be a triangle inscribed in a circle, where the tangents to the circle at points 
.
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 
,
.
and radius 
intersects 
.
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 Amir Hossein, Adventure10
Let 
be the incenter of 
Let 
be the line through and perpendicular to 
The perpendicular to 
through 
and that to 
through 
meet at 
and 
respectively. The feet of perpendiculars from and to onto 
are 
and 
Prove that 
and 
are tangent together.
be the incenter of 
Let 
be the line through and perpendicular to 
The perpendicular to 
through 
and that to 
through 
meet at 
and 
respectively. The feet of perpendiculars from and to onto 
are 
and 
Prove that 
and 
are tangent together.This post has been edited 1 time. Last edited by Orkhan-Ashraf_2002, Feb 25, 2017, 5:02 PM
