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be a triangle with circumcircle 
and incentre 
. A line 
intersects the lines 
, 
, and 
at points 
, 
, and 
, respectively, distinct from the points 
, 
, 
, and . The perpendicular bisectors 
, 
, and 
of the segments 
, 
, and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to .
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)
. Let be a subset of 
such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
that satisfy the equation 
.
be a circle, and let a line intersect at two points, 
and 
Circles 
and 
are internally tangent to at points 
and 
respectively, and both are tangent to at a common point 
Similarly, circles 
and 
are externally tangent to at and respectively, and are tangent to at points and 
respectively. is 
the segment 
has a length of 
and 
find the length of segment 
with right angle . On side 
, a point 
is chosen such that the angle 
is 
. A point 
is marked on the ray 
such that that the angle 
is 
. Prove that 
.
with right angle . On side , a point is chosen such that the angle is . A point is marked on the ray such that that the angle is . Prove that .
and 
Prove that
.
Prove that
and 
Prove that
.