[/quote]
,
with circumcircle 
is given. Let 
,
be the intersection point of 
.
lies on the arc 
.
.
with diameter 
and radius 
meets 
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
In the parallelogram 
, 
. Let 
be the intersection point of the diagonals. The circle circumscribed to the triangle 
intersects the line 
at the point 
(different from 
), the line 
at the point 
(different from 
), and to the line 
in 
(different from 
). The line 
intersects the circumscribed circle of the triangle 
at the points 
and 
. Show that the triangles 
and 
are congruent.
, 
. Let 
be the intersection point of the diagonals. The circle circumscribed to the triangle 
intersects the line 
at the point 
(different from 
), the line 
at the point 
(different from 
), and to the line 
in 
(different from 
). The line 
intersects the circumscribed circle of the triangle 
at the points 
and 
. Show that the triangles 
and 
are congruent.This post has been edited 1 time. Last edited by parmenides51, Nov 8, 2021, 6:26 PM
