[/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 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
Parallelogram 
is given with 
, and 
intersection point of 
and 
. Circle with center at and radius 
intersects extensions of 
and 
at points 
and 
, respectively. Let 
be intersection point of lines and 
. Prove that 
.
is given with 
, and 
intersection point of 
and 
. Circle with center at and radius 
intersects extensions of 
and 
at points 
and 
, respectively. Let 
be intersection point of lines and 
. Prove that 
.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 1:32 AM
perpendicular to 
and 
at points 
and 
,
,
,
.
,
.
,![\[
\angle KCN = \angle KLN.
\]](http://latex.artofproblemsolving.com/8/1/7/8172e797dd8148dbe5548c68c2ecdbde354347c8.png)
Now, since 
,![\[
\angle KLN = \angle KZO.
\]](http://latex.artofproblemsolving.com/5/4/5/5456dedfb5ec7c925093f1025db396bbf027849d.png)
It implies that quadrilateral 
is cyclic, and![\[
\angle OKZ = \angle OCZ.
\]](http://latex.artofproblemsolving.com/9/5/c/95cb1cd504069f9bfcb89b3033c92b47e2efb2ca.png)
Since 
,
.
