[/quote]
,
and 
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
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 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
,
intersect 
at 
.
and 
are either parallel or coincide.
and 
.
Where 
such that :
.
Y by
Is it always possible that 
points marked on a plane, none of which lie on the same line, can be divided into 
groups of 
points each, so that each of the four points of the group is the vertices of a convex quadrilateral and the interior of any two of did these quadrilaterals have no common points?
points marked on a plane, none of which lie on the same line, can be divided into 
groups of 
points each, so that each of the four points of the group is the vertices of a convex quadrilateral and the interior of any two of did these quadrilaterals have no common points?This post has been edited 1 time. Last edited by parmenides51, May 26, 2022, 3:53 AM
Reason: title typo
Reason: title typo
