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, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals 
and 
. The circle with center 
and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
such that 
and 
for any real numbers 
and 
.
, let 
denote the sum of squares of digits of . Prove that the sequence 
is eventually periodic.
grid, where is a composite integer.
unit squares are divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
. unit squares are again divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
.
unit squares such that no two unit squares are in the same set of or the same set of ?
be a triangle, and let 
be an arbitrary point on line 
, where 
is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of .
be a right triangle in with 
. Let 
be the midpoint of 
and 
a point on such that 
. Let 
.
and the ratio 
.
be positive real numbers satisfying 
. Prove that![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
and 
Prove that
and 
Prove that
