be a parallelogram of center 
.
,
and 
such that 
.
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 
Determine all the local minima of 
Y by Rounak_iitr
An acute-angled triangle 
with an altitude 
and orthocenter 
are given. intersects the circumcircle of 
at 
. 
is a point on segment 
such that 
. The circumcircle of 
intersects at 
and at 
. A line perpendicular to 
and passing through intersects at 
. Prove that the center of lies on line 
.
U. Maksimenkau
with an altitude 
and orthocenter 
are given. intersects the circumcircle of 
at 
. 
is a point on segment 
such that 
. The circumcircle of 
intersects at 
and at 
. A line perpendicular to 
and passing through intersects at 
. Prove that the center of lies on line 
.U. Maksimenkau
This post has been edited 1 time. Last edited by nAalniaOMliO, Oct 31, 2024, 10:11 AM
be the circumcenter of triangle 
hence 
and 
we have 
thus 
is parallel to 
and 
(By SAS)
so 
we get 
hence 
.
,
and 
is cyclic.
also passes through 
.
also passes through 
.
,
.
