[/quote]
,
be a trapezium inscribed in a circle 
with diameter 
be the intersection point of the diagonals 
. 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
In a right triangle 
, angle 
is a right , 
. A circle 
is circumscribed around the triangle . The tangent to at point intersects the line 
at point 
. Point 
is the midpoint of the arc 
of circle not containing point 
. The line 
intersects the circle at points and 
. The tangent to at intersects the line 
at point 
. Prove that the angle 
is right .
, angle 
is a right , 
. A circle 
is circumscribed around the triangle . The tangent to at point intersects the line 
at point 
. Point 
is the midpoint of the arc 
of circle not containing point 
. The line 
intersects the circle at points and 
. The tangent to at intersects the line 
at point 
. Prove that the angle 
is right .
,
we get that 
are collinear. Now since 
we get 
so we get that 
is cyclic, now easy angle chasing leads to 
and the conclusion follows.
