and incentre 
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.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
Y by Adventure10
Suppose that 
is the incenter of triangle 
. The perpendicular to line 
from point intersects sides 
and 
at points 
and 
respectively. Points 
and 
are placed on half lines 
and 
respectively, in such a way that 
and 
. If 
is the second intersection point of the circumcircles of triangles 
and 
, prove that the circumcenter of triangle 
lies on the line
is the incenter of triangle 
. The perpendicular to line 
from point intersects sides 
and 
at points 
and 
respectively. Points 
and 
are placed on half lines 
and 
respectively, in such a way that 
and 
. If 
is the second intersection point of the circumcircles of triangles 
and 
, prove that the circumcenter of triangle 
lies on the line 
and not 
.
