[/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 
,
be a point on the extension of 
such that 
and 
are tangent to 
and the line 
at 
.
be the second point of intersection between 
and 
be a triangle with incenter 
,
and 
at 
.
and 
the reflections of 
with 
,
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.
be tangent to the side 
.
on 
on 
lies on the circumcircle of triangle 
.
and 
and 
are on the same side of 
with 
and 
Y by samrocksnature
Let 
be an acute scalene triangle, and let 
be the center of the circle which pass through the feet of altitudes. Let 
be the intersection of tangents to the circumcircle of at 
and 
. Prove that 
and are collinear iff 
.
be an acute scalene triangle, and let 
be the center of the circle which pass through the feet of altitudes. Let 
be the intersection of tangents to the circumcircle of at 
and 
. Prove that 
and are collinear iff 
.This post has been edited 1 time. Last edited by parmenides51, Apr 10, 2021, 7:47 PM
be the orthocenter, circumcenter of 
and 
.
is a parallelogram. Thus 
and 
and 
is rectangle because of the tangency. Thus 
.
and then 
