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 parallelogram of center 
. Prove that for any point 
, there exist unique points 
and 
such that is the center of mass of 
.
Compute -![\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*}](http://latex.artofproblemsolving.com/8/6/f/86fb48db976012106bd30f74cb6ada8c6e6d7800.png)
For 
define 
Determine the open interval(s) (if any) where 
is decreasing and all the open interval(s) (if any) where is increasing.
Determine all the local minima of (if any) and all the local maxima of (if any) 
, angle 
. Point 
lies on 
such that 
. Line parallel to 
passing trough intersects 
at 
. Prove that 
.
of positive integers such that
, then it must not contain edge 
.
with circumcenter and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line such that 
. Let and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be nonnegative real numbers such that 
. Find min
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
of a permutation 
of 
and a subset 
of such that![\[ \forall x\in S: \alpha(x)\not\in S\]](http://latex.artofproblemsolving.com/3/b/5/3b5b87de58ac757a2eb09ddeb7989dcbd6b8270d.png)
in which 
is the Fibonacci sequence such that 
elements we should pick these elements and put them in 
place and fill other blanks with other elements so the nimber is:
=
=
=
and 
Prove that
Prove that