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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) 
in 
.
to 
, so the same problem with orthocenter
, construct its circumcenter
then we can draw the antipode of 
by first constructing 
then 
and 
, we can draw midpoint of 
by 
, we can also construct the intersection of symmedian with circumcircle by 
then draw 
s.t. 
by 
. 
to , so the same problem with orthocenter
, construct its circumcenter then we can draw the antipode of by first constructing then and , we can draw midpoint of by , we can also construct the intersection of symmedian with circumcircle by then draw s.t. by . 
.
as the second intersection between 
and . Construct 
, then 
. Since 
, 
. Construct 
as the second intersection between 
and . Since 
, 
is a diameter of so it passes through the circumcenter .
, and 
so we are done.
meet again at the midpoint 
of arc , and construct 
, 
similarly, then 
is the midpoint of (as 
is the perpendicular bisector of ). Construct 
, 
similarly. Then 
is the centroid of 
, so 
is the midpoint of and allows us to construct the perpendicular bisector 
of . Construct the other perpendicular bisectors similarly, then they meet at the circumcenter.
,
are drawn on the plane. Construct the circumcenter of 