i Cos i Dodecahedron 13
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http://www.mathematica.gr. mathematica.gr. LaTEX--: , : . LTEX. .ISSN: 2241-7133
Leonardo da Vinci
(32-) . 30
. -
- - quasiregular ,
( - ).
(0,0,),( 12 ,
2 ,
1+2
), 1+
5
2 .
:http://en.wikipedia.org/wiki/Icosidodecahedron
:
mathematica.gr - (http://www.mathematica.gr) .
mathematica.gr
1. ()
2. (matha)
3. (nsmavrogiannis)
4. (chris_gatos)
5. (Mihalis_Lambrou)
6. (m.papagrigorakis) -
7. (Demetres)
1. (grigkost)
2. (cretanman)
1. (stranton)
2. (vittasko)
3. ()
4. (s.kap)
5. ( )
6. ( )
7. (chris_gatos)
8. (grigkost)
9. (emouroukos)
10. ( )
11. (m.papagrigorakis)
12. ( )
13. (cretanman)
14. (achilleas)
15. (xr.tsif)
16. (polysot)
1. (stranton)
2. ( )
3. (spyros)
4. (vittasko)
5. ()
6. (KAKABASBASILEIOS)
7. (exdx)
8. (s.kap)
9. ( )
http://www.mathematica.grhttp://en.wikipedia.org/wiki/Icosidodecahedronhttp://www.mathematica.gr -
10. (nkatsipis)
11. ( )
12. ( )
13. ( )
14. (emouroukos)
15. (gbaloglou)
16. (R BORIS)
17. ( )
18. ( )
19. ( )
20. (dement)
21. ( )
22. (swsto)
23. (achilleas)
24. ( )
25. ()
26. (xr.tsif)
27. (polysot)
1. ( )
2. (mathfinder)
3. ( )
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.
http://www.mathematica.gr/forum/viewtopic.php?f=44&t=31494
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 ;
http://www.mathematica.gr/forum/viewtopic.php?f=33&t=41194
( ) 10 , 40. 10
212
360 = 60.
10
4060
112
360 = 20.
80.
4 ( PetranOmayromixalis) 3 , 4 , 5 . ;
http://www.mathematica.gr/forum/viewtopic.php?f=33&t=10369
(Geopa)) ,
:
+ = 3. + = 4. + = 5.
2( + + ) = 12. + + = 6.)
:
+ + = 3. + + = 4. + + = 5. :
= = = 1. 2 = 2. 3 = 3., .
12
http://www.mathematica.gr/forum/viewtopic.php?f=33&t=41194http://www.mathematica.gr/forum/viewtopic.php?f=33&t=10369 -
:
5 ( KARKAR) - AB,CS .
http://www.mathematica.gr/forum/viewtopic.php?f=34&t=40719
( )
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.
http://www.mathematica.gr/forum/viewtopic.php?f=34&t=40296
( )
() 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
http://www.mathematica.gr/forum/viewtopic.php?f=34&t=40719http://www.mathematica.gr/forum/viewtopic.php?f=34&t=40296 -
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)
http://www.mathematica.gr/forum/viewtopic.php?f=35&t=33893
( ) 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.
http://www.mathematica.gr/forum/viewtopic.php?f=35&t=25144
( ) 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 = .
http://www.mathematica.gr/forum/viewtopic.php?f=18&t=5879
( ) :
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
http://www.mathematica.gr/forum/viewtopic.php?f=18&t=15652http://www.mathematica.gr/forum/viewtopic.php?f=18&t=5879 -
|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| .
http://www.mathematica.gr/forum/viewtopic.php?f=51&t=40335
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.
http://www.mathematica.gr/forum/viewtopic.php?f=51&t=27420
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
http://www.mathematica.gr/forum/viewtopic.php?f=51&t=40335http://www.mathematica.gr/forum/viewtopic.php?f=51&t=27420 -
() 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 .
http://www.mathematica.gr/forum/viewtopic.php?f=52&t=38484
( ) 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 .
http://www.mathematica.gr/forum/viewtopic.php?f=52&t=39318
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
http://www.mathematica.gr/forum/viewtopic.php?f=52&t=38484http://www.mathematica.gr/forum/viewtopic.php?f=52&t=39318 -
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.
http://www.mathematica.gr/forum/viewtopic.php?f=53&t=41418
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
http://www.mathematica.gr/forum/viewtopic.php?f=53&t=41418 -
http://www.mathematica.gr/forum/viewtopic.php?f=53&t=40954
( )
() 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
http://www.mathematica.gr/forum/viewtopic.php?f=53&t=40954 -
:
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).
http://www.mathematica.gr/forum/viewtopic.php?f=54&t=37941
( )
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.
http://www.mathematica.gr/forum/viewtopic.php?p=195414
( ) 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
http://www.mathematica.gr/forum/viewtopic.php?p=195414 -
.
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.
http://www.mathematica.gr/forum/viewtopic.php?f=55&t=40496
( )
(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
http://www.mathematica.gr/forum/viewtopic.php?f=55&t=40496 -
:
31 ( )
f : R R (
f (x) x2) (
f (x) 1)= 1, x R
f (0) = 1, f (0) = 0.
f .
http://www.mathematica.gr/forum/viewtopic.php?f=56&t=41822
( ) 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
http://www.mathematica.gr/forum/viewtopic.php?f=56&t=41822http://www.mathematica.gr/forum/viewtopic.php?f=56&t=38844 -
b
a
f (x)
x2dx 0. - f M(x0, f (x0) Ox A.
() AB 1
| ln x0|.
() AB 1, - M f .
http://www.mathematica.gr/forum/viewtopic.php?f=69&t=41743
(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.
http://www.mathematica.gr/forum/viewtopic.php?f=109&t=41776
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
.
http://www.mathematica.gr/forum/viewtopic.php?f=109&t=40916
( )
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
http://www.mathematica.gr/forum/viewtopic.php?f=109&t=41776http://www.mathematica.gr/forum/viewtopic.php?f=109&t=40916 -
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.
http://www.mathematica.gr/forum/viewtopic.php?f=110&t=37822
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
http://www.mathematica.gr/forum/viewtopic.php?f=110&t=37822 -
http://www.mathematica.gr/forum/viewtopic.php?f=110&t=38382
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
http://www.mathematica.gr/forum/viewtopic.php?f=110&t=38382 -
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
-
50
-
, , : -
39 ( ) f : Z+ Z+,
f (x2 + f (y)) = x f (x) + y,
x, y Z+.
http://www.mathematica.gr/forum/viewtopic.php?f=111&t=35047
( )
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
http://www.mathematica.gr/forum/viewtopic.php?p=165244 -
:
41 ( parmenides51.) - ABCD (O,R) - (I, r). OI = x
1
r2=
1
(R + x)2+
1
R x)2.
http://www.mathematica.gr/forum/viewtopic.php?f=112&t=42144
( .) 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
http://www.mathematica.gr/forum/viewtopic.php?f=112&t=42144 -
42 ( .) -
ABC (O) (K), AD. - BE, CF, DZ , E (K)AC, F (K)AB, Z (K) (O) Z , A.
http://www.mathematica.gr/forum/viewtopic.php?f=112&t=42240
( .) -
(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
http://www.mathematica.gr/forum/viewtopic.php?f=112&t=42240 -
:
43 ( ) - 100 . 30% - , .
http://www.mathematica.gr/forum/viewtopic.php?p=190087
( )
: 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
;
http://www.mathematica.gr/forum/viewtopic.php?p=192844
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
http://www.mathematica.gr/forum/viewtopic.php?p=185428 -
:
45 ( Kyiv Taras 2013) n n A, B n n C
AX + YB = C
. n n C
A2013X + YB2013 = C
.
http://www.mathematica.gr/forum/viewtopic.php?f=59&t=40087
( )
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
http://www.mathematica.gr/forum/viewtopic.php?f=59&t=39843
( )
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
http://www.mathematica.gr/forum/viewtopic.php?f=59&t=40087http://www.mathematica.gr/forum/viewtopic.php?f=59&t=39843 -
:
47 ( ) V - Rnn , W Rnn - Rnn V,W .
http://www.mathematica.gr/forum/viewtopic.php?f=10&t=42265
( )
() V : -
.
() W : ( - ).
48 ( ) p(x) - 2.
A = {x R \ Q : p(x) Q}
R.
http://www.mathematica.gr/forum/viewtopic.php?f=10&t=40333
( ) -
:
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
http://www.mathematica.gr/forum/viewtopic.php?f=10&t=42265http://www.mathematica.gr/forum/viewtopic.php?f=10&t=40330 -
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].
http://www.mathematica.gr/forum/viewtopic.php?f=9&t=37825
( ) -
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,+).
http://www.mathematica.gr/forum/viewtopic.php?f=9&t=41661
( ) 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 =( ( ( ))) ( ) .
http://www.mathematica.gr/forum/viewtopic.php?f=11&t=41083
( )
( ) = ( ) ( ) =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 .
http://www.mathematica.gr/forum/viewtopic.php?f=11&t=42673
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 .
http://www.mathematica.gr/forum/viewtopic.php?f=63&t=41369#p192805
( ) 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.
http://www.mathematica.gr/forum/viewtopic.php?f=63&t=31929#p147794
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.
http://www.mathematica.gr/forum/viewtopic.php?f=47&t=40995
( )
:
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
http://www.mathematica.gr/forum/viewtopic.php?f=47&t=40995 -
56 ( vz f ) - W - B , - h.
http://www.mathematica.gr/forum/viewtopic.php?f=47&t=6626
( ) -
,
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
http://www.mathematica.gr/forum/viewtopic.php?f=47&t=6626 -
:
57 ( ) -
K := cot 70 + 4 cos 70.
http://www.mathematica.gr/forum/viewtopic.php?f=27&t=41829
( )
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.
http://www.mathematica.gr/forum/viewtopic.php?f=27&t=39505
( ) -
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
http://www.mathematica.gr/forum/viewtopic.php?f=27&t=41829http://www.mathematica.gr/forum/viewtopic.php?f=27&t=39505 -
:
59 ( - . ) (an)nN . :
n=1
an < ,
n=1
an1
nn < .
, ,:
n=2
an = s < ,
n=2
an1
nn s + 2
s.
http://www.mathematica.gr/forum/viewtopic.php?f=61&t=31976
( ) 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
http://www.mathematica.gr/forum/viewtopic.php?f=61&t=31976http://www.mathematica.gr/forum/viewtopic.php?f=61&t=40597 -
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.
http://www.mathematica.gr/forum/viewtopic.php?f=62&t=39416
( ) 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).
http://www.mathematica.gr/forum/viewtopic.php?f=62&t=38914
1 ( ) -
.
O M.
-
: A (t, 0) t > 0, B (p, q) q > 0, C (t, 0) D (r, s) s < 0. O AC.
69
http://www.mathematica.gr/forum/viewtopic.php?f=62&t=39416http://www.mathematica.gr/forum/viewtopic.php?f=62&t=38914 -
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.
http://www.mathematica.gr/forum/viewtopic.php?f=60&t=41670
( )
()
(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
.
http://www.mathematica.gr/forum/viewtopic.php?f=60&t=31191
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
http://www.mathematica.gr/forum/viewtopic.php?f=60&t=41670http://www.mathematica.gr/forum/viewtopic.php?f=60&t=31191 -
.
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