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