[/quote]
,
is given. Let 
,
to 
,
and 
.
lies on the arc 
.
.
be a trapezium inscribed in a circle 
with diameter 
. The circle with center 
and radius 
meets 
and 
(where 
)
at 
.
is perpendicular to 
.
![\[ x + y^2 = p^m \]](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 
.
at 
intersect again at 
.
Equality holds when 
intersect its circumcircle 
.
determine a triangle 
,
is relative to 
are concurrent at a point 
and that 
and 
.
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
be reals such that 
Prove that 
Let 
Prove that 
Y by Adventure10, Mango247
The altitudes 
, 
and 
are drawn in the acute triangle 
. The bisector of the angle 
intersects the segments and 
at 
and 
respectively. The bisector of the angle 
intersects the segments and 
at 
and 
respectively. The circumcircles of the triangles 
and 
intersect at 
and 
.
Prove that
.
, 
and 
are drawn in the acute triangle 
. The bisector of the angle 
intersects the segments and 
at 
and 
respectively. The bisector of the angle 
intersects the segments and 
at 
and 
respectively. The circumcircles of the triangles 
and 
intersect at 
and 
.Prove that
.
.
.
.
.
