,
is expressed by the sum of 
consecutive odd numbers. Express the minimum term in terms of 
.
If the value of 
can be written as 
where 
and 
are co-prime positive integers then compute the value of 
and 
where each summation runs over 
variables 
and 
Then find the value of 
Prove that
[/quote]
and 
Prove that 
be real numbers such that 
.
Prove that 
and ![$e\in [-2,2]$](http://latex.artofproblemsolving.com/b/b/7/bb78c9dedb18af5e6a904f5492917921e41a7ebc.png)
such that 
Find the value of 
:
be the altitude and 
be the intersection point of the altitudes of triangle 
.
is symmetric to 
.
intersects 
and 
.
.
where 
and 
where 
.
lie on the same circle. If 
and 
,
,
or 11. However, I didn't know how to find the radius. Can anyone give me a hint?
Y by
In a circumference 
a chord 
is considered such that the segment that joins the midpoint of the smallest arc and the midpoint of the segment measures 
. Let 
and 
be three tangent circumferences to the chord that are in the same half plane than the center of with respect to the line . Furthermore, 
and are internally tangent to and externally tangent to
, and the centers of and are on different halfplanes with respect to the line determined by the centers of and 
.
If the sum of the radii of and is equal to the radius of , calculate the radius of .
a chord 
is considered such that the segment that joins the midpoint of the smallest arc and the midpoint of the segment measures 
. Let 
and 
be three tangent circumferences to the chord that are in the same half plane than the center of with respect to the line . Furthermore, 
and are internally tangent to and externally tangent to
, and the centers of and are on different halfplanes with respect to the line determined by the centers of and 
.If the sum of the radii of
and is equal to the radius of , calculate the radius of .
