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)
with another positive integer 
,
is a perfect square. For example, if the number on the blackboard is 
,
,
,
,
.
,
and 
,
to be the unique polynomial 
such that we can write 
,
is a polynomial such that either 
or the degree of 
with real coefficients such that 
be non-negative real numbers such that 
Prove that
Y by
Let 
be an acute triangle with the orthocenter 
. On the segments 
and 
, we choose three points 
and 
respectively such that 
. Denote by 
, 
and 
.
a) Prove that the sides of the hexagon
are tangent to the one circle.
b) Let
be a incenter of the hexagon . Prove that the lines 
and 
are concurrent.
be an acute triangle with the orthocenter 
. On the segments 
and 
, we choose three points 
and 
respectively such that 
. Denote by 
, 
and 
.a) Prove that the sides of the hexagon
are tangent to the one circle.b) Let
be a incenter of the hexagon . Prove that the lines 
and 
are concurrent.
