[/quote]
,
has side lengths 
and 
.
on 
.
is rotated about 
to 
so that 
,
lie on the same line and 
is perpendicular to 
.
,
and 
are relatively prime positive integers, find 
.
,
,
cut three common chords concurrent at 
,
which are internally tangent to all 
.
.
,
and 
such that 
.
then prove 
and 
.
be given . On the extension of the segment 
beyond 
is chosen arbitrarily. Let the line through 
.
and 
.
is tangent to the circumcircle of the triangle 
.
be any point. Let 
be projection point from 
.
cuts circumcircle 
at 
.
be antipode of 
are cyclic.
,
be real numbers, such that 
.
.
.
.
passing trough 
at 
.
.
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
In a convex quadrilateral 
we draw the bisectors 
, 
, 
, 
of external angles 
, 
, 
, 
respectively. The intersection points of the lines and , and , and , and are designated by 
, 
, 
, 
. It is known that 
perpendiculars drawn from on 
, from om 
, from on 
intersect at one point. Prove that the quadrilateral is cyclic.
we draw the bisectors 
, 
, 
, 
of external angles 
, 
, 
, 
respectively. The intersection points of the lines and , and , and , and are designated by 
, 
, 
, 
. It is known that 
perpendiculars drawn from on 
, from om 
, from on 
intersect at one point. Prove that the quadrilateral is cyclic.
,
.
is the midpoint of 
are among 
.
,
at 
respectively. There is a circle centered at 
touching 
,
is that of 
,
