![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
.
Equality holds when 
or 
and 
lie on a circle 
such that 
.
at 
and 
.
intersect again at 
.
.
Equality holds when 
Y by
Two disjoint circles 
and 
and points 
and 
on them so that the segment 
has the greatest possible length. Let 
and 
be tangents drawn from point to circle , 
and 
be tangents from point to the circle . The circle 
touches the circle internally and the rays and . The circle 
touches circle internally and the rays and . Prove that the radii of the circles and are equal.
and 
and points 
and 
on them so that the segment 
has the greatest possible length. Let 
and 
be tangents drawn from point to circle , 
and 
be tangents from point to the circle . The circle 
touches the circle internally and the rays and . The circle 
touches circle internally and the rays and . Prove that the radii of the circles and are equal.
