Ημέρα 1η
2.In an acute triangle 
the angle bisector of 
intersects the side 
at 
Let 
be the midpoints of the line segments 
respectively and 
be the circumcenters of triangles 
respectively. If 
and 
are midpoints of the line segments 
and 
respectively, prove that 
is an isosceles trapezoid.
the angle bisector of intersects the side at Let be the midpoints of the line segments respectively and be the circumcenters of triangles respectively. If and are midpoints of the line segments and respectively, prove that is an isosceles trapezoid.
1. In a triangle incenter touches the sides 
at 
respectively. A circle 
passing through and tangent to line at 
intersects the line segments 
and 
at 
and 
respectively. The line passing through 
and parallel to 
intersects the line passing through 
and parallel to 
at 
If 
denotes the circumradius of the triangles 
respectively, prove that
2. A positive integer incenter touches the sides at respectively. A circle passing through and tangent to line at intersects the line segments and at and respectively. The line passing through and parallel to intersects the line passing through and parallel to at If denotes the circumradius of the triangles respectively, prove that 
is called good if for all positive integers 
which can be written as
where 
are integers, it is possible to express as
are integers, it is possible to express as
where 
are integers with none of them is divisible by 
Find all good numbers.
are integers with none of them is divisible by Find all good numbers. 3. Two players and 
play a game on a 
board, using 
pieces numbered from 
to 
At each turn, chooses a piece and places it to an empty place. After 
turns, if all pieces are placed on the board increasingly, then wins, otherwise wins. For which values of 
pairs can guarantee to win?
and play a game on a board, using pieces numbered from to At each turn, chooses a piece and places it to an empty place. After turns, if all pieces are placed on the board increasingly, then wins, otherwise wins. For which values of pairs can guarantee to win? Ημέρα 3η
1. Let 
where is a positive integer and 
is a rational number. If 
for all positive integers where 
are positive rationals and 
is positive integer then we call 
as nice triple. Find all nice triples.
where is a positive integer and is a rational number. If for all positive integers where are positive rationals and is positive integer then we call as nice triple. Find all nice triples. 2. In a plane, the six different points 
are given such that triangles 
and 
are congruent, i.e. 
Let 
be the centroid of and 
be an intersection point of the circle with diameter 
and the circle with center 
and passing through 
Define 
and 
similarly. Prove that
are given such that triangles and are congruent, i.e. Let be the centroid of and be an intersection point of the circle with diameter and the circle with center and passing through Define and similarly. Prove that 
3. Let 
and 
denote the set of positive integers and the set of prime numbers, respectively. A set is called 
where 
if there exists a positive integer 
such that for all 
and for all 
there exist 
satisfying 
and 
and denote the set of positive integers and the set of prime numbers, respectively. A set is called where if there exists a positive integer such that for all and for all there exist satisfying and Find a subset of for which is but is not.
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