[/quote]
,
?
be a prime number and 
an integer polynomial of degree 
such that 
and 
is congruent to 
or 
modulo 
.
.
such that 
is an integer for all 
,
are all the positive divisors of 
.
be three positive reals such that 
.![\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]](http://latex.artofproblemsolving.com/a/1/4/a14adf0f1e35df57c734c7df701d5a67da22dc3c.png)
where 
are lengths of parallel sides and 
are other two sides.
be the feet of the perpendiculars from 
to the lines 
,
,
if and only if the bisectors of 
and 
are concurrent with 
.
.
be the feet of the altitudes from 
to 
to 
and 
as the reflections of 
be the circumcircle of 
.
with 
with 
is the circumcenter of 
can be inscribed within a circle.
be a triangle. A circle passing through 
intersect at 
,
and 
intersect at 
if and only if 
.
and 
be points on segment 
denote the circle passing through 
denote the circle passing through 
.
)
.
spheres centered at 
and 
and 
,
points 
outside of 
as the number of sets 
such that the midpoint of 
lies entirely inside of 
and 
.
.
has circumcenter 
.
and 
intersect at 
,
and 
intersect at 
and 
and 
intersect at 
.
are similar(with vertices in that order), than 
be the feet of the altitudes from 
respectively. Let 
be the projections of 
.
.
Y by celilcelil, Adventure10, Mango247
Let 
and 
be the midpoints of the sides 
and 
, respectively in a rectangle 
. Let 
and 
be the intersections of the line 
with the lines 
and 
, respectively, and let 
be the intersection of the lines 
and . Let 
, 
and 
be the midpoints of the segments 
, 
and 
, respectively. Let 
be the line passing through and perpendicular to 
, 
be the line passing through and perpendicular to and 
the line passing through and perpendicular to . Prove that the lines , and are concurrent.
and 
be the midpoints of the sides 
and 
, respectively in a rectangle 
. Let 
and 
be the intersections of the line 
with the lines 
and 
, respectively, and let 
be the intersection of the lines 
and . Let 
, 
and 
be the midpoints of the segments 
, 
and 
, respectively. Let 
be the line passing through and perpendicular to 
, 
be the line passing through and perpendicular to and 
the line passing through and perpendicular to . Prove that the lines , and are concurrent.
is cyclic
and 
and you get that 
.
