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.
with the set 
such that each element in the permutations has at most one immediate neighbor greater than itself.
be reals such that 
Prove that
Let be reals such that 
Prove that
Let be reals such that 
Prove that
such that there exists 
satisfying 
.
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
. Prove that exists many 
such that every equation: 
has no real roots
meet at point 
. Points 
and 
lie on 
, 
respectively in such a way that 
and 
. Prove that the line joining the common points of circles 
and 
passes through the mass-center of .
lies inside a triangle 
on the bisector of angle 
. Let 
and 
be the circles touching 
and 
at and passing through ; and 
be the common points of and with the circumcircle of distinct from . Prove that the circumcircles of the triangles 
and 
are tangent.
)
)
). I would also highly recommend taking Helen Yang's classes(almost all NJ ppl have gone through her, with the exceptions of dotted and advait).
:
![$\mathbb Z[\sqrt n]$](http://latex.artofproblemsolving.com/7/3/8/738c449809946e13bc9a16c316a2233f0a2db4f9.png)
is a UFD.
such that 
.
