[/quote]
,
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 
with integer coefficients. Find all functions ![$\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$](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)
that verify the following equation for all 
:
is a prime number
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
,
such that 
and 
are tangent to 
intersects 
.
be the second point of intersection between 
and 
be a triangle with incenter 
,
and 
at 
.
and 
the reflections of 
,
be the midpoint of 
and 
are perpendicular.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the circle with diameter 
be the intersection of 
and 
be the intersection of 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
be a function from the Euclidean plane to the real numbers such that 
for any acute triangle 
with circumcenter 
is constant.
.
on 
on 
lies on the circumcircle of triangle 
.
and 
and 
are on the same side of 
be midpoint of segment 
with 
and 
Y by
Circles 
and 
intersect at points 
and 
. Through point 
of circle , lying inside , a chord 
of circle is drawn. Chord 
of circle passes through point . A straight line passing through points 
and intersects circle at a point 
different from point . Let 
be a point symmetric to point wrt point . Prove that 
is the diameter of the circle if and only if the points 
and lie on the same circle.
and 
intersect at points 
and 
. Through point 
of circle , lying inside , a chord 
of circle is drawn. Chord 
of circle passes through point . A straight line passing through points 
and intersects circle at a point 
different from point . Let 
be a point symmetric to point wrt point . Prove that 
is the diameter of the circle if and only if the points 
and lie on the same circle.
