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and 
. Prove that
Where 
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 
.
is the altitude of acute triangle 
. The line 
touches the circumscribed circle of the triangle at point 
. A perpendicular 
is drawn from 
on line . Prove that the lines 
and 
are parallel.
,
is cyclic. Thus, 
,