Three concurrent chords
by v_Enhance, Mar 21, 2025, 8:45 PM
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Three distinct circles 
, 
, 
cut three common chords concurrent at 
. Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true.
(1) is the insimilicenter of and
(2) is the exsimilicenter of and .
, 
, 
cut three common chords concurrent at 
. Consider two distinct circles 
, 
which are internally tangent to all 
. Determine, with proof, which of the following two statements is true.(1)
is the insimilicenter of and (2)
is the exsimilicenter of and .
,
.
lies on 
such that 
.
passing trough 
at 
.
.
.
,
and 
such that 
.
be given . On the extension of the segment 
beyond 
,
is chosen arbitrarily. Let the line through 
intersect the line 
at the point 
.
be the intersection of the lines 
and 
.
is tangent to the circumcircle of the triangle 
.
be projection point from 
.
cuts circumcircle 
at 
.
be antipode of 
are cyclic.
,
be real numbers, 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 
May 2018
April 2018