be positive real numbers such that : 
. Prove that : 
candies. A starts first. On a move, if in the box there are 
candies, the player chooses positive integer 
so that 
and 
,
.
such that 
and 
.
![$\ [\text{Blog Post 141}] $](http://latex.artofproblemsolving.com/e/2/e/e2e1563c16c5d526310409de610870dc615964d8.png)
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 .
