i Cos i Dodecahedron 13

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Leonardo da Vinci

(32-) . 30

. -

- - quasiregular ,

( - ).

(0,0,),( 12 ,

2 ,

1+2

), 1+

5

2 .

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

:

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4. (rek2)

5. (hsiodos)

6. (bilstef)

2

1 ( ) - . , , .

() ;

() -, ;

2 ( KARKAR) - a, b,

1a+

1b=

29.

3 ( ) - 11 20 ;

4 ( PetranOmayromixalis) 3 , 4 -, 5 . ;

5 ( KARKAR) AB,CS .

6 ( KARKAR) ABCD - 8 3 M,N AB,AD.

() S DC ,

S MN 13

ABCD.

() BT B MS , 3.

7 ( ) AB = A, B = = P + 2. (AB)

8 ( vzf) - a, b, c, m > 2, am + bm < cm.

,

9 ( qwerty) :

A =3

108 + 10 3

108 10.

10 ( ) -

(3x 1)(4x 1)(6x 1)(12x 1) = 5.

3

,

11 ( )

BC ABCD E, F, : BAE = CDF EAF = EDF. FAC = EDB.

12 ( )

ABC(AB = AC). - (O) O. S - O, S AC. (O) B, S P. C, S , P, BP T ( P T ).

() S O, C, T ;

() - = ACS = 420, ABC CT PS .

,

13 ( )

2x2 + xy = 19x2

2(1x)4 = 1 +3xy

2(1x)2

14 ( )

) R :

{x + y = 4

x4 + y4 = 82

) R :441 x + 4

41 + x = 4

,

15 ( )

ABC M BC. S A, AB,AC, , S EAB S ZAC. T AM EZ, : T SBC.

16 ( ) :

PA,PBE EC A, B,C - () ABC -.

BHAC. HB PHE,

,

17 ( ) -

x2 2x y2 4y 32 = 0 (Q) .

4

() - R.

() .

() (Q) (3,0) 0 . 0 Q0 x = 1.

() - Q0 1 - - 2 - - . 1, 2 3

2 .

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

18 ( ) A(1, 1), B(2,5) AB I(2, 2) - . .

,

19 ( parmenides51) f (x) = ex x 0 .

() f (x) - A

P(A) = f (x) = ex

x 0.() ()

.

20 ( ) : x1, x2, ..., x( 2). - :

f (x) = |x x1 | + |x x2 | + ... + |x x|

x = .

, ,

21 ( ) z K(0, 1) 2,

|z2 (2z + 1)i + z 5| .

22 ( ) z

11z10 + 10iz9 + 10iz 11 = 0.

:

() |z| = 1.

()

(2z zi 2i + 1

1 + zi

)2 0.

() Re (z) =|z + 1|2 2

2.

() |z2 3z + 1| = 5 |z + 1|2.

, , ,

23 ( ) - :

f (x) =

1, x < 00, x = 01, x > 0

g(x) = x2 2x + 4 .

f g g f .

24 ( ) x > 0 f

f (e f (x)) = ln x.

f .

, ,

25 ( ) - f : R R :f (x) = ex+ f (x), x R f (0) = 0.

26 ( ) f : R R f (R) = R. C f + y = x + :

() > 0.

() ( ) C f1 +.() f 1(x) + < x x R.

, ,

27 ( ) -

:

1x

1

ext

t2dt e (1 x) x > 0.

28 ( ) f : [1,+) (0,+) :

f (1) = 1,

f (1,+), x > 1 xx f (x) = f (x) f

(x) ln x,

5

x [1,+) x+1

x

f (t) dt < f (x).

, ,

29 ( )

f (x) =

x2 ln

x +73 c , x 1

x log1x+ c , 0 < x < 1

76

, x = 0

f x0 = 1.

1. , .

2. .

3. f x0 = 0 - C f x = 0, x = 1 xx.

30 ( ) x, y R,

(2x + 3y)

e24y + y2 + (1 x)ex + 1 =

(ex + y)

x2 + y2 + 2xy + x + 3,

x + 2y = 1.

, ,

31 ( ) f : R R

(f (x) x2

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

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

f .

32 ( ) 0 < a < b f : [a, b] R ,

ab

b

a

f (x)

x2dx 0. f M(x0, f (x0) Ox A.

() AB 1

| ln x0|.

() AB 1, M f - .

34 ( ) x, y R, :

A =2014

x2 + y2 4x + 2y + 58.

Juniors,- -

35 ( ) a, b > 0, :

(1 +

a

b

)2014+

(1 +

b

a

)2014 22015.

36 ( )

xn =

n +

n2 1 n = 1, 2, . . . .

1x1+

1x2+ . . . +

1x49

.

Juniors,

37 ( KARKAR) A, B CD. B , CD, - P,T . PC T D S , CA DA L,N. S L = S N.

38 ( ) ABC AB , AC . (C) ABC, H , O (C). M BC. AM (C) N (C) AM P.

6

() AP, BC,OH AH = HN.

() ;

Seniors,- -

39 ( ) f : Z+ Z+ ,

f (x2 + f (y)) = x f (x) + y,

x, y Z+.

40 ( ) a, b, c > 0

(a + b + c)

(1a+

1b+

1c

) 9

a2 + b2 + c2

ab + bc + ca.

Seniors,

41 ( parmenides51.) ABCD (O,R) (I, r). OI = x

1r2=

1(R + x)2

+1

R x)2.

42 ( .)

ABC (O) (K), - AD. BE, CF, DZ , E (K)AC, F (K)AB, Z (K) (O) Z , A.

43 ( ) - 100 . 30% - , .

44 ( ) p1 < p2 < ... < p99

p1 + p2, p2 + p3, . . . , p98 + p99, p99 + p1

;

45 ( Kyiv Taras 2013) n n A, B n n C

AX + Y B = C

. n n C

A2013X + Y B2013 = C

.

46 ( ) a R f : (a,+) R, (a, b) b R a < b.

limx+

f (x + 1) f (x)xk

,

k N,

limx+

f (x)

xk+1=

1k + 1

limx+

f (x + 1) f (x)xk

7

47 ( ) V Rnn -, W Rnn Rnn V,W.

48 ( ) p(x) - 2. -

A = {x R \Q : p(x) Q}

R.

49 ( )

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

0sin

(x + f 2(t)

)dt,

x [0, 1].

50 ( ) an - . {

an

m: n.m N

} (0,+).

51 ( )

= OA , = OB = O

3,

(~, ~

)=

(~, ~

)=

(~, ~

)=

3.

A =( ( ( ))) ( ) .

52 ( ) - M

x2

2+

y2

2= 1 , , R ,

, MA MB, -. A B,

2

x2+2

y2= 1 .

53 ( ) 2 , 2 .

54 ( vzf) . , 9.

55 ( petros r). a - .

I =

+

0

sin x cos x

x(x2 + a2)dx.

56 ( vz f ) - W B , h.

()

57 ( )

K := cot 70 + 4 cos 70.

58 ( )

f (x) =| sin 2x|

| sin x| + | cos x| + 1.

,

59 ( - . ) (an)nN . :

n=1

an < ,

n=1

an1

nn < .

, , :

n=2

an = s < ,

n=2

an1

nn s + 2

s.

60 ( ) - f : R R ,

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

x, y R.

,

8

61 ( ) y2 = 2px E . n 3 A1A2A3 An E - . EA1, EA2, . . . , EAn B1, B2, . . . , Bn - :

EB1 + EB2 + EB3 + + EBn np.

62 ( ) - ABCD. , O :

(AOB) = (BOC) = (COD) = (DOA).

,

63 ( ) n 1 a C R |a| = 1.

n

k=0

(nk

)(1 + ak)xk = 0.

:

() a

() a, b a + b = 0.

64 ( )

arccos

(15

)= 2 arctan

23

.

9

:

1 ( ) - . , , .

() ;

() , ;

http://www.mathematica.gr/forum/viewtopic.php?f=44&t=31906

( ) .

:

() , -

.

() ,

.

()

-

,

.

-

.

Nk

k. (Nk) .

Nk+1 < Nk,

k + 1 ().

()

m , m .

(Nk(m+1))

, -

.

-

,

.

2 ( KARKAR) - a, b,

1a+

1b=

29.


1 ( )

1a+

1b=

29 2ab 9a 9b = 0

4ab 18a 18b = 0 (2a 9) (2b 9) = 81.

: (a, b) = (5, 45) , (a, b) =(6, 18) , (a, b) = (9, 9) , (a, b) = (18, 6) , (a, b) = (45, 5) .

2 ( ) -

Rh-ind, -

2

2n + 1 n = 2, 3, . . . , 50 -

, 1.

29=

16+

118.

-

:

10
http://www.mathematica.gr/forum/viewtopic.php?f=44&t=31906http://www.mathematica.gr/forum/viewtopic.php?f=44&t=31494

22n + 1

=1

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

1n + 1

.

:

29=

145+

15.

:

- G. Loria, , .

3 ( )

1a+

1b=

29 a = 1

2

(9 +

812b 9

)

a -

:

2b 9 = 1 b = 5, a = 45, (a, b) = (45, 5).2b 9 = 3 b = 6, a = 18, (a, b) = (18, 6).2b 9 = 9 b = 9, a = 9, (a, b) = (9, 9).2b 9 = 27 b = 18, a = 6, (a, b) = (6, 18).2b 9 = 81 b = 45, a = 5, (a, b) = (5, 45). -

.

11

:

3 ( ) - 11 20 ;


( ) 10 , 40. 10

212

360 = 60.

10

4060

112

360 = 20.

80.

4 ( PetranOmayromixalis) 3 , 4 , 5 . ;


(Geopa)) ,

:

+ = 3. + = 4. + = 5.

2( + + ) = 12. + + = 6.)

:

+ + = 3. + + = 4. + + = 5. :

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

12

:

5 ( KARKAR) - AB,CS .


( )

COS = 20. BO CO COB

OCB = OBC = 40.

OCS = 140 CS O = 20. S CO S C = CO. CO = BO

:

S C = BO (1)

BAO

BOK, K AO .

BOK = 120 :

BAO =BOK

2= 60

, BOA :

AB = BO (2)

(1) (2) .

6 ( KARKAR) ABCD 8 3 M,N AB, AD.

() S DC , -

S MN 13

ABCD.

() BT - B MS , 3.


( )

() DS = x, S C = 8 x. (ABCD) = 24, (MAN) = 3, (MNS ) = 8

(DNS ) + (S MBC) = 13.

3x4+

3(12 x)2

= 13

x =203.

() S E = CB = 3, EB = S C =43

ME =83.

S ME

S M2 =

(83

)2+ 32 S M =

1453

.

13

S MB

BT S M2

=S E MB

2 BT = 36

145.

BT 2 =1296145

< 9 = 32 1296 < 1305,

BT < 3.

14

:

7 ( ) AB = A, B = = P + 2. (AB)


( ) A -

B , A

B.

AP = 6 .

2= P2 + P2 (P + 2)2 = 64 + P2

4P = 60 P = 15

P = BP = 8.

(AB) = (AB) + (B) E =6 16 + 15 16

2= 168.

8 ( vzf) a, b, c, m > 2, - am + bm < cm.


( ) ac< 1, b

c< 1 m > 2

:

(a

c

)m+

(b

c

)m 0 , 0 P(A) 1. ex 0 . f (x) = ex 1 . f (x) = ex < 0 f .

x 0

f (x) f (0) ex 1 0 ex 1.

() -

P(A) =N(A)

N() N() =

N(A)

P(A)

N() =N(A)

ex.

N()

N(A) ex. -

N(A) , N(A)

. ..

.

20 ( ) :x1, x2, ..., x( 2). :

f (x) = |x x1| + |x x2| + ... + |x x|

x = .


( ) :

x1 x2 ... x.

() : = 2 + 1 ( N).:

x1 ... x x+1 = x+2 ... x2 x2+1 (I)

x < x1, :

|x x1| +x x2+1

= x + x1 x + x2+1= x2+1 x1 + 2 (x1 x)

> x2+1 x1 .

x > x2+1, :

|x x1| +x x2+1

= x x1 + x x2+1= x2+1 x1 + 2

(x x2+1

)

> x2+1 x1 .

x1 x x2+1, :

|x x1| +x x2+1

= x x1 x + x2+1= x2+1 x1.

:

|x x1| +x x2+1

x2+1 x1 (1),

= x1 x x2+1.:

29

|x x2| +x x2

x2 x2 (2),

= x2 x x2.x x

+x x+2

x+2 x (),

= x x x+2.x x+1

0 ( + 1) ,

= x = x+1.

-

:

f (x) (x2+1 + x2 + ... + x+2

)

(x1 + x2 + ... + x

)(II)

, x , x+1, (+1) - > (II) >.

x = x+1 = , (I) (+ 1) = (II)

= . , x = f

.

() : = 2 ( N). :

x1 ... x x + x+1

2= x+1

... x21 x2 (I)

:

|x x1| +x x2

x2 x1 (1),

(=) x1 x x2.

|x x2| +x x21

x21 x2 (2),

= x2 x x21.

x x +

x x+1 x+1 x (),

= x x x+1. -

:

f (x) (x2 + x21 + ... + x+1

)

(x1 + x2 + ... + x

)(II)

x =x + x+1

2= ,

(I), () = (II) =. x = f .

30

:

21 ( ) z - K(0, 1) 2,

|z2 (2z + 1)i + z 5| .


1 ( ) w = z i - :

|w| = 2, |w2 + w 4|. w = x + yi, x, y R x2 + y2 = 4

|w|2 = (x2 y2 + x 4)2 + (2xy + y)2

= (2x2 + x 8)2 + (4 x2)(2x + 1)2

= ..........................................

= 68 16x2 .

|w|2 68 = |w| 217 -

.. x = 0, y = 2. 2

17.

2 ( )

|z2 (2z + 1)i + z 5| = |z2 2iz + i2 + z i 4|

=

(z i)2 + (z i) 4

=

(z i)2 + (z i) (z i) (z i)

= 2 |(z i) + 1 (z + i)|

= 2 |z z + 1 2i|

= 21 + 4(Im (z) 1)2

z=3i 2

1 + 4(3 1)2

= 217 .

3 ( )

2 |z z + 1 2i| = 2 |(z z) (1 + 2i)| , , -

(1, 2) (0,2) (0, 6). |(z z) (1 + 2i)|max =

17, z

(0, 3) (0,1).

22 ( ) - z

11z10 + 10iz9 + 10iz 11 = 0.

:

() |z| = 1.

()

(2z zi 2i + 1

1 + zi

)2 0.

() Re (z) =|z + 1|2 2

2.

() |z2 3z + 1| = 5 |z + 1|2.


1 ( )

()

z9 =11 10zi11z + 10i

,

|z|9 = |11 10iz||11z + 10i|

.

z = x + yi, x, y R ,

(x2 + y2)9 =121 + 220y + 100y2 + 100x2

121x2 + 121y2 + 220y + 100.

x2 + y2 > 1 x2 + y2 < 1, . - x2 + y2 < 1. x2 + y2 = 1.

31

() z =1z,

2z zi 2i + 11 + zi

2z+

i

z+ 2i + 1

1 iz

=2 + i + 2iz + z

z i

, i

2z zi 2i + 1

1 + zi.

.

()

|z + 1|2 22

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

2

=|z|2 + z + z + 1 2

2

=z + z

2

= Re(z) .

() ,

|z + 1|2 = z + z + 2 (1)

|z2 3z + 1|2 = (z2 3z + 1)(z2 3z + 1)

= (z2 3z + 1)( 1z2

3z+ 1

)

=(z2 3z + 1)2

z2

=

(z +

1z 3

)2

= (z + z 3)2 .

|z2 3z + 1| = |z + z 3|,

,

z + z . ,

z + z < 3, z + z = 2Re(z) 2|z| = 2.

2 ( )

() |z| > 1.

|z| > 1 z9

> 1

11 10zi11z + 10i

> 1

|11 10zi|2 > |11z + 10i|2

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

21 > 21|z|2

|z| < 1 .

|z| < 1 - . 1.

32

:

23 ( ) - :

f (x) =

1, x < 00, x = 01, x > 0

g(x) = x2 2x + 4 .

f g g f .


( ) D ( f ) = D (g) =

R . ,

D ( f g) ,D (g f ) R (I) .

x R . x D (g) x D ( f ) f (x) {1, 0, 1} R

f (x) R = D (g) .

g(x) = x2 2x + 4 R = D ( f ) .

R D ( f g) , D (g f ) (II) .

(I) (II)

D ( f g) = D (g f ) = R .

x R ( f g) (x) = f (g(x)) .

g(x) = x2 2x + 4 = (x 1)2 + 3 > 0 , x R

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

f g . (g f ) (x) = g ( f (x)) , x R . x R : x < 0, (g f ) (x) = g ( f (x)) = g(1) = 7 . x = 0, (g f ) (x) = g ( f (0)) = g(0) = 4 . x > 0, (g f ) (x) = g ( f (x)) = g(1) = 3 .,

(g f ) (x) =

7 , x < 04 , x = 03 , x > 0

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

limx0

(g f ) (x) = 7 , 4 = g(0) , 3 = limx0+

(g f ) (x).

24 ( ) x > 0 f

f (e f (x)) = ln x.

f .


1 ( )

g (x) = f (x) + ln x, x > 0

1 - 1.

g(e f (x)

)= f

(e f (x)

)+ ln e f (x)

= ln x + f (x)

= g (x)11

e f (x) = x

f (x) = ln x .

2 ( ) -

g ( ) g = g1, g (, .. g(x) > x g1

x = g1(g(x)) > g1(x) = g(x) > x,

). : E -

,

33

f E f = E1.

E f = f 1 E1 = (E f )1,

E f = , f = E1 = ln .

3 ( )

x0 > 0 f (x0) > ln x0,

e f (x0) > eln x0 e f (x0) > x0 .

e f (x0) > 0 , x0 > 0 f (0,+).

f(e f (x0)

)> f (x0) ln x0 > f (x0)

.

x0 > 0 f (x0) 0.

4 ( ) ex = k(x),

ln x = k1(x).

f (k( f (x))) = k1(x) .

x = k(x)

f (k( f (k(x)))) = x .

f (k(x)) = g(x), g(g(x)) = x . g(g(x))

f (ex) , 1 - 1,

. g(x) = g1(x) - , g(x) = x .

f (k(x)) = x f (x) = k1(x) f (x) = ln x .

34

:

25 ( ) - f : R R : f (x) = ex+ f (x), x R f (0) = 0.


1 ( ) -

g (x) = f (x) x, x R, -

g (0) = f (0) 0 = 0.

f (x) = ex+ f (x) (g (x) + x) = ex+g(x)x g (x) = eg(x) 1 : (1) g (x) = eg(x) 1 : (2).(1) (2) g (x)

(eg(x) 1

)= g (x)

(eg(x) 1

)

eg(x)g (x) + eg(x)(g (x)) = g (x) + (g (x)) (eg(x) + eg(x)

)= (g (x) + g (x))

eg(x) + eg(x) = g (x) + g (x) + c. x = 0 eg(0) + eg(0) = g (0) + g (0) + c c = 2. x R eg(x) + eg(x) = g (x) + g (x) + 2 : (3) ex x + 1,x R

x = 0. eg(x) g (x)+ 1

g (x) = 0. eg(x) g (x) + 1

g (x) = 0. eg(x) + eg(x) g (x) + g (x) + 2

g (x) = g (x) = 0. (3) x R

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

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

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

= ( f (x) + 1) f (x) = (e f (x)x + 1) f (x)= (1 eg(x))(1 + g(x)), g(0) = 0 (1)

g , -

2 r > 0 (a, b) a > 0, b > 0 g(a) = 0, g(b) = 0 g(x) , 0, x (a, b). ( g [0, a] a = 0).

g(x) , 0 - g(x) , 0 g(a) = 0, g(c) = 0 ( Rolle g (a, b) Rolle g (a, c)). - r < 0. g x > 0 x < 0.

g(x) > 0 R+ g(x) < 0, (2)

g(x) < g(0) = 0 g(x) < 0 . - R

g g(0) = 0. g(x) = 0, x R

:

1.

a = 0.

2. g(0) = f (0) 1 = e0+ f (0) 1 = 0.

3. (2) (1)

g(x) = (e0 eg(x))(1 + g(x))= g(x)eu(1 + g(x))

-

ex

(1 + g(x)) = ex+ f (x) > 0.

26 ( ) f : R R f (R) = R. C f + y = x + :

() > 0.

() ( ) C f 1 +.

() f 1(x)+ < x x R.

35


( )

() f : R R f

x0 R f (x0) = 0 x < x0 f (x) < f (x0) = 0 f - (, x0] f R f (x) > 0, x R.

f

(x0, f (x0))

y f (x0) = f (x0)(xx0) y = f (x0)(xx0)+ f (x0)

limx+

f (x0)(x x0) = + ( f (x0) > 0) g(x) = f (x0)(x x0)+ f (x0) lim

x+g(x) = +

, -

f (x) g(x), x R, lim

x+f (x) = +.

limx+

( f (x) x ) = 0 h(x) = f (x)x lim

x+h(x) = 0 -

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

( f (x)h(x)) =+, lim

x(x) = + -

> 0.

() f 1( f (x)) = x, x R x > 0

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

1( f (x))f (x)

f (x)

x= 1

limx+

f (x)

x= > 0 f

1( f (x))f (x)

=1

f (x)x

limx+

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

= limx+

1f (x)

x

=1.

limx+

f (x) = + u = f (x)

limu+

f 1(u)u=

1 .

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

f (x) =x f (x)

limx+

( f (x)x) = -

limx+

( f 1( f (x)) 1 f (x)) = limx+x f (x)

=

limu+

( f 1(u) 1u) = ,

y = 1 x C f 1 +.

() x f (x),

f 1( f (x)) + < f (x) x + < f (x)

h(x) = f (x) x

limx+

h(x) = 0

h(x) = f (x) f - h h R, h(x) > 0. x0 R h(x0) > 0 - h lim

x+h(x) = +

h(x) 0 , x R (1)

x0 R h(x0) = 0 - h x > x0 h(x) > h(x0) = 0, (1) h(x) < 0 , x R h - R lim

x+h(x) = 0

h(x) > 0, x R.

36

:

27 ( )

:

1x

1

ext

t2dt e (1 x) x > 0.

http://www.mathematica.gr/forum/viewtopic.php?p=191552

( )

1x

1

ext

t2dt e (1 x)

1x

1

ext

(xt)2x dt e

(1 x)x

xt=u, du=xdt

1

x

eu

u2du e

(1 x)x

0

e (1 x)x

+

x

1

eu

u2du 0

F(x) =e (1 x)

x+

x

1

eu

u2du x > 0.

F

F(x) =

e (1 x)

x+

x

1

eu

u2du

= e

x2+

ex

x2=

ex ex2

F(x) 0 x 1. F - (0, 1] [1,+). - x = 1 F(1) = 0. F(x) 0 x > 0.

1. ( Fermat) Chebyshev:

b

a

f (x)g(x) dx 1b a

b

a

f (x) dx b

a

g(x) dx,

[a, b].

2. ( ) ( -

):

ext

t2

e

t2

.

28 ( ) f : [1,+)(0,+) :

f (1) = 1,

f (1,+),

x > 1 xx f (x) = f (x) f(x) ln x,

x [1,+) x+1

x

f (t) dt < f (x).


( )

xx f (x) = f (x) f(x) ln x x > 1

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

f (x) > 0 ,

x =f (x)

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

(x)

f (x)ln f (x)

(x2) = (ln2 f (x)),

c R, ln2 f (x) = x2 + c, x > 1.

f ,

limx1+

ln2 f (x) = ln2 f (1) = 0

37
http://www.mathematica.gr/forum/viewtopic.php?p=191552http://www.mathematica.gr/forum/viewtopic.php?f=54&t=37941http://www.mathematica.gr/forum/viewtopic.php?f=54&t=37941

limx1+

(x2 + c) = 1 + c, c = 1

ln2 f (x) = x2 1, x > 1. x > 1 x2 1 , 0, ln2 f (x) , 0 .

ln f (x) =

x2 1 ln f (x) =

x2 1,

f (x) = e

x21 f (x) = e

x21 x 1.

g(x) =

x

1f (t)dt [x, x + 1]

(x, x + 1) g(x + 1) g(x) = g() x+1

x

f (t)dt = f ()

x+1

x

f (t) dt < f (x)

f () < f (x), (1)

x < .

f (x) = e

x21: f [1,+)

f (x) = e

x21 xx2 1

> 0, x > 1

(1).

f (x) = e

x21: f [1,+)

f (x) = e

x21 xx2 1

< 0, x > 1

.

f (x) = e

x21, x 1.

38

:

29 ( )

f (x) =

x2 ln

x +73 c , x 1

x log1x+ c , 0 < x < 1

76

, x = 0

f x0 = 1.

1. , - .

2. .

3. f x0 = 0 C f x = 0, x = 1 x

x.


( ) D( f ) = [0,+). f x = 1 , lim

x1f (x) = f (1) R .

limx1

f (x) = limx1+

f (x)

limx1

(x log

1x+ c

)= lim

x1+

(x2 ln

x +

73 c

)

c =73 c

c =76

,

,

f (x) =

x2 ln

x + 7/6 , x [1,+)

(x ln x/ ln 10) + 7/6 , x (0, 1)

7/6 , x = 0

() x > 1

f (x) = 2 x ln

x + x2 (

x)

x

= 2 x ln

x +x

2> 0

f

[1,+) . , x (0, 1)

f (x) = ln x + 1

ln 10

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

)

f (x) = 0 x =1e

f (x) > 0 x (0, 1

e

).

limx0

x ln x = limx0

ln x

1/xD.L.H==== lim

x0

1/x

1/x2

= limx0

(x) = 0 .

, f x = 0 f -

[1e, 1

]

[0, 1

e

] -

x = 1e

f(1e

)=

1e ln 10 +

76

x = 1 f (1) = 76 .

, f (x) f (0) = 76 , x 0 , f x = 0 , f (0) = 76 .

: f - x = 1 [0,+) , f (1) = 0 f

39

.

limx1

f (x) f (1)x 1

= limx1

x ln x

ln 10 (x 1)

= 1

ln 10limx1

x ln x 1 ln 1x 1

= 1

ln 10

[d

dx(x ln x)

]

x=1

= 1ln 10

[ln x + 1

]x=1

= 1

ln 10,

limx1+

f (x) f (1)x 1

= limx1+

x2 ln

x

x 1

= limx1+

x2 ln

x 12 ln1

x 1

=

[d

dx

(x2 ln

x)]

x=1

=

[2 x ln

x +

x

2

]

x=1

=12.

x = 0 [0,+). -, x = 1

e [0,+), f

f ( 1

e

)= 0 ( Fermat) .

, x = 1e -

f (0, 1) f [0, 1], (0, 1), f (0) = f (1) = 76 f (0, 1). (-, , -

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

()

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

(1e

)]

[f (1), f

(1e

)]

[f (1), lim

x+f (x)

)

f (0) =76, f

(1e

)=

1e ln 10

+76, f (1) =

76,

limx+

f (x) = limx+

(x2 ln

x +

76

)= + .

f ([0,+)) =[76 ,+

).

1. ), f -

x = 0 .

E =

1

0f (x) dx =

1/2

0f (x) dx +

1

1/2f (x) dx

= I1 + I2 .

, , -

76 .

I1 =

1/2

0f (x) dx = lim

a0+

1/2

a

f (x) dx

1/2

a

f (x) dx =

1/2

a

( x ln x

ln 10+

76

)dx

=

[ 1/2

a

x ln x

ln 10dx +

1/2

a

76

dx

]=

[ x

2 ln x

2 ln 10

]1/2

a

+

1/2

a

x2

2 ln 10 1

xdx +

76

(12 a

) =[

ln 28 ln 10

+a2 ln a

2 ln 10+

1/2

a

x

2 ln 10dx +

76

(12 a

)]=

ln 2

8 ln 10+

a2 ln a

2 ln 10+

[x2

4 ln 10

]1/2

a

+76

(12 a

)

I1 =2 ln 2 + 116 ln 10

+712

I2 =

1

1/2f (x) dx

= limb1

b

1/2f (x) dx

= limb1

b

1/2

( x ln x

ln 10+

76

)dx

= limb1

[ b

1/2

x ln x

ln 10dx +

b

1/2

76

dx

]

= limb1

[ x

2 ln x

2 ln 10

]b

1/2+

b

1/2

x2

2 ln 10 1

xdx +

76

(b 1

2

)

= limb1

[

ln 28 ln 10

b2 ln b

2 ln 10+

b

1/2

x

2 ln 10dx +

76

(b

12

)]

= limb1

ln 2

8 ln 10 b

2 ln b

2 ln 10+

[x2

4 ln 10

]b

1/2+

76

(b 1

2

)

= limb1

[ ln 28 ln 10

b2 ln b

2 ln 10 1

16 ln 10+

b2

4 ln 10+

76

(b 1

2

)]

= 2 ln 2 + 116 ln 10

+712.

40

30 ( ) - x, y R,

(2x + 3y)

e24y + y2 + (1 x)ex + 1 =

(ex + y)

x2 + y2 + 2xy + x + 3,

x + 2y = 1.


( )

(3, 2) ( y , 2) x = 1 2y, :

(e12y + y

)2+ 1

e12y + y=

(2 y)2 + 12 y

(1)

f (x) =

x2+1x

,

R f (x) = 1x2

x2+1< 0

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

(1) e12y + y, 2 y (1) :

f (e12y + y) = f (2 y)

f

, 1 1

e12y + y = 2 y

e12y = 1 + (1 2y).

ex 1 + x - x = 0 1 2y = 0 y = 12 x = 0 (x, y) =

(0, 12

)

.

41

:

31 ( )

f : R R (

f (x) x2) (

f (x) 1)= 1, x R

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

f .


( ) g(x) = f (x) x2 :

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

g(x) , 0, g(0) > 0

g . g(x) > 0 .

g(x) + 1 =1

g(x)= g(x)g(x) + g(x) =

g(x)

g(x)

= 12

(g(x))2 + g(x) = ln g(x) + c.

c = 1. , ln A A 1,

12

(g(x))2 0 = g(x) 0 = g ,

g(x) = 1 x. f (x) = x2 + 1, .

32 ( ) 0 < a < b f : [a, b] R -,

ab

b

a

f (x)

x2dx 0.

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

g(b) > g(a) 1b

b

a

f (t)dt a b

a

f (t)

t2dt > 0

b

a

f (t)dt > ab

b

a

f (t)

t2dt.

2 ( ) -

Chebychev. -

( f ) ( 1x2

).

:

42

b

a

f (x)

x2dx 0. - f M(x0, f (x0) Ox A.

() AB 1

| ln x0|.

() AB 1, - M f .


(dr.tasos)

()

y f (x0) = f (x0)(x x0),

A(x0 1ln x0 , 0

)

(AB) =

(x0 x0

1ln x0

)2 =1

| ln x0|.

()

| ln x0| = 1 x = e x = e1.

M1(e, ee) M2

(( 1

e)1e ), ( 1

e)1e

). -

y ee = ee(x x0)

y = (1e

) 1e

(x x0) +(1e

) 1e

.

34 ( )

x, y R, -:

A =2014

x2 + y2 4x + 2y + 58.

http://www.mathematica.gr/...php?f=69&t=42368&p=198072#p198072

( )

A =2014

(x2 2 2x + 4) + (y2 + 1 2y + 1) + 53

=2014

(x 2)2 + (y + 1)2 + 53

201453= 38

x = 2 y = 1

44
http://www.mathematica.gr/forum/viewtopic.php?f=69&t=41743http://www.mathematica.gr/forum/viewtopic.php?f=69&t=42368&p=198072

, , :

35 ( ) a, b > 0, :

(1 +

a

b

)2014+

(1 +

b

a

)2014 22015.


1 ( )

- :

(1 +

b

)2014+

(1 +

b

a

)2014 2

[(1 +

b

) (1 +

b

a

)]2014

= 2

(a

b+

b

a+ 2

)2014

= 2

a

b+

b

a

2

2014

= 2

a

b+

b

a

2014

2 22014 = 22015.

2 ( ) a

b= k

b

a=

1k

k > 0. -

(1 + k)2014 +(1 + 1

k

)2014

2

(1 + k)2014(1 +

1k

)2014.

:

(1 + k)2014(1 +

1k

)2014 22014

((1 + k)

(1 +

1k

))1007 22014

(1 + k)

(1 +

1k

) 22

(k + 1)2 4k,

(a + b)2 4ab.

3( ) :

(1 +

a

b

)2014+

(1 +

b

a

) 2

(1 +

a

b

)1007 (1 +

b

a

)1007

= 2[(1 +

a

b

) (1 +

b

a

)]1007

= 2

(1 +

a

b+

b

a+

ab

ab

)1007

= 2

(2 +

a

b+

b

a

)1007

2 (2 + 2)1007

= 2 41007 = 2 22014 = 22015.

a = b,

a, b , 0.

36 ( )

xn =

n +

n2 1 n = 1, 2, . . . .

1x1+

1x2+ . . . +

1x49

.


( )

n +

n2 1 = n + 12+

n 12+ 2

n 12

n + 12

=

n 12+

n + 12

2

.

xn =

n+1+

n12

,

1xn=

2

n + 1 +

n 1

=

22

(n + 1

n 1

).

45

1x1+

1x2+ . . . +

1x49=

22

((2 0) + (

3 1) + + (

50

48)

)=

22

(52 1 + 7

)=

5 + 32.

46

:

37 ( KARKAR) - A, B - CD. B , CD, P, T . PC T D S , CA DA L,N. S L = S N.


1 ( ) M ABCD M CD ( M -

AB ( ) (O) , (K)

MC, MD (O) , (K) ).

LAD, BAM,NAC CD LN, M CD, B

LN, S B S LN.

S DC = DT B (, )

= CBD (. )

S CD = CPB (, )

= BCD (. ).

S DC = BDC (DS ) = (DB) S DB - DS S DB DCS B

CDPT S BPT .

S LN S B -

, (S L) = (S N)

.

38 ( ) ABC AB , AC . (C) ABC, H , O (C). M BC. AM (C) N (C) AM P.

() AP, BC,OH AH = HN.

() ;

47


1 ( )

:

: O

. AN .

O

( ) -

H

HG = 2 OG. AH -

O

AN M.

OM

(O) B,C

A .

:

AH -

D AD OM

BC. (AHG), (GOM) - : AH = 2 OM. H ABC. -

H AN :

AH = HN.

:

.

AN -

.

AN

-

. ,

.

:

BC,OH, AP

,

, .

2 ( )

.

: ABC (AB , AC)

M BC. P (

A) (O)

ABC (K) AM, - PM H

ABC.

48

H PM AD (ADBC) E PM (O) .

MPA ============== 900

EPA = 900 AE (O)

:

ECAC (1)

OMBC ( BC (O)) ADBC

OM ADAHE ( O AE)============ M

M BC======= BHCE

( )

BH EC(1) BHAC

ADBC H ABC

.

P,H, M .

AH = HNOA=ON=RO====== AHNO ()

OHAN ( )

OHAM. (1)

APMHP, AM - (K) MBAH, H ABC,

AP,OH,CB AHM

T .

:

OM, AP, BC (

T ), AHT M ( H - ABC) MHPAT ( AM - (K)) H

AT M (

) T HO -

, T HOAN O ABC AN -

(O) T HO AN

HA = HN .

3 ( ) ,

H1 MP AH AK = KH1, KH1MO . -

AH1 = 2KH1 = 2OM = AH H H1 TOAN HA = HN.

AH = HN T OH,CB. AT M MH

, P MH AT MPA = 90.

49

, , : -

39 ( ) f : Z+ Z+,

f (x2 + f (y)) = x f (x) + y,

x, y Z+.


( )

x = 1, f (1 + f (y)) = f (1) + y y Z+. f (1) = a.

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

x Z+.

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

n N. n = 1

f (a + 1) = a + 1 f (1 + f (1)) = f (1) + 1

(1). f ((a + 1)n) = (a + 1)n. (1)

f (1 + f ((a + 1)n)) = a + (a + 1)n

f (na + n + 1) = a + na + n

(1) :

f (1 + f (na + n + 1)) = a + na + n + 1

f (1 + a + na + n) = a(n + 1) + n + 1

f ((a + 1)n) = (a + 1)n.

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

x = a , y = a + 2 ,

f (a2 + f (a + 2)) = a f (a) + a + 2

f (a2 + 2a + 1) = a f (a) + a + 2

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

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

f (a) = a + 1 1a.

a, f (a) N, a = 1, f (1) = 1. f (n) = n n N. n = 1 f (1) = 1, . f (n) = n. (1)

f (1 + f (n)) = f (1) + n f (n + 1) = n + 1.

f (x) = x , x Z+.

40 ( ) a, b, c > 0 -

(a + b + c)

(1a+

1b+

1c

) 9

a2 + b2 + c2

ab + bc + ca.

http://www.mathematica.gr/forum/viewtopic.php?f=111&t=12183#p66050

( ) -

, a + b + c = 9.

a2 + b2 + c2 = 27 + 6t2

t [0, 1]. t = 0, , a = b = c = 3. 0 < t 1. (x, y, z) a, b, c, (.

)

3 2t x 3 + 2t (2)

x = 3 + 2t y = z = 3 t x = 3 2t y = z = 3 + t. ,

1a+

1b+

1c

9 + 2t2

9 t2.

, ,

1x=

13 + 2t

+3 + 2t x(3 + 2t)x

51
http://www.mathematica.gr/forum/viewtopic.php?f=111&t=35047http://www.mathematica.gr/forum/viewtopic.php?f=111&t=12183#p66080

x (0, 9)

33 + 2t

+1

3 + 2t

cyclic

3 + 2t aa

9 + 2t2

9 t2.

(2),

Cauchy-Schwarz

cyclic

3 + 2t aa

cyclic

(3 + 2t a)2

cyclic

a(3 + 2t a)=

6t3 t

.

, ,

33 + 2t

+6t

(3 + 2t)(3 t)

9 + 2t2

9 t2,

3t + 9(3 + 2t)(3 t)

9 + 2t2

9 t2.

, -

9(t + 3)2(9 t2) (9 + 2t2)(3 + 2t)2(3 t)2.

,

t2(3 t)(8t3 9t + 27) 0

0 < t 1.:

Cauchy-Schwarz

:

http://www.mathematica.gr/forum/viewtopic.php?p=165244#p165244

52

:

41 ( parmenides51.) - ABCD (O,R) - (I, r). OI = x

1

r2=

1

(R + x)2+

1

R x)2.


( .) ABCD

,

B + D = 180 IBC + IDC = B2+D

2= 90 (1)

Z L, C, LZ = KB KIB,KIZ

(1), IDZ -

DIZ = 90 (2)

,

( )

1

(IL)2=

1

(IZ)2+

1

(ID)2(3)

(3), IL = r IZ = IB

1

r2=

1

(IB)2+

1

(ID)2(4)

E (O) DI F (O) BI., AOE = D AOF = B

AOE + AOF = 180(5)

(5) E,O, F -

(IB)(IF) = (ID)(IE) = R2 x2 (6)

(6)

1

(IB)2=

(IF)2

(R2 x2)2(7)

1

(ID)2=

(IE)2

(R2 x2)2(8)

(4), (7), (8)

1

r2=

(IE)2 + (IF)2

(R2 x2)2(9)

IEF,

,

(IE)2 + (IF)2 = 2(OI)2 +(EF)2

2= 2(R2 + x2) (10)

(9), (10)

1

r2=

2(R2 + x2)

(R2 x2)2=

(R + x)2 + (R x)2

(R + x)2 (R x)2(11)

(11)

1

r2=

1

(R + x)2+

1

(R x)2

.

53

42 ( .) -

ABC (O) (K), AD. - BE, CF, DZ , E (K)AC, F (K)AB, Z (K) (O) Z , A.


( .) -

(K) AEDF AFE = ADE (1) DAC DE AC

ADE = C (2) (1), (2) AFE = C , -

BCEF (L).

AZ, EF, BC, -

(O), (K), (L), ,

, S .

AEDFBC

A. S BPC , P BE CF.

CF , M, F, P, C ,

M AS CF. N (K) CZ ZNF =

ZAF ZAB = ZCB FN BC FN AD (3) (3) , -

(K) F, N, T, AD.

AFDN

FN, T

(K) , Z. AFDN -

.

CM M, F, P, C,

ZD P BE CF, M F, C .

54

:

43 ( ) - 100 . 30% - , .


( )

: 5 3 , - 3 , , .

: ,

, .

() ABC - D, E -

.

ADB, BDC,CDA

D. , -

AEB, BEC,CEA . -

.

() ABCD

E .

-

. E ABC CDA.

, ,

.

.

() ,

. -

,

-.

A, B,C .

ACDE. -

,

. ,

.

-

.

:

-

. (1005

)

,

. -

/ (972

).

3

(1005

)

(972

)

100 -.

100 (1003

).

:

3

(1005

)

(972

)(1003

) = 3100 99 98 97 96 2 6120 97 96 100 99 98

= 30%

.

44 ( ) - p1 < p2 < ... < p99

p1 + p2, p2 + p3, . . . , p98 + p99, p99 + p1

;


1 ( ) . -

p1 < p2 < < p99

55
http://www.mathematica.gr/forum/viewtopic.php?p=190087http://www.mathematica.gr/forum/viewtopic.php?p=192844

p1 + p2 = n21 , p2 + p3 = n

22, . . . , p99 + p1 = n

299

n1, . . . , n99 .

p1 + p2 + + p99 =n21 + n

22 + + n

299

2 pi . ,

ni

.

pi . p1.

p1 = 2, , n1, n99 , n2, ..., n98 .

p1 + n22 + n

24 + ... + n

298 =

n21 + n22 + + n

299

2

p1 =n21 n

22 + n

23 n

24 + n

298 + n

299

2.

n21 n22 + n

23 n

24 + n

298 + n

299 = 4.

n22 + n23 n

24 + n

298

4, n21 n299 1 mod 4.

4 - 2, .

2 ( ) ,

2 4k + 1, 4k + 3, k N.

, 4n 4n + 1, n N. :

1: p1 = 2 p1 + p2 ,

p2 = 4m2 + 3 ( p2 = 4m2 + 1, p1 + p2 = 2 + 4m2 + 1 =4 + 3 - ).

, p2 = 4m2 + 3, p2 + p3 , p3 = 4m3 + 1 (- p3 = 4m3 + 3 , p2 + p3 =4 + 2 ).

, :

p3 = 4m3 + 1, p4 = 4m4 + 3, p5 = 4m5 + 1, . . . , p99 = 4m99 + 1.

p99 + p1 = 4m99 + 1 + 2 = .4 + 3

.

2: p1 = 4m1 + 1. p1 + p2 ,

p2 = 4m2 + 3, p2 + p3 , p3 = 4m3 + 1 , , p99 = 4m99 + 1. -

p99 + p1 = 4m99 + 1 + 4m1 + 1 = .4 + 2,

.

3: p1 = 4m1 + 3. p1 + p2 ,

p2 = 4m2 + 1, p2 + p3 , p3 = 4m3 + 3, -, p99 = 4m99 + 3.

p99 + p1 = 4m99 + 3 + 4m1 + 3 = 4(m99 + m1) + 4 + 2

= .4 + 2

.

.

3 ( )

.

p1 > 2. .

4a1, 4a2, . . . , 4a99 (

4).

(p1 + p2) + (p2 + p3) + + (p99 + p1)= 4a1 + 4a2 + + 4a99,

p1 + p2 + p3 + + p99 = 2(a1 + a2 + a3 + + a99)

99 .

p1 = 2. p1 + p2, p99 + p1.

4a1 + 1, 4a99 + 1 ( -

8k + 1 4m + 1).

56

,

(p1 + p2) + (p2 + p3) + + (p99 + p1)= 4a1 + 1 + 4a2 + 4a3 + 4a4 + + 4a99 + 1.

p1 + p2 + + p99 = 2(a1 + a2 + a99) + 1

, 98 -

.

: - ,

-

viewtopic.php?p=185428 ( 2 3 )

.

57

:

45 ( Kyiv Taras 2013) n n A, B n n C

AX + YB = C

. n n C

A2013X + YB2013 = C

.


( )

A, B . -

r1, r2

n.

A = Q1

(Ir1 O

O O

)P1 B = Q2

(Ir2 O

O O

)P2

P1, P2,Q1,Q2 .

S 1 =

(O O

O Inr1

)Q11 S 2 = P

12

(O O

O Inr2

).

S 1A = BS 2 = O. C = Q1P2 -

X1, Y1

AX1 + Y1B = Q1P2,

S 1 S 2

O = (S 1Q1)(P2S 2) =

(O O

O Inr1

) (O O

O Inr2

).

n r1, n r2 > 0. ,

A ( B). C Z

A2013Z = C. (Z,O)

.

46 ( ) - a R f : (a,+) R, (a, b) b R a < b.

limx+

f (x + 1) f (x)xk

,

k N,

limx+

f (x)

xk+1=

1k + 1

limx+

f (x + 1) f (x)xk


( )

an = f (n)

limn

an

nk+1.

Stolz

limn

an

nk+1= lim

n

an an1nk+1 (n 1)k+1

= limn an an1k

t=1

(k+1

t

)(1)k+1tnt

=1(

k+1k

) limn

an an1nk

=1

k + 1limn

an an1nk

.

. ( )

58

:

47 ( ) V - Rnn , W Rnn - Rnn V,W .


( )

() V : -

.

() W : ( - ).

48 ( ) p(x) - 2.

A = {x R \ Q : p(x) Q}

R.


( ) -

:

1: f Q [x] n xn f

(1x

)

.

:. h, s m, k

xn f

(1x

)= h (x) s (x)

1xn

f (x) = h

(1x

)s

(1x

)

f (x) = xmh

(1x

)xk s

(1x

).

f (x) = h1 (x) s1 (x)

h1 (x) = xmh

(1x

) s1 (x) = x

k s

(1x

).

2:

f (x) = anxn+ + a0

Z[x] p

an

g (x) = anxn+ + a0 +

1p

.

: 1

xng

(1x

)= an + an1x + +

(a0 +

1p

)xn

.

xng

(1x

)=

1p

((pa0 + 1) x

n+ pa1x

n1+ + pan

)

(pa0 + 1) xn+ pa1x

n1+ + pan

Eisenstein: - p -

p2 p

an.

: (a, b). p(x) -

c, d a < c < d < b p (c) , p (d).

59

s = p(c)+p(d)

2 .

p(c), p(d)

(p (c) s) (p (d) s) < 0.

q(x) = p(x) s. - :

q (x) =1r

(anx

n+ ... + a0

)

a0, . . . , an . -

p1 < p2 < < pk <

an. 2 -

anxn+ + a0 +

1pk

.

qk (x) =1r

(anx

n+ + a0 +

1pk

)= q (x) +

1rpk Q [x]

.

1rpk

qk (c) = q (c) +1

rpk q (c)

qk (d) = q (d) +1

rpk q (d) .

k qk (c) , qk (d)

q(c), q(d)

Bolzano qk(x) t c, d (a, b).

:

() qk(x)

1 t ,

() qk(t) = 0

p (t) s +1

rpk= 0

p (t) = s 1rpk Q

p -

.

60

: -

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

f (x) =

1

0sin

(x + f 2(t)

)dt, x [0, 1].


( ) -

K [0, 1] sin(.), cos(.) ||.||.

T : K C[0, 1], (T f )(x) = 1

0sin

(x + f 2(t)

)dt.

T .

,

1

0sin

(x + f 2(t)

)dt =

( 1

0cos f 2(t) dt

)sin x +

( 1

0sin f 2(t) dt

)cos x =

a sin x + b cos x

T f K. |a| 1, |b| 1

||T f || = ||a sin(.) + b cos(.)|| =

a2 + b2 2,

{||T f || : f : [0, 1] R }

-

.

T -

K.

( Schauder), T , .

50 ( ) an .

{

an

m: n.m N

}

(0,+).


( ) x > 0 0 < < x. = x = x + .

bn = n cn = n.

k n k bn+1 < cn. k >

.

A = (bk, ck)

(bk+1, ck+1)

A = (bk,+) . A an (an) , - m an (bm, cm) .

m (x ) < an < m (x + ) ,

an

m x

< . .

61
http://www.mathematica.gr/forum/viewtopic.php?f=9&t=37825http://en.wikipedia.org/wiki/Schauder_fixed_point_theoremhttp://www.mathematica.gr/forum/viewtopic.php?f=9&t=41661

:

51 ( )

= OA , = OB = O

3,

(~, ~

)=

(~, ~ ) = (~, ~ ) = 3.

A =( ( ( ))) ( ) .


( )

( ) = ( ) ( ) =12 .

( ( )) = ( 12

)

=12

= .

A = ( ) ( )

= (( ) )

= ( ( ))

=

[( ) ( )

]

=

( 12 1

2

)

=12 1

2

=14 1

2= 1

4.

52 ( ) M

x2

2+

y2

2= 1 , , R ,

, MA MB, . - A B,

2

x2+2

y2= 1 .


1 ( ) M(a cos t, b sin t), t [0, 2).

A(a cos t, 0), B(0, b sin t). P(x0, y0)

A B,

AB :xx0

a2+

yy0

a2= 1.

A, B AB, a cos tx0

a2= 1 cos t =

a

x0

b sin tx0

b2= 1 sin t =

b

y0.

cos2 t + sin2 t = 1 a2

x20

+b2

y20

= 1,

P 2

x2+2

y2= 1.

62
http://www.mathematica.gr/forum/viewtopic.php?f=11&t=41083 http://www.mathematica.gr/forum/viewtopic.php?f=11&t=42673

:

53 ( ) 2 -, 2 .


( ) k

,

k2 = (n + 1)3 n3 = 3n2 + 3n + 1,

n. , :

4k2 = 12n2 + 12n + 4

4k2 1 = 12n2 + 12n + 3 (2k 1)(2k + 1) = 3(2n + 1)2

2k 1 2k + 1 . , -

() ,

.

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

m,

2k = (2m + 1)2 1 = 4m (m + 1)

k ,

k2 = 3n (n + 1) 1,

n (n + 1) .,

2k 1 = (2m + 1)2,

k = (m + 1)2 + m2,

.

54 ( vzf) -. , 9.


1 ( )

a1, a2, . . . , a17.

,

a1, a2, . . . , a5,

3, 0,1 2 3.

, , -

a1, a2, a3.

, a4, a5, a6, a7, a8,

3, ... , 17

5 3.

0, 3 6 9., 3

5, - 9.: ( )

Erdos-Ginzburg: - 2n 1 n n.

63
http://www.mathematica.gr/forum/viewtopic.php?f=63&t=41369#p192805http://www.mathematica.gr/forum/viewtopic.php?f=63&t=31929#p147794

:

55 ( petros r). a - .

I =

+

0

sin x cos x

x(x2 + a2)dx.


( )

:

0

sin 2xx

dx =

0

sin y

ydy =

2.

:

0

sin x cos x

x(x2 + a2

) dx = 12

0

sin 2x

x(x2 + a2

) dx

=1

2a2

0

sin 2x

(1x x

x2 + a2

)dx

=1

2a2

0

sin 2xx

dx 1

2a2

0

x sin 2x

x2 + a2dx

=

4a2 1

2a2

0

x sin 2x

x2 + a2dx

=

4a2 1

4a2

x sin 2x

x2 + a2dx

f (z) =ze2iz

z2 + a2,

. -

| f (z)| =ze2i(x+iy)

z2 + a2

=|z|

e2y (cos 2x + i sin 2x)

z2 + a2

=|z| e2yz2 + a2

z 0.

( )

z = ia

Res ( f (z) , z = ia) = limzia

(z ia) f (z) =e2a

2.

f (z)

C -

xx x1 = M x2 = M = M :

C

f (z) dz = 2i Res ( f (z) , z = ia) = ie2a.

M +

xe2ix

x2 + a2dx = ie2a

x (cos 2x + i sin 2x)

x2 + a2dx = ie2a

x sin 2x

x2 + a2dx = e2a

0

sin x cos x

x(x2 + a2

) dx = 4a2

1

4a2

x sin 2x

x2 + a2dx

=

4a2 1

4a2e2a =

4a2(1 e2a

).

64

56 ( vz f ) - W - B , - h.


( ) -

,

B = GMm

R2,

m =BR2

GM.

h :

GM

R G

M

R + h,

-

:

(G

M

RG M

R + h

)BR2

GM= B

Rh

R + h.

65

:

57 ( ) -

K := cot 70 + 4 cos 70.


( )

K =sin 20

cos 20+ 4 sin 20

=sin 20 + 2 sin 40

cos 20

=

3

cos 20

12

sin 10 +

32

cos 10

=

3

cos 20sin 70

=

3

58 ( )

f (x) =| sin 2x|

| sin x| + | cos x| + 1.


( ) -

0 x =k

2, k Z.

| sin 2x| = 2| sin x|| cos x| | sin x|2 + | cos x|2 = 1

x = k +

4, k Z. -

2| sin 2x| | sin x| + | cos x|

2| sin 2x|2 1 + | sin 2x| 2| sin 2x|2 | sin 2x| 1 0 (| sin 2x| 1)(2| sin 2x| + 1) 0

.

| sin 2x| = 1 x = k +

4, k Z.

| sin 2x|+2| sin 2x| | sin x|+ | cos x|+ 1 f (x)

1

1 +2

x = k +

4, k Z.

1

1 +2.

66

:

59 ( - . ) (an)nN . :

n=1

an < ,

n=1

an1

nn < .

, ,:

n=2

an = s < ,

n=2

an1

nn s + 2

s.


( ) n > 2 -

an1

nn 6

(n 2)an + 2

an

n< an +

2n

an.

n=2

an1

nn 6 s + 2

n=2

an

n.

Cauchy Schwarz

n=2

an

n

2

6

n=2

1

n2

n=2

an

=2 6

6s < s.

-

s.

n=1

1

n2=2

6

n=2

1

n2 0, lim

xf (x) = (*) -

x0 < 0 f (x0) = 0, (3) u = x0

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

. f (0) = 0.*[ lim

xf (x) = a R, ,

(3) f (a) = , , lim

xf (x) = ].

(3)

f ( f (x)) = x (4),

x (, 0]. x 0 f (x) < x.

f ( f (x)) < f (x) < x, .

x 0 f (x) > x. f ( f (x)) > f (x) > x, .

67

f (x) = x (5),

x 0. (1) y = 1

2

f

(x 1

2+ f (x)

)= x + f (x 1

2)

f (x 12

) = x 12,

x > 0, x = x +12

f (x) = x,

x > 0,

f (x) = x x R,

, (1).2 : f

(, 0].

limx

f (x) = + ( , -) a > f (0) f (h(a)), h(a) < 0.

1: f (0) > 0 . f x

, x0 > 0 f (x0) = 0.

(1) x, y

x0

2

f

(2

x0

2+ f (x0)

)= x0 + f

(2

x0

2

) x0 = 0,

. f x

, (1) y = x > 0

f (2x + f (2x2) = 2x2 + f (2x) > 2x2.

.

f (x1) .

h(x1)

f (x1) = f (h(x1))

f ( f (x1)) = f ( f (h(x1))) f ( f (x1)) = h(x1) + f (0) < 0,

. 2: f (0) < 0 ., , f ( f (0)) > f (0), -

, f ( f (0)) = f (0).

f (0) = 0,

f ( f (x)) = x (a),

x 0. x1, x2 > 0 f (x1) = f (x2).

f ( f (h(x1))) = f ( f (h(x2)) h(x1), h(x2) < 0 -

h(x1) = h(x2) f (h(x1)) = f (h(x2)) x1 = x2. f 1 1 [0,+),

.

, f (x) > 0 x > 0.

x1 > 0 x2 < 0

f (x1) = f (x2) f ( f (x1)) = f ( f (x2)) = x2 < 0,

.

f -

, , R,

f (x) < 0 x > 0. x > 0 f ( f (x)) , x.

f ( f ( f (x))) , f (x) f (x) , f (x),

.

f ( f (x)) = x (b),

x R. (b) f -

R f (x) = f 1(x) x R. , a > 0, b 0

x, y x + y = a 2xy = b, (1)

f (a + f (b)) = b + f (a) ().

a, b > 0,

f (a + b) = f ( f (h(a)) + f (h(b)) = h(b) + f ( f (h(a)) =

= h(b) + h(a) = f 1(b) + f 1(a) = f (b) + f (a).

f Cauchy (0,+), - f (x) = kx x > 0, k < 0. - f (x) = x x > 0.

f (x) = x x < 0,

f (x) = x x R,

, (1).

68

:

61 ( ) y2 = 2px E . n 3 - A1A2A3 An E - . EA1, EA2, . . . , EAn B1, B2, . . . , Bn :

EB1 + EB2 + EB3 + + EBn np.


( ) p > 0, ( p < 0).

Ak

(z

p

2

)n= rn(cos + i sin ),

(0,

2n

) r > 0 (r ).

wn rn(cos + i sin ) = 0

wk = r

(cos

2(k 1) + n

+ i sin2(k 1) +

n

)

= r(cos k + i sin k)

.

n

k=1

wk = 0n

k=1

cos k = 0 (1)

Bk k = 1, 2, . . . , n

zk Arg(zk

p

2

)= k.

zk p

2= rk(cos k + i sin k) zk = rk cos k +

p

2+ rk sin k

k = 1, . . . , n. zk -,

r2k sin2 k = 2p

(rk cos k +

p

2

)

r2k sin2 k + r

2k cos

2 k = r2k cos

2 k + 2prk cos k + p2

r2k = (rk cos k + p)2

rk>0,p>0 rk =p

1 cos k

rk =p(1 + cos k)

sin2 k p(1 + cos k).

1 cos k > 0 Ak x

x.

EBk =

zk p

2

= rk p(1 + cos k).

EB1 + EB2 + + EBn np + pn

k=1

cos k(1)= np.

62 ( ) - ABCD. , O - :

(AOB) = (BOC) = (COD) = (DOA).


1 ( ) -

.

O M.

-

: A (t, 0) t > 0, B (p, q) q > 0, C (t, 0) D (r, s) s < 0. O AC.

69

CB

qx + (p t) y + tq = 0

O .

BA

qx + (p + t) y tq = 0

O .

DC

sx + (r t) y + ts = 0

O .

DA

sx + (r + t) y ts = 0

O .

M

M DA, CB -

BA, DC .

-

.

(MCB) =12

p q 1t 0 1x y 1

, (MAB) =

12

p q 1t 0 1x y 1

,

(MCD) = 12

r s 1t 0 1x y 1

, (MAD) =

12

r s 1t 0 1x y 1

.

qx py = 0 (1)

sx ry = 0 (2)

(q + s) x + (p r 2t) y = t (s + q) (3)

:

q p 0s r 0

q + s p r 2t t (s + q)

= 0

t (s + q) (qr ps) = 0,

t > 0

(s + q) (qr ps) = 0.

:

()

s + q = 0

BD AC ,

p+ r+2t , 0 (B,C,D ), y = 0, x = 0, M AC.

() p q

r s

= 0. (4)

(4) OB,

OD .

BD AC.

(1) M - y = q

px

BD (CMD) = (CMB)

BD (*).

.

.

:

.

-

.(*) :

. (1), (2) (4)

x =p

q

s + q

2, y =

q + s

2

M BD.

70

2 ( ) E

ABCD.

O - ABCD

(OBA) = (OAD) = (ODC) = (OCB) (1)

O E

(1), ABCD -

.

(1) - O ,

BD. -

(1)

BF1AO, DG1AO (2)

(1), (2) BF1= DG1. -

: BE1 = E1D E1 DB OA. BF2OC, DG2OC -

(BOC) = (DOC)

BE2 = E2D, E2 DB OC. E1, E2 L BD. -

A OL, C OL L AC L E.

A OL, C OL (1) O AC.

(1) O BD

AC -

BD O

AC. (1) O AC

BD

AC O

BD.

(1) - O ABCD,

: .

.

3 ( ) .

xOy

OA,OB

, OM M

AB.

(OBA) = (OBC) (ODA) = (ODC) -

O BM, DM, M

AC. (OAB) = (OAD) (OCB) = (OCD)

O AN, CN, N

BD. O M N AC O N M BD ( , ABCD -).

71

:

63 ( ) n 1 a C R |a| = 1.

n

k=0

(nk

)(1 + ak)xk = 0.

:

() a

() a, b a + b = 0.


( )

()

(x + 1)n + (ax + 1)n = 0.

x0 , (x0 + 1)

n+ (ax0 + 1)

n= 0

|x0 + 1| = |ax0 + 1| .

|a| = 1,

|x0 + 1| = |x0 + a| .

x0

(1, 0) a.

() a = xa + yai a -

a =ya

1 xa b =

yb

1 xb b = xb + ybi x

2a + y

2a = x

2b+ y2

b= 1.

a b yayb

(1 xa) (1 xb)= 1

yayb

(1 xa) (1 xb)=

(1 + xa) (1 + xb)

yayb

xaxb + yayb = xa xb 1 (xa + xb)2 + (ya + yb)2 = 2xa 2xb = 2xa + 2xb xa + xb = ya + yb = 0 a + b = 0

64 ( )

arccos

(15

)= 2 arctan

23

.


1 ( )

2 arctan x = arctan2x

1 x2

( tan 2a =2 tan a

1 tan2 a),

2

23

1 23= 6

23.

cos a = 1tan2 a + 1

cos C =a2 + b2 c2

2ab=

52 + 62 72

2 5 6=

1260=

15

C = arccos(15

)

( 6

23 280

).

72

.

2 ( ) ABC a = 5, b = 6 c = 7. ,

cos C =a2 + b2 c2

2ab=

52 + 62 72

2 5 6=

1260=

15

C = arccos

(15

).

s =a + b + c

2= 9 -

,

tan

(C

2

)=

(s a) (s b)

s (s c)

tan

(C

2

)=

4 39 2

=

23 C = 2 arctan

23

!

73

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