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 
.
inside the convex quadrilateral 
such that the triangles 
, 
, 
, 
have the same area, then one of the diagonals of bisects the area of the quadrilateral.
doesn't lie on 
(if it did, we'd be done). Let 
and 
. Clearly, 
are both midpoints of , so lies on 
, and we are done.
and 
.
Where 
