AoPS Academy
and 
be prime numbers such that 
is perfect square. Prove that 
is prime
such that for all positive real numbers 
and 
:
that satisfy the equation![\[
f(x) + 3f(-x) + f\left( \frac{1}{x} \right) = x, \quad \text{for all } x \in \mathbb{R}^*.
\]](http://latex.artofproblemsolving.com/6/a/2/6a2e35c42644036a62451b5e1794b36ffa728591.png)
be a metric space and 
be a function such that 
.
Prove that for all 
, 
exists, where 
is 
.
Prove that the amount of the limit does not depend on choosing .
![\[\iint_{R} \sin(xy) \,dx\,dy, \quad R = \left[0, \frac{\pi}{2}\right] \times \left[0, \frac{\pi}{2}\right]\]](http://latex.artofproblemsolving.com/a/2/0/a205868dd99f4794de53ad1db50f34b17c70a923.png)
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
, there exists some constant 
, for any 
, among the sequence 
and 
could be chosen to satisfy 
and 
.
be external direct sum of vector spaces 
and 
over a field 
.let 
and 
and 
is subspaces.
. Let 
be the external direct sum of and . show that is isomorphic to under the correspondence 
and whose roots are all real.
. Is it possible to equip 
with a finitely generated k-algebra with one generator such that make be an affine variety?
be a ring with unity such that for every 
there exist 
such that 
. Prove that
then 
such that 
then the result from a) may no longer hold.
be a positive integer and let 
be an irrational number. Show that for every integer 
, there exists a positive integer 
such that 
.
is increasing , because 
is increasing and unbounded, and 
so take all non-negative values is an arbitrary irrational number” xD. I was confused.
Then 
is
is dense in 
so we split the interval into