AoPS Academy
![\[ x + y^2 = p^m \]](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 
. Prove that
Equality holds when 
or 
, 
, 
and 
lie on a circle 
such that 
. The tangent line to at point intersects the tangent line to at and the line 
at 
and 
. The circle and the circumcircle of triangle 
intersect again at 
. Prove that 
. and . Prove that
Equality holds when or 
and 
. Points 
and 
are on line segment 
so that 
and 
. Lines 
and 
intersect at 
. What is the area of 
?
now do find area of quaderilateral 
we simply subtract the areas of 
and 
from the area of rectangle 
and we get area=
now 
let 
so we have 
cross multiplying we get 
or 
now
now do find area of quaderilateral we simply subtract the areas of and from the area of rectangle and we get area= now let so we have cross multiplying we get or now
with 
on 
. So, 
. Let 
Thus, 
and 
. So we get 
. No apply 
to get the answer as 
.
Line joining 
.
.
.