[/quote]
,
be a trapezium inscribed in a circle 
with diameter 
and 
. 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 
,
such that 
.
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 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
Y by Adventure10
Let 
and 
be two circles intersecting at 
and 
. Let 
and 
be the centres of 
and , respectively. Let 
be a point on the segment 
such that 
and 
. We draw a line perpendicular to 
through and denote by 
and 
the points at which this line intersects . We likewise draw a line perpendicular to 
through and denote by 
and F the points at which this line intersects . Show that 
and 
are the vertices of a rectangle.
and 
be two circles intersecting at 
and 
. Let 
and 
be the centres of 
and , respectively. Let 
be a point on the segment 
such that 
and 
. We draw a line perpendicular to 
through and denote by 
and 
the points at which this line intersects . We likewise draw a line perpendicular to 
through and denote by 
and F the points at which this line intersects . Show that 
and 
are the vertices of a rectangle.
and 
,
.
are the vertices of a parallelogram.
.
