be a triangle with circumcircle 
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)
.
such that no sum of three distinct elements of 
.
.
that satisfy the equation 
.
be a circle, and let a line 
and 
Circles 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
respectively.
the segment 
has a length of 
and 
find the length of segment 
and 
Prove that
.
Prove that
and 
Prove that
Y by
Let 
and 
be two different points of intersection of two circles 
and 
, 
is the point of intersection of the tangent to the circle , drawn through the point , with the tangent to the circle , drawn through the point . The straight line 
intersects the circle at the point 
, different from A. On the circle , a point 
different from and is arbitrarily chosen so that the line 
intersects the circle at the point 
, other than . Let the line 
intersect the line 
at the point 
. Prove that the lines 
and 
are parallel.
and 
be two different points of intersection of two circles 
and 
, 
is the point of intersection of the tangent to the circle , drawn through the point , with the tangent to the circle , drawn through the point . The straight line 
intersects the circle at the point 
, different from A. On the circle , a point 
different from and is arbitrarily chosen so that the line 
intersects the circle at the point 
, other than . Let the line 
intersect the line 
at the point 
. Prove that the lines 
and 
are parallel.This post has been edited 1 time. Last edited by parmenides51, Jul 9, 2021, 9:02 AM
