[/quote]
,
and 
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
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 
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 
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 pasionfruit, Mango247
The incircle of the triangle 
touches the sides 
and 
at points 
and 
, respectively. the 
-excircle touches the side of this triangle at point 
. The segment 
intersects the inscribed circle for the second time at point 
. Line 
intersects the circumscribed circle of triangle 
for the second at point 
. Prove that 
.
touches the sides 
and 
at points 
and 
, respectively. the 
-excircle touches the side of this triangle at point 
. The segment 
intersects the inscribed circle for the second time at point 
. Line 
intersects the circumscribed circle of triangle 
for the second at point 
. Prove that 
.
,
.
implies 
.
and 
.
and 
.
.
and from that we get 
