[/quote]
,
and 
.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
and all positive integers 
and 
satisfying 
is interesting if 2018 divides 
(the number of positive divisors of 
such that there exists an infinite arithmetic progression with common difference 
,
.
that satisfies:
satisfy 
and 
Prove that always exists positive integer 
.
is "not congruence"
and 
Prove that
Solution:
and 
then 
and 
.
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
Prove that
Prove that
and 
intersect at points 
and 
,
intersects 
,
.
,
.
,
.
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
,
past 
past 
Y by Adventure10, Mango247
Let 
be a triangle with 
and let 
the interior bisectrix of 
with 
. Let 
be the midpoint of the arc 
that contains the point 
in the circumcircle of the triangle .The circumcircle of the triangle 
intersects the segment 
in 
. Denote by 
the symmetric of 
with respect to 
.If 
intersects 
in 
then prove that 
are concyclic.
be a triangle with 
and let 
the interior bisectrix of 
with 
. Let 
be the midpoint of the arc 
that contains the point 
in the circumcircle of the triangle .The circumcircle of the triangle 
intersects the segment 
in 
. Denote by 
the symmetric of 
with respect to 
.If 
intersects 
in 
then prove that 
are concyclic.This post has been edited 1 time. Last edited by ComplexPhi, May 15, 2015, 5:06 AM
and 
after finding 
(which I have not yet found), to get 
.
be the midpoint of 
lie on 
.
are concyclic.
are concyclic.
are concyclic. Q.E.D
Q.E.D
are collinear with 
?
