![\[ x + y^2 = p^m \]](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 Adventure10, Mango247
Let 
be a convex quadrilateral. The midpoints of 
and 
are 
and 
, respectively. Given that the quadrilateral 
has area 
, prove that the area of the quadrilateral is 
.
be a convex quadrilateral. The midpoints of 
and 
are 
and 
, respectively. Given that the quadrilateral 
has area 
, prove that the area of the quadrilateral is 
.
,
,
( 
is the angle between the diagonals 
and 
).
