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.
be a group, 
a subgroup of 
and 
an endomorphism with the following property:
such that for any 
there exist 
with 
and 
, for any 
.
is abelian.
is a subgroup of , prove that 
be positive integers satisfying the conditions 
and 
Show that there exists a real number 
with the property that all the three numbers 
have their fractional parts lying in the interval ![$ \left(\frac {1}{3}, \frac {2}{3} \right].$](http://latex.artofproblemsolving.com/4/2/b/42ba58ddd6032a3176122f1c9b2015cb6f4ca925.png)
for which each cell of 
table can be filled with one of the letters 
and 
in such a way that:
, one third are 
and one third are ; and [/*], one third are and one third are .[/*] table are each labelled 
to in a natural order. Thus each cell corresponds to a pair of positive integer 
with 
. For 
, the table has 
diagonals of two types. A diagonal of first type consists all cells for which 
is a constant, and the diagonal of this second type consists all cells for which 
is constant.
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
of 
touch 
at 
, 
, let 
denote the line tangent to through . Define 
such that 
. Prove that the circumcenter of lies on 
.
be a cubic polynomial given by 
, where 
are integers and 
. Suppose that 
for infinitely many pairs 
of integers with 
. Prove that the equation 
has an integer root.
be real numbers such that 
. Determine the minimum value of 
and determine all values of 
such that the minimum value is achived.
![$f:[1,8]\to R$](http://latex.artofproblemsolving.com/c/e/c/cec97ea0372e37977e8a77e8e77518a2344c1c43.png)
that:
satisfying
Prove
