[/quote]
,
denote the set of positive integers
such that for all 
it holds that 
Note: consider 
and 
such that
for all 
.
and 
Prove that
Where 
Where 
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
.
such that every equation: 
has no real roots
with 
.
and 
respectively such that 
.
and 
is the incenter of 
and 
is the orthocenter of 
.
and 
are circles with diameter 
,
and 
.
.
such that 
.
.
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
Y by
Let a rhombus 
with 
be given . On the extension of the segment 
beyond 
, a point 
is chosen arbitrarily. Let the line through and 
intersect the line 
at the point 
. Let 
be the intersection of the lines 
and 
. Prove that the line 
is tangent to the circumcircle of the triangle 
.
with 
be given . On the extension of the segment 
beyond 
, a point 
is chosen arbitrarily. Let the line through and 
intersect the line 
at the point 
. Let 
be the intersection of the lines 
and 
. Prove that the line 
is tangent to the circumcircle of the triangle 
.
.
.
.
.
,
.
.
