[/quote]
,
meet at point 
.
and 
lie on 
,
respectively in such a way that 
and 
.
and 
passes through the mass-center of 
Prove that
and 
be concentric circles, with 
on 
to 
)
be the second point of intersection of 
be the midpoint of 
and 
in such a way that the perpendicular bisectors of 
and 
intersect at a point 
on 
.
that is tangent to the four sides 
,
,
,
.
,
,
,
be the circumcenters of the triangles 
,
,
,
,
,
,
,
.
and 
for all 
be real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
be an acute non-isosceles triangle with circumcircle 
,
and orthocenter 
.
through 
through 
.
.
,
intersect the sides 
respectively. Finally, let the rays 
,
meet the circumcircle of 
in 
,
meet on the circumcircle of 
such that there exists exactly 
different functions 
such that ![$f:[1,5]\rightarrow[1,5]$](http://latex.artofproblemsolving.com/6/2/0/620e1621905a4670627d46fcb367e1a8c461bb49.png)
satisfying 
.
Y by
An and Binh play a game on a square board of size (2n+1)x(2n+1) with An going first. Initially, all the squares on the board are white. In each turn, An colors a white square blue and Binh colors a white square red. The game ends after both players have colored all the squares on the board. An wins if, for any two blue squares, there exists at least one chain of neighboring blue squares connecting them (two squares are called neighboring if they have at least one vertex in common). Otherwise, Binh wins. Determine the player with the winning strategy in the following cases:
a) with n=1
b) with n>=2
a) with n=1
b) with n>=2
