is inscribed in circle 
.
denote its incenter and 
its 
-
denote the midpoint of arc 
.
meets 
.
at 
respectively. The circumcircle of 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
intersects the lines 
,
at points 
,
,
of the segments 
,
,
,
.
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
denote the set of single-variable polynomials in ![$\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/3/2/032a637d4ae21da2c0ddd8dc127723807f11cc5f.png)
(i.e. functions taking polynomials to polynomials)![$p, q \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/7/e/7/7e7eb925258a33e456ef4ff722c1f943dc155d10.png)
,
;![$p \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/2/6/0267b98c16b400dda1f081877e9b84fe672ce0e7.png)
,
has an integer root if and only if 
does.
Y by Mango247
In an acute-angled triangle 
through the center 
of the circumscribed circle and the vertices 
and 
a circle is drawn. Let 
be the diameter of this circle, and 
and 
be the points of its intersection with lines 
and 
, respectively. Prove that 
is a parallelogram.
through the center 
of the circumscribed circle and the vertices 
and 
a circle is drawn. Let 
be the diameter of this circle, and 
and 
be the points of its intersection with lines 
and 
, respectively. Prove that 
is a parallelogram.
