AoPS Academy
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
be a regular polygon, and let 
be its set of vertices. Each point in is colored red, white, or blue. A subset of is patriotic if it contains an equal number of points of each color, and a side of is dazzling if its endpoints are of different colors. is patriotic and the number of dazzling edges of is even. Prove that there exists a line, not passing through any point in , dividing into two nonempty patriotic subsets.
meet 
at 
.
if 
has an incircle . we have 
so 
and 
so 
so 
and 
but since 
, we have 
and with 
and 
we have 
.
if then 
and 
and 
so 
and in this case 
so has an incircle .
,
and the cevians 
and 
concur at a point 
inside the triangle. Show that the quadrilateral 
has an incircle if and only if 
.
