1η Ημέρα
1. Let 
be positive reals such that 
. Prove the inequality
2. There were finitely many persons at a party among whom some were friends. Among any
of them there were either 
who were all friends among each other or who weren't friend with each other. Prove that you can separate all the people at the party in two groups in such a way that in the first group everyone is friends with each other and that all the people in the second group are not friends to anyone else in second group. (Friendship is a mutual relation).
is given with its centroid 
and cicumcentre 
is such that 
is perpendicular to 
. Let 
be the second intersection of with circumcircle of triangle . Let 
be the intersection of lines 
and 
and 
the intersection of lines 
and 
. Prove that the circumcentre of triangle 
is on the circumcircle of triangle . 4. We define the sequence
so that
Where
are relatively prime numbers. Show that is not an integer for 
.2η Ημέρα
1. We define a sequence 
so that 
and
for all postive integers
.Find all positive integers
such that there is some positive integer 
for which 
. 2. There are lamps in every field of
table. At start all the lamps are off. A move consists of chosing 
consecutive fields in a row or a column and changing the status of that lamps. Prove that you can reach a state in which all the lamps are on only if divides 
3. Let
and 
be the points on the semicircle with diameter . Denote intersection of 
and 
as 
and let 
be the point such that is on segment and line 
is perpendicular to . If 
is the intersection of perpendicular bisector of an 
and 
is a point on such that angles 
and 
are equal. Prove that 
is perpendicular to . 4. Find all pairs of integers
for which 
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