is interesting if 2018 divides 
(the number of positive divisors of 
such that there exists an infinite arithmetic progression with common difference 
such that![\[
2^a \cdot 3^b + 1 = 5^c.
\]](http://latex.artofproblemsolving.com/0/a/c/0ac1f22162eacf698847bceb33f9a337bd53c44b.png)
be an isosceles trapezoid with 
.
be the midpoint of 
.
and 
the circumcircles of the triangles 
and 
,
be the crossing point of the tangent to 
with the tangent to 
.
is tangent to 
,
denote the feet of the altitudes from 
,
.
and 
be the circles through 
,
other than 
Y by
A polynomial 
with degree 
is reflexive if there is an integer 
such that 
for every 
, where 
for 
. Let 
be an integer and 
be a polynomial with integer coefficients. Prove that there exist reflexive polynomials 
, 
with integer coefficients such that
![\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]](//latex.artofproblemsolving.com/6/6/7/66729b54a8b87c6ddbfdf7e7e5932dbb8b262439.png)
with degree 
is reflexive if there is an integer 
such that 
for every 
, where 
for 
. Let 
be an integer and 
be a polynomial with integer coefficients. Prove that there exist reflexive polynomials 
, 
with integer coefficients such that![\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]](http://latex.artofproblemsolving.com/6/6/7/66729b54a8b87c6ddbfdf7e7e5932dbb8b262439.png)
such that 
is a polynomial. Now we state that taking:
works. First we see that both are polynomials because the numerators have roots at 
.
Finally we compute:
