1η Ημέρα – 26/11/2011
1: Let a sequence 
defined by:
defined by:
Denote 
and 
. Prove that 
, 
have finite limit.
and . Prove that , have finite limit.
2: Let 
be a set of finite distinct positive real numbers. Two other sets 
, 
are defined by:
be a set of finite distinct positive real numbers. Two other sets , are defined by:
. Prove that 
.3: Two circles 
and 
intersect at and . Take two points 
on and , respectively, such that 
. The line 
intersects and respectively at 
. Let 
respectively be the centers of the two arcs 
and 
(which don’t contains ). Prove that 
is a cyclic quadrilateral.
and intersect at and . Take two points on and , respectively, such that . The line intersects and respectively at . Let respectively be the centers of the two arcs and (which don’t contains ). Prove that is a cyclic quadrilateral.4: For a table 
(
rows and 
columns), determine the maximum of that we can write one number in the set 
in each cell such that these conditions are satisfied:
( rows and columns), determine the maximum of that we can write one number in the set in each cell such that these conditions are satisfied:a. Each row contains enough numbers of the set .
numbers of the set .b. Any two rows are distinct.
c. For any two rows, we can find at least one column such that the two intersecting cells between it and the two rows contain the same number.
2η Ημέρα – 26/11/2011
5: Determine all values of satisfied the following condition: there’s exist a cyclic 
of 
such that 
is a complete residue systems modulo .
satisfied the following condition: there’s exist a cyclic of such that is a complete residue systems modulo . 6: Find all function 
such that
such that
for all 
.
.7: There are students. Denoted the number of the selections to select two students (with their weights are 
and 
) such that 
(kg) and 
(kg) by 
and 
, respectively. Prove that 
.
students. Denoted the number of the selections to select two students (with their weights are and ) such that (kg) and (kg) by and , respectively. Prove that .
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