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)
,
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, Mango247
On the sides 
and 
of a triangle 
of area 
are taken points 
respectively such that 
, where 
is variable.
(a) Find the area of triangle
in terms of 
.
(b) Find the value of
which minimizes this area.
(c) Find the locus of the intersection point
of the lines through 
and 
parallel to and 
respectively.
and 
of a triangle 
of area 
are taken points 
respectively such that 
, where 
is variable.(a) Find the area of triangle
in terms of 
.(b) Find the value of
which minimizes this area.(c) Find the locus of the intersection point
of the lines through 
and 
parallel to and 
respectively.
