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in reals such that:
. Prove that 
and 
for all 
.
be an acute triangle, and let 
be its incenter. Let 
be the incircle of triangle 
, which touches 
at points 
, respectively. Let 
intersect again at point 
, and let the circumcircle of triangle 
intersect again at point 
. The circumcircles of triangles 
and 
intersect again at a point 
. Let 
be the midpoint of arc 
. Prove that points 
are collinear.
be a prime number. Let 
denote the integers modulo , and let ![$\mathbb F_p[x]$](http://latex.artofproblemsolving.com/1/6/3/1633a0f1c6691e5e3ecf9189e677d06ca10465e7.png)
be the set of polynomials with coefficients in . Define ![$\Psi : \mathbb F_p[x] \to \mathbb F_p[x]$](http://latex.artofproblemsolving.com/b/2/3/b235991d65027382c6916d1466228099becc77a3.png)
by ![\[ \Psi\left( \sum_{i=0}^n a_i x^i \right) = \sum_{i=0}^n a_i x^{p^i}. \]](http://latex.artofproblemsolving.com/1/a/f/1af46a31b71c32f3c194a7afeea5d5be3d23547d.png)
Prove that for nonzero polynomials ![$F,G \in \mathbb F_p[x]$](http://latex.artofproblemsolving.com/5/b/a/5ba6c808e48ff00cb83533b007a12711d9bde772.png)
, ![\[ \Psi(\gcd(F,G)) = \gcd(\Psi(F), \Psi(G)). \]](http://latex.artofproblemsolving.com/1/3/f/13ff0c15075cdac64c52ec9c62429e0e7d55a966.png)
Here, a polynomial 
divides 
if there exists ![$R \in \mathbb F_p[x]$](http://latex.artofproblemsolving.com/2/c/5/2c5975abb489c4833da46845ade615919cce8169.png)
such that 
is the polynomial with all coefficients 
(with all addition and multiplication in the coefficients taken modulo ), and the gcd of two polynomials is the highest degree polynomial with leading coefficient 
which divides both of them. A non-zero polynomial is a polynomial with not all coefficients . As an example of multiplication, 
in ![$\mathbb F_5[x]$](http://latex.artofproblemsolving.com/a/f/4/af4a4857b833e03853a81db34fe5e915d127c142.png)
.
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
.
and 
where 
and 
.
.