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)
Y by
A set of 
light bulbs are arranged around a circle, and are consecutively numbered with
. Each bulb can be in one of two states: either it is on or off. In the initial configuration,
at least one bulb is turned on. On each one of
days we change the current on/off configuration as
follows: for
, on the 
-th day we start from the -th bulb and moving in clockwise direction
along the circle, we change the state of every traversed bulb until we switch on a bulb which was
previously off.
Prove that the final configuration, reached on the-th day, coincides with the initial one.
light bulbs are arranged around a circle, and are consecutively numbered with
. Each bulb can be in one of two states: either it is on or off. In the initial configuration,at least one bulb is turned on. On each one of
days we change the current on/off configuration asfollows: for
, on the 
-th day we start from the -th bulb and moving in clockwise directionalong the circle, we change the state of every traversed bulb until we switch on a bulb which was
previously off.
Prove that the final configuration, reached on the
-th day, coincides with the initial one.
