1

5

)+ arctan

(1

7

)

+ arctan(1

11

)+ arctan

(1

13

)+ arctan

(111

121

) )= 1.

57

:

59 ( KARKAR)

A ABC BC. (O) A AB, AC P, Q .

PQ S , S T . AT M BC.


1 ( ) AQT P , AT AQP. BPQC ( ) BC PQ. AT BC.

2 ( ) AM S . PQ ( S ) S . - A.PRQS . BC// BC AT .

3 ( )

A = AK AN = AP AB = AT AM =AQ AC. (o) BC. S T A, M, S ,(S S , AS ) BC M. S A M MM AS , = PBCQ, = QCS S , = ()//BC, ABCS , M AS , M BC.

4 ( ) N PQ, NAP = MAC, ABC, APQ . S , A,O, N, T S O, S NA = S T A = S AT , NPA = S AQ NAP = MAC( ).

58

60 (

[ ] ) ABC A , BAC BC .

ABC .


1 ( )

,

p, c, Q = (0, p), KQ KA = c(1). BKC = 2A, KBC ,

> 1,

KA = y0 x20 = 2y20 y20 2 + 2y0(2). (1)

x20 + (y0 p)2 = y0 + c (2)

2y0( p c) = c2 + 2 p2. c = p, c2 + 2 = p2

c =2

2 1 > 0, p =(2 + 1

)1 2 < 0.

,

(Q, 2

2 1

).

2 ( )

AW, AS , BAC = AEZ = AZE (), c ,

.

: BAC = AEZ = AZE KBFC, QZFE. L, F,. Q QA = r2

QH QA = KQ2 R2 = 2Rr + r2, QH

Q =2Rr + r2

r2

H

Q =2Rr.

Q

r=

KLR

=AH2R

H

r=

AH2R

2Rr H = AH D = AD,

QZ = A, ct, Q - .

, r2 = Q QA.

3 ( ) AD A = AD2 () (c) AD B,C B,C .

59

(o) (ABC) BC. A , BC (n) (c). ( BC (n)) , (ABC) (n) (n).

60

:

61 ( )

C :z + 1z =

z2 + 1z2 = 2


1 ( ) |z + 1z| = 2

|z2 + 1z2+ 2| = 4. w = z2 + 1

z2.

|w + 2| = 4 |w| = 2.

w = 2 z2 = 1. z = 1 z = 1. 2 ( ) z + 1z

= 2 z + 1z

2 = 4 z2 + 1z2 + 2

= 4 : (1) z2 + 1z2

= 2 : (2)

z2 +1

z2= x + yi, (x, y R)

(2)

x2 + y2 = 2 x2 + y2 = 4 : (3)

(1) |(x + 2) + yi| = 4 (x + 2)2 + y2 = 4 (x + 2)2 + y2 = 16 : (4)

(3) (4) (x, y) = (2, 0).

z2 +1

z2= 2 z4 2z2 + 1 = 0

(z2 1

)2= 0 (z = 1 z = 1)

3 ( )

z +1

z= w

|w| = |w2 2| = 2

2 =ww

2

|w2 2| = 2 |w2 ww2| = 2 |2w w| = 2

w = x + yi |w| = 2 |2w w| = 2 w = 2 w = 2 (x = 2 y = 0). .

62 ( )

P(x) , n x1, x2, ..., xn.

ni=1

P(xi)P(xi) = 0


1 ( ) an = 1, -

Q(x) = 1an

P(x).P(x) = (x x1)(x x2)(x x3)...(x xn1)(x xn)P(x) = (x x2)(x x3)...(x xn1)(x xn)+(x x1)(x x3)...(x xn1)(x xn)...

+(x x1)(x x2)(x x3)...(x xn2)(x xn)+(x x1)(x x2)(x x3)...(x xn1)P(x) = (xx3)...(xxn1)(xxn)+(xx2)(xx4)...(x

xn1)(x xn) +...+ (x x2)(x x3)...(x xn2)(x xn)+ (xx2)(x x3)...(x xn1)

+(x x3)...(x xn1)(x xn) + (x x1)(x x4)...(x xn1)(x xn)+ (x x1)(x x3)...(x xn2)(x xn)+ ...+ (xx1)(x x3)...(x xn1)

...

+(x x2)...(x xn2)(x xn) + (x x1)(x x3)...(x xn2)(x xn) +..+ (x x1)(x x2)...(x xn3)(x xn)+ (xx1)(x x2)...(x xn2)

+(x x2)...(x xn1) + (x x1)(x x3)...(x xn1)+... + (x x1)(x x2)...(x xn2)

: P(xi) = (xi x1)...(xi xi1)(xi xi+1)...(xi xn)

61

P(xi) = 2(xi x2)...(xi xi1)(xi xi+1)...(xi xn)+...+ 2(xi x1)...(xi xi2)(xi xi+1)...(xi xn)+

2(xi x1)...(xi xi1)(xi xi+2)...(xi xn) + ...+ 2(xi x1)...(xi xi1)(xi xi+1)...(xi xn1)

:P(xi)P(xi) =

2

(1

xi x1+ ... +

1

xi xi1+

1

xi xi+1+ ... +

1

xi xn

):

ni=1

P(xi)P(xi) = 2

(1

x1 x2+

1

x1 x3+ ... +

1

x1 xn

)+

2

(1

x2 x1+

1

x2 x3+ ... +

1

x2 xn

)+ ...

+2

(1

xn x1+

1

xn x2+ ... +

1

xn xn1

)= 0

2 ( )

( p

p) = 1

x r1+ ... +

1

x rn= S ,x , r1, ..., rn

( pp

) = 1S,x , u1, ..., un1 u1, ..., un1

p ()

(p)2 pp(p)2 =

S S 2

p

p= 1 +

S

S 2,x , u1, ..., un1

p

p=

( 1xr1 + ... +

1xrn )

2 ( 1(xr1)2 + ... +1

(xrn)2

( 1xr1 + ... +

1xrn )

2=

2

ni, j

1xix j

( 1xr1 + ... +

1xrn )

2

P(ri)P(ri) =

2

(1

ri r1+ ... +

1

ri ri1+

1

ri ri+1+ ... +

1

ri rn

)...

62

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