be a triangle with incenter 
,
.
intersects 
and 
.
be the midpoint arc 
and let 
intersect 
.
intersect the circumcircle of 
at a point 
.
.
and 
intersect on 
satisfy,
Where 
}
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
countries 
Each country has a flag 
units wide and one unit high composed of 
each field being either yellow or blue. No two countries have the same flag. We say that a set of 
square so that all 
Y by Adventure10
Consider a sequence of circles 
of radii 
, respectively, situated inside a triangle 
. The circle 
is tangent to 
and 
; 
is tangent to , 
, and 
; 
is tangent to , 
, and 
; 
is tangent to , , and ; etc.
(a) Prove the relation
![\[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \]](//latex.artofproblemsolving.com/4/b/5/4b58ea2f504c79401d29ff70f96fb43d1ef9fb02.png)
where
is the radius of the incircle of the triangle . Deduce the existence of a 
such that
![\[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\]](//latex.artofproblemsolving.com/4/3/2/4324c7c175cd2615abffcdf495eaba738dc8ff9b.png)
(b) Prove that the sequence of circles
is periodic.
of radii 
, respectively, situated inside a triangle 
. The circle 
is tangent to 
and 
; 
is tangent to , 
, and 
; 
is tangent to , 
, and 
; 
is tangent to , , and ; etc.(a) Prove the relation
![\[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \]](http://latex.artofproblemsolving.com/4/b/5/4b58ea2f504c79401d29ff70f96fb43d1ef9fb02.png)
where
is the radius of the incircle of the triangle . Deduce the existence of a 
such that![\[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\]](http://latex.artofproblemsolving.com/4/3/2/4324c7c175cd2615abffcdf495eaba738dc8ff9b.png)
(b) Prove that the sequence of circles
is periodic.
