[/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 
.
passing trough 
.
.
of positive integers such that
,
.
and 
Prove that
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
be the point on the circumcircle of triangle 
is perpendicular to 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
.
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
Let 
be the circumscribed circle of the triangle 
, in which 
, 
is the center of the arc 
, 
is the diameter of the circle . The circle 
is inscribed in the curvilinear triangle formed by the segments 
, 
and the arc 
of the circle . It turned out that circle also touches at point 
. Let 
be the center of the triangle , 
is the midpoint of the smaller arc 
, and 
is the second intersection point of 
with the circle . Prove that lines 
, 
and intersect at one point.
(Mykhailo Sydorenko)
be the circumscribed circle of the triangle 
, in which 
, 
is the center of the arc 
, 
is the diameter of the circle . The circle 
is inscribed in the curvilinear triangle formed by the segments 
, 
and the arc 
of the circle . It turned out that circle also touches at point 
. Let 
be the center of the triangle , 
is the midpoint of the smaller arc 
, and 
is the second intersection point of 
with the circle . Prove that lines 
, 
and intersect at one point.(Mykhailo Sydorenko)
This post has been edited 1 time. Last edited by parmenides51, Feb 26, 2024, 7:04 PM
touches 
respectively at 
-
,
are colinear.
is the tangent line of 
is the center of 
,
are concyclic, and 
.
are concyclic.
we have 
are concurrent.
are concyclic:
be the center of 
is the tangential point of 
is a tangent of 
.
and 
,
intersect 
again at 
.
and 
is an isosceles trapezoid
.
.
is the external bisector of 
intersects 
is the external bisector of 
are collinear. (Q. E. D)
