[/quote]
,
.
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
be a cyclic quadrilateral. Let 
be the feet of the perpendiculars from 
,
if and only if the bisectors of 
and 
are concurrent with 
be reals such that 
. Prove that
where 
are lengths of parallel sides and 
are other two sides.
and 
Prove that 
be a triangle. A circle passing through 
intersect at 
,
and 
intersect at 
.
if and only if 
.
and 
.
be reals such that 
Prove that
Prove that
such that 
.
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
.
?
then
where 
Also, 
and 
is largest real root of the equation
then 
Equality holds iff 
and any cyclic permutations.
and 
be points on segment 
denote the circle passing through 
denote the circle passing through 
.
)
.
Y by Adventure10
Give a square of side 
. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to 
.
. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to 
.
,
points to each vertex must always be greater than or equal to 
.
.
