Ημέρα 1η
1 In acute-angled triangle 
, let 
be the foot of perpendicular from 
to 
and also suppose that 
and 
are excenters oposite to the side 
in triangles 
and 
. If 
is the point that incircle touches , prove that 
are concyclic.
, let be the foot of perpendicular from to and also suppose that and are excenters oposite to the side in triangles and . If is the point that incircle touches , prove that are concyclic. 
2 Find the maximum number of subsets from 
such that for any two of them like 
if 
then 
. (Here 
is the number of elements of the set 
.)
such that for any two of them like if then . (Here is the number of elements of the set .)
3 For nonnegative integers 
and 
, define the sequence 
of real numbers as follows. Set 
and for every natural number , set 
and 
. Then for 
, define
Prove that for every natural number 
, all the roots of the polynomial
and , define the sequence of real numbers as follows. Set and for every natural number , set and . Then for , defineProve that for every natural number , all the roots of the polynomial
are real.
Ημέρα 2η
4 and are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:
and are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:
Starting from one of the hexagons, the Donkey moves cells in one of the 
directions, then it turns 
degrees clockwise and after that moves cells in this new direction until it reaches it's final cell.
cells in one of the directions, then it turns degrees clockwise and after that moves cells in this new direction until it reaches it's final cell.
At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?
6 Points 
and 
lie on line 
in this order. Two circular arcs 
and 
, which both lie on one side of line , pass through points and 
and two circular arcs 
and 
pass through points 
and such that is tangent to and is tangent to . Prove that the common external tangent of and and the common external tangent of and meet each other on line .
and lie on line in this order. Two circular arcs and , which both lie on one side of line , pass through points and and two circular arcs and pass through points and such that is tangent to and is tangent to . Prove that the common external tangent of and and the common external tangent of and meet each other on line .
Πηγή: artofproblemsolving
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