[/quote]
,
with circumcircle 
is given. Let 
be the midpoint of 
,
be the foot of altitude from 
to 
,
be the intersection point of 
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 
.
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
Let 
be a triangle and let 
be a point in the plane, not situated on the circle . The line 
meets again the circle at point 
. A line 
through the point meets the line 
at point 
. Show that the lines 
, are concurrent and determine the locus of their concurrency point as the line turns about the point .
be a triangle and let 
be a point in the plane, not situated on the circle . The line 
meets again the circle at point 
. A line 
through the point meets the line 
at point 
. Show that the lines 
, are concurrent and determine the locus of their concurrency point as the line turns about the point .
