be a triangle inscribed in a circle, where the tangents to the circle at points 
and 
intersect at the point 
be a point on the arc 
(not containing 
and 
.
and 
intersect at point 
.
be the reflection of 
and 
intersect at point 
.
.
be circumscribed about circle 
,
be the orthocenter of 
.
at 
be the midpoint of 
,
meet 
.
at 
is tangent to 
,
and 
,
is the intersection point of the diagonals 
.
and radius 
intersects 
and 
.
is tangent to the circles with diameters 
and 
.
such that 
and 
for any real numbers 
and 
.
,
denote the sum of squares of digits of 
is eventually periodic.
grid, where 
unit squares are divided up into 
disjoint sets of 
unit squares arbitrarily such that 
.
disjoint sets of 
unit squares arbitrarily such that 
.
unit squares such that no two unit squares are in the same set of 
be a triangle, and let 
,
is the circumcenter of 
and 
as the orthocenters of triangles 
and 
.
passes through a fixed point which is independent of the choice of 
and 
Prove that
.
.
.
and 
.
and 
Prove that
be positive real numbers satisfying 
.![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
Y by
In a rectangle 
two not intersecting circles 
and 
are drawn such that is tangent to 
and 
at points 
and 
respectively, and is tangent to 
and 
at 
and 
respectively. It is known that 
.
Find the distance between centers of circles and .
two not intersecting circles 
and 
are drawn such that is tangent to 
and 
at points 
and 
respectively, and is tangent to 
and 
at 
and 
respectively. It is known that 
.Find the distance between centers of circles
and .
