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Language: Greek
Day: 1
, 23 2013
1. k n, k m1,m2, . . . ,mk ( v )
1 +2k 1
n=
(1 +
1
m1
)(1 +
1
m2
)
(1 +
1
mk
).
2. 4027 , 2013 2014 . . , :
,
.
k 4027 , k .
3. ABC A BC A1. B1 CA C1 AB, B C,. A1B1C1 ABC. ABC .
ABC A BC, AB B AC C. B C .
Language : Greek : 4 30 7
Language: Greek
Day: 2
, 24 2013
4. ABC H W BC, B C. M N B C, . 1 BWN X 1 WX 1. , 2 CWM Y 2 WY 2. X, Y H .
5. Q>0 . f : Q>0 R :
(i) x, y Q>0, f(x)f(y) f(xy),
(ii) x, y Q>0, f(x+ y) f(x) + f(y),
(iii) a > 1 f(a) = a.
f(x) = x, x Q>0.
6. n 3 . n + 1 n + 1 . 0, 1, . . . , n . , . , , a < b < c < d a + d = b + c, a d b c.
M N (x, y), x, y , x+ y n MK(x, y) = 1.
M = N + 1.
Language : Greek : 4 30 7