AoPS Academy
satisfies 
. If 
, find the maximum possible value of 
.
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
, two tangents are drawn touching at points 
and 
. A point 
is chosen on the segment 
. Let points 
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects again at point 
(
). Prove that the line 
passes through the centroid of triangle 
.
and 
are the bisectors of the angles 
of triangle 
and suppose 
are the perpendiculars from 
to 
. Prove that 
is parallel to 
.
and 
.
Where 
.
.
.