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
Chords 
and 
, which do not intersect, are drawn in a circle. On the chord or on its extension is taken the point 
. Using a compass and construct the point 
on the arc , such that 
, where 
are given natural numbers, 
is the point of intersection of the chord with the chord 
, 
is the point of intersection of the chord with the chord 
. Consider cases where 
and 
.
and 
, which do not intersect, are drawn in a circle. On the chord or on its extension is taken the point 
. Using a compass and construct the point 
on the arc , such that 
, where 
are given natural numbers, 
is the point of intersection of the chord with the chord 
, 
is the point of intersection of the chord with the chord 
. Consider cases where 
and 
.
