[/quote]
,
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
is tangent to 
to be a point such that it lies on both the circles 
and 
are perpendicular.
be an acute triangle. Let 
be a point on side 
and 
be a point on side 
such that lines 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
.
and 
intersect at a point 
.
,
are collinear.
.
and 
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
to 
lies on the arc 
.
.
be a trapezium inscribed in a circle 
with diameter 
. 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 
.
Y by jhu08
Let 
be four different points on a line 
, such that 
. In one of the semiplanes determined by the line , the points 
and 
are chosen in such a way that the triangle 
is equilateral with its vertices named clockwise. Let 
and 
be two points on the plane such that the triangles 
and 
are equilateral (the vertices are also named clockwise). Find the measure of the angle 
.
be four different points on a line 
, such that 
. In one of the semiplanes determined by the line , the points 
and 
are chosen in such a way that the triangle 
is equilateral with its vertices named clockwise. Let 
and 
be two points on the plane such that the triangles 
and 
are equilateral (the vertices are also named clockwise). Find the measure of the angle 
.
