[/quote]
,
be a point on side 
be a point on side 
are parallel. Let 
.
and 
meet side 
and 
.
and 
intersect at a point 
.
,
.
and 
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
to 
be the intersection point of 
lies on the arc 
.
.
be a trapezium inscribed in a circle 
. The circle with center 
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 
lie in the interiors of segments 
respectively, such that 
Let 
intersect 
at 
intersect 
at 
and 
are either parallel or coincide.
and 
.
Where 
Y by
Circle 
touches the circumcircle of triangle 
at point 
, and it touches the sides 
and 
at points 
and 
, respectively. Rays 
and 
intersect at point 
. The tangent line at point to the circle meets the segment 
at point 
. The line segment 
intersects the circumcircle of triangle 
at point 
. Prove that the circumscribed circles of triangles and 
are tangent.
touches the circumcircle of triangle 
at point 
, and it touches the sides 
and 
at points 
and 
, respectively. Rays 
and 
intersect at point 
. The tangent line at point to the circle meets the segment 
at point 
. The line segment 
intersects the circumcircle of triangle 
at point 
. Prove that the circumscribed circles of triangles and 
are tangent.
is the intersection of 
and 
.
.
is external bisector of 
,
