x ABC c)(a c m x m · PDF fileΜαθηματικά...

To {Eikosidwdekedron} parousizei jmata pou qoun suzhthje ston isttopo http://www.mathematica.gr.H epilog kai h frontda tou perieqomnou gnetai ap tou Epimelht tou http://www.mathematica.gr.Metatrop se LATEX: Fwtein Kald, Anastsh Kotrnh, Leutrh Prwtopap, Aqilla Sunefakpouo , Sqmata: Miqlh Nnno, Qrsto Tsifkh Seli-dopohsh: Anastsh Kotrnh, Nko Mauroginnh, Exfullo: Grhgrh Kwstko. Stoiqeiojetetai me to LATEX.Mpore na anaparaqje kai na dianemhje elejera.Eikosidwekedro filoteqnhmno ap ton Leonardo da VinciTo eikosidwdekedro enai na poledro (32-edro) me ekosi trigwnik dre kai ddeka pentagwnik. 'Eqei 30 panomoitupe koruf st opoe sunantntai dotrgwna kai do pentgwna kai exnta se akm pou h kje ma tou qwrzei na trgwno ap na pentgwno. Enai arqimdeio stere - dhlad na hmikanonikkurt poledro pou do perissteroi tpoi polugnwn sunantntai me ton dio trpo sti koruf tou - kai eidiktera enai to na ap ta do oiwne kanonik- quasiregular poledra pou uprqoun, dhlad stere pou mpore na qei do tpou edrn oi opoe enallssontai sthn koin koruf (To llo enai to kubo -oktedro). To eikosidwdekedro qei eikosiedrik summetra kai oi suntetagmne twn korufn en eikosadrou me monadiae akm enai oi kuklik metajseitwn (0, 0,), ( 1
2,
2,
1+2
), pou o qrus lgo 1+52
en to duadik tou poledro enai to rombik triakontedro.Phg:http://en.wikipedia.org/wiki/IcosidodecahedronApdosh: Pno GiannpouloO diktuak tpo mathematica.gr ankei kai dieujnetai smfwna me ton kanonism tou pou uprqei sthn arqiktou selda (http://www.mathematica.gr) ap omda Dieujunntwn Meln.Dieujnonta Mlh tou mathematica.grSuntoniste Airet Mlh1. Fwtein Kald (Fwtein) Genik Suntonstria2. Miqlh Lmprou (Mihalis Lambrou) Genik Sun-tonist3. Nko Mauroginnh (nsmavrogiannis) Genik Sun-tonist4. Spro Kardamtsh (Kardamtsh Spro)Upejuno Enhmrwsh5. Qrsto Kuriaz (chris gatos)Upejuno Programmatismo6. Mlto Papagrhgorkh (m.papagrigorakis)Upejuno Oikonomikn7. Girgo Rzo (Girgo Rzo)Upejuno Ekdsewn Mnima Mlh1. Grhgrh Kwstko (grigkost) Diaqeirist2. Alxandro Sugkelkh (cretanman) DiaqeiristEpimelhte1. Strth Antwna (stranton)2. Andra Barberkh (ANDREAS BARBERAKHS)3. Kwnstantno Btta (vittasko)4. Nko Katsph (nkatsipis)5. Anastsio Kotrnh (Kotrnh Anastsio)6. Jno Mgko (matha)
7. Girgo Mpalglou (gbaloglou)8. Rodlfo Mprh (R BORIS)9. Miqlh Nnno (Miqlh Nnno)10. Leutrh Prwtotopap (Prwtopap Leutrh)11. Dhmtrh Skoutrh (dement)12. Mpmph Stergou (Mpmph Stergou)13. Swtrh Stgia (swsto)14. Aqilla Sunefakpoulo (achilleas)15. Kwnstantno Thlgrafo (Thlgrafo Ksta)16. Serafem Tsiplh (Serafem)17. Qrsto Tsifkh (xr.tsif)18. Dhmtrh Qristofdh (Demetres)Melh1. Spro Basilpoulo (spyros)2. Ksta Zugorh (kostas.zig)3. Girgh Kalajkh (exdx)4. Qrsto Kardsh (QRHSTOS KARDASHS)5. Jansh Mpelhginnh (mathfinder)6. Jwm Rakftsalh (Jwm Rakftsalh)7. Kwnstantno Rekomh (rek2)8. Girgo Rodpoulo (hsiodos)9. Staro Staurpoulo (Staro Staurpoulo)10. Baslh Stefandh (bilstef)
http://www.mathematica.grhttp://en.wikipedia.org/wiki/Icosidodecahedronhttp://www.mathematica.gr

1 ( ) 100 , 99% .
98% .
2 ( )
() -
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()
.
() .
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.
A : ! B : ! . :
!
( -
) B
7 ( )
:
P(x) = x15 2012x14 + 2012x13 ... 2012x2 + 2012x.
P(2011).
8 ( ) ABC A = 900
ab, bc, (a + c)(a c) .
,
9 ( )
m (m 1)x4 5x2 + 3m 2 = 0 .
1

10 ( ) :
x = 111...12
y = 111...1+1
z = 666...6
,
: x + y + z + 8 .
,
11 ( )
AB O . A O E, Z EO 24, Z.
12 ( KARKAR) S CD ABCD. BS AD T . AM,T M,CN, S N . MDN = A = .
,
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+1 =8 9 + 2
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()
=1
3 N .
().
- ().
14 ( KARKAR) : 2x = 3a = 6b ,
: x =ab
a b.
,
15 ( )
(O,R) , (K, r) E. AB, , OK AB .
.
16 ( )
ABCD AC, BD CAB,BCA,CDB,BDA 70, 30, (50 a), a , , . BD CBA a.
,
17 ( KARKAR) ,
A(1, 3) B(4, 2), .
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1 +12+
13+ + 1
n>
n.
,
19 ( )
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x
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k .
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()
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lit
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() .
()
,
1000 2 lit.
, ,
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f : R R z z , 1/2
f 2(x) + sin2(x) = 2x f (x)
x R limx0
f (x)x= m,
m = |z2||2z1| .
() |z 2| = |2z 1|.() z
() limx0
f (sin x)x2 x .
() g(x) = f (x) x (,0) (0,+).
() f .
() (|z + 3 4i| + 5)x = x3 + 10 [1, 2].
22
z = (k t) + (k t)i t R k > 1.
(1) z.
(2) w
y = x (k 1) , k |z w|min = 52
2 1.
(3) k (2) z |z z|.
(4) k (2) |w 3 + 4i|.
(5) u
u = (1 + mt) + (1 + mt)i,
m u .
(6) k,m (2) (5) , |z u|.
, , ,
23 ( )
f (x) = ex ln x + 2 1 1, ex
21 ln(x2 + 1) = 1e ,
e1x + 2 ln x > ex.
24 ( ) f
h(x) = e f (x) f 3(x) + 2 .
f (x) :
(12
) f (x2x)
(12
) f (4x)> 0
, ,
25 ( )
f [a, b], (a, b) f (a) , f (b). 1, 2 (a, b) 1 , 2
: f(1) f
(2) =
(f (b) f (a)
b a
)2.
26 ( )
f : [1, e] R f (1) = 0, f (e) = 1 f
(x) + e f (x) = x +
1
x, x [1, e] . :
f (x) = ln x x [1, e] .
, ,
27 ( ) :2
0
x10 + 210
x15 + 215dx 0,x [a, b].
(x a) f (x) = (x b)g(x),x [a, b]
1)
g(x) > 0 f (x) < 0, x (a, b)
2) b
af (x)dx =
ba
g(x)dx
) f (b) = g(a)
) (a, b) : f () = g()
30 ( parmenides51)
z1 = a + eai, z2 = b + bi, z3 = c + i ln c a, b R, c > 0.
1.
z1, z2, z3,
2. |z1 z2 | - z1, z2 ,
3. |z1 z3| - z1, z3 .
, ,
31 ( )
f : [0,+) R
f (x) + x f (x) =1
2f( x2
) x [0,+) .
32 ( )
f (x) =ln(x2)x2+1
, N , N ,
.
33 ( )
f : [0, 1] R
f (0) = 0 1
0
e f(x) f (x) dx = f (1).
f (x) = 0,x [0, 1]
34 ( )
f : (0,+) R
f
(xy
)=
f (x)y x f (y)
y2 x, y (0,+)
1, f (1) = 1
Juniors,- -
35 ( )
n, sin2n x + cos2n x + n sin2 x cos2 x = 1,
x R.
36 ( ) a, b, c a + b + c = 3,
a2
(b + c)3+
b2
(c + a)3+
c2
(a + b)3 3
8.
Juniors,
37 ( )
ABC BC, I . BI AC D E, D CI. EI AB Z. DZ CI.
38 ( )
ABC E M, .
M, ABC ,
E1, E2 , E3. 1
E1+
1
E2+
1
E3 18
E.
Seniors,- -
39 ( )
f :
(x y) f (x + y) (x + y) f (x y) = 4xy(x2 y2)
4

40 ( ) -
f : {1, 2, . . . , 10} {1, 2, . . . , 100}
x + y|x f (x) + y f (y), x, y {1, 2, . . . , 10}.
Seniors,
41 ( )
ABCD (O) P AC BD.
ABCD Q,
QAB + QCB = QBC + QDC = 90o.
P, Q, O , O (O).
42 ( )
P ABCD, AB, BC, CD, DA, . ABCD -.
43 ( - 1
1995) 6-
5. ,
6 -
44 ( )
ABC AB AC D E , DE
. :ADDB+
AEEC= 1
45 ( IMC 1996) n .
sin nx(1 + 2x) sin x
dx.
46 ( ) (an)nN an > 1n
n. +
n=1
an .
47 ( ) M = R \ {3} x y = 3(xy3x3y)+m, m R. m (M, ) .
48 (vzf) m n n .
, n , - .
49 ( )
ln(1 aix)x2 + m
dx , a > 0, m > 0
50 ( ) an = 1 +n
j=21
ln j
lim( n
n)an
51 ( ) An = {n, 2n, 3n, ...} n N
J N,
iJAi
52 ( )
A, B AB = (A \ B) (B \ A), - .
A1A2...An Ai
53 ( ) p Pp > 5 (p 1)! + 1 p( pk k N).
5

54 ( dimtsig) -
.
55 ( ) S ,
s1, s2, ..., sn s1+s2+...+sn b ab > b2, a > c ab > bc, ab.
a2b2, b2c2, b4
a2b2 = (b2 + c2)b2 = b4 + c2b2, , .
10
http://www.mathematica.gr/forum/viewtopic.php?f=35&t=18552http://www.mathematica.gr/forum/viewtopic.php?f=35&t=10214

:
9 ( ) -
m (m 1)x4 5x2 + 3m 2 = 0 - .
http://www.mathematica.gr/forum/viewtopic.php?f=19&t=24722
( ) m = 1
: 5x2 + 1 = 0 x = 15.
.
m , 1 x2 = z

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