[/quote]
,
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 
,
.
lies on 
such that 
.
.
.
of positive integers such that
,
.
and 
Prove that
with circumcenter 
and orthocenter 
and that line 
is not parallel to line 
is perpendicular to 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be nonnegative real numbers such that 
.
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
Prove that
denote the sum of the sum of the factors of all positive 
,
is the sum of the factors of 
added to the sum of the factors of 
and so on until the sum of the factors of 
,
.
denote 
.
or find a counterexample
Y by Rounak_iitr
Let 
and 
be points on the sides 
and 
of the triangle 
, respectively. The circumcircle of 
meets the circumcircle of 
for the second time at 
. The line 
meets the line 
at 
, while the line 
meets the line 
at 
. Prove that the lines 
, 
and 
pass through the same point.
Authored by Petar Filipovski
and 
be points on the sides 
and 
of the triangle 
, respectively. The circumcircle of 
meets the circumcircle of 
for the second time at 
. The line 
meets the line 
at 
, while the line 
meets the line 
at 
. Prove that the lines 
, 
and 
pass through the same point.Authored by Petar Filipovski
This post has been edited 2 times. Last edited by steppewolf, Feb 9, 2025, 10:35 PM
![[asy]
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draw(circle((-2.207672768100939,-1.2676980623519316), 4.18320952285643), linewidth(0.7) + linetype("4 4") + green);
/* dots and labels */
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/* end of picture */
[/asy]](http://latex.artofproblemsolving.com/4/d/e/4de3f2b4e05844d80f528f91b7b58bd6cd906027.png)
be intersections of 
as a circumcircle of 
, 
as a circumcircle of 
be intersections of the circumcircle of 
with respect to 
are concyclic ( 
as well)
done. 
(Similarly, 
(by 
and 
we get 
,
are isogonal w.r.t. 
.
are isogonal w.r.t. 
and 
,
are isogonal w.r.t. 
and 
,
and 
and 
,
are also isogonal w.r.t. 
and 
we get that 
,
and 
are totally not needed, the point 
and 
gives that 
,
and 
are collinear, done. 
