invertible, for an infinite number of 
:
such that 
is a scalar
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
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 
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
,
is never a square of an integer.
are complex numbers with absolute value 
.
define 
.
converges. Show that,
.
Y by
Today I was browsing the forum High School Olympiads. There's a user whose username is the_universe6626, but then, when I flicked down to browse earlier answers today(note: I didn't 'search for this user's posts'), I found that one of them was labeled '23 posts'. I browsed his earlier answers today, and the number of posts increases, from 25 posts in the morning all the way to 36 posts right now. Is this a feature or a bug? If it is a feature, then certainly I haven't seen it elsewhere.
