AoPS Academy
be an acute triangle. Let 
be a point on side 
and 
be a point on side 
such that lines 
and 
are parallel. Let 
be an interior point of 
. Suppose rays 
and 
meet side at points 
and 
, respectively, such that both and lie between 
and 
. Suppose that the circumcircles of triangles 
and 
intersect at a point 
. Prove that the points 
, and 
are collinear.
. Prove that:
and 
, define 
as the greatest exponent 
such that 
divides 
. Prove that there are infinitely many such that 
and infinitely many 
such that 
.
-gons inscribed in a circle such that all sides of the -gon are of different length. Determine the maximal number of 
angles in this kind of a -gon.
angles.
. If two adjacent angles are equal, then the sides are symmetric with respect to their perpendicular bisector, contradicting our original condition of different-length sides.
non-adjacent angles are equal to 
, say 
. If 
is the circumcenter of our circle, then 
, but this means 
, contradiction. 
and 
.
