Icosidodecahedron All

http://www.mathematica.gr. .

Leonardo da Vinci

(32-) . 30 . - - - quasiregular , ( - ). (0, 0,),

( 1

2,

2, 1+

2

), 1+

5

2 .

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

mathematica.gr (http://www.mathematica.gr) .

mathematica.gr

1. (Mihalis_Lambrou) 2. (nsmavrogiannis) 3. ( ) 4. (k-ser) 5. ( ) 6. (m.papagrigorakis) 7. ( )

1. (grigkost) 2. (cretanman)

1. ( )

2. (lonis)

3. ()

4. (nkatsipis)

5. ( )

6. (chris_gatos)

7. (gbaloglou)

8. (R BORIS)

9. (Rigio)

10. (dement)

11. (swsto)

12. (achilleas)

13. ( )

14. (Demetres)

1. (spyros)

2. (vittasko)

3. (p_gianno)

4. (kostas.zig)

5. (exdx)

6. ( )

7. (mathxl)

8. (mathnder)

9. (mathematica)

10. (rek2)

11. (hsiodos)

12. (A.Spyridakis)

13. ( )

14. (bilstef)

15. ()

16. (xr.tsif )

1

1 ( ) x1, x2, x3. - y1, y2, y3 x1y1 + x2y2 + x3y3. y1, y2, y3 x1, x2, x3. ;

2 ( ) : , , 10. . ;

3 ( ) - - 2001, 2002, 2003, , 2010 - , , 40 ; ;

4 (qwerty) - 417 530 ;

5 ( ) :

= 102450 , = 5200 , = 24360

6 ( ) 360 36h = 123 (1 h) 34567 .

7 ( ) :

1.

2

526

2

5+26

2.

4 +15+

415 2

35

8 ( ) 2 =2 + 2, :A=3 + 3 + 3

,

9 ( ) x+ 1

y= 1 y+ 1

z= 1

xyz.

10 ( )

|x2 + x 2| = x3 1 |x+ 1|.

,

11 ( ) - ( ), . ,, ,,.

12 ( ) AB B = 15o, = 30o. B , B =A. BA.

,

13 ( ) a >0

a RQ, -

1,a, a

().

14 ( ) P (x) = x5 + x3 + x2 6x 12 x2 1

1. +

2. P (x) x2 1

,

15 ( ) E = 8, ZB = 6, ZB = 90. AB.

16 ( ) 4 1. .

,

17 ( ) . (

AB A) A =

(A B

) AB

.

18 ( ) 2 + = 6 +2 = 18 24

+ = 22 ,

:

1. + .

2.

2

http://www.mathematica.gr

3. .

,

19 ( ) t = 0 x(t) =t(t 4)2 + 3 t sec x(t) m. :

1. t .

2. .

3. .

4. .

20 ( )

f(x) =

2 (x )2 , x [0, )

x2+ 3

2, x [, 3]

1. f

2.

, ,

21 (math68) C :

z2 ( + 2)3z + 5+ 6 = 0

, .

1. z1, z2 .

2. :

K = z20121 z20152 + z1

2015z20122

3. - a =z20121 z

20152 b = z1

2015 z20122 .

22 ( ) z

|z 4i| 2 |z 2i| = 2 .

z.

, ,,

23 ( ) f : R R 1-1 :

f(x2) (f(x))2 14;

24 ( ) x 0 1, f(x) + f

(1

1x)= 2(12x)

x(1x) . f .

, ,

25 ( ) f : R (0,+) :f = f f ;

26 ( ) f [a, b], f(a) = a, f(b) = b f (x) 1 x (a, b), f(x) = x x [a, b].

, ,

27 ( ) - /2

0

sin x1 k2 sin2 x

dx

k (0, 1).

28 (xgastone) f , (0, ) f(/2) = 0 f (x) = (1 + f2(x)). f . , ,

29 ( ) f, g [0, 1] : f(x) > 0, 0 < g(x) < 1, x [0, 1]. :F , G, H :

F (x) = x0f(t)dt , x [0, 1]

G(x) = x0g(t)dt , x [0, 1]

H(x) = x0f(t)G(t)dt , x [0, 1]

1. x (0, 1] :

H(x) < F (x)G(x) .2. x (0, 1)

:H(x)F (x)

< H(1)F (1)

< 1 .

3. H(1)+10

g(t)F (t)dt = F (1) +G(1),

(0, 1) : f()

g()= 1F ()

1G()4. :

limx0+

H2(x)F (x)G(x)

30 ( )

f(x) = (x ) ( x) , x R, < , .

1. x (, )

f (x)f(x)

=

x +

x .

2. f Rolle [, ] f

= +

+ +

.

3. I(, ) =

f(x) dx.

I(, ) =

+ 1I( + 1, 1) .

4. , , , , , , : Cf , x = , x = xx.

, ,

31 ( ) z1, z2 C |z1| = |z2| = 0. : {

xy = z1xy = z2

} 32 ( ) f : (0, +) R f(x) > 0

f(x) = x+

f(x)x

t

2tdt

x (0, +).

3


. Juniors

33 ( ) 9 ;

34 ( ) : a, b, c - , : a+ 1

b= b+ 1

c=

c + 1a

= t, t R, :t = abc.

. Seniors

35 ( ) ( ), .

36 ( ) 0, a1a2a3... -. a1, a2, a3, ... ;

37 ( 2010) AB AB = B = . . BA A AE = AO B B BZ = BO. EZ , :

(i) = 3

(ii) AZ = EO

(iii) EOZ

38 ( 2010) ABC (O) I . AI, BI, CI (O), D, E, F, . (O1), (O2), (O3), ID, IE, IF , BC, AC, AB, A1, A2 B1, B2 C1, C2, . , (O).

39 (Putnam 2009) A,B,C A -. (AB)C = BA1 C(AB) = A1B.

40 ( ) f : [0, 1] R :

(1) f [a, b] [0, 1].

(2) c R, f1(c) = {x [0, 1] : f(x) = c} .

f .

41 ( ) n N, ()

k=0

(n

6k

).

42 (giannis1990) A M33(R) A2 + I = 0.

43 ( )

+n=1

(1)nn

ln(n!)

ln(Hn+1

) ,, Hn =

nk=1

1

k n-

.

44 (MoV ) : N N .

45 ( ) 51985 1 , 5100 .

46 ( )

xy + y = yx + x .

()

47 ( )

x2

a2+

y2

2= 1.

A(x1, y1) B(x2, y2) :x1x2a2

+y1y22

1+x1x2

a2+

y1y22

+ 1 = 2 .

48 ( ) :(1 + tan 1)(1 + tan 3) (1 + tan 43) 0 123 (1 h) 123 34567 123 34567 (1 h) 34567 123 3456734690 (1 h) 34567 34444

360 36h = 123 (1 h) 34567

6


, (1h) 34567

-34690 -34444. .

:

7 http://www.mathematica.gr/forum/viewtopic.php?f=35&t=8760

:

1.

2526

2

5+26

2.4 +

15 +

415 2

35

1 Stavros11 1

2

526

2

5+26=

2(

5+26

526)(5+2

6)(526) =

2(

5+26

526)2524 =

=2(5 + 2

6

5 26) =

=2((3)2 + 2 3 2 + (2)2

(3)2 2 3 2 + (2)2) =

=2((3 +

2)2

(32)2 =

=2(3 +

23 +2) = 24 = 4

2 Stauroulitsa

1: :a

b =

a+ c

2

a c2

a2 b = c2 x =

4 +

15 =

4+12 +

412 =

52 +

32

y =415 =

4+12

412 =

52

32

z = 235 =

1280 =

12+82

1282 =

102x + y z =

52 +

32 +

52

32

10 +

2 =

2

52

10 +

2 =

2

2: x =4 +

15 =

=

(52

)2+(

32

)2+ 2

52 32 =

(52 +

32

)2=

52 +

32

y =415 =

(52

)2+(

32

)2 2

52 32 =

(52

32

)2=

52

32

z = 235 = 2

(52

)2+(

12

)2 2

52 12 =

2

(52

12

)2= 2

52 2

12

x+yz =

52+

32+

52

322

52+2

12 =

2

3

1: x =4 +

15 +

415,

x2 = 10 x = 10 2: y =

10 2

35, y2 = 22 45

430 105 = 22454

52 2 5 5 + (5)2 =

= 22454 (55) = 2, y =

2


2 = 2 + 2, :

A=3 + 3 + 3

1

a3 + b3 + c3 = a(b2 + c2) + b3 + c3 =

ab2 + ac2 + b3 + c3 = b2(a+ b) + c2(a+ c) =

(a c)(a+ c)(a+ b) + (a b)(a+ b)(a+ c) =(a+ b)(a+ c)(2a b c) . 2

c2 = (a b)(a+ b) c3 = c(a b)(a+ b). :a3 + b3 + c3 = a3 + b3 + c(a b)(a+ b) =(a+ b)(a2 ab+ b2) + c(a b)(a+ b) =(a+ b)(a2 ab+ b2 + ac bc) =(a+ b)(b2 + c2 ab+ b2 + ac bc) =(a+ b)[2b2 + c(a+ c) b(a+ c)] =(a+ b)[2(a2 c2) + c(a+ c) b(a+ c)] =(a+ b)[2(a c)(a+ c) + c(a+ c) b(a+ c)] =(a+ b)(a+ c)(2a c b)

7


:

9 http://whttp://www.mathematica.gr/forum/viewtopic.php?f=19&t=9469

x+ 1y = 1 y +1z = 1 xyz.

1 xy + y + 1z = xy + 1 = y(x +

1y ) = y,

xy + 1z = 0, xyz = 1. 2 ... y+ 1z =1 yz + 1 = z yz z = 1 z(y 1) = 1 y 1 = 1z x + 1y = 1 xy + 1 = y xy =y 1 xy = 1z xyz = 1 : y + 1z = 1 x+ 1y = 1 xy + 1 = y xy + y + 1z = y xy + 1z =0 xyz = 1 3 , :

1 x = 1y 1 y = 1z 1yz = (1 x)(1 y) 1yz = (1 x) y(1 x). 1yz = 1 x y 1y 1yz = x. xyz = 1. 10 http://www.mathematica.gr/forum/viewtopic.php?f=19&t=4858

|x2 + x 2| = x3 1 |x+ 1|. , x31 = |x2+x2|+|x+1| 0 x 1. x2 + x 2 = (x 1)(x+ 2) 0 |x2 + x 2| = x2 + x 2, |x + 1| = x + 1. x3 1 = |x2 + x 2| + |x + 1| x 1 x3 1 (x2 + x 2) (x + 1) = x(x2 x 2) =x(x+ 1)(x 2) = 0. , -1, 0, 2, 2.

:

11 http://mathematica.gr/forum/viewtopic.php?p=41418

( ), . ,, ,,.

A1 =

A2 =

K1 =

K2 =

= A = B. (= ) .

1 =

2 =

1.

= B,O = O 1 =

1

1 =

B1 . .

12 http://www.mathematica.gr/forum/viewtopic.php?p=44281

AB B = 15o, = 30o. B , B = A. - BA.

1 B = 15, = 30 , = . , > , . AE = 120 EAB = 135 = 120 = 15.

8


= . , BA = 15.

2 . : :A

15=

BE

A

BE=

15

.

:A

30=

A

(15 + ) A

A=

30

(15 + ),

: 15

=

30

(15 + ) 1

=

215

(15 + ) 15 + 15 = 2 15 15 15 = 0 (15 ) = 0 . :

= 360 + 15 = 360 165, Z 0 < < 135 = 15.

3 . AK = 30 AKB = 60 . .

AE = 75, . (--), . (--), , BA = 15. : : , (2 ), ,

:


a > 0 a R Q, -

1,a, a

( ).

1,

a, a k +

1, l + 1,m + 1 . a1 + kd = 1 (1)a1 + ld =

a (2)

a1 +md = a (3). (2) (1) (3) (1) (k l)d = a 1 (m k)d = a 1. , k l

m k =a 1a 1

a =

k lm k 1,

a .

14 http://www.mathematica.gr/forum/viewtopic.php?p=57479 P (x) = x5 + x3 + x2 6x 12 x2 1

1. +

2. P (x) x2 1

1, x2 1 =(x 1) (x+ 1) 1 1 x = 1 : x5+ ax3+

9


x2 6x 12 = 0 1 + a+ 6 12 = 0 a+ =17, (1). x = 1 : x5+ax3+x26x12 =0 1 a + + 6 12 = 0 a = 7, (2). (1) (2) a + b = 17 b a = 7 b = 12 a = 5. x5 + 5x3 + 12x2 6x 12 = (x2 1)(x3 + 6x+ 12) x3 + 6x+ 12.

2, (x) = ( 5)x+ 12 (x) = x3 + (+ 1)x+ . , = 5 =12. + = 5 + 12 = 17 (x) = x3 + (+ 1)x+ = x3 + 6x+ 12.

:

15 http://mathematica.gr/forum/viewtopic.php?p=45740

E = 8, ZB = 6, ZB = 90. AB.

1 , AE = = EBZ EB = 45 .

(45 ) = 68 + ZE

, (1)

=ZE

6, (2)

ZE = 27 4

E = a2 = 40 + 8

7

2 . =, =, =. BE =

2 AE = a 2.

:

8 = 2( 2) 642 = 22 (64 2)

64(22 36) = 22 (64 2) 22 = 1152 (1).

:2 +

( 2)2 = 64 2 + 2 = 80 (2).

(1), (2) 2, 2 : t2 80t + 1152 = 0 2 < 2 :2 = 40 + 4

28.

3 x . B =2x ZB

ZB = B = 45 ZB = ZB = . . - ZH = H = 6 + y Z =

2 (6 + y).

: x2 = y2 + (6 + y)2(1). :2 (6 + y)

2x=

x

8 x2 = 8 (6 + y) (2).

(1), (2) : y2 + 2y 6 = 0 y =

7 1, (2)

E. = x2 = 40 + 87 ..


4 1. .

1 Eukleidis

10


. (AB)2 = 1+42 2 (12) (AB) = 7 2

,

B

(5

2,

3

2

), (AB) =

25

4+

3

4=

7.

3 , . ( ). ,,

3,

3

. 1 =

72 . , =

7.

4

(AB)2 =(A)2 + (B)2 = 22 +

(2

32

)2=

4 + 3 = 7 AB =7

5 .

T1 = T2, T3 = T4 7- . (AB)

23

4 = 712

3

4 (AB) =7.

6 . , , - ( ). 2,, 3,. 1 : 3 2+ = x x, 7a2 = x2 x = a7. 7 . - . () = 2..120

2 =

23

2 , (1). () =

t(t a)(t 2a)(t x)

t t = 3a+x2 , :() =

(9a2x2)(x2a2)

4 (2) x > 3. (1) (2) x4 10a2x2 + 21a4 = 0, x2 = 3a2 x2 = 7a2. x2 = 7a2 , x = a

7.

8 - , Euler: - -

11


. . . 2 ( )

3,

. Euler : 22 + 2(2a)2 =(a3)2 + x2 10a2 = 3a2 + x2 7a2 = x2 x = 7.

9 -.

(ABC) = 7(DEZ),

ABE, BCZ, CAD, .

(ABE) = (BCZ) = (CAD) =2(DEZ).

, ( =

), (ABC)(DEZ)

=(AB)2

(DE)2= 7 =

AB =7, ( DE = 1 ) .

10

- (

). AP =7

3 =

3

27

( ) AP60

=

P

7332

=P3

27

P = 13, = 2

:

:


. (AB A

) A =

(A B

) AB

.

- AB A = A B = : A = AB ( : = , , = = 0).

= 0 A = AB A//AB

, = 0. = 0, AB A = 0 ABA A B = 0 AB. .


2 + = 6 +2 = 18 24

+ = 22 ,12


:

1. + .

2.

3. .

1. 62 =

3+ 3 = (2+ ) + (+ 2)2 + + + 2 24 24

62 24 + 24 0 6( 2)2 0 = 2

+ = 82.

=2 :2 + =

12, +2 = 12, + = 8 2 + =

12 2 + 2 = 144 (2 + )2 = 144

4 2 + 4 + 2 = 144 (1) +2 = 12 +22 = 144 ( +2)2 = 144 2 +

4 + 4 2 = 144 (2) (1) (2): 4 2 + 4 + 2 = 2 +4 + 4 2 3 2 = 3 2 2 = 2 | |2 =

2 | | = (3) + = 8 + 2 = 64 ( + )2 =64 2+2 + 2 = 64

2= 2 2 2+2 = 64 2 + = 32(1)

2= 2 5 2 + 4 = 144 : 2 = 16 | |2 = 16 | | = 4

3. (3) | | = 4 =

+ = 8 = | | + .

:


t 0 x(t) = t(t 4)2 + 3 t sec x(t) m. :

1. t .

2. .

3. .

4. t0 .

: x (t) =t3 8t2 + 16t+ 3, t 0.

1. : (t) = x (t) = 3t2 16t + 16, t 0, : (0) = 16 m/sec

2. (t) = 0 ... t = 4

3sec t = 4 sec

: (t) = (t) = 6t 16, t 0, ,

: (4

3

)= 8 m/sec2, (4) = 8 m/sec2

3. v > 0 v < 0. : (t) > 0 0 t 0 (t) < 0 4

3< t < 4,

t [0,

4

3

) (4, +)

t (4

3, 4

).

4. (t) = 0 t =

8

3sec.

(4) , 0, , () t 0.

13



f(x) =

2 (x )2 , x [0, )

x2 + 32 , x [, 3]1. f

2.

-

1. .

2. , 1.

a2

4 +122a a = 1 a2 = 44+ a = 2

4+

4+

E =a2

4

E =122a a = a2 = 4a

2

4 >a2

4 = E

[, 3]. x = . - 3

=3a

2

1/2. 12 (3a ) = 12 ( 3a)2 = 2 =3a2 = 3a2 = 3 2

4+

4+ 2

:

21 math68http://www.mathematica.gr/forum/viewtopic.php?f=51&t=9819

C :

z2 ( + 2)3z + 5 + 6 = 0

, .

1. z1, z2 .

2. :

K = z20121 z20152 + z1

2015z20122

3. a = z20121 z

20152 b = z1

2015z20122 .

1. < 0. z2 3( + 2)z + 5 + 6 = 0. = 32 8 12 = 1 = 4. = 1 z23z+1 = 0, z1,2 =

32 12 i

2. z1 =32 +

12 i z2 =

32 12 i. (

z1 =32 12 i z2 =

32 +

12 i ) z

31 = i

z32 = i K = z20121 z

20152 + z

20151 z

20122 =

1z20121 z

20152

+ 1z20151 z

20122

=z31+z

32

z20151 z20152

= ii12015

= 0

3. a = z20121 z20152 =

1z20121 z

20152

= 1z20121 z

20122 z

32=

1z32(z1z2)

2012 =1i = i

= z20151 z20122 =

1z20122 z

20151

= 1z20121 z

20122 z

31=

1z31(z1z2)

2012 =1i = i

A(0,1)

14


B(0, 1) O(0, 0).


z

|z 4i| 2

|z 2i| = 2 z.

chris |z 4i| 2 z C1 K1(0, 4) R1 = 2 |z 2i| = 2 z K2(0, 2) R2 = 2 . z

AB

A,B. |z|max =(OK1) = 4 z = 4i |z|min = (OA) = (OB) =(OK1)2 R21 =

16 4 = 23

:


f : R R 1-1 : f(x2) (f(x))2 14 ; giannisn1990 f : R R 1-1 f(x2) f2(x) 14 , - 0 1 f2(1) f(1) + 14 0 f2(0) f(0) +14 0 2 (f(0) 12)2 +(f(1) 12)2 0 f (0) = f (1) = 12 1-1 1=0


x 0

1, f(x) + f(

11x

)= 2(12x)x(1x) .

f .

x 11x ( ) :

f

(1

1 x)+ f

(x

x 1)= 2

1 x2x

(1)

x 11x (1) ( ) :

f

(x

x 1)+ f(x) = 2

x(x 2)1 x (2)

(2) (1) f(x)f

(1

1x)= 2x2x+1x(1x)

f(x) = x+1x1 .

15


:


f : R (0,+) : f = f f ;, f(x) > 0 x R. x f(x) f(f(x)) > 0,x R f (x) > 0 f R [1] f(x) > 0 [1] f(f(x)) > f(0) f (x) > f(0) (f(x) xf(0)) >0 f(x) xf(0) R x < 0 f(x) xf(0) < f(0) 0f(0) f(x) < (1 + x)f(0) x < 0 [2] f(x) > 0 f(0) > 0. x < 1 [2], 26 http://www.mathematica.gr/forum/viewtopic.php?f=53&t=2081

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

1, g(x) = f(x) x x g g () g (x) g () g (x) = 0 f (x) = x,x [, ] 2, x0 f(x0) = k = x0. k > x0 [a, x0] f () 1 f(x0)f()x0 1 kx0 1 k xo k < x0 [x0, b] f () 1 f()f(x0)x0 1 kx0 1 k x0 x0 [a, b] f(x0) = x0, f(x) = x

:

27 http://www.mathematica.gr/forum /viewtopic.php?f=54&t=8057&p=46212

/20

sinx1k2 sin2 x

dx,

k (0, 1).

1 I =

/20

sinx(1k2)+k2 cos2 x dx =

11k2

/20

sinx1+ k

2

1k2 cos2 x

dx

k1k2 cos x = tanu, u [0, ],

(0, /2) tan = k

1k2 (0,+). I = 1

1k2 0

11+tan2 u

1k2k

1cos2 u

du =

1k

0

cosucos2 u

du = 1k 0

cos u1sin2 u du = 12k

0

cosusinu1

cosusinu+1 du = 12k ln 1sin1+sin

tan2 = k21k2 sin = k

I =1

2kln

1 + k

1 k 2

20

xdx1k2+k22x

k(0,1)=

1k

20

xdx1k2k2

+2x

x=t

1k2k2=

xdx=

1k2k2

dt

1k

k2

1k20

1k2k2

dt1k2k2

+ 1k2k2

t2=

1k

k2

1k20

dt1+t2

=

1k

[ln(t+

1 + t2

)] k21k2

0=

16


12k ln

1+k1k

28 xgastonehttp://www.mathematica.gr/forum/viewtopic.php?f=54&t=8003

f , (0, ) f(/2) = 0 f (x) = (1 + f2(x)). f .

f (x) = (1 + f2(x)) f (x)1 + f2(x)

= 1

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

(2

)= f

(2

)= 0

f (x) = g(x)2g(x) f (x) =

(1 + 2g (x)

)g (x)

f (x) = (1 + f2 (x)) g (x) f (x)1+f2(x)

= g (x) 1 = g (x) g (x) = x + c x = /2 g(2

)=2 + c g

(2

)=

(2 + c

)0 = c c =0 g f (x) = g (x) f (x) = (x), - .

:

29 http://www.mathematica.gr/forum/viewtopic.php?f=55&p=39775

f, g [0, 1] : f(x) > 0, 0 < g(x) < 1, x [0, 1]. : F , G, H :

F (x) = x0 f(t) dt , x [0, 1] G(x) = x0 g(t) dt , x [0, 1] H(x) = x0 f(t)G(t) dt , x [0, 1]1. x (0, 1] :

H(x) < F (x)G(x) .

2. x (0, 1) :H(x)

F (x) 0 x [0, 1] F [0, 1] F (0) = 0 x (0, 1].

G(x) = x0 g(t)dt G(x) = g(x) > 0 x [0, 1] G [0, 1] G(0) = 0 x (0, 1].

H(x) = x0 f(t)G(t)dt H (x) = f(x)G(x) > 0 x (0, 1], H [0, 1] H(0) = 0 x (0, 1].

1. w(x) = H(x) F (x)G(x),x [0, 1], : w(x) =H (x) F (x)G(x) F (x)G(x) = g(x)F (x). F (x) =

x0 f(t)dt

[0, 1], x > 0 F (x) > F (0) F (x) >

00 f(t)dt F (x) > 0 0 0 w(x) < w(0) H(x) F (x)G(x) < 0 H(x) 0, (0, 1], x < 1 u(x) < u(1) H(x)F (x) 0

f(x) = x+

f(x)x

t

2tdt

x (0, +).

1

f(x) f(x)1

sin t

2tdt = x

x1

sin t

2tdt.

g g(x) = 1 sinx

2x> 0, x > 0.

g 1 1 ( . ) g[f(x)] = g(x) x, f(x) = x, x > 0. . 2 |x| < |x| ,x = 0 1 < xx < 1,x (0,+) x2x < 12 < 1,x > 0, (1)

g (x) =x

a

t

2tdt, a, t, x

(0,+). f (x) =x+g (f (x))g (x) f (x)x = g (f (x))g (x), (2)

19


x0 (0,+) f (x0) = x0, f (x0) , x0 ... g,

, g () = g (f (x0)) g (x0)f (x0) x0

2

(2)= 1, (1) f(x) = x, x > 0.

3

|f(x) x| =f(x)x t2t dt

f(x)x

t2t

dt f(x)x

12dt 12 |f(x) x|

.: f(x) > 0 .

:

33 http://www.mathematica.gr/forum/viewtopic.php?f=49&t=8906&start=0

9 ;

stavros11 A = abcd N .

: 9A = dcba 9abcd = dcba a = 0, d = 0

, A 1111 (1112 9 ). a d : 1001 A 1111 a = 1 d = 9.

b 0 1. dcba

9 ( dcba = 9A A N ), 9. 9/a+ b+ c+ d. :9/10 +b+ c. b = 1 c = 7 A = 1179. ,

A < 1112. b = 0, c = 8 A = 1089.

1089 1089 9 = 9801. 34 http://www.mathematica.gr/forum/viewtopic.php?f=49&t=8079&start=0

: a, b, c - , . : a+ 1b = b+

1c = c+

1a =

t, t R, : t = abc chris

t + abc = a + 1b+ abc =

ab2c+ ab+ 1

b=

abc(b+ 1c ) + 1

b=

abct+ 1

b(1)

:abct+ 1

b=

abct+ 1

a=

abct+ 1

c

a = b = c = 0 abct = 1 (1) .

:


( ), - .

1

, , . , ( ) - , .

,

20


. , , (). , .

, , = = .

2 , , . = = . AB +MN AM +BN (1)., , + 3.


0, a1a2a3... . a1, a2, a3, ... ;

- Hilbert. a1, a2, a3, . . ., A = {0, 1, 2, 3, . . . , 9}. , , . , , - . b1, b2 . . . , bn b1 0 (xn), (n) lim dn = 0,|x xn| < n |f(x) f(xn)| n. n f(xn) f(x) + n f(xn) f(x) . f(xn) f(x) + n. (1) Dn x, xn sn Dn f(sn) = f(x) + . lim sn = limxn = x sn f1(f(x)+ ) (2) lim sn = x f1(f(x)+), f(x) = f(x)+,. f .

:


n N, () k=0

(n

6k

).

( ), :

(1 + x)n =

(n

0

)+

(n

1

)x+

(n

2

)x2 + +

(n

n

)xn.

x x, x 2x ,

23


. 1 + + 2 = 0. xk k 3, 3 = 1. (1 + x)n + (1 + x)n + (1 + 2x)n =

3((

n0

)+(n3

)x3 +

(n6

)x6 + ) .

x = +1 x = 1 .

2n + (1 + )n + (1 + 2)n + (1 )n + (1 2)n6

.

42 giannisn1990http://www.mathematica.gr/forum/viewtopic.php?f=10&t=1005

A M33(R) A2 + I = 0.

. A2 = I, det(A2) =det(I) = 1 (det(A))2 = 1, (det(A))2 0. .

:

43 http://www.mathematica.gr/forum/memberlist.php?mode=viewprole&u=54

+n=1

(1)nnln(n!)

ln(Hn+1

) ,, Hn =

nk=1

1

k n-

.

-

an :=nln(n!)

ln(Hn+1

) , n N, ,

Hn+1 ln(n+ 1) n+ Hn+1ln(n+ 1)

1

Stirling

limn+ an = limn+

n

ln(

2n(ne

)n)ln(ln(n+ 1)

) , LHospital 0. ln

(Hn+1

)

, an, n N,, n

ln(n!),

n N, n. sn := ln(n!), n+1sn+1 =

n+1sn + ln(n+ 1) =

n+1sn

(1 + ln(n+1)sn

) 1n+1

()

n+1sn

(1 +

ln(n+ 1)

(n+ 1)sn

) n+1sn + ln(n+ 1)

(n+ 1)sn

ln(n+ 1)

(n+ 1)sn n+1sn+1 n+1sn (1).

, ..., sxn

[1

n+1 ,1n

] ( 1n+1 , 1n),

nsn n+1sn = s

n ln(sn)

n(n+ 1)

n+1sn ln(sn)

n(n+ 1) ln(sn)

(n+ 1)2(2).

n

sn ln(sn) (n+ 1) ln(n+ 1) ln(sn)(n+ 1)2

ln(n+ 1)(n+ 1)sn

(2) (1)

nsn n+1sn n+1sn+1 n+1sn nsn n+1sn+1.

Leibniz .

() Bernoulli, 0 1.

44 MoVhttp://www.mathematica.gr/forum/memberlist.php?mode=viewprole&u=89

: N N .

1 x = 0, 12... [0, 1) . x :

21+1 , 21+1 31+2 , 21+1 31+2 51+3 , . . .

24


. 1-1 [0, 1) . - [0, 1) Cantor -. 2 X . g : X [0, 1] X -.

g() = 0, (1)(2) . . ., n n 10.

x = 0, x1x2 . . . [0, 1], :N N; (n) = 10n + xn X g() = x, g . 3 MoV Cantor :

- : N N :

(0)n := (0,0, 0,1, 0,2, . . .)

(1)n := (1,0, 1,1, 1,2, . . .)

(2)n := (2,0, 2,1, 2,2, . . .)

.............................................

n := n,n + 1 + n1 0 = 0,0,

n = n +n

i=0

i,i

n+1 n = n+1,n+1 + 1 > 0 - (ai)n. . 4 N., N ( N ). N. P(N) N , .

:


(51985 1) -, 5100.

x5 1 = (x 1)(x4 + x3 + x2 + x+ 1).

x4 + x3 + x2 + x+ 1 = (x2 + 3x+ 1)2 5x(x+ 1)2. x = 5397, x4 +x3 +x2 +x+1 = (x2 +3x+1)2

5398(x+ 1)2 = (x2 + 3x+ 1)2 (5199(x+ 1))2 =(x2 + 3x+ 1 5199(x+ 1))(x2 + 3x+ 1 + 5199(x+ 1))

,x5 1 = (x 1)(x2 + 3x+ 1 5199(x+ 1))(x2 + 3x+ 1 + 5199(x+ 1)

)

x 1, (x2 + 3x+ 1 + 5199(x+ 1)) 5100.,(x2 + 3x+ 1 5199(x+ 1)) =

x(x 5199) + 3x 5199 + 1 x+ 0 + 1 5100 () 1985.


xy + y = yx + x .

(x, y) = (c, c), (c, 1), (1, c), c N (x, y) = (2, 3), (3, 2). .

, x = 2 y > x, xy yx ( f(x) = lnxx x = e f(2) = f(4)) y > x, xy + y > yx+ x, .

3 x < y, f(x) xy > yx y > x .

25


:


x2

a2+

y2

2= 1.

A(x1, y1) B(x2, y2) :x1x2

a2+

y1y22

1+ x1x2

a2+

y1y22

+ 1 = 2 .

1 (*) : (x1, y1),(x2, y2) x

2

a2+ y

2

b2= 1

1 x1x2a2 + y1y2b2 1. : - x2 x2, y2 y2 ( (0, 0)). (x1, y1) x1xa2 +

y1yb2

= 1, x1x

a2+ y1y

b2< 1 (x, y)

(**) - ( (0, 0) (x1, y1), (x2, y2)).

(*) 1 c 1 |c1|+|c+1| = (1c)+(c+1) = 2(**) Ax + By = 1 ( Ax + By > 1) - ( Ax+By < 1). 2 manos66x1x2a2

+ y1y22

1 = 12(2x1x2a2

+ 2y1y22

2)=

12

(2x1x2a2

+ 2y1y22

x21a2

y212} x22

a2 y22

2

)=

12(x212x1x2+x22

a2+

y212y1y2+y222

)=

12((x1x2)2

a2 +(y1y2)2

2

) 0

x1x2a2 +

y1y22 + 1 0

x1x2a2 + y1y22 1+ x1x2a2 + y1y22 + 1 =x1x2a2 y1y22 + 1 + x1x2a2 + y1y22 + 1 = 2 3

Cauchy-Schwarz.... :|x1x2a2 + y1y22 | = |x1a x2a + y1 y2 |

x21a2 +

y212

x22a2 +

y222 =

1 1 = 1 : |x1x2a2 + y1y22 | 1 1 x1x2a2 + y1y22 1:x1x2a2

+ y1y22

1x1x2a2 +

y1y22 1

:|x1x2

a2+ y1y2

2 1|+ |x1x2

a2+ y1y2

2+ 1| =

1 x1x2a2 + y1y22 + 1 + x1x2a2 + y1y22 = 2 48 http://www.mathematica.gr/forum/viewtopic.php?f=27&t=9347

:(1 + tan 1)(1 + tan 3) (1 + tan 43) < 211 211 2 :(1 + tan 1)(1 + tan 3) (1 + tan 43)

A

< 211B

0 f . g(x) = 12 x2,

f (x) = 1 > 0 f . f(x) =g(x) x = k k Z. 2 ( f g):

f(x) 0 g(x) =

(x+ 1)2 x < 10 1 x 1(x 1)2 x > 1

3 : f(x) = x2, g(x) =x2 + sin2 x. . .. g(x) = 2x + 2 sin x cos x = 2x + sin 2x, g(x) =2 + 2 cos x 0. sinx = 0, = . g(k) = 2k + sin 2k = 2k =f (k)

, . .... , , , , , ... . ... (- ) , , .

50 http://www.mathematica.gr/forum/viewtopic.php?f=61&p=16318#p16318

f (, ) lim

xa+f(x) = lim

xf(x) = k, k

, (, ) f () = 0.

Rolle: g : [, ] R g (x) = f (x) , x (, ) g () = g () = k. g (, ) [, ] Rolle. - - .

27


:


z C |z1| = 1, z2.

z C |z 1| = 1, z2. z = 1 + (cos(t) + i sin(t)) t[0, 2 ). z = 2 cos( t

2) (cos( t

2) + i sin( t

2))

z2 = 4 cos2( t2) (cos(t) + i sin(t))

x = 4 cos2( t

2) cos(t)),

y = 4 cos2( t2) sin(t)) t[0, 2 )

Geogebra :

- :

1. z (x 1)2 + y2 = 1

2. . Geogebra .

3.

4. z

5. w =z2 w

6. ( 4 ) w z.

52 http://www.mathematica.gr/forum/viewtopic.php?f=60&p=47538#p47538

z1, z2, z3 C |z1| = |z2| = |z3| = r > 0, |z1 z2|2 + |z2 z3|2 + |z3 z1|2 9r2. =, z1, z2, z3 .

1 , , , r. |z1 z2|2 + |z2 z3|2 +|z3 z1|2 = (AB)2 + (BA)2 + (A)2 =(OB OA

)2+(OOB

)2+(OAO

)2= 6r2

2(OA

OB +

OB

O +

O

OA) =

6r2 2r2(a+ b+ c) =r2 [6 2(a + b+ c)], a, b, c , a+ b+ c = 360, : a+b+c+(a+ b+ c) =4 a+b2

b+c2

c+a2

a+ b+ c = 1 4 a2 b2 c2 : a2

b2

c2 18 4 a2 b2 c2 12

a+ b+ c 1 12 = 32 ,a2 +

b2 +

c2 = 180

. , :|z1 z2|2 + |z2 z3|2 + |z3 z1|2 6r2 + 3r2 = 9r2 a = b = c = 120, . 2 |z|2 = zz |z1 z2|2 +|z2z3|2+|z3z1|2+|z1+z2+z3|2 = 3(|z1|2+|z2|2+|z3|2),|z1z2|2+ |z2z3|2+ |z3z1|2 3(|z1|2+ |z2|2+ |z3|2) =

28


9r2 = z1+ z2+ z3 = 0, . . 3 A,B,C z1, z2, z3 , ABC O r. G ABC M , MA2 +MB2 +MC2 = 3MG2 + 13 (a

2 + b2 + c2), BC =a,CA = b,AB = c. , M O 3r2 = 3OC2+ 13(a

2+b2+c2) 9r2 = 9OG2+a2+b2+c2, 9r2 |z3 z2|2 + |z3 z1|2 + |z1 z2|2. = OG = 0, . 4 (0, 0) A(, 0) > 0,

B(, ) (, ). = , , r. OA+OB >AB 2 > AB, 42 > (a)2+2. OB = 2 + 2 = p2. < 2 (1). OB +O > B, 2 > B, 42 > ( )2 + ( )2. OB = 2 + 2 = p2. O = 2 + 2 = p2. < p2(2). < p2 (3). AB2 +B2 +A2. - 6p2 2a 2 2 2a AB2 +B2 +A2 6p2 + 3p2 = 9p2: , , - > , > > , > , (3).

29


http://www.mathematica.gr . .

Leonardo da Vinci

(32-) . 30 . - - - quasiregular , ( - ). (0, 0,),

( 12,

2, 1+

2

), 1+

5

2 .



mathematica.gr

1. (Mihalis_Lambrou) 2. (nsmavrogiannis) 3. ( ) 4. (k-ser) 5. ( ) 6. (m.papagrigorakis) 7. ( )


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3. (nkatsipis)

4. ( )

5. (chris_gatos)

6. (gbaloglou)

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13. ( )

14. (bilstef)

15. ()

16. (xr.tsif )

1 (maths-!!) , - a 1 a 10 a b 1 b 10. ... 100. , - ; ;

2 ( ) 9 13 16( 16) 22 9 6. 17 5 13 19. 15 24 27. 20 15.

3 ( ) . , , , . , ( );

4 ( ) 170 330. , - 1

3

, 27 . ,

14 .

;

5 ( ) x y . 110 .

6 ( ) 20 20 cm. . 15 .

7 ( ) x2 + y2 = 3xy, x, y , (

x

x y)2

+

(y

x y)2

8 ( ) 0 < b < a a2 + b2 = 6ab

a+ b

a b

,

9 ( ) :x2

x 1 +x 1 +

x 1x2

=x 1x2

+

1x 1 +

x2x 1

10 ( )

| 1| x2 + |3 2|x+ |1 | = 0 = 1 x .) .) .) ( 1) ( 2) 1 2 ) , .

,

11 ( ) . , = + .

12 ( ) ==5.

, , , : =2=6, 3

5.

.

,

13 ( )

|x 1|+|x 5| = 2

x

14 ( ) - - , .

,

15 ( ) K . A :A1 A A2 A1 KA3 A2 B A3 K AB 4- K .

1

16 ( ) , . . . .

,

17 ( ) OA =

OB = -

, =

+ || AOB

18 () 3KA+2

KB+

A

=5K

KA=6 , KB=8 , K=5 . :) .

) v =KB+K A = 32

7

,

19 ( ) 200 5 , . 4, 20, 60, 50, 40, 30 20.

1. .

2. 5 20 .

() .

() ( ), 5.

() - ) :20, 60 , 50 , 40 ,30 .

20 ( ) 2 + 2 2 2 , 2. .

1. - : : .

2. : P () = P ();

3. .

, ,

21 ( ) - (

1 +3

22

+

3 122

i

)72

22 ( )

2z2 = 6iz + 3, z C

, ,,

23 ( ) f f(f(x)) = x2x+1,x R. g : R R g(x) + xf(x) x2 = 1, x R

24 ( ) f, g R .

3f(x) + 5g(x) + 8f(x)g(x) = 0 (1)

R.

, ,

25 ( ) R

(f (x))2 = f(x)

x.

26 ( ) f : R (0,+) , :f =f f ;

, ,

27 ( ) f [0,+) lim

x+[xf (x)] = 0, :

limx+

2x

2x2

f (t) dt

= 0

28 ( ) -

I :=

0

cos(2x+ 2 sin(3x)) dx

, ,

2

29 (Math Rider) f : (0,+) R f (x) + 1

xf(x) = 1

x2,

x > 0 A

(e, 1

e

).

) f .)

323

exdx 3

23

xedx

x > 0.

) g(x) =x1

f(t)dt,

x > 0. h :(0,+) R

h(x) = g(x) + g

(1

x

) ln2x

(0,+).)

x1

ln t

tdt+ 2 2xf(x) = 2

x

1x1

ln t

tdt

x > 0.

30 ( ) f : R R f R. x0 R f (x0) > 0, lim

x+f(x).

. Juniors

31 ( ) :

2x = y + 2

y

2y = z + 2z

2z = x+ 2x

32 ( )

A = {[ 2x2 + 10x+ 19

x2 + 5x+ 7]/x R}

( )

. Seniors

33 ( ) x, y, z > 0 x+ y + z = 1

3xyz(xy+yz+zx)+2xyz (xy+yz+zx)2

34 ( ) f : R R

f(x+ y) + f(x)f(y) = f(xy) + 2xy + 1

x y.

35 ( 2010) c1(O1, r1) c2(O2, r2) , . c1(O1, r1), c2(O2, r2) r1 < r2, MA < MB.

36 ( 2010) f : R R : f

(f(x)

f(y)

)=

1

y f(f(x)),

x, y R (0,+).

37 ( ) - ( ) :

limn

nk=1(k 1)knk=1(k + 1)

k

38 ( ) An = {k {1, 2, . . . , n} : 2k 1}

an = |An|

limn

ann

.

39 ( ) -

K1 = {a+ bp : a, b Q}

K2 = {a + bq : a, b Q} , p, q , .

40 ( ) -

nk=1

(k,n)=1

e2ikn

41 ( ) f : R R , x0 R,

f(x0) limxx0

f(x)

f(x0+) lim

xx+0f(x)

R {}. f .

42 ( )

+n=0

(3n

n

)1

8n

43 ( ) , ( ).

44 ( ) ( ) X Rn x X. Y X :

|Y | n+ 1 x Y .

45 ( ) - ()

()| ()

()

3.

3

46 ( )

an = [n2] nN

2.

()

47 ( ) f : [a, b] R. n

1 < 2 < ... < n

[a, b] :

f(b) f(a)b a =

f (1) + f (2) + ....+ f (n)n

48 ( ) A

A2 A + I = O, B = A kI k R A.

.

49 ( ) (, ) limxa+ f(x) =limx f(x) = , (, ) f () = 0.

50 ( ) f : (0,+) (0,+) g g(x) = f2(x) + f(x),x > 0, lim

x+f(x)

x=

1

2

.

51 ( ) Lagrange : , , 5, 1974 f(x) 1, x1, x2, ..., x x, y1, y2, ..., y , :

f(x1) = y1, f(x2) = y2, ..., f(x) = y

52 ( ) ( 17 Hilbert). f(x) , f(x) 0 x R. A(x),B(x) ,

f(x) = (A(x))2 + (B(x))2

x R.

4

:

1 (maths-!!) , a 1 a 10 a b 1 b 10. ... 100. , ; ;

http://www.mathematica.gr/forum/viewtopic.php?f=44&t=9776

1 ( ) ( 100), 89 (11 100). 89 11 89 ( 78) ... , . :1,12,23,34,45,56,67,78,89 ( ) , . - 67 ( 78) . ( ) .

2 ( ) 9 13 16 ( 16) 22 9 6. 17 5 13 19. 15

24 27. 20 15.


1 ( ) 25 20 , 15. 16 13 , 9. 3 17 , 22 5. 2 ( ) 5 .

.1 25 164 23 312 2 8

10 20 1321 5 1711 18 7

27 15 926 24 2214 19 6

5

:

3 ( ) . , , , . , ( );

http://www.mathematica.gr/forum/viewtopic.php?f=33&p=62943

(chris t) 2 , 1 4 ( 1 4 - ). .

, - 1 1x 4x .

1x +4x = 1 x = 5.

15 ,

25 ,

35 ,

45 , 4 . (

55 -

). ( ) . (

) . .

4 ( ) - 170 330. , - 13 , 27 . ,

14

. ;


( ):

1/3 , 3,

2/7 . , 7,

1/4 , 4.

3, 7, 4 170 330, (3,4,7)=84 170 330, 252.

6

:

5 ( ) x y . 110 .


() , a .

220 + a. ( 360 360/12=30 ): 30360 =

a220+a

220 + a = 12a a = 22011 a = 20 20 240, : 60360 =

x240 x = 40 .

6 ( ) 20 20 cm. . 15 .


(Broly) 4 10 cm . . 200 14.14cm < 15cm

:

7 ( ) x2+y2 = 3xy, x, y , (

x

x y)2

+

(y

x y)2


( ):(x y)2 = x2 2xy + y2 = 3xy 2xy = xy :

7

(x

x y)2

+

(y

x y)2

=x2

(x y)2 +y2

(x y)2 =

=x2 + y2

(x y)2 =3xy

xy= 3

8 ( ) 0 < b < a

a2 + b2 = 6ab a+ b

a b


( )

I =a+ b

a b I2 =

a2 + b2 + 2ab

a2 + b2 2ab =8ab

4ab= 2 I =

2.

:

9 ( ) - :x2

x 1 +x 1 +

x 1x2

=x 1x2

+1x 1 +

x2x 1

http://www.mathematica.gr/forum/viewtopic.php?f=19&p=44884#p44884

( ) x > 1. a =x2x1 , b =

x 1, c =

x1x2

. abc = 1, , :

a+ b+ c =1

a+

1

b+

1

c= ab+ bc+ ca

q = a+ b + c, a, b, c

y3 qy2 + qy 1 = 0 1. a, b, c 1. a 1. c. ,

x4 + 1 = x. x4 + 1 2x2 > x, x > 1. b = 1, x = 2.

10 ( )

| 1| x2 + |3 2|x+ |1 | = 0 = 1 x .) .) . ) ( 1) ( 2) 1 2 ) , .


( )) | 1| x2 + |3 2| x+ |1 | = 0

- :

> 0 |32a|24|a1||1a| > 0 (3 2a)24(1 a)2 > 0 (3 2a 2 + 2a) (3 2a+ 2 2a) >0 5 4a > 0 a < 5

4, a = 1

a (, 1) (1, 54)) 1, 2 = 0 1 = 2 :

12 =|1a||a1| =

|1a||1a| = 1 > 0

1 .

1 + 2 = |32a||a1| < 0 .) 1 = 42

2 =0 12 = 422 422 = 1 22 =14 2 = 12 .

12 = 1 1 = 2 (1, 2) =

(2, 1

2

)) 1 + 2 = |3 2a||a 1|

5

2=

|3 2a||a 1| |6 4a| =

|5a 5|

6 4a = 5a 5

6 4a = 5a+ 5

a = 119 4 |x 5| > 2 5 .) [1,5].: 1 x 5 x 1 0 5x 0 :

x 1 +5 x = 2

x1+5x+2x 15 x = 4 x 15 x = 0: x = 1 x = 5

14 ( ) - , .


1 ( ) b, c, a a. : 2c = a+b, E = tr, t = a+b+c2E = bc2 , 2tr = bc, (a + b + c)r = bc 3cr = bc, b = 3r,w = a c = 2r + x (r + x) = r x .3, 4, 5 2 ( ) x ,x, x + (x + ) : (x+ )2 = x2 + (x )2 x2 4x = 0 x(x 4) = 0 x x = 4 :

E = 12x(x) = x+ x + x+

2 x = 3

4 = 3 =

10

:

15 ( ) K . A :A1 A A2 A1 KA3 A2 B A3 K AB 4- K .


1 ( ) ...

A1, A,B K, ( ), . A1KB A3KA2 , A1B = A3A2. A

K (), AA1= 2. , = = + , A2A3 = 2A2Z= 2( + 2) = 2 +4.

AB = A1B A1A= 2 +4 -2 =4. 2 ( ) P ()A1A2 , A, P, A3 K , - K (), APA3.

, KA3 = KB K A3 = K A ( KA2 = KA1 ), AB KK .

, A3AB, AB = 2 KK = 4 KQ, Q () KK . 3 ( ) AA1A3A2 A1BA2A3 . MK = K2 =

A3A2A1A4 MK =

A1BA1A4 =

AB4

AB = 4MK.

.

16 ( ) , . . . .

11


1 ( ) 4 , , , : , K2 KN2 = M2 MN2. , AB2 AO2 = EB2 OE2. 2 O2 = E2 OE2. = AB2 2 = EB2 E2. AB2 2 =OB2 O2. EB2 E2 = OB2 O2.. . 2 ( ) : EA OB = 0 ( EO + OA)

OB = 0. EO OB + OA OB = 0.(1)

EO O + O O = 0.(2) O O = O2 =

OA

2= OA OB. (1),(2)

EO ( O OB) = 0 EO B = 0 , .

:

17 ( ) OA =

OB = ,

= + ||

AOB


1 ( ) |||| =|| +

|| , , . 2 ( Papel ) ao = a|a| , o =

|| 1 (). : = ... =||(o + o) 1, 2

: 2 =

ao(ao+o)|ao||ao+o| =

1+aoo|ao+o| =

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

(a,) = |||| =||||+

||

(,) = |||| =||||+

|| . (,

) =

(, ). , . , .

18 () 3KA+2

KB+

A =5

K KA=6 , KB=8 , K=5 . :

) .

) v =KB+K A = 327


( ) )

3KA+ 2

KB+

A = 5

K

3KA+ 2

KB+

A 3K 2K = 0

3(KAK

)+ 2

(KBK

)+A = 0

12

3A+ 2

B +

A = 0

2A+ 2

B = 0

A =

B

, . 4K2 = 2AK2 + 2KB2 AB24 52 = 2 62 + 2 82 AB2AB2 = 100AB = 10

.) (

KB+ K)A = 0

KB A + KA = 0 (1)

KB A =

KB A (KB,A) = 8 5 810 =32 (2) (

(KB,

A

)= B) K A =

K A (K,A) = 5 5 725 = 7 (3) (

(K,

A

)= 2B = 2B 2B = 1625 925 =

725 ) (1), (2), (3) 32 + 7 = 0 = 327 ( ) 0 = (KB +K)

A =

KB

A +

K

A =

Ao

A

KB +

Ao

A

K =

A(o

A

KB + o

A

K) =

A(

PB +

P) .

A,PB+P ,,

PB +

P = 0

PB = P || = PBPPB = P < 0 =

PBP}

, 82 62 = 2.10.(P) (P) = 75(PB) = 75 + 5 =

325 : = 327

:

19 ( ) 200 5 , . 4, 20, 60, 50, 40, 30 20.

1. .

2. 5 20 .

() .

() (), 5.

() ) : 20, 60 , 50, 40 ,30 .


( )

1. a . : [a, a + 4), [a + 4, a + 8), [a +8, a + 12), [a + 12, a + 16), [a + 16, a + 20). 20, :20(a+2)+60(a+6)+50(a+10)+40(a=14)+30(a+18)

200 = 20 2a+4+6a+36+5a+5a+4a+56+3a+54 =200 20a = 200 a = 10. :[10, 14), [14, 18), [18, 22), [22, 26), [26, 30).

2. X Y : Y = (X+5)0, 8 Y = 0, 8X + 4

() Ry = YmaxYmin = (0, 8 30+4) (0, 8 10+4) = 16

() :0, 8 10 + 4 = 12, :[12, 16), [16, 20), [20, 24), [24, 28), [28, 32)

() ( ) Y = 0, 8X+4, ,

13

.

20 ( ) 2 + 2 2 2 , 2. .

1. : : .

2. : P () = P ();

3. .


(.)

1. :P (A) =

2+2(2+2)+(22) =

2+222 .

2. : P () = P () P () = 1P () 2

2+222 = 1 = 4

3. : f(x) = x2x+22x2x , x

[2,+). , : P (A) = f(), = 2, 3, ... x 2, :f (x) = (2x1)(2x

2x)(x2x+2)(4x1)(2x2x)2 =

x24x+1(2x2x)2

> 0, x > 2+

3

= 0, x = 2+3

< 0, x < 2+3

f

[2, 2 +

3]

[2 +

3,+

). , : 2 f(3)

f(4) < f(5) < ..., : P (A2) > P (A3) P (A4) < P (A5) < ...: P (A3) P (A4) = 815 1428 = 130 > 0 P (A3) > P (A4). : P (A4) P (A), = 2, 3, 4, .... = 4. = 4 P () .

:

21 ( ) (

1 +3

22

+

3 122

i

)72


( ) z iz = 1+

3

22

+3122

z2 =(1+

3

22+

3122i)2

= 1+23+3

8 +2318 i 12

3+3

8 =3+i2

z4=(

3+i2

)2= 3+2

3i1

4 =3i+12 = i

3i2

z6 = z4z2 = i3i2

3+i2 = i

3+14 = i..

z72 =

(z6)12

= i12 =(i4)3

= 1

22 ( )

2z2 = 6iz + 3, z C


1 ( )2z2 = 6iz + 3 4z2 = 12iz + 6 [(2z)2 12iz + (3i)2] + 3 = 0 (2z 3i)2 (

3i)2 = 0

(2z 3i3i)(2z 3i+

3i) = 0

z = (3+

3)i

2

z = (33)i

2

2 ( ) w = iz : 2w2 = 6w + 3, .

14

, az2 + bz + c = 0, R, C ( , R -

, , C, ). C , .

:

23 ( ) f f(f(x)) = x2 x+ 1, x R. g : R R g(x) + xf(x) x2 = 1, x R


( ) f g .

f(f(x)) = x2 x+ 1 x f (x) :

f(f(f(x))) = f2 (x) f (x) + 1

f(x2 x+ 1) = f2 (x) f (x) + 1 x = 1 : f (1) = f2 (1) f (1) + 1 f (1) = 1.

g(x) + xf(x) x2 = 1 x = 0

g(0) = 1

x = 1

g(1) + f(1) 12 = 1

g(1) = 1

.

24 ( ) f, g R .

3f(x) + 5g(x) + 8f(x)g(x) = 0 (1)

R.


1 ( ) 3f(x) + 5g(x) + 8f(x)g(x) = 0 (8f(x)+5)(8g(x)+3) = 15. f(x), g(x) < 0 8f(x) + 5 < 5 8g(x) + 3 < 3 x1 < x2 . f, g , 8f(x1) + 5 < 8f(x2) + 5 8g(x1) + 3 < 8g(x1) + 3 (2) 8f(x1)+5 > 0 v, (2) 15 < 15. 8f(x1) + 5 < 0 8g(x1) + 3 < 0, :- 8f(x2) + 5 < 0 8g(x2) + 3 < 0, (2) 15 > 15, - (5 >)8f(x2) + 5 > 0 (3 >)8g(x2) + 3 > 0, 15 < 15, . 2 ( ) (1) - x1, x2 x1 < x2. :f(x1) < f(x2) < 0 g(x1) < g(x2) < 0 (2) (1) 3f(x1)+5g(x1)+8f(x1)g(x1) = 0 g(x1)(5+8f(x1)) =3f(x1) g(x1) = 3f(x1)5+8f(x1) , 5 + 8f(x1) < 0 (2) g(x2) = 3f(x2)5+8f(x2) , 5 + 8f(x2) < 0 g(x1) < g(x2) 3f(x1)5+8f(x1) f(x2)

5+8f(x2) f(x1)(5 + 8f(x2)) > f(x2)(5 + 8f(x1))

5f(x1) + 8f(x1)f(x2)) > 5f(x2) + 8f(x1)f(x2)) f(x1) > f(x2) , . (1) .

15

:

25 ( ) R

(f (x))2 = f(x)

x.


( ). y . (a, b) y(x) > 0, x (a, b). y a, b Rolle y y (a, b) (). y k . y(k) = 0 x > k |y| = y > 0 () y 2y = x+ c1, x > k 2y = x+ c2, x > k k c1 = c2 = k c1 (-)=(+) y(x) = 1/4(x k)2, x k x < k c2 y(x) = 1

4(x k)2, x R

y(x) = 0, x R ( y ) ( y Darboux). (c, d) : y(x) = 0,x (c, d)

c d y y ,

y(x) =

{0 x < c14 (x c)2 x c

y(x) =

{0 x > d14 (x d)2 x d

y(x) =

14 (x c)2 x c

0 c < x < d14 (x d)2 x d

c, d Rc < d

26 ( ) f :R (0,+) , :f = f f ;


( ) f(x) >0 x R.

x f(x) f(f(x)) > 0,x R f (x) > 0 f R [1]

f(x) > 0 [1] f(f(x)) >f(0) f (x) > f(0) (f(x) xf(0)) > 0 f(x) xf(0) R

x < 0 f(x)xf(0) < f(0) 0f(0) f(x) < (1 + x)f(0) x < 0 [2]

f(x) > 0 f(0) > 0. x < 1 [2],

16

:

27 ( ) f [0,+) lim

x+ [xf (x)] = 0, :

limx+

2x2

x2

f (t) dt

= 0


( ) xf(x) = g(x), :

limx+ g(x) = 0,

limx+

2x2x2

g(x)

xdx = 0.

2x2x2

g(t)

tdt

2x2x2

g(t)t dt 2x2

x2

g(t)x2 dt =

1

x2

2x2x2

|g(t)| dt = 1x2

x2 |g()| = |g()|, (x2, 2x2).

, x +, + lim

x+ |g()| = 0. , . G(y) :=

yx2

|g(t)| dt [x2, 2x2] x > 0.

28 ( )

I :=

0

cos(2x+ 2 sin(3x)) dx


(James Merryeld) cos(2x+2 sin 3x) ,

I =

0

cos(2x+ 2 sin 3x) dx =

1

2

cos(2x+ 2 sin 3x) dx :=I1

t=x+=

1

2

20

cos(2t 2 + 2 sin(3t 3)) dt =

1

2

20

cos(2t 2 sin 3t) dt =

1

2

cos(2t 2 sin 3t) dt := I2,

2I = I1 + I2=

cos(2x) cos(2 sin 3x) dx= 5/3

/3cos(2x) cos(2 sin 3x) dx (1).

2Iu=x+2/3

= 5/3/3

cos

(2u 4

3

)cos(2 sin(3u 2)) du =

5/3/3

cos

(2u 4

3

)cos(2 sin 3u) du (2)

2Iu=x+4/3

=

7/3/3

cos

(2v 8

3

)cos(2 sin(3v4)) dx =

5/3/3

cos

(2v 2

3

)cos(2 sin 3v) dv (3).

(1) + (2) + (3)

17

6I = 5/3/3

(cos 2x+ cos

(2x 2

3

)+ cos

(2x 4

3

))

=0

cos(2 sin 3x) dx =

0. f : R R T ,

a+Ta

f(x) dx =

T0

f(x) dx a R.

cosA + cosB =2cos

A+B

2cos

AB2

.

: a, c Z c a, 20

cos(ax +

b sin(cx)) dx = 0.

:

29 (Math Rider) f : (0,+) R f (x) + 1xf(x) =

1x2

, x > 0 A

(e, 1e

).

) f .)

323

exdx 3

23

xedx

x > 0.

) g(x) =x1

f(t)dt, x > 0.

h : (0,+) R

h(x) = g(x) + g

(1

x

) ln2x

(0,+).)

x1

ln t

tdt+ 2 2xf(x) = 2

x

1x

1

ln t

tdt

x > 0.


) x > 0 : f (x) + 1xf(x) =

1x2

xf (x) + f(x) = 1x xf (x) + (x)f(x) = 1x (xf(x)) = (lnx) xf(x) = lnx + c f(x) = lnx+cx ,x > 0.

A Cf f(e) = 1e ln e+ce = 1e 1+ce = 1e 1 + c =1 c = 0 f(x) = lnxx , x > 0. ( ) f (0,+) f (x) = (lnx)

xlnx(x)x2 =

1lnxx2

f , f :

f (0, e], [e,+) f(e) = 1e .

( limx0+

f(x) = limx0+

lnxx = lim

x0+(1x lnx

)=

(+) () = limx+ f(x) = limx+

lnxx =(

++

)(DLH)

= limx+

(lnx)(x) = limx+

1x = 0 ).

) ) x > 0 :f(x) f(e) lnxx ln ee e lnx x ln e lnxe ln ex xe ex (1) 0 < 3 < 2 < e f (0, e] f(

3) < f(2) ln

3

3< ln 22 2 ln

3 0. , ( x), 3 2

3 :

23

3

(ex xe)dx 0 23

3

exdx 23

3

xedx 0

18

23

3

xedx 23

3

exdx 23

3

xedx 23

3

exdx

23

3

exdx 23

3

xedx 3

23

exdx 3

23

xedx

x > 0.) g(x) =

x1

f(t)dt =x1

ln tt dt , x > 0.

g (0,+)[ f(t) = ln tt (0,+) ].

g(1x

)=

1x1

f(t)dt =

1x1

ln tt dt

(0,+) (0,+) (x) = 1x g(x). h (0,+) h(x) =

(g(x) + g

(1x

) ln2x) = x1

ln tt dt+

1x1

ln tt dt+ ln

2x

= lnxx + ln 1x1x

(1x

) 2 lnx(lnx) = lnxx + x ( lnx)

( 1x2

) 2 ln x 1x =lnxx +

lnxx 2 lnxx = 2 lnxx 2 lnxx = 0

h(x) = 0 x > 0. h (0,+).) ) h(x) = c, c R x > 0.

h(1) = g(1) + g(11

) ln21 = 2g(1) 0 = 2 11

f(t)dt =

2 0 = 0. h(1) = c 0 = c. h(x) = 0 g(x)+g ( 1x)ln2x = 0 g(x)+g ( 1x) = ln2x, x > 0 (2) :x1

ln tt dt+2 2xf(x) = 2x

1x1

ln tt dt

x1

ln tt dt+

1x1

ln tt dt+

2 2x lnxx = 2x g(x) + g(1x

)+ 2 2 lnx = 2x

(2)ln2x + 2 2 ln x = 2x xln2x + 2x 2x lnx = 2 xln2x+ 2x 2x lnx 2 = 0 (3) t(x) = xln2x+2x 2x lnx 2, x > 0. = 1 t(1) =1 ln21+2 1 2 1 ln 1 2 = 0+2 0 2 = 0.

x = 1 t(x) = 0 ( (3. t (0,+) t(x) = (x)ln2x+x(ln2x)+(2x)2(x) lnx2x(ln x)(2) =

ln2x+ 2x lnx(lnx) + 2 2 lnx 2x 1x 0 =ln2x+ 2x lnx 1x + 2 2 ln x 2 =ln2x+ 2 lnx+ 2 2 lnx 2 = ln2x > 0

x > 0 x = 1. t (0,+) ( t(x) (0, 1) (1,+) t(x) x = 1). x = 1 t(x) = 0.

30 ( ) f : R R f R. x0 R f (x0) > 0, lim

x+ f(x).

http://www.mathematica.gr/forum/viewtopic.php?f=55&t=7807&p=44481#p44481

( ) x > x0. f [x, x0] , (x, x0) f () =f(x)f(x0)

xx0 . f R :

> x0 f () > f (x0) f(x)f(x0)

xx0 > f(x0)

xx0>0f(x) f(x0) > (x x0) f (x0)

f(x) > f(x0) + (x x0) f (x0) x R :f(x0) + (x x0) f (x0) 0 f(x0) + xf (x0) x0f

(x0) 0 f(x0) + xf

(x0) x0f (x0) 0 x x0f(x0)f(x0)f (x0) ,

, R f(x0) + (x x0) f (x0) > 0 x (,+). f(x) > f(x0) + (x x0) f (x0) > 0, x (,+), 0 < 1f(x) 0 im

x+ f(x) = +.

19

:

31 ( ) :2x = y + 2y2y = z + 2z2z = x+ 2x


( ) x, y, z = 0 x, y, z > 0. :

y + 2y 2y 2y = 2

2 2x 22 x 2

: y 2, z 2 :

x+ y + z =2

x+

2

y+

2

z

(x 2, 2x

2),(y 2, 2y

2),(z 2, 2z

2)

x = 2x , y =2y , z =

2z : (x, y, z) =

(2,2,2)

(x, y, z) =(

2,

2,

2)

32 ( )

A = {[2x2 + 10x+ 19

x2 + 5x+ 7]/x R}

( )


( ) : K(x) = 2x2+10x+19x2+5x+7

, x R : x2+5x+7 :D = 3 < 0 : K(x) = 2 + 5

x2+5x+7, x R

: x2 + 5x + 7 =(x+ 52

)2+ 34 , x R :

x2 + 5x + 7 34 , x R : 0 < 1x2+5x+7 43 0 < 5x2+5x+7 203 , x R : 2 0 x+ y + z = 1

3xyz(xy + yz + zx) + 2xyz (xy + yz + zx)2


1 ( ) (: xyz (xyz)2). :

3

(1

x+

1

y+

1

z

)+

2

xyz (

(1

x+

1

y+

1

z

)220

a = 1x .

1

a+

1

b+

1

c= 1 ab+ bc+ ca = abc,

3(a+ b+ c)+2abc (a+ b+ c)2. ( abc )

abc = ab + bc + ca (a+ b+ c)2

3.

3(a+ b+ c) (a+ b+ c)2

3 9 a+ b+ c,

: a+ b+ c = (a+ b+ c)(1a+

1

b+

1

c) 9

- .

2 ( )

(

xy)2 =

x2y2 + 2

xyz(

x)

=

x2y2

x+ 2xyz

=

(x3y2 + xy2z2 + x3z2) + 2xyz.

(x3y2 + xy2z2 + x3z2) 3xyz(

xy)

,x3(y2 + z2) 2xyz(

xy).

- x3(y2+z2)

2x3yz = 2xyz(

x2) 2xyz

xy

.

.

34 ( ) f : R R

f(x+ y) + f(x)f(y) = f(xy) + 2xy + 1

x y.


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

f(1) = 1, x = 1 f(x) = 2x1 . f(1) = a = 1, f(1) = 0 (x, y) = (z, 1) (x, y) =(z,1) f(z + 1) = (1 a)f(z) + 2z + 1 f(z 1) = f(z) + 2z + 1. f(z + 1) =(1 a)f(z 1) + a(2z + 1)

f(x) = (1 a)f(x) + a(2x 1) ()

x x

f(x) = (1 a)f(x) + a(2x 1).

:

(a2 2a)f(x) = 2a2x (a2 2a)

a 0 2, f(x) =2axa2 1 a = 2 f(x) = x 1. a = 2 a = 0. a = 0 (*) f(x) = f(x) (x, y) = (z, z) (x, y) = (z,z) - : f(2z)+ f2(z) = f(z2)+2z2+1 1 + f2(z) = f(z2) 2z2 + 1. :f(2z) = 4z21, f(x) = x21. f(x) = 2x 1, f(x) = x2 + 1, f(x) = x2 1.

:

35 ( 2010) c1(O1, r1) c2(O2, r2) - , .

c1(O1, r1), c2(O2, r2) r1 < r2, MA < MB.

21

http://www.mathematica.gr/forum/viewtopic.php?p=57805

( ) : O1ABO2 O1A, O2B O1A < O2B. M O1A = O1M O2M =O2B, AM < MB. - O1(0, 0), A(0, a), B(b, a), O2(b,c) a, b, c > 0 M(x, y). |OA| = |OM | , |O2M | =|O2B| : a2 = x2 + y2 (1),(bx)2+(y+ c)2 = (a+ c)2 (2). (1) (2) 2c(a c) = b(b 2c). M a > y b > 2x (3)., AM < BM : x2 + (a y)2 < (b x)2 + (y a)2 a2 = x2 + y2, 0 < b(b 2x), (3).: (O1, r1) (O2, r2) () , - .

36 ( 2010) f : R R : f

(f(x)

f(y)

)=

1

y f(f(x)),

x, y R (0,+).

http://www.mathematica.gr/forum/viewtopic.php?p=35560

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

xf(f(x)) f(f(x)) = f(1)x, x

R. a = f(1) f(f(x)) = ax, x R. x = 1

a

f(f

(1

a

))= 1 (1).

y = f(1

a

) (1)

: f(f(x)) = 1f(1a

)f(f(x)), x R ax =

1

f(1a

)ax, x R x = 1 :f

(1

a

)= 1 (1): f(1) = 1

a = 1. (1) : f(f(x)) = x, x R( 1-1 R)

: f(f(x)

f(y)

)=

x

y, x, y R (2).

x 1 y f(y)

: f(1

y

)=

1

f(y),y R (3)

(2) y 1y

(3)

: f(xy) = f(x)f(y), x, y R . x =y = 1 f(1) = 1 f(1) = 1. f(1) = 1 f 1 1. f(1) = 1 . (0,+) f . g(x) = ln f(ex), g(x + y) = g(x) + g(y) g . ( . ), g(x) = ax a R. x > 0 ln f(ex) = ax f(x) = xa a R. f(f(x)) = x a2 = 1 a = 1 a = 1 f(x) = x f(x) =

1

x, x R. f .

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

1

x, x (0,+).

:

(i) f(x) = x, x (0,+) f(x) = x, x R. x < 0 y < 0. xy > 0 f(x)f(y) = f(xy) f(x)f(y) = xy y = 1 f(x)(1) = x f(x) = x x (, 0). ( x > 0 y < 0. f(x)f(y) = f(xy) xf(y) = f(xy) y = 1 : f(x) = x x > 0 f(x) = x x < 0 . x < 0 y > 0.)

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

f(x) = 1x, x R

.

f(x) = x, x R f(x) = 1x, x R

2 ( ) (1) y = x :f(f(x)) = x f(1) (2). (2) 1 1 f . (2) x = 1 : f(f(1)) = f(1) 11 f f(1) = 1. f(f(x)) = x x, R (3) f

(f(x)

f(y)

)=

x

y, x, y R

(4)

22

(4) x = 1, y = 1 : f(

f(1)

f(1))

=

1 f(

1

f(1))

= f(f(1)) , , f(1) = 1.

(4) y = x : f(

f(x)

f(x))

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

(4) x = 1, y = x :f(

1

f(x)

)=

1

x

f

[f

(1

f(x)

)]= f

(1

x

) f

(1

x

)=

1

f(x)(5).

f (0,+) : x > 1 f(x) > f(1) = 1 > 0 0 < x < 1 1

x> 1 f

(1

x

)> f(1) = 1 > 0 (5) f(x) > 0.

x > 0 f(x) > 0. f (, 0) f(x) < 0 x < 0. , f R. , (3), :

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

: f(x) = 1x

x R.

:

37 ( ) () :

limn

nk=1(k 1)knk=1(k + 1)

k


( ) an =

nk=1(k1)k bn =

nk=1(k+1)

k. (bn) , . ,

limn

an+1 anbn+1 bn =

limn

nn+1

(n+ 2)n+1= lim

n

(1 +

2

n

)(1 2

n+ 2

)n+2= e2

Cesaro-Stolz, limn

nk=1(k1)knk=1(k+1)

k e2. : Cesaro-Stolz :

(an), (bn) (bn) , . limn an+1anbn+1bn , limn anbn .

38 ( ) An = {k {1, 2, . . . , n} : 2k 1}

an = |An|

limn

ann

.


( ) 2k 1 k [ ln 10ln 2 , ln 10ln 2 + 1) N.

an =

n ln 2

ln 10

+O(1)

limn

ann

=ln 2

ln 10.

( ): ln 2/ ln 10 Weyl :

, 0 a b 1 An = {k {1, 2, . . . , n} : a {k} b}, lim

n|An|n

= b a.

23

:

39 ( )

K1 = {a+ bp : a, b Q}

K2 = {a+ bq : a, b Q} , p, q , .


( ) . : Q

(p) Q (q)

, = p ()2 = p. = () Q

(q) 2 = p. = x + yq

x, y . (x2 + y2q

)+ (2xy)

q = p xy = 0

p,p/q .

40 ( )

nk=1

(k,n)=1

e2ikn


( ) sn. n . - n- 1 sn = 1. n = pk k > 1 p . n- pk1- , 0, sn = 0. n = pq gcd(p, q) = 1. k 1 s q 1 r p (s, q) = 1 (r, p) = 1 k sp+rq mod pq . ,

spq =

pqk=1

(k,pq)=1

e2ikpq =

qs=1

(s,q)=1

pr=1

(r,p)=1

e2i(sp+rq)

pq = sp + sq.

sn n (1)r, n r . (n) Mobius.

:

41 ( ) f : R R , x0 R,

f(x0) limxx0

f(x)

24

f(x0+) lim

xx+0f(x)

R {}. f .


1 ( ) x f n N. f(x) f(x+) = (x, n) :

() y1 (x2, x) f(y1) f(x)

12>

k + 1/2

2m.

2m

k + 1 1 > 0)(x (a, a + 1))(f(x) < 0) x1 (a, a + 1) f(x1) = 2f(a + 1). , lim

xbf(x) = :

( ba2 > 2 > 0)(x (b 2, b))(f(x) < 0) x2 (b 2, b) f(x2) = 2f(b 2). 2f(a + 1) = 2f(b 2) Rolle [x1, x2] ( x1, x2 1, 2 a+ b

2) . ..

2f(a + 1) > 2f(b 2), - (a, x1) x3 f(x3) = 2f(b 2) . Rolle

[x3, x2]. 2f(a+ 1) < 2f(b 2). 1 ( ) , . .1) g , . ..g(x) = 1

1+f2(x) a < x < b g(a) = g(b) = 0.

( arctan)g(x) = arctan f(x) a < x < b g(a) = g(b) =/2.2) : y = c y = f((a + b)/2). f , - x = (a + b)/2. , x = (a + b)/2 f(x) > c limxa+f(x) c, . . p, q p < (a + b)/2 < q f(p) = f(q) = c. Rolle [p, q]. . , : ( + - ) . 3 ( ) . : f 1-1 ( ) (, )

29

. f 1-1 x1, x2 (, ) x1 = x2 f(x1) = f(x2) Rolle [x1, x2] [x2, x1] .

50 ( ) f : (0,+) (0,+) g g(x) = f2(x)+f(x),x >0, lim

x+f(x)

x=

1

2


1 ( ) f = g = (f2 + f) = 2ff + f (*). f(x) = 1/2 (*) 1/2 = 0. (*) f = f2f+1 > 0. (**) f x c +.

c (**) f c2c+1 , x f 12 c2c+1 = , () f(x) . ,f(x) , (**) f 1/2. . De lHospital f

(x) = f

, 1/2. 2 ( ) - , f : , g g , , lim

x+ g(x) = k > 0 g(x) < k,x > 0 g , g = f = f2f+1 > 0,

g(1+n2 ) 0.

P (x) =

mi=1

(x zi)(z zi)

i=1

(x i)2mi K

=

(a2m

ki=1

(x zi))(a2m

ki=1

(x zi))

L

=

(A(x) + iB(x)) (A(x) iB(x)) = A2(x) +B2(x)1. K zi, zi

i .

2. L zi, zi -.

3. A(x) , B(x) R[x].4. (x i)

f(x) = (x )2k+1g(x) g() = 0, g, f ,

Prasolov Polynomials, Springer, 2004 . 1967 T. Motzkin F (x, y) = x2y2(x2 + y2 3) + 1. - .

31

http://www.mathematica.gr . .

Leonardo da Vinci

(32-) . 30 . - - - quasiregular , ( - ). (0, 0,),

( 12,

2, 1+

2

), 1+

5

2 .



mathematica.gr

1. (Mihalis_Lambrou) 2. (nsmavrogiannis) 3. ( ) 4. (m.papagrigorakis)

5. ( ) 6. (Rigio) 7. ()


1. ( )

2. ()

3. (nkatsipis)

4. ( )

5. (chris_gatos)

6. (gbaloglou)

7. (R BORIS)

8. (dement)

9. (swsto)

10. (achilleas)

11. ( )

12. (Demetres)

1. (spyros)

2. (vittasko)

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11. (rek2)

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14. (bilstef)

15. (xr.tsif )

1 ( ) : . . , , , . , - , , . . - , . , ., . , . , , - . . , ,

2 (papel) . ;

3 ( ) 30% , 20 30% .

;

4 ( ) 9 1, 2, 3, ..., 8, 9 . . x;

5 ( KARKAR ) , E1. - , E2.

E2E1

6 ( ) . 7 . 2 . : 1...6 .

7 ( ) a, b, c

a+ b+ c = 0

1

b2 + c2 a2+1

c2 + a2 b2+1

a2 + b2 c2 = 0

8 ( ) a b

a

b+

a+ 1

b+ 1+ ...+

a+ 2009

b+ 2009= 2010

a = b

,

9 ( ) > 2 > 2 > + .

10 ( ) m

|2x |2x 1|| = m2x

,

11 ( ) B = 45 = 30. B = 15.

12 ( ) , , , .

1

,

13 ( ) :

(2 + 1)(x+1)

2

+ (2 1)(x+1)2 = 2

14 ( )

3x+1 9x + 3 5x 25x = 15x + 3

,

15 (KARKAR) AB a, O, M O. M A N . N N, A . A a .

16 ( ) 28 . .

,

17 ( ) AB. E Z AB B, , E Z - A.

18 ( )

, , - x

x2 + x + = 0

) 2 4 0

)

2 4 = 0

//

,

19 ( ) f : R R :

f (x) =

{ax2+x+b

x+3, x = 3

5 , x = 3 f yy 2 , .. a = 1 b = 6.. f x = 3.. f xx

20 ( ) x, y, z, x < y < z < 3, 3 4.) x = 1 = 5)

5

2 y z.

) 6 x1, x2, x3, x4, x5, x6

6i=1

xi = 38 6

i=1

x2i = 244

10 .

, ,

21 ( ) - a, b, c 1 : a + b + c = 1 :) ab+ bc+ ca = abc) (1 a)(1 b)(1 c) = 0) 1

a2009+ 1

b2009+ 1

c2009= 1

) a, b, c, , - .

22 ( ) x2 x + = 0, , R z1, z2 C R. , : z31 + z

22 = 1

, ,,

23 ( Iason Pap.) f : R R f (f(x)) = x3 x R

24 ( ) f : R R

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

x, y R. :) f (0) = 0 f (x) = f (x) x R) N x1, x2, ..... , x R :f (x1 + x2 + ..... + x) = f (x1) + f (x2) +..... + f (x) (2)

) f(1

)= f(1)

N.

) R, f (q) = q q Q.

, ,

25 (Stelmarg )

P (x) = x2n+1 2x+ 1, n N, n 2 (0, 1)( ) : (0, 1) ( , - ). xn , (0, 1), (xn) limxn = 12 .

26 ( ) f : R R f(x) > 0,x R y R y(a) = k R y , a, k, f(t) ,

y(x) f(x)y2(x) = 0, x R

, ,

27 ( )

F (t) =

t0

sin x

1 + x2dx

t > 0.

28 ( )

I =

10

xn(2 x)ndx = 22n1

0

xn(1 x)ndx = J

2

. Juniors

29 ( ) ABC, (AB = AC)

A = 20. AC D,

DBC = 60 AB E, ECB = 30.

EDB

30 ( ) f, g :[0, 1] R , x, y [0, 1] ,

|f(x) + g(y) xy| 14

. Seniors

31 ( ) a, b, c

a+ b+ c = 3

:

a

b2 + 1+

b

c2 + 1+

c

a2 + 1 3

2

32 ( ) ( , ) ABCD E, F, AB, CD ,

AE

EB=

DF

FC= p

BC, EF, AD, - K, L, M,

BK

KC=

EL

LF=

AM

MD= q

KL

LM= p

33 ( ) f : R R f(f(x)) + f(x) = x x R. 34 ( )

k=0

(1)k(2k + 1)3

=3

32

35 ( ) A Q(x) = 1

2(x,Ax) (x, b).

Q x0 Ax0 = b 1

2(b, A1b). ( n

n A x Rn (x,Ax) > 0.)

36 ( ) f Z[x] x1, . . . , xn M = max {|ai| : 1 i n}. m Z |m| > M + 1 |f(m)| . f Z[x].

37 ( ) - 1

2

0

ln(1 x) ln xx (1 x) dx .

38 ( )

limn

n2( 1

0

n1 + xn dx 1

)=

2

12

-

39 ( ) .

40 ( )

(An)nN, (Bn)nN

X

An X = Bn n N

41 ( )

111...111 91

. : , , , . , 1998.

42 ( )

A = {2n 3|n N} .

()

43 ( )

1 +1 x

2x=1 x2 + x

44 ( ) ABC

a2 cos2A+ b2 cos2B = c2 cos2 C.

;

.

45 ( ) f R f(x)sin3x, f(x)cos3x -, f .

46 ( ) P (x) - n n x1, x2, ..., xn. Q(x) - n 1,

nk=1

Q(xk)

P (xk)= lim

xxQ(x)

P (x)

.

3

47 ( irakleios) P (x) R[x] :

(x 1)143 + (x+ 1)2002 = [P (x)]13

48 ( ) p(x) n, n , xi, xj xixj >1.( : )

: p(x) - n n - 1 , xi, xj p

xi xj > 1

4

:

1 ( ) : - . . , , , ., , , . - . - , . , - . , - . , . , , . . , ,


( ) . 6 18 . .

,,,,,,,, . (

.)[ 1 3 , 2 1 3 2 .] . 7 . . ( .) - .

2 (papel) . ;


( ) , ;.

5

:

3 ( ) 30% , 20 30% .

;

http://www.mathematica.gr/forum/posting.php?mode=edit&f=33&p=65774

( 70% 20 30% . 40% 20l. 20% 10l, 100% , 50l.

4 ( ) 9 1, 2, 3, ..., 8, 9 . . x;


1 ( ) 1:

1 + 2 + 3 + ...+ 9 = 45

2: 4 + 7 + 9 = 20. 3: x x x 20 23 45. 4: 21, 1 45 21 21 = 3 x = 3 2, 5, 6, 8 x - 22, 2 45 22 22 = 31 x = 1 3, 5, 6, 8 x . 1 : , , 2 ( ) A ,

A+A = 1 + 2 + 3...+ 9+ x = 45 + x

x . x , 1, 3, 5 ( 7, 9 - ). x = 1 ,x = 3 , x = 5 .

:

5 ( KARKAR ) , E1.

, E2. E2E1


1 ( ) (K, R) . E1

E1 = 2R2

= R2.

6

x. KH, .H = x

2K = HZ = x HZK.

:

K2 + H2 = KH2

, x2 +

x2

4= R2

x2 =4R2

5

E2 =4R2

5

E2E1

=4R2

5

2R2=

2

5

6 ( ) . 7 . 2. : 1...6 .


1 ( ) . 2+7 6+2 . 9+9.9+9=1..6. 9+9196 9+9106 . , 1023. 9+9 6. 11 22 9+9 6 =13. 33 80. 2 ( ) , : , 9+9, 9(+1). 9. 1..6 9, 2 (1+2+6=9 , , 9) 126. 9+9=126 =13.

:

7 ( ) - a, b, c

a+ b+ c = 0

1

b2 + c2 a2 +1

c2 + a2 b2 +1

a2 + b2 c2 = 0


1 ( () )

a+ b+ c = 0

(a+ b)2 = c2 a2 + b2 c2 = 2ab

:b2 + c2 a2 = 2bc

a2 + c2 b2 = 2ac

1b2+c2a2 +

1c2+a2b2 +

1a2+b2c2 =

12bc +

12ac +

12ab = 12

[a2bc+b2ac+c2ab

(abc)2

]=

= 12 (a+ b+ c) = 0

7

8 ( ) a b

a

b+

a+ 1

b+ 1+ ...+

a+ 2009

b+ 2009= 2010

a = b


1 ( ) a > b, a+ kb+ k

>

1 k = 0, 1, 2, . . . , 2009 ( 2010 ). k = 0, 1, 2, . . . , 2009,

2010 =a

b+

a+ 1

b + 1+ ...+

a+ 2009

b+ 2009> 1 + 1 + + 1 = 2010

() a b. ( ), b a, a = b. 2 ( )

a

b+

a+ 1

b+ 1+ ...+

a+ 2009

b+ 2009= 2010

a

b+

a+ 1

b + 1+ ...+

a+ 2009

b + 2009=

b

b+

b+ 1

b+ 1+ ...+

b+ 2009

b+ 2009a bb

+a bb+ 1

+ ...+a b

b+ 2009= 0

(a b) (1

b+

1

b+ 1+ ...+

1

b+ 2009

)= 0

a b = 0a = b

:

9 ( ) > 2 > 2 > + .


( ) 1 : > 2, > 2. :

> 4 (1)

> 2 2 > 0, > 2 2 > 0 :

2 2 + 4 > 0 (2) (1) (2) :

2 2 2 > 0 > +

2 = : 2 > 2 2 2 > 0 ( 2) > 0 , > 2. > . :

2 > + (1) ( ) :

> 2 > 2 (2) ( ) (1) (2):

2+ > + + 2 > + > .3 > + : 2 > 2+ 2 : + 2 2 > 0 : ( 2) + ( 2) > 0, > 2 > 2.4

f(x) = ( 1)x

> 2 A = [0,+). x1, x2A f(x1) < f(x2) ( 1)x1 < ( 1)x2 ( 1)(x1 x2) < 0 x1 x2 < 0 x1 < x2,( > 2), f A.f(2) = ( 1) 2 = 2 > 0 , x > 2 :f(x) > f(2) > 0 = x x > 0 x > x+ x = , > 2 : > +

10 ( ) m

|2x |2x 1|| = m2x


( ) m = 0, 2x =|2x 1| x = 1/4. m = 0,

m2x = |2x |2x 1|| 0

x 0. , |2x 1| = 1 2x |4x 1| = 1 4x

1 4x = m2x

(4m2)x = 1

m2 4 , m2 > 4, x = 14m2 .

8

:

11 ( ) B = 45 = 30. B = 15.


1 ( ) =45

= (1)

=60 =/2= (1) =

= (2)

==30 (3) (2,3) =15 . 2 ( ) B = x :

b

2. sinx=

B

sin 30

:

b

2. sin(45 x) =B

sin 75

sin(45 x)sinx

=sin 75

sin 30, (1)

x = 15 (1),

f(x) =sin(45 x)

sinx

3 ( ) :

45=

105

=

45

75

:2

=

(150 )

=

2

(30 + )

:22

6+2

4

=2

(30 + ) 2

2

6 +2=

4

+3

2 +

6 = 2

6 + 2

2

2 =

(6 + 2

2) =

3 + 2

0 < < 45, = 15.

( : 15 = 15

15=

6+

2

4624

= ... =3 + 2)

4 ( ) - .

, -. - A(0, 1) B(1, 0)

(3, 0

).

A = 105, B = 45 = 30.

(

3

2,1

2

). :

B =12

32 + 1

=13 + 2

=3 2

0 < B < 45

B = 15

9

12 ( ) , , , .


1 ( ) ABC ACD : BAH1 = //2(OM), DH2 = //2(OM) AH1 =//DH2 :

BD = //H1H2

. .

:

13 ( ) :

(2 + 1)(x+1)

2

+ (2 1)(x+1)2 = 2

http://www.mathematica.gr/forum/viewtopic.php?f=21&t=12239#wrapheader

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

2 1 = 1

(2 + 1)(x+1)

2

+ (12 + 1

)(x+1)2

= 2 az + az = 2 az = (

2 + 1)(x+1)

2

2. az = 1 (x + 1)2 = 0 x = 1. 2 ( ) a + b 2ab, a, b > 0 : (2 + 1)(x+1)2 +(2 1)(x+1)2

2[(2 1)(2 + 1)](x+1)2 = 2

:(2 + 1)(x+1)

2

= (2 1)(x+1)2

(2 + 1)2(x+1)

2

= 1 = (2 + 1)0

(x+ 1)2 = 0 x = 1 3 () :(2 1)(2 + 1) = 1 (2 1) = (2 + 1)1

: (2 + 1)(x+1)

2

+ (2 + 1)(x+1)

2

= 2 a+ b 2ab, a, b > 0 (2 + 1)(x+1)

2

+ (2 + 1)(x+1)

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

(2 + 1)(x+1)

2

= (2 + 1)(x+1)

2 x = 1 4 ( ) :av + ( 1a )

v = 2 a2v 2av +1 = 0 (av 1)2 = 0 av = 1 a =

2 + 1, v = (x+ 1)2 :

(2 + 1)(x+1)

2

= 1 (x+ 1)2 = 0 x = 1

14 ( ) 3x+1 9x + 3 5x 25x = 15x + 3

http://www.mathematica.gr/forum/viewtopic.php?f=21&t=12190#wrapheader

1 ( ) 3(3x + 5x) = 9x + 25x + 3x5x + 3 (1) - 9x + 25x 29x25x = 2.5x3x (1) 3(3x + 5x 3x5x 1) 0 (3x 1)(1 5x) 0 (3x 1 5x 1) (3x 1 5x 1) x 0 x 0 x = 0

0 2 () 3x = w 5x = y. w2 + y2 + wy 3y 3w + 3 = 0 y2+(w3)y+w23w+3 = 0 y D = 3(w 1)2 w = 1 y = 1. x = 0

10

( ) : (+ 2)2 + ( 1)2 + ( 1)2 = 0, (1) -

= = 1 (1) = 3x = 5x

.

:

15 (KARKAR) AB a, O, M O. M A N . N N, A . A a .


1 ( ) , , : A

2

4

=

x

(1)

, :

x

=

2

42

2+

2

4

(2) (1), (2)

A =2

6.

2 ( ) ( ). (0, 0) (1, 0), (1, 1), (0, 1).

O(1

2,1

2

).

M(3

4,1

4

).

N

=1

3 :

y =1

3x N

(1,

1

3

).

: y = x

N=

3, : y 13

= 3 (x 1)

(5

6,5

6

),

(A) =

2 (1 5

6

)2=

2

6 , :

(A) =

2

6 3 ( ) OM:( 45o) = 12 = 3 N :(90o ) = Na N = a3 AN = 2a3 AN :

A(90) =

AN(135)

A =

2a3(45+) A = a

2

6

11

16 ( ) 28 . .


1 ( ) :

+ + = 28 (1)

:

AZ2 = (AK)(A) ( )2 = 13

2

3 =

2

92 (2)

:M2 = (M)(MK) =

2

92 (3)

(2) (3) :

AZ = M = ( ) 2 = 2 (4)

(2) :

(14 )2 = 29.22 + 22 2

4

(1) (4) : 212+35 = 0 : 1 = 7, 2 = 5 : 1 = 7, 1 = 14, 1 = 7 2 = 5, 2 = 10, 2 = 131. .2. 14, . C(A, 0). - - - . .

:

17 ( ) AB. E Z AB B, , E Z A.


( ) AB = , A = , AM = xA, ME = yEAM = x

A = x( + ), (1)

AM =AE ME = 12

AB yE = 12 y(

+ 12

) =(1y)

2 + y , (2)

(1), (2) x = y = 13

AM = 13

A

CN

18 ( ) , , x

x2 + x + = 0

) 2 4 0

12

)

2 4 = 0

//


( ) x = 0 x = 0 = 0, .) 2 4 0 x22 4x2 0 ( x2 )2 4x2 0 ( x2 )2 0, .) x22 = ( x2 )2 4x2 = ( x2 )2 ...( x2a)2 = 0 = x2a :2x2+ x = 0 = 2x . //

:

19 ( ) f : R R :

f (x) =

{ax2+x+b

x+3 , x = 35 , x = 3

f yy 2 , .. a = 1 b = 6.. f x = 3.. f xx

http://www.mathematica.gr/forum/viewtopic.php?f=18&t=11229&p=61361#p61361

( ) f yy -2,

f(0) = 2 a 02 + 0 + b

0 + 3= 2 b = 6 (I)

f R, -3, : lim

x3f(x) = f(3). x = 3,

: f(x) = ax2+x+bx+3 f(x)(x + 3) = ax2 +

x + b, limx3

[f(x)(x + 3)] = limx3

(x2 + x+ b)

0 = (3)2 + (3) + b) 9+ b = 3 (II). () () : a = 1 b = 6.. a = 1 b = 6, :f (x) =

{x2+x6x+3 , x = 35 , x = 3

=

{x 2 , x = 35 , x = 3 = x 2

f R , f (x) =1, f (3) = 1.. f B(0,2) A(0, 2). f OA = OB = 2, (OAB) = 2..

f xx, = 1, = 4 , [0, ) x (0, /2), . 20 ( ) x, y, z, x < y < z < 3, 3 4.) x = 1 = 5

) 5

2

y z.) 6 x1, x2, x3, x4, x5, x6

6i=1

xi = 38 6

i=1

x2i = 244

10 .


( ) i. :

x = 4 (1) :

+ z

2= 3 + z = 6 (2)

:x++z+

4 = 3 x + + z + = 12(2) x + 6 + = 12 x+ = 6 (3) (1) (3) : x = 1 = 5ii.

2 =(1 3)2 + ( 3)2 + (z 3)2 + (5 3)2

4

5

2=

4 + ( 3)2 + (z 3)2 + 44

5

2=

( 3)2 + (z 3)24

+ 2

13

12=

( 3)2 + (z 3)24

( 3)2 + (z 3)2 = 2 (4) (2),(4) :

( 3)2 + (z 3)2 = 2(2)

( 3)2 + (3 )2 = 2 ( 3)2 = 1 3 = 1 = 4 z = 2 3 = 1 = 2 z = 4 x = 1, = 2,z = 4, = 5

iii.

=

6i=1

xi + 12

10=

38 + 12

10= 5

12 + 22 + 42 + 52 = 1 + 4 + 16 + 25 = 4610i=1

xi = 50

10i=1

x2i = 244 + 46 = 290

2 =110 (

10i=1

x2i (10

i=1

xi)2

10 ) =110 (290 50

2

10 ) =110 (290

250010 ) =

110 (290 250) = 4

=4 = 2

:

21 ( ) a, b, c 1 : a + b + c = 1 :) ab+ bc+ ca = abc) (1 a)(1 b)(1 c) = 0) 1a2009 +

1b2009 +

1c2009 = 1

) a, b, c, , .


1 (giannisn1990)) a = 1

a b = 1

b c = 1

c

a+ b+ c = 1 a+ b+ c = 1 a+ b+ c = 1 1a+

1

b+

1

c=

1 ab+ bc+ ca = abc) ab+ bc+ ca = abc ab+ bc+ ca abc = 0 b(a+ c) + ca(1 b) = 0 b(1 b) + ca(1 b) = 0 (1 b)(b+ ca) = 0 (1 b)(1 a c+ ca) = 0 (1 a)(1 b)(1 c) = 0) (1a)(1b)(1c) = 0 a = 1 b = 1 c = 1 a = 1 . 1

b2009+

1

c2009=

0 b2009 = c2009 a + b + c = 1 b+ c = 0 b = c b2009 = c2009 b2009 + c2009 = 0) |b c|2 = |a c|2+ |b a|2 . a = 1 b+ c = 0 |2c|2 = |c+ 1|2 + |c 1|2 |c+ 1|2 + |c 1|2 = 2(|c|2 + 1) = 4 |2c|2 = 4 22 ( ) x2 x+ = 0,, R z1, z2 C R. - , : z31 + z

22 = 1


1 ( ) z21z1+ =0 z22 z2 + = 0 z2 = z1 - z31 + z22 = 1 z21 z22. a2z1ab z1b+az1 b = 1 (1). z1, z2 C R D = a2 4b

z1 =a

2+

4b a22

i

(1) a2 b a = 0 (2) ( a2) a ab 1 = 0 (3) (2), (3) (a, b) = (1, 0) (a, b) =(1, 2). (1, 0) (1, 2) . 2 ( )

z31 +

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