AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
+1 w
and all positive integers 
and 
satisfying 
is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers 
such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
be an odd prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer satisfying for all positive integers , then 
.
is "not congruence"
and 
Prove that
Solution:
and 
then 
and 
intersect at points 
and 
. A line through intersects and at points 
and 
, respectively. Line 
intersects at point 
, and line 
intersects at point 
. If 
is the circumcenter of 
, prove that 
.
be a triangle with incenter 
, and let be a point on side 
. Points 
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side such that 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
, the extension of 
past , and the extension of 
past . Prove that is tangent to the circumcircle of triangle .
is cyclic, this is due to simple angle chase, 
.
. There is quite a well known property of harmonic quadruples,
, then if 
is the midpoint of 
, then 
, now with PoP we conclude that,
is cyclic.
is cyclic, so 
is harmonic so by Newton 
,
is cyclic!
and 
.
,
.
that satisfies:
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
being zero. Then because 
we know that,
,
,
is cyclic
M, T, X, Y are concyclic (By PoP)