[/quote]
,
![\[ x + y^2 = p^m \]](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 
,
,
lie on a circle 
such that 
.
at 
and 
.
intersect again at 
.
.
Equality holds when 
intersect its circumcircle 
at 
.
determine a triangle 
,
is relative to 
are concurrent at a point 
and that 
and 
.
be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points 
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.
This easy via the binomial theorem for the quantity is just 
,
Y by Adventure10, Mango247
Let 
be a triangle inscribed in circle 
. Circles 
are tangent internally with circle in 
and tangent to sides ![$[BC], [CA], [AB]$](//latex.artofproblemsolving.com/b/6/8/b680af4154b81dbe0e0c26ea61c2150b5f9b34cd.png)
in points 
respectively, so that 
are on one side of 
and so on. Lines 
and 
intersect the circle for second time at points 
and 
, respectively. If 
, prove that 
.
be a triangle inscribed in circle 
. Circles 
are tangent internally with circle in 
and tangent to sides ![$[BC], [CA], [AB]$](http://latex.artofproblemsolving.com/b/6/8/b680af4154b81dbe0e0c26ea61c2150b5f9b34cd.png)
in points 
respectively, so that 
are on one side of 
and so on. Lines 
and 
intersect the circle for second time at points 
and 
, respectively. If 
, prove that 
.
