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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.
invertible, for an infinite number of 
:
such that 
is a scalar
and 
be positive integers. Prove that![\[
a^2 + \left\lceil \frac{4a^2}{b} \right\rceil
\]](http://latex.artofproblemsolving.com/5/6/2/562d003cf06ca46031cec9e4e968f8b67784225f.png)
is not a perfect square.
matrices
matrices matrices
, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
. How many complex numbers 
are there such that 
and 
is an integer.
then 
and 
. Note that 
and we can only have equality when 
. Then 
. This 
gives 120 solutions for , from now on 
.
then 
hence 
where 
. If 
then 
impossible. If 
then consider 
. We have 
and 
. This is impossible as we just saw.
gives no solutions. The final case is 
hence 
. This gives 600 solutions. Hence in total we have 720 solutions.
Prove that
Where 
. Prove that
Then 
is
![\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}}
\, \mathrm{d}u
\]](http://latex.artofproblemsolving.com/9/1/1/91165e6b531233c56b799ed32ded982d57b6c87a.png)
.
and
Since this quantity is an integer, it must be real, so 
From here we have two cases.
(side note: is it an abuse of notation to use "mod" for non-integer values?), or, put simply, 
and 
are the same angle. In this case, 
is already 
,
possible values for 
or that 
Rearranging, we get that this is equivalent to 
which also means that 
so 
is an integer if and only if 
is. From here, we have several cases:
Hence, the only 
,
total possible 
different 
)
for 
.
such